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CHAPTER I.

EUCLID AND THE TRADITIONS ABOUT HIM.

As in the case of the other great mathematicians of Greece, so in Euclid's case, we have only the most meagre particulars of the life and personality of the man.

Most of what we have is contained in the passage of Proclus' summary relating to him, which is as follows1:

“Not much younger than these (sc. Hermotimus of Colophon and Philippus of Medma) is Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived2 in the time of the first Ptolemy. For Archimedes, who came immediately after the first (Ptolemy)3, makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry4. He is then younger than the pupils of Plato but older than Eratosthenes and Archimedes; for the latter were contemporary with one another, as Eratosthenes somewhere says.”

This passage shows that even Proclus had no direct knowledge of Euclid's birthplace or of the date of his birth or death. He proceeds by inference. Since Archimedes lived just after the first Ptolemy, and Archimedes mentions Euclid, while there is an anecdote about some Ptolemy and Euclid, therefore Euclid lived in the time of the first Ptolemy.

We may infer then from Proclus that Euclid was intermediate between the first pupils of Plato and Archimedes. Now Plato died in 347/6, Archimedes lived 287-212, Eratosthenes c. 284-204 B.C. Thus Euclid must have flourished c. 300 B.C., which date agrees well with the fact that Ptolemy reigned from 306 to 283 B.C.

It is most probable that Euclid received his mathematical training in Athens from the pupils of Plato; for most of the geometers who could have taught him were of that school, and it was in Athens that the older writers of elements, and the other mathematicians on whose works Euclid's Elements depend, had lived and taught. He may himself have been a Platonist, but this does not follow from the statements of Proclus on the subject. Proclus says namely that he was of the school of Plato and in close touch with that philosophy5. But this was only an attempt of a New Platonist to connect Euclid with his philosophy, as is clear from the next words in the same sentence, “for which reason also he set before himself, as the end of the whole Elements, the construction of the so-called Platonic figures.” It is evident that it was only an idea of Proclus' own to infer that Euclid was a Platonist because his Elements end with the investigation of the five regular solids, since a later passage shows him hard put to it to reconcile the view that the construction of the five regular solids was the end and aim of the Elements with the obvious fact that they were intended to supply a foundation for the study of geometry in general, “to make perfect the understanding of the learner in regard to the whole of geometry6.” To get out of the difficulty he says7 that, if one should ask him what was the aim (σκοπός) of the treatise, he would reply by making a distinction between Euclid's intentions (1) as regards the subjects with which his investigations are concerned, (2) as regards the learner, and would say as regards (1) that “the whole of the geometer's argument is concerned with the cosmic figures.” This latter statement is obviously incorrect. It is true that Euclid's Elements end with the construction of the five regular solids; but the planimetrical portion has no direct relation to them, and the arithmetical no relation at all; the propositions about them are merely the conclusion of the stereometrical division of the work.

One thing is however certain, namely that Euclid taught, and founded a school, at Alexandria. This is clear from the remark of Pappus about Apollonius8 : “he spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought.”

It is in the same passage that Pappus makes a remark which might, to an unwary reader, seem to throw some light on the personality of Euclid. He is speaking about Apollonius' preface to the first book of his Conics, where he says that Euclid had not completely worked out the synthesis of the “three- and four-line locus,” which in fact was not possible without some theorems first discovered by himself. Pappus says on this9: “Now Euclid— regarding Aristaeus as deserving credit for the discoveries he had already made in conics, and without anticipating him or wishing to construct anew the same system (such was his scrupulous fairness and his exemplary kindliness towards all who could advance mathematical science to however small an extent), being moreover in no wise contentious and, though exact, yet no braggart like the other [Apollonius] —wrote so much about the locus as was possible by means of the conics of Aristaeus, without claiming completeness for his demonstrations.” It is however evident, when the passage is examined in its context, that Pappus is not following any tradition in giving this account of Euclid: he was offended by the terms of Apollonius' reference to Euclid, which seemed to him unjust, and he drew a fancy picture of Euclid in order to show Apollonius in a relatively unfavourable light.

Another story is told of Euclid which one would like to believe true. According to Stobaeus10, “some one who had begun to read geometry with Euclid, when he had learnt the first theorem, asked Euclid, ’But what shall I get by-learning these things?’ Euclid called his slave and said ’Give him threepence, since he must make gain out of what he learns.’”

In the middle ages most translators and editors spoke of Euclid as Euclid of Megara. This description arose out of a confusion between our Euclid and the philosopher Euclid of Megara who lived about 400 B.C. The first trace of this confusion appears in Valerius Maximus (in the time of Tiberius) who says11 that Plato, on being appealed to for a solution of the problem of doubling the cubical altar, sent the inquirers to “Euclid the geometer.” There is no doubt about the reading, although an early commentator on Valerius Maximus wanted to correct “Eucliden” into “Eudoxum,” and this correction is clearly right. But, if Valerius Maximus took Euclid the geometer for a contemporary of Plato, it could only be through confusing him with Euclid of Megara. The first specific reference to Euclid as Euclid of Megara belongs to the 14th century, occurring in the ὑπομνηματισμοί of Theodorus Metochita (d. 1332) who speaks of “Euclid of Megara, the Socratic philosopher, contemporary of Plato,” as the author of treatises on plane and solid geometry, data, optics etc. : and a Paris MS. of the 14th century has “Euclidis philosophi Socratici liber elementorum.” The misunderstanding was general in the period from Campanus' translation (Venice 1482) to those of Tartaglia (Venice 1565) and Candalla (Paris 1566). But one Constantinus Lascaris (d. about 1493) had already made the proper distinction by saying of our Euclid that “he was different from him of Megara of whom Laertius wrote, and who wrote dialogues” 12; and to Commandinus belongs the credit of being the first translator13 to put the matter beyond doubt: “Let us then free a number of people from the error by which they have been induced to believe that our Euclid is the same as the philosopher of Megara” etc.

Another idea, that Euclid was born at Gela in Sicily, is due to the same confusion, being based on Diogenes Laertius' description14 of the philosopher Euclid as being “of Megara, or, according to some, of Gela, as Alexander says in the Διαδοχαί.”

In view of the poverty of Greek tradition on the subject even as early as the time of Proclus (410-485 A.D.), we must necessarily take cum grano the apparently circumstantial accounts of Euclid given by Arabian authors; and indeed the origin of their stories can be explained as the result (1) of the Arabian tendency to romance, and (2) of misunderstandings.

We read15 that “Euclid, son of Naucrates, grandson of Zenarchus16, called the author of geometry, a philosopher of somewhat ancient date, a Greek by nationality domiciled at Damascus, born at Tyre, most learned in the science of geometry, published a most excellent and most useful work entitled the foundation or elements of geometry, a subject in which no more general treatise existed before among the Greeks: nay, there was no one even of later date who did not walk in his footsteps and frankly profess his doctrine. Hence also Greek, Roman and Arabian geometers not a few, who undertook the task of illustrating this work, published commentaries, scholia, and notes upon it, and made an abridgment of the work itself. For this reason the Greek philosophers used to post up on the doors of their schools the well-known notice: ’Let no one come to our school, who has not first learned the elements of Euclid.’” The details at the beginning of this extract cannot be derived from Greek sources, for even Proclus did not know anything about Euclid's father, while it was not the Greek habit to record the names of grandfathers, as the Arabians commonly did. Damascus and Tyre were no doubt brought in to gratify a desire which the Arabians always showed to connect famous Greeks in some way or other with the East. Thus Nas<*>īraddīn, the translator of the Elements, who was of T<*>ūs in Khurāsān, actually makes Euclid out to have been “Thusinus” also17. The readiness of the Arabians to run away with an idea is illustrated by the last words of the extract. Everyone knows the story of Plato's inscription over the porch of the Academy: “let no one unversed in geometry enter my doors” ; the Arab turned geometry into Euclid's geometry, and told the story of Greek philosophers in general and “their Academies.”

Equally remarkable are the Arabian accounts of the relation of Euclid and Apollonius18. According to them the Elements were originally written, not by Euclid, but by a man whose name was Apollonius, a carpenter, who wrote the work in 15 books or sections19. In the course of time some of the work was lost and the rest became disarranged, so that one of the kings at Alexandria who desired to study geometry and to master this treatise in particular first questioned about it certain learned men who visited him and then sent for Euclid who was at that time famous as a geometer, and asked him to revise and complete the work and reduce it to order. Euclid then re-wrote it in 13 books which were thereafter known by his name. (According to another version Euclid composed the 13 books out of commentaries which he had published on two books of Apollonius on conics and out of introductory matter added to the doctrine of the five regular solids.) To the thirteen books were added two more books, the work of others (though some attribute these also to Euclid) which contain several things not mentioned by Apollonius. According to another version Hypsicles, a pupil of Euclid at Alexandria, offered to the king and published Books XIV. and XV., it being also stated that Hypsicles had “discovered” the books, by which it appears to be suggested that Hypsicles had edited them from materials left by Euclid.

We observe here the correct statement that Books XIV. and XV. were not written by Euclid, but along with it the incorrect information that Hypsicles, the author of Book XIV., wrote Book XV. also.

The whole of the fable about Apollonius having preceded Euclid and having written the Elements appears to have been evolved out of the preface to Book XIV. by Hypsicles, and in this way; the Book must in early times have been attributed to Euclid, and the inference based upon this assumption was left uncorrected afterwards when it was recognised that Hypsicles was the author. The preface is worth quoting:

“Basilides of Tyre, O Protarchus, when he came to Alexandria and met my father, spent the greater part of his sojourn with him on account of their common interest in mathematics. And once, when examining the treatise written by Apollonius about the comparison between the dodecahedron and the icosahedron inscribed in the same sphere, (showing) what ratio they have to one another, they thought that Apollonius had not expounded this matter properly, and accordingly they emended the exposition, as I was able to learn from my father. And I myself, later, fell in with another book published by Apollonius, containing a demonstration relating to the subject, and I was greatly interested in the investigation of the problem. The book published by Apollonius is accessible to all— for it has a large circulation, having apparently been carefully written out later—but I decided to send you the comments which seem to me to be necessary, for you will through your proficiency in mathematics in general and in geometry in particular form an expert judgment on what I am about to say, and you will lend a kindly ear to my disquisition for the sake of your friendship to my father and your goodwill to me.”

The idea that Apollonius preceded Euclid must evidently have been derived from the passage just quoted. It explains other things besides. Basilides must have been confused with βασιλεύς, and we have a probable explanation of the “Alexandrian king,” and of the “learned men who visited” Alexandria. It is possible also that in the “Tyrian” of Hypsicles' preface we have the origin of the notion that Euclid was born in Tyre. These inferences argue, no doubt, very defective knowledge of Greek: but we could expect no better from those who took the Organon of Aristotle to be “instrumentum musicum pneumaticum,” and who explained the name of Euclid, which they variously pronounced as Uclides or Icludes, to be compounded of Ucli a key, and Dis a measure, or, as some say, geometry, so that Uclides is equivalent to the key of geometry!

Lastly the alternative version, given in brackets above, which says that Euclid made the Elements out of commentaries which he wrote on two books of Apollonius on conics and prolegomena added to the doctrine of the five solids, seems to have arisen, through a like confusion, out of a later passage20 in Hypsicles' Book XIV.: “And this is expounded by Aristaeus in the book entitled ’Comparison of the five figures,’ and by Apollonius in the second edition of his comparison of the dodecahedron with the icosahedron.” The “doctrine of the five solids” in the Arabic must be the “Comparison of the five figures” in the passage of Hypsicles, for nowhere else have we any information about a work bearing this title, nor can the Arabians have had. The reference to the two books of Apollonius on conics will then be the result of mixing up the fact that Apollonius wrote a book on conics with the second edition of the other work mentioned by Hypsicles. We do not find elsewhere in Arabian authors any mention of a commentary by Euclid on Apollonius and Aristaeus: so that the story in the passage quoted is really no more than a variation of the fable that the Elements were the work of Apollonius.

CHAPTER II.

EUCLID'S OTHER WORKS.

In giving a list of the Euclidean treatises other than the Elements, I shall be brief: for fuller accounts of them, or speculations with regard to them, reference should be made to the standard histories of mathematics21.

I will take first the works which are mentioned by Greek authors.

I. The Pseudaria.

I mention this first because Proclus refers to it in the general remarks in praise of the Elements which he gives immediately after the mention of Euclid in his summary. He says22: “But, inasmuch as many things, while appearing to rest on truth and to follow from scientific principles, really tend to lead one astray from the principles and deceive the more superficial minds, he has handed down methods for the discriminative understanding of these things as well, by the use of which methods we shall be able to give beginners in this study practice in the discovery of paralogisms, and to avoid being misled. This treatise, by which he puts this machinery in our hands, he entitled (the book) of Pseudaria, enumerating in order their various kinds, exercising our intelligence in each case by theorems of all sorts, setting the true side by side with the false, and combining the refutation of error with practical illustration. This book then is by way of cathartic and exercise, while the Elements contain the irrefragable and complete guide to the actual scientific investigation of the subjects of geometry.”

The book is considered to be irreparably lost. We may conclude however from the connexion of it with the Elements and the reference to its usefulness for beginners that it did not go outside the domain of elementary geometry23.

2. The Data.

The Data (δεδομένα) are included by Pappus in the Treasury of Analysis (τόπος ἀναλυόμενος), and he describes their contents24 They are still concerned with elementary geometry, though forming part of the introduction to higher analysis. Their form is that of propositions proving that, if certain things in a figure are given (in magnitude, in species, etc.), something else is given. The subjectmatter is much the same as that of the planimetrical books of the Elements, to which the Data are often supplementary. We shall see this later when we come to compare the propositions in the Elements which give us the means of solving the general quadratic equation with the corresponding propositions of the Data which give the solution. The Data may in fact be regarded as elementary exercises in analysis.

It is not necessary to go more closely into the contents, as we have the full Greek text and the commentary by Marinus newly edited by Menge and therefore easily accessible25.

3. The book On divisions (of figures).

This work (περὶ διαιρέσεων βιβλίον) is mentioned by Proclus26. In one place he is speaking of the conception or definition (λόγος) of figure, and of the divisibility of a figure into others differing from it in kind; and he adds: “For the circle is divisible into parts unlike in definition or notion (ἀνόμοια τῷ λόγῳ), and so is each of the rectilineal figures; this is in fact the business of the writer of the Elements in his Divisions, where he divides given figures, in one case into like figures, and in another into unlike27.” “Like” and “unlike” here mean, not “similar” and “dissimilar” in the technical sense, but “like” or “unlike in definition or notion” (λόγῳ): thus to divide a triangle into triangles would be to divide it into “like” figures, to divide a triangle into a triangle and a quadrilateral would be to divide it into “unlike” figures.

The treatise is lost in Greek but has been discovered in the Arabic. First John Dee discovered a treatise De divisionibus by one Muhammad Bagdadinus28 and handed over a copy of it (in Latin) in 1563 to Commandinus, who published it, in Dee did not himself translate the tract from the Arabic; he in 157029. Dee did not himself translate the tract from the Arabic; he found it in Latin in a MS. which was then in his own possession but was about 20 years afterwards stolen or destroyed in an attack by a mob on his house at Mortlake30. Dee, in his preface addressed to Commandinus, says nothing of his having translated the book, but only remarks that the very illegible MS. had caused him much trouble and (in a later passage) speaks of “the actual, very ancient, copy from which I wrote out...” (in ipso unde descripsi vetustissimo exemplari). The Latin translation of this tract from the Arabic was probably made by Gherard of Cremona (1114-1187), among the list of whose numerous translations a “liber divisionum” occurs. The Arabic original cannot have been a direct translation from Euclid, and probably was not even a direct adaptation of it; it contains mistakes and unmathematical expressions, and moreover does not contain the propositions about the division of a circle alluded to by Proclus. Hence it can scarcely have contained more than a fragment of Euclid's work.

But Woepcke found in a MS. at Paris a treatise in Arabic on the division of figures, which he translated and published in 185131. It is expressly attributed to Euclid in the MS. and corresponds to the description of it by Proclus. Generally speaking, the divisions are divisions into figures of the same kind as the original figures, e.g. of triangles into triangles; but there are also divisions into “unlike” figures, e.g. that of a triangle by a straight line parallel to the base. The missing propositions about the division of a circle are also here: “to divide into two equal parts a given figure bounded by an arc of a circle and two straight lines including a given angle” and “to draw in a given circle two parallel straight lines cutting off a certain part of the circle.” Unfortunately the proofs are given of only four propositions (including the two last mentioned) out of 36, because the Arabic translator found them too easy and omitted them. To illustrate the character of the problems dealt with I need only take one more example: “To cut off a certain fraction from a (parallel-) trapezium by a straight line which passes through a given point lying inside or outside the trapezium but so that a straight line can be drawn through it cutting both the parallel sides of the trapezium.” The genuineness of the treatise edited by Woepcke is attested by the facts that the four proofs which remain are elegant and depend on propositions in the Elements, and that there is a lemma with a true Greek ring: “to apply to a straight line a rectangle equal to the rectangle contained by AB, AC and deficient by a square.” Moreover the treatise is no fragment, but finishes with the words “end of the treatise,” and is a well-ordered and compact whole. Hence we may safely conclude that Woepcke's is not only Euclid's own work but the whole of it. A restoration of the work, with proofs, was attempted by Ofterdinger32, Who however does not give Woepcke's props. 30, 31, 34, 35, 36. We have now a satisfactory restoration, with ample notes and an introduction, by R. C. Archibald, who used for the purpose Woepcke's text and a section of Leonardo of Pisa's Practica geometriae (1220)33.

4.The Porisms.

It is not possible to give in this place any account of the controversies about the contents and significance of the three lost books of Porisms, or of the important attempts by Robert Simson and Chasles to restore the work. These may be said to form a whole literature, references to which will be found most abundantly given by Heiberg and Loria, the former of whom has treated the subject from the philological point of view, most exhaustively, while the latter, founding himself generally on Heiberg, has added useful details, from the mathematical side, relating to the attempted restorations, etc.34 It must suffice here to give an extract from the only original source of information about the nature and contents of the Porisms, namely Pappus35. In his general preface about the books composing the Treasury of Analysis (τόπος ἀναλυόμενος) he says:

“After the Tangencies (of Apollonius) come, in three books, the Porisms of Euclid, [in the view of many] a collection most ingeniously devised for the analysis of the more weighty problems, [and] although nature presents and unlimited number of such porisms36, [they have added nothing to what was written originally by Euclid, except that some before my time have shown their want of taste by adding to a few (of the propositions) second proofs, each (proposition) admitting of a definite number of demonstrations, as we have shown, and Euclid having given one for each, namely that which is the most lucid. These porisms embody a theory subtle, natural, necessary, and of considerable generality, which is fascinating to those who can see and produce results].

“Now all the varieties of porisms belong, neither to theorems nor problems, but to a species occupying a sort of intermediate position [so that their enunciations can be formed like those of either theorems or problems], the result being that, of the great number of geometers, some regarded them as of the class of theorems, and others of problems, looking only to the form of the proposition. But that the ancients knew better the difference between these three things is clear from the definitions. For they said that a theorem is that which is proposed with a view to the demonstration of the very thing proposed, a problem that which is thrown out with a view to the construction of the very thing proposed, and a porism that which is proposed with a view to the producing of the very thing proposed. [But this definition of the porism was changed by the more recent writers who could not produce everything, but used these elements and proved only the fact that that which is sought really exists, but did not produce it37 and were accordingly confuted by the definition and the whole doctrine. They based their definition on an incidental characteristic, thus: A porism is that which falls short of a locustheorem in respect of its hypothesis38. Of this kind of porisms loci are a species, and they abound in the Treasury of Analysis; but this species has been collected, named and handed down separately from the porisms, because it is more widely diffused than the other species]. But it has further become characteristic of porisms that, owing to their complication, the enunciations are put in a contracted form, much being by usage left to be understood; so that many geometers understand them only in a partial way and are ignorant of the more essential features of their contents.

“[Now to comprehend a number of propositions in one enunciation is by no means easy in these porisms, because Euclid himself has not in fact given many of each species, but chosen, for examples, one or a few out of a great multitude39. But at the beginning of the first book he has given some propositions, to the number of ten, of one species, namely that more fruitful species consisting of loci.] Consequently, finding that these admitted of being comprehended in one enunciation, we have set it out thus: “ If, in a system of four straight lines40 which cut each other two and two, three points on one straight line be given while the rest except one lie on different straight lines given in position, the remaining point also will lie on a straight line given in position.41.

“This has only been enunciated of four straight lines, of which not more than two pass through the same point, but it is not known (to most people) that it is true of any assigned number of straight lines if enunciated thus: “ If any number of straight lines cut one another, not more than two (passing) through the same point, and all the points (of intersection situated) on one of them be given, and if each of those which are on another (of them) lie on a straight line given in position—

” or still more generally thus: “if any number of straight lines cut one another, not more than two (passing) through the same point, and all the points (of intersection situated) on one of them be given, while of the other points of intersection in multitude equal to a triangular number a number corresponding to the side of this triangular number lie respectively on straight lines given in position, provided that of these latter points no three are at the angular points of a triangle (sc. having for sides three of the given straight lines)—each of the remaining points will lie on a straight line given in position42.”

“It is probable that the writer of the Elements was not unaware of this but that he only set out the principle; and he seems, in the case of all the porisms, to have laid down the principles and the seed only [of many important things], the kinds of which should be distinguished according to the differences, not of their hypotheses, but of the results and the things sought. [All the hypotheses are different from one another because they are entirely special, but each of the results and things sought, being one and the same, follow from many different hypotheses.]

“We must then in the first book distinguish the following kinds of things sought:

“At the beginning of the book43 is this proposition: “ I.If from two given points straight lines be drawn meeting on a straight line given in position, and one cut off from a straight line given in position (a segment measured) to a given point on it, the other will also cut off from another (straight line a segment) having to the first a given ratio.’

“Following on this (we have to prove) “ II. that such and such a point lies on a straight line given in position;

III. that the ratio of such and such a pair of straight lines is given;” etc. etc. (up to XXIX.).

“The three books of the porisms contain 38 lemmas; of the theorems themselves there are 171.”

Pappus further gives lemmas to the Porisms (pp. 866-918, ed. Hultsch).

With Pappus' account of Porisms must be compared the passages of Proclus on the same subject. Proclus distinguishes two senses in which the word πόρισμα is used. The first is that of corollary where something appears as an incidental result of a proposition, obtained without trouble or special seeking, a sort of bonus which the investigation has presented us with44. The other sense is that of Euclid's Porisms45. In this sense46porism is the name given to things which are sought, but need some finding and are neither pure bringing into existence nor simple theoretic argument. For (to prove) that the angles at the base of isosceles triangles are equal is a matter of theoretic argument, and it is with reference to things existing that such knowledge is (obtained). But to bisect an angle, to construct a triangle, to cut off, or to place—all these things demand the making of something; and to find the centre of a given circle, or to find the greatest common measure of two given commensurable magnitudes, or the like, is in some sort between theorems and problems. For in these cases there is no bringing into existence of the things sought, but finding of them, nor is the procedure purely theoretic. For it is necessary to bring that which is sought into view and exhibit it to the eye. Such are the porisms which Euclid wrote, and arranged in three books of Porisms.”

Proclus' definition thus agrees well enough with the first, “older,” definition of Pappus. A porism occupies a place between a theorem and a problem: it deals with something already existing, as a theorem does, but has to find it (e.g. the centre of a circle), and, as a certain operation is therefore necessary, it partakes to that extent of the nature of a problem, which requires us to construct or produce something not previously existing. Thus, besides III. I of the Elements and X. 3, 4 mentioned by Proclus, the following propositions are real porisms: III. 25, VI. 11-13, VII. 33, 34, 36, 39, VIII. 2, 4, X. 10, XIII. 18. Similarly in Archimedes On the Sphere and Cylinder I. 2-6 might be called porisms.

The enunciation given by Pappus as comprehending ten of Euclid's propositions may not reproduce the form of Euclid's enunciations; but, comparing the result to be proved, that certain points lie on straight lines given in position, with the class indicated by II. above, where the question is of such and such a point lying on a straight line given in position, and with other classes, e.g. (V.) that such and such a line is given in position, (VI.) that such and such a line verges to a given point, (XXVII.) that there exists a given point such that straight lines drawn from it to such and such (circles) will contain a triangle given in species, we may conclude that a usual form of a porism was “to prove that it is possible to find a point with such and such a property” or “a straight line on which lie all the points satisfying given conditions” etc.

Simson defined a porism thus: “Porisma est propositio in qua proponitur demonstrare rem aliquam, vel plures datas esse, cui, vel quibus, ut et cuilibet ex rebus innumeris, non quidem datis, sed quae ad ea quae data sunt eandem habent relationem, convenire ostendendum est affectionem quandam communem in propositione descriptam47.”

From the above it is easy to understand Pappus' statement that loci constitute a large class of porisms. A locus is well defined by Simson thus: “Locus est proposition in qua propositum est datam esse demonstrare, vel invenire lineam aut superficiem cuius quodlibet punctum, vel superficiem in qua quaelibet linea data lege descripta, communem quandam habet proprietatem in propositione descriptam.” Heiberg cites an excellent instance of a locus which is a porism, namely the following proposition quoted by Eutocius48 from the Plane Loci of Apollonius:

“Given two points in a plane, and a ratio between unequal straight lines, it is possible to draw, in the plane, a circle such that the straight lines drawn from the given points to meet on the circumference of the circle have (to one another) a ratio the same as the given ratio.”

A difficult point, however, arises on the passage of Pappus, which says that a porism is “that which, in respect of its hypothesis, falls short of a locus-theorem” (τοπικοῦ θεωρήματος). Heiberg explains it by comparing the porism from Apollonius' Plane Loci just given with Pappus' enunciation of the same thing, to the effect that, if from two given points two straight lines be drawn meeting in a point, and these straight lines have to one another a given ratio, the point will lie on either a straight line or a circumference of a circle given in position. Heiberg observes that in this latter enunciation something is taken into the hypothesis which was not in the hypothesis of the enunciation of the porism, viz. “that the ratio of the straight lines is the same.” I confess this does not seem to me satisfactory: for there is no real difference between the enunciations, and the supposed difference in hypothesis is very like playing with words. Chasles says: “Ce qui constitue le porisme est ce qui manque à l' hypothèse d'un théorème local (en d'autres termes, le porisme est inférieur, par l'hypothèse, au théorème local; c'est-à-dire que quand quelques parties d'une proposition locale n'ont pas dans l'énoncé la détermination qui leur est propre, cette proposition cesse d'être regardée comme un theéorème et devient un porisme).” But the subject still seems to require further elucidation.

While there is so much that is obscure, it seems certain (1) that the Porisms were distinctly part of higher geometry and not of elementary geometry, (2) that they contained propositions belonging to the modern theory of transversals and to projective geometry. It should be remembered too that it was in the course of his researches on this subject that Chasles was led to the idea of anharmonic ratios.

Lastly, allusion should be made to the theory of Zeuthen49 on the subject of the porisms. He observes that the only porism of which Pappus gives the complete enunciation, “If from two given points straight lines be drawn meeting on a straight line given in position, and one cut off from a straight line given in position (a segment measured) towards a given point on it, the other will also cut off from another (straight line a segment) bearing to the first a given ratio,” is also true if there be substituted for the first given straight line a conic regarded as the “locus with respect to four lines,” and that this extended porism can be used for completing Apollonius' exposition of that locus. Zeuthen concludes that the Porisms were in part byproducts of the theory of conics and in part auxiliary means for the study of conics, and that Euclid called them by the same name as that applied to corollaries because they were corollaries with respect to conics. But there appears to be no evidence to confirm this conjecture.

5. The Surface-loci (τόποι πρὸς ἐπιφανείᾳ).

The two books on this subject are mentioned by Pappus as part of the Treasury of Analysis50. As the other works in the list which were on plane subjects dealt only with straight lines, circles, and conic sections, it is a priori likely that among the loci in this treatise (loci which are surfaces) were included such loci as were cones, cylinders and spheres. Beyond this all is conjecture based on two lemmas given by Pappus in connexion with the treatise.

(1) The first of these lemmas51 and the figure attached to it are not satisfactory as they stand, but a possible restoration is indicated by Tannery52. If the latter is right, it suggests that one of the loci contained all the points on the elliptical parallel sections of a cylinder and was therefore an oblique circular cylinder. Other assumptions with regard to the conditions to which the lines in the figure may be subject would suggest that other loci dealt with were cones regarded as containing all points on particular elliptical parallel sections of the cones53.

(2) In the second lemma Pappus states and gives a complete proof of the focus-and-directrix property of a conic, viz. that the locus of a point whose distance from a given point is in a given ratio to its distance from a fixed line is a conic section, which is an ellipse, a parabola or a hyperbola according as the given ratio is less than, equal to, or greater than unity54. Two conjectures are possible as to the application of this theorem in Euclid's Surface-loci. (a) It may have been used to prove that the locus of a point whose distance from a given straight line is in a given ratio to its distance from a given plane is a certain cone. (b) It may have been used to prove that the locus of a point whose distance from a given point is in a given ratio to its distance from a given plane is the surface formed by the revolution of a conic about its major or conjugate axis55. Thus Chasles may have been correct in his conjecture that the Surface-loci dealt with surfaces of revolution of the second degree and sections of the same56.

6. The Conics.

Pappus says of this lost work: “The four books of Euclid's Conics were completed by Apollonius, who added four more and gave us eight books of Conics 57.” It is probable that Euclid's work was lost even by Pappus'time, for he goes on to speak of “Aristaeus, who wrote the still extant five books of Solid Loci connected with the conics.” Speaking of the relation of Euclid's work to that of Aristaeus on conics regarded as loci, Pappus says in a later passage (bracketed however by Hultsch) that Euclid, regarding Aristaeus as deserving credit for the discoveries he had already made in conics, did not (try to) anticipate him or construct anew the same system. We may no doubt conclude that the book by Aristaeus on solid loci preceded Euclid's on conics and was, at least in point of originality, more important. Though both treatises dealt with the same subject-matter, the object and the point of view were different; had they been the same, Euclid could scarcely have refrained, as Pappus says he did, from attempting to improve upon the earlier treatise. No doubt Euclid wrote on the general theory of conics as Apollonius did, but confined himself to those properties which were necessary for the analysis of the Solid Loci of Aristaeus. The Conics of Euclid were evidently superseded by the treatise of Apollonius.

As regards the contents of Euclid's Conics, the most important source of our information is Archimedes, who frequently refers to propositions in conics as well known and not needing proof, adding in three cases that they are proved in the “elements of conics” or in “the conics,” which expressions must clearly refer to the works of Aristaeus and Euclid58

Euclid still used the old names for the conics (sections of a rightangled, acute-angled, or obtuse-angled cone), but he was aware that an ellipse could be obtained by cutting a cone in any manner by a plane not parallel to the base (assuming the section to lie wholly between the apex of the cone and its base) and also by cutting a cylinder. This is expressly stated in a passage from the Phaenomena of Euclid about to be mentioned59.

7. The Phaenomena.

This is an astronomical work and is still extant. A much interpolated version appears in Gregory's Euclid. An earlier and better recension is however contained in the MS. Vindobonensis philos. Gr. 103, though the end of the treatise, from the middle of prop. 16 to the last (18), is missing. The book, now edited by Menge60,consists of propositions in spheric geometry. Euclid based it on Autolycus' work περὶ κινουμένης σφαίρας, but also, evidently, on an earlier textbook of Sphaerica of exclusively mathematical content. It has been conjectured that the latter textbook may have been due to Eudoxus61.

8. The Optics.

This book needs no description, as it has been edited by Heiberg recently62,both in its genuine form and in the recension by Theon. The Catoptrica published by Heiberg in the same volume is not genuine, and Heiberg suspects that in its present form it may be Theon's. It is not even certain that Euclid wrote Catoptrica at all, as Proclus may easily have had Theon's work before him and inadvertently assigned it to Euclid63.

9. Besides the above-mentioned works, Euclid is said to have written the Elements of Music64 (αἱ κατὰ μουσικὴν στοιχειώσεις). Two treatises are attributed to Euclid in our MSS. of the Musici, the κατατομὴ κανόνος, Sectio canonis (the theory of the intervals), and the εἰσαγωγὴ ἁρμονική (introduction to harmony)65.The first, resting on the Pythagorean theory of music, is mathematical, and the style and diction as well as the form of the propositions mostly agree with what we find in the Elements. Jan thought it genuine, especially as almost the whole of the treatise (except the preface) is quoted in extenso, and Euclid is twice mentioned by name, in the commentary on Ptolemy's Harmonica published by Wallis and attributed by him to Porphyry. Tannery was of the opposite opinion66.The latest editor, Menge, suggests that it may be a redaction by a less competent hand from the genuine Euclidean Elements of Music. The second treatise is not Euclid's, but was written by Cleonides, a pupil of Aristoxenus67.

Lastly, it is worth while to give the Arabians' list of Euclid's works. I take this from Suter's translation of the list of philosophers and mathematicians in the Fihrist, the oldest authority of the kind that we possess68.“To the writings of Euclid belong further [in addition to the Elements]: the book of Phaenomena; the book of Given Magnitudes [Data]; the book of Tones, known under the name of Music, not genuine; the book of Division, emended by Thābit; the book of Utilisations or Applications [Porisms], not genuine; the book of the Canon; the book of the Heavy and Light; the book of Synthesis, not genuine; and the book of Analysis, not genuine.”

It is to be observed that the Arabs already regarded the book of Tones (by which must be meant the εἰσαγωγὴ ἁρμονική) as spurious. The book of Division is evidently the book on Divisions (of figures). The next book is described by Casiri as “liber de utilitate suppositus.” Suter gives reason for believing the Porisms to be meant69,but does not apparently offer any explanation of why the work is supposed to be spurious. The book of the Canon is clearly the κατατομὴ κανόνος. The book on “the Heavy and Light” is apparently the tract De levi et ponderoso, included in the Basel Latin translation of 1537, and in Gregory's edition. The fragment, however, cannot safely be attributed to Euclid, for (1) we have nowhere any mention of his having written on mechanics, (2) it contains the notion of specific gravity in a form so clear that it could hardly be attributed to anyone earlier than Archimedes70.Suter thinks71that the works on Analysis and Synthesis (said to be spurious in the extract) may be further developments of the Data or Porisms, or may be the interpolated proofs of Eucl. XIII. 1-5, divided into analysis and synthesis, as to which see the notes on those propositions.

CHAPTER III.

GREEK COMMENTATORS ON THE ELEMENTS OTHER THAN PROCLUS.

That there was no lack of commentaries on the Elements before the time of Proclus is evident from the terms in which Proclus refers to them; and he leaves us in equally little doubt as to the value which, in his opinion, the generality of them possessed. Thus he says in one place (at the end of his second prologue)72:

“Before making a beginning with the investigation of details, I warn those who may read me not to expect from me the things which have been dinned into our ears ad nauseam (διατεθρύληται) by those who have preceded me, viz. lemmas, cases, and so forth. For I am surfeited with these things and shall give little attention to them. But I shall direct my remarks principally to the points which require deeper study and contribute to the sum of philosophy, therein emulating the Pythagoreans who even had this common phrase for what I mean ’a figure and a platform, but not a figure and sixpence73.’”

In another place74 he says: “Let us now turn to the elucidation of the things proved by the writer of the Elements, selecting the more subtle of the comments made on them by the ancient writers, while cutting down their interminable diffuseness, giving the things which are more systematic and follow scientific methods, attaching more importance to the working-out of the real subject-matter than to the variety of cases and lemmas to which we see recent writers devoting themselves for the most part.”

At the end of his commentary on Eucl. I. Prochis remarks75 that the commentaries then in vogue were full of all sorts of confusion, and contained no account of causes, no dialectical discrimination, and no philosophic thought.

These passages and two others in which Proclus refers to “the commentators76” suggest that these commentators were numerous. He does not however give many names; and no doubt the only important commentaries were those of Heron, Porphyry, and Pappus.

1. Heron.

Proclus alludes to Heron twice as Heron mechanicus77, in another place78 he associates him with Ctesibius, and in the three other passages79 where Heron is mentioned there is no reason to doubt that the same person is meant, namely Heron of Alexandria. The date of Heron is still a vexed question. In the early stages of the controversy much was made of the supposed relation of Heron to Ctesibius. The best MS. of Heron's Belopoeica has the heading Ηρωνος ΚτησιΒίου βελοποιϊκά, and an anonymous Byzantine writer of the tenth century, evidently basing himself on this title, speaks of Ctesibius as Heron's καθηγητής, “master” or “teacher.” We know of two men of the name of Ctesibius. One was a barber who lived in the time of Ptolemy Euergetes II, i.e. Ptolemy VII, called Physcon (died 117 B.C.), and who is said to have made an improved water-organ80. The other was a mechanician mentioned by Athenaeus as having made an elegant drinking-horn in the time of Ptolemy Philadelphus (285-247 B.C.)81. Martin82 took the Ctesibius in question to be the former and accordingly placed Heron at the beginning of the first century B.C., say 126-50 B.C. But Philo of Byzantium83, who repeatedly mentions Ctesibius by name, says that the first mechanicians had the advantage of being under kings who loved fame and supported the arts. Hence our Ctesibius is more likely to have been the earlier Ctesibius who was contemporary with Ptolemy II Philadelphus.

But, whatever be the date of Ctesibius, we cannot safely conclude that Heron was his immediate pupil. The title “Heron's (edition of) Ctesibius's Belopoeica” does not, in fact, justify any inferenee as to the interval of time between the two works.

We now have better evidence for a terminus post quem. The Metrica of Heron, besides quoting Archimedes and Apollonius, twice refers to “the books about straight lines (chords) in a circle” (ἐν τοῖς περὶ τῶν ἐν κύκλῳ εὐθειῶν). Now we know of no work giving a Table of Chords earlier than that of Hipparchus. We get, therefore, at once, 150 B.C. or thereabouts as the terminus post quem. But, again, Heron's Mechanica quotes a definition of “centre of gravity” as given by “Posidonius, a Stoic” : and, even if this Posidonius lived before Archimedes, as the context seems to imply, it is certain that another work of Heron's, the Definitions, owes something to Posidonius of Apamea or Rhodes, Cicero's teacher (135-51 B.C.). This brings Heron's date down to the end of the first century B.C., at least.

We have next to consider the relation, if any, between Heron and Vitruvius. In his De Architectura, brought out apparently in 14 B.C., Vitruvius quotes twelve authorities on machinationes including Archytas (second), Archimedes (third), Ctesibius (fourth) and Philo of Byzantium (sixth), but does not mention Heron. Nor is it possible to establish inter-dependence between Vitruvius and Heron; the differences between them seem on the whole more numerous and important than the resemblances (e.g. Vitruvius uses 3 as the value of π, while Heron always uses the Archimedean value 3 1/7). The inference is that Heron can hardly have written earlier than the first century A.D.

The most recent theory of Heron's date makes him later than Claudius Ptolemy the astronomer (100-178 A.D.). The arguments are mainly these. (1) Ptolemy claims as a discovery of his own a method of measuring the distance between two places (as an arc of a great circle on the earth's surface) in the case where the places are neither on the same meridian nor on the same parallel circle. Heron, in his Dioptra, speaks of this method as of a thing generally known to experts. (2) The dioptra described in Heron's work is a fine and accurate instrument, much better than anything Ptolemy had at his disposal. (3) Ptolemy, in his work Περὶ ῥοπῶν, asserted that water with water round it has no weight and that the diver, however deep he dives, does not feel the weight of the water above him. Heron, strangely enough, accepts as true what Ptolemy says of the diver, but is dissatisfied with the explanation given by “some,” namely that it is because water is uniformly heavy—this seems to be equivalent to Ptolemy's dictum that water in water has no weight—and he essays a different explanation based on Archimedes. (4) It is suggested that the Dionysius to whom Heron dedicated his Definitions is a certain Dionysius who was praefectus urbi in 301 A.D.

On the other hand Heron was earlier than Pappus, who was writing under Diocletian (284-305 A.D.), for Pappus alludes to and draws upon the works of Heron. The net result, then, of the most recent research is to place Heron in the third century A.D. and perhaps little earlier than Pappus. Heiberg84 accepts this conclusion, which may therefore, perhaps, be said to hold the field for the present.85.

That Heron wrote a systematic commentary on the Elements might be inferred from Proclus, but it is rendered quite certain by references to the commentary in Arabian writers, and particularly in an-Nairīzī's commentary on the first ten Books of the Elements. The Fihrist says, under Euclid, that “Heron wrote a commentary on this book [the Elements], endeavouring to solve its difficulties86” ; and under Heron, “He wrote: the book of explanation of the obscurities in Euclid87....” An-Nairīzī's commentary quotes Heron by name very frequently, and often in such a way as to leave no doubt that the author had Heron's work actually before him. Thus the extracts are given in the first person, introduced by “Heron says” (“Dixit Yrinus” or “Heron” ); and in other places we are told that Heron “says nothing,” or “is not found to have said anything,” on such and such a proposition. The commentary of an-Nairīzī is in part edited by Besthorn and Heiberg from a Leiden MS. of the translation of the Elements by al-Hajjāj with the commentary attached88. But this MS. only contains six Books, and several pages in the first Book, which contain the comments of Simplicius on the first twenty-two definitions of the first Book, are missing. Fortunately the commentary of an-Nairīzī has been discovered in a more complete form, in a Latin translation by Gherardus Cremonensis of the twelfth century, which contains the missing comments by Simplicius and an-Nairīzī's comments on the first ten Books. This valuable work has recently been edited by Curtze89.

Thus from the three sources, Proclus, and the two versions of an-Nairīzī, which supplement one another, we are able to form a very good idea of the character of Heron's commentary. In some cases observations given by Proclus without the name of their author are seen from an-Nairīzī to be Heron's; in a few cases notes attributed by Proclus to Heron are found in an-Nairīzī without Heron's name; and, curiously enough, one alternative proof (of I. 25) given as Heron's by Proclus is introduced by the Arab with the remark that he has not been able to discover who is the author.

Speaking generally, the comments of Heron do not seem to have contained much that can be called important. We find

(1) A few general notes, e.g. that Heron would not admit more than three axioms.

(2) Distinctions of a number of particular cases of Euclid's propositions according as the figure is drawn in one way or in another.

Of this class are the different cases of I. 35, 36, III. 7, 8 (where the chords to be compared are drawn on different sides of the diameter instead of on the same side), III. 12 (which is not Euclid's, but Heron's own, adding the case of external contact to that of internal contact in III. 11), VI. 19 (where the triangle in which an additional line is drawn is taken to be the smaller of the two), VII. 19 (where he gives the particular case of three numbers in continued proportion, instead of four proportionals).

(3) Alternative proofs. Of these there should be mentioned (a) the proofs of II. 1-10 “without a figure,” being simply the algebraic forms of proof, easy but uninstructive, which are so popular nowadays, the proof of III. 25 (placed after III. 30 and starting from the arc instead of the chord), III. 10 (proved by III. 9), III. 13 (a proof preceded by a lemma to the effect that a straight line cannot meet a circle in more than two points). Another class of alternative proof is (b) that which is intended to meet a particular objection (ἔνστασις) which had been or might be raised to Euclid's construction. Thus in certain cases he avoids producing a particular straight line, where Euclid produces it, in order to meet the objection of any one who should deny our right to assume that there is any space available90. Of this class are Heron's proofs of I. II, 1.20, and his note on 1. 16. Similarly on 1. 48 he supposes the right-angled triangle which is constructed to be constructed on the same side of the common side as the given triangle is. A third class (c) is that which avoids reductio ad absurdum. Thus, instead of indirect proofs, Heron gives direct proofs of I. 19 (for which he requires, and gives, a preliminary lemma), and of I. 25.

(4) Heron supplies certain converses of Euclid's propositions, e.g. converses of II. 12, 13, VIII. 27.

(5) A few additions to, and extensions of, Euclid's propositions are also found. Some are unimportant, e.g. the construction of isosceles and scalene triangles in a note on I. 1, the construction of two tangents in III. 17, the remark that VII. 3 about finding the greatest common measure of three numbers can be applied to as many numbers as we please (as Euclid tacitly assumes in VII. 31). The most important extension is that of III. 20 to the case where the angle at the circumference is greater than a right angle, and the direct deduction from this extension of the result of III. 22. Interesting also are the notes on I. 37 (on I. 24 in Proclus), where Heron proves that two triangles with two sides of one equal to two sides of the other and with the included angles supplementary are equal, and compares the areas where the sum of the two included angles (one being supposed greater than the other) is less or greater than two right angles, and on I. 47, where there is a proof (depending on preliminary lemmas) of the fact that, in the figure of the proposition, the straight lines AL, BK, CF meet in a point. After iv. 16 there is a proof that, in a regular polygon with an even number of sides, the bisector of one angle also bisects its opposite, and an enunciation of the corresponding proposition for a regular polygon with an odd number of sides.

Van Pesch91 gives reason for attributing to Heron certain other notes found in Proclus, viz. that they are designed to meet the same sort of points as Heron had in view in other notes undoubtedly written by him. These are (a) alternative proofs of I. 5, I. 17, and I. 32, which avoid the producing of certain straight lines, (b) an alternative proof of 1.9 avoiding the construction of the equilateral triangle on the side of BC opposite to A; (c) partial converses of I. 35-38, starting from the equality of the areas and the fact of the parallelograms or triangles being in the same parallels, and proving that the bases are the same or equal, may also be Heron's. Van Pesch further supposes that it was in Heron's commentary that the proof by Menelaus of I. 25 and the proof by Philo of I. 8 were given.

The last reference to Heron made by an-Nairīzī occurs in the note on VIII. 27, so that the commentary of the former must at least have reached that point.

II. Porphyry.

The Porphyry here mentioned is of course the Neo-Platonist who lived about 232-304 A.D. Whether he really wrote a systematic commentary on the Elements is uncertain. The passages in Proclus which seem to make this probable are two in which he mentions him (1) as having demonstrated the necessity of the words “not on the same side” in the enunciation of I. 1492, and (2) as having pointed out the necessity of understanding correctly the enunciation of I. 26, since, if the particular injunctions as to the sides of the triangles to be taken as equal are not regarded, the student may easily fall into error93. These passages, showing that Porphyry carefully anaiysed Euclid's enunciations in these cases, certainly suggest that his remarks were part of a systematic commentary. Further, the list of mathematicians in the Fihrist gives Porphyry as having written “a book on the Elements.” It is true that Wenrich takes this book to have been a work by Porphyry mentioned by Suidas and Proclus (Theolog. Platon.), περὶ ἀρχῶν libri II.94.

There is nothing of importance in the notes attributed to Porphyry by Proclus.

(1) Three alternative proofs of I. 20, which avoid producing a side of the triangle, are assigned to Heron and Porphyry without saying which belonged to which. If the first of the three was Heron's, I agree with van Pesch that it is more probable that the two others were both Porphyry's than that the second was Heron's and only the third Porphyry's. For they are similar in character, and the third uses a result obtained in the second95.

(2) Porphyry gave an alternative proof of I. 18 to meet a childish objection which is supposed to require the part of AC equal to AB to be cut off from CA and not from AC.

Proclus gives a precisely similar alternative proof of I. 6 to meet a similar supposed objection; and it may well be that, though Proclus mentions no name, this proof was also Porphyry's, as van Pesch suggests96.

Two other references to Porphyry found in Proclus cannot have anything to do with commentaries on the Elements. In the first a work called the Συμμικτά is quoted, while in the second a philosophical question is raised.

III. Pappus.

The references to Pappus in Proclus are not numerous; but we have other evidence that he wrote a commentary on the Elements. Thus a scholiast on the definitions of the Data uses the phrase “as Pappus says at the beginning of his (commentary) on the 10th (book) of Euclid97.” Again in the Fihrist we are told that Pappus wrote a commentary to the tenth book of Euclid in two parts98. Fragments of this still survive in a MS. described by Woepcke99, Paris. No. 952. 2 (supplément arabe de la Bibliothèque impériale), which contains a translation by Abū `Uthmān (beginning of 10th century) of a Greek commentary on Book X. It is in two books, and there can now be no doubt that the author of the Greek commentary was Pappus100. Again Eutocius, in his note on Archimedes, On the Sphere and Cylinder I. 13, says that Pappus explained in his commentary on the Elements how to inscribe in a circle a polygon similar to a polygon inscribed in another circle; and this would presumably come in his commentary on Book XII., just as the problem is solved in the second scholium on Eucl. XII. I. Thus Pappus' commentary on the Elements must have been pretty complete, an additional confirmation of this supposition being forthcoming in the reference of Marinus (a pupil and follower of Proclus) in his preface to the Data to “the commentaries of Pappus on the book101.”

The actual references to Pappus in Proclus are as follows:

(1) On the Postulate (4) that all right angles are equal, Pappus is quoted as saying that the converse, viz. that all angles equal to a right angle are right, is not true102, since the angle included between the arcs of two semicircles which are equal, and have their diameters at right angles and terminating at one point, is equal to a right angle, but is not a right angle.

(2) On the axioms Pappus is quoted as saying that, in addition to Euclid's axioms, others are on record as well (συναναγράφεσθαι) about unequals added to equals and equals added to unequals103; these, says Proclus, follow from the Euclidean axioms, while others given by Pappus are involved by the definitions, namely those which assert that “all parts of the plane and of the straight line coincide with one another,” that “a point divides a straight line, a line a surface, and a surface a solid,” and that “the infinite is (obtained) in magnitudes both by addition and diminution104.”

(3) Pappus gave a pretty proof of 1. 5. This proof has, I think, been wrongly understood; on this point see my note on the proposition.

(4) On 1. 47 Proclus says105: “As the proof of the writer of the Elements is manifest, I think that it is not necessary to add anything further, but that what has been said is sufficient, since indeed those who have added more, like Heron and Pappus, were obliged to make use of what is proved in the sixth book, without attaining any important result.” We shall see what Heron's addition consisted of; what Pappus may have added we do not know, unless it was something on the lines of his extension of 1. 47 found in the Synagoge (IV. p. 176, ed. Hultsch).

We may fairly conclude, with van Pesch106, that Pappus is drawn upon in various other passages of Proclus where he quotes no authority, but where the subject-matter reminds us of other notes expressly assigned to Pappus or of what we otherwise know to have been favourite questions with him. Thus:

1. We are reminded of the curvilineal angle which is equal to but not a right angle by the note on 1. 32 to the effect that the converse (that a figure with its interior angles together equal to two right angles is a triangle) is not true unless we confine ourselves to rectilineal figures. This statement is supported by reference to a figure formed by four semicircles whose diameters form a square, and one of which is turned inwards while the others are turned outwards. The figure forms two angles “equal to” right angles in the sense described by Pappus on Post. 4, while the other curvilineal angles are not considered to be angles at all, and are left out in summing the internal angles. Similarly the allusions in the notes on 1. 4, 23 to curvilineal angles of which certain moon-shaped angles (μηνοειδεῖς) are shown to be “equal to” rectilineal angles savour of Pappus.

2. On 1. 9 Proclus says107 that “Others, starting from the Archimedean spirals, divided any given rectilineal angle in any given ratio.” We cannot but compare this with Pappus iv. p. 286, where the spiral is so used; hence this note, including remarks immediately preceding about the conchoid and the quadratrix, which were used for the same purpose, may very well be due to Pappus.

3. The subject of isoperimetric figures was a favourite one with Pappus, who wrote a recension of Zenodorus' treatise on the subject108. Now on I. 35 Proclus speaks109 about the paradox of parallelograms having equal area (between the same parallels) though the two sides between the parallels may be of any length, adding that of parallelograms with equal perimeter the rectangle is greatest if the base be given, and the square greatest if the base be not given etc. He returns to the subject on 1. 37 about triangles110. Compare111 also his note on 1. 4. These notes may have been taken from Pappus.

4. Again, on 1. 21, Proclus remarks on the paradox that straight lines may be drawn from the base to a point within a triangle which are (1) together greater than the two sides, and (2) include a less angle, provided that the straight lines may be drawn from points in the base other than its extremities. The subject of straight lines satisfying condition (1) was treated at length, with reference to a variety of cases, by Pappus112, after a collection of “paradoxes” by Erycinus, of whom nothing more is known. Proclus gives Pappus' first case, and adds a rather useless proof of the possibility of drawing straight lines satisfying condition (2) alone, adding that “the proposition stated has been proved by me without using the parallels of the commentators113.” By “the commentators” Pappus is doubtless meant.

5. Lastly, the “four-sided triangle,” called by Zenodorus the “hollow-angled,” 114 is mentioned in the notes on 1. Def. 24-29 and I. 21. As Pappus wrote on Zenodorus' work in which the term occurred115, Pappus may be responsible for these notes.

IV. Simplicius.

According to the Fihrist116, Simplicius the Greek wrote “a commentary to the beginning of Euclid's book, which forms an introduction to geometry.” And in fact this commentary on the definitions, postulates and axioms (including the postulate known as the ParallelAxiom) is preserved in the Arabic commentary of an-Nairĩzĩ117. On two subjects this commentary of Simplicius quotes a certain “.Aganis,” the first subject being the definition of an angle, and the second the definition of parallels and the parallel-postulate. Simplicius gives word for word, in a long passage placed by an-Nairīzī after 1. 29, an attempt by “Aganis” to prove the parallel-postulate. It starts from a definition of parallels which agrees with Geminus' view of them as given by Proclus118, and is closely connected with the definition given by Posidonius119. Hence it has been assumed that “Aganis” is none other than Geminus, and the historical importance of the commentary of Simplicius has been judged accordingly. But it has been recently shown by Tannery that the identification of “Aganis” with Geminus is practically impossible120 In the translation of Besthorn-Heiberg Aganis is called by Simplicius in one place “philosophus Aganis,” in another “magister noster Aganis,” in Gherard's version he is “socius Aganis” and “socius noster Aganis.” These expressions seem to leave no doubt that Aganis was a contemporary and friend, if not master, of Simplicius; and it is impossible to suppose that Simplicius (fl. about 500 A.D.) could have used them of a man who lived four and a half centuries before his time. A phrase in Simplicius' word-forword quotation from Aganis leads to the same conclusion. He speaks of people who objected “even in ancient times” (iam antiquitus) to the use by geometers of this postulate. This would not have been an appropriate phrase had Geminus been the writer. I do not think that this difficulty can he got over by Suter's suggestion121 that the passages in question may have been taken out of Heron's commentary, and that an-Nairĩzĩ may have forgotten to name the author; it seems clear that Simplicius is the person who described “Aganis.” Hence we are driven to suppose that Aganis was not Geminus, but some unknown contemporary of Simplicius122 Considerable interest will however continue to attach to the comments of Simplicius so fortunately preserved.

Proclus tells us that one Aegaeas (? Aenaeas) of Hierapolis wrote an epitome of the Elements123; but we know nothing more of him or of it.

CHAPTER IV.

PROCLUS AND HIS SOURCES124.

It is well known that the commentary of Proclus on Eucl. Book I. is one of the two main sources of information as to the history of Greek geometry which we possess, the other being the Collection of Pappus. They are the more precious because the original works of the forerunners of Euclid, Archimedes and Apollonius are lost, having probably been discarded and forgotten almost immediately after the appearance of the masterpieces of that great trio.

Proclus himself lived 410-485 A.D., so that there had already passed a sufficient amount of time for the tradition relating to the pre-Euclidean geometers to become obscure and defective. In this connexion a passage is quoted from Simplicius125 who, in his account of the quadrature of certain lunes by Hippocrates of Chios, while mentioning two authorities for his statements, Alexander Aphrodisiensis (about 220 A.D.) and Eudemus, says in one place126, “As regards Hippocrates of Chios we must pay more attention to Eudemus, since he was nearer the times, being a pupil of Aristotle.”

The importance therefore of a critical examination of Proclus' commentary with a view to determining from what original sources he drew need not be further emphasised.

Proclus received his early training in Alexandria, where Olympiodorus was his instructor in the works of Aristotle, and mathematics was taught him by one Heron127 (of course a different Heron from the “mechanicus Hero” of whom we have already spoken). He afterwards went to Athens where he was imbued by Plutarch, and by Syrianus, with the Neo-Platonic philosophy, to which he then devoted heart and soul, becoming one of its most prominent exponents. He speaks everywhere with the highest respect of his masters, and was in turn regarded with extravagant veneration by his contemporaries, as we learn from Marinus his pupil and biographer. On the death of Syrianus he was put at the head of the Neo-Platonic school. He was a man of untiring industry, as is shown by the number of books which he wrote, including a large number of commentaries, mostly on the dialogues of Plato. He was an acute dialectician, and pre-eminent among his contemporaries in the range of his learning128; he was a competent mathematician; he was even a poet. At the same time he was a believer in all sorts of myths and mysteries and a devout worshipper of divinities both Greek and Oriental.

Though he was a competent mathematician, he was evidently much more a philosopher than a mathematician129. This is shown even in his commentary on Eucl. I., where, not only in the Prologues (especially the first), but also in the notes themselves, he seizes any opportunity for a philosophical digression. He says himself that he attaches most importance to “the things which require deeper study and contribute to the sum of philosophy130” ; alternative proofs, cases, and the like (though he gives many) have no attraction for him; and, in particular, he attaches no value to the addition of Heron to 1. 47131, which is of considerable mathematical interest. Though he esteemed mathematics highly, it was only as a handmaid to philosophy. He quotes Plato's opinion to the effect that “mathematics, as making use of hypotheses, falls short of the non-hypothetical and perfect science132” ... “Let us then not say that Plato excludes mathematics from the sciences, but that he declares it to be secondary to the one supreme science133.” And again, while “mathematical science must be considered desirable in itself, though not with reference to the needs of daily life,” “if it is necessary to refer the benefit arising from it to something else, we must connect that benefit with intellectual knowledge (νοερὰν γνῶσιν), to which it leads the way and is a propaedeutic, clearing the eye of the soul and taking away the impediments which the senses place in the way of the knowledge of universals (τῶν ὅλων134.”

We know that in the Neo-Platonic school the younger pupils learnt mathematics; and it is clear that Proclus taught this subject, and that this was the origin of the commentary. Many passages show him as a master speaking to scholars. Thus “we have illustrated and made plain all these things in the case of the first problem, but it is necessary that my hearers should make the same inquiry as regards the others as well135,” and “I do not indicate these things as a merely incidental matter but as preparing us beforehand for the doctrine of the Timaeus136.” Further, the pupils whom he was addressing were beginners in mathematics; for in one place he says that he omits “for the present” to speak of the discoveries of those who employed the curves of Nicomedes and Hippias for trisecting an angle, and of those who used the Archimedean spiral for dividing an angle in any given ratio, because these things would be too difficult for beginners (δυσθεωρήτους τοῖς εἰσαγομένοις137. Again, if his pupils had not been beginners, it would not have been necessary for Proclus to explain what is meant by saying that sides subtend certain angles138, the difference between adjacent and uertical angles139 etc., or to exhort them, as he often does, to work out other particular cases for themselves, for practice (γυμνασίας ἕνεκα140.

The commentary seems then to have been founded on Proclus' lectures to beginners in mathematics. But there are signs that it was revised and re-edited for a larger public; thus he gives notice in one place141 “to those who shall come upon” his work (τοῖς ἐντευξομένοις). There are also passages which could not have been understood by the beginners to whom he lectured, e.g. passages about the cylindrical helix142, conchoids and cissoids143. These passages may have been added in the revised edition, or, as van Pesch conjectures, the explanations given in the lectures may have been much fuller and more comprehensible to beginners, and they may have been shortened on revision.

In his comments on the propositions of Euclid, Proclus generally proceeds in this way: first he gives explanations regarding Euclid's proofs, secondly he gives a few different cases, mainly for the sake of practice, and thirdly he addresses himself to refuting objections raised by cavillers to particular propositions. The latter class of note he deems necessary because of “sophistical cavils” and the attitude of the people who rejoiced in finding paralogisms and in causing annoyance to scientific men144. His commentary does not seem to have been written for the purpose of correcting or improving Euclid. For there are very few passages of mathematical content in which Proclus can be supposed to be propounding anything of his own; nearly all are taken from the works of others, mostly earlier commentators, so that, for the purpose of improving on or correcting Euclid, there was no need for his commentary at all. Indeed only in one place does he definitely bring forward anything of his own to get over a difficulty which he finds in Euclid145; this is where he tries to prove the parallel-postulate, after first giving Ptolemy's attempt and then pointing out objections to it. On the other hand, there are a number of passages in which he extols Euclid; thrice146 also he supports Euclid against Apollonius where the latter had given proofs which he considered better than Euclid's (1. 10, 11, and 23).

Allusion must be made to the debated question whether Proclus continued his commentaries beyond Book 1. His intention to do so is clear from the following passages. Just after the words above quoted about the trisection etc. of an angle by means of certain curves he says, “For we may perhaps more appropriately examine these things on the third book, where the writer of the Elements bisects a given circumference147.” Again, after saying that of all parallelograms which have the same perimeter the square is the greatest “and the rhomboid least of all,” he adds: “But this we will prove in another place; for it is more appropriate to the (discussion of the) hypotheses of the second book148.” Lastly, when alluding (on I. 45) to the squaring of the circle, and to Archimedes' proposition that any circle is equal to the right-angled triangle in which the perpendicular is equal to the radius of the circle and the base to its perimeter, he adds, “But of this elsewhere149” ; this may imply an intention to treat of the subject on Eucl. XII., though Heiberg doubts it150. But it is clear that, at the time when the commentary on Book 1. was written, Proclus had not yet begun to write on the other Books and was uncertain whether he would be able to do so: for at the end he says151, “For my part, if I should be able to discuss the other books152 in the same manner, I should give thanks to the gods; but, if other cares should draw me away, I beg those who are attracted by this subject to complete the exposition of the other books as well, following the same method, and addressing themselves throughout to the deeper and better defined questions involved” (τὸ πραγματειῶδες πανταχοῦ καὶ εὐδιαίρετον μεταδιώκοντας).

There is in fact no satisfactory evidence that Proclus did actually write any more commentaries than that on Book 1.153 The contrary view receives support from two facts pointed out by Heiberg, viz. (1) that the scholiast's copy of Proclus was not so much better than our MSS. as to suggest that the scholiast had further commentaries of Proclus which have vanished for us154; (2) that there is no trace in the scholia of the notes which Proclus promised in the passages quoted above.

Coming now to the question of the sources of Proclus, we may say that everything goes to show that his commentary is a compilation, though a compilation “in the better sense” of the term155. He does not even give us to understand that we shall find in it much of his own; “let us,” he says, “now turn to the exposition of the theorems proved by Euclid, selecting the more subtle of the comments made on them by the ancient writers, and cutting down their interminable diffuseness...156” : not a word about anything of his own. At the same time, he seems to imply that he will not necessarily on each occasion quote the source of each extract from an earlier commentary; and, in fact, while he quotes the name of his authority in many places, especially where the subject is important, in many others, where it is equally certain that he is not giving anything of his own, he mentions no authority. Thus he quotes Heron by name six times; but we now know, from the commentary of an-Nairīzī, that a number of other passages, where he mentions no name, are taken from Heron, and among them the not unimportant addition of an alternative proof to 1. 19. Hence we can by no means conclude that, where no authority is mentioned, Proclus is giving notes of his own. The presumption is génerally the other way; and it is often possible to arrive at a conclusion, either that a particular note is not Proclus' own, or that it is definitely attributable to someone else, by applying the ordinary principles of criticism. Thus, where the note shows an unmistakable affinity to another which Proclus definitely attributes to some commentator by name, especially when both contain some peculiar and distinctive idea, we cannot have much doubt in assigning both to the same commentator157. Again, van Pesch finds a criterion in the form of a note, where the explanation is so condensed as to be only just intelligible; the note is that in which a converse of 1. 32 is proved158 the proposition namely that a rectilineal figure which has all its interior angles together equal to two right angles is a triangle.

It is not safe to attribute a passage to Proclus himself because he uses the first person in such expressions as “I say” or “I will prove” —for he was in the habit of putting into his own words the substance of notes borrowed from others—nor because, in speaking of an objection raised to a particular proposition, he uses such expressions as “perhaps someone may object” (ἴσως δ̓ ἄν τινες ἐνσταῖεν...): for sometimes other words in the same passage indicate that the objection had actually been taken by someone159. Speaking generally, we shall not be justified in concluding that Proclus is stating something new of his own unless he indicates this himself in express terms.

As regards the form of Proclus' references to others by name, van Pesch notes that he very seldom mentions the particular work from which he is borrowing. If we leave out of account the references to Plato's dialogues, there are only the following references to books: the Bacchae of Philolaus160, the Symmikta of Porphyry161, Archimedes On the Sphere and Cylinder162, Apollonius On the cochlias163, a book by Eudemus on The Angle164, a whole book of Posidonius directed against Zeno of the Epicurean sect165, Carpus' Astronomy166, Eudemus' History of Geometry167, and a tract by Ptolemy on the parallel-postulate168.

Again, Proclus does not always indicate that he is quoting something at second-hand. He often does so, e.g. he quotes Heron as the authority for a statement about Philippus, Eudemus as attributing a certain theorem to Oenopides etc.; but he says on 1. 12 that “Oenopides first investigated this problem, thinking it useful for astronomy” when he cannot have had Oenopides' work before him.

It has been said above that Proclus was in the habit of stating in his own words the substance of the things which he borrowed. We are prepared for this when we find him stating that he will select the best things from ancient commentaries and “cut short their interminable diffuseness,” that he will “briefly describe” (συντομως ἱστορῆσαι) the other proofs of 1. 20 given by Heron and Porphyry and also the proofs of 1. 25 by Menelaus and Heron. But the best evidence is of course to be found in the passages where he quotes works still extant, e.g. those of Plato, Aristotle and Plotinus. Examination of these passages shows great divergences from the original; even where he purports to quote textually, using the expressions “Plato says,” or “Plotinus says,” he by no means quotes word for word169. In fact, he seems to have had a positive distaste for quoting textually from other works. He cannot conquer this even when quoting from Euclid; he says in his note on 1. 22, “we will follow the words of the geometer” but fails, nevertheless, to reproduce the text of Euclid unchanged170.

We now come to the sources themselves from which Proclus drew in writing his commentary. Three have already been disposed of, viz. Heron, Porphyry and Pappus, who had all written commentaries on the Elements171. We go on to

Eudemus, the pupil of Aristotle, who, among other works, wrote a history of arithmetic, a history of astronomy, and a history of geometry. The importance of the last mentioned work is attested by the frequent use made of it by ancient writers. That there was no other history of geometry written after the time of Eudemus seems to be proved by the remark of Proclus in the course of his famous summary: “Those who compiled histories bring the development of this science up to this point. Not much younger than these is Euclid172....” The loss of Eudemus' history is one of the gravest which fate has inflicted upon us, for it cannot be doubted that Eudemus had before him a number of the actual works of earlier geometers, which, as before observed, seem to have vanished completely when they were superseded by the treatises of Euclid, Archimedes and Apollonius. As it is, we have to be thankful for the fragments from Eudemus which such writers as Proclus have preserved to us.

I agree with van Pesch173 that there is no sufficient reason for doubting that the work of Eudemus was accessible to Proclus at first hand. For the later writers Simplicius and Eutocius refer to it in terms such as leave no room for doubt that they had it before them. I have already quoted a passage from Simplicius' account of the lunes of Hippocrates to the effect that Eudemus must be considered the best authority since he lived nearer the times174. In the same place Simplicius says175, “I will set out what Eudemus says word for word (κατὰ λέξιν λεγόμενα), adding only a little explanation in the shape of reference to Euclid's Elements owing to the memorandum-like style of Eudemus (διὰ τὸν ὑπομνηματικὸν τρόπον τοῦ Εὐδήμου) who sets out his explanations in the abbreviated form usual with ancient writers. Now in the second book of the history of geometry he writes as follows176.” It is not possible to suppose that Simplicius would have written in this way about the style of Eudemus if he had merely been copying certain passages second-hand out of some other author and had not the original work itself to refer to. In like manner, Eutocius speaks of the paralogisms handed down in connexion with the attempts of Hippocrates and Antiphon to square the circle177, “with which I imagine that those are accurately acquainted who have examined (ἐπεσκεμμένους) the geometrical history of Eudemus and know the Ceria Aristotelica.” How could the contemporaries of Eutocius have examined the work of Eudemus unless it was still extant in his time?

The passages in which Proclus quotes Eudemus by name as his authority are as follows:

(1) On 1. 26 he says that Eudemus in his history of geometry referred this theorem to Thales, inasmuch as it was necessary to Thales' method of ascertaining the distance of ships from the shore178.

(2) Eudemus attributed to Thales the discovery of Eucl. 1. 15179, and

(3) to Oenopides the problem of 1. 23180.

(4) Eudemus referred the discovery of the theorem in 1. 32 to the Pythagoreans, and gave their proof of it, which Proclus reproduees181.

(5) On 1. 44 Proclus tells us182 that Eudemus says that “these things are ancient, being discoveries of the Pythagorean muse, the application (παραβολή) of areas, their exceeding (ὑπερβολή) and their falling short (ἔλλειψις).” The next words about the appropriation of these terms (parabola, hyperbola and ellipse) by later writers (i.e. Apollonius) to denote the conic sections are of course not due to Eudemus.

Coming now to notes where Eudemus is not named by Proclus, we may fairly conjecture, with van Pesch, that Eudemus was really the authority for the statements (1) that Thales first proved that a circle is bisected by its diameter183 (though the proof by reductio ad absurdum which follows in Proclus cannot be attributed to Thales184), (2) that “Plato made over to Leodamas the analytical method, by means of which it is recorded (ἱστόρηται) that the latter too made many discoveries in geometry185,” (3) that the theorem of 1. 5 was due to Thales, and that for equal angles he used the more archaic expression “similar” angles186, (4) that Oenopides first investigated the problem of 1. 12, and that he called the perpendicular the gnomonic line (κατὰ γνώμονα187, (5) that the theorem that only three sorts of polygons can fill up the space round a point, viz. the equilateral triangle, the square and the regular hexagon, was Pythagorean188. Eudemus may also be the authority for Proclus' description of the two methods, referred to Plato and Pythagoras respectively, of forming right-angled triangles in whole numbers189.

We cannot attribute to Eudemus the beginning of the note on 1. 47 where Proclus says that “if we listen to those who like to recount ancient history, we may find some of them referring this theorem to Pythagoras and saying that he sacrificed an ox in honour of his discovery190.” As such a sacrifice was contrary to the Pythagorean tenets, and Eudemus could not have been unaware of this, the story cannot rest on his authority. Moreover Proclus speaks as though he were not certain of the correctness of the tradition; indeed, so far as the story of the sacrifice is concerned, the same thing is told of Thales in connexion with his discovery that the angle in a semicircle is a right angle191, and Plutarch is not certain whether the ox was sacrificed on the discovery of I. 47 or of the problem about application of areas192. Plutarch's doubt suggests that he knew of no evidence for the story beyond the vague allusion in the distich of Apollodorus “Logisticus” (the “calculator” ) cited by Diogenes Laertius also193; and Proclus may have had in mind this couplet with the passages of Plutarch.

We come now to the question of the famous historical summary given by Proclus194. No one appears to maintain that Eudemus is the author of even the early part of this summary in the form in which Proclus gives it. It is, as is well known, divided into two distinct parts, between which comes the remark, “Those who compiled histories195 bring the development of this science up to this point. Not much younger than these is Euclid, who put together the Elements, collecting many of the theorems of Eudoxus, perfecting many others by Theaetetus, and bringing to irrefragable demonstration the things which had only been somewhat loosely proved by his predecessors.” Since Euclid was later than Eudemus, it is impossible that Eudemus can have written this. Yet the style of the summary after this point does not show any such change from that of the former portion as to suggest different authorship. The author of the earlier portion recurs frequently to the question of the origin of the elements of geometry in a way in which no one would be likely to do who was not later than Euclid; and it must be the same hand which in the second portion connects Euclid's Elements with the work of Eudoxus and Theaetetus196.

If then the summary is the work of one author, and that author not Eudemus, who is it likely to have been? Tannery answers that it is Geminus197; but I think, with van Pesch, that he has failed to show why it should be Geminus rather than another. And certainly the extracts which we have from Geminus' work suggest that the sort of topics which it dealt with was quite different; they seem rather to have been general questions of the content of mathematics, and even Tannery admits that historical details could only have come incidentally into the work198.

Could the author have been Proclus himself? Circumstances which seem to suggest this possibility are (1) that, as already stated, the question of the origin of the Elements is kept prominent, (2) that there is no mention of Democritus, whom Eudemus would not be likely to have ignored, while a follower of Plato would be likely enough to do him the injustice, following the example of Plato who was an opponent of Democritus, never once mentions him, and is said to have wished to burn all his writings199, and (3) the allusion at the beginning to the “inspired Aristotle” ( δαιμονιος Ἀριστοτέλης200, though this may easily have been inserted by Proclus in a quotation made by him from someone else. On the other hand there are considerations which suggest that Proclus himself was not the writer. (1) The style of the whole passage is not such as to point to him as the author. (2) If he wrote it, it is hardly conceivable that he would have passed over in silence the discovery of the analytical method, the invention of Plato to which he attached so much importance201.

There is nothing improbable in the conjecture that Proclus quoted the summary from a compendium of Eudemus' history made by some later writer: but as yet the question has not been definitely settled. All that is certain is that the early part of the summary must have been made up from scattered notices found in the great work of Eudemus.

Proclus refers to another work of Eudemus besides the history, viz. a book on The Angle (βιβλίον περὶ γωνίας202. Tannery assumes that this must have been part of the history, and uses this assumption to confirm his idea that the history was arranged according to subjects, not according to chronological order203. The phraseology of Proclus however unmistakably suggests a separate work; and that the history was chronologically arranged seems to be clearly indicated by the remark of Simplicius that Eudemus “also counted Hippocrates among the more ancient writers” (ἐν τοῖς παλαιοτέροις204.

The passage of Simplicius about the lunes of Hippocrates throws considerable light on the style of Eudemus' history. Eudemus wrote in a memorandum-like or summary manner (τὸν ὑπομνηματικὸν τρόπον τοῦ Εὐδήμου205 when reproducing what he found in the ancient writers; sometimes it is clear that he left out altogether proofs or constructions of things by no means easy206.

Geminus.

The discussions about the date and birthplace of Geminus form a whole literature, as to which I must refer the reader to Manitius and Tittel207. Though the name looks like a Latin name (Gem[icaron]nus), Manitius concluded that, since it appears as Γεμῖνος in all Greek MSS. and as Γεμεῖνος in some inscriptions, it is Greek and possibly formed from γεμ as Ἐργῖνος is from ἐργ and Ἀλεξῖνος from ἀλεξ (cf. also Ἰκτῖνος, Κρατῖνος). Tittel is equally positive that it is Gem[icaron]nus and suggests that Γεμῖνος is due to a false analogy with Ἀλεξῖνος etc. and Γεμεῖνος wrongly formed on the model of Ἀντωνεῖνος, Ἀγριππεῖνα. Geminus, a Stoic philosopher, born probably in the island of Rhodes, was the author of a comprehensive work on the classification of mathematics, and also wrote, about 73-67 B.C., a not less comprehensive commentary on the meteorological textbook of his teacher Posidonius of Rhodes.

It is the former work in which we are specially interested here. Though Proclus made great use of it, he does not mention its title, unless we may suppose that, in the passage (p. 177, 24) where, after quoting from Geminus a classification of lines which never meet, he says, “these remarks I have selected from the φιλοκαλία of Geminus,” φιλοκαλία is a title or an alternative title. Pappus however quotes a work of Geminus “on the classification of the mathematics” (ἐν τῷ περὶ τῆς τῶν μαθημάτων τάξεως208, while Eutocius quotes from “the sixth book of the doctrine of the mathematics” (ἐν τῷ ἕκτῳ τῆς τῶν μαθημάτων θεωρίας209. Tannery210 pointed out that the former title corresponds well enough to the long extract211 which Proclus gives in his first prologue, and also to the fragments contained in the Anonymi variae collectiones published by Hultsch at the end of his edition of Heron212; but it does not suit most of the óther passages borrowed by Proclus. The correct title was therefore probably that given by Eutocius, The Doctrine, or Theory, of the Mathematics; and Pappus probably refers to one particular portion of the work, say the first Book. If the sixth Book treated of conics, as we may conclude from Eutocius, there must have been more Books to follow, because Proclus has preserved us details about higher curves, which must have come later. If again Geminus finished his work and wrote with the same fulness about the other branches of mathematics as he did about geometry, there must have been a considerable number of Books altogether. At all events it seems to have been designed to give a complete view of the whole science of mathematics, and in fact to be a sort of encyclopaedia of the subject.

I shall now indicate first the certain, and secondly the probable, obligations of Proclus to Geminus, in which task I have only to follow van Pesch, who has embodied the results of Tittel's similar inquiry also213. I shall only omit the passages as regards which a case for attributing them to Geminus does not seem to me to have been made out.

First come the following passages which must be attributed to Geminus, because Proclus mentions his name:

(1) (In the first prologue of Proclus214) on the division of mathematical sciences into arithmetic, geometry, mechanics, astronomy, optics, geodesy, canonic (science of musical harmony), and logistic (apparently arithmetical problems);

(2) (in the note on the definition of a straight line) on the classification of lines (including curves) as simple (straight or circular) and mixed, composite and incomposite, uniform (ὁμοιομερεῖς) and non-uniform (ἀνομοιομερεῖς), lines “about solids” and lines produced by cutting solids, including conic and spiric sections215;

(3) (in the note on the definition of a plane surface) on similar distinctions extended to surfaces and solids216;

(4) (in the note on the definition of parallels) on lines which do not meet (ἀσύμπτωτοι) but which are not on that account parallel, e.g. a curve and its asymptote, showing that the property of not meeting does not make lines parallel—a favourite observation of Geminus—and, incidentally, on bounded lines or those which enclose a figure and those which do not217;

(5) (in the same note) the definition of parallels given by Posidonius218;

(6) on the distinction between postulates and axioms, the futility of trying to prove axioms, as Apollonius tried to prove Axiom 1, and the equal incorrectness of assuming what really requires proof, “as Euclid did in the fourth postulate [equality of right angles] and in the fifth postulate [the parallel-postulate]219” ;

(7) on Postulates 1, 2, 3, which Geminus makes depend on the idea of a straight line being described by the motion of a point220;

(8) (in the note on Postulate 5) on the inadmissibility in geometry of an argument which is merely plausible, and the danger in this particular case owing to the existence of lines which do converge ad infinitum and yet never meet221;

(9) (in the note on I. 1) on the subject-matter of geometry, theorems, problems and διορισμοί (conditions of possibility) for problems222;

(10) (in the note on 1. 5) on a generalisation of 1. 5 by Geminus through the substitution for the rectilineal base of “one uniform line (curve),” by means of which he proved that the only “uniform lines” (alike in all their parts) are a straight line, a circle, and a cylindrical helix223;

(11) (in the note on I. 10) on the question whether a line is made up of indivisible parts (ἀμερῆ), as affecting the problem of bisecting a given straight line224;

(12) (in the note on I. 35) on topical, or locus-theorems225, where the illustration of the equal parallelograms described between a hyperbola and its asymptotes may also be due to Geminus226.

Other passages which may fairly be attributed to Geminus, though his name is not mentioned, are the following:

(1) in the prologue, where there is the same allusion as in the passage (8) above to a remark of Aristotle that it is equally absurd to expect scientific proofs from a rhetorician and to accept mere plausibilities from a geometer227;

(2) a passage in the prologue about the subject-matter, methods, and bases of geometry, the latter including axioms and postulates228;

(3) another on the definition and nature of elemenis229;

(4) a remark on the Stoic use of the term axiom for every simple statement (ἀπόφανσις ἁπλῆ230;

(5) another discussion on theorems and problems231 in the middle of which however there are some sentences by Proclus himself232.

(6) another passage, in connexion with Def. 3, on lines including or not including a figure (with which cf. part of the passage (4) above)233;

(7) a classification of different sorts of angles according as they are contained by simple or mixed lines (or curves)234;

(8) a similar classification of figures235, and of plane figures236;

(9) Posidonius' definition of a figure237;

(10) a classification of triangles into seven kinds238;

(11) a note distinguishing lines (or curves) producible indefinitely or not so producible, whether forming a figure or not forming a figure (like the “single-turn spiral” )239;

(12) passages distinguishing different sorts of problems240, different sorts of theorems241, and two sorts of converses (complete and partial)242;

(13) the definition of the term “porism” as used in the title of Euclid's Porisms, as distinct from the other meaning of “corollary” 243;

(14) a note on the Epicurean objection to I. 20 as being obvious even to an ass244;

(15) a passage on the properties of parallels, with allusions to Apollonius' Conics, and the curves invented by Nicomedes, Hippias and Perseus245;

(16) a passage on the parallel-postulate regarded as the converse of I. 17246.

Of the authors to whom Proclus was indebted in a less degree the most important is Apollonius of Perga. Two passages allude to his Conics247, one to a work on irrationals248, and two to a treatise On the cochlias (apparently the cylindrical helix) by Apollonius249. But more important for our purpose are six references to Apollonius in connexion with elementary geometry.

(1) He appears as the author of an attempt to explain the idea of a line (possessing length but no breadth) by reference to daily experience, e.g. when we tell someone to measure, merely, the length of a road or of a wall250; and doubtless the similar passage showing how we may in like manner get a notion of a surface (without depth) is his also251.

(2) He gave a new general definition of an angle252.

(3) He tried to prove certain axioms253, and Proclus gives his attempt to prove Axiom I, word for word254.

Proclus further quotes:

(4) Apollonius' solution of the problem in Eucl. I. 10, avoiding Euclid's use of I. 9255,

(5) his solution of the problem in I. 11, differing only slightly from Euclid's256, and

(6) his solution of the problem in I. 23257.

Heiberg258 conjectures that Apollonius departed from Euclid's method in these propositions because he objected to solving problems of a more general, by means of problems of a more particular, character. Proclus however considers all three solutions inferior to Euclid's; and his remarks on Apollonius' handling of these elementary matters generally suggest that he was nettled by criticisms of Euclid in the work containing the things which he quotes from Apollonius, just as we conclude that Pappus was offended by the remarks of Apollonius about Euclid's incomplete treatment of the “three- and four-line locus259.” If this was the case, Proclus can hardly have got his information about these things at second-hand; and there seems to be no reason to doubt that he had the actual work of Apollonius before him. This work may have been the treatise mentioned by Marinus in the words “Apollonius in his general treatise” (Ἀπολλώνιος ἐν τῇ καθόλου πραγματείᾳ260. If the notice in the Fihrist261 stating, on the authority of Thābit b. Qurra, that Apollonius wrote a tract on the parallel-postulate be correct, it may have been included in the same work. We may conclude generally that, in it, Apollonius tried to remodel the beginnings of geometry, reducing the number of axioms, appealing, in his definitions of lines, surfaces etc., more to experience than to abstract reason, and substituting for certain proofs others of a more general character.

The probabilities are that, in quoting from the tract of Ptolemy in which he tried to prove the parallel-postulate, Proclus had the actual work before him. For, after an allusion to it as “a certain book262” he gives two long extracts263, and at the beginning of the second indicates the title of the tract, “in the (book) about the meeting of straight lines produced from (angles) less than two right angles,” as he has very rarely done in other cases.

Certain things from Posidonius are evidently quoted at secondhand, the authority being Geminus (e.g. the definitions of figure and parallels); but besides these we have quotations from a separate work which he wrote to controvert Zeno of Sidon, an Epicurean who had sought to destroy the whole of geometry264. We are told that Zeno had argued that, even if we admit the fundamental principles (ἀρχαί) of geometry, the deductions from them cannot be proved without the admission of something else as well, which has not been included in the said principles265. On I. 1 Proclus gives at some length the arguments of Zeno and the reply of Posidonius as regards this proposition266. In this case Zeno's “something else” which he considers to be assumed is the fact that two straight lines cannot have a common segment, and then, as regards the “proof” of it by means of the bisection of a circle by its diameter, he objects that it has been assumed that two circumferences (arcs) of circles cannot have a common part. Lastly, he makes up, for the purpose of attacking it, another supposed “proof” of the fact that two straight lines cannot have a common part. Proclus appears, more than once, to be quoting the actual words of Zeno and Posidonius; in particular, two expressions used by Posidonius about “the acrid Epicurean” (τὸν δριμὺν Ἐπικούρειον267 and his “misrepresentations” (Ποσειδώνιός φησι τὸν Ζήνωνα συκοφαντεῖν268. It is not necessary to suppose that Proclus had the original work of Zeno before him, because Zeno's arguments may easily have been got from Posidonius' reply; but he would appear to have quoted direct from the latter at all events.

The work of Carpus mechanicus (a treatise on astronomy) quoted from by Proclus269 must have been accessible to him at first-hand, because a portion of the extract from it about the relation of theorems and problems270 is reproduced word for word. Moreover, if he were not using the book itself, Proclus would hardly be in a position to question whether the introduction of the subject of theorems and problems was opportune in the place where it was found (εἰ μὲν κατὰ καιρὸν μή, παρείσθω πρὸς τὸ παρόν271.

It is of course evident that Proclus had before him the original works of Plato, Aristotle, Archimedes and Plotinus, as well as the Συμμικτά of Porphyry and the works of his master Syrianus ( ἡμέτερος καθηγεμών272, from whom he quotes in his note on the definition of an angle. Tannery also points out that he must have had before him a group of works representing the Pythagorean tradition on its mystic, as distinct from its mathematical, side, from Philolaus downwards, and comprising the more or less apocryphal ἱερὸς λόγος of Pythagoras, the Oracles (λόγια), and Orphic verses273.

Besides quotations from writers whom we can identify with more or less certainty, there are many other passages which are doubtless quoted from other commentators whose names we do not know. A list of such passages is given by van Pesch274, and there is no need to cite them here.

Van Pesch also gives at the end of his work275 a convenient list of the books which, as the result of his investigation, he deems to have been accessible to and directly used by Proclus. The list is worth giving here, on the same ground of convenience. It is as follows: “Eudemus: history of geometry. Geminus: the theory of the mathematical sciences. Heron: commentary on the Elements of Euclid. Porphyry: commentary on the Elements of Euclid. Pappus: commentary on the Elements of Euclid. Apollonius of Perga: a work relating to elementary geometry. Ptolemy: on the parallel-postulate. Posidonius: a book controverting Zeno of Sidon. Carpus: astronomy. Syrianus: a discussion on the angle. Pythagorean philosophical tradition. Plato's works. Aristotle's works. Archimedes' works. Plotinus: Enneades.”

Lastly we come to the question what passages, if any, in the commentary of Proclus represent his own contributions to the subject. As we have seen, the onus probandi must be held to rest upon him who shall maintain that a particular note is original on the part of Proclus. Hence it is not enough that it should be impossible to point to another writer as the probable source of a note; we must have a positive reason for attributing it to Proclus. The criterion must therefore be found either (1) in the general terms in which Proclus points out the deficiencies in previous commentaries and indicates the respects in which his own will differ from them, or (2) in specific expressions used by him in introducing particular notes which may indicate that he is giving his own views. Besides indicating that he paid more attention than his predecessors to questions requiring deeper study (τὸ πραγματειῶδες) and “pursued clear distinctions” (τὸ εὐδιαίρετον μεταδιώκοντας276— by which he appears to imply that his predecessors had confused the different departments of their commentaries, viz. lemmas, cases, and objections (ἐνστάσεις277—Proclus complains that the earlier commentators had failed to indicate the ultimate grounds or causes of propositions278. Although it is from Geminus that he borrowed a passage maintaining that it is one of the proper functions of geometry to inquire into causes (τὴν αἰτίαν καὶ τὸ διὰ τί279, yet it is not likely that Geminus dealt with Euclid's propositions one by one; and consequently, when we find Proclus, on I. 8, 16, 17, 18, 32, and 47280, endeavouring to explain causes, we have good reason to suppose that the explanations are his own.

Again, his remarks on certain things which he quotes from Pappus can scarcely be due to anyone else, since Pappus is the latest of the commentators whose works he appears to have used. Under this head come

(1) his objections to certain new axioms introduced by Pappus281,

(2) his conjecture as to how Pappus came to think of his alternative proof of I. 5282,

(3) an addition to Pappus' remarks about the curvilineal angle which is equal to a right angle without being one283.

The defence of Geminus against Carpus, who combated his view of theorems and problems, is also probably due to Proclus284, as well as an observation on I. 38 to the effect that I. 35-38 are really comprehended in VI. 1 as particular cases285.

Lastly, we can have no hesitation in attributing to Proclus himself (1) the criticism of Ptolemy's attempt to prove the parallel-postulate286, and (2) the other attempted proof given ín the same note287 (on I. 29) and assuming as an axiom that “if from one point two straight lines forming an angle be produced ad infinitum the distance between them when so produced ad infinitum exceeds any finite magnitude (i.e. length),” an assumption which purports to be the equivalent of a statement in Aristotle288. It is introduced by words in which the writer appears to claim originality for his proof: “To him who desires to see this proved (κατασκευαζόμενον) let it be said by us (λεγέσθω παῤ η<*>μῶν)” etc.289 Moreover, Philoponus, in a note on Aristotle's Anal. post. I. 10, says that “the geometer (Euclid) assumes this as an axiom, but it wants a great deal of proof, insomuch that both Ptolemy and Proclus wrote a whole book upon it290.”

CHAPTER V.

THE TEXT291.

It is well known that the title of Simson's edition of Euclid (first brought out in Latin and English in 1756) claims that, in it, “the errors by which Theon, or others, have long ago vitiated these books are corrected, and some of Euclid's demonstrations are restored” ; and readers of Simson's notes are familiar with the phrases used, where anything in the text does not seem to him satisfactory, to the effect that the demonstration has been spoiled, or things have been interpolated or omitted, by Theon “or some other unskilful editor.” Now most of the MSS. of the Greek text prove by their titles that they proceed from the recension of the Elements by Theon; they purport to be either “from the edition of Theon” (ἐκ τῆς Θέωνος ἐκδόσεως) or “from the lectures of Theon” (ἀπὸ συνουσιῶν τοῦ Θέωνος). This was Theon of Alexandria (4th c. A.D.) who also wrote a commentary on Ptolemy, in which there occurs a passage of the greatest importance in this connexion292: “But that sectors in equal circles are to one another as the angles on which they stand has been proved by me in my edition of the Elements at the end of the sixth book.” Thus Theon himself says that he edited the Elements and also that the second part of VI. 33, found in nearly all the MSS., is his addition.

This passage is the key to the whole question of Theon's changes in the text of Euclid; for, when Peyrard found in the Vatican the MS. 190 which contained neither the words from the titles of the other MSS. quoted above nor the interpolated second part of VI. 33, he was justified in concluding, as he did, that in the Vatican MS. we have an edition more ancient than Theon's. It is also clear that the copyist of P, or rather of its archetype, had before him the two recensions and systematically gave the preference to the earlier one; for at XIII. 6 in P the first hand has added a note in the margin: “This theorem is not given in most copies of the new edition, but is found in those of the old.” Thus we are more fortunate than Simson, since our judgment of Theon's recension can be formed on the basis, not of mere conjecture, but of the documentary evidence afforded by a comparison of the Vatican MS. just mentioned with what we may conveniently call, after Heiberg, the Theonine MSS.

The MSS. used for Heiberg's edition of the Elements are the following:

(1) P = Vatican MS. numbered 190, 4to, in two volumes (doubtless one originally); 10th c.

This is the MS. which Peyrard was able to use; it was sent from Rome to Paris for his use and bears the stamp of the Paris Imperial Library on the last page. It is well and carefully written. There are corrections some of which are by the original hand, but generally in paler ink, others, still pretty old, by several different hands, or by one hand with different ink in different places (P m. 2), and others again by the latest hand (P m. rec.). It contains, first, the Elements I.—XIII. with scholia, then Marinus' commentary on the Data (without the name of the author), followed by the Data itself and scholia, then the Elements XIV., XV. (so called), and lastly three books and a part of a fourth of a commentary by Theon εἰς τοὺς προχείρους κανόνας Πτολεμαίου.

The other MSS. are “Theonine.”

(2) F = MS. XXVIII, 3, in the Laurentian Library at Florence, 4to; 10th c.

This MS. is written in a beautiful and scholarly hand and contains the Elements I.—XV., the Optics and the Phaenomena, but is not well preserved. Not only is the original writing renewed in many places, where it had become faint, by a later hand of the 16th c., but the same hand has filled certain smaller lacunae by gumming on to torn pages new pieces of parchment, and has replaced bodily certain portions of the MS., which had doubtless become illegible, by fresh leaves. The larger gaps so made good extend from Eucl. VII. 12 to IX. 15, and from XII. 3 to the end; so that, besides the conclusion of the Elements, the Optics and Phaenomena are also in the later hand, and we cannot even tell what in addition to the Elements I.—XIII. the original MS. contained. Heiberg denotes the later hand by φ and observes that, while in restoring words which had become faint and filling up minor lacunae the writer used no other MS., yet in the two larger restorations he used the Laurentian MS. XXVIII, 6, belonging to the 13th—14th c. The latter MS. (which Heiberg denotes by f) was copied from the Viennese MS. (V) to be described below.

(3) B = Bodleian MS., D'Orville X. 1 inf. 2, 30, 4to; A.D. 888.

This MS. contains the Elements I.—XV. with many scholia. Leaves 15-118 contain I. 14 (from about the middle of the proposition) to the end of Book VI., and leaves 123-387 (wrongly numbered 397) Books VII.—XV. in one and the same elegant hand (9th c.). The leaves preceding leaf 15 seem to have been lost at some time, leaves 6 to 14 (containing Elem. I. to the place in I. 14 above referred to) being carelessly written by a later hand on thick and common parchment (13th c.). On leaves 2 to 4 and 122 are certain notes in the hand of Arethas, who also wrote a two-line epigram on leaf 5, the greater part of the scholia in uncial letters, a few notes and corrections, and two sentences on the last leaf, the first of which states that the MS. was written by one Stephen clericus in the year of the world 6397 (= 888 A.D.), while the second records Arethas' own acquisition of it. Arethas lived from, say, 865 to 939 A.D. He was Archbishop of Caesarea and wrote a commentary on the Apocalypse. The portions of his library which survive are of the greatest interest to palaeography on account of his exact notes of dates, names of copyists, prices of parchment etc. It is to him also that we owe the famous Plato MS. from Patmos (Cod. Clarkianus) which was written for him in November 895293.

(4) V = Viennese MS. Philos. Gr. No. 103; probably 12th c.

This MS. contains 292 leaves, Eucl. Elements I.—XV. occupying leaves 1 to 254, after which come the Optics (to leaf 271), the Phaenomena (mutilated at the end) from leaf 272 to leaf 282, and lastly scholia, on leaves 283 to 292, also imperfect at the end. The different material used for different parts and the varieties of handwriting make it necessary for Heiberg to discuss this MS. at some length294. The handwriting on leaves 1 to 183 (Book I. to the middle of X. 105) and on leaves 203 to 234 (from XI. 31, towards the end of the proposition, to XIII. 7, a few lines down) is the same; between leaves 184 and 202 there are two varieties of handwriting, that of leaves 184 to 189 and that of leaves 200 (verso) to 202 being the same. Leaf 235 begins in the same handwriting, changes first gradually into that of leaves 184 to 189 and then (verso) into a third more rapid cursive writing which is the same as that of the greater part of the scholia, and also as that of leaves 243 and 282, although, as these leaves are of different material, the look of the writing and of the ink seems altered. There are corrections both by the first and a second hand, and scholia by many hands. On the whole, in spite of the apparent diversity of handwriting in the MS., it is probable that the whole of it was written at about the same time, and it may (allowing for changes of material, ink etc.) even have been written by the same man. It is at least certain that, when the Laurentian MS. XXVIII, 6 was copied from it, the whole MS. was in the condition in which it is now, except as regards the later scholia and leaves 283 to 292 which are not in the Laurentian MS., that MS. coming to an end where the Phaenomena breaks off abruptly in V. Hence Heiberg attributes the whole MS. to the 12th c.

But it was apparently in two volumes originally, the first consisting of leaves 1 to 183; and it is certain that it was not all copied at the same time or from one and the same original. For leaves 184 to 202 were evidently copied from two MSS. different both from one another and from that from which the rest was copied. Leaves 184 to the middle of leaf 189 (recto) must have been copied from a MS. similar to P, as is proved by similarity of readings, though not from P itself. The rest, up to leaf 202, were copied from the Bologna MS. (b) to be mentioned below. It seems clear that the content of leaves 184 to 202 was supplied from other MSS. because there was a lacuna in the original from which the rest of V was copied.

Heiberg sums up his conclusions thus. The copyist of V first copied leaves 1 to 183 from an original in which two quaterniones were missing (covering from the middle of Eucl. X. 105 to near the end of XI. 31). Noticing the lacuna he put aside one quaternio of the parchment used up to that point. Then he copied onwards from the end of the lacuna in the original to the end of the Phaenomena. After this he looked about him for another MS. from which to fill up the lacuna; finding one, he copied from it as far as the middle of leaf 189 (recto). Then, noticing that the MS. from which he was copying was of a different class, he had recourse to yet another MS. from which he copied up to leaf 202. At the same time, finding that the lacuna was longer than he had reckoned for, he had to use twelve more leaves of a different parchment in addition to the quaternio which he had put aside. The whole MS. at first formed two volumes (the first containing leaves 1 to 183 and the second leaves 184 to 282); then, after the last leaf had perished, the two volumes were made into one to which two more quaterniones were also added. A few leaves of the latter of these two have since perished.

(5) b = MS. numbered 18-19 in the Communal Library at Bologna, in two volumes, 4to; 11th c.

This MS. has scholia in the margin written both by the first hand and by two or three later hands; some are written by the latest hand, Theodorus Cabasilas (a descendant apparently of Nicolaus Cabasilas, 14th c.) who owned the MS. at one time. It contains (a) in 14 quaterniones the definitions and the enunciations (without proofs) of the Elements I.—XIII. and of the Data, (b) in the remainder of the volumes the Proem to Geometry (published among the Variae Collectiones in Hultsch's edition of Heron, pp. 252, 24 to 274, 14) followed by the Elements I.—XIII. (part of XIII. 18 to the end being missing), and then by part of the Data (from the last three words of the enunciation of Prop. 38 to the end of the penultimate clause in Prop. 87, ed. Menge). From XI. 36 inclusive to the end of XII. this MS. appears to represent an entirely different recension. Heiberg is compelled to give this portion of b separately in an appendix. He conjectures that it is due to a Byzantine mathematician who thought Euclid's proofs too long and tiresome and consequently contented himself with indicating the course followed295. At the same time this Byzantine must have had an excellent MS. before him, probably of the ante-Theonine variety of which the Vatican MS. 190 (P) is the sole representative.

(6) p = Paris MS. 2466, 4to; 12th c.

This manuscript is written in two hands, the finer hand occupying leaves 1 to 53 (recto), and a more careless hand leaves 53 (verso) to 64, which are of the same parchment as the earlier leaves, and leaves 65 to 239, which are of a thinner and rougher parchment showing traces of writing of the 8th—9th c. (a Greek version of the Old Testament). The MS. contains the Elements I.—XIII. and some scholia after Books XI., XII. and XIII.

(7) q = Paris MS. 2344, folio; 12th c.

It is written by one hand but includes scholia by many hands. On leaves 1 to 16 (recto) are scholia with the same title as that found by Wachsmuth in a Vatican MS. and relied upon by him to prove that Proclus continued his commentaries beyond Book I.296 Leaves 17 to 357 contain the Elements I.—XIII. (except that there is a lacuna from the middle of VIII. 25 to the ἔκθεσις of IX. 14); before Books VII. and X. there are some leaves filled with scholia only, and leaves 358 to 366 contain nothing but scholia.

(8) Heiberg also used a palimpsest in the British Museum (Add. 17211). Five pages are of the 7th—8th c. and are contained (leaves 49-53) in the second volume of the Syrian MS. Brit. Mus. 687 of the 9th c.; half of leaf 50 has perished. The leaves contain various fragments from Book X. enumerated by Heiberg, Vol. III., p. v, and nearly the whole of XIII. 14.

Since his edition of the Elements was published, Heiberg has collected further material bearing on the history of the text297. Besides giving the results of further or new examination of MSS., he has collected the fresh evidence contained in an-Nairīzī's commentary, and particularly in the quotations from Heron's commentary given in it (often word for word), which enable us in several cases to trace differences between our text and the text as Heron had it, and to identify some interpolations which actually found their way into the text from Heron's commentary itself; and lastly he has dealt with some valuable fragments of ancient papyri which have recently come to light, and which are especially important in that the evidence drawn from them necessitates some modification in the views expressed in the preface to Vol. V. as to the nature of the changes made in Theon's recension, and in the principles laid down for differentiating between Theon's recension and the original text, on the basis of a comparison between P and the Theonine MSS. alone.

The fragments of ancient papyri referred to are the following.

1. Papyrus Herculanensis No. 1061.298

This fragment quotes Def. 15 of Book I. in Greek, and omits the words καλεῖται περιφέρεια, “which is called the circumference,” found in all our MSS., and the further addition πρὸς τὴν τοῦ κύκλου περιφέρειαν also found in practically all the MSS. Thus Heiberg's assumption that both expressions are interpolations is now confirmed by this oldest of all sources.

2. The Oxyrhynchus Papyri I. p. 58, No. XXIX. of the 3rd or 4th c. This fragment contains the enunciation of Eucl. II. 5 (with figure, apparently without letters, immediately following, and not, as usual in our MSS., at the end of the proof) and before it the part of a word περιεχομε belonging to II. 4 (with room for -νῳ ὀρθογωνίῳ: ὅπερ ἔδει δεῖξαι and a stroke to mark the end), showing that the fragment had not the Porism which appears in all the Theonine MSS. and (in a later hand) in P, and thereby confirming Heiberg's assumption that the Porism was due to Theon.

3. A fragment in Fayum towns and their papyri, p. 96, No. IX. of 2nd or 3rd c.

This contains I. 39 and I. 41 following one another and almost complete, showing that I. 40 was wanting, whereas it is found in all the MSS. and is recognised by Proclus. Moreover the text of the beginning of I. 39 is better than ours, since it has no double διορισμός but omits the first (“I say that they are also in the same parallels” ) and has “and” instead of “for let AD be joined” in the next sentence. It is clear that I. 40 was interpolated by someone who thought there ought to be a proposition following I. 39 and related to it as I. 38 is related to I. 37 and I. 36 to I. 35, although Euclid nowhere uses I. 40, and therefore was not likely to include it. The same interpolator failed to realise that the words “let AD be joined” were part of the ἔκθεσις or setting-out, and took them for the κατασκευή or “construction” which generally follows the διορισμός or “particular statement” of the conclusion to be proved, and consequently thought it necessary to insert a διορισμός before the words.

The conclusions drawn by Heiberg from a consideration of particular readings in this papyrus along with those of our MSS. will be referred to below.

We now come to the principles which Heiberg followed, when preparing his edition, in differentiating the original text from the Theonine recension by means of a comparison of the readings of P and of the Theonine MSS. The rules which he gives are subject to a certain number of exceptions (mostly in cases where one MS. or the other shows readings due to copyists' errors), but in general they may be relied upon to give conclusive results.

The possible alternatives which the comparison of P with the Theonine MSS. may give in particular passages are as follows:

I. There may be agreement in three different degrees.

(1) P and all the Theonine MSS. may agree.

In this case the reading common to all, even if it is corrupt or interpolated, is more ancient than Theon, i.e. than the 4th c.

(2) P may agree with some (only) of the Theonine MSS.

In this case Heiberg considered that the latter give the true reading of Theon's recension, and the other Theonine MSS. have departed from it.

(3) P and one only of the Theonine MSS may agree.

In this case too Heiberg assumed that the one Theonine MS. which agrees with P gives the true Theonine reading, and that this rule even supplies a sort of measure of the quality and faithfulness of the Theonine MSS. Now none of them agrees alone with P in preserving the true reading so often as F. Hence F must be held to have preserved Theon's recension more faithfully than the other Theonine MSS.; and it would follow that in those portions where F fails us P must carry rather more weight even though it may differ from the Theonine MSS. BVpq. (Heiberg gives many examples in proof of this, as of his main rules generally, for which reference must be made to his Prolegomena in Vol. v.) The specially close relation of F and P is also illustrated by passages in which they have the same errors; the explanation of these common errors (where not due to accident) is found by Heiberg in the supposition that they existed, but were not noticed by Theon, in the original copy in which he made his changes.

Although however F is by far the best of the Theonine MSS., there are a considerable number of passages where one of the others (B, V, p or q) alone with P gives the genuine reading of Theon's recension.

As the result of the discovery of the papyrus fragment containing I. 39, 41, the principles above enunciated under (2) and (3) are found by Heiberg to require some qualification. For there is in some cases a remarkable agreement between the papyrus and the Theonine MSS. (some or all) as against P. This shows that Theon took more trouble to follow older MSS., and made fewer arbitrary changes of his own, than has hitherto been supposed. Next, when the papyrus agrees with some of the Theonine MSS. against P, it must now be held that these MSS. (and not, as formerly supposed, those which agree with P) give the true reading of Theon. If it were otherwise, the agreement between the papyrus and the Theonine MSS. would be accidental: but it happens too often for this. It is clear also that there must have been contamination between the two recensions; otherwise, whence could the Theonine MSS. which agree with P and not with the papyrus have got their readings? The influence of the P class on the Theonine F is especially marked.

II. There may be disagreement between P and all the Theonine MSS.

The following possibilities arise.

(1) The Theonine MSS. differ also among themselves.

In this case Heiberg considered that P nearly always has the true reading, and the Theonine MSS. have suffered interpolation in different ways after Theon's time.

(2) The Theonine MSS. all combine against P.

In this case the explanation was assumed by Heiberg to be one or other of the following. “ (α) The common reading is due to an error which cannot be imputed to Theon (though it may have escaped him when putting together the archetype of his edition); such error may either have arisen accidentally in all alike, or (more frequently) may be referred to a common archetype of all the MSS.

” “ (β) There may be an accidental error in P; e.g. something has dropped out of P in a good many places, generally through ὁμοιοτέλευτον

” “ (γ) There may be words interpolated in P.

” “ (δ) Lastly, we may have in the Theonine MSS. a change made by Theon himself.

(The discovery of the ancient papyrus showing readings agreeing with some, or with all, of the Theonine MSS. against P now makes it necessary to be very cautious in applying these criteria.)

It is of course the last class (δ) of changes which we have to investigate in order to get a proper idea of Theon's recension.

Heiberg first observes, as regards these, that we shall find that Theon, in editing the Elements, altered hardly anything without some reason, often inadequate according to our ideas, but still some reason which seemed to him sufficient. Hence, in cases of very slight differences where both the Theonine MSS. and P have readings good and probable in themselves, Heiberg is not prepared to put the differences down to Theon. In those passages where we cannot see the least reason why Theon, if he had the reading of P before him, should have altered it, Heiberg would not at once assume the superiority of P unless there was such a consistency in the differences as would indicate that they were due not to accident but to design. In the absence of such indications, he thinks that the ordinary principles of criticism should be followed and that proper weight should be attached to the antiquity of the sources. And it cannot be denied that the sources of the Theonine version are the more ancient. For not only is the British Museum palimpsest (L), which is intimately connected with the rest of our MSS., át least two centuries older than P, but the other Theonine MSS. are so nearly allied that they must be held to have had a common archetype intermediate between them and the actual edition of Theon; and, since they themselves are as old as, or older than P, their archetype must have been much older. Heiberg gives (pp. xlvi, xlvii) a list of passages where, for this reason, he has followed the Theonine MSS. in preference to P.

It has been mentioned above that the copyist of P or rather of its archetype wished to give an ancient recension. Therefore (apart from clerical errors and interpolations) the first hand in P may be relied upon as giving a genuine reading even where a correction by the first hand has been made at the same time. But in many places the first hand has made corrections afterwards; on these occasions he must have used new sources, e.g. when inserting the scholia to the first Book which P alone has, and in a number of passages he has made additions from Theonine MSS.

We cannot make out any “family tree” for the different Theonine MSS. Although they all proceeded from a common archetype later than the edition of Theon itself, they cannot have been copied one from the other; for, if they had been, how could it have come about that in one place or other each of them agrees alone with P in preserving the genuine reading? Moreover the great variety in their agreements and disagreements indicates that they have all diverged to about the same extent from their archetype. As we have seen that P contains corrections from the Theonine family, so they show corrections from P or other MSS. of the same family. Thus V has part of the lacuna in the MS. from which it was copied filled up from a MS. similar to P, and has corrections apparently derived from the same; the copyist, however, in correcting V, also used another MS. to which he alludes in the additions to IX. 19 and 30 (and also on X. 23 Por.): “in the book of the Ephesian (this) is not found.” Who this Ephesian of the 12th c. was, we do not know.

We now come to the alterations made by Theon in his edition of the Elements. I shall indicate classes into which these alterations may be divided but without details (except in cases where they affect the mathematical content as distinct from form or language pure and simple).299.

I. Alterations made by Theon where he found, or thought he found, mistakes in the original.

1. Real blots in the original which Theon saw and tried to remove.

a) Euclid has a porism (corollary) to VI. 19, the enunciation of which speaks of similar and similarly described figures though the proposition itself refers only to triangles, and therefore the porism should have come after VI. 20. Theon substitutes triangle for figure and proves the more general porism after VI. 20.

b) In IX. 19 there is a statement which is obviously incorrect. Theon saw this and altered the proof by reducing four alternatives to two, with the result that it fails to correspond to the enunciation even with Theon's substitution of “if” for “when” in the enunciation.

c) Theon omits a porism to IX. II, although it is necessary for the proof of the succeeding proposition, apparently because, owing to an error in the text (κατὰ τὸν corrected by Heiberg into ἐπὶ τὸ), he could not get out of it the right sense.

d) I should also put into this category a case which Heiberg classifies among those in which Theon merely fancied that he found mistakes, viz. the porism to V. 7 stating that, if four magnitudes are proportional, they are proportional inversely. Theon puts this after V. 4 with a proof, which however has no necessary connexion with V. 4 but is obvious from the definition of proportion.

e) I should also put under this head XI. 1, where Euclid's argument to prove that two straight lines cannot have a common segment is altered.

2. Passages which seemed to Theon to contain blots, and which he therefore set himself to correct, though more careful consideration would have shown that Euclid's words are right or at least may be excused and offer no difficulty to an intelligent reader. Under this head come:

a) an alteration in III. 24.

b) a perfectly unnecessary alteration, in Vi. 14, of “equiangular parallelograms” into “parallelograms having one angle equal to one angle,” where Theon followed the false analogy of VI. 15.

c) an omission of words in V. 26, owing to his having been misled by a wrong figure.

d) an alteration of the order of XI. Deff. 27, 28.

e) the substitution of “parallelepipedal solid” for “cube” in XI. 38, because Theon observed, correctly enough, that it was true of the parallelepipedal solid in general as well as of the cube, but failed to give weight to the fact that Euclid must have given the particular case of the cube for the simple reason that that was all he wanted for use in XIII. 17.

f) the substitution of the letter Φ for Ω (V for Z in my figure) because he saw that the perpendicular from Κ to ΒΦ would fall on Φ itself, so that Φ, Ω coincide. But, if the substitution is made, it should be proved that Φ, Ω coincide. Euclid can hardly have failed to notice the fact, but it may be that he deliberately ignored it as unnecessary for his purpose, because he did not want to lengthen his proposition by giving the proof.

II. Emendations intended to improve the form or diction of Euclid.

Some of these emendations of Theon affect passages of appreciable length. Heiberg notes about ten such passages; the longest is in Eucl. XII. 4 where a whole page of Heiberg's text is affected and Theon's version is put in the Appendix. The kind of alteration may be illustrated by that in IX. 15 where Euclid uses successively the propositions VII. 24, 25, quoting the enunciation of the former but not of the latter; Theon does exactly the reverse. In a few of the cases here quoted by Heiberg, Theon shortened the original somewhat.

But, as a rule, the emendations affect only a few words in each sentence. Sometimes they are considerable enough to alter the conformation of the sentence, sometimes they are trifling alterations “more magistellorum ineptorum” and unworthy of Theon. Generally speaking, they were prompted by a desire to change anything which was out of the common in expression or in form, in order to reduce the language to one and the same standard or norm. Thus Theon changed the order of words, substituted one word for another where the latter was used in a sense unusual with Euclid (e.g. ἐπειδήπερ, “since,” for ὅτι in the sense of “because” ), or one expression for another in like circumstances (e.g. where, finding “that which was enjoined would be done” in a theorem, VII. 31, and deeming the phrase more appropriate to a problem, he substituted for it “that which is sought would be manifest” ; probably also and for similar reasons he made certain variations between the two expressions usual at the end of propositions ὅπερ ἔδει δεῖξαι and ὅπερ ἔδει ποιῆσαι, quod erat demonstrandum and quod erat faciendum). Sometimes his alterations show carelessness in the use of technical terms, as when he uses ἅπτεσθαι (to meet) for ἐφάπτεσθαι (to touch) although the ancients carefully distinguished the two words. The desire of keeping to a standard phraseology also led Theon to omit or add words in a number of cases, and also, sometimes, to change the lettering of figures.

But Theon seems, in editing the Elements, to have bestowed the most attention upon

III. Additions designed to supplement or explain Euclid.

First, he did not hesitate to interpolate whole propositions where he thought there was room or use for them. We have already mentioned the addition to VI. 33 of the second part relating to sectors, for which Theon himself takes credit in his commentary on Ptolemy. Again, he interpolated the proposition commonly known as VII. 22 (ex aequo in proportione perturbata for numbers, corresponding to V. 23), and perhaps also VII. 20, a particular case of VII. 19 as VI. 17 is of VI. 16. He added a second case to VI. 27, a porism to II. 4, a second porism to III. 16, and a lemma after X. 12; perhaps also the porism to V. 19 and the first porism to VI. 20. He also inserted alternative proofs here and there, e.g. in II. 4 (where the alternative differs little from the original) and in VII. 31; perhaps also in X. 1, 6, and 9.

Secondly, he sometimes repeats an argument where Euclid had said “For the same reason,” adds specific references to points, straight lines etc. in the figures in order to exclude the possibility of mistake arising from Euclid's reference to them in general terms, or inserts words to make the meaning of Euclid more plain, e.g. componendo and alternately, where Euclid had left them out. Sometimes he thought to increase by his additions the mathematical precision of Euclid's language in enunciations or elsewhere, sometimes to make smoother and clearer things which Euclid had expressed with unusual brevity and harshness or carelessness, in reliance on the intelligence of his readers.

Thirdly, he supplied intermediate steps where Euclid's argument seemed too rapid and not easy enough to follow. The form of these additions varies; they are sometimes placed as a definite intermediate step with “therefore” or “so that,” sometimes they are additions to the statement of premisses, sometimes phrases introduced by “since,” “for” and the like, after the inference.

Lastly, there is a very large class of additions of a word, or one or two words, for the sake of clearness or consistency. Heiberg gives a number of examples of the addition of such nouns as “triangle,” “square,” “rectangle,” “magnitude,” “number,” “point,” “side,” “circle,” “straight line,” “area” and the like, of adjectives such as “remaining,” “right,” “whole,” “proportional,” and of other parts of speech, even down to words like “is” (ἐστί) which is added 600 times, δή, ἄρα, μέν, γάρ, καί and the like.

IV. Omissions by Theon.

Heiberg remarks that, Theon's object having been, as above shown, to amplify and explain Euclid, we should not natuially have expected to find him doing much in the contrary process of compression, and it is only owing to the recurrence of a certain sort of omissions so frequently (especially in the first Books) as to exclude the hypothesis of their being all due to chance that we are bound to credit him with alterations making for greater brevity. We have seen, it is true, that he made omissions as well as additions for the purpose of reducing the language to a certain standard form. But there are also a good number of cases where in the enunciation of propositions, and in the exposition (the re-statement of them with reference to the figure), he has left out words because, apparently, he regarded Euclid's language as being too careful and precise. Again, he is apparently responsible for the frequent omission of the words ὅπερ ἔδει δεῖξαι (or ποιῆσαι), Q.E.D. (or F.), at the end of propositions. This is often the case at the end of porisms, where, in omitting the words, Theon seems to have deliberately departed from Euclid's practice. The MS. P seems to show clearly that, where Euclid put a porism at the end of a proposition, he omitted the Q.E.D. at the end of the proposition but inserted it at the end of the porism, as if he regarded the latter as being actually a part of the proposition itself. As in the Theonine MSS. the Q.E.D. is generally omitted, the omission would seem to have been due to Theon. Sometimes in these cases the Q.E.D. is interpolated at the end of the proposition.

Heiberg summed up the discussion of Theon's edition by the remark that Theon evidently took no pains to discover and restore from MSS. the actual words which Euclid had written, but aimed much more at removing difficulties that might be felt by learners in studying the book. His edition is therefore not to be compared with the editions of the Alexandrine grammarians, but rather with the work done by Eutocius in editing Apollonius and with an interpolated recension of some of the works of Archimedes by a certain Byzantine, Theon occupying a position midway between these two editors, being superior to the latter in mathematical knowledge but behind Eutocius in industry (these views now require to be somewhat modified, as above stated). But however little Theon's object may be approved by those of us who would rather know the ipsissima verba of Euclid, there is no doubt that his work was approved by his pupils at Alexandria for whom it was written; and his edition was almost exclusively used by later Greeks, with the result that the more ancient text is only preserved to us in one MS.

As the result of the above investigation, we may feel satisfied that, where P and the Theonine MSS. agree, they give us (except in a few accidental instances) Euclid as he was read by the Greeks of the 4th c. But even at that time the text had been passed from hand to hand through more than six centuries, so that it is certain that it had already suffered changes, due partly to the fault of copyists and partly to the interpolations of mathematicians. Some errors of copyists escaped Theon and were corrected in some MSS. by later hands. Others appear in all our MSS. and, as they cannot have arisen accidentally in all, we must put them down to a common source more ancient than Theon. A somewhat serious instance is to be found in III. 8; and the use of ἁπτέσθω for ἐφαπτέσθω in the sense of “touch” may also be mentioned, the proper distinction between the words having been ignored as it was by Theon also. But there are a number of imperfections in the ante-Theonine text which it would be unsafe to put down to the errors of copyists, those namely where the good MSS. agree and it is not possible to see any motive that a copyist could have had for altering a correct reading. In these cases it is possible that the imperfections are due to a certain degree of carelessness on the part of Euclid himself; for it is not possible “Euclidem ab omni naevo vindicare,” to use the words of Saccheri300, and consequently Simson is not right in attributing to Theon and other editors all the things in Euclid to which mathematical objection can be taken. Thus, when Euclid speaks of “the ratio compounded of the sides” for “the ratio compounded of the ratios of the sides,” there is no reason for doubting that Euclid himself is responsible for the more slip-shod expression. Again, in the Books XI.—XIII. relating to solid geometry there are blots neither few nor altogether unimportant which can only be attributed to Euclid himself301; and there is the less reason for hesitation in so attributing them because solid geometry was then being treated in a thoroughly systematic manner for the first time. Sometimes the conclusion (συμπέρασμα) of a proposition does not correspond exactly to the enunciation, often it is cut short with the words καὶ τὰ ἑξῆς “and the rest” (especially from Book X. onwards), and very often in Books VIII., IX. it is omitted. Where all the MSS. agree, there is no ground for hesitating to attribute the abbreviation or omission to Euclid; though, of course, where one or more MSS. have the longer form, it must be retained because this is one of the cases where a copyist has a temptation to abbreviate.

Where the true reading is preserved in one of the Theonine MSS. alone, Heiberg attributes the wrong reading to a mistake which arose before Theon's time, and the right reading of the single MS. to a successful correction.

We now come to the most important question of the Interpolations introduced before Theon's time.

I. Alternative proofs or additional cases.

It is not in itself probable that Euclid would have given two proofs of the same proposition; and the doubt as to the genuineness of the alternatives is increased when we consider the character of some of them and the way in which they are introduced. First of all, we have those of VI. 20 and XII. 17 introduced by “we shall prove this otherwise more readily (προχειρότερον)” or that of X. 90 “it is possible to prove more shortly (συντομώτερον).” Now it is impossible to suppose that Euclid would have given one proof as that definitely accepted by him and then added another with the express comment that the latter has certain advantages over the former. Had he considered the two proofs and come to this conclusion, he would have inserted the latter in the received text instead of the former. These alternative proofs must therefore have been interpolated. The same argument applies to alternatives introduced with the words “or even thus” ( καὶ οὕτως), “or even otherwise” ( καὶ ἄλλως). Under this head come the alternatives for the last portions of III. 7, 8; and Heiberg also compares the alternatives for parts of III. 31 (that the angle in a semicircle is a right angle) and XIII. 18, and the alternative proof of the lemma after X. 32. The alternatives to X. 105 and 106, again, are condemned by the place in which they occur, namely after an alternative proof to X. 115. The above alternatives being all admitted to be spurious, suspicion must necessarily attach to the few others which are in themselves unobjectionable. Heiberg instances the alternative proofs to III. 9, III. 10, VI. 30, VI. 31 and XI. 22, observing that it is quite comprehensible that any of these might have occurred to a teacher or editor and seemed to him, rightly or wrongly, to be better than the corresponding proofs in Euclid. Curiously enough, Simson adopted the alternatives to III. 9, 10 in preference to the genuine proofs. Since Heiberg's preface was written, his suspicion has been amply confirmed as regards III. 10 by the commentary of an-Nairīzī (ed. Curtze) which shows not only that this alternative is Heron's, but also that the substantive proposition III. 12 in Euclid is also Heron's, having been given by him to supplement III. II which must originally have been enunciated of circles “touching one another” simply, i.e. so as to include the case of external as well as internal contact, though the proof covered the case of internal contact only. “Euclid, in the 11th proposition,” says Heron, “supposed two circles touching one another internally and wrote the proposition on this case, proving what it was required to prove in it. But I will show how it is to be proved if the contact be external.302.” This additional proposition of Heron's is by way of adding another case, which brings us to that class of interpolation. It was the practice of Euclid and the ancients to give only one case (generally the most difficult one) and to leave the others to be investigated by the reader for himself. One interpolation of a second case (VI. 27) is due, as we have seen, to Theon. The two extra cases of XI. 23 were manifestly interpolated before Theon's time, for the preliminary distinction of three cases, “(the centre) will either be within the triangle LMN, or on one of the sides, or outside. First let it be within,” is a spurious addition (B and V only). Similarly an unnecessary case is interpolated in III. 11.

II. Lemmas.

Heiberg has unhesitatingly placed in his Appendix to Vol. III. certain lemmas interpolated either by Theon (on X. 13) or later writers (on X. 27, 29, 31, 32, 33, 34, where V only has the lemmas). But we are here concerned with the lemmas found in all the MSS., which however are, for different reasons, necessarily suspected. We will deal with the Book X. lemmas last.

(1) There is an a priori ground of objection to those lemmas which come after the propositions to which they relate and prove properties used in those propositions; for, if genuine, they would be a sign of faulty arrangement such as would not be likely in a systematic work so carefully ordered as the Elements. The lemma to VI. 22 is one of this class, and there is the further objection to it that in VI. 28 Euclid makes an assumption which would equally require a lemma though none is found. The lemma after XII. 4 is open to the further objections that certain altitudes are used but are not drawn in the figure (which is not in the manner of Euclid), and that a peculiar expression “parallelepipedal solids described on (ἀναγραφόμενα ἀπό) prisms” betrays a hand other than Euclid's. There is an objection on the score of language to the lemma after XIII. 2. The lemmas on XI. 23, XIII. 13, XIII. 18, besides coming after the propositions to which they relate, are not very necessary in themselves and, as regards the lemma to XIII. 13, it is to be noticed that the writer of a gloss in the proposition could not have had it, and the words “as will be proved afterwards” in the text are rightly suspected owing to differences between the MS. readings. The lemma to XII. 2 also, to which Simson raised objection, comes after the proposition; but, if it is rejected, the words “as was proved before” used in XII. 5 and 18, and referring to this lemma, must be struck out.

(2) Reasons of substance are fatal to the lemma before X. 60, which is really assumed in X. 44 and therefore should have appeared there if anywhere, and to the lemma on X. 20, which tries to prove what is already stated in X. Def. 4.

We now come to the remaining lemmas in Book X., eleven in number, which come before the propositions to which they relate and remove difficulties in the way of their demonstration. That before X. 42 introduces a set of propositions with the words “that the said irrational straight lines are uniquely divided ... we will prove after premising the following lemma,” and it is not possible to suppose that these words are due to an interpolator; nor are there any objections to the lemmas before X. 14, 17, 22, 33, 54, except perhaps that they are rather easy. The lemma before X. 10 and X. 10 itself should probably be removed from the Elements; for X. 10 really uses the following proposition X. 11, which is moreover numbered 10 by the first hand in P, and the words in X. 10 referring to the lemma “for we learnt (how to do this)” betray the interpolator. Heiberg gives reason also for rejecting the lemmas before X. 19 and 24 with the words “in any of the aforesaid ways” (omitted in the Theonine MSS.) in the enunciations of X. 19, 24 and in the exposition of X. 20. Lastly, the lemmas before X. 29 may be genuine, though there is an addition to the second of them which is spurious.

Heiberg includes under this heading of interpolated lemmas two which purport to be substantive propositions, XI. 38 and XIII. 6. These must be rejected as spurious for reasons which will be found in detail in my notes on XI. 37 and XIII. 6 respectively. The latter proposition is only quoted once (in XIII. 17); probably the words quoting it (with γραμμή instead of εὐθεῖα) are themselves interpolated, and Euclid thought the fact stated a sufficiently obvious inference from XIII. 1.

III. Porisms (or corollaries).

Most of the porisms in the text are both genuine and necessary; but some are shown by differences in the MSS. not to be so, e.g. those to I. 15 (though Proclus has it), III. 31 and VI. 20 (Por. 2). Sometimes parts of porisms are interpolated. Such are the last few lines in the porisms to IV. 5, VI. 8; the latter addition is proved later by means of VI. 4, 8, so that the writer of these proofs could not have had the addition to VI. 8 Por. before him. Lastly, interpolators have added a sort of proof to some porisms, as though they were not quite obvious enough; but to add a demonstration is inconsistent with the idea of a porism, which, according to Proclus, is a by-product of a proposition appearing without our seeking it.

IV. Scholia.

Several interpolated scholia betray themselves by their wording, e.g. those given by Heiberg in the Appendix to Book X. and containing the words καλεῖ, ἐκάλεσε (“he calls” or “called” ); these scholia were apparently written as marginal notes before Theon's time, and, being adopted as such by Theon, found their way into the text in P and some of the Theonine MSS. The same thing no doubt accounts for the interpolated analyses and syntheses to XIII. 1-5, as to which see my note on XIII. 1.

V. Interpolations in Book X.

First comes the proposition “Let it be proposed to us to show that in square figures the diameter is incommensurable in length with the side,” which, with a scholium after it, ends the tenth Book. The form of the enunciation is suspicious enough and the proposition, the proof of which is indicated by Aristotle and perhaps was Pythagorean, is perfectly unnecessary when X. 9 has preceded. The scholium ends with remarks about commensurable and incommensurable solids, which are of course out of place before the Books on solids. The scholiast on Book X. alludes to this particular scholium as being due to “Theon and some others.” But it is doubtless much more ancient, and may, as Heiberg conjectures have been the beginning of Apollonius' more advanced treatise on incommensurables. Not only is everything in Book X. after X. 115 interpolated, but Heiberg doubts the genuineness even of X. 112-115, on the ground that X. 111 rounds off the theory of incommensurables as we want it in the Books on solid geometry, while X. 112-115 are not really connected with what precedes, nor wanted for the later Books, but seem to form the starting-point of a new and more elaborate theory of irrationals.

VI. Other minor interpolations are found of the same character as those above attributed to Theon. First there are two places (XI. 35 and XI. 26) where, after “similarly we shall prove” and “for the same reason,” an actual proof is nevertheless given. Clearly the proofs are interpolated; and there are other similar interpolations. There are also interpolations of intermediate steps in proofs, unnecessary explanations and so on, as to which I need not enter into details.

Lastly, following Heiberg's order, I come to

VII. Interpolated definitions, axioms etc.

Apart from VI. Def. 5 (which may have been interpolated by Theon although it is found written in the margin of P by the first hand), the definition of a segment of a circle in Book I. is interpolated, as is clear from the fact that it occurs in a more appropriate place in Book III. and Proclus omits it. VI. Def. 2 (reciprocal figures) is rightly condemned by Simson—perhaps it was taken from Heron—and Heiberg would reject VII. Def. 10, as to which see my note on that definition. Lastly the double definition of a solid angle (XI. Def. 11) constitutes a difficulty. The use of the word ἐπιφάνεια suggests that the first definition may have been older than Euclid, and he may have quoted it from older elements, especially as his own definition which follows only includes solid angles contained by planes, whereas the other includes other sorts (cf. the words γραμμῶν, γραμμαῖς) which are also distinguished by Heron (Def. 22). If the first definition had come last, it could have been rejected without hesitation: but it is not so easy to reject the first part up to and including “otherwise” (ἄλλως). No difficulty need be felt about the definitions of “oblong,” “rhombus,” and “rhomboid,” which are not actually used in the Elements; they were no doubt taken from earlier elements and given for the sake of completeness.

As regards the axioms or, as they are called in the text, common notions (κοιναὶ ἔννοιαι), it is to be observed that Proclus says303 that Apollonius tried to prove “the axioms,” and he gives Apollonius' attempt to prove Axiom I. This shows at all events that Apollonius had some of the axioms now appearing in the text. But how could Apollonius have taken a controversial line against Euclid on the subject of axioms if these axioms had not been Euclid's to his knowledge? And, if they had been interpolated between Euclid's time and his own, how could Apollonius, living so comparatively short a time after Euclid, have been ignorant of the fact? Therefore some of the axioms are Euclid's (whether he called them common notions, or axioms, as is perhaps more likely since Proclus calls them axioms): and we need not hesitate to accept as genuine the first three discussed by Proclus, viz. (1) things equal to the same equal to one another, (2) if equals be added to equals, wholes equal, (3) if equals be subtracted from equals, remainders equal. The other two mentioned by Proclus (whole greater than part, and congruent figures equal) are more doubtful, since they are omitted by Heron, Martianus Capella, and others. The axiom that “two lines cannot enclose a space” is however clearly an interpolation due to the fact that I. 4 appeared to require it. The others about equals added to unequals, doubles of the same thing, and halves of the same thing are also interpolated; they are connected with other interpolations, and Proclus clearly used some source which did not contain them.

Euclid evidently limited his formal axioms to those which seemed to him most essential and of the widest application; for he not unfrequently assumes other things as axiomatic, e.g. in VII. 28 that, if a number measures two numbers, it measures their difference.

The differences of reading appearing in Proclus suggest the question of the comparative purity of the sources used by Proclus, Heron and others, and of our text. The omission of the definition of a segment in Book I. and of the old gloss “which is called the circumference” in I. Def. 15 (also omitted by Heron, Taurus, Sextus Empiricus and others) indicates that Proclus had better sources than we have; and Heiberg gives other cases where Proclus omits words which are in all our MSS. and where Proclus' reading should perhaps be preferred. But, except in these instances (where Proclus may have drawn from some ancient source such as one of the older commentaries), Proclus' MS. does not seem to have been among the best. Often it agrees with our worst MSS., sometimes it agrees with F where F alone has a certain reading in the text, so that (e.g. in I. 15 Por.) the common reading of Proclus and F must be rejected, thrice only does it agree with P alone, sometimes it agrees with P and some Theonine MSS., and once it agrees with the Theonine MSS. against P and other sources.

Of the other external sources, those which are older than Theon generally agree with our best MSS., e.g. Heron, allowing for the difference in the plan of his definitions and the somewhat free adaptation to his purpose of the Euclidean definitions in Books X., XI.

Heiberg concludes that the Elements were most spoiled by interpolations about the 3rd c., for Sextus Empiricus had a correct text, while Iamblichus had an interpolated one; but doubtless the purer text continued for a long time in circulation, as we conclude from the fact that our MSS. are free from interpolations already found in Iamblichus' MS.

CHAPTER VI.

THE SCHOLIA.

Heiberg has collected scholia, to the number of about 1500, in Vol. v. of his edition of Euclid, and has also discussed and classified them in a separate short treatise, in which he added a few others.304

These scholia cannot be regarded as doing much to facilitate the reading of the Elements. As a rule, they contain only such observations as any intelligent reader could make for himself. Among the few exceptions are XI. Nos. 33, 35 (where XI. 22, 23 are extended to solid angles formed by any number of plane angles), XII. No. 85 (where an assumption tacitly made by Euclid in XII. 17 is proved), IX. Nos. 28, 29 (where the scholiast has pointed out the error in the text of IX. 19).

Nor are they very rich in historical information; they cannot be compared in this respect with Proclus' commentary on Book I. or with those of Eutocius on Archimedes and Apollonius. But even under this head they contain some things of interest, e.g. II. No. 11 explaining that the gnomon was invented by geometers for the sake of brevity, and that its name was suggested by an incidental characteristic, namely that “from it the whole is known (γνωρίζεται), either of the whole area or of the remainder, when it (the γνώμων) is either placed round or taken away” ; II. No. 13, also on the gnomon; IV. No. 2 stating that Book IV. was the discovery of the Pythagoreans; V. No. 1 attributing the content of Book V. to Eudoxus; X. No. 1 with its allusion to the discovery of incommensurability by the Pythagoreans and to Apollonius' work on irrationals; X. No. 62 definitely attributing X. 9 to Theaetetus; XIII. No. 1 about the “Platonic” figures, which attributes the cube, the pyramid, and the dodecahedron to the Pythagoreans, and the octahedron and icosahedron to Theaetetus.

Sometimes the scholia are useful in connexion with the settlement of the text, (1) directly, e.g. III. No. 16 on the interpolation of the word “within” (ἐντός) in the enunciation of III. 6, and X. No. 1 alluding to the discussion by “Theon and some others” of irrational “surfaces” and “solids,” as well as “lines,” from which we may conclude that the scholium at the end of Book X. is not genuine; (2) indirectly in that they sometimes throw light on the connexion of certain MSS.

Lastly, they have their historical importance as enabling us to judge of the state of mathematical science at the times when they were written.

Before passing to the classification of the scholia, Heiberg remarks that we must separate from them a number of additions in the nature of scholia which are found in the text of our MSS. but which can, in one way or another, be proved to be spurious. As they are found both in P and in the Theonine MSS., they must have been in the MSS. anterior to Theon (4th c.). But they are, in great part, only found in the margin of P and the Theonine MSS.; in V they are half in the text and half in the margin. This can hardly be explained except on the supposition that these additions were originally (in the MSS. before Theon) in the margin, and that Theon kept them there in his edition, but that they afterwards found their way gradually into the text of P as well as of the Theonine MSS., or were omitted altogether, while particular MSS. have in certain places preserved the old arrangement. Of such spurious additions Heiberg enumerates the following: the axiom about equals subtracted from unequals, the last lines of the porism to VI. 8, second porisms to V. 19 and to VI. 20, the porism to III. 31, VI. Def. 5, various additions in Book X., the analyses and syntheses of XIII. 1-5, and the proposition XIII. 6.

The two first classes of scholia distinguished by Heiberg are denoted by the convenient abbreviations “Schol. Vat.” and “Schol. Vind.”

I. Schol. Vat.

It is first necessary to set out the letters by which Heiberg denotes certain collections of scholia.

P=Scholia in P written by the first hand.

B=Scholia in B by a hand of the same date as the MS. itself, generally that of Arethas.

F=Scholia in F by the first hand.

Vat.=Scholia of the Vatican MS. 204 of the 10th c., which has these scholia on leaves 198-205 (the end is missing) as an independent collection. It does not contain the text of the Elements.

V^{c}=Scholia found on leaves 283-292 of V and written in the same hand as that part of the MS. itself which begins at leaf 235.

Vat. 192=a Vatican MS. of the 14th c. which contains, after (1) the Elements I.—XIII. (without scholia), (2) the Data with scholia, (3) Marinus on the Data, the Schol. Vat. as an independent collection and in their entirety, beginning with 1. No. 88 and ending with XIII. No. 44.

The Schol. Vat., the most ancient and important collection of scholia, comprise those which are found in PBF Vat. and, from VII. 12 to IX. 15, in PB Vat. only, since in that portion of the Elements F was restored by a later hand without scholia; they also include 1. No. 88 which only happens to be erased in F, and IX. Nos. 28, 29 which may be left out because F. here has a different text. In F and Vat. the collection ends with Book X.; but it must also include Schol. PB of Books XI.—XIII., since these are found along with Schol. Vat. to Books I.—X. in several MSS. (of which Vat. 192 is one) as a separate collection. The Schol. Vat. to Books X.—XIII. are also found in the collection V^{c} (where, curiously enough, XIII. Nos. 43, 44 are at the beginning). The Schol. Vat. accordingly include Schol. PBV^{c} Vat. 192, and doubtless also those which are found in two of these sources. The total number of scholia classified by Heiberg as Schol. Vat. is 138.

As regards the contents of Schol. Vat. Heiberg has the following observations. The thirteen scholia to Book I. are extracts made from Proclus by a writer thoroughly conversant with the subject, and cleverly recast (with some additions). Their author does not seem to have had the two lacunae which our text of Proclus has (at the end of the note on I. 36 and the beginning of the next note, and at the beginning of the note on I. 43), for the scholia I. Nos. 125 and 137 seem to fill the gaps appropriately, at least in part. In some passages he had better readings than our MSS. have. The rest of Schol. Vat. (on Books II.—XIII.) are essentially of the same character as those on Book I., containing prolegomena, remarks on the object of the propositions, critical remarks on the text, converses, lemmas; they are, in general, exact and true to tradition. The reason of the resemblance between them and Proclus appears to be due to the fact that they have their origin in the commentary of Pappus, of which we know that Proclus also made use. In support of the view that Pappus is the source, heiberg places some of the Schol. Vat. to Book X. side by side with passages from the commentary of Pappus in the Arabic translation discovered by Woepcke;305; he also refers to the striking confirmation afforded by the fact that XII. No. 2 contains the solution of the problem of inscribing in a given circle a polygon similar to a polygon inscribed in another circle, which problem Eutocius says306 that Pappus gave in his commentary on the Elements.

But, on the other hand, Schol. Vat. contain some things which cannot have come from Pappus, e.g. the allusion in X. No. 1 to Theon and irrational surfaces and solids, Theon being later than Pappus; III. No. 10 about porisms is more like Proclus' treatment of the subject than Pappus', though one expression recalls that of Pappus about forming (σχηματίζεσθαι) the enunciations of porisms like those of either theorems or problems.

The Schol. Vat. give us important indications as regards the text of the Elements as Pappus had it. In particular, they show that he could not have had in his text certain of the lemmas in Book X. For example, three of these are identical with what we find in Schol. Vat. (the lemma to X. 17=Schol. X. No. 106, and the lemmas to X. 54, 60 come in Schol. X. No. 328); and it is not possible to suppose. that these lemmas, if they were already in the text, would also be given as scholia. Of these three lemmas, that before X. 60 has already been condemned for other reasons; the other two, unobjectionable in themselves, must be rejected on the ground now stated. There were four others against which Heiberg found nothing to urge when writing his prolegomena to Vol. v., viz. the lemmas before X. 42, X. 14, X. 22 and X. 33. Of these, the lemma to X. 22 is not reconcilable with Schol. X. No. 161, which takes up the assumption in the text of Eucl. X. 22 as if no lemma had gone before. The lemma to X. 42, which, on account of the words introducing it (see p. 60 above), Heiberg at first hesitated to regard as an interpolation, is identical with Schol. X. No. 270. It is true that in Schol. X. No. 269 we find the words “this lemma has been proved before (ἐν τοῖς ἔμπροσθεν), but it shall also be proved now for convenience' sake (τοῦ ἑτοίμου ἕνεκα),” and it is possible to suppose that “before” may mean in Euclid's text before X. 42; but a proof in that place would surely have been as “convenient” as could be desired, and it is therefore more probable that the proof had been given by Pappus in some earlier place. (It may be added that the lemma to X. 14, which is identical with the lemma to XI. 23, condemned on other grounds, is for that reason open to suspicion.)

Heiberg's conclusion is that all the lemmas are spurious, and that most or all of them have found their way into the text from Pappus' commentary, though at a time anterior to Theon's edition, since they are found in all our MSS. This enables us to fix a date for these interpolations, namely the first half of the 4th c.

Of course Pappus had not in his text the interpolations which, from the fact of their appearing only in some of our MSS., are seen to be later than those above-mentioned. Such are the lemmas which are found in the text of V only after X. 29 and X. 31 respectively and are given in Heiberg's Appendix to Book X. (numbered 10 and 11). On the other hand it appears from Woepcke's tract307 that Pappus already had X. 115 in his text: though it does not follow from this that the proposition is genuine but only that interpolations began very early.

Theon interpolated a proposition (or lemma) between X. 12 and X. 13 (No. 5 in Heiberg's Appendix). Schol. Vat. has the same thing (X. No. 125). The writer of the scholia therefore did not find this lemma in the text. Schol. Vat. IX. Nos. 28, 29 show that neither did he find in his text the alterations which Theon made in Eucl. IX. 19; the scholia in fact only agree with the text of P, not with Theon's. This suggests that Schol. Vat. were written for use with a MS. of the ante-Theonine recension such as P is. This probability is further confirmed by a certain independence which P shows in several places when compared with the Theonine MSS. Not only has P better readings in some passages, but more substantial divergences; and, in particular, the absence in P of three notes of a historical character which are added, wholly or partly from Proclus, in the Theonine MSS. attests an independent and more primitive point of view in P.

In view of the distinctive character of P, it is possible that some of the scholia found in it in the first hand, but not in the other sources of Schol. Vat., also belong to that collection; and several circumstances confirm this. Schol. XIII. No. 45, found in P only, which relates to a passage in Eucl. XIII. 13, shows that certain words in the text, though older than Theon, are interpolated; and, as the scholium is itself older than Theon, is headed “third lemma,” and follows a “second lemma” relating to a passage in the text immediately preceding, which “second lemma” belongs to Schol. Vat. and is taken from Pappus, the “third” in all probability came from Pappus also. The same is true of Schol. XII. No. 72 and XIII. No. 69, which are respectively identical with the propositions vulgo XI. 38 (Heiberg, App. to Book XI., No. 3) and XIII. 6; for both of these interpolations are older than Theon. Moreover most of the scholia which P in the first hand alone has are of the same character as Schol. Vat. Thus VII. No. 7 and XIII. No. I introducing Books VII. and XIII. respectively are of the same historical character as several of Schol. Vat.; that VII. No. 7 appears in the text of P at the beginning of Book VII. constitutes no difficulty. There are a number of converses, remarks on the relation of propositions to one another, explanations such as XII. No. 89 in which it is remarked that Φ, Ω in Euclid's figure to XII. 17 (Z, V in my figure) are really the same point but that this makes no difference in the proof. Two other Schol. P on XII. 17 are connected by their headings with XII. No. 72 mentioned above. XI. No. 10 (P) is only another form of XI. No. 11 (B); and B often, alone with P, has preserved Schol. Vat. On the whole Heiberg considers some 40 scholia found in P alone to belong to Schol. Vat.

The history of Schol. Vat. appears to have been, in its main outlines, the following. They were put together after 500 A.D., since they contain extracts from Proclus, to which we ought not to assign a date too near to that of Proclus' work itself; and they must at least be earlier than the latter half of the 9th c., in which B was written. As there must evidently have been several intermediate links between the archetype and B, we must assign them rather to the first half of the period between the two dates, and it is not improbable that they were a new product of the great development of mathematical studies at the end of the 6th c. (Isidorus of Miletus). The author extracted what he found of interest in the commentary of Proclus on Book I. and in that of Pappus on the rest of the work, and put these extracts in the margin of a MS. of the class of P. As there are no scholia to I. 1-22, the first leaves of the archetype or of one of the earliest copies must have been lost at an early date, and it was from that mutilated copy that partly P and partly a MS. of the Theonine class were taken, the scholia being put in the margin in both. Then the collection spread through the Theonine MSS., gradually losing some scholia which could not be read or understood, or which were accidentally or deliberately omitted. Next it was extracted from one of these MSS. and made into a separate work which has been preserved, in part, in its entirety (Vat. 192 etc.) and, in part, divided into sections, so that the scholia to Books X.—XIII. were detached (V^{c}). It had the same fate in the MSS. which kept the original arrangement (in the margin), and in consequence there are some MSS. where the scholia to the stereometric Books are missing, those Books having come to be less read in the period of decadence. It is from one of these MSS. that the collection was extracted as a separate work such as we find it in Vat. (10th c.).

II. The second great division of the scholia is Schol. Vind.

This title is taken from the Viennese MS. (V), and the letters used by Heiberg to indicate the sources here in question are as follows.

V^{a}=scholia in V written by the same hand that copied the MS. itself from fol. 235 onward.

q=scholia of the Paris MS. 2344 (q) written by the first hand.

l=scholia of the Florence MS. Laurent. XXVIII, 2 written in the 13th—14th c., mostly in the first hand, but partly in two later hands.

V^{b}=scholia in V written by the same hand as the first part (leaves 1-183) of the MS. itself; V^{b} wrote his scholia after V^{a}.

q^{1}=scholia of the Paris MS. (q) found here and there in another hand of early date.

Schol. Vind. include scholia found in V^{a}q. 1 is nearly related to q; and in fact the three MSS. which, so far as Euclid's text is concerned, show no direct interdependence, are, as regards their scholia, derived from one original. Heiberg proves this by reference to the readings of the three in two passages (found in Schol. 1. No. 109 and X. No. 39 respectively). The common source must have contained, besides the scholia found in the three MSS. V^{a}ql, those also which are contained in two of them, for it is more unlikely that two of the three should contain common interpolations than that a particular scholium should drop out of one of them. Besides V^{a} and q, the scholia V^{b} and q^{1} must equally be referred to Schol. Vind., since the greater part of their scholia are found in 1. There is a lacuna in q from Eucl. VIII. 25 to IX. 14, so that for this portion of the Elements Schol. Vind. are represented by Vl only. Heiberg gives about 450 numbers in all as belonging to this collection.

Schol. Vind. did not all come from one source; this is shown by differences of substance, e.g. between X. Nos. 36 and 39, and by differences of time of writing: e.g. VI. No. 52 refers at the beginning to No. 55 with the words “as the scholium has it” and is therefore later than that scholium; X. No. 247 is also later than X. No. 246.

The scholia to Book I. are here also extracts from Proclus, but more copious and more verbatim than in Schol. Vat. The author has not always understood Proclus; and he had a text as bad as that of our MSS., with the same lacunae. The scholia to the other Books are partly drawn (1) from Schol. Vat., the MSS. representing Schol. Vind. and Schol. Vat. in these cases showing nearly all possible combinations; but there is no certain trace in Schol. Vind. of the scholia peculiar to P. The author used a copy of Schol. Vat. in the form in which they were attached to the Theonine text; thus Schol. Vind. correspond to BF Vat., where these diverge from P, and especially closely to B. Besides Schol. Vat., the editors of Schol. Vind. used (2) other old collections of scholia of which we find traces in B and F; Schol. Vind. have also some scholia common with b. The scholia which Schol. Vind. have in common with BF come from two different sources, and were apparently afterwards introduced into the other MSS.; one result of this is that several scholia are reproduced twice.

But, besides the scholia derived from these sources, Schol. Vind. contain a large number of others of late date, characterised by incorrect language or by triviality of content (there are many examples in numbers, citations of propositions used, absurd ἀπορίαι, and the like). Unlike Schol. Vat., these scholia often quote words from Euclid as a heading (in one case a heading is inserted in Schol. Vind. where a scholium without the heading is quoted from Schol. Vat., see V. No. 14). The explanations given often presuppose very little knowledge on the part of the reader and frequently contain obscurities and gross errors.

Schol. Vind. were collected for use with a MS. of the Theonine class; this follows from the fact that they contain a note on the proposition vulgo VII. 22 interpolated by Theon (given in Heiberg's App. to Vol. 11. p. 430). Since the scholium to VII. 39 given in V and p in the text after the title of Book VIII. quotes the proposition as VII. 39, it follows that this scholium must have been written before the interpolation of the two propositions vulgo VII. 20, 22; Schol. Vind. contain (VII. No. 80) the first sentence of it, but without the heading referring to VII. 39. Schol. VII. No. 97 quotes VII. 33 as VII. 34, so that the proposition vulgo VII. 22 may have stood in the scholiast's text but not the later interpolation vulgo VII. 20 (later because only found in B in the margin by the first hand). Of course the scholiast had also the interpolations earlier than Theon.

For the date of the collection we have a lower limit in the date (12th c.) of MSS. in which the scholia appear. That it was not much earlier than the 12th c. is indicated (1) by the poverty of its contents, (2) by the quality of the MS. of Proclus which was used in the compilation of it (the Munich MS. used by Friedlein with which the scholiast's excerpts are essentially in agreement belongs to the 11th— 12th c.), (3) by the fact that Schol. Vind. appear only in MSS. of the 12th c. and no trace of them is found in our MSS. belonging to the 9th—10th c. in which Schol. Vat. are found. The collection may therefore probably be assigned to the 11th c. Perhaps it may be in part due to Psellus who lived towards the end of that century: for in a Florence MS. (Magliabecch. XI, 53 of the 15th c.) containing a mathematical compendium intended for use in the reading of Aristotle the scholia 1. Nos. 40 and 49 appear with the name of Psellus attached.

Schol. Vind. are not found without the admixture of foreign elements in any of our three sources. In 1 there are only very few such in the first hand. In q there are several new scholia in the first hand, for the most part due to the copyist himself. The collection of scholia on Book X. in q (Heiberg's q^{c}) is also in the first hand; it is not original, and it may perhaps be due to Psellus (Maglb. has some definitions of Book X. with a heading “scholia of...Michael Psellus on the definitions of Euclid's 10th Element” and Schol. X. No. 9), whose name must have been attached to it in the common source of Maglb. and q; to a great extent it consists of extracts from Schol. Vind. taken from the same source as Vl. The scholia q^{1} (in an ancient hand in q), confined to Book II., partly belong to Schol. Vind. and partly correspond to b^{1} (Bologna MS.). q^{a} and q^{b} are in one hand (Theodorus Antiochita), the nearest to the first hand of q; they are doubtless due to an early possessor of the MS. of whom we know nothing more.

V^{a} has, besides Schol. Vind., a number of scholia which also appear in other MSS., one in BFb, some others in P, and some in v (Codex Vat. 1038, 13th c.); these scholia were taken from a source in which many abbreviations were used, as they were often misunderstood by V^{a}. Other scholia in V^{a} which are not found in the older sources—some appearing in V^{a} alone—are also not original, as is proved by mistakes or corruptions which they contain; some others may be due to the copyist himself.

V^{b} seldom has scholia common with the other older sources; for the most part they either appear in V^{b} alone or only in the later sources as v or F^{2} (later scholia in F), some being original, others not. In Book X. V^{b} has three series of numerical examples, (1) with Greek numerals, (2) alternatives added later, also mostly with Greek numerals, (3) with Arabic numerals. The last class were probably the work of the copyist himself. These examples (cf. p. 74 below) show the facility with which the Byzantines made calculations at the date of the MS. (12th c.). They prove also that the use of the Arabic numerals (in the East-Arabian form) was thoroughly established in the 12th c.; they were actually known to the Byzantines a century earlier, since they appear, in the first hand, in an Escurial MS. of the 11th c.

Of collections in other hands in V distinguished by Heiberg (see preface to Vol. v.), V^{1} has very few scholia which are found in other sources, the greater part being original; V^{2}, V^{3} are the work of the copyist himself; V^{4} are so in part only, and contain several scholia from Schol. Vat. and other sources. V^{3} and V^{4} are later than 13th —14th c., since they are not found in f (cod. Laurent. XXVIII, 6) which was copied from V and contains, besides V^{a} V^{b}, the greater part of V^{1} and VI. No. 20 of V^{2} (in the text).

In P there are, besides P^{3} (a quite late hand, probably one of the old Scriptores Graeci at the Vatican), two late hands (P^{2}), one of which has some new and independent scholia, while the other has added the greater part of Schol. Vind., partly in the margin and partly on pieces of leaves stitched on.

Our sources for Schol. Vat. also contain other elements. In P there were introduced a certain number of extracts from Proclus, to supplement Schol. Vat. to Book I.; they are all written with a different ink from that used for the oldest part of the MS., and the text is inferior. There are additions in the other sources of Schol. Vat. (F and B) which point to a common source for FB and which are nearly all found in other MSS., and, in particular, in Schol. Vind., which also used the same source; that they are not assignable to Schol. Vat. results only from their not being found in Vat. Of other additions in F, some are peculiar to F and some common to it and b; but they are not original. F^{2} (scholia in a later hand in F) contains three original scholia; the rest come from V. B contains, besides scholia common to it and F, b or other sources, several scholia which seem to have been put together by Arethas, who wrote at least a part of them with his own hand.

Heiberg has satisfied himself, by a closer study of b, that the scholia which he denotes by b, β and b^{1} are by one hand; they are mostly to be found in other sources as well, though some are original. By the same hand (Theodorus Cabasilas, 15th c.) are also the scholia denoted by b^{2}, B^{2}, b^{3} and B^{3}. These scholia come in great part from Schol. Vind., and in making these extracts Theodorus probably used one of our sources, l, mistakes in which often correspond to those of Theodorus. To one scholium is attached the name of Demetrius (who must be Demetrius Cydonius, a friend of Nicolaus Cabasilas, 14th c.); but it could not have been written by him, since it appears in B and Schol. Vind. Nor are all the scholia which bear the name of Theodorus due to Theodorus himself, though some are so.

As B^{3} (a late hand in B) contains several of the original scholia of b^{2}, B^{3} must have used b itself as his source, and, as all the scholia in B^{3} are in b, the latter is also the source of the scholia in B^{3} which are found in other MSS. B and b were therefore, in the 15th c., in the hands of the same person; this explains, too, the fact that b in a late hand has some scholia which can only come from B. We arrive then at the conclusion that Theodorus Cabasilas, in the 15th c., owned both the MSS. B and b, and that he transferred to B scholia which he had before written in b, either independently or after other sources, and inversely transferred some scholia from B to b. Further, B^{2} are earlier than Theodorus Cabasilas, who certainly himself wrote B^{3} as well as b^{2} and b^{3}.

An author's name is also attached to the scholia VI. No. 6 and X. No. 223, which are attributed to Maximus Planudes (end of 13th c.) along with scholia on I. 31, X. 14 and X. 18 found in 1 in a quite late hand and published on pp. 46, 47 of Heibėrg's dissertation. These seem to have been taken from lectures of Planudes on the Elements by a pupil who used l as his copy.

There are also in l two other Byzantine scholia, written by a late hand, and bearing the names Ioannes and Pediasimus respectively; these must in like manner have been written by a pupil after lectures of Ioannes Pediasimus (first half of 14th c.), and this pupil must also have used l.

Before these scholia were edited by Heiberg, very few of them had been published in the original Greek. The Basel editio princeps has a few (V. No. I, VI. Nos. 3, 4 and some in Book X.) which are taken, some from the Paris MS. (Paris. Gr. 2343) used by Grynaeus, others probably from the Venice MS. (Marc. 301) also used by him; one published by Heiberg, not in his edition of Euclid but in his paper on the scholia, may also be from Venet. 301, but appears also in Paris. Gr. 2342. The scholia in the Basel edition passed into the Oxford edition in the text, and were also given by August in the Appendix to his Vol. II.

Several specimens of the two series of scholia (Vat. and Vind.) were published by C. Wachsmuth (Rhein. Mus. XVIII. p. 132 sqq.) and by Knoche (Untersuchungen über die neu aufgefundenen Scholien des Proklus, Herford, 1865).

The scholia published in Latin were much more numerous. G. Valla (De expetendis et fugiendis rebus, 1501) reproduced apparently some 200 of the scholia included in Heiberg's edition. Several of these he obtained from two Modena MSS. which at one time were in his possession (Mutin. III B, 4 and II E, 9, both of the 15th c.); but he must have used another source as well, containing extracts from other series of scholia, notably Schol. Vind. with which he has some 87 scholia in common. He has also several that are new.

Commandinus included in his translation under the title “Scholia antiqua” the greater part of the Schol. Vat. which he certainly obtained from a MS. of the class of Vat. 192; on the whole he adhered closely to the Greek text. Besides these scholia Commandinus has the scholia and lemmas which he found in the Basel editio princeps, and also three other scholia not belonging to Schol. Vat., as well as one new scholium (to XII. 13) not included in Heiberg's edition, which are distinguished by different type and were doubtless taken from the Greek MS. used by him along with the Basel edition.

In Conrad Dasypodius' Lexicon mathematicum published in 1573 there is (on fol. 42-44) “Graecum scholion in definitiones Euclidis libri quinti elementorum appendicis loco propter pagellas vacantes annexum.” This contains four scholia, and part of two others, published in Heiberg's edition, with some variations of readings, and with some new matter added (for which see pp. 64-6 of Heiberg's pamphlet). The source of these scholia is revealed to us by another work of Dasypodius, Isaaci Monachi Scholia in Euclidis elementorum geometriae sex priores libros per C. Dasypodium in latinum sermonem translata et in lucem edita (1579). This work contains, besides excerpts from Proclus on Book I. (in part closely related to Schol. Vind.), some 30 scholia included in Heiberg's edition, several new scholia, and the above-mentioned scholia to the definitions of Book V. published in Greek in 1573. After the scholia follow “Isaaci Monachi prolegomena in Euclidis Elementorum geometriae libros” (two definitions of geometry) and “Varia miscellanea ad geometriae cognitionem necessaria ab Isaaco Monacho collecta” (mostly the same as pp. 252, 24-272, 27 in the Variae Collectiones included in Hultsch's Heron); lastly, a note of Dasypodius to the reader says that these scholia were taken “ex clarissimi viri Joannis Sambuci antiquo codice manu propria Isaaci Monachi scripto.” Isaak Monachus is doubtless Isaak Argyrus, 14th c.; and Dasypodius used a MS. in which, besides the passage in Hultsch's Variae Collectiones, there were a number of scholia marked in the margin with the name of Isaak (cf. those in b under the name of Theodorus Cabasilas). Whether the new scholia are original cannot be decided until they are published in Greek; but it is not improbable that they are at all events independent arrangements of older scholia. All but five of the others, and all but one of the Greek scholia to Book v., are taken from Schol. Vat.; three of the excepted ones are from Schol. Vind., and the other three seem to come from F (where some words of them are illegible, but can be supplied by means of Mut. III B, 4, which has these three scholia and generally shows a certain likeness to Isaak's scholia).

Dasypodius also published in 1564 the arithmetical commentary of Barlaam the monk (14th c.) on Eucl. Book II., which finds a place in Appendix IV. to the Scholia in Heiberg's edition.

Hultsch has some remarks on the origin of the scholia308. He observes that the scholia to Book I. contain a considerable portion of Geminus' commentary on the definitions and are specially valuable because they contain extracts from Geminus only, whereas Proclus, though drawing mainly upon him, quotes from others as well. On the postulates and axioms the scholia give more than is found in Proclus. Hultsch conjectures that the scholium on Book v., No. 3, attributing the discovery of the theorems to Eudoxus but their arrangement to Euclid, represents the tradition going back to Geminus, and that the scholium XIII., No. 1, has the same origin.

A word should be added about the numerical illustrations of Euclid's propositions in the scholia to Book X. They contain a large number of calculations with sexagesimal fractions309; the fractions go as far as fourth-sixtieths (1/60^{4}). Numbers expressed in these fractions are handled with skill and include some results of surprising accuracy310

CHAPTER VII.

EUCLID IN ARABIA.

We are told by [Hnull ]ājī Khalfa311 that the Caliph al-Mansūr (754-775) sent a mission to the Byzantine Emperor as the result of which he obtained from him a copy of Euclid among other Greek books, and again that the Caliph al-Ma'mūn (813-833) obtained manuscripts of Euclid, among others, from the Byzantines. The version of the Elements by al-[Hnull ]ajjāj b. Yūsuf b. Matar is, if not the very first, at least one of the first books translated from the Greek into Arabic312. According to the Fihrist313 it was translated by al-[Hnull ]ajjāj twice; the first translation was known as “Hārūni” (“for Hārūn” ), the second bore the name “Ma'mūni” (“for al-Ma'mūn” ) and was the more trustworthy. Six Books of the second of these versions survive in a Leiden MS. (Codex Leidensis 399, 1) now in part published by Besthorn and Heiberg314. In the preface to this MS. it is stated that, in the reign of Hārūn ar-Rashīd (786-809), al-[Hnull ]ajjāj was commanded by Ya[hnull ]yā b. Khālid b. Barmak to translate the book into Arabic. Then, when al-Ma'mūn became Caliph, as he was devoted to learning, al-[Hnull ]ajjāj saw that he would secure the favour of al-Ma'mūn “if he illustrated and expounded this book and reduced it to smaller dimensions. He accordingly left out the superfluities, filled up the gaps, corrected or removed the errors, until he had gone through the book and reduced it, when corrected and explained, to smaller dimensions, as in this copy, but without altering the substance, for the use of men endowed with ability and devoted to learning, the earlier edition being left in the hands of readers.”

The Fihrist goes on to say that the work was next translated by Ishāq b. Hunain, and that this translation was improved by Thābit b. Qurra. This Abū Ya`qūb Is[hnull ]āq b. [Hnull ]unain b. Is[hnull ]āq al-`Ibādī (d. 910) was the son of the most famous of Arabic translators, Hunain b. Ishāq al-`Ibādī (809-873), a Christian and physician to the Caliph alMutawakkil (847-861). There seems to be no doubt that Is[hnull ]āq, who must have known Greek as well as his father, made his translation direct from the Greek. The revision must apparently have been the subject of an arrangement between Is[hnull ]āq and Thābit as the latter died in 901 or nine years before Is[hnull ]āq. Thābit undoubtedly consulted Greek MSS. for the purposes of his revision. This is expressly stated in a marginal note to a Hebrew version of the Elements, made from Ishāq's, attributed to one of two scholars belonging to the same family, viz. either to Moses b. Tibbon (about 1244-1274) or to Jakob b. Machir (who died soon after 1306)315. Moreover Thābit observes, on the proposition which he gives as IX. 31, that he had not found this proposition and the one before it in the Greek but only in the Arabic; from which statement Klamroth draws two conclusions, (1) that the Arabs had already begun to interest themselves in the authenticity of the text and (2) that Thābit did not alter the numbers of the propositions in Ishāq's translation316. The Fihrist also says that Yu[hnull ]annā al-Qass (i.e. “the Priest” ) had seen in the Greek copy in his possession the proposition in Book I. which Thābit took credit for, and that this was confirmed by Na[znull ]īf, the physician, to whom Yuhannā had shown it. This proposition may have been wanting in Ishāq, and Thābit may have added it, but without claiming it as his own discovery317. As a fact, I. 45 is missing in the translation by al-[Hnull ]ajjāj.

The original version of Is[hnull ]āq without the improvements by Thābit has probably not survived any more than the first of the two versions by al-Hajjāj; the divergences between the MSS. are apparently due to the voluntary or involuntary changes of copyists, the former class varying according to the degree of mathematical knowledge possessed by the copyists and the extent to which they were influenced by considerations of practical utility for teaching purposes318. Two MSS. of the Ishāq-Thābit version exist in the Bodleian Library (No. 279 belonging to the year 1238, and No. 280 written in 1260-1)319; Books I.—XIII. are in the Is[hnull ]āq-Thābit version, the non-Euclidean Books XIV., XV. in the translation of Qustā b. L'ūqā al-Ba`labakkī (d. about 912). The first of these MSS. (No. 279) is that (O) used by Klamroth for the purpose of his paper on the Arabian Euclid. The other MS. used by Klamroth is (K) Kjobenhavn LXXXI, undated but probably of the 13th c., containing Books V.—XV., Books V.—X. being in the Is[hnull ]āq-Thābit version, Books XI.—XIII. purporting to be in al-Hajjāj's translation, and Books XIV., XV. in the version of Qus[tnull ]ā b. Lūqā. In not a few propositions K and O show not the slightest difference, and, even where the proofs show considerable differences, they are generally such that, by a careful comparison, it is possible to reconstruct the common archetype, so that it is fairly clear that we have in these cases, not two recensions of one translation, but arbitrarily altered and shortened copies of one and the same recension320. The Bodleian MS. No. 280 contains a preface, translated by Nîcoll, which cannot be by Thābit himself because it mentions Avicenna (980-1037) and other later authors. The MS. was written at Marāġa in the year 1260-1 and has in the margin readings and emendations from the edition of Na[snull ]ĩraddĩn a[tnull ]-[Tnull ]ĩsī (shortly to be mentioned) who was living at Marāġa at the time. Is it possible that a[tnull ]-[Tnull ]ūsī himself is the author of the preface321? Be this as it may, the preface is interesting because it throws light on the liberties which the Arabians allowed themselves to take with the text. After the observation that the book (in spite of the labours of many editors) is not free from errors, obscurities, redundancies, omissions etc., and is without certain definitions necessary for the proofs, it goes on to say that the man has not yet been found who could make it perfect, and next proceeds to explain (1) that Avicenna “cut out postulates and many definitions” and attempted to clear up difficult and obscure passages, (2) that Abū'l Wafā al-Būzjānĩ (939-997) “introduced unnecessary additions and left out many things of great importance and entirely necessary,” inasmuch as he was too long in various places in Book VI. and too short in Book X. where he left out entirely the proofs of the apotomae, while he made an unsuccessful attempt to emend XII. 14, (3) that Abū Ja`far al-Khāzin (d. between 961 and 971) arranged the postulates excellently but “disturbed the number and order of the propositions, reduced several propositions to one” etc. Next the preface describes the editor's own claims322 and then ends with the sentences, “But we have kept to the order of the books and propositions in the work itself (i.e. Euclid's) except in the twelfth and thirteenth books. For we have dealt in Book XIII. with the (solid) bodies and in Book XII. with the surfaces by themselves.”

After Thābit the Fihrist mentions Abū ’Uthmān ad-Dimashqī as having translated some Books of the Elements including Book X. (It is Abū ’Uthmān's translation of Pappus’ commentary on Book X. which Woepcke discovered at Paris.) The Fihrist adds also that “Na[znull ]ĩf the physician told me that he had seen the tenth Book of Euclid in Greek, that it had 40 propositions more than the version in common circulation which had 109 propositions, and that he had determined to translate it into Arabic.”

But the third form of the Arabian Euclid actually accessible to us is the edition of Abū Ja`far Mu[hnull ]. b. Mu[hnull ]. b. al-[Hnull ]asan Na[snull ]raddĩn aţ-Ţĩsī (whom we shall call aţ-Ţĩsī for short), born at Tĩs (in Khurāsān) in 1201 (d. 1274). This edition appeared in two forms, a larger and a smaller. The larger is said to survive in Florence only (Pal. 272 and 313, the latter MS. containing only six Books); this was published at Rome in 1594, and, remarkably enough, some copies of this edition are to be found with 12 and some with 13 Books, some with a Latin title and some without323. But the book was printed in Arabic, so that Kästner remarks that he will say as much about it as can be said about a book which one cannot read324. The shorter form, which however, in most MSS., is in 15 Books, survives at Berlin, Munich, Oxford, British Museum (974, 1334325, 1335), Paris (2465, 2466), India Office, and Constantinople; it was printed at Constantinople in 1801, and the first six Books at Calcutta in 1824326.

A[tnull ]-[Tnull ]ũsĩ's work is however not a translation of Euclid's text, but a re-written Euclid based on the older Arabic translations. In this respect it seems to be like the Latin version of the Elements by Campanus (Campano), which was first published by Erhard Ratdolt at Venice in 1482 (the first printed edition of Euclid327). Campanus (13th c.) was a mathematician, and it is likely enough that he allowed himself the same liberty as a[tnull ]-[Tnull ]ĩsī in reproducing Euclid. Whatever may be the relation between Campanus’ version and that of Athelhard of Bath (about 1120), and whether, as Curtze thinks328, they both used one and the same Latin version of 10th—11th c., or whether Campanus used Athelhard's version in the same way as a[tnull ]-[Tnull ]ĩsī used those of his predecessors329,it is certain that both versions came from an Arabian source, as is evident from the occurrence of Arabic words in them330. Campanus’ version is not of much service for the purpose of forming a judgment on the relative authenticity of the Greek and Arabian tradition ; but it sometimes preserves traces of the purer source, as when it omits Theon's addition to VI. 33331. A curious circumstance is that, while Campanus’ version agrees with aţ-Ţĩsī's in the number of the propositions in all the genuine Euclidean Books except V. and IX., it agrees with Athelhard's in having 34 propositions in Book V. (as against 25 in other versions), which confirms the view that the two are not independent, and also leads, as Klamroth says, to this dilemma: either the additions to Book V. are Athelhard's own, or he used an Arabian Euclid which is not known to us332. Heiberg also notes that Campanus’ Books XIV., XV. show a certain agreement with the preface to the Thābit-Is[hnull ]āq version, in which the author claims to have (1) given a method of inscribing spheres in the five regular solids, (2) carried further the solution of the problem how to inscribe any one of the solids in any other and (3) noted the cases where this could not be done.333

With a view to arriving at what may be called a common measure of the Arabian tradition, it is necessary to compare, in the first place, the numbers of propositions in the various Books. [Hnull ]ājĩ Khalfa says that al-[Hnull ]ajjāj's translation contained 468 propositions, and Thābit's 478; this is stated on the authority of a[tnull ]-[Tnull ]ĩsĩ, whose own edition contained 468334. The fact that Thābit's version had 478 propositions is confirmed by an index in the Bodleian MS. 279 (called O by Klamroth). A register at the beginning of the Codex Leidensis 399, 1 which gives Is[hnull ]aq's numbers (although the translation is that of al-[Hnull ]ajjāj) apparently makes the total 479 propositions (the number in Book XIV. being apparently 11, instead of the 10 of O335). I subjoin a table of relative numbers taken from Klamroth, to which I have added the corresponding numbers in August's and Heiberg's editions of the Greek text.

The numbers in the case of Heiberg include all propositions which he has printed in the text; they include therefore XIII. 6 and III. 12 now to be regarded as spurious, and X. 112-115 which he brackets as doubtful. He does not number the propositions in Books XIV., XV., but I conclude that the numbers in P reach at least 9 in XIV., and 9 in XV.

The Fihrist confirms the number 109 for Book X., from which Klamroth concludes that Is[hnull ]āq's version was considered as by far the most authoritative.

In the text of O, Book IV. consists of 17 propositions and Book XIV. of 12, differing in this respect from its own table of contents; IV. 15, 16 in O are really two proofs of the same proposition.

In al-[Hnull ]ajjāj's version Book I. consists of 47 propositions only, I. 45 being omitted. It has also one proposition fewer in Book III., the Heronic proposition III. 12 being no doubt omitted.

In speaking of particular propositions, I shall use Heiberg's numbering, except where otherwise stated.

The difference of 10 propositions between Thābit-Is[hnull ]āq and a[tnull ]-[Tnull ]ĩsī is accounted for thus:

(1) The three propositions VI. 12 and X. 28, 29 which both Is[hnull ]āq and the Greek text have are omitted in a[tnull ]-[Tnull ]ĩsī.

(2) Is[hnull ]āq divides each of the propositions XIII. 1-3 into two, making six instead of three in a[tnull ]-[Tnull ]ĩsī and in the Greek.

(3) Is[hnull ]āq has four propositions (numbered by him VIII. 24, 25, IX. 30, 31) which are neither in the Greek Euclid nor in aţ-Ţĩsī.

Apart from the above differences al-[Hnull ]ajjāj (so far as we know), Ishāq and a[tnull ]-[Tnull ]ĩsī agree; but their Euclid shows many differences from our Greek text. These differences we will classify as follows336.

1. Propositions.

The Arabian Euclid omits VII. 20, 22 of Gregory's and August's editions (Heiberg, App. to Vol. II. pp. 428-32); VIII. 16, 17; X. 7, 8, 13, 16, 24, 112, 113, 114, besides a lemma vulgo X. 13, the proposition X. 117 of Gregory's edition, and the scholium at the end of the Book (see for these Heiberg's Appendix to Vol. III. pp. 382, 408-416); XI. 38 in Gregory and August (Heiberg, App. to Vol. IV. p. 354); XII. 6, 13, 14; (also all but the first third of Book XV.).

The Arabian Euclid makes III. 11, 12 into one proposition, and divides some propositions (X. 31, 32; XI. 31, 34; XIII. 1-3) into two each.

The order is also changed in the Arabic to the following extent. V. 12, 13 are interchanged and the order in Books VI., VII., IX.— XIII. is:

VI. 1-8, 13, 11, 12, 9, 10, 14-17, 19, 20, 18, 21, 22, 24, 26, 23, 25, 27-30, 32, 31, 33.

VII. 1-20, 22, 21, 23-28, 31, 32, 29, 30, 33-39.

IX. 1-13, 20, 14-19, 21-25, 27, 26, 28-36, with two new propositions coming before prop. 30.

X. 1-6, 9-12, 15, 14, 17-23, 26-28, 25, 29-30, 31, 32, 33— 111, 115.

XI. 1-30, 31, 32, 34, 33, 35-39.

XII. 1-5, 7, 9, 8, 10, 12, 11, 15, 16-18.

XIII. 1-3, 5, 4, 6, 7, 12, 9, 10, 8, 11, 13, 15, 14, 16-18.

2. Definitions.

The Arabic omits the following definitions : IV. Deff. 3-7, VII. Def. 9 (or 10), XI. Deff. 5-7, 15, 17, 23, 25-28; but it has the spurious definitions VI. Deff. 2, 5, and those of proportion and ordered proportion in Book V. (Deff. 8, 19 August), and wrongly interchanges V. Deff. 11, 12 and also VI. Deff. 3, 4.

The order of the definitions is also different in Book VII. where, after Def. 11, the order is 12, 14, 13, 15, 16, 19, 20, 17, 18, 21, 22, 23, and in Book XI. where the order is 1, 2, 3, 4, 8, 10, 9, 13, 14, 16, 12, 21, 22, 18, 19, 20, 11, 24.

3. Lemmas and porisms.

All are omitted in the Arabic except the porisms to VI. 8, VIII. 2, X. 3; but there are slight additions here and there, not found in the Greek, e.g. in VIII. 14, 15 (in K).

4. Alternative proofs.

These are all omitted in the Arabic, except that in X. 105, 106 they are substituted for the genuine proofs; but one or two alternative proofs are peculiar to the Arabic (VI. 32 and VIII. 4, 6).

The analyses and syntheses to XIII. 1-5 are also omitted in the Arabic.

Klamroth is inclined, on a consideration of all these differences, to give preference to the Arabian tradition over the Greek (1) “on historical grounds,” subject to the proviso that no Greek MS. as ancient as the 8th c. is found to contradict his conclusions, which are based generally (2) on the improbability that the Arabs would have omitted so much if they had found it in their Greek MSS., it being clear from the Fihrist that the Arabs had already shown an anxiety for a pure text, and that the old translators were subjected in this matter to the check of public criticism. Against the “historical grounds,” Heiberg is able to bring a considerable amount of evidence337. First of all there is the British Museum palimpsest (L) of the 7th or the beginning of the 8th c. This has fragments of propositions in Book X. which are omitted in the Arabic; the numbering of one proposition, which agrees with the numbering in other Greek MS., is not comprehensible on the assumption that eight preceding propositions were omitted in it, as they are in the Arabic; and lastly, the readings in L are tolerably like those of our MSS., and surprisingly like those of B. It is also to be noted that, although P dates from the 10th c. only, it contains, according to all appearance, an ante-Theonine recension.

Moreover there is positive evidence against certain omissions by the Arabians. A[tnull ]-[Tnull ]ĩsī omits VI. 12, but it is scarcely possible that, if Eutocius had not had it, he would have quoted VI. 23 by that number338. This quotation of VI. 23 by Eutocius also tells against Is[hnull ]āq who has the proposition as VI. 25. Again, Simplicius quotes VI. 10 by that number, whereas it is VI. 13 in Is[hnull ]āq; and Pappus quotes, by number, XIII. 2 (Is[hnull ]āq 3, 4), XIII. 4 (Is[hnull ]āq 8), XIII. 16 (Is[hnull ]āq 19). On the other hand the contraction of III. 11, 12 into one proposition in the Arabic tells in favour of the Arabic.

Further, the omission of certain porisms in the Arabic cannot be supported; for Pappus quotes the porism to XIII. 17339, Proclus those to II. 4, III. 1, VII. 2340, and Simplicius that to IV. 15.

Lastly, some propositions omitted in the Arabic are required in later propositions. Thus X. 13 is used in X. 18, 22, 23, 26 etc.; X. 17 is wanted in X. 18, 26, 36; XII. 6, 13 are required for XII. 11 and XII. 15 respectively.

It must also be remembered that some of the things which were properly omitted by the Arabians are omitted or marked as doubtful in Greek MSS. also, especially in P, and others are rightly suspected for other reasons (e.g. a number of alternative proofs, lemmas, and porisms, as well as the analyses and syntheses of XIII. 1-5). On the other hand, the Arabic has certain interpolations peculiar to our inferior MSS. (cf. the definition VI. Def. 2 and those of proportion and ordered proportion).

Heiberg comes to the general conclusion that, not only is the Arabic tradition not to be preferred offhand to that of the Greek MSS., but it must be regarded as inferior in authority. It is a question how far the differences shown in the Arabic are due to the use of Greek MSS. differing from those which have been most used as the basis of our text, and how far to the arbitrary changes made by the Arabians themselves. Changes of order and arbitrary omissions could not surprise us, in view of the preface above quoted from the Oxford MS. of Thābit-Is[hnull ]āq, with its allusion to the many important and necessary things left out by Abĩ'l Wafā and to the author's own rearrangement of Books XII., XIII. But there is evidence of differences due to the use by the Arabs of other Greek MSS. Heiberg341 is able to show considerable resemblances between the Arabic text and the Bologna MS. b in that part of the MS. where it diverges so remarkably from our other MSS. (see the short description of it above, p.49); in illustration he gives a comparison of the proofs of XII. 7 in b and in the Arabic respectively, and points to the omission in both of the proposition given in Gregory's edition as XI. 38, and to a remarkable agreement between them as regards the order of the propositions of Book XII. As above stated, the remarkable divergence of b only affects Books XI. (at end) and XII.; and Book XIII. in b shows none of the transpositions and other peculiarities of the Arabic. There are many differences between b and the Arabic, especially in the definitions of Book XI., as well as in Book XIII. It is therefore a question whether the Arabians made arbitrary changes, or the Arabic form is the more ancient, and b has been altered through contact with other MSS. Heiberg points out that the Arabians must be alone responsible for their definition of a prism, which only covers a prism with a triangular base. This could not have been Euclid's own, for the word prism already has the wider meaning in Archimedes, and Euclid himself speaks of prisms with parallelograms and polygons as bases (XI. 39; XII. 10). Moreover, a Greek would not have been likely to leave out the definitions of the “Platonic” regular solids.

Heiberg considers that the Arabian translator had before him a MS. which was related to b, but diverged still further from the rest of our MSS. He does not think that there is evidence of the existence of a redaction of Books I.—X. similar to that of Books XI., XII. in b; for Klamroth observes that it is the Books on solid geometry (XI.—XIII.) which are more remarkable than the others for omissions and shorter proofs, and it is a noteworthy coincidence that it is just in these Books that we have a divergent text in b.

An advantage in the Arabic version is the omission of VII. Def. 10, although, as Iamblichus had it, it may have been deliberately omitted by the Arabic translator. Another advantage is the omission of the analyses and syntheses of XIII. 1-5; but again these may have been omitted purpōsely, as were evidently a number of porisms which are really necessary.

One or two remarks may be added about the Arabic versions as compared with one another. Al-[Hnull ]ajjāj's object seems to have been less to give a faithful reflection of the original than to write a useful and convenient mathematical text-book. One characteristic of it is the careful references to earlier propositions when their results are used. Such specific quotations of earlier propositions are rare in Euclid; but in al-[Hnull ]ajjāj we find not only such phrases as “by prop. so and so,” “which was proved” or “which we showed how to do in prop. so and so,” but also still longer phrases. Sometimes he kepeats a construction, as in I. 44 where, instead of constructing “the parallelogram BEFG equal to the triangle C in the angle EBG which is equal to the angle D” and placing it in a certain position, he produces AB to G, making BG equal to half DE (the base of the triangle CDE in his figure), and on GB so constructs the parallelogram BHKG by I. 42 that it is equal to the triangle CDE, and its angle GBH is equal to the given angle.

Secondly, al-[hnull ]ajjāj, in the arithmetical books, in the theory of proportion, in the applications of the Pythagorean I. 47, and generally where possible, illustrates the proofs by numerical examples. It is true, observes Klamroth, that these examples are not apparently separated from the commentary of an-Nairīzī, and might not therefore have been due to al-[Hnull ]ajjāj himself; but the marginal notes to the Hebrew translation in Municn MS. 36 show that these additions were in the copy of al-[Hnull ]ajjāj used by the translator, for they expressly give these proofs in numbers as variants taken from al-[Hnull ]ajjāj342.

These characteristics, together with al-[Hnull ]ájjāj's freer formulation of the propositions and expansion of the proofs, constitute an intelligible reason why Is[hnull ]āq should have undertaken a fresh translation from the Greek. Klamroth calls Is[hnull ]āq's version a model of a good translation of a mathematical text; the introductory and transitional phrases are stereotyped and few in number, the technical terms are simply and consistently rendered, and the less formal expressions connect themselves as closely with the Greek as is consistent with intelligibility and the character of the Arabic language. Only in isolated cases does the formulation of definitions and enunciations differ to any considerable extent from the original. In general, his object seems to have been to get rid of difficulties and unevennesses in the Greek text by next devices, while at the same time giving a faithful reproduction of it.343.

There are curious points of contact between the versions of al-[Hnull ]ajjāj and T[hnull ]amacr;bit-Is[hnull ]āq. For example, the definitions and enunciations of propositions are often word for word the same. Presumably this is owing to the fact that Is[hnull ]āq found these definitions and enunciations already established in the schools in his time, where they would no doubt be learnt by heart, and refrained from translating them afresh, merely adopting the older version with some changes344. Secondly, there is remarkable agreement between the Arabic versions as regards the figures, which show considerable variations from the figures of the Greek text, especially as regards the letters; this is also probably to be explained in the same way, all the later translators having most likely borrowed al-[hnull ]ajjāj's adaptation of the Greek figures345. Lastly, it is remarkable that the version of Books XI.—XIII. in the Kjφbenhavn MS. (K), purporting to be by al-[Hnull ]ajjāj, is almost exactly the same as the Thābit-Is[hnull ]āq version of the same Books in O. Klamroth conjectures that Is[hnull ]āq may not have translated the Books on solid geometry at all, and that Thābit took them from al-[Hnull ]ajjāj, only making some changes in order to fit them to the translation of Is[hnull ]āq346.

From the facts (1) that a[tnull ]-[Tnull ]ĩsī's edition had the same number of propositions (468) as al-[hnull ]ajjāj's version, while Thābit-Is[hnull ]āq's had 478, and (2) that a[tnull ]-[Tnull ]ĩsī has the same careful references to earlier propositions, Klamroth concludes that a[tnull ]-[Tnull ]ĩsī deliberately preferred al-[hnull ]ajjāj's version to that of Is[hnull ]āq347. Heiberg, however, points out (1) that a[tnull ]-[Tnull ]ĩsī left out VI. 12 which, if we may judge by Klamroth's silence, al-[hnull ]ajjāj had, and (2) al-[hnull ]ajjāj's version had one proposition less in Books I. and III. than a[tnull ]-[Tnull ]ĩsī has. Besides, in a passage quoted by [hnull ]ājī Khalfa348 from a[tnull ]-[Tnull ]ĩsī, the latter says that “he separated the things which, in the approved editions, were taken from the archetype from the things which had been added thereto,” indicating that he had compiled his edition from both the earlier translations349.

There were a large number of Arabian commentaries on, or reproductions of, the Elements or portions thereof, which will be found fully noticed by Steinschneider350. I shall mention here the commentators etc. referred to in the Fihrist, with a few others.

1. Abĩ 'l `Abbās al-Fa[dnull ]l b. [hnull ]ātim an-Nairīzī (born at Nairīz, died about 922) has already been mentioned351. His commentary survives, as regards Books I.—VI., in the Codex Leidensis 399, I, now edited, as to four Books, by Besthorn and Heiberg, and as regards Books I.—X. in the Latin translation made by Gherard of Cremona in the 12th c. and now published by Curtze from a Cracow MS352. Its importance lies mainly in the quotations from Heron and Simplicius.

2. Ahmad b. `Umar al-Karābīsī (date uncertain, probably 9th— 10th c.), “who was among the most distinguished geometers and arithmeticians353.”

3. Al-'Abbās b. Sa`īd al-Jauharī (fl. 830) was one of the astronomical observers under al-Ma'mĩn, but devoted himself mostly to geometry. He wrote a commentary to the whole of the Elements, from the beginning to the end; also the “Book of the propositions which he added to the first book of Euclid354.”

4. Muh. b. `Īsā Abĩ `Abdallāh al-Māhānī (d. between 874 and 884) wrote, according to the Fihrist, (I) a commentary on Eucl. Book v., (2) “On proportion,” (3) “On the 26 propositions of the first Book of Euclid which are proved without reductio ad absurdum355.” The work “On proportion” survives and is probably identical with, or part of, the commentary on Book v.356 He also wrote, what is not mentioned by the Fihrist, a commentary on Eucl. Book X., a fragment of which survives in a Paris MS.357

5. Abĩ Ja`far al-Khāzin (i.e. “the treasurer” or “librarian” ), one of the first mathematicians and astronomers of his time, was born in Khurāsān and died between the years 961 and 971. The Fihrist speaks of him as having written a commentary on the whole of the Elements358, but only the commentary on the beginning of Book X. survives (in Leiden, Berlin and Paris); therefore either the notes on the rest of the Books have perished, or the Fihrist is in error359. The latter would seem more probable, for, at the end of his commentary, al-Khāzin remarks that the rest had already been commented on by Sulaimān b. 'Usma (Leiden MS.)360 or 'Oqba (Suter), to be mentioned below. Al-Khāzin's method is criticised unfavourably in the preface to the Oxford MS. quoted by Nicoll (see p. 77 above).

6. Abĩ'l Wafā al-Bĩzjānī (940-997), one of the greatest Arabian mathematicians, wrote a commentary on the Elements, but did not complete it361. His method is also unfavourably regarded in the same preface to the Oxford MS. 280. According to Hājī Khalfa, he also wrote a book on geometrical constructions, in thirteen chapters. Apparently a book answering to this description was compiled by a gifted pupil from lectures by Abĩ 'l Wafā, and a Paris MS. (Anc. fonds 169) contains a Persian translation of this work, not that of Abĩ 'l Wafā himself. An analysis of the work was given by Woepcke362, and some particulars will be found in Cantor363. Abĩ 'l Wafā also wrote a commentary on Diophantus, as well as a separate “book of proofs to the propositions which Diophantus used in his book and to what he (Abĩ 'l Wafā) employed in his commentary364.”

7. Ibn Rāhawaihi al-Arjānī also commented on Eucl. Book X.365.

8. `Alī b. Ahmad Abĩ 'l-Qāsim al-An[tnull ]ākī (d. 987) wrote a commentary on the whole book366; part of it seems to survive (from the 5th Book onwards) at Oxford (Catal. MSS. orient. II. 281)367.

9. Sind b. `Alī Abĩ '[tnull ]-[Tnull ]aiyib was a Jew who went over to Islam in the time of al-Ma'mĩn, and was received among his astronomical observers, whose head he became368 (about 830); he died after 864. He wrote a commentary on the whole of the Elements; “Abĩ `Alī saw nine books of it, and a part of the tenth369.” His book “On the Apotomae and the Medials,” mentioned by the Fihrist, may be the same as, or part of, his commentary on Book X.

10. Abĩ Yĩsuf Ya`qĩb b. Muh. ar-Rāzī “wrote a commentary on Book X., and that an excellent one, at the instance of Ibn al`Amīd370.”

11. The Fihrist next mentions al-Kindī (Abĩ Yĩsuf Ya'qĩb b. Ishāq b. as-Sabbāh al-Kindī, d. about 873), as the author (1) of a work “on the objects of Euclid's book,” in which occurs the statement that the Elements were originally written by Apollonius, the carpenter (see above, p. 5 and note), (2) of a book “on the improvement of Euclid's work,” and (3) of another “on the improvement of the 14th and 15th Books of Euclid.” “He was the most distinguished man of his time, and stood alone in the knowledge of the old sciences collectively; he was called ’the philosopher of the Arabians’; his writings treat of the most different branches of knowledge, as logic, philosophy, geometry, calculation, arithmetic, music, astronomy and others371.” Among the other geometrical works of al-Kindī mentioned by the Fihrist372 are treatises on the closer investigation of the results of Archimedes concerning the measure of the diameter of a circle in terms of its circumference, on the construction of the figure of the two mean proportionals, on the approximate determination of the chords of the circle, on the approximate determination of the chord (side) of the nonagon, on the division of triangles and quadrilaterals and constructions for that purpose, on the manner of construction of a circle which is equal to the surface of a given cylinder, on the division of the circle, in three chapters etc.

12. The physician Na[znull ]ĩf b. Yumn (or Yaman) al-Qass (“the priest” ) is mentioned by the Fihrist as having seen a Greek copy of Eucl. Book X. which had 40 more propositions than that which was in general circulation (containing 109), and having determined to translate it into Arabic373. Fragments of such a translation exist at Paris, Nos. 18 and 34 of the MS. 2457 (952, 2 Suppl. Arab. in Woepcke's tract); No. 18 contains “additions to some propositions of the 10th Book, existing in the Greek language374.” Nazĩf must have died about 990375.

13. Yũ[hnull ]annã b. Yūsuf b. al-[Hnull ]ãrith b. al-Bitrĩq al-Qass (d. about 980) lectured on the Elements and other geometrical books, made translations from the Greek, and wrote a tract on the “proof” of the case of two straight lines both meeting a third and making with it, on one side, two angles together less than two right angles376. Nothing of his appears to survive, except that a tract “on rational and irrational magnitudes,” No. 48 in the Paris MS. just mentioned, is attributed to him.

14. Abũ Mu[hnull ]. al-[Hnull ]asan b. 'Ubaidallãh b. Sulaimãn b. Wahb (d. 901) was a geometer of distinction, who wrote works under the two distinct titles “A commentary on the difficult parts of the work of Euclid” and “The Book on Proportion377.” Suter thinks that another reading is possible in the case of the second title, and that it may refer to the Euclidean work “on the divisions (of figures)378.”

15. Qustã b. Lũqã al-Ba'labakkĩ (d. about 912), a physician, philosopher, astronomer, mathematician and translator, wrote “on the difficult passages of Euclid's book” and “on the solution of arithmetical problems from the third book of Euclid379” ; also an “introduction to geometry,” in the form of question and answer380.

16. Thãbit b. Qurra (826-901), besides translating some parts of Archimedes and Books V.mdashVII. of the Conics of Apollonius, and revising Ishãq's translation of Euclid's Elements, also revised the translation of the Data by the same Ishãq and the book On divisions of figures translated by an anonymous writer. We are told also that he wrote the following works: (I) On the Premisses (Axioms, Postulates etc.) of Euclid, (2) On the Propositions of Euclid, (3) On the propositions and questions which arise when two straight lines are cut by a third (or on the “proof” of Euclid's famous postulate). The last tract is extant in the MS. discovered by Woepcke (Paris 2457, 32^{o}). He is also credited with “an excellent work” in the shape of an “Introduction to the Book of Euclid,” a treatise on Geometry dedicated to Ismã'il b. Bulbul, a Compendium of Geometry, and a large number of other works for the titles of which reference may be made to Suter, who also gives particulars as to which are extant381.

17. Abũ Sa'ĩd Sinãn b. Thãbit b. Qurra, the son of the translator of Euclid, followed in his father's footsteps as geometer, astronomer and physician. He wrote an “improvement of the book of ...... on the Elements of Geometry, in which he made various additions to the original.” It is natural to conjecture that Euclid is the name missing in this description (by Ibn abĩ U[snull ]aibi'a); Casiri has the name Aqãton382. The latest editor of the Ta'rĩkh al-[Hnull ]ukamã, however, makes the name to be Iflãton (=Plato), and he refers to the statement by the Fihrist and Ibn al-Qiftĩ attributing to Plato a work on the Elements of Geometry translated by Qust<*>ã. It is just possible, therefore, that at the time of Qus[tnull ]ã the Arabs were acquainted with a book on the Elements of Geometry translated from the Greek, which they attributed to Plato383. Sinãn died in 943.

18. Abũ Sahl Wĩjan (or Waijan) b. Rustam al-Kũhĩ (fl. 988), born at Kũh in [Tnull ]abaristãn, a distinguished geometer and astronomer, wrote, according to the Fihrist, a “Book of the Elements” after that of Euclid384; the 1st and 2nd Books survive at Cairo, and a part of the 3rd Book at Berlin (5922)385. He wrote also a number of other geometrical works: Additions to the 2nd Book of Archimedes on the Sphere and Cylinder (extant at Paris, at Leiden, and in the India Office), On the finding of the side of a heptagon in a circle (India Office and Cairo), On two mean proportionals (India Office), which last may be only a part of the Additions to Archimedes' On the Sphere and Cylinder, etc.

19. Abũ Na[snull ]r Mu[hnull ]. b. Mu[hnull ]. b. [Tnull ]arkhãn b. Uzlaġ al-Fãrãbĩ (870-950) wrote a commentary on the difficulties of the introductory matter to Books I. and V.386 This appears to survive in the Hebrew translation which is, with probability, attributed to Moses b. Tibbon387.

20. Abũ 'Alĩ al-Hasan b. al-Hasan b. al-Haitham (about 9651039), known by the name Ibn al-Haitham or Abũ 'Alĩ al-Basrĩ, was a man of great powers and knowledge, and no one of his time approached him in the field of mathematical science. He wrote several works on Euclid the titles of which, as translated by Woepcke from Usaibi'a, are as follows388: “ 1. Commentary and abridgment of the Elements.

2. Collection of the Elements of Geometry and Arithmetic, drawn from the treatises of Euclid and Apollonius.

3. Collection of the Elements of the Calculus deduced from the principles laid down by Euclid in his Elements.

4. Treatise on “measure” after the manner of Euclid's Elements.

5. Memoir on the solution of the difficulties in Book I.

6. Memoir for the solution of a doubt about Euclid, relative to Book V.

7. Memoir on the solution of a doubt about the stereometric portion.

8. Memoir on the solution of a doubt about Book XII.

9. Memoir on the division of the two magnitudes mentioned in X. I (the theorem of exhaustion).

10. Commentary on the definitions in the work of Euclid (where Steinschneider thinks that some more general expression should be substituted for “definitions” ).

The last-named work (which Suter calls a commentary on the Postulates of Euclid) survives in an Oxford MS. (Catal. MSS. orient. I. 908) and in Algiers (1446, 1^{o}).

A Leiden MS. (966) contains his Commentary “on the difficult places” up to Book V. We do not know whether in this commentary, which the author intended to form, with the commentary on the Musãdarãt, a sort of complete commentary, he had collected the separate memoirs on certain doubts and difficult passages mentioned in the above list.

A commentary on Book V. and following Books found in a Bodleian MS. (Catal. II. p. 262) with the title “Commentary on Euclid and solution of his difficulties” is attributed to b. Haitham; this might be a continuation of the Leiden MS.

The memoir on X. 1 appears to survive at St Petersburg, MS. de l'Institut des langues orient. 192, 5^{o} (Rosen, Catal. p. 125).

21. Ibn Sĩnã, known as Avicenna (980-1037), wrote a Compendium of Euclid, preserved in a Leiden MS. No. 1445, and forming the geometrical portion of an encyclopaedic work embracing Logic, Mathematics, Physics and Metaphysics389.

22. Ahmad b. al-Husain al-Ahwãzĩ al-Kãtib wrote a commentary on Book X., a fragment of which (some 10 pages) is to be found at Leiden (970), Berlin (5923) and Paris (2467, 18^{o})390.

23. Na[snull ]ĩraddĩn a[tnull ]-[Tnull ]ũsĩ (1201-1274) who, as we have seen, brought out a Euclid in two forms, wrote: “ 1. A treatise on the postulates of Euclid (Paris, 2467, 5^{o}).

2. A treatise on the 5th postulate, perhaps only a part of the foregoing (Berlin, 5942, Paris, 2467, 6^{o}).

3. Principles of Geometry taken from Euclid, perhaps identical with No. 1 above (Florence, Pal. 298).

4. 105 problems out of the Elements (Cairo). He also edited the Data (Berlin, Florence, Oxford etc.)391.

24. Muh. b. Ashraf Shamsaddĩn as-Samarqandĩ (fl. 1276) wrote “Fundamental Propositions, being elucidations of 35 selected propositions of the first Books of Euclid,” which are extant at Gotha (1496 and 1497), Oxford (Catal. I. 967, 2^{o}), and Brit. Mus.392.

25. Mũsã b. Mu[hnull ]. b. Ma[hnull ]mũd, known as Qã[dnull ]ĩzãde ar-Rũmĩ (i.e. the son of the judge from Asia Minor), who died between 1436 and 1446, wrote a commentary on the “Fundamental Propositions” just mentioned, of which many MSS. are extant393. It contained biographical statements about Euclid alluded to above (p. 5, note).

26. Abũ Dã'ũd Sulaimãn b. ’Uqba, a contemporary of al-Khãzin (see above, No. 5), wrote a commentary on the second half of Book X., which is, at least partly, extant at Leiden (974) under the title “On the binomials and apotomae found in the 10th Book of Euclid394.”

27. The Codex Leidensis 399, 1 containing al-[Hnull ]ajjãj's translation of Books I.—VI. is said to contain glosses to it by Sa`ĩd b. Mas'ũd b. al-Qass, apparently identical with Abũ Nasr Gars al-Na'ma, son of the physician Mas'ũd b. al-Qass al-Bagdãdĩ, who lived in the time of the last Caliph al-Musta'sim (d. 1258)395.

28. Abũ Mu[hnull ]ammad b. Abdalbãqĩ al-Bag[dnull ]ãdĩ al-Fara[dnull ]ĩ (d. 1141, at the age of over 70 years) is stated in the Ta'rĩkh al-[Hnull ]ukamã to have written an excellent commentary on Book X. of the Elements, in which he gave numerical examples of the propositions396. This is published in Curtze's edition of an-Nairĩzĩ where it occupies pages 252-386397.

29. Ya[hnull ]yã b. Mu[hnull ]. b. 'Abdãn b. 'Abdalwã[hnull ]id, known by the name of Ibn al-Lubũdĩ (1210-1268), wrote a Compendium of Euclid, and a short presentation of the postulates398.

30. Abũ 'Abdallãh Mu[hnull ]. b. Mu'ãdh al-Jayyãnĩ wrote a commentary on Eucl. Book V. which survives at Algiers (1446, 3^{o})399.

31. Abũ Na[snull ]r Mansũr b. 'Aliã b.'Irãq wrote, at the instance of Mu[hnull ]. b. A[hnull ]mad Abũ ’r-Rai[hnull ]ãn al-Bĩrũnĩ (973-1048), a tract “on a doubtful (difficult) passage in Eucl. Book XIII.” (Berlin, 5925)400.

CHAPTER VIII.

PRINCIPAL TRANSLATIONS AND EDITIONS OF THE ELEMENTS.

Cicero is the first Latin author to mention Euclid401; but it is not likely that in Cicero's time Euclid had been translated into Latin or was studied to any considerable extent by the Romans; for, as Cicero says in another place402, while geometry was held in high honour among the Greeks, so that nothing was more brilliant than their mathematicians, the Romans limited its scope by having regard only to its utility for measurements and calculations. How very little theoretical geometry satisfied the Roman agrimensores is evidenced by the work of Balbus de mensuris403, where some of the definitions of Eucl. Book I. are given. Again, the extracts from the Elements found in the fragment attributed to Censorinus (fl. 238 A.D)404 are confined to the definitions, postulates, and common notions. But by degrees the Elements passed even among the Romans into the curriculum of a liberal education; for Martianus Capella speaks of the effect of the enunciation of the proposition “how to construct an equilateral triangle on a given straight line” among a company of philosophers, who, recognising the first proposition of the Elements, straightway break out into encomiums on Euclid405. But the Elements were then (c. 470 A.D.) doubtless read in Greek; for what Martianus Capella gives406 was drawn from a Greek source, as is shown by the occurrence of Greek words and by the wrong translation of I. def. 1 (“punctum vero est cuius pars nihil est” ). Martianus may, it is true, have quoted, not from Euclid himself, but from Heron or some other ancient source.

But it is clear from a certain palimpsest at Verona that some scholar had already attempted to translate the Elements into Latin. This palimpsest407 has part of the “Moral reflections on the Book of Job” by Pope Gregory the Great written in a hand of the 9th c. above certain fragments which in the opinion of the best judges date from the 4th c. Among these are fragments of Vergil and of Livy, as well as a geometrical fragment which purports to be taken from the 14th and 15th Books of Euclid. As a matter of fact it is from Books XII. and XIII. and is of the nature of a free rendering, or rather a new arrangement, of Euclid with the propositions in different order408. The MS. was evidently the translator's own copy, because some words are struck out and replaced by synonyms. We do not know whether the translator completed the translation of the whole, or in what relation his version stood to our other sources.

Magnus Aurelius Cassiodorus (b. about 475 A.D.) in the geometrical part of his encyclopaedia De artibus ac disciplinis liberalium literarum says that geometry was represented among the Greeks by Euclid, Apollonius, Archimedes, and others, “of whom Euclid was given us translated into the Latin language by the same great man Boethius” ; also in his collection of letters409 is a letter from Theodoric to Boethius containing the words, “for in your translations...Nicomachus the arithmetician, and Euclid the geometer, are heard in the Ausonian tongue.” The so-called Geometry of Boethius which has come down to us by no means constitutes a translation of Euclid. The MSS. variously give five, four, three or two Books, but they represent only two distinct compilations, one normally in five Books and the other in two. Even the latter, which was edited by Friedlein, is not genuine410,but appears to have been put together in the 11th c., from various sources. It begins with the definitions of Eucl. I., and in these are traces of perfectly correct readings which are not found even in the MSS. of the 10th c., but which can be traced in Proclus and other ancient sources; then come the Postulates (five only), the Axioms (three only), and after these some definitions of Eucl. II., III., IV. Next come the enunciations of Eucl. I., of ten propositions of Book II., and of some from Books III., IV., but always without proofs; there follows an extraordinary passage which indicates that the author will now give something of his own in elucidation of Euclid, though what follows is a literal translation of the proofs of Eucl. I. 1-3. This latter passage, although it affords a strong argument against the genuineness of this part of the work, shows that the Pseudoboethius had a Latin translation of Euclid from which he extracted the three propositions.

Curtze has reproduced, in the preface to his edition of the translation by Gherard of Cremona of an-Nairĩzĩ's Arabic commentary on Euclid, some interesting fragments of a translation of Euclid taken from a Munich MS. of the 10th c. They are on two leaves used for the cover of the MS. (Bibliothecae Regiae Universitatis Monacensis 2^{o} 757) and consist of portions of Eucl. I. 37, 38 and II. 8, translated literally word for word from the Greek text. The translator seems to have been an Italian (cf. the words “capitolo nono” used for the ninth prop. of Book II.) who knew very little Greek and had moreover little mathematical knowledge. For example, he translates the capital letters denoting points in figures as if they were numerals: thus τὰ ΑΒΓ, ΔΕΖ is translated “que primo secundo et tertio quarto quinto et septimo,” T becomes “tricentissimo” and so on. The Greek MS. which he used was evidently written in uncials, for ΔΕΖΘ becomes in one place “quod autem septimo nono,” showing that he mistook ΔΕ for the particle δέ, and καὶ ΣΤΥ is rendered “sicut tricentissimo et quadringentissimo,” showing that the letters must have been written KAIOCTU.

The date of the Englishman Athelhard (Æthelhard) is approximately fixed by some remarks in his work Perdifficiles Quaestiones Naturales which, on the ground of the personal allusions they contain, must be assigned to the first thirty years of the 12th c.411 He wrote a number of philosophical works. Little is known about his life. He is said to have studied at Tours and Laon, and to have lectured at the latter school. He travelled to Spain, Greece, Asia Minor and Egypt, and acquired a knowledge of Arabic, which enabled him to translate from the Arabic into Latin, among other works, the Elements of Euclid. The date of this translation must be put at about 1120. MSS. purporting to contain Athelhard's version are extant in the British Museum (Harleian No. 5404 and others), Oxford (Trin. Coll. 47 and Ball. Coll. 257 of 12th c.), Nürnberg (Johannes Regiomontanus' copy) and Erfurt.

Among the very numerous works of Gherard of Cremona (1114— 1187) are mentioned translations of “15 Books of Euclid” and of the Data412. Till recently this translation of the Elements was supposed to be lost; but Axel Anthon Björnbo has succeeded (1904) in discovering a translation from the Arabic which is different from the two others known to us (those by Athelhard and Campanus respectively), and which he, on grounds apparently convincing, holds to be Gherard's. Already in 1901 Björnbo had found Books X.—XV. of this translation in a MS. at Rome (Codex Reginensis lat. 1268 of 14th c.)413; but three years later he had traced three MSS. containing the whole of the same translation at Paris (Cod. Paris. 7216, 15th c.), Boulogne-sur-Mer (Cod. Bononiens. 196, 14th c.), and Bruges (Cod. Brugens. 521, 14th c.), and another at Oxford (Cod. Digby 174, end of 12th c.) containing a fragment, XI. 2 to XIV. The occurrence of Greek words in this translation such as rombus, romboides (where Athelhard keeps the Arabic terms), ambligonius, orthogonius, gnomo, pyramis etc., show that the translation is independent of Athelhard's. Gherard appears to have had before him an old translation of Euclid from the Greek which Athelhard also often followed, especially in his terminology, using it however in a very different manner. Again, there are some Arabic terms, e.g. meguar for axis of rotation, which Athelhard did not use, but which is found in almost all the translations that are with certainty attributed to Gherard of Cremona; there occurs also the 414expression “superficies equidistantium laterum et rectorum angulorum,” found also in Gherard's translation of an-Nairĩzĩ, where Athelhard says “parallelogrammum rectangulum.” The translation is much clearer than Athelhard's: it is neither abbreviated nor “edited” as Athelhard's appears to have been; it is a word-for-word translation of an Arabic MS. containing a revised and critical edition of Thãbit's version. It contains several notes quoted from Thãbit himself (Thebit dixit), e.g. about alternative proofs etc. which Thãbit found “in another Greek MS.,” and is therefore a further testimony to Thãbit's critical treatment of the text after Greek MSS. The new editor also added critical remarks of his own, e.g. on other proofs which he found in other Arabic versions, but not in the Greek: whence it is clear that he compared the Thãbit version before him with other versions as carefully as Thãbit collated the Greek MSS. Lastly, the new editor speaks of “Thebit qui transtulit hunc librum in arabicam linguam” and of “translatio Thebit,” which may tend to confirm the statement of al-Qiftĩ who credited Thãbit with an independent translation, and not (as the Fihrist does) with a mere improvement of the version of Is[hnull ]ãq b. Hunain.

Gherard's translation of the Arabic commentary of an-Nairĩzĩ on the first ten Books of the Elements was discovered by Maximilian Curtze in a MS. at Cracow and published as a supplementary volume to Heiberg and Menge's Euclid415: it will often be referred to in this work.

Next in chronological order comes Johannes Campanus (Campano) of Novara. He is mentioned by Roger Bacon (1214-1294) as a prominent mathematician of his time416, and this indication of his date is confirmed by the fact that he was chaplain to Pope Urban IV, who was Pope from 1261 to 1281417. His most important achievement was his edition of the Elements including the two Books XIV. and XV. which are not Euclid's. The sources of Athelhard's and Campanus' translations, and the relation between them, have been the subject of much discussion, which does not seem to have led as yet to any definite conclusion. Cantor (II_{1}, p. 91) gives references418 and some particulars. It appears that there is a MS. at Munich (Cod. lat. Mon. 13021) written by Sigboto in the 12th c. at Prüfning near Regensburg, and denoted by Curtze by the letter R, which contains the enunciations of part of Euclid. The Munich MSS. of Athelhard and Campanus' translations have many enunciations textually identical with those in R, so that the source of all three must, for these enunciations, have been the same; in others Athelhard and Campanus diverge completely from R, which in these places follows the Greek text and is therefore genuine and authoritative. In the 32nd definition occurs the word “elinuam,” the Arabic term for “rhombus,” and throughout the translation are a number of Arabic figures. But R was not translated from the Arabic, as is shown by (among other things) its close resemblance to the translation from Euclid given on pp. 377 sqq. of the Gromatici Veteres and to the so-called geometry of Boethius. The explanation of the Arabic figures and the word “elinuam” in Def. 32 appears to be that R was a late copy of an earlier original with corruptions introduced in many places; thus in Def. 32 a part of the text was completely lost and was supplied by some intelligent copyist who inserted the word “elinuam,” which was known to him, and also the Arabic figures. Thus Athelhard certainly was not the first to translate Euclid into Latin; there must have been in existence before the 11th c. a Latin translation which was the common source of R, the passage in the Gromatici, and “Boethius.” As in the two latter there occur the proofs as well as the enunciations of I. 1mdash;3, it is possible that this translation originally contained the proofs also. Athelhard must have had before had before him this translation of the enunciations, as well as the Arabic source from which he obtained his proofs. That some sort of translation, or at least fragments of one, were available before Athelhard's time even in England is indicated by some old English verses419:

The clerk Euclide on this wyse hit fonde Thys craft of gemetry yn Egypte londe Yn Egypte he tawghte hyt ful wyde, In dyvers londe on every syde. Mony erys afterwarde y understonde Yer that the craft com ynto thys londe. Thys craft com into England, as y yow say, Yn tyme of good kyng Adelstone's day,
which would put the introduction of Euclid into England as far back as 924-940 A.D.

We now come to the relation between Athelhard and Campanus. That their translations were not independent, as Weissenborn would have us believe, is clear from the fact that in all MSS. and editions, apart from orthographical differences and such small differences as are bound to arise when MSS. are copied by persons with some knowledge of the subject-matter, the definitions, postulates, axioms, and the 364 enunciations are word for word identical in Athelhard and Campanus; and this is the case not only where both have the same text as R but where they diverge from it. Hence it would seem that Campanus used Athelhard's translation and only developed the proofs by means of another redaction of the Arabian Euclid. It is true that the difference between the proofs of the propositions in the two translations is considerable; Athelhard's are short and compressed, Campanus' clearer and more complete, following the Greek text more closely, though still at some distance. Further, the arrangement in the two is different; in Athelhard the proofs regularly precede the enunciations, Campanus follows the usual order. It is a question how far the differences in the proofs, and certain additions in each, are due to the two translators themselves or go back to Arabic originals. The latter supposition seems to Curtze and Cantor the more probable one. Curtze's general view of the relation of Campanus to Athelhard is to the effect that Athelhard's translation was gradually altered, from the form in which it appears in the two Erfurt MSS. described by Weissenborn, by successive copyists and commentators who had Arabic originals before them, until it took the form which Campanus gave it and in which it was published. In support of this view Curtze refers to Regiomontanus' copy of the Athelhard-Campanus translation. In Regiomontanus' own preface the title is given, and this attributes the translation to Athelhard; but, while this copy agrees almost exactly with Athelhard in Book I., yet, in places where Campanus is more lengthy, it has similar additions, and in the later Books, especially from Book III. onwards, agrees absolutely with Campanus; Regiomontanus, too, himself implies that, though the translation was Athelhard's, Campanus had revised it; for he has notes in the margin such as the following, “Campani est hec,” “dubito an demonstret hic Campanus” etc.

We come now to the printed editions of the whole or of portions of the Elements. This is not the place for a complete bibliography, such as Riccardi has attempted in his valuable memoir issued in five parts between 1887 and 1893, which makes a large book in itself420. I shall confine myself to saying something of the most noteworthy translations and editions. It will be convenient to give first the Latin translations which preceded the publication of the editio princeps of the Greek text in 1533, next the most important editions of the Greek text itself, and after them the most important translations arranged according to date of first appearance and languages, first the Latin translations after 1533, then the Italian, German, French and English translations in order.

It may be added here that the first allusion, in the West, to the Greek text as still extant is found in Boccaccio's commentary on the Divina Commedia of Dante421. Next Johannes Regiomontanus, who intended to publish the Elements after the version of Campanus, but with the latter's mistakes corrected, saw in Italy (doubtless when staying with his friend Bessarion) some Greek MSS. and noticed how far they differed from the Latin version (see a letter of his written in the year 1471 to Christian Roder of Hamburg)422.


I. Latin translations prior to 1533.

1482. In this year appeared the first printed edition of Euclid, which was also the first printed mathematical book of any importance. This was printed at Venice by Erhard Ratdolt and contained Campanus' translation423. Ratdolt belonged to a family of artists at Augsburg, where he was born about 1443. Having learnt the trade of printing at home, he went in 1475 to Venice, and founded there a famous printing house which he managed for II years, after which he returned to Augsburg and continued to print important books until 1516. He is said to have died in 1528. Kästner424 gives a short description of this first edition of Euclid and quotes the dedication to Prince Mocenigo of Venice which occupies the page opposite to the first page of text. The book has a margin of 2 1/2 inches, and in this margin are placed the figures of the propositions. Ratdolt says in his dedication that at that time, although books by ancient and modern authors were printed every day in Venice, little or nothing mathematical had appeared: a fact which he puts down to the difficulty involved by the figures, which no one had up to that time succeeded in printing. He adds that after much labour he had discovered a method by which figures could be produced as easily as letters425. Experts are in doubt as to the nature of Ratdolt's discovery. Was it a method of making figures up out of separate parts of figures, straight or curved lines, put together as letters are put together to make words? In a life of Joh. Gottlob Immanuel Breitkopf, a contemporary of Kästner's own, this member of the great house of Breitkopf is credited with this particular discovery. Experts in that same house expressed the opinion that Ratdolt's figures were woodcuts, while the letters denoting points in the figures were like the other letters in the text; yet it was with carved wooden blocks that printing began. If Ratdolt was the first to print geometrical figures, it was not long before an emulator arose; for in the very same year Mattheus Cordonis of Windischgrätz employed woodcut mathematical figures in printing Oresme's De latitudinibus426. How eagerly the opportunity of spreading geometrical knowledge was seized upon is proved by the number of editions which followed in the next few years. Even the year 1482 saw two forms of the book, though they only differ in the first sheet. Another edition came out in 1486 (Ulmae, apud Io. Regerum) and another in 1491 (Vincentiae per Leonardum de Basilea et Gulielmum de Papia), but without the dedication to Mocenigo who had died in the meantime (1485). If Campanus added anything of his own, his additions are at all events not distinguished by any difference of type or otherwise; the enunciations are in large type, and the rest is printed continuously in smaller type. There are no superscriptions to particular passages such as Euclides ex Campano, Campanus, Campani additio, or Campani annotatio, which are found for the first time in the Paris edition of 1516 giving both Campanus' version and that of Zamberti (presently to be mentioned).

1501. G. Valla included in his encyclopaedic work De expetendis et fugiendis rebus published in this year at Venice (in aedibus Aldi Romani) a number of propositions with proofs and scholia translated from a Greek MS. which was once in his possession (cod. Mutin. III B, 4 of the 15th c.).

1505. In this year Bartolomeo Zamberti (Zambertus) brought out at Venice the first translation, from the Greek text, of the whole of the Elements. From the title427, as well as from his prefaces to the Catoptrica and Data, with their allusions to previous translators “who take some things out of authors, omit some, and change some,” or “to that most barbarous translator” who filled a volume purporting to be Euclid's “with extraordinary scarecrows, nightmares and phantasies,” one object of Zamberti's translation is clear. His animus against Campanus appears also in a number of notes, e.g. when he condemns the terms “helmuain” and “helmuariphe” used by Campanus as barbarous, un-Latin etc., and when he is roused to wrath by Campanus' unfortunate mistranslation of v. Def. 5. He does not seem to have had the penetration to see that Campanus was translating from an Arabic, and not from a Greek, text. Zamberti tells us that he spent seven years over his translation of the thirteen Books of the Elements. As he seems to have been born in 1473, and the Elements were printed as early as 1500, though the complete work (including the Phaenomena, Optica, Catoptrica, Data etc.) has the date 1505 at the end, he must have translated Euclid before the age of 30. Heiberg has not been able to identify the MS. of the Elements which Zamberti used; but it is clear that it belonged to the worse class of MSS., since it contains most of the interpolations of the Theonine variety. Zamberti, as his title shows, attributed the proofs to Theon.

1509. As a counterblast to Zamberti, Luca Paciuolo brought out an edition of Euclid, apparently at the expense of Ratdolt, at Venice (per Paganinum de Paganinis), in which he set himself to vindicate Campanus. The title-page of this now very rare edition428 begins thus: “The works of Euclid of Megara, a most acute philosopher and without question the chief of all mathematicians, translated by Campanus their most faithful interpreter.” It proceeds to say that the translation had been, through the fault of copyists, so spoiled and deformed that it could scarcely be recognised as Euclid. Luca Paciuolo accordingly has polished and emended it with the most critical judgment, has corrected 129 figures wrongly drawn and added others, besides supplying short explanations of difficult passages. It is added that Scipio Vegius of Milan, distinguished for his knowledge “of both languages” (i.e. of course Latin and Greek), as well as in medicine and the more sublime studies, had helped to make the edition more perfect. Though Zamberti is not once mentioned, this latter remark must have reference to Zamberti's statement that his translation was from the Greek text; and no doubt Zamberti is aimed at in the wish of Paciuolo's “that others too would seek to acquire knowledge instead of merely showing off, or that they would not try to make a market of the things of which they are ignorant, as it were (selling) smoke 429.” Weissenborn observes that, while there are many trivialities in Paciuolo's notes, they contain some useful and practical hints and explanations of terms, besides some new proofs which of course are not difficult if one takes the liberty, as Paciuolo does, of divering from Euclid's order and assuming for the proof of a proposition results not arrived at till later. Two not inapt terms are used in this edition to describe the figures of III. 7, 8, the former of which is called the goose's foot (pes anseris), the second the peacock's tail (cauda pavonis) Paciuolo as the castigator of Campanus' translation, as he calls himself, failed to correct the mistranslation of V. Def. 5430. Before the fifth Book he inserted a discourse which he gave at Venice on the 15th August, 1508, in S. Bartholomew's Church, before a select audience of 500, as an introduction to his elucidation of that Book.

1516. The first of the editions giving Campanus' and Zamberti's translations in conjunction was brought out at Paris (in officina Henrici Stephani e regione scholae Decretorum). The idea that only the enunciations were Euclid's, and that Campanus was the author of the proofs in his translation, while Theon was the author of the proofs in the Greek text, reappears in the title of this edition; and the enunciations of the added Books XIV., XV. are also attributed to Euclid, Hypsicles being credited with the proofs431. The date is not on the title-page nor at the end, but the letter of dedication to François Briconnet by Jacques Lefèvre is dated the day after the Epiphany, 1516. The figures are in the margin. The arrangement of the propositions is as follows: first the enunciation with the heading Euclides ex Campano, then the proof with the note Campanus, and after that, as Campani additio, any passage found in the edition of Campanus' translation but not in the Greek text; then follows the text of the enunciation translated from the Greek with the heading Euclides ex Zamberto, and lastly the proof headed Theo ex Zamberto. There are separate figures for the two proofs. This edition was reissued with few changes in 1537 and 1546 at Basel (apud Iohannem Hervagium), but with the addition of the Phaenomena, Optica, Catoptrica etc. For the edition of 1537 the Paris edition of 1516 was collated with “a Greek copy” (as the preface says) by Christian Herlin, professor of mathematical studies at Strassburg, who however seems to have done no more than correct one or two passages by the help of the Basel editio princeps (1533), and add the Greek word in cases where Zamberti's translation of it seemed unsuitable or inaccurate.

We now come to


II. Editions of the Greek text.

1533 is the date of the editio princeps, the title-page of which reads as follows:
*e*u*k*l*e*i*d*o*u *s*t*o*i*x*e*i*w*n *b*i*b*l<*> *i*e<*><*> *e*k *t*w*n *q*e*w*n*o*s *s*u*n*o*u*s*i*w*n. *ei)s tou= au)tou= to\ prw=ton, e)chghma/twn *pro/klou bibl. d_. Adiecta praefatiuncula in qua de disciplinis Mathematicis nonnihil. BASILEAE APVD IOAN. HERVAGIVM ANNO M.D.XXXIII. MENSE SEPTEMBRI.

The editor was Simon Grynaeus the elder (d. 1541), who, after working at Vienna and Ofen, Heidelberg and Tübingen, taught last of all at Basel, where theology was his main subject. His “praefatiuncula” is addressed to an Englishman, Cuthbert Tonstall (14741559), who, having studied first at Oxford, then at Cambridge, where he became Doctor of Laws, and afterwards at Padua, where in addition he learnt mathematics—mostly from the works of Regiomontanus and Paciuolo—wrote a book on arithmetic432 as “a farewell to the sciences,” and then, entering politics, became Bishop of London and member of the Privy Council, and afterwards (1530) Bishop of Durham. Grynaeus tells us that he used two MSS. of the text of the Elements, entrusted to friends of his, one at Venice by “Lazarus Bayfius” (Lazare de Baïf, then the ambassador of the King of France at Venice), the other at Paris by “Ioann. Rvellius” (Jean Ruel, a French doctor and a Greek scholar), while the commentaries of Proclus were put at the disposal of Grynaeus himself by “Ioann. Claymundus” at Oxford. Heiberg has been able to identify the two MSS. used for the text; they are (1) cod. Venetus Marcianus 301 and (2) cod. Paris. gr. 2343 of the 16th c., containing Books I.—XV., with some scholia which are embodied in the text. When Grynaeus notes in the margin the readings from “the other copy,” this “other copy” is as a rule the Paris MS., though sometimes the reading of the Paris MS. is taken into the text and the “other copy” of the margin is the Venice MS. Besides these two MSS. Grynaeus consulted Zamberti, as is shown by a number of marginal notes referring to “Zampertus” or to “latinum exemplar” in certain propositions of Books IX.—XI. When it is considered that the two MSS. used by Grynaeus are among the worst, it is obvious how entirely unauthoritative is the text of the editio princeps. Yet it remained the source and foundation of later editions of the Greek text for a long period, the editions which followed being designed, not for the purpose of giving, from other MSS., a text more nearly representing what Euclid himself wrote, but of supplying a handy compendium to students at a moderate price.

1536. Orontius Finaeus (Oronce Fine) published at Paris (apud Simonem Colinaeum) “demonstrations on the first six books of Euclid's elements of geometry,” “in which the Greek text of Euclid himself is inserted in its proper places, with the Latin translation of Barth. Zamberti of Venice,” which seems to imply that only the enunciations were given in Greek. The preface, from which Kästner quotes433, says that the University of Paris at that time required, from all who aspired to the laurels of philosophy, a most solemn oath that they had attended lectures on the said first six Books. Other editions of Fine's work followed in 1544 and 1551.

1545. The enunciations of the fifteen Books were published in Greek, with an Italian translation by Angelo Caiani, at Rome (apud Antonium Bladum Asulanum). The translator claims to have corrected the books and “purged them of six hundred things which did not seem to savour of the almost divine genius and the perspicuity of Euclid434

1549. Joachim Camerarius published the enunciations of the first six Books in Greek and Latin (Leipzig). The book, with preface, purports to be brought out by Rhaeticus (1514-1576), a pupil of Copernicus. Another edition with proofs of the propositions of the first three Books was published by Moritz Steinmetz in 1577 (Leipzig); a note by the printer attributes the preface to Camerarius himself.

1550. Ioan. Scheubel published at Basel (also per Ioan. Hervagium) the first six Books in Greek and Latin “together with true and appropriate proofs of the propositions, without the use of letters” (i.e. letters denoting points in the figures), the various straight lines and angles being described in words435.

1557 (also 1558). Stephanus Gracilis published another edition (repeated 1573, 1578, 1598) of the enunciations (alone) of Books I.—XV. in Greek and Latin at Paris (apud Gulielmum Cavellat). He remarks in the preface that for want of time he had changed scarcely anything in Books I.—VI., but in the remaining Books he had emended what seemed obscure or inelegant in the Latin translation, while he had adopted in its entirety the translation of Book X. by Pierre Mondoré (Petrus Montaureus), published separately at Paris in 1551. Gracilis also added a few “scholia.”

1564. In this year Conrad Dasypodius (Rauchfuss), the inventor and maker of the clock in Strassburg cathedral, similar to the present one, which did duty from 1571 to 1789, edited (Strassburg, Chr. Mylius) (1) Book I. of the Elements in Greek and Latin with scholia, (2) Book II. in Greek and Latin with Barlaam's arithmetical version of Book II., and (3) the enunciations of the remaining Books III.—XIII. Book I. was reissued with “vocabula quaedam geometrica” of Heron, the enunciations of all the Books of the Elements, and the other works of Euclid, all in Greek and Latin. In the preface to (1) he says that it had been for twenty-six years the rule of his school that all who were promoted from the classes to public lectures should learn the first Book, and that he brought it out, because there were then no longer any copies to be had, and in order to prevent a good and fruitful regulation of his school from falling through. In the preface to the edition of 1571 he says that the first Book was generally taught in all gymnasia and that it was prescribed in his school for the first class. In the preface to (3) he tells us that he published the enunciations of Books III.—XIII. in order not to leave his work unfinished, but that, as it would be irksome to carry about the whole work of Euclid in extenso, he thought it would be more convenient to students of geometry to learn the Elements if they were compressed into a smaller book.

1620. Henry Briggs (of Briggs' logarithms) published the first six Books in Greek with a Latin translation after Commandinus, “corrected in many places” (London, G. Jones).

1703 is the date of the Oxford edition by David Gregory which, until the issue of Heiberg and Menge's edition, was still the only edition of the complete works of Euclid436. In the Latin translation attached to the Greek text Gregory says that he followed Commandinus in the main, but corrected numberless passages in it by means of the books in the Bodleian Library which belonged to Edward Bernard (1638-1696), formerly Savilian Professor of Astronomy, who had conceived the plan of publishing the complete works of the ancient mathematicians in fourteen volumes, of which the first was to contain Euclid's Elements I.—XV. As regards the Greek text, Gregory tells us that he consulted, as far as was necessary, not a few MSS. of the better sort, bequeathed by the great Savile to the University, as well as the corrections made by Savile in his own hand in the margin of the Basel edition. He had the help of John Hudson, Bodley's Librarian, who punctuated the Basel text before it went to the printer, compared the Latin version with the Greek throughout, especially in the Elements and Data, and, where they differed or where he suspected the Greek text, consulted the Greek MSS. and put their readings in the margin if they agreed with the Latin and, if they did not agree, affixed an asterisk in order that Gregory might judge which reading was geometrically preferable. Hence it is clear that no Greek MS., but the Basel edition, was the foundation of Gregory's text, and that Greek MSS. were only referred to in the special passages to which Hudson called attention.

1814-1818. A most important step towards a good Greek text was taken by F. Peyrard, who published at Paris, between these years, in three volumes, the Elements and Data in Greek, Latin and French437. At the time (1808) when Napoleon was having valuable MSS. selected from Italian libraries and sent to Paris, Peyrard managed to get two ancient Vatican MSS. (190 and 1038) sent to Paris for his use (Vat. 204 was also at Paris at the time, but all three were restored to their owners in 1814). Peyrard noticed the excellence of Cod. Vat. 190, adopted many of its readings, and gave in an appendix a conspectus of these readings and those of Gregory's edition; he also noted here and there readings from Vat. 1038 and various Paris MSS. He therefore pointed the way towards a better text, but committed the error of correcting the Basel text instead of rejecting it altogether and starting afresh.

1824-1825. A most valuable edition of Books I.—VI. is that of J. G. Camerer (and C. F. Hauber) in two volumes published at Berlin438. The Greek text is based on Peyrard, although the Basel and Oxford editions were also used. There is a Latin translation and a collection of notes far more complete than any other I have seen and well nigh inexhaustible. There is no editor or commentator of any mark who is not quoted from; to show the variety of important authorities drawn upon by Camerer, I need only mention the following names: Proclus, Pappus, Tartaglia, Commandinus, Clavius, Peletier, Barrow, Borelli, Wallis, Tacquet, Austin, Simson, Playfair. No words of praise would be too warm for this veritable encyclopaedia of information.

1825. J. G. C. Neide edited, from Peyrard, the text of Books I.—VI., XI. and XII. (Halis Saxoniae).

1826-9. The last edition of the Greek text before Heiberg's is that of E. F. August, who followed the Vatican MS. more closely than Peyrard did, and consulted at all events the Viennese MS. Gr. 103 (Heiberg's V). August's edition (Berlin, 1826-9) contains Books I.-XIII.


III. Latin versions or commentaries after 1533.

1545. Petrus Ramus (Pierre de la Ramée, 1515-1572) is credited with a translation of Euclid which appeared in 1545 and again in 1549 at Paris439. Ramus, who was more rhetorician and logician than geometer, also published in his Scholae mathematicae (1559, Frankfurt; 1569, Basel) what amounts to a series of lectures on Euclid's Elements, in which he criticises Euclid's arrangement of his propositions, the definitions, postulates and axioms, all from the point of view of logic.

1557. Demonstrations to the geometrical Elements of Euclid, six Books, by Peletarius (Jacques Peletier). The second edition (1610) contained the same with the addition of the “Greek text of Euclid” ; but only the enunciations of the propositions, as well as the definitions etc., are given in Greek (with a Latin translation), the rest is in Latin only. He has some acute observations, for instance about the “angle” of contact.

1559. Johannes Buteo, or Borrel (1492-1572), published in an appendix to his book De quadratura circuli some notes “on the errors of Campanus, Zambertus, Orontius, Peletarius, Pena, interpreters of Euclid.” Buteo in these notes proved, by reasoned argument based on original authorities, that Euclid himself and not Theon was the author of the proofs of the propositions.

1566. Franciscus Flussates Candalla (François de Foix, Comte de Candale, 1502-1594) “restored” the fifteen Books, following, as he says, the terminology of Zamberti's translation from the Greek, but drawing, for his proofs, on both Campanus and Theon (i.e. Zamberti) except where mistakes in them made emendation necessary. Other editions followed in 1578, 1602, 1695 (in Dutch).

1572. The most important Latin translation is that of Commandinus (1509-1575) of Urbino, since it was the foundation of most translations which followed it up to the time of Peyrard, including that of Simson and therefore of those editions, numerous in England, which give Euclid “chiefly after the text of Simson.” Simson's first (Latin) edition (1756) has “ex versione Latina Federici Commandini” on the title-page. Commandinus not only followed the original Greek more closely than his predecessors but added to his translation some ancient scholia as well as good notes of his own. The title of his work is “ Euclidis elementorum libri XV, una cum scholiis antiquis. A Federico Commandino Urbinate nuper in latinum conversi, commentariisque quibusdam illustrati (Pisauri, apud Camillum Francischinum).

He remarks in his preface that Orontius Finaeus had only edited six Books without reference to any Greek MS., that Peletarius had followed Campanus' version from the Arabic rather than the Greek text, and that Candalla had diverged too far from Euclid, having rejected as inelegant the proofs given in the Greek text and substituted faulty proofs of his own. Commandinus appears to have used, in addition to the Basel editio princeps, some Greek MS., so far not identified; he also extracted his “scholia antiqua” from a MS. of the class of Vat. 192 containing the scholia distinguished by Heiberg as “Schol. Vat.” New editions of Commandinus' translation followed in 1575 (in Italian), 1619, 1749 (in English, by Keill and Stone), 1756 (Books I.—VI., XI., XII. in Latin and English, by Simson), 1763 (Keill). Besides these there were many editions of parts of the whole work, e.g. the first six Books.

1574. The first edition of the Latin version by Clavius440 (Christoph Klau [?]. born at Bamberg 1537, died 1612) appeared in 1574, and new editions of it in 1589, 1591, 1603, 1607, 1612. It is not a translation, as Clavius himself states in the preface, but it contains a vast amount of notes collected from previous commentators and editors, as well as some good criticisms and elucidations of his own. Among other things, Clavius finally disposed of the error by which Euclid had been identified with Euclid of Megara. He speaks of the differences between Campanus who followed the Arabic tradition and the “commentaries of Theon,” by which he appears to mean the Euclidean proofs as handed down by Theon; he complains of predecessors who have either only given the first six Books, or have rejected the ancient proofs and substituted worse proofs of their own, but makes an exception as regards Commandinus, “a geometer not of the common sort, who has lately restored Euclid, in a Latin translation, to his original brilliancy.” Clavius, as already stated, did not give a translation of the Elements but rewrote the proofs, compressing them or adding to them, where he thought that he could make them clearer. Altogether his book is a most useful work.

1621. Henry Savile's lectures (Praelectiones tresdecim in principium Elementorum Euclidis Oxoniae habitae MDC.XX., Oxonii 1621), though they do not extend beyond I. 8, are valuable because they grapple with the difficulties connected with the preliminary matter, the definitions etc., and the tacit assumptions contained in the first propositions.

1654. André Tacquet's Elementa geometriae planae et solidae containing apparently the eight geometrical Books arranged for general use in schools. It came out in a large number of editions up to the end of the eighteenth century.

1655. Barrow's Euclidis Elementorum Libri XV breviter demonstrati is a book of the same kind. In the preface (to the edition of 1659) he says that he would not have written it but for the fact that Tacquet gave only eight Books of Euclid. He compressed the work into a very small compass (less than 400 small pages, in the edition of 1659, for the whole of the fifteen Books and the Data) by abbreviating the proofs and using a large quantity of symbols (which, he says, are generally Oughtred's). There were several editions up to 1732 (those of 1660 and 1732 and one or two others are in English).

1658. Giovanni Alfonso Borelli (1608-1679) published Euclides restitutus, on apparently similar lines, which went through three more editions (one in Italian, 1663).

1660. Claude François Milliet Dechales' eight geometrical Books of Euclid's Elements made easy. Dechales' versions of the Elements had great vogue, appearing in French, Italian and English as well as Latin. Riccardi enumerates over twenty editions.

1733. Saccheri's Euclides ab omni naevo vindicatus sive conatus geometricus quo stabiliuntur prima ipsa geometriae principia is important for his elaborate attempt to prove the parallel-postulate, forming an important stage in the history of the development of nonEuclidean geometry.

1756. Simson's first edition, in Latin and in English. The Latin title is “ Euclidis elementorum libri priores sex, item undecimus et duodecimus, ex versione latina Federici Commandini; sublatis iis quibus olim libri hi a Theone, aliisve, vitiati sunt, et quibusdam Euclidis demonstrationibus restitutis. A Roberto Simson M.D. Glasguae, in aedibus Academicis excudebant Robertus et Andreas Foulis, Academiae typographi.

1802. Euclidis elementorum libri priores XII ex Commandini et Gregorii versionibus latinis. In usum juventutis Academicae...by Samuel Horsley, Bishop of Rochester. (Oxford, Clarendon Press.)


IV. Italian versions or commentaries.

1543. Tartaglia's version, a second edition of which was published in 1565441, and a third in 1585. It does not appear that he used any Greek text, for in the edition of 1565 he mentions as available only “the first translation by Campano,” “the second made by Bartolomeo Zamberto Veneto who is still alive,” “the editions of Paris or Germany in which they have included both the aforesaid translations,” and “our own translation into the vulgar (tongue).”

1575. Commandinus' translation turned into Italian and revised by him.

1613. The first six Books “reduced to practice” by Pietro Antonio Cataldi, re-issued in 1620, and followed by Books VII.—IX. (1621) and Book X. (1625).

1663. Borelli's Latin translation turned into Italian by Domenico Magni.

1680. Euclide restituto by Vitale Giordano.

1690. Vincenzo Viviani's Elementi piani e solidi di Euclide (Book V. in 1674).

1731. Elementi geometrici piani e solidi di Euclide by Guido Grandi. No translation, but an abbreviated version, of which new editions followed one another up to 1806.

1749. Italian translation of Dechales with Ozanam's corrections and additions, re-issued 1785, 1797.

1752. Leonardo Ximenes (the first six Books). Fifth edition, 1819.

1818. Vincenzo Flauti's Corso di geometria elementare e sublime (4 vols.) contains (Vol. I.) the first six Books, with additions and a dissertation on Postulate 5, and (Vol. II.) Books XI., XII. Flauti also published the first six Books in 1827 and the Elements of geometry of Euclid in 1843 and 1854.


V. German.

1558. The arithmetical Books VII.-IX. by Scheubel442 (cf. the edition of the first six Books, with enunciations in Greek and Latin, mentioned above, under date 1550).

1562. The version of the first six Books by Wilhelm Holtzmann (Xylander)443. This work has its interest as the first edition in German, but otherwise it is not of importance. Xylander tells us that it was written for practical people such as artists, goldsmiths, builders etc., and that, as the simple amateur is of course content to know facts, without knowing how to prove them, he has often left out the proofs altogether. He has indeed taken the greatest possible liberties with Euclid, and has not grappled with any of the theoretical difficulties, such as that of the theory of parallels.

1651. Heinrich Hoffmann's Teutscher Euclides (2nd edition 1653), not a translation.

1694. Ant. Ernst Burkh. v. Pirckenstein's Teutsch Redender Euclides (eight geometrical Books), “for generals, engineers etc.” “proved in a new and quite easy manner.” Other editions 1699, 1744.

1697. Samuel Reyher's In teutscher Sprache vorgestellter Euclides (six Books), “made easy, with symbols algebraical or derived from the newest art of solution.”

1714. Euclidis XV Bücher teutsch, “treated in a special and brief manner, yet completely,” by Chr. Schessler (another edition in 1729).

1773. The first six Books translated from the Greek for the use of schools by J. F. Lorenz. The first attempt to reproduce Euclid in German word for word.

1781. Books XI., XII. by Lorenz (supplementary to the preceding). Also Euklid's Elemente fünfzehn Bücher translated from the Greek by Lorenz (second edition 1798; editions of 1809, 1818, 1824 by Mollweide, of 1840 by Dippe). The edition of 1824, and I presume those before it, are shortened by the use of symbols and the compression of the enunciation and “setting-out” into one.

1807. Books I.—VI., XI., XII. “newly translated from the Greek,” by J. K. F. Hauff.

1828. The same Books by Joh. Jos. Ign. Hoffmann “as guide to instruction in elementary geometry,” followed in 1832 by observations on the text by the same editor.

1833. Die Geometrie des Euklid und das Wesen derselben by E. S. Unger; also 1838, 1851.

1901. Max Simon, Euclid und die sechs planimetrischen Bücher.


VI. French.

1564-1566. Nine Books translated by Pierre Forcadel, a pupil and friend of P. de la Ramée.

1604. The first nine Books translated and annotated by Jean Errard de Bar-le-Duc; second edition, 1605.

1615. Denis Henrion's translation of the 15 Books (seven editions up to 1676).

1639. The first six Books “demonstrated by symbols, by a method very brief and intelligible,” by Pierre Hérigone, mentioned by Barrow as the only editor who, before him, had used symbols for the exposition of Euclid.

1672. Eight Books “rendus plus faciles” by Claude FranÇis Milliet Dechales, who also brought out Les élémens d'Euclide expliqués d'une manière nouvelle et très facile, which appeared in many editions, 1672, 1677, 1683 etc. (from 1709 onwards revised by Ozanam), and was translated into Italian (1749 etc.) and English (by William Halifax, 1685).

1804. In this year, and therefore before his edition of the Greek text, F. Peyrard published the Elements literally translated into French. A second edition appeared in 1809 with the addition of the fifth Book. As this second edition contains Books I.—VI. XI., XII. and X. I, it would appear that the first edition contained Books I.—IV., VI., XI., XII. Peyrard used for this translation the Oxford Greek text and Simson.


VII. Dutch.

1606. Jan Pieterszoon Dou (six Books). There were many later editions. Kästner, in mentioning one of 1702, says that Dou explains in his preface that he used Xylander's translation, but, having afterwards obtained the French translation of the six Books by Errard de Bar-le-Duc (see above), the proofs in which sometimes pleased him more than those of the German edition, he made his Dutch version by the help of both.

1617. Frans van Schooten, “The Propositions of the Books of Euclid's Elements” ; the fifteen Books in this version “enlarged” by Jakob van Leest in 1662.

1695. C. J. Vooght, fifteen Books complete, with Candalla's “16th.”

1702. Hendrik Coets, six Books (also in Latin, 1692); several editions up to 1752. Apparently not a translation. but an edition for school use.

1763. Pybo Steenstra, Books I.—VI., XI., XII., likewise an abberviated version, several times reissued until 1825.


VIII. English.

1570 saw the first and the most important translation, that of Sir Henry Billingsley. The title-page is as follows:
THE ELEMENTS OF GEOMETRIE of the most auncient Philosopher EVCLIDE of Megara
Faithfully (now first) translated into the Englishe toung, by H. Billingsley, Citizen of London. Whereunto are annexed certaine Scholies, Annotations, and Inuentions, of the best Mathematiciens, both of time past, and in this our age.

With a very fruitfull Preface by M. I. Dee, specifying the chiefe Mathematicall Sciēces, what they are, and whereunto commodious: where, also, are disclosed certaine new Secrets Mathematicall and Mechanicall, vntill these our daies, greatly missed.

Imprinted at London by John Daye.

The Preface by the translator, after a sentence observing that without the diligent study of Euclides Elementes it is impossible to attain unto the perfect knowledge of Geometry, proceeds thus. “Wherefore considering the want and lacke of such good authors hitherto in our Englishe tounge, lamenting also the negligence, and lacke of zeale to their countrey in those of our nation, to whom God hath geuen both knowledge and also abilitie to translate into our tounge, and to publishe abroad such good authors and bookes (the chiefe instrumentes of all learninges): seing moreouer that many good wittes both of gentlemen and of others of all degrees, much desirous and studious of these artes, and seeking for them as much as they can, sparing no paines, and yet frustrate of their intent, by no meanes attaining to that which they seeke: I haue for their sakes, with some charge and great trauaile, faithfully translated into our vulgare toũge, and set abroad in Print, this booke of Euclide. Whereunto I haue added easie and plaine declarations and examples by figures, of the definitions. In which booke also ye shall in due place finde manifolde additions, Scholies, Annotations, and Inuentions: which I haue gathered out of many of the most famous and chiefe Mathematiciēs, both of old time, and in our age: as by diligent reading it in course, ye shall well perceaue....”

It is truly a monumental work, consisting of 464 leaves, and therefore 928 pages, of folio size, excluding the lengthy preface by Dee. The notes certainly include all the most important that had ever been written, from those of the Greek commentators, Proclus and the others whom he quotes, down to those of Dee himself on the last books. Besides the fifteen Books, Billingsley included the “sixteenth” added by Candalla. The print and appearance of the book are worthy of its contents; and, in order that it may be understood how no pains were spared to represent everything in the clearest and most perfect form, I need only mention that the figures of the propositions in Book XI. are nearly all duplicated, one being the figure of Euclid, the other an arrangement of pieces of paper (triangular, rectangular etc.) pasted at the edges on to the page of the book so that the pieces can be turned up and made to show the real form of the solid figures represented.

Billingsley was admitted Lady Margaret Scholar of St John's College, Cambridge, in 1551, and he is also said to have studied at Oxford, but he did not take a degree at either University. He was afterwards apprenticed to a London haberdasher and rapidly became a wealthy merchant. Sheriff of London in 1584, he was elected Lord Mayor on 31st December, 1596, on the death, during his year of office, of Sir Thomas Skinner. From 1589 he was one of the Queen's four “customers,” or farmers of customs, of the port of London. In 1591 he founded three scholarships at St John's College for poor students, and gave to the College for their maintenance two messuages and tenements in Tower Street and in Mark Lane, Allhallows, Barking. He died in 1606.

1651. Elements of Geometry. The first VI Boocks: In a compendious form contracted and demonstrated by Captain Thomas Rudd, with the mathematicall preface of John Dee (London).

1660. The first English edition of Barrow's Euclid (published in Latin in 1655), appeared in London. It contained “the whole fifteen books compendiously demonstrated” ; several editions followed, in 1705, 1722, 1732, 1751.

1661. Euclid's Elements of Geometry, with a supplement of divers Propositions and Corollaries. To which is added a Treatise of regular Solids by Campane and Flussat; likewise Euclid's Data and Marinus his Preface. Also a Treatise of the Divisions of Superficies, ascribed to Machomet Bagdedine, but published by Commandine at the request of J. Dee of London. Published by care and industry of John Leeke and Geo. Serle, students in the Math. (London). According to Potts this was a second edition of Billingsley's translation.

1685. William Halifax's version of Dechales' “Elements of Euclid explained in a new but most easy method” (London and Oxford).

1705. The English Euclide; being the first six Elements of Geometry, translated out of the Greek, with annotations and usefull supplements by Edmund Scarburgh (Oxford). A noteworthy and useful edition.

1708. Books I.—VI., XI., XII., translated from Commandinus' Latin version by Dr John Keill, Savilian Professor of Astronomy at Oxford.

Keill complains in his preface of the omissions by such editors as Tacquet and Dechales of many necessary propositions (e.g. VI. 27-29), and of their substitution of proofs of their own for Euclid's. He praises Barrow's version on the whole, though objecting to the “algebraical” form of proof adopted in Book II., and to the excessive use of notes and symbols, which (he considers) make the proofs too short and thereby obscure; his edition was therefore intended to hit a proper mean between Barrow's excessive brevity and Clavius' prolixity.

Keill's translation was revised by Samuel Cunn and several times reissued. 1749 saw the eighth edition, 1772 the eleventh, and 1782 the twelfth.

1714. W. Whiston's English version (abridged) of The Elements of Euclid with select theorems out of Archimedes by the learned Andr. Tacquet.

1756. Simson's first English edition appeared in the same year as his Latin version under the title: “ The Elements of Euclid, viz. the first six Books together with the eleventh and twelfth. In this Edition the Errors by which Theon or others have long ago vitiated these Books are corrected and some of Euclid's Demonstrations are restored. By Robert Simson (Glasgow).

As above stated, the Latin edition, by its title, purports to be “ex versione latina Federici Commandini,” but to the Latin edition, as well as to the English editions, are appended “ Notes Critical and Geometrical; containing an Account of those things in which this Edition differs from the Greek text; and the Reasons of the Alterations which have been made. As also Observations on some of the Propositions.

Simson says in the Preface to some editions (e.g. the tenth, of 1799) that “the translation is much amended by the friendly assistance of a learned gentleman.”

Simson's version and his notes are so well known as not to need any further description. The book went through some thirty successive editions. The first five appear to have been dated 1756, 1762, 1767, 1772 and 1775 respectively; the tenth 1799, the thirteenth 1806, the twenty-third 1830, the twenty-fourth 1834, the twenty-sixth 1844 The Data “in like manner corrected” was added for the first time in the edition of 1762 (the first octavo edition).

1781, 1788. In these years respectively appeared the two volumes containing the complete translation of the whole thirteen Books by James Williamson, the last English translation which reproduced Euclid word for word. The title is “ The Elements of Euclid, with Disserlations intended to assist and encourage a critical examination of these Elements, as the most effectual means of establishing a juster taste upon mathematical subjects than that which at present prevails. By James Williamson.

In the first volume (Oxford, 1781) he is described as “M.A. Fellow of Hertford College,” and in the second (London, printed by T. Spilsbury, 1788) as “B.D.” simply. Books V., VI. with the Conclusion in the first volume are paged separately from the rest.

1781. An examination of the first six Books of Euclid's Elements, by William Austin (London).

1795. John Playfair's first edition, containing “the first six Books of Euclid with two Books on the Geometry of Solids.” The book reached a fifth edition in 1819, an eighth in 1831, a ninth in 1836, and a tenth in 1846.

1826. Riccardi notes under this date Euclid's Elements of Geometry containing the whole twelve Books translated into English, from the edition of Peyrard, by George Phillips. The editor, who was President of Queens' College, Cambridge, 1857-1892, was born in 1804 and matriculated at Queens' in 1826, so that he must have published the book as an undergraduate.

1828. A very valuable edition of the first six Books is that of Dionysius Lardner, with commentary and geometrical exercises, to which he added, in place of Books XI., XII., a Treatise on Solid Geometry mostly based on Legendre. Lardner compresses the propositions by combining the enunciation and the setting-out, and he gives a vast number of riders and additional propositions in smaller print. The book had reached a ninth edition by 1846, and an eleventh by 1855. Among other things, Lardner gives an Appendix “on the theory of parallel lines,” in which he gives a short history of the attempts to get over the difficulty of the parallel-postulate, down to that of Legendre.

1833. T. Perronet Thompson's Geometry without axioms, or the first Book of Euclid's Elements with alterations and notes; and an intercalary book in which the straight line and plane are derived from properties of the sphere, with an appendix containing notices of methods proposed for getting over the difficulty in the twelfth axiom of Euclid.

Thompson (1783-1869) was 7th wrangler 1802, midshipman 1803, Fellow of Queens' College, Cambridge, 1804, and afterwards general and politician. The book went through several editions, but, having been well translated into French by Van Tenac, is said to have received more recognition in France than at home.

1845. Robert Potts' first edition (and one of the best) entitled: “ Euclid's Elements of Geometry chiefly from the text of Dr Simson with explanatory notes...to which is prefixed an introduction containing a brief outline of the History of Geometry. Designed for the use of the higher forms in Public Schools and students in the Universities (Cambridge University Press, and London, John W. Parker), to which was added (1847) An Appendix to the larger edition of Euclid's Elements of Geometry, containing additional notes on the Elements, a short tract on transversals, and hints for the solution of the problems etc

”.

1862. Todhunter's edition.

The later English editions I will not attempt to enumerate; their name is legion and their object mostly that of adapting Euclid for school use, with all possible gradations of departure from his text and order.


IX. Spanish.

1576. The first six Books translated into Spanish by Rodrigo Çamorano.

1637. The first six Books translated, with notes, by L. Carduchi.

1689. Books I.—VI., XI., XII., translated and explained by Jacob Knesa.


X. Russian.

1739. Ivan Astaroff (translation from Latin).

1789. Pr. Suvoroff and Yos. Nikitin (translation from Greek).

1880. Vachtchenko-Zakhartchenko.

(1817. A translation into Polish by Jo. Czecha.)


XI. Swedish.

1744. Mårten Strömer, the first six Books; second edition 1748. The third edition (1753) contained Books XI.—XII. as well; new editions continued to appear till 1884.

1836. H. Falk, the first six Books.

1844, 1845, 1859. P. R. Bråkenhjelm, Books I.—VI., XI., XII.

1850. F. A. A. Lundgren.

1850. H. A. Witt and M. E. Areskong, Books I.—VI., XI., XII.


XII. Danish.

1745. Ernest Gottlieb Ziegenbalg.

1803. H. C. Linderup, Books I.—VI.


XIII. Modern Greek.

1820. Benjamin of Lesbos.

I should add a reference to certain editions which have appeared in recent years.

A Danish translation (Euklid's Elementer oversat af Thyra Eibe) was completed in 1912; Books I.—II. were published (with an Introduction by Zeuthen) in 1897, Books III.—IV. in 1900, Books V.—VI. in 1904, Books VII.—XIII. in 1912.

The Italians, whose great services to elementary geometry are more than once emphasised in this work, have lately shown a noteworthy disposition to make the ipsissima verba of Euclid once more the object of study. Giovanni Vacca has edited the text of Book I. (Il primo libro degli Elementi. Testo greco, versione italiana, introduzione e note, Firenze 1916.) Federigo Enriques has begun the publication of a complete Italian translation (Gli Elementi d' Euclide e la critica antica e moderna); Books I.—IV. appeared in 1925 (Alberto Stock, Roma).

An edition of Book I. by the present writer was published in 1918 (Euclid in Greek, Book I., with Introduction and Notes, Camb. Univ. Press).

CHAPTER IX.


§ 1. ON THE NATURE OF ELEMENTS.

It would not be easy to find a more lucid explanation of the terms element and elementary, and of the distinction between them, than is found in Proclus444, who is doubtless, here as so often, quoting from Geminus. There are, says Proclus, in the whole of geometry certain leading theorems, bearing to those which follow the relation of a principle, all-pervading, and furnishing proofs of many properties. Such theorems are called by the name of elements; and their function may be compared to that of the letters of the alphabet in relation to language, letters being indeed called by the same name in Greek (στοιχεῖα).

The term elementary, on the other hand, has a wider application: it is applicable to things “which extend to greater multiplicity, and, though possessing simplicity and elegance, have no longer the same dignity as the elements, because their investigation is not of general use in the whole of the science, e.g. the proposition that in triangles the perpendiculars from the angles to the transverse sides meet in a point.”

“Again, the term element is used in two senses, as Menaechmus says. For that which is the means of obtaining is an element of that which is obtained, as the first proposition in Euclid is of the second, and the fourth of the fifth. In this sense many things may even be said to be elements of each other, for they are obtained from one another. Thus from the fact that the exterior angles of rectilineal figures are (together) equal to four right angles we deduce the number of right angles equal to the internal angles (taken together)445, and vice versa. Such an element is like a lemma. But the term element is otherwise used of that into which, being more simple, the composite is divided; and in this sense we can no longer say that everything is an element of everything, but only that things which are more of the nature of principles are elements of those which stand to them in the relation of results, as postulates are elements of theorems. It is according to this signification of the term element that the elements found in Euclid were compiled, being partly those of plane geometry, and partly those of stereometry. In like manner many writers have drawn up elementary treatises in arithmetic and astronomy.

“Now it is difficult, in each science, both to select and arrange in due order the elements from which all the rest proceeds, and into which all the rest is resolved. And of those who have made the attempt some were able to put together more and some less; some used shorter proofs, some extended their investigation to an indefinite length; some avoided the method of reductio ad absurdum, some avoided proportion; some contrived preliminary steps directed against those who reject the principles; and, in a word, many different methods have been invented by various writers of elements.

“It is essential that such a treatise should be rid of everything superfluous (for this is an obstacle to the acquisition of knowledge); it should select everything that embraces the subject and brings it to a point (for this is of supreme service to science); it must have great regard at once to clearness and conciseness (for their opposites trouble our understanding); it must aim at the embracing of theorems in general terms (for the piecemeal division of instruction into the more partial makes knowledge difficult to grasp). In all these ways Euclid's system of elements will be found to be superior to the rest; for its utility avails towards the investigation of the primordial figures446, its clearness and organic perfection are secured by the progression from the more simple to the more complex and by the foundation of the investigation upon common notions, while generality of demonstration is secured by the progression through the theorems which are primary and of the nature of principles to the things sought. As for the things which seem to be wanting, they are partly to be discovered by the same methods, like the construction of the scalene and isosceles (triangle), partly alien to the character of a selection of elements as introducing hopeless and boundless complexity, like the subject of unordered irrationals which Apollonius worked out at length447, and partly developed from things handed down (in the elements) as causes, like the many species of angles and of lines. These things then have been omitted in Euclid, though they have received full discussion in other works; but the knowledge of them is derived from the simple (elements).”

Proclus, speaking apparently on his own behalf, in another place distinguishes two objects aimed at in Euclid's Elements. The first has reference to the matter of the investigation, and here, like a good Platonist, he takes the whole subject of geometry to be concerned with the “cosmic figures,” the five regular solids, which in Book XIII. are constructed, inscribed in a sphere and compared with one another. The second object is relative to the learner; and, from this standpoint, the elements may be described as “a means of perfecting the learner's understanding with reference to the whole of geometry. For, starting from these (elements), we shall be able to acquire knowledge of the other parts of this science as well, while without them it is impossible for us to get a grasp of so complex a subject, and knowledge of the rest is unattainable. As it is, the theorems which are most of the nature of principles, most simple, and most akin to the first hypotheses are here collected, in their appropriate order; and the proofs of all other propositions use these theorems as thoroughly well known, and start from them. Thus Archimedes in the books on the sphere and cylinder, Apollonius, and all other geometers, clearly use the theorems proved in this very treatise as constituting admitted principles448

Aristotle too speaks of elements of geometry in the same sense. Thus: “in geometry it is well to be thoroughly versed in the elements 449 ” ; “in general the first of the elements are, given the definitions, e.g. of a straight line and of a circle, most easy to prove, although of course there are not many data that can be used to establish each of them because there are not many middle terms 450 ”; “among geometrical propositions we call those ’elements’ the proofs of which are contained in the proofs of all or most of such propositions 451.”; “(as in the case of bodies), so in like manner we speak of the elements of geometrical propositions and, generally, of demonstrations; for the demonstrations which come first and are contained in a variety of other demonstrations are called elements of those demonstrations... the term element is applied by analogy to that which, being one and small, is useful for many purposes 452.”


§ 2. ELEMENTS ANTERIOR TO EUCLID'S.

The early part of the famous summary of Proclus was no doubt drawn, at least indirectly, from the history of geometry by Eudemus; this is generally inferred from the remark, made just after the mention of Philippus of Medma, a disciple of Plato, that “those who have written histories bring the development of this science up to this point.” We have therefore the best authority for the list of writers of elements given in the summary. Hippocrates of Chios (fl. in second half of 5th c.) is the first; then Leon, who also discovered diorismi, put together a more careful collection, the propositions proved in it being more numerous as well as more serviceable453. Leon was a little older than Eudoxus (about 408-355 B.C.) and a little younger than Plato (428/7-347/6 B.C.), but did not belong to the latter's school. The geometrical text-book of the Academy was written by Theudius of Magnesia, who, with Amyclas of Heraclea, Menaechmus the pupil of Eudoxus, Menaechmus' brother Dinostratus and Athenaeus of Cyzicus consorted together in the Academy and carried on their investigations in common. Theudius “put together the elements admirably, making many partial (or limited) propositions more general454.” Eudemus mentions no text-book after that of Theudius, only adding that Hermotimus of Colophon “discovered many of the elements455.” Theudius then must be taken to be the immediate precursor of Euclid, and no doubt Euclid made full use of Theudius as well as of the discoveries of Hermotimus and all other available material. Naturally it is not in Euclid's Elements that we can find much light upon the state of the subject when he took it up; but we have another source of information in Aristotle. Fortunately for the historian of mathematics, Aristotle was fond of mathematical illustrations; he refers to a considerable number of geometrical propositions, definitions etc., in a way which shows that his pupils must have had at hand some textbook where they could find the things he mentions; and this text-book must have been that of Theudius. Heiberg has made a most valuable collection of mathematical extracts from Aristotle456, from which much is to be gathered as to the changes which Euclid made in the methods of his predecessors; and these passages, as well as others not included in Heiberg's selection, will often be referred to in the sequel.


§ 3. FIRST PRINCIPLES: DEFINITIONS, POSTULATES, AND AXIOMS.

On no part of the subject does Aristotle give more valuable information than on that of the first principles as, doubtless, generally accepted at the time when he wrote. One long passage in the Posterior Analytics is particularly full and lucid, and is worth quoting in extenso. After laying it down that every demonstrative science starts from necessary principles457, he proceeds458:

“By first principles in each genus I mean those the truth of which it is not possible to prove. What is denoted by the first (terms) and those derived from them is assumed; but, as regards their existence, this must be assumed for the principles but proved for the rest. Thus what a unit is, what the straight (line) is, or what a triangle is (must be assumed); and the existence of the unit and of magnitude must also be assumed, but the rest must be proved. Now of the premisses used in demonstrative sciences some are peculiar to each science and others common (to all), the latter being common by analogy, for of course they are actually useful in so far as they are applied to the subject-matter included under the particular science. Instances of first principles peculiar to a science are the assumptions that a line is of such and such a character, and similarly for the straight (line); whereas it is a common principle, for instance, that, if equals be subtracted from equals, the remainders are equal. But it is enough that each of the common principles is true so far as regards the particular genus (subject-matter); for (in geometry) the effect will be the same even if the common principle be assumed to be true, not of everything, but only of magnitudes, and, in arithmetic, of numbers.

“Now the things peculiar to the science, the existence of which must be assumed, are the things with reference to which the science investigates the essential attributes, e.g. arithmetic with reference to units, and geometry with reference to points and lines. With these things it is assumed that they exist and that they are of such and such a nature. But, with regard to their essential properties, what is assumed is only the meaning of each term employed: thus arithmetic assumes the answer to the question what is (meant by) ’odd’ or ’even,’ ’a square’ or ’a cube,’ and geometry to the question what is (meant by) ’the irrational’ or ’deflection’ or (the so-called) ’verging’ (to a point); but that there are such things is proved by means of the common principles and of what has already been demonstrated. Similarly with astronomy. For every demonstrative science has to do with three things, (1) the things which are assumed to exist, namely the genus (subject-matter) in each case, the essential properties of which the science investigates, (2) the common axioms so-called, which are the primary source of demonstration, and (3) the properties with regard to which all that is assumed is the meaning of the respective terms used. There is, however, no reason why some sciences should not omit to speak of one or other of these things. Thus there need not be any supposition as to the existence of the genus, if it is manifest that it exists (for it is not equally clear that number exists and that cold and hot exist); and, with regard to the properties, there need be no assumption as to the meaning of terms if it is clear: just as in the common (axioms) there is no assumption as to what is the meaning of subtracting equals from equals, because it is well known. But none the less is it true that there are three things naturally distinct, the subject-matter of the proof, the things proved, and the (axioms) from which (the proof starts).

“Now that which is per se necessarily true, and must necessarily be thought so, is not a hypothesis nor yet a postulate. For demonstration has not to do with reasoning from outside but with the reason dwelling in the soul, just as is the case with the syllogism. It is always possible to raise objection to reasoning from outside, but to contradict the reason within us is not always possible. Now anything that the teacher assumes, though it is matter of proof, without proving it himself, is a hypothesis if the thing assumed is believed by the learner, and it is moreover a hypothesis, not absolutely, but relatively to the particular pupil; but, if the same thing is assumed when the learner either has no opinion on the subject or is of a contrary opinion, it is a postulate. This is the difference between a hypothesis and a postulate; for a postulate is that which is rather contrary than otherwise to the opinion of the learner, or whatever is assumed and used without being proved, although matter for demonstration. Now definitions are not hypotheses, for they do not assert the existence or non-existence of anything, while hypotheses are among propositions. Definitions only require to be understood: a definition is therefore not a hypothesis, unless indeed it be asserted that any audible speech is a hypothesis. A hypothesis is that from the truth of which, if assumed, a conclusion can be established. Nor are the geometer's hypotheses false, as some have said: I mean those who say that ’you should not make use of what is false, and yet the geometer falsely calls the line which he has drawn a foot long when it is not, or straight when it is not straight.’ The geometer bases no conclusion on the particular line which he has drawn being that which he has described, but (he refers to) what is illustrated by the figures. Further, the postulate and every hypothesis are either universal or particular statements; definitions are neither” (because the subject is of equal extent with what is predicated of it).

Every demonstrative science, says Aristotle, must start from indemonstrable principles: otherwise, the steps of demonstration would be endless. Of these indemonstrable principles some are (a) common to all sciences, others are (b) particular, or peculiar to the particular science; (a) the common principles are the axioms, most commonly illustrated by the axiom that, if equals be subtracted from equals, the remainders are equal. Coming now to (b) the principles peculiar to the particular science which must be assumed, we have first the genus or subject-matter, the existence of which must be assumed, viz. magnitude in the case of geometry, the unit in the case of arithmetic. Under this we must assume definitions of manifestations or attributes of the genus, e.g. straight lines, triangles, deflection etc. The definition in itself says nothing as to the existence of the thing defined: it only requires to be understood. But in geometry, in addition to the genus and the definitions, we have to assume the existence of a few primary things which are defined, viz. points and lines only: the existence of everything else, e.g. the various figures made up of these, as triangles, squares, tangents, and their properties, e.g. incommensurability etc., has to be proved (as it is proved by construction and demonstration). In arithmetic we assume the existence of the unit: but, as regards the rest, only the definitions, e.g. those of odd, even, square, cube, are assumed, and existence has to be proved. We have then clearly distinguished, among the indemonstrable principles, axioms and definitions. A postulate is also distinguished from a hypothesis, the latter being made with the assent of the learner, the former without such assent or even in opposition to his opinion (though, strangely enough, immediately after saying this, Aristotle gives a wider meaning to “postulate” which would cover “hypothesis” as well, namely whatever is assumed, though it is matter for proof, and used without being proved). Heiberg remarks that there is no trace in Aristotle of Euclid's Postulates, and that “postulate” in Aristotle has a different meaning. He seems to base this on the alternative description of postulate, indistinguishable from a hypothesis; but, if we take the other description in which it is distinguished from a hypothesis as being an assumption of something which is a proper subject of demonstration without the assent or against the opinion of the learner, it seems to fit Euclid's Postulates fairly well, not only the first three (postulating three constructions), but eminently also the other two, that all right angles are equal, and that two straight lines meeting a third and making the internal angles on the same side of it less than two right angles will meet on that side. Aristotle's description also seems to me to suit the “postulates” with which Archimedes begins his book On the equilibrium of planes, namely that equal weights balance at equal distances, and that equal weights at unequal distances do not balance but that the weight at the longer distance will prevail.

Aristotle's distinction also between hypothesis and definition, and between hypothesis and axiom, is clear from the following passage: “Among immediate syllogistic principles, I call that a thesis which it is neither possible to prove nor essential for any one to hold who is to learn anything; but that which it is necessary for any one to hold who is to learn anything whatever is an axiom: for there are some principles of this kind, and that is the most usual name by which we speak of them. But, of theses, one kind is that which assumes one or other side of a predication, as, for instance, that something exists or does not exist, and this is a hypothesis; the other, which makes no such assumption, is a definition. For a definition is a thesis: thus the arithmetician posits (τίθεται) that a unit is that which is indivisible in respect of quantity; but this is not a hypothesis, since what is meant by a unit and the fact that a unit exists are different things459.”

Aristotle uses as an alternative term for axioms “common (things),” τὰ κοινά, or “common opinions” (κοιναὶ δόξαι), as in the following passages. “That, when equals are taken from equals, the remainders are equal is (a) common (principle) in the case of all quantities, but mathematics takes a separate department (ἀπολαβοῦσα) and directs its investigation to some portion of its proper subject-matter, as e.g. lines or angles, numbers, or any of the other quantities460.” “The common (principles), e.g. that one of two contradictories must be true, that equals taken from equals etc., and the like461....” “With regard to the principles of demonstration, it is questionable whether they belong to one science or to several. By principles of demonstration I mean the common opinions from which all demonstration proceeds, e.g. that one of two contradictories must be true, and that it is impossible for the same thing to be and not be462.” Similarly “every demonstrative (science) investigates, with regard to some subject-matter, the essential attributes, starting from the common opinions463.” We have then here, as Heiberg says, a sufficient explanation of Euclid's term for axioms, viz. common notions (κοιναὶ ἔννοιαι), and there is no reason to suppose it to be a substitution for the original term due to the Stoics: cf. Proclus' remark that, according to Aristotle and the geometers, axiom and common notion are the same thing464.

Aristotle discusses the indemonstrable character of the axioms in the Metaphysics. Since “all the demonstrative sciences use the axioms465,” the question arises, to what science does their discussion belong466? The answer is that, like that of Being (οὐσία), it is the province of the (first) philosopher467. It is impossible that there should be demonstration of everything, as there would be an infinite series of demonstrations: if the axioms were the subject of a demonstrative science, there would have to be here too, as in other demonstrative sciences, a subject-genus, its attributes and corresponding axioms468; thus there would be axioms behind axioms, and so on continually. The axiom is the most firmly established of all principles469. It is ignorance alone that could lead any one to try to prove the axioms470; the supposed proof would be a petitio principii471. If it is admitted that not everything can be proved, no one can point to any principle more truly indemonstrable472. If any one thought he could prove them, he could at once be refuted; if he did not attempt to say anything, it would be ridiculous to argue with him: he would be no better than a vegetable473. The first condition of the possibility of any argument whatever is that words should signify something both to the speaker and to the hearer: without this there can be no reasoning with any one. And, if any one admits that words can mean anything to both hearer and speaker, he admits that something can be true without demonstration. And so on474.

It was necessary to give some sketch of Aristotle's view of the first principles, if only in connexion with Proclus' account, which is as follows. As in the case of other sciences, so “the compiler of elements in geometry must give separately the principles of the science, and after that the conclusions from those principles, not giving any account of the principles but only of their consequences. No science proves its own principles, or even discourses about them: they are treated as self-evident....Thus the first essential was to distinguish the principles from their consequences. Euclid carries out this plan practically in every book and, as a preliminary to the whole enquiry, sets out the common principles of this science. Then he divides the common principles themselves into hypotheses, postulates, and axioms. For all these are different from one another: an axiom, a postulate and a hypothesis are not the same thing, as the inspired Aristotle somewhere says. But, whenever that which is assumed and ranked as a principle is both known to the learner and convincing in itself, such a thing is an axiom, e.g. the statement that things which are equal to the same thing are also equal to one another. When, on the other hand, the pupil has not the notion of what is told him which carries conviction in itself, but nevertheless lays it down and assents to its being assumed, such an assumption is a hypothesis. Thus we do not preconceive by virtue of a common notion, and without being taught, that the circle is such and such a figure, but, when we are told so, we assent without demonstration. When again what is asserted is both unknown and assumed even without the assent of the learner, then, he says, we call this a postulate, e.g. that all right angles are equal. This view of a postulate is clearly implied by those who have made a special and systematic attempt to show, with regard to one of the postulates, that it cannot be assented to by any one straight off. According then to the teaching of Aristotle, an axiom, a postulate and a hypothesis are thus distinguished475.”

We observe, first, that Proclus in this passage confuses hypotheses and definitions, although Aristotle had made the distinction quite plain. The confusion may be due to his having in his mind a passage of Plato from which he evidently got the phrase about “not giving an account of” the principles. The passage is476: “I think you know that those who treat of geometries and calculations (arithmetic) and such things take for granted (ὑποθέμενοι) odd and even, figures, angles of three kinds, and other things akin to these in each subject, implying that they know these things, and, though using them as hypotheses, do not even condescend to give any account of them either to themselves or to others, but begin from these things and then go through everything else in order, arriving ultimately, by recognised methods, at the conclusion which they started in search of.” But the hypothesis is here the assumption, e.g. ’that there may be such a thing as length without breadth, henceforward called a line477,’ and so on, without any attempt to show that there is such a thing; it is mentioned in connexion with the distinction between Plato's ’superior’ and ’inferior’ intellectual method, the former of which uses successive hypotheses as stepping-stones by which it mounts upwards to the idea of Good.

We pass now to Proclus' account of the difference between postulates and axioms. He begins with the view of Geminus, according to which “they differ from one another in the same way as theorems are also distinguished from problems. For, as in theorems we propose to see and determine what follows on the premisses, while in problems we are told to find and do something, in like manner in the axioms such things are assumed as are manifest of themselves and easily apprehended by our untaught notions, while in the postulates we assume such things as are easy to find and effect (our understanding suffering no strain in their assumption), and we require no complication of machinery478.” ...“Both must have the characteristic of being simple and readily grasped, I mean both the postulate and the axiom; but the postulate bids us contrive and find some subject-matter (ὕλη) to exhibit a property simple and easily grasped, while the axiom bids us assert some essential attribute which is self-evident to the learner, just as is the fact that fire is hot, or any of the most obvious things479.”

Again, says Proclus, “some claim that all these things are alike postulates, in the same way as some maintain that all things that are sought are problems. For Archimedes begins his first book on Inequilibrium480 with the remark ’I postulate that equal weights at equal distances are in equilibrium,’ though one would rather call this an axiom. Others call them all axioms in the same way as some regard as theorems everything that requires demonstration481.”

“Others again will say that postulates are peculiar to geometrical subject-matter, while axioms are common to all investigation which is concerned with quantity and magnitude. Thus it is the geometer who knows that all right angles are equal and how to produce in a straight line any limited straight line, whereas it is a common notion that things which are equal to the same thing are also equal to one another, and it is employed by the arithmetician and any scientific person who adapts the general statement to his own subject482.”

The third view of the distinction between a postulate and an axiom is that of Aristotle above described483.

The difficulties in the way of reconciling Euclid's classification of postulates and axioms with any one of the three alternative views are next dwelt upon. If we accept the first view according to which an axiom has reference to something known, and a postulate to something done, then the 4th postulate (that all right angles are equal) is not a postulate; neither is the 5th which states that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, will meet on that side on which are the angles less than two right angles. On the second view, the assumption that two straight lines cannot enclose a space, “which even now,” says Proclus, “some add as an axiom,” and which is peculiar to the subject-matter of geometry, like the fact that all right angles are equal, is not an axiom. According to the third (Aristotelian) view, “everything which is confirmed (πιστοῦται) by a sort of demonstration will be a postulate, and what is incapable of proof will be an axiom484.” This last statement of Proclus is loose, as regards the axiom, because it omits Aristotle's requirement that the axiom should be a selfevident truth, and one that must be admitted by any one who is to learn anything at all, and, as regards the postulate, because Aristotle calls a postulate something assumed without proof though it is “matter of demonstration” (ἀποδεικτὸν ὄν), but says nothing of a quasi-demonstration of the postulates. On the whole I think it is from Aristotle that we get the best idea of what Euclid understood by a postulate and an axiom or common notion. Thus Aristotle's account of an axiom as a principle common to all sciences, which is self-evident, though incapable of proof, agrees sufficiently with the contents of Euclid's common notions as reduced to five in the most recent text (not omitting the fourth, that “things which coincide are equal to one another” ). As regards the postulates, it must be borne in mind that Aristotle says elsewhere485 that, “other things being equal, that proof is the better which proceeds from the fewer postulates or hypotheses or propositions.” If then we say that a geometer must lay down as principles, first certain axioms or common notions, and then an irreducible minimum of postulates in the Aristotelian sense concerned only with the subject-matter of geometry, we are not far from describing what Euclid in fact does. As regards the postulates we may imagine him saying: “Besides the common notions there are a few other things which I must assume without proof, but which differ from the common notions in that they are not self-evident. The learner may or may not be disposed to agree to them; but he must accept them at the outset on the superior authority of his teacher, and must be left to convince himself of their truth in the course of the investigation which follows. In the first place certain simple constructions, the drawing and producing of a straight line, and the drawing of a circle, must be assumed to be possible, and with the constructions the existence of such things as straight lines and circles; and besides this we must lay down some postulate to form the basis of the theory of parallels.” It is true that the admission of the 4th postulate that all right angles are equal still presents a difficulty to which we shall have to recur.

There is of course no foundation for the idea, which has found its way into many text-books, that “the object of the postulates is to declare that the only instruments the use of which is permitted in geometry are the rule and compass486.”


§ 4. THEOREMS AND PROBLEMS.

“Again the deductions from the first principles,” says Proclus, “are divided into problems and theorems, the former embracing the generation, division, subtraction or addition of figures, and generally the changes which are brought about in them, the latter exhibiting the essential attributes of each487.”

“Now, of the ancients, some, like Speusippus and Amphinomus, thought proper to call them all theorems, regarding the name of theorems as more appropriate than that of problems to theoretic sciences, especially as these deal with eternal objects. For there is no becoming in things eternal, so that neither could the problem have any place with them, since it promises the generation and making of what has not before existed, e.g. the construction of an equilateral triangle, or the describing of a square on a given straight line, or the placing of a straight line at a given point. Hence they say it is better to assert that all (propositions) are of the same kind, and that we regard the generation that takes place in them as referring not to actual making but to knowledge, when we treat things existing eternally as if they were subject to becoming: in other words, we may say that everything is treated by way of theorem and not by way of problem488 (πάντα θεωρηματικῶς ἀλλ̓ οὐ προβληματικῶς λαμβάνεσθαι).

“Others on the contrary, like the mathematicians of the school of Menaechmus, thought it right to call them all problems, describing their purpose as twofold, namely in some cases to furnish (πορίσασθαι) the thing sought, in others to take a determinate object and see either what it is, or of what nature, or what is its property, or in what relations it stands to something else.

“In reality both assertions are correct. Speusippus is right because the problems of geometry are not like those of mechanics, the latter being matters of sense and exhibiting becoming and change of every sort. The school of Menaechmus are right also because the discoveries even of theorems do not arise without an issuing-forth into matter, by which I mean intelligible matter. Thus forms going out into matter and giving it shape may fairly be said to be like processes of becoming. For we say that the motion of our thought and the throwing-out of the forms in it is what produces the figures in the imagination and the conditions subsisting in them. It is in the imagination that constructions, divisions, placings, applications, additions and subtractions (take place), but everything in the mind is fixed and immune from becoming and from every sort of change489.”

“Now those who distinguish the theorem from the problem say that every problem implies the possibility, not only of that which is predicated of its subject-matter, but also of its opposite, whereas every theorem implies the possibility of the thing predicated but not of its opposite as well. By the subject-matter I mean the genus which is the subject of inquiry, for example, a triangle or a square or a circle, and by the property predicated the essential attribute, as equality, section, position, and the like. When then any one enunciates thus, To inscribe an equilateral triangle in a circle, he states a problem; for it is also possible to inscribe in it a triangle which is not equilateral. Again, if we take the enunciation On a given limited straight line to construct an equilateral triangle, this is a problem; for it is possible also to construct one which is not equilateral. But, when any one enunciates that In isosceles triangles the angles at the base are equal, we must say that he enunciates a theorem; for it is not also possible that the angles at the base of isosceles triangles should be unequal. It follows that, if any one were to use the form of a problem and say In a semicircle to describe a right angle, he would be set down as no geometer. For every angle in a semicircle is right490.”

“Zenodotus, who belonged to the succession of Oenopides, but was a disciple of Andron, distinguished the theorem from the problem by the fact that the theorem inquires what is the property predicated of the subject-matter in it, but the problem what is the cause of what effect (τίνος ὄντος τί ἐστιν). Hence too Posidonius defined the one (the problem) as a proposition in which it is inquired whether a thing exists or not (εἰ ἔστιν μή), the other (the theorem491) as a proposition in which it is inquired what (a thing) is or of what nature (τί ἐστιν ποῖόν τι); and he said that the theoretic proposition must be put in a declaratory form, e.g., Any triangle has two sides (together) greater than the remaining side and In any isosceles triangle the angles at the base are equal, but that we should state the problematic propositi\on as if inquiring whether it is possible to construct an equilateral triangle upon such and such a straight line. For there is a difference between inquiring absolutely and indeterminately (ἁπλῶς τε καὶ ἀορίστως) whether there exists a straight line from such and such a point at right angles to such and such a straight line and investigating which is the straight line at right angles492.”

“That there is a certain difference between the problem and the theorem is clear from what has been said; and that the Elements of Euclid contain partly problems and partly theorems will be made manifest by the individual propositions, where Euclid himself adds at the end of what is proved in them, in some cases, ’that which it was required to do,’ and in others, ’that which it was required to prove,’ the latter expression being regarded as characteristic of theorems, in spite of the fact that, as we have said, demonstration is found in problems also. In problems, however, even the demonstration is for the purpose of (confirming) the construction: for wė bring in the demonstration in order to show that what was enjoined has been done; whereas in theorems the demonstration is worthy of study for its own sake as being capable of putting before us the nature of the thing sought. And you will find that Euclid sometimes interweaves theorems with problems and employs them in turn, as in the first book, while at other times he makes one or other preponderate. For the fourth book consists wholly of problems, and the fifth of theorems493.”

Again, in his note on Eucl. 1. 4, Proclus says that Carpus, the writer on mechanics, raised the question of theorems and problems in his treatise on astronomy. Carpus, we are told, “says that the class of problems is in order prior to theorems. For the subjects, the properties of which are sought, are discovered by means of problems. Moreover in a problem the enunciation is simple and requires no skilled intelligence; it orders you plainly to do such and such a thing, to construct an equilateral triangle, or, given two straight lines, to cut off from the greater (a straight line) equal to the lesser, and what is there obscure or elaborate in these things? But the enunciation of a theorem is a matter of labour and requires much exactness and scientific judgment in order that it may not turn out to exceed or fall short of the truth; an example is found even in this proposition (1. 4), the first of the theorems. Again, in the case of problems, one general way has been discovered, that of analysis, by following which we can always hope to succeed; it is this method by which the more obscure problems are investigated. But, in the case of theorems, the method of setting about them is hard to get hold of since ’up to our time,’ says Carpus, ’no one has been able to hand down a general method for their discovery. Hence, by reason of their easiness, the class of problems would naturally be more simple.’ After these distinctions, he proceeds: ’Hence it is that in the Elements too problems precede theorems, and the Elements begin from them; the first theorem is fourth in order, not because the fifth494 is proved from the problems, but because, even if it needs for its demonstration none of the propositions which precede it, it was necessary that they should be first because they are problems, while it is a theorem. In fact, in this theorem he uses the common notions exclusively, and in some sort takes the same triangle placed in different positions; the coincidence and the equality proved thereby depend entirely upon sensible and distinct apprehension. Nevertheless, though the demonstration of the first theorem is of this character, the problems properly preceded it, because in general problems are allotted the order of precedence495.’”

Proclus himself explains the position of Prop. 4 after Props. 1-3 as due to the fact that a theorem about the essential properties of triangles ought not to be introduced before we know that such a thing as a triangle can be constructed, nor a theorem about the equality of sides or straight lines until we have shown, by constructing them, that there can be two straight lines which are equal to one another496. It is plausible enough to argue in this way that Props. 2 and 3 at all events should precede Prop. 4. And Prop. 1 is used in Prop. 2, and must therefore precede it. But Prop. I showing how to construct an equilateral triangle on a given base is not important, in relation to Prop. 4, as dealing with the “production of triangles” in general: for it is of no use to say, as Proclus does, that the construction of the equilateral triangle is “common to the three species (of triangles)497,” as we are not in a position to know this at such an early stage. The existence of triangles in general was doubtless assumed as following from the existence of straight lines and points in one plane and from the possibility of drawing a straight line from one point to another.

Proclus does not however seem to reject definitely the view of Carpus, for he goes on498: “And perhaps problems are in order before theorems, and especially for those who need to ascend from the arts which are concerned with things of sense to theoretical investigation. But in dignity theorems are prior to problems....It is then foolish to blame Geminus for saying that the theorem is more perfect than the problem. For Carpus himself gave the priority to problems in respect of order, and Geminus to theorems in point of more perfect dignity,” so that there was no real inconsistency between the two.

Problems were classified according to the number of their possible solutions. Amphinomus said that those which had a unique solution (μοναχῶς) were called “ordered” (the word has dropped out in Proclus, but it must be τεταγμένα, in contrast to the third kind, ἄτακτα); those which had a definite number of solutions “intermediate” (μέσα); and those with an infinite variety of solutions “unordered” (ἄτακτα499. Proclus gives as an example of the last the problem To divide a given straight line into three parts in continued proportion500. This is the same thing as solving the equations . Proclus' remarks upon the problem show that it was solved, like all quadratic equations, by the method of “application of areas.” The straight line a was first divided into any two parts, (x+z) and y, subject to the sole limitation that (x+z) must not be less than 2y, which limitation is the διορισμός, or condition of possibility. Then an area was applied to (x+z), or (a-y), “falling short by a square figure” (ἐλλεῖπον εἴδει τετραγώνῳ) and equal to the square on y. This determines x and z separately in terms of a and y. For, if z be the side of the square by which the area (i.e. rectangle) “falls short,” we have whence And y may be chosen arbitrarily, provided that it is not greater than a/3. Hence there are an infinite number of solutions. If then, as Proclus remarks, the three parts are equal.

Other distinctions between different kinds of problems are added by Proclus. The word “problem,” he says, is used in several senses. In its widest sense it may mean anything “propounded” (προτεινόμενον), whether for the purpose of instruction (μαθήσεως) or construction (ποιήσεως). (In this sense, therefore, it would include a theorem.) But its special sense in mathematics is that of something “propounded with a view to a theoretic construction501.”

Again you may apply the term (in this restricted sense) even to something which is impossible, although it is more appropriately used of what is possible and neither asks too much nor contains too little in the shape of data. According as a problem has one or other of these defects respectively, it is called (1) a problem in excess (πλεονάζον) or (2) a deficient problem (ἐλλιπὲς πρόβλημα). The problem in excess (1) is of two kinds, (a) a problem in which the properties of the figure to be found are either inconsistent (ἀσύμβατα) or non-existent (ἀνύπαρκτα), in which case the problem is called impossible, or (b) a problem in which the enunciation is merely redundant: an example of this would be a problem requiring us to construct an equilateral triangle with its vertical angle equal to two-thirds of a right angle; such a problem is possible and is called “more than a problem” (μεῖζον πρόβλημα). The deficient problem (2) is similarly called “less than a problem” (ἔλασσον πρόβλημα), its characteristic being that something has to be added to the enunciation in order to convert it from indeterminateness (ἀοριστία) to order (τάξις) and scientific determinateness (ὅρος ἐπιστημονικός): such would be a problem bidding you “to construct an isosceles triangle,” for the varieties of isosceles triangles are unlimited. Such “problems” are not problems in the proper sense (κυρίως λεγόμενα προβλήματα), but only equivocally502.


§ 5. THE FORMAL DIVISIONS OF A PROPOSITION.

“Every problem,” says Proclus503, “and every theorem which is complete with all its parts perfect purports to contain in itself all of the following elements: enunciation (πρότασις), setting-out (ἔκθεσις), definition or specification (διορισμός), construction or machinery (κατασκευή), proof (ἀπόδειξις), conclusion (συμπέρασμα). Now of these the enunciation states what is given and what is that which is sought, the perfect enunciation consisting of both these parts. The setting-out marks off what is given, by itself, and adapts it beforehand for use in the investigation. The definition or specification states separately and makes clear what the particular thing is which is sought. The construction or machinery adds what is wanting to the datum for the purpose of finding what is sought. The proof draws the required inference by reasoning scientifically from acknowledged facts. The conclusion reverts again to the enunciation, confirming what has been demonstrated. These are all the parts of problems and theorems, but the most essential and those which are found in all are enunciation, proof, conclusion. For it is equally necessary to know beforehand what is sought, to prove this by means of the intermediate steps, and to state the proved fact as a conclusion; it is impossible to dispense with any of these three things. The remaining parts are often brought in, but are often left out as serving no purpose. Thus there is neither setting-out nor definition in the problem of constructing an isosceles triangle having each of the angles at the base double of the remaining angle, and in most theorems there is no construction because the setting-out suffices without any addition for proving the required property from the data. When then do we say that the setting-out is wanting? The answer is, when there is nothing given in the enunciation; for, though the enunciation is in general divided into what is given and what is sought, this is not always the case, but sometimes it states only what is sought, i.e. what must be known or found, as in the case of the problem just mentioned. That problem does not, in fact, state beforehand with what datum we are to construct the isosceles triangle having each of the equal angles double of the remaining angle, but (simply) that we are to find such a triangle.... When, then, the enunciation contains both (what is given and what is sought), in that case we find both definition and setting-out, but, whenever the datum is wanting, they too are wanting. For not only is the setting-out concerned with the datum, but so is the definition also, as, in the absence of the datum, the definition will be identical with the enunciation. In fact, what could you say in defining the object of the aforesaid problem except that it is required to find an isosceles triangle of the kind referred to? But that is what the enunciation stated. If then the enunciation does not include, on the one hand, what is given and, on the other, what is sought, there is no setting-out in virtue of there being no datum, and the definition is left out in order to avoid a mere repetition of the enunciation.”

The constituent parts of an Euclidean proposition will be readily identified by means of the above description. As regards the definition or specification (διορισμός) it is to be observed that we have here only one of its uses. Here it means a closer definition or description of the object aimed at, by means of the concrete lines or figures set out in the ἔκθεσις instead of the general terms used in the enunciation; and its purpose is to rivet the attention better, as Proclus indicates in a later passage (τρόπον τινὰ προσεχείας ἐστὶν αἴτιος διορισμός504.

The other technical use of the word to signify the limitations to which the possible solutions of a problem are subject is also described by Proclus, who speaks of διορισμοί determining “whether what is sought is impossible or possible, and how far it is practicable and in how many ways505” ; and the διορισμός in this sense appears in Euclid as well as in Archimedes and Apollonius. Thus we have in Eucl. 1. 22 the enunciation “From three straight lines which are equal to three given straight lines to construct a triangle,” followed immediately by the limiting condition (διορισμός). “Thus two of the straight lines taken together in any manner must be greater than the remaining one.” Similarly in VI. 28 the enunciation “To a given straight line to apply a parallelogram equal to a given rectilineal figure and falling short by a parallelogrammic figure similar to a given one” is at once followed by the necessary condition of possibility: “Thus the given rectilineal figure must not be greater than that described on half the line and similar to the defect.”

Tannery supposed that, in giving the other description of the διορισμός as quoted above, Proclus, or rather his guide, was using the term incorrectly. The διορισμός in the better known sense of the determination of limits or conditions of possibility was, we are told, invented by Leon. Pappus uses the word in this sense only. The other use of the term might, Tannery thought, be due to a confusion occasioned by the use of the same words (δεῖ δή) in introducing the parts of a proposition corresponding to the two meanings of the word διορισμός506. On the other hand it is to be observed that Eutocius distinguishes clearly between the two uses and implies that the difference was well known507. The διορισμός in the sense of condition of possibility follows immediately on the enunciation, is even part of it; the διορισμός in the other sense of course comes immediately after the setting-out.

Proclus has a useful observation respecting the conclusion of a proposition508. “The conclusion they are accustomed to make double in a certain way: I mean, by proving it in the given case and then drawing a general inference, passing, that is, from the partial conclusion to the general. For, inasmuch as they do not make use of the individuality of the subjects taken, but only draw an angle or a straight line with a view to placing the datum before our eyes, they consider that this same fact which is established in the case of the particular figure constitutes a conclusion true of every other figure of the same kind. They pass accordingly to the general in order that we may not conceive the conclusion to be partial. And they are justified in so passing, since they use for the demonstration the particular things set out, not quâ particulars, but quâ typical of the rest. For it is not in virtue of such and such a size attaching to the angle which is set out that I effect the bisection of it, but in virtue of its being rectilineal and nothing more. Such and such size is peculiar to the angle set out, but its quality of being rectilineal is common to all rectilineal angles. Suppose, for example, that the given angle is a right angle. If then I had employed in the proof the fact of its being right, I should not have been able to pass to every species of rectilineal angle; but, if I make no use of its being right, and only consider it as rectilineal, the argument will equally apply to rectilineal angles in general.”


§ 6. OTHER TECHNICAL TERMS.

1. Things said to be given.

Proclus attaches to his description of the formal divisions of a proposition an explanation of the different senses in which the word given or datum (δεδομένον) is used in geometry. “Everything that is given is given in one or other of the following ways, in position, in ratio, in magnitude, or in species. The point is given in position only, but a line and the rest may be given in all the senses509.”

The illustrations which Proclus gives of the four senses in which a thing may be given are not altogether happy, and, as regards things which are given in position, in magnitude, and in species, it is best, I think, to follow the definitions given by Euclid himself in his book of Data. Euclid does not mention the fourth class, things given in ratio, nor apparently do any of the great geometers.

(1) Given in position really needs no definition; and, when Euclid says (Data, Def. 4) that “Points, lines and angles are said to be given in position which always occupy the same place,” we are not really the wiser.

(2) Given in magnitude is defined thus (Data, Def. 1): “Areas, lines and angles are called given in magnitude to which we can find equals.” Proclus' illustration is in this case the following: when, he says, two unequal straight lines are given from the greater of which we have to cut off a straight line equal to the lesser, the straight lines are obviously given in magnitude, “for greater and less, and finite and infinite are predications peculiar to magnitude.” But he does not explain that part of the implication of the term is that a thing is given in magnitude only, and that, for example, its position is not given and is a matter of indifference

(3) Given in species. Euclid's definition (Data, Def. 3) is: “Rectilineal figures are said to be given in species in which the angles are severally given and the ratios of the sides to one another are given” And this is the recognised use of the term (cf. Pappus, passim) Proclus uses the term in a much wider sense for which I am not aware of any authority. Thus, he says, when we speak of (bisecting) a given rectilineal angle, the angle is given in species by the word rectilineal, which prevents our attempting, by the same method, to bisect a curvilineal angle! On Eucl. 1. 9, to which he here refers, he says that an angle is given in species when e.g. we say that it is right or acute or obtuse or rectilineal or “mixed,” but that the actual angle in the proposition is given in species only. As a matter of fact, we should say that the actual angle in the figure of the proposition is given in magnitude and not in species, part of the implication of given in species being that the actual magnitude of the thing given in species is indifferent; an angle cannot be given in species in this sense at all. The confusion in Proclus' mind is shown when, after saying that a right angle is given in species, he describes a third of a right angle as given in magnitude.

No better example of what is meant by given in species, in its proper sense, as limited to rectilineal figures, can be quoted than the given parallelogram in Eucl. VI. 28, to which the required parallelogram has to be made similar; the former parallelogram is in fact given in species, though its actual size, or scale, is indifferent.

(4) Given in ratio presumably means something which is given by means of its ratio to some other given thing. This we gather from Proclus' remark (in his note on 1. 9) that an angle may be given in ratio “as when we say that it is double and treble of such and such an angle or, generally, greater and less.” The term, however, appears to have no authority and to serve no purpose. Proclus may have derived it from such expressions as “in a given ratio” which are common enough.

2. Lemma.

“The term lemma,” says Proclus 510, “is often used of any proposition which is assumed for the construction of something else: thus it is a common remark that a proof has been made out of such and such lemmas. But the special meaning of lemma in geometry is a proposition requiring confirmation. For when, in either construction or demonstration, we assume anything which has not been proved but requires argument, then, because we regard what has been assumed as doubtful in itsėlf and therefore worthy of investigation, we call it a lemma 511, differing as it does from the postulate and the axiom in being matter of demonstration, whereas they are immediately taken for granted, without demonstration, for the purpose of confirming other things. Now in the discovery of lemmas the best aid is a mental aptitude for it. For we may see many who are quick at solutions and yet do not work by method; thus Cratistus in our time was able to obtain the required result from first principles, and those the fewest possible, but it was his natural gift which helped him to the discovery. Nevertheless certain methods have been handed down. The finest is the method which by means of analysis carries the thing sought up to an acknowledged principle, a method which Plato, as they say, communicated to Leodamas 512, and by which the latter, too, is said to have discovered many things in geometry. The second is the method of division 513, which divides into its parts the genus proposed for consideration and gives a starting-point for the demonstration by means of the elimination of the other elements in the construction of what is proposed, which method also Plato extolled as being of assistance to all sciences. The third is that by means of the reductio ad absurdum, which does not show what is sought directly; but refutes its opposite and discovers the truth incidentally.”

3. Case.

“The case514 (πτῶσις),” Proclus proceeds515, “announces different ways of construction and alteration of positions due to the transposition of points or lines or planes or solids. And, in general, all its varieties are seen in the figure, and this is why it is called case, being a transposition in the construction.”

4. Porism.

“The term porism is used also of certain problems such as the Porisms written by Euclid. But it is specially used when from what has been demonstrated some other theorem is revealed at the same time without our propounding it, which theorem has on this very account been called a porism (corollary) as being a sort of incidental gain arising from the scientific demonstration516.” Cf. the note on I. 15.

5. Objection.

“The objection (ἔνστασις) obstructs the whole course of the argument by appearing as an obstacle (or crying ’halt,’ ἀπαντῶσα) either to the construction or to the demonstration. There is this difference between the objection and the case, that, whereas he who propounds the case has to prove the proposition to be true of it, he who makes the objection does not need to prove anything: on the contrary it is necessary to destroy the objection and to show that its author is saying what is false517.”

That is, in general the objection endeavours to make it appear that the demonstration is not true in every case; and it is then necessary to prove, in refutation of the objection, either that the supposed case is impossible, or that the demonstration is true even for that case. A good instance is afforded by Eucl. 1. 7. The text-books give a second case which is not in the original text of Euclid. Proclus remarks on the proposition as given by Euclid that the objection may conceivably be raised that what Euclid declares to be impossible may after all be possible in the event of one pair of stiaight lines falling completely within the other pair. Proclus then refutes the objection by proving the impossibility in that case also. His proof then came to be given in the text-books as part of Euclid's proposition.

The objection is one of the technical terms in Aristotle's logic and its nature is explained in the Prior Analytics518. “An objection is a proposition contrary to a proposition.... Objections are of two sorts, general or partial.... For when it is maintained that an attribute belongs to every (member of a class), we object either that it belongs to none (of the class) or that there is some one (member of the class) to which it does not belong.”

6. Reduction.

This is again an Aristotelian term, explained in the Prior Analytics519. It is well described by Proclus in the following passage:

Reduction (ἀπαγωγή) is a transition from one problem or theorem to another, the solution or proof of which makes that which is propounded manifest also. For example, after the doubling of the cube had been investigated, they transformed the investigation into another upon which it follows, namely the finding of the two means; and from that time forward they inquired how between two given straight lines two mean proportionals could be discovered. And they say that the first to effect the reduction of difficult constructions was Hippocrates of Chios, who also squared a lune and discovered many other things in geometry, being second to none in ingenuity as regards constructions520.”

7. Reductio ad absurdum.

This is variously called by Aristotle “reductio ad absurdum” ( εἰς τὸ ἀδύνατον ἀπαγωγή521, “proof per impossibile” ( διὰ τοῦ ἀδυνάτου δεῖξις or ἀπόδειξις522, “proof leading to the impossible” ( εἰς τὸ ἀδύνατον ἄγουσα ἀπόδειξις523. It is part of “proof (starting) from a hypothesis524” (ἐξ ὑποθέσεως). “All (syllogisms) which reach the conclusion per impossibile reason out a conclusion which is false, and they prove the original contention (by the method starting) from a hypothesis, when something impossible results from assuming the contradictory of the original contention, as, for example, when it is proved that the diagonal (of a square) is incommensurable because, if it be assumed commensurable, it will follow that odd (numbers) are equal to even (numbers)525.” Or again, “proof (leading) to the impossible differs from the direct (δεικτικῆς) in that it assumes what it desires to destroy [namely the hypothesis of the falsity of the conclusion] and then reduces it to something admittedly false, whereas the direct proof starts from premisses admittedly true526.”

Proclus has the following description of the reductio ad absurdum. “Proofs by reductio as absurdum in every case reach a conclusion manifestly impossible, a conclusion the contradictory of which is admitted. In some cases the conclusions are found to conflict with the common notions, or the postulates, or the hypotheses (from which we started); in others they contradict propositions previously established527” ...“Every reductio ad absurdum assumes what conflicts with the desired result, then, using that as a basis, proceeds until it arrives at an admitted absurdity, and, by thus destroying the hypothesis, establishes the result originally desired. For it is necessary to understand generally that all mathematical arguments either proceed from the first principles or lead back to them, as Porphyry somewhere says. And those which proceed from the first principles are again of two kinds, for they start either from common notions and the clearness of the self-evident alone, or from results previously proved; while those which lead back to the principles are either by way of assuming the principles or by way of destroying them. Those which assume the principles are called analyses, and the opposite of these are syntheses— for it is possible to start from the said principles and to proceed in the regular order to the desired conclusion, and this process is synthesis—while the arguments which would destroy the principles are called reductiones ad absurdum. For it is the function of this method to upset something admitted as clear528.”

8. Analysis and Synthesis.

It will be seen from the note on Eucl. XIII. I that the MSS. of the Elements contain definitions of Analysis and Synthesis followed by alternative proofs of XIII. 1-5 after that method. The definitions and alternative proofs are interpolated, but they have great historical interest because of the possibility that they represent an ancient method of dealing with these propositions, anterior to Euclid. The propositions give properties of a line cut “in extreme and mean ratio,” and they are preliminary to the construction and comparison of the five regular solids. Now Pappus, in the section of his Collection dealing with the latter subject529, says that he will give the comparisons between the five figures, the pyramid, cube, octahedron, dodecahedron and icosahedron, which have equal surfaces, “not by means of the so-called analytical inquiry, by which some of the ancients worked out the proofs, but by the synthetical method530....” The conjecture of Bretschneider that the matter interpolated in Eucl. XIII. is a survival of investigations due to Eudoxus has at first sight much to commend it531. In the first place, we are told by Proclus that Eudoxus “greatly added to the number of the theorems which Plato originated regarding the section, and employed in them the method of analysis532.” It is obvious that “the section” was some particular section which by the time of Plato had assumed great importance; and the one section of which this can safely be said is that which was called the “golden section,” namely, the division of a straight line in extreme and mean ratio which appears in Eucl. II. 11 and is therefore most probably Pythagorean. Secondly, as Cantor points out533, Eudoxus was the founder of the theory of proportions in the form in which we find it in Euclid V., VI., and it was no doubt through meeting, in the course of his investigations, with proportions not expressible by whole numbers that he came to realise the necessity for a new theory of proportions which should be applicable to incommensurable as well as commensurable magnitudes. The “golden section” would furnish such a case. And it is even mentioned by Proclus in this connexion. He is explaining534 that it is only in arithmetic that all quantities bear “rational” ratios (ῥητὸς λόγος) to one another, while in geometry there are “irrational” ones (ἄρρητος) as well. “Theorems about sections like those in Euclid's second Book are common to both [arithmetic and geometry] except that in which the straight line is cut in extreme and mean ratio535.”

The definitions of Analysis and Synthesis interpolated in Eucl. XIII. are as follows (I adopt the reading of B and V, the only intelligible one, for the second).

Analysis is an assumption of that which is sought as if it were admitted <and the passage> through its consequences to something admitted (to be) true.

Synthesis is an assumption of that which is admitted <and the passage> through its consequences to the finishing or attainment of what is sought.”

The language is by no means clear and has, at the best, to be filled out.

Pappus has a fuller account536:

“The so-called ἀναλυόμενος (’Treasury of Analysis’) is, to put it shortly, a special body of doctrine provided for the use of those who, after finishing the ordinary Elements, are desirous of acquiring the power of solving problems which may be set them involving (the construction of) lines, and it is useful for this alone. It is the work of three men, Euclid the author of the Elements, Apollonius of Perga, and Aristaeus the elder, and proceeds by way of analysis and synthesis.

Analysis then takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis: for in analysis we assume that which is sought as if it were (already) done (γεγονός), and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards (ἀνάπαλιν λύσιν).

“But in synthesis, reversing the process, we take as already done that which was last arrived at in the analysis and, by arranging in their natural order as consequences what were before antecedents, and successively connecting them one with another, we arrive finally at the construction of what was sought; and this we call synthesis.

“Now analysis is of two kinds, the one directed to searching for the truth and called theoretical, the other directed to finding what we are told to find and called problematical. (1) In the theoretical kind we assume what is sought as if it were existent and true, after which we pass through its successive consequences, as if they too were true and established by virtue of our hypothesis, to something admitted: then (a), if that something admitted is true, that which is sought will also be true and the proof will correspond in the reverse order to the analysis, but (b), if we come upon something admittedly false, that which is sought will also be false. (2) In the problematical kind we assume that which is propounded as if it were known, after which we pass through its successive consequences, taking them as true, up to something admitted: if then (a) what is admitted is possible and obtainable, that is, what mathematicians call given, what was originally proposed will also be possible, and the proof will again correspond in reverse order to the analysis, but if (b) we come upon something admittedly impossible, the problem will also be impossible.”

The ancient Analysis has been made the subject of careful studies by several writers during the last half-century, the most complete being those of Hankel, Duhamel and Zeuthen; others by Ofterdinger and Cantor should also be mentioned537.

The method is as follows. It is required, let us say, to prove that a certain proposition A is true. We assume as a hypothesis that A is true and, starting from this we find that, if A is true, a certain other proposition B is true; if B is true, then C; and so on until we arrive at a proposition K which is admittedly true. The object of the method is to enable us to infer, in the reverse order, that, since K is true, the proposition A originally assumed is true. Now Aristotle had already made it clear that false hypotheses might lead to a conclusion which is true. There is therefore a possibility of error unless a certain precaution is taken. While, for example, B may be a necessary consequence of A, it may happen that A is not a necessary consequence of B. Thus, in order that the reverse inference from the truth of K that A is true may be logically justified, it is necessary that each step in the chain of inferences should be unconditionally convertible. As a matter of fact, a very large number of theorems in elementary geometry are unconditionally convertible, so that in practice the difficulty in securing that the successive steps shall be convertible is not so great as might be supposed. But care is always necessary. For example, as Hankel says538, a proposition may not be unconditionally convertible in the form in which it is generally quoted. Thus the proposition “The vertices of all triangles having a common base and constant vertical angle lie on a circle” cannot be converted into the proposition that “All triangles with common base and vertices lying on a circle have a constant vertical angle” ; for this is only true if the further conditions are satisfied (1) that the circle passes through the extremities of the common base and (2) that only that part of the circle is taken as the locus of the vertices which lies on one side of the base. If these conditions are added, the proposition is unconditionally convertible. Or again, as Zeuthen remarks539, K may be obtained by a series of inferences in which A or some other proposition in the series is only apparently used; this would be the case e.g. when the method of modern algebra is being employed and the expressions on each side of the sign of equality have been inadvertently multiplied by some composite magnitude which is in reality equal to zero.

Although the above extract from Pappus does not make it clear that each step in the chain of argument must be convertible in the case taken, he almost implies this in the second part of the definition of Analysis where, instead of speaking of the consequences B, C... successively following from A, he suddenly changes the expression and says that we inquire what it is (B) from which A follows (A being thus the consequence of B, instead of the reverse), and then what (viz. C) is the antecedent cause of B; and in practice the Greeks secured what was wanted by always insisting on the analysis being confirmed by subsequent synthesis, that is, they laboriously worked backwards the whole way from K to A, reversing the order of the analysis, which process would undoubtedly bring to light any flaw which had crept into the argument through the accidental neglect of the necessary precautions.

Reductio ad absurdum a variety of analysis.

In the process of analysis starting from the hypothesis that a proposition A is true and passing through B, C... as successive consequences we may arrive at a proposition K which, instead of being admittedly true, is either admittedly false or the contradictory of the original hypothesis A or of some one or more of the propositions B, C... intermediate between A and K. Now correct inference from a true proposition cannot lead to a false proposition; and in this case therefore we may at once conclude, without any inquiry whether the various steps in the argument are convertible or not, that the hypothesis A is false, for, if it were true, all the consequences correctly inferred from it would be true and no incompatibility could arise. This method of proving that a given hypothesis is false furnishes an indirect method of proving that a given hypothesis A is true, since we have only to take the contradictory of A and to prove that it is false. This is the method of reductio ad absurdum, which is therefore a variety of analysis. The contradictory of A, or not-A, will generally include more than one case and, in order to prove its falsity, each of the cases must be separately disposed of: e.g., if it is desired to prove that a certain part of a figure is equal to some other part, we take separately the hypotheses (1) that it is greater, (2) that it is less, and prove that each of these hypotheses leads to a conclusion either admittedly false or contradictory to the hypothesis itself or to some one of its consequences.

Analysis as applied to problems.

It is in relation to problems that the ancient analysis has the greatest significance, because it was the one general method which the Greeks used for solving all “the more abstruse problems” (τὰ ἀσαφέστερα τῶν προβλημάτων540.

We have, let us suppose, to construct a figure satisfying a certain set of conditions If we are to proceed at all methodically and not by mere guesswork, it is first necessary to “analyse” those conditions. To enable this to be done we must get them clearly in our minds, which is only possible by assuming all the conditions to be actually fulfilled, in other words, by supposing the problem solved. Then we have to transform those conditions, by all the means which practice in such cases has taught us to employ, into other conditions which are necessarily fulfilled if the original conditions are, and to continue this transformation until we at length arrive at conditions which we are in a position to satisfy541. In other words, we must arrive at some relation which enables us to construct a particular part of the figure which, it is true, has been hypothetically assumed and even drawn, but which nevertheless really requires to be found in order that the problem may be solved. From that moment the particular part of the figure becomes one of the data, and a fresh relation has to be found which enables a fresh part of the figure to be determined by means of the original data and the new one together. When this is done, the second new part of the figure also belongs to the data; and we proceed in this way until all the parts of the required figure are found542. The first part of the analysis down to the point of discovery of a relation which enables us to say that a certain new part of the figure not belonging to the original data is given, Hankel calls the transformation; the second part, in which it is proved that all the remaining parts of the figure are “given,” he calls the resolution. Then follows the synthesis, which also consists of two parts, (1) the construction, in the order in which it has to be actually carried out, and in general following the course of the second part of the analysis, the resolution; (2) the demonstration that the figure obtained does satisfy all the given conditions, which follows the steps of the first part of the analysis, the transformation, but in the reverse order. The second part of the analysis, the resolution, would be much facilitated and shortened by the existence of a systematic collection of Data such as Euclid's book bearing that title, consisting of propositions proving that, if in a figure certain parts or relations are given, other parts or relations are also given. As regards the first part of the analysis, the transformation, the usual rule applies that every step in the chain must be unconditionally convertible; and any failure to observe this condition will be brought to light by the subsequent synthesis. The second part, the resolution, can be directly turned into the construction since that only is given which can be constructed by the means provided in the Elements.

It would be difficult to find a better illustration of the above than the example chosen by Hankel from Pappus543.

Given a circle ABC and two points D, E external to it, to draw straight lines DB, EB from D, E to a point B on the circle such that, if DB, EB produced meet the circle again in C, A, AC shall be parallel to DE.

Analysis.

Suppose the problem solved and the tangent at A drawn, meeting ED produced in F.

(Part I. Transformation.)

Then, since AC is parallel to DE, the angle at C is equal to the angle CDE.

But, since FA is a tangent, the angle at C is equal to the angle FAE.

Therefore the angle FAE is equal to the angle CDE, whence A, B, D, F are concyclic.

Therefore the rectangle AE, EB is equal to the rectangle FE, ED.

(Part II. Resolution.)

But the rectangle AE, EB is given, because it is equal to the square on the tangent from E.

Therefore the rectangle FE, ED is given; and, since ED is given, FE is given (in length). [Data, 57.]

But FE is given in position also, so that F is also given. [Data, 27.]

Now FA is the tangent from a given point F to a circle ABC given in position; therefore FA is given in position and magnitude. [Data, 90.]

And F is given; therefore A is given.

But E is also given; therefore the straight line AE is given in position. [Data, 26.]

And the circle ABC is given in position; therefore the point B is also given. [Data, 25.]

But the points D, E are also given; therefore the straight lines DB, BE are also given in position.

Synthesis.

(Part I. Construction.)

Suppose the circle ABC and the points D, E given.

Take a rectangle contained by ED and by a certain straight line EF equal to the square on the tangent to the circle from E.

From F draw FA touching the circle in A; join ABE and then DB, producing DB to meet the circle at C. Join AC.

I say then that AC is parallel to DE.

(Part II. Demonstration.)

Since, by hypothesis, the rectangle FE, ED is equal to the square on the tangent from E, which again is equal to the rectangle AE, EB, the rectangle AE, EB is equal to the rectangle FE, ED.

Therefore A, B, D, F are concyclic, whence the angle FAE is equal to the angle BDE.

But the angle FAE is equal to the angle ACB in the alternate segment; therefore the angle ACB is equal to the angle BDE.

Therefore AC is parallel to DE.

In cases where a διορισμός is necessary, i.e. where a solution is only possible under certain conditions, the analysis will enable those conditions to be ascertained. Sometimes the διορισμός is stated and proved at the end of the analysis, e.g. in Archimedes, On the Sphere and Cylinder, II. 7; sometimes it is stated in that place and the proof postponed till after the end of the synthesis, e.g. in the solution of the problem subsidiary to On the Sphere and Cylinder, II. 4, preserved in Eutocius' commentary on that proposition. The analysis should also enable us to determine the number of solutions of which the problem is susceptible.


§ 7. THE DEFINITIONS.


General. “Real” and “Nominal” Definitions.

It is necessary, says Aristotle, whenever any one treats of any whole subject, to divide the genus into its primary constituents, those which are indivisible in species respectively: e.g. number must be divided into triad and dyad; then an attempt must be made in this way to obtain definitions, e.g. of a straight line, of a circle, and of a right angle544.

The word for definition is ὅρος. The original meaning of this word seems to have been “boundary,” “landmark.” Then we have it in Plato and Aristotle in the sense of standard or determining principle (“id quo alicuius rei natura constituitur vel definitur,” Index Aristotelicus545; and closely connected with this is the sense of definition. Aristotle uses both ὅρος and ὁρισμός for definition, the former occurring more frequently in the Topics, the latter in the Metaphysics.

Let us now first be clear as to what a definition does not do. There is nothing in connexion with definitions which Aristotle takes more pains to emphasise than that a definition asserts nothing as to the existence or non-existence of the thing defined. It is an answer to the question what a thing is (τί ἐστι), and does not say that it is (ὅτι ἔστι). The existence of the various things defined has to be proued, except in the case of a few primary things in each science, the existence of which is indemonstrable and must be assumed among the first principles of each science; e.g. points and lines in geometry must be assumed to exist, but the existence of everything else must be proved. This is stated clearly in the long passage quoted above under First Principles546. It is reasserted in such passages as the following. “The (answer to the question) what is a man and the fact that a man exists are different things547.” “It is clear that, even according to the view of definitions now current, those who define things do not prove that they exist548.” “We say that it is by demonstration that we must show that everything exists, except essence (εἰ μὴ οὐσία εἴη). But the existence of a thing is never essence; for the existent is not a genus. Therefore there must be demonstration that a thing exists. Thus, what is meant by triangle the geometer assumes, but that it exists he has to prove549.” “Anterior knowledge of two sorts is necessary: for it is necessary to presuppose, with regard to some things, that they exist; in other cases it is necessary to understand what the thing described is, and in other cases it is necessary to do both. Thus, with the fact that one of two contradictories must be true, we must know that it exists (is true); of the triangle we must know that it means such and such a thing; of the unit we must know both what it means and that it exists550.” What is here so much insisted on is the very fact which Mill pointed out in his discussion of earlier views of Definitions, where he says that the so-called real definitions or definitions of things do not constitute a different kind of definition from nominal definitions, or definitions of names; the former is simply the latter plus something else, namely a covert assertion that the thing defined exists. “This covert assertion is not a definition but a postulate. The definition is a mere identical proposition which gives information only about the use of language, and from which no conclusion affecting matters of fact can possibly be drawn. The accompanying postulate, on the other hand, affirms a fact which may lead to consequences of every degree of importance. It affirms the actual or possible existence of Things possessing the combination of attributes set forth in the definition: and this, if true, may be foundation sufficient on which to build a whole fabric of scientific truth551.” This statement really adds nothing to Aristotle's doctrine552: it has even the slight disadvantage, due to the use of the word “postulate” to describe “the covert assertion” in all cases, of not definitely pointing out that there are cases where existence has to be proued as distinct from those where it must be assumed. It is true that the existence of a definiend may have to be taken for granted provisionally until the time comes for proving it; but, so far as regards any case where existence must be proved sooner or later, the provisional assumption would be for Aristotle, not a postulate, but a hypothesis. In modern times, too, Mill's account of the true distinction between real and nominal definitions had been fully anticipated by Saccheri553, the editor of Euclides ab omni naevo vindicatus (1733), famous in the history of non-Euclidean geometry. In his Logica Demonstrativa (to which he also refers in his Euclid) Saccheri lays down the clear distinction between what he calls definitiones quid nominis or nominales, and definitiones quid rei or reales, namely that the former are only intended to explain the meaning that is to be attached to a given term, whereas the latter, besides declaring the meaning of a word, affirm at the same time the existence of the thing defined or, in geometry, the possibility of constructing it. The definitio quid nominis becomes a definitio quid rei “by means of a postulate, or when we come to the question whether the thing exists and it is answered affirmatively 554. ” Definitiones quid nominis are in themselves quite arbitrary, and neither require nor are capable of proof; they are merely provisional and are only intended to be turned as quickly as possible into definitiones quid rei, either (1) by means of a postulate in which it is asserted or conceded that what is defined exists or can be constructed, e.g. in the case of straight lines and circles, to which Euclid's first three postulates refer, or (2) by means of a demonstration reducing the construction of the figure defined to the successive carrying-out of a certain number of those elementary constructions, the possibility of which is postulated. Thus definitiones quid rei are in general obtained as the result of a series of demonstrations. Saccheri gives as an instance the construction of a square in Euclid I. 46. Suppose that it is objected that Euclid had no right to define a square, as he does at the beginning of the Book, when it was not certain that such a figure exists in nature; the objection, he says, could only have force if, before proving and making the construction, Euclid had assumed the aforesaid figure as given. That Euclid is not guilty of this error is clear from the fact that he never presupposes the existence of the square as defined until after I. 46.

Confusion between the nominal and the real definition as thus described, i.e. the use of the former in demonstration before it has been turned into the latter by the necessary proof that the thing defined exists, is according to Saccheri one of the most fruitful sources of illusory demonstration, and the danger is greater in proportion to the “complexity” of the definition, i.e. the number and variety of the attributes belonging to the thing defined. For the greater is the possibility that there may be among the attributes some that are incompatible, i.e. the simultaneous presence of which in a given figure can be proved, by means of other postulates etc. forming part of the basis of the science, to be impossible.

The same thought is expressed by Leibniz also. “If,” he says, “we give any definition, and it is not clear from it that the idea, which we ascribe to the thing, is possible, we cannot rely upon the demonstrations which we have derived from that definition, because, if that idea by chance involves a contradiction, it is possible that even contradictories may be true of it at one and the same time, and thus our demonstrations will be useless. Whence it is clear that definitions are not arbitrary. And this is a secret which is hardly sufficiently known555.” Leibniz' favourite illustration was the “regular polyhedron with ten faces,” the impossibility of which is not obvious at first sight.

It need hardly be added that, speaking generally, Euclid's definitions, and his use of them, agree with the doctrine of Aristotle that the definitions themselves say nothing as to the existence of the things defined, but that the existence of each of them must be proved or (in the case of the “principles” ) assumed. In geometry, says Aristotle, the existence of points and lines only must be assumed, the existence of the rest being proved. Accordingly Euclid's first three postulates declare the possibility of constructing straight lines and circles (the only “lines” except straight lines used in the Elements). Other things are defined and afterwards constructed and proved to exist: e.g. in Book I., Def. 20, it is explained what is meant by an equilateral triangle; then (I. 1) it is proposed to construct it, and, when constructed, it is proved to agree with the definition. When a square is defined (I. Def. 22), the question whether such a thing really exists is left open until, in I. 46, it is proposed to construct it and, when constructed, it is proved to satisfy the definition556. Similarly with the right angle (I. Def. 10, and I. 11) and parallels (I. Def. 23, and I. 27-29). The greatest care is taken to exclude mere presumption and imagination. The transition from the subjective definition of names to the objective definition of things is made, in geometry, by means of constructions (the first principles of which are postulated), as in other sciences it is made by means of experience557.


Aristotle's requirements in a definition.

We now come to the positive characteristics by which, according to Aristotle, scientific definitions must be marked.

First, the different attributes in a definition, when taken separately, cover more than the notion defined, but the combination of them does not. Aristotle illustrates this by the “triad,” into which enter the several notions of number, odd and prime, and the last “in both its two senses (a) of not being measured by any (other) number (ὡς μὴ μετρεῖσθαι ἀριθμῷ) and (b) of not being obtainable by adding numbers together” (ὡς μὴ συγκεῖσθαι ἐξ ἀριθμῶν), a unit not being a number. Of these attributes some are present in all other odd numbers as well, while the last [primeness in the second sense] belongs also to the dyad, but in nothing but the triad are they all present558.” The fact can be equally well illustrated from geometry. Thus, e.g. into the definition of a square (Eucl. I., Def. 22) there enter the several notions of figure, four-sided, equilateral, and right-angled, each of which covers more than the notion into which all enter as attributes559.

Secondly, a definition must be expressed in terms of things which are prior to, and better known than, the things defined560. This is clear, since the object of a definition is to give us knowledge of the thing defined, and it is by means of things prior and better known that we acquire fresh knowledge, as in the course of demonstrations. But the terms “prior” and “better known” are, as usual susceptible of two meanings; they may mean (1) absolutely or logically prior and better known, or (2) better known relatively to us. In the absolute sense, or from the standpoint of reason, a point is better known than a line, a line than a plane, and a plane than a solid, as also a unit is better known than number (for the unit is prior to, and the first principle of, any number). Similarly, in the absolute sense, a letter is prior to a syllable. But the case is sometimes different relatively to us; for example, a solid is more easily realised by the senses than a plane, a plane than a line, and a line than a point. Hence, while it is more scientific to begin with the absolutely prior, it may, perhaps, be permissible, in case the learner is not capable of following the scientific order, to explain things by means of what is more intelligible to him. “Among the definitions framed on this principle are those of the point, the line and the plane; all these explain what is prior by means of what is posterior, for the point is described as the extremity of a line, the line of a plane, the plane of a solid.” But, if it is asserted that such definitions by means of things which are more intelligible relatively only to a particular individual are really definitions, it will follow that there may be many definitions of the same thing, one for each individual for whom a thing is being defined, and even different definitions for one and the same individual at different times, since at first sensible objects are more intelligible, while to a better trained mind they become less so. It follows therefore that a thing should be defined by means of the absolutely prior and not the relatively prior, in order that there may be one sole and immutable definition. This is further enforced by reference to the requirement that a good definition must state the genus and the differentiae, for these are among the things which are, in the absolute sense, better known than, and prior to, the species (τῶν ἁπλῶς γνωριμωτέρων καὶ προτέρων τοῦ εἴδους ἐστίν). For to destroy the genus and the differentia is to destroy the species, so that the former are prior to the species; they are also better known, for, when the species is known, the genus and the differentia must necessarily be known also, e.g. he who knows “man” must also know “animal” and “land-animal,” but it does not follow, when the genus and differentia are known, that the species is known too, and hence the species is less known than they are561. It may be frankly admitted that the scientific definition will require superior mental powers for its apprehension; and the extent of its use must be a matter of discretion. So far Aristotle; and we have here the best possible explanation why Euclid supplemented his definition of a point by the statement in I. Def. 3 that the extremities of a line are points and his definition of a surface by I. Def. 6 to the effect that the extremities of a surface are lines. The supplementary explanations do in fact enable us to arrive at a better understanding of the formal definitions of a point and a line respectively, as is well explained by Simson in his note on Def. 1. Simson says, namely, that we must consider a solid, that is, a magnitude which has length, breadth and thickness, in order to understand aright the definitions of a point, a line and a surface. Consider, for instance, the boundary common to two solids which are contiguous or the boundary which divides one solid into two contiguous parts; this boundary is a surface. We can prove that it has no thickness by taking away either solid, when it remains the boundary of the other; for, if it had thickness, the thickness must either be a part of one solid or of the other, in which case to take away one or other solid would take away the thickness and therefore the boundary itself: which is impossible. Therefore the boundary or the surface has no thickness. In exactly the same way, regarding a line as the boundary of two contiguous surfaces, we prove that the line has no breadth; and, lastly, regarding a point as the common boundary or extremity of two lines, we prove that a point has no length, breadth or thickness.


Aristotle on unscientific definitions.

Aristotle distinguishes three kinds of definition which are unscientific because founded on what is not prior (μὴ ἐκ προτέρων). The first is a definition of a thing by means of its opposite, e.g. of “good” by means of “bad” ; this is wrong because opposites are naturally evolved together, and the knowledge of opposites is not uncommonly regarded as one and the same, so that one of the two opposites cannot be better known than the other. It is true that, in some cases of opposites, it would appear that no other sort of definition is possible: e.g. it would seem impossible to define double apart from the half and, generally, this would be the case with things which in their very nature (καθ̓ αὑτά) are relative terms (πρός τι λέγεται), since one cannot be known without the other, so that in the notion of either the other must be comprised as well562. The second kind of definition which is based on what is not prior is that in which there is a complete circle through the unconscious use in the definition itself of the notion to be defined though not of the name563. Trendelenburg illustrates this by two current definitions, (1) that of magnitude as that which can be increased or diminished, which is bad because the positive and negative comparatives “more” and “less” presuppose the notion of the positive “great,” (2) the famous Euclidean definition of a straight line as that which “lies evenly with the points on itself” (ἐξ ἴσου τοῖς ἐφ̓ ἑαυτῆς σημείοις κεῖται), where “lies evenly” can only be understood with the aid of the very notion of a straight line which is to be defined564. The third kind of vicious definition from that which is not prior is the definition of one of two coordinate species by means of its coordinate (ἀντιδιῃρημένον), e.g. a definition of “odd” as that which exceeds the even by a unit (the second alternative in Eucl. VII. Def. 7); for “odd” and “even” are coordinates, being differentiae of number565. This third kind is similar to the first. Thus, says Trendelenburg, it would be wrong to define a square as “a rectangle with equal sides.”


Aristotle's third requirement.

A third general observation of Aristotle which is specially relevant to geometrical definitions is that “to know what a thing is (τί ἐστιν) is the same as knowing why it is (διὰ τί ἐστιν566.” “What is an eclipse? A deprivation of light from the moon through the interposition of the earth. Why does an eclipse take place? Or why is the moon eclipsed? Because the light fails through the earth obstructing it. What is harmony? A ratio of numbers in high or low pitch. Why does the high-pitched harmonise with the low-pitched? Because the high and the low have a numerical ratio to one another567.” “We seek the cause (τὸ διότι) when we are already in possession of the fact (τὸ ὅτι). Sometimes they both become evident at the same time, but at all events the cause cannot possibly be known [as a cause] before the fact is known568.” “It is impossible to know what a thing is if we do not know that it is569” Trendelenburg paraphrases: “The definition of the notion does not fulfil its purpose until it is made genetic. It is the producing cause which first reveals the essence of the thing.... . The nominal definitions of geometry have only a provisional significance and are superseded as soon as they are made genetic by means of construction.” E.g. the genetic definition of a parallelogram is evolved from Eucl. I. 31 (giving the construction for parallels) and I. 33 about the lines joining corresponding ends of two straight lines parallel and equal in length. Where existence is proved by construction, the cause and the fact appear together570.

Again, “it is not enough that the defining statement should set forth the fact, as most definitions do; it should also contain and present the cause; whereas in practice what is stated in the definition is usually no more than a conclusion (συμπέρασμα). For example, what is quadrature? The construction of an equilateral right-angled figure equal to an oblong. But such a definition expresses merely the conclusion. Whereas, if you say that quadrature is the discovery of a mean proportional, then you state the reason571.” This is better understood if we compare the statement elsewhere that “the cause is the middle term, and this is what is sought in all cases572,” and the illustration of this by the case of the proposition that the angle in a semicircle is a right angle. Here the middle term which it is sought to establish by means of the figure is that the angle in the semi-circle is equal to the half of two right angles. We have then the syllogism: Whatever is half of two right angles is a right angle; the angle in a semi-circle is the half of two right angles; therefore (conclusion) the angle in a semi-circle is a right angle573. As with the demonstration, so it should be with the definition. A definition which is to show the genesis of the thing defined should contain the middle term or cause; otherwise it is a mere statement of a conclusion. Consider, for instance, the definition of “quadrature” as “making a square equal in area to a rectangle with unequal sides.” This gives no hint as to whether a solution of the problem is possible or how it is solved: but, if you add that to find the mean proportional between two given straight lines gives another straight line such that the square on it is equal to the rectangle contained by the first two straight lines, you supply the necessary middle term or cause574.


Technical terms not defined by Euclid.

It will be observed that what is here defined, “quadrature” or “squaring” (τετραγωνισμός), is not a geometrical figure, or an attribute of such a figure or a part of a figure, but a technical term used to describe a certain problem. Euclid does not define such things; but the fact that Aristotle alludes to this particular definition as well as to definitions of deflection (κεκλάσθαι) and of verging (νεύειν) seems to show that earlier text-books included among definitions explanations of a number of technical terms, and that Euclid deliberately omitted these explanations from his Elements as surplusage. Later the tendency was again in the opposite direction, as we see from the much expanded Definitions of Heron, which, for example, actually include a definition of a deflected line (κεκλασμένη γραμμή575. Euclid uses the passive of κλᾶν occasionally576, but evidently considered it unnecessary to explain such terms, which had come to bear a recognised meaning.

The mention too by Aristotle of a definition of verging (νεύειν) suggests that the problems indicated by this term were not excluded from elementary text-books before Euclid. The type of problem (νεῦσις) was that of placing a straight line across two lines, e.g. two straight lines, or a straight line and a circle, so that it shall verge to a given point (i.e. pass through it if produced) and at the same time the intercept on it made by the two given lines shall be of given length. In general, the use of conics is required for the theoretical solution of these problems, or a mechanical contrivance for their practical solution577. Zeuthen, following Oppermann, gives reasons for supposing, not only that mechanical constructions were practically used by the older Greek geometers for solving these problems, but that they were theoretically recognised as a permissible means of solution when the solution could not be effected by means of the straight line and circle, and that it was only in later times that it was considered necessary to use conics in every case where that was possible578. Heiberg579 suggests that the allusion of Aristotle to νεύσεις perhaps confirms this supposition, as Aristotle nowhere shows the slightest acquaintance with conics. I doubt whether this is a safe inference, since the problems of this type included in the elementary text-books might easily have been limited to those which could be solved by “plane” methods (i.e. by means of the straight line and circle). We know, e.g., from Pappus that Apollonius wrote two Books on plane νεύσεις580. But one thing is certain, namely that Euclid deliberately excluded this class of problem, doubtless as not being essential in a book of Elements.


Definitions not afterwards used.

Lastly, Euclid has definitions of some terms which he never afterwards uses, e.g. oblong (ἑτερόμηκες), rhombus, rhomboid. The “oblong” occurs in Aristotle; and it is certain that all these definitions are survivals from earlier books of Elements.

BOOK I.


DEFINITIONS.


1

A point is that which has no part.


2

A line is breadthless length.


3

The extremities of a line are points.


4

A straight line is a line which lies evenly with the points on itself.


5

A surface is that which has length and breadth only.


6

The extremities of a surface are lines.


7

A plane surface is a surface which lies evenly with the straight lines on itself.


8

A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.


9

And when the lines containing the angle are straight, the angle is called rectilineal.


10

When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.


11

An obtuse angle is an angle greater than a right angle.


12

An acute angle is an angle less than a right angle.


13

A boundary is that which is an extremity of anything.


14

A figure is that which is contained by any boundary or boundaries.


15

A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;


16

And the point is called the centre of the circle.


17

A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.


18

A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle.


19

Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.


20

Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.


21

Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acuteangled triangle that which has its three angles acute.


22

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.


23

Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.


POSTULATES.


1

Let the following be postulated:

To draw a straight line from any point to any point.


2

To produce a finite straight line continuously in a straight line.


3

To describe a circle with any centre and distance.


4

That all right angles are equal to one another.


5

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.


COMMON NOTIONS.


1

Things which are equal to the same thing are also equal to one another.


2

If equals be added to equals, the wholes are equal.


3

If equals be subtracted from equals, the remainders are equal.


4

[7] Things which coincide with one another are equal to one another.


5

[8] The whole is greater than the part.


BOOK I. PROPOSITIONS.


Proposition 1.


On a given finite straight line to construct an equilateral triangle.


Let AB be the given finite straight line.

Thus it is required to construct
an equilateral triangle on the straight line AB.

With centre A and distance AB let the circle BCD be described; [Post. 3]
again, with centre B and distance BA let the circle ACE be described; [Post. 3] and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined. [Post. 1]

Now, since the point A is the centre of the circle CDB,

AC is equal to AB. [Def. 15]

Again, since the point B is the centre of the circle CAE,

BC is equal to BA. [Def. 15]

But CA was also proved equal to AB;
therefore each of the straight lines CA, CB is equal to AB.

And things which are equal to the same thing are also equal to one another; [C.N. 1]

therefore CA is also equal to CB.

Therefore the three straight lines CA, AB, BC are
equal to one another.

Therefore the triangle ABC is equilateral; and it has been constructed on the given finite straight line AB.


(Being) what it was required to do.

581 582 583 584 585


Proposition 2.


To place at a given point (as an extremity) a straight line equal to a given straight line.


Let A be the given point, and BC the given straight line.

Thus it is required to place at the point A (as an extremity)
a straight line equal to the given straight line BC.

From the point A to the point B let the straight line AB be joined; [Post. 1] and on it let the equilateral triangle
DAB be constructed. [I. 1]

Let the straight lines AE, BF be produced in a straight line with DA, DB; [Post. 2] with centre B and distance BC let the
circle CGH be described; [Post. 3] and again, with centre D and distance DG let the circle GKL be described. [Post. 3]

Then, since the point B is the centre of the circle CGH,

BC is equal to BG.

Again, since the point D is the centre of the circle GKL,

DL is equal to DG.

And in these DA is equal to DB;

therefore the remainder AL is equal to the remainder BG. [C.N. 3]

But BC was also proved equal to BG;

therefore each of the straight lines AL, BC is equal to BG.

And things which are equal to the same thing are also equal to one another; [C.N. 1]

therefore AL is also equal to BC.

Therefore at the given point A the straight line AL is placed equal to the given straight line BC.


(Being) what it was required to do.

586 587 588


Proposition 3.


Given two unequal straight lines, to cut off from the greater a straight line equal to the less.


Let AB, C be the-two given unequal straight lines, and let AB be the greater of them.

Thus it is required to cut off from AB the greater a straight line equal to C the less.

At the point A let AD be placed equal to the straight line C; [I. 2] and with centre A and distance AD let the circle DEF be described. [Post. 3]

Now, since the point A is the centre of the circle DEF, AE is equal to AD. [Def. 15] But C is also equal to AD. Therefore each of the straight lines AE, C is equal to AD; so that AE is also equal to C. [C.N. 1]

Therefore, given the two straight lines AB, C, from AB the greater AE has been cut off equal to C the less.


(Being) what it was required to do.


Proposition 4.


If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles
will be equal to the remaining angles respectively, namely those which the equal sides subtend.


Let ABC, DEF be two triangles having the two sides AB, AC equal to the two sides DE, DF respectively, namely AB to DE and AC to DF, and the angle BAC equal to the
angle EDF.

I say that the base BC is also equal to the base EF, the triangle ABC will be equal to the triangle DEF, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend, that
is, the angle ABC to the angle DEF, and the angle ACB to the angle DFE.

For, if the triangle ABC be applied to the triangle DEF,

and if the point A be placed on the point D and the straight line AB on DE,
then the point B will also coincide with E, because AB is equal to DE.

Again, AB coinciding with DE, the straight line AC will also coincide with DF, because the angle BAC is equal to the angle EDF;

hence the point C will also coincide with the point F, because AC is again equal to DF.

But B also coincided with E; hence the base BC will coincide with the base EF.

[For if, when B coincides with E and C with F, the base BC does not coincide with the base EF, two straight lines will enclose a space: which is impossible.

Therefore the base BC will coincide with EF] and will be equal to it. [C.N. 4]

Thus the whole triangle ABC will coincide with the whole triangle DEF,

and will be equal to it.

And the remaining angles will also coincide with the remaining angles and will be equal to them, the angle ABC to the angle DEF, and the angle ACB to the angle DFE.

Therefore etc.


(Being) what it was required to prove.

589 590 591 592 593 594 595 596


Proposition 5.


In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another.


Let ABC be an isosceles triangle having the side AB
equal to the side AC; and let the straight lines BD, CE be produced further in a straight line with AB, AC. [Post. 2]

I say that the angle ABC is equal to the angle ACB, and the angle CBD to the angle BCE.

Let a point F be taken at random on BD; from AE the greater let AG be cut off equal to AF the less; [I. 3] and let the straight lines FC, GB be joined. [Post. 1]

Then, since AF is equal to AG and AB to AC,

the two sides FA, AC are equal to the two sides GA, AB, respectively;
and they contain a common angle, the angle FAG.
Therefore the base FC is equal to the base GB, and the triangle AFC is equal to the triangle AGB,
and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend,
that is, the angle ACF to the angle ABG, and the angle AFC to the angle AGB. [I. 4]

And, since the whole AF is equal to the whole AG,

and in these AB is equal to AC, the remainder BF is equal to the remainder CG.

But FC was also proved equal to GB;
therefore the two sides BF, FC are equal to the two sides CG, GB respectively; and the angle BFC is equal to the angle CGB,

while the base BC is common to them;
therefore the triangle BFC is also equal to the triangle CGB,
and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend;
therefore the angle FBC is equal to the angle GCB, and the angle BCF to the angle CBG.

Accordingly, since the whole angle ABG was proved equal to the angle ACF,

and in these the angle CBG is equal to the angle BCF,
the remaining angle ABC is equal to the remaining angle ACB;
and they are at the base of the triangle ABC.
But the angle FBC was also proved equal to the angle GCB;
and they are under the base.

Therefore etc.


Q. E. D.

597 598 599 600 601


Proposition 6.


If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another.


Let ABC be a triangle having the angle ABC equal to the angle ACB;

I say that the side AB is also equal to the side AC.

For, if AB is unequal to AC, one of them is greater.

Let AB be greater; and from AB the greater let DB be cut off equal to AC the less;

let DC be joined.

Then, since DB is equal to AC, and BC is common,

the two sides DB, BC are equal to the two sides AC, CB respectively;
and the angle DBC is equal to the angle ACB;
therefore the base DC is equal to the base AB, and the triangle DBC will be equal to the triangle ACB, the less to the greater: which is absurd. Therefore AB is not unequal to AC; it is therefore equal to it.

Therefore etc.


Q. E. D.


Proposition 7.


Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be constructed on the same straight line (from its extremities), and on the same side of it, two other straight lines meeting in
another point and equal to the former two respectively, namely each to that which has the same extremity with it.


For, if possible, given two straight lines AC, CB constructed on the straight line AB and meeting at the point C, let two other straight lines
AD, DB be constructed on the same straight line AB, on the same side of it, meeting in another point D and equal to the former two respectively, namely each to that which has the same extremity with it, so that CA is
equal to DA which has the same extremity A with it, and CB to DB which has the same extremity B with it; and let CD be joined.

Then, since AC is equal to AD,

the angle ACD is also equal to the angle ADC; [I. 5] therefore the angle ADC is greater than the angle DCB;
therefore the angle CDB is much greater than the angle DCB.

Again, since CB is equal to DB,

the angle CDB is also equal to the angle DCB.

But it was also proved much greater than it:
which is impossible.

Therefore etc.


Q. E. D.

602 603 604


Proposition 8.


If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines.


Let ABC, DEF be two triangles having the two sides AB, AC equal to the two sides DE, DF respectively, namely AB to DE, and AC to DF; and let them have the base BC equal
to the base EF;

I say that the angle BAC is also equal to the angle EDF.

For, if the triangle ABC be applied to the triangle DEF, and if the point B be placed on
the point E and the straight line BC on EF,

the point C will also coincide with F, because BC is equal to EF.

Then, BC coinciding with EF,

BA, AC will also coincide with ED, DF;

for, if the base BC coincides with the base EF, and the sides BA, AC do not coincide with ED, DF but fall beside them as EG, GF,
then, given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there will have been constructed on the same straight line (from its extremities), and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it. But they cannot be so constructed. [I. 7]

Therefore it is not possible that, if the base BC be applied to the base EF, the sides BA, AC should not coincide with ED, DF;

they will therefore coincide,

so that the angle BAC will also coincide with the angle EDF, and will be equal to it.

If therefore etc.


Q. E. D.

605 606


Proposition 9.


To bisect a given rectilineal angle.


Let the angle BAC be the given rectilineal angle.

Thus it is required to bisect it.

Let a point D be taken at random on AB; let AE be cut off from AC equal to AD; [I. 3] let DE be joined, and on DE let the equilateral triangle DEF be constructed; let AF be joined.

I say that the angle BAC has been bisected by the straight line AF.

For, since AD is equal to AE, and AF is common,

the two sides DA, AF are equal to the two sides EA, AF respectively.

And the base DF is equal to the base EF;

therefore the angle DAF is equal to the angle EAF. [I. 8]

Therefore the given rectilineal angle BAC has been bisected by the straight line AF.


Q. E. F.


Proposition 10.


To bisect a given finite straight line.


Let AB be the given finite straight line.

Thus it is required to bisect the finite straight line AB.

Let the equilateral triangle ABC be constructed on it, [I. 1] and let the angle ACB be bisected by the straight line CD; [I. 9]

I say that the straight line AB has been bisected at the point D.

For, since AC is equal to CB, and CD is common,

the two sides AC, CD are equal to the two sides BC, CD respectively;
and the angle ACD is equal to the angle BCD;
therefore the base AD is equal to the base BD. [I. 4]

Therefore the given finite straight line AB has been bisected at D.


Q. E. F.


Proposition 11.


To draw a straight line at right angles to a given straight line from a given point on it.


Let AB be the given straight line, and C the given point on it.

Thus it is required to draw from the point C a straight line at right angles to the straight line AB.

Let a point D be taken at random on AC;
let CE be made equal to CD; [I. 3] on DE let the equilateral triangle FDE be constructed, [I. 1] and let FC be joined;

I say that the straight line FC has been drawn at right
angles to the given straight line AB from C the given point on it.

For, since DC is equal to CE, and CF is common,

the two sides DC, CF are equal to the two sides EC, CF respectively;
and the base DF is equal to the base FE;
therefore the angle DCF is equal to the angle ECF; [I. 8]
and they are adjacent angles.

But, when a straight line set up on a straight line makes
the adjacent angles equal to one another, each of the equal angles is right; [Def. 10]

therefore each of the angles DCF, FCE is right.

Therefore the straight line CF has been drawn at right angles to the given straight line AB from the given point
C on it.


Q. E. F.

607


Proposition 12.


To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line.


Let AB be the given infinite straight line, and C the given point which is not on it;
thus it is required to draw to the given infinite straight line AB, from the given point C which is not on it, a perpendicular straight line.

For let a point D be taken
at random on the other side of the straight line AB, and with centre C and distance CD let the circle EFG be described; [Post. 3]

let the straight line EG be bisected at H, [I. 10] and let the straight lines CG, CH, CE be joined. [Post. 1]

I say that CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it.

For, since GH is equal to HE, and HC is common,

the two sides GH, HC are equal to the two sides EH, HC respectively;
and the base CG is equal to the base CE;
therefore the angle CHG is equal to the angle EHC. [I. 8] And they are adjacent angles.

But, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is
called a perpendicular to that on which it stands. [Def. 10]

Therefore CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it.


Q. E. F.

608 609


Proposition 13.


If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles.


For let any straight line AB set up on the straight line
CD make the angles CBA, ABD;

I say that the angles CBA, ABD are either two right angles or equal to two right angles.

Now, if the angle CBA is equal to
the angle ABD,

they are two right angles. [Def. 10]

But, if not, let BE be drawn from the point B at right angles to CD; [I. 11]

therefore the angles CBE, EBD are two right angles.

Then, since the angle CBE is equal to the two angles CBA, ABE,

let the angle EBD be added to each; therefore the angles CBE, EBD are equal to the three angles CBA, ABE, EBD. [C. N. 2]

Again, since the angle DBA is equal to the two angles DBE, EBA,

let the angle ABC be added to each; therefore the angles DBA. ABC are equal to the three angles DBE, EBA, ABC. [C. N. 2]

But the angles CBE, EBD were also proved equal to the same three angles;

and things which are equal to the same thing are also equal to one another; [C. N. 1] therefore the angles CBE, EBD are also equal to the angles DBA, ABC.
But the angles CBE, EBD are two right angles;
therefore the angles DBA, ABC are also equal to two right angles.

Therefore etc.


Q. E. D.

610


Proposition 14.


If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be in a straight line with one another.


For with any straight line AB, and at the point B on it, let the two straight lines BC, BD not lying on the same side make the adjacent angles ABC, ABD equal to two right angles;

I say that BD is in a straight line with CB.

For, if BD is not in a straight line with BC, let BE be in a straight line with CB.

Then, since the straight line AB stands on the straight line CBE,

the angles ABC, ABE are equal to two right angles. [I. 13]
But the angles ABC, ABD are also equal to two right angles;
therefore the angles CBA, ABE are equal to the angles CBA, ABD. [Post. 4 and C.N. 1]

Let the angle CBA be subtracted from each;
therefore the remaining angle ABE is equal to the remaining angle ABD, [C.N. 3]

the less to the greater: which is impossible. Therefore BE is not in a straight line with CB.

Similarly we can prove that neither is any other straight
line except BD.

Therefore CB is in a straight line with BD.

Therefore etc.


Q. E. D.

611 612 613


Proposition 15.


If two straight lines cut one another, they make the vertical angles equal to one another.


For let the straight lines AB, CD cut one another at the point E;

I say that the angle AEC is equal to the angle DEB,

and the angle CEB to the angle AED.

For, since the straight line AE stands
on the straight line CD, making the angles CEA, AED,

the angles CEA, AED are equal to two right angles [I. 13]

Again, since the straight line DE stands on the straight line AB, making the angles AED, DEB,

the angles AED, DEB are equal to two right angles. [I. 13]

But the angles CEA, AED were also proved equal to two right angles;

therefore the angles CEA, AED are equal to the angles AED DEB. [Post. 4 and C. N. 1] Let the angle AED be subtracted from each; therefore the remaining angle CEA is equal to the remaining angle BED. [C. N. 3]

Similarly it can be proved that the angles CEB, DEA are also equal.

Therefore etc. Q. E. D.


Porism.

[From this it is manifest that, if two straight lines cut one another, they will make the angles at the point of section equal to four right angles.

614 615


Proposition 16.


In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles.


Let ABC be a triangle, and let one side of it BC be produced to D;

I say that the exterior angle ACD is greater than either of the interior and opposite angles CBA, BAC.

Let AC be bisected at E [I. 10], and let BE be joined and produced
in a straight line to F;

let EF be made equal to BE[I. 3], let FC be joined [Post. 1], and let AC be drawn through to G [Post. 2].

Then, since AE is equal to EC,
and BE to EF,

the two sides AE, EB are equal to the two sides CE, EF respectively; and the angle AEB is equal to the angle FEC, for they are vertical angles. [I. 15] Therefore the base AB is equal to the base FC, and the triangle ABE is equal to the triangle CFE, and the remaining angles are equal to the remaining angles respectively, namely those which the equal sides subtend; [I. 4] therefore the angle BAE is equal to the angle ECF.

But the angle ECD is greater than the angle ECF; [C. N. 5]

therefore the angle ACD is greater than the angle BAE.

Similarly also, if BC be bisected, the angle BCG, that is, the angle ACD [I. 15], can be proved greater than the angle ABC as well.

Therefore etc.


Q. E. D.

616 617 618 619


Proposition 17.


In any triangle two angles taken together in any manner are less than two right angles.


Let ABC be a triangle; I say that two angles of the triangle ABC taken together in any manner are less than two right angles.

For let BC be produced to D. [Post. 2]

Then, since the angle ACD is an exterior angle of the triangle ABC,

it is greater than the interior and opposite angle ABC. [I. 16]

Let the angle ACB be added to each;
therefore the angles ACD, ACB are greater than the angles ABC, BCA. But the angles ACD, ACB are equal to two right angles. [I. 13]

Therefore the angles ABC, BCA are less than two right angles.

Similarly we can prove that the angles BAC, ACB are also less than two right angles, and so are the angles CAB, ABC as well.

Therefore etc.


Q. E. D.

620


Proposition 18.


In any triangle the greater side subtends the greater angle.


For let ABC be a triangle having the side AC greater than AB;

I say that the angle ABC is also greater than the angle BCA.

For, since AC is greater than AB, let AD be made equal to AB [I. 3], and let BD bejoined.

Then, since the angle ADB is an exterior angle of the triangle BCD,

it is greater than the interior and opposite angle DCB. [I. 16]

But the angle ADB is equal to the angle ABD,

since the side AB is equal to AD; therefore the angle ABD is also greater than the angle ACB; therefore the angle ABC is much greater than the angle ACB.

Therefore etc.


Q. E. D.

621


Proposition 19.


In any triangle the greater angle is subtended by the greater side.


Let ABC be a triangle having the angle ABC greater than the angle BCA;

I say that the side AC is also greater than the side AB.

For, if not, AC is either equal to AB or less.

Now AC is not equal to AB; for then the angle ABC would also have been equal to the angle ACB; [I. 5] but it is not;

therefore AC is not equal to AB.

Neither is AC less than AB, for then the angle ABC would also have been less than the angle ACB; [I. 18] but it is not;

therefore AC is not less than AB.

And it was proved that it is not equal either.

Therefore AC is greater than AB.

Therefore etc.


Q. E. D.


Proposition 20.


In any triangle two sides taken together in any manner are greater than the remaining one.


For let ABC be a triangle; I say that in the triangle ABC two sides taken together in any manner are greater than the remaining one, namely
BA, AC greater than BC, AB, BC greater than AC, BC, CA greater than AB.

For let BA be drawn through to the point D, let DA be made equal to CA, and let DC be joined.

Then, since DA is equal to AC, the angle ADC is also equal to the angle ACD; [I. 5]

therefore the angle BCD is greater than the angle ADC. [C.N. 5]

And, since DCB is a triangle having the angle BCD greater than the angle BDC,

and the greater angle is subtended by the greater side, [I. 19] therefore DB is greater than BC.

But DA is equal to AC;

therefore BA, AC are greater than BC.

Similarly we can prove that AB, BC are also greater than CA, and BC, CA than AB.

Therefore etc.


Q. E. D.

622


Proposition 21.


If on one of the sides of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a greater
angle.


On BC, one of the sides of the triangle ABC, from its extremities B, C, let the two straight lines BD, DC be constructed meeting within the triangle;

I say that BD, DC are less than the remaining two sides
of the triangle BA, AC, but contain an angle BDC greater than the angle BAC.

For let BD be drawn through to E.

Then, since in any triangle two sides are greater than the remaining
one, [I. 20]

therefore, in the triangle ABE, the two sides AB, AE are greater than BE.

Let EC be added to each;

therefore BA, AC are greater than BE, EC.

Again, since, in the triangle CED,

the two sides CE, ED are greater than CD, let DB be added to each; therefore CE, EB are greater than CD, DB.

But BA, AC were proved greater than BE, EC;

therefore BA, AC are much greater than BD, DC.

Again, since in any triangle the exterior angle is greater than the interior and opposite angle, [I. 16] therefore, in the triangle CDE,

the exterior angle BDC is greater than the angle CED.

For the same reason, moreover, in the triangle ABE also,

the exterior angle CEB is greater than the angle BAC.
But the angle BDC was proved greater than the angle CEB;
therefore the angle BDC is much greater than the angle BAC.

Therefore etc.


Q. E. D.

623 624


Proposition 22.


Out of three straight lines, which are equal to three given straight lines, to construct a triangle: thus it is necessary that two of the straight lines taken together in any manner should be greater than the remaining one. [I. 20]


Let the three given straight lines be A, B, C, and of these let two taken together in any manner be greater than the remaining one, namely A, B greater than C,
A, C greater than B,
and B, C greater than A; thus it is required to construct a triangle out of straight lines equal to A, B, C.

Let there be set out a straight line DE, terminated at D but of infinite length in the direction of E, and let DF be made equal to A, FG equal to B, and GH equal to C. [I. 3]

With centre F and distance FD let the circle DKL be described; again, with centre G and distance GH let the circle KLH be described; and let KF, KG be joined;

I say that the triangle KFG has been constructed out of three straight lines equal to A, B, C.

For, since the point F is the centre of the circle DKL,

FD is equal to FK.

But FD is equal to A;

therefore KF is also equal to A.

Again, since the point G is the centre of the circle LKH,

GH is equal to GK.

But GH is equal to C;

therefore KG is also equal to C. And FG is also equal to B;
therefore the three straight lines KF, FG, GK are equal to the three straight lines A, B, C.

Therefore out of the three straight lines KF, FG, GK, which are equal to the three given straight lines A, B, C, the triangle KFG has been constructed.


Q. E. F.

625 626 627


Proposition 23.


On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.


Let AB be the given straight line, A the point on it, and the angle DCE the given rectilineal angle;

thus it is required to construct on the given straight line AB, and at the point A on it, a rectilineal angle equal to the given rectilineal angle DCE.

On the straight lines CD, CE respectively let the points D, E be taken at random; let DE be joined, and out of three straight lines which are equal to the three straight lines CD, DE, CE let the triangle AFG be constructed in such a way that CD is equal to AF, CE to AG, and further DE to FG.

Then, since the two sides DC, CE are equal to the two sides FA, AG respectively,

and the base DE is equal to the base FG, the angle DCE is equal to the angle FAG. [I. 8]

Therefore on the given straight line AB, and at the point A on it, the rectilineal angle FAG has been constructed equal to the given rectilineal angle DCE.


Q. E. F.


Proposition 24.


If two triangles have the two sides equal to two sides respectively, but have the one of the angles contained by the equal straight lines greater than the other, they will also have the base greater than the base.


Let ABC, DEF be two triangles having the two sides AB, AC equal to the two sides DE, DF respectively, namely AB to DE, and AC to DF, and let the angle at A be greater than the angle at D;

I say that the base BC is also greater than the base EF.

For, since the angle BAC is greater than the angle EDF, let there be constructed, on the straight line DE, and at the point D on it, the angle EDG
equal to the angle BAC; [I. 23] let DG be made equal to either of the two straight lines AC, DF, and let EG, FG be joined.

Then, since AB is equal to DE, and AC to DG,
the two sides BA, AC are equal to the two sides ED, DG, respectively;

and the angle BAC is equal to the angle EDG; therefore the base BC is equal to the base EG. [I. 4]

Again, since DF is equal to DG,

the angle DGF is also equal to the angle DFG; [I. 5] therefore the angle DFG is greater than the angle EGF.

Therefore the angle EFG is much greater than the angle EGF.

And, since EFG is a triangle having the angle EFG
greater than the angle EGF,

and the greater angle is subtended by the greater side, [I. 19] the side EG is also greater than EF.

But EG is equal to BC.

Therefore BC is also greater than EF.

Therefore etc.


Q. E. D.

628


Proposition 25.


If two triangles have the two sides equal to two sides respectively, but have the base greater than the base, they will also have the one of the angles contained by the equal straight lines greater than the other.


Let ABC, DEF be two triangles having the two sides AB, AC equal to the two sides DE, DF respectively, namely AB to DE, and AC to DF; and let the base BC be greater than the base EF;

I say that the angle BAC is also greater than the angle EDF.

For, if not, it is either equal to it or less.

Now the angle BAC is not equal to the angle EDF; for then the base BC would also have been equal to the base EF, [I. 4]

but it is not; therefore the angle BAC is not equal to the angle EDF.

Neither again is the angle BAC less than the angle EDF; for then the base BC would also have been less than the base EF, [I. 24]

but it is not; therefore the angle BAC is not less than the angle EDF.

But it was proved that it is not equal either;

therefore the angle BAC is greater than the angle EDF.

Therefore etc.


Q. E. D.


Proposition 26.


If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to
the remaining sides and the remaining angle to the remaining angle.


Let ABC, DEF be two triangles having the two angles ABC, BCA equal to the two angles DEF, EFD respectively, namely the angle ABC to the angle DEF, and the angle
BCA to the angle EFD; and let them also have one side equal to one side, first that adjoining the equal angles, namely BC to EF;

I say that they will also have the remaining sides equal to the remaining sides respectively, namely AB to DE and
AC to DF, and the remaining angle to the remaining angle, namely the angle BAC to the angle EDF.

For, if AB is unequal to DE, one of them is greater.

Let AB be greater, and let BG be made equal to DE; and let GC be joined.

Then, since BG is equal to DE, and BC to EF, the two sides GB, BC are equal to the two sides DE, EF respectively; and the angle GBC is equal to the angle DEF;

therefore the base GC is equal to the base DF, and the triangle GBC is equal to the triangle DEF, and the remaining angles will be equal to the remaining angles, namely those which the equal sides subtend; [I. 4] therefore the angle GCB is equal to the angle DFE.
But the angle DFE is by hypothesis equal to the angle BCA;
therefore the angle BCG is equal to the angle BCA, the less to the greater: which is impossible. Therefore AB is not unequal to DE, and is therefore equal to it.

But BC is also equal to EF;

therefore the two sides AB, BC are equal to the two sides DE, EF respectively, and the angle ABC is equal to the angle DEF; therefore the base AC is equal to the base DF, and the remaining angle BAC is equal to the remaining angle EDF. [I. 4]

Again, let sides subtending equal angles be equal, as AB to DE;

I say again that the remaining sides will be equal to the remaining sides, namely AC to DF and BC to EF, and
further the remaining angle BAC is equal to the remaining angle EDF.

For, if BC is unequal to EF, one of them is greater.

Let BC be greater, if possible, and let BH be made equal to EF; let AH be joined.

Then, since BH is equal to EF, and AB to DE, the two sides AB, BH are equal to the two sides DE, EF respectively, and they contain equal angles;

therefore the base AH is equal to the base DF,
and the triangle ABH is equal to the triangle DEF,
and the remaining angles will be equal to the remaining angles, namely those which the equal sides subtend; [I. 4]
therefore the angle BHA is equal to the angle EFD.

But the angle EFD is equal to the angle BCA; therefore, in the triangle AHC, the exterior angle BHA is
equal to the interior and opposite angle BCA:

which is impossible. [I. 16]

Therefore BC is not unequal to EF,

and is therefore equal to it.

But AB is also equal to DE;
therefore the two sides AB, BC are equal to the two sides DE, EF respectively, and they contain equal angles;

therefore the base AC is equal to the base DF, the triangle ABC equal to the triangle DEF,
and the remaining angle BAC equal to the remaining angle
EDF. [I. 4]

Therefore etc.


Q. E. D.

629 630


Proposition 27.


If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.


For let the straight line EF falling on the two straight
lines AB, CD make the alternate angles AEF, EFD equal to one another;

I say that AB is parallel to CD.

For, if not, AB, CD when produced will meet either in the direction
of B, D or towards A, C.

Let them be produced and meet, in the direction of B, D, at G.

Then, in the triangle GEF, the exterior angle AEF is equal to the interior and opposite
angle EFG: which is impossible. [I. 16]

Therefore AB, CD when produced will not meet in the direction of B, D.

Similarly it can be proved that neither will they meet
towards A, C.

But straight lines which do not meet in either direction are parallel; [Def. 23]

therefore AB is parallel to CD.

Therefore etc.


Q. E. D.

631 632 633


Proposition 28.


If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another.


For let the straight line EF falling on the two straight lines AB, CD make the exterior angle EGB equal to the interior and opposite angle GHD, or the interior angles on the same side, namely BGH, GHD, equal to two right angles;

I say that AB is parallel to CD.

For, since the angle EGB is equal to the angle GHD, while the angle EGB is equal to the angle AGH, [I. 15] the angle AGH is also equal to the angle GHD; and they are alternate;

therefore AB is parallel to CD. [I. 27]

Again, since the angles BGH, GHD are equal to two right angles, and the angles AGH, BGH are also equal to two right angles, [I. 13] the angles AGH, BGH are equal to the angles BGH, GHD.

Let the angle BGH be subtracted from each; therefore the remaining angle AGH is equal to the remaining angle GHD; and they are alternate;

therefore AB is parallel to CD. [I. 27]

Therefore etc.


Q. E. D.


Proposition 29.


A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles.


For let the straight line EF fall on the parallel straight lines AB, CD;

I say that it makes the alternate angles AGH, GHD equal, the exterior angle EGB equal to the interior and opposite angle GHD, and the interior angles on the same
side, namely BGH, GHD, equal to two right angles.

For, if the angle AGH is unequal to the angle GHD, one of them is greater.

Let the angle AGH be greater.

Let the angle BGH be added to each; therefore the angles AGH, BGH are greater than the angles BGH, GHD.

But the angles AGH, BGH are equal to two right angles; [I. 13]

therefore the angles BGH, GHD are less than two right angles.

But straight lines produced indefinitely from angles less than two right angles meet; [Post. 5]

therefore AB, CD, if produced indefinitely, will meet; but they do not meet, because they are by hypothesis parallel.

Therefore the angle AGH is not unequal to the angle GHD,

and is therefore equal to it.

Again, the angle AGH is equal to the angle EGB; [I. 15]

therefore the angle EGB is also equal to the angle GHD. [C.N. 1]

Let the angle BGH be added to each;

therefore the angles EGB, BGH are equal to the angles BGH, GHD. [C.N. 2]

But the angles EGB, BGH are equal to two right angles; [I. 13]

therefore the angles BGH, GHD are also equal to two right angles.

Therefore etc.


Q. E. D.

634 635


Proposition 30.


Straight lines parallel to the same straight line are also parallel to one another.


Let each of the straight lines AB, CD be parallel to EF; I say that AB is also parallel to CD.

For let the straight line GK fall upon them;

Then, since the straight line GK has fallen on the parallel straight lines AB, EF,

the angle AGK is equal to the angle GHF. [I. 29]

Again, since the straight line GK has fallen on the parallel straight lines EF, CD,

the angle GHF is equal to the angle GKD. [I. 29]

But the angle AGK was also proved equal to the angle GHF;

therefore the angle AGK is also equal to the angle GKD; [C.N. 1]
and they are alternate.

Therefore AB is parallel to CD.


Q. E. D.

636


Proposition 31.


Through a given point to draw a straight line parallel to a given straight line.


Let A be the given point, and BC the given straight line; thus it is required to draw through the point A a straight line parallel to the straight line BC.

Let a point D be taken at random on BC, and let AD be joined; on the straight line DA, and at the point A on it, let the angle DAE be constructed equal to the angle ADC [I. 23]; and let the straight line AF be produced in a straight line with EA.

Then, since the straight line AD falling on the two straight lines BC, EF has made the alternate angles EAD, ADC equal to one another,

therefore EAF is parallel to BC. [I. 27]

Therefore through the given point A the straight line EAF has been drawn parallel to the given straight line BC.


Q. E. F.


Proposition 32.


In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles.


Let ABC be a triangle, and let one side of it BC be produced to D;

I say that the exterior angle ACD is equal to the two interior and opposite angles CAB, ABC, and the three interior angles of the triangle ABC, BCA, CAB are equal to two right angles.

For let CE be drawn through the point C parallel to the straight line AB. [I. 31]

Then, since AB is parallel to CE,

and AC has fallen upon them, the alternate angles BAC, ACE are equal to one another. [I. 29]

Again, since AB is parallel to CE,

and the straight line BD has fallen upon them,
the exterior angle ECD is equal to the interior and opposite angle ABC. [I. 29]

But the angle ACE was also proved equal to the angle BAC;

therefore the whole angle ACD is equal to the two interior and opposite angles BAC, ABC.

Let the angle ACB be added to each;

therefore the angles ACD, ACB are equal to the three angles ABC, BCA, CAB.

But the angles ACD, ACB are equal to two right angles; [I. 13]

therefore the angles ABC, BCA, CAB are also equal to two right angles.

Therefore etc.


Q. E. D.


Proposition 33.


The straight lines joining equal and parallel straight lines (at the extremities which are) in the same directions (respectively) are themselves also equal and parallel.


Let AB, CD be equal and parallel, and let the straight
lines AC, BD join them (at the extremities which are) in the same directions (respectively); I say that AC, BD are also equal and parallel.

Let BC be joined.

Then, since AB is parallel to CD,
and BC has fallen upon them,

the alternate angles ABC, BCD are equal to one another. [I. 29]

And, since AB is equal to CD,

and BC is common, the two sides AB, BC are equal to the two sides DC, CB; and the angle ABC is equal to the angle BCD; therefore the base AC is equal to the base BD, and the griangle ABC is equal to the triangle DCB, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend; [I. 4] therefore the angle ACB is equal to the angle CBD.

And, since the straight line BC falling on the two straight lines AC, BD has made the alternate angles equal to one another,

AC is parallel to BD. [I. 27]

And it was also proved equal to it.

Therefore etc.


Q. E. D.

637 638


Proposition 34.


In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.


Let ACDB be a parallelogrammic area, and BC its diameter;
I say that the opposite sides and angles of the parallelogram ACDB are equal to one another, and the diameter BC bisects it.

For, since AB is parallel to CD, and the straight line BC has fallen
upon them,

the alternate angles ABC, BCD are equal to one another. [I. 29]

Again, since AC is parallel to BD, and BChas fallen upon them,

the alternate angles ACB, CBD are equal to one another. [I. 29]

Therefore ABC, DCB are two triangles having the two angles ABC, BCA equal to the two angles DCB, CBD respectively, and one side equal to one side, namely that
adjoining the equal angles and common to both of them, BC;

therefore they will also have the remaining sides equal to the remaining sides respectively, and the remaining angle to the remaining angle; [I. 26] therefore the side AB is equal to CD, and AC to BD,
and further the angle BAC is equal to the angle CDB.

And, since the angle ABC is equal to the angle BCD,

and the angle CBD to the angle ACB, the whole angle ABD is equal to the whole angle ACD. [C.N. 2]

And the angle BAC was also proved equal to the angle CDB.

Therefore in parallelogrammic areas the opposite sides and angles are equal to one another.

I say, next, that the diameter also bisects the areas.

For, since AB is equal to CD,
and BC is common, the two sides AB, BC are equal to the two sides DC, CB respectively;

and the angle ABC is equal to the angle BCD; therefore the base AC is also equal to DB, and the triangle ABC is equal to the triangle DCB. [I. 4]

Therefore the diameter BC bisects the parallelogram ACDB.


Q. E. D.

639 640


Proposition 35.


Parallelograms which are on the same base and in the same parallels are equal to one another.


Let ABCD, EBCF be parallelograms on the same base BC and in the same parallels AF, BC;
I say that ABCD is equal to the parallelogram EBCF.

For, since ABCD is a parallelogram,

AD is equal to BC. [I. 34]

For the same reason also

EF is equal to BC, so that AD is also equal to EF; [C.N. 1]
and DE is common;
therefore the whole AE is equal to the whole DF. [C.N. 2]

But AB is also equal to DC; [I. 34] therefore the two sides EA, AB are equal to the two sides
FD, DC respectively,

and the angle FDC is equal to the angle EAB, the exterior to the interior; [I. 29] therefore the base EB is equal to the base FC, and the triangle EAB will be equal to the triangle FDC. [I. 4]

Let DGE be subtracted from each; therefore the trapezium ABGD which remains is equal to the trapezium EGCF which remains. [C.N. 3]

Let the triangle GBC be added to each; therefore the whole parallelogram ABCD is equal to the whole parallelogram EBCF. [C.N. 2]

Therefore etc.


Q. E. D.

641 642


Proposition 36.


Parallelograms which are on equal bases and in the same parallels are equal to one another.


Let ABCD, EFGH be parallelograms which are on equal bases BC, FG and in the same parallels AH, BG; I say that the parallelogram ABCD is equal to EFGH.

For let BE, CH be joined.

Then, since BC is equal to FG while FG is equal to EH,

BC is also equal to EH. [C.N. 1]

But they are also parallel.

And EB, HC join them; but straight lines joining equal and parallel straight lines (at the extremities which are) in the same directions (respectively) are equal and parallel. [I. 33]

Therefore EBCH is a parallelogram. [I. 34]

And it is equal to ABCD; for it has the same base BC with it, and is in the same parallels BC, AH with it. [I. 35]

For the same reason also EFGH is equal to the same EBCH; [I. 35] so that the parallelogram ABCD is also equal to EFGH. [C.N. 1]

Therefore etc. Q. E. D.


Proposition 37.


Triangles which are on the same base and in the same parallels are equal to one another.


Let ABC, DBC be triangles on the same base BC and in the same parallels AD, BC;
I say that the triangle ABC is equal to the triangle DBC.

Let AD be produced in both directions to E, F; through B let BE be drawn parallel to CA, [I. 31]
and through C let CF be drawn parallel to BD. [I. 31]

Then each of the figures EBCA, DBCF is a parallelogram; and they are equal,

for they are on the same base BC and in the same parallels BC, EF. [I. 35]

Moreover the triangle ABC is half of the parallelogram EBCA; for the diameter AB bisects it. [I. 34]

And the triangle DBC is half of the parallelogram DBCF;
for the diameter DC bisects it. [I. 34]

[But the halves of equal things are equal to one another.]

Therefore the triangle ABC is equal to the triangle DBC.

Therefore etc.


Q. E. D.

643


Proposition 38.


Triangles which are on equal bases and in the same parallels are equal to one another.


Let ABC, DEF be triangles on equal bases BC, EF and in the same parallels BF, AD; I say that the triangle ABC is equal to the triangle DEF.

For let AD be produced in both directions to G, H; through B let BG be drawn parallel to CA, [I. 31] and through F let FH be drawn parallel to DE.

Then each of the figures GBCA, DEFH is a parallelogram; and GBCA is equal to DEFH;

for they are on equal bases BC, EF and in the same parallels BF, GH. [I. 36]

Moreover the triangle ABC is half of the parallelogram GBCA; for the diameter AB bisects it. [I. 34]

And the triangle FED is half of the parallelogram DEFH; for the diameter DF bisects it. [I. 34]

[But the halves of equal things are equal to one another.]

Therefore the triangle ABC is equal to the triangle DEF.

Therefore etc.


Q. E. D.


Proposition 39.


Equal triangles which are on the same base and on the same side are also in the same parallels.


Let ABC, DBC be equal triangles which are on the same base BC and on the same side of it;
[I say that they are also in the same parallels.]

And [For] let AD be joined; I say that AD is parallel to BC.

For, if not, let AE be drawn through the point A parallel to the straight line
BC, [I. 31] and let EC be joined.

Therefore the triangle ABC is equal to the triangle EBC; for it is on the same base BC with it and in the same
parallels. [I. 37]

But ABC is equal to DBC;

therefore DBC is also equal to EBC, [C.N. 1] the greater to the less: which is impossible.

Therefore AE is not parallel to BC.

Similarly we can prove that neither is any other straight line except AD;

therefore AD is parallel to BC.

Therefore etc.


Q. E. D.

644


[Proposition 40.

Equal triangles which are on equal bases and on the same side are also in the same parallels.

Let ABC, CDE be equal triangles on equal bases BC, CE and on the same side.

I say that they are also in the same parallels.

For let AD be joined; I say that AD is parallel to BE.

For, if not, let AF be drawn through A parallel to BE [I. 31], and let FE be joined.

Therefore the triangle ABC is equal to the triangle FCE; for they are on equal bases BC, CE and in the same parallels BE, AF. [I. 38]

But the triangle ABC is equal to the triangle DCE;

therefore the triangle DCE is also equal to the triangle FCE, [C.N. 1] the greater to the less: which is impossible. Therefore AF is not parallel to BE.

Similarly we can prove that neither is any other straight line except AD;

therefore AD is parallel to BE.

Therefore etc. Q. E. D.]


Proposition 41.


If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle.


For let the parallelogram ABCD have the same base BC with the triangle EBC, and let it be in the same parallels BC, AE;

I say that the parallelogram ABCD is double of the triangle BEC.

For let AC be joined.

Then the triangle ABC is equal to the triangle EBC; for it is on the same base BC with it and in the same parallels BC, AE. [I. 37]

But the parallelogram ABCD is double of the triangle ABC;

for the diameter AC bisects it; [I. 34]
so that the parallelogram ABCD is also double of the triangle EBC.

Therefore etc.


Q. E. D.


Proposition 42.


To construct, in a given rectilineal angle, a parallelogram equal to a given triangle.


Let ABC be the given triangle, and D the given rectilineal angle; thus it is required to construct in the rectilineal angle D a parallelogram equal to the triangle ABC.

Let BC be bisected at E, and let AE be joined; on the straight line EC, and at the point E on it, let the angle CEF be constructed equal to the angle D; [I. 23] through A let AG be drawn parallel to EC, and [I. 31] through C let CG be drawn parallel to EF.

Then FECG is a parallelogram.

And, since BE is equal to EC,

the triangle ABE is also equal to the triangle AEC, for they are on equal bases BE, EC and in the same parallels BC, AG; [I. 38] therefore the triangle ABC is double of the triangle AEC.

But the parallelogram FECG is also double of the triangle AEC, for it has the same base with it and is in the same parallels with it; [I. 41]

therefore the parallelogram FECG is equal to the triangle ABC.

And it has the angle CEF equal to the given angle D.

Therefore the parallelogram FECG has been constructed equal to the given triangle ABC, in the angle CEF which is equal to D. Q. E. F.


Proposition 43.


In any parallelogram the complements of the parallelograms about the diameter are equal to one another.


Let ABCD be a parallelogram, and AC its diameter; and about AC let EH, FG be parallelograms, and BK, KD
the so-called complements;

I say that the complement BK is equal to the complement KD.

For, since ABCD is a parallelogram, and AC its diameter,

the triangle ABC is equal to the triangle ACD. [I. 34]

Again, since EH is a parallelogram, and AK is its diameter,

the triangle AEK is equal to the triangle AHK.

For the same reason
the triangle KFC is also equal to KGC.

Now, since the triangle AEK is equal to the triangle AHK,

and KFC to KGC,

the triangle AEK together with KGC is equal to the triangle AHK together with KFC. [C.N. 2]

And the whole triangle ABC is also equal to the whole ADC; therefore the complement BK which remains is equal to the
complement KD which remains. [C.N. 3]

Therefore etc.


Q. E. D.

645 646


Proposition 44.


To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle.


Let AB be the given straight line, C the given triangle and D the given rectilineal angle;
thus it is required to apply to the given straight line AB, in an angle equal to the angle D, a parallelogram equal to the given triangle C.

Let the parallelogram BEFG be constructed equal to the triangle C, in the angle EBG which is equal to D [I. 42];
let it be placed so that BE is in a straight line with AB; letFG be drawn through to H, and let AH be drawn through A parallel to either BG or EF. [I. 31]

Let HB be joined.

Then, since the straight line HF falls upon the parallels
AH, EF,

the angles AHF, HFE are equal to two right angles. [I. 29]
Therefore the angles BHG, GFE are less than two right angles; and straight lines produced indefinitely from angles less than
two right angles meet; [Post. 5]
therefore HB, FE, when produced, will meet.

Let them be produced and meet at K; through the point K let KL be drawn parallel to either EA or FH, [I. 31] and let HA, GB be produced to the points L, M.

Then HLKF is a parallelogram, HK is its diameter, and AG, ME are parallelograms. and LB, BF the so-called complements, about HK;

therefore LB is equal to BF. [I. 43]

But BF is equal to the triangle C;

therefore LB is also equal to C. [C.N. 1]

And, since the angle GBE is equal to the angle ABM, [I. 15]

while the angle GBE is equal to D, the angle ABM is also equal to the angle D.

Therefore the parallelogram LB equal to the given triangle
C has been applied to the given straight line AB, in the angle ABM which is equal to D.


Q. E. F.

647


Proposition 45.


To construct, in a given rectilineal angle, a parallelogram equal to a given rectilineal figure.


Let ABCD be the given rectilineal figure and E the given rectilineal angle;
thus it is required to construct, in the given angle E, a parallelogram equal to the rectilineal figure ABCD.

Let DB be joined, and let the parallelogram FH be constructed equal to the triangle ABD, in the angle HKF which is equal to E; [I. 42]
let the parallelogram GM equal to the triangle DBC be applied to the straight line GH, in the angle GHM which is equal to E. [I. 44]

Then, since the angle E is equal to each of the angles HKF, GHM,

the angle HKF is also equal to the angle GHM. [C.N. 1]

Let the angle KHG be added to each; therefore the angles FKH, KHG are equal to the angles KHG, GHM.

But the angles FKH, KHG are equal to two right angles; [I. 29]
therefore the angles KHG, GHM are also equal to two right angles.

Thus, with a straight line GH, and at the point H on it, two straight lines KH, HM not lying on the same side make the adjacent angles equal to two right angles;

therefore KH is in a straight line with HM. [I. 14]

And, since the straight line HG falls upon the parallels KM, FG, the alternate angles MHG, HGF are equal to one another. [I. 29]

Let the angle HGL be added to each;
therefore the angles MHG, HGL are equal to the angles HGF, HGL. [C.N. 2]

But the angles MHG, HGL are equal to two right angles; [I. 29] therefore the angles HGF, HGL are also equal to two right angles. [C.N. 1]

Therefore FG is in a straight line with GL. [I. 14]

And, since FK is equal and parallel to HG, [I. 34]

and HG to ML also,
KF is also equal and parallel to ML; [C.N. 1; I. 30] and the straight lines KM, FL join them (at their extremities);
therefore KM, FL are also equal and parallel. [I. 33]
Therefore KFLM is a parallelogram.

And, since the triangle ABD is equal to the parallelogram FH,

and DBC to GM,

the whole rectilineal figure ABCD is equal to the whole parallelogram KFLM.

Therefore the parallelogram KFLM has been constructed equal to the given rectilineal figure ABCD, in the angle FKM which is equal to the given angle E.


Q. E. F.

648


Proposition 46.


On a given straight line to describe a square.


Let AB be the given straight line; thus it is required to describe a square on the straight line AB.

Let AC be drawn at right angles to the straight line AB from the point A on it [I. 11], and let AD be made equal to AB; through the point D let DE be drawn
parallel to AB, and through the point B let BE be drawn parallel to AD. [I. 31]

Therefore ADEB is a parallelogram;

therefore AB is equal to DE, and AD to BE. [I. 34]

But AB is equal to AD;

therefore the four straight lines BA, AD, DE, EB are equal to one another;
therefore the parallelogram ADEB is equilateral.

I say next that it is also right-angled.

For, since the straight line AD falls upon the parallels
AB, DE,

the angles BAD, ADE are equal to two right angles. [I. 29]

But the angle BAD is right;

therefore the angle ADE is also right.

And in parallelogrammic areas the opposite sides and
angles are equal to one another; [I. 34]

therefore each of the opposite angles ABE, BED is also right. Therefore ADEB is right-angled.

And it was also proved equilateral.

Therefore it is a square; and it is described on the straight line AB.


Q. E. F.

649


Proposition 47.


In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.


Let ABC be a right-angled triangle having the angle
BAC right;

I say that the square on BC is equal to the squares on BA, AC.

For let there be described on BC the square BDEC,
and on BA, AC the squares GB, HC; [I. 46] through A let AL be drawn parallel to either BD or CE, and let AD, FC be joined.

Then, since each of the angles BAC, BAG is right, it follows that with a straight line BA, and at the point A on it, the two straight lines
AC, AG not lying on the same side make the adjacent angles equal to two right angles;

therefore CA is in a straight line with AG. [I. 14]

For the same reason

BA is also in a straight line with AH.

And, since the angle DBC is equal to the angle FBA: for each is right: let the angle ABC be added to each;

therefore the whole angle DBA is equal to the whole angle FBC. [C.N. 2]

And, since DB is equal to BC, and FB to BA, the two sides AB, BD are equal to the two sides FB, BC respectively,

and the angle ABD is equal to the angle FBC; therefore the base AD is equal to the base FC, and the triangle ABD is equal to the triangle FBC. [I. 4]

Now the parallelogram BL is double of the triangle ABD, for they have the same base BD and are in the same parallels
BD, AL. [I. 41]

And the square GB is double of the triangle FBC, for they again have the same base FB and are in the same parallels FB, GC. [I. 41]

[But the doubles of equals are equal to one another.]

Therefore the parallelogram BL is also equal to the square GB.

Similarly, if AE, BK be joined, the parallelogram CL can also be proved equal to the square HC;

therefore the whole square BDEC is equal to the two squares GB, HC. [C.N. 2]

And the square BDEC is described on BC,

and the squares GB, HC on BA, AC.

Therefore the square on the side BC is equal to the
squares on the sides BA, AC.

Therefore etc.


Q. E. D.

650 651 652 653


Proposition 48.


If in a triangle the square on one of the sides be equal to the squares on the remaining two sides of the triangle, the angle contained by the remaining two sides of the triangle is right.


For in the triangle ABC let the square on one side BC be equal to the squares on the sides BA, AC;

I say that the angle BAC is right.

For let AD be drawn from the point A at right angles to the straight line AC, let AD be made equal to BA, and let DC be joined.

Since DA is equal to AB, the square on DA is also equal to the square on AB.

Let the square on AC be added to each;

therefore the squares on DA, AC are equal to the squares on BA, AC.

But the square on DC is equal to the squares on DA, AC, for the angle DAC is right; [I. 47] and the square on BC is equal to the squares on BA, AC, for this is the hypothesis;

therefore the square on DC is equal to the square on BC, so that the side DC is also equal to BC.

And, since DA is equal to AB, and AC is common,

the two sides DA, AC are equal to the two sides BA, AC;
and the base DC is equal to the base BC;
therefore the angle DAC is equal to the angle BAC. [I. 8]
But the angle DAC is right;
therefore the angle BAC is also right.

Therefore etc.


Q. E. D.

BOOK II.


DEFINITIONS.


1

Any rectangular parallelogram is said to be contained by the two straight lines containing the right angle.


2

And in any parallelogrammic area let any one whatever of the parallelograms about its diameter with the two complements be called a gnomon.


BOOK II. PROPOSITIONS.


Proposition 1.

If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.

Let A, BC be two straight lines, and let BC be cut at random at the points D, E; I say that the rectangle contained by A, BC is equal to the rectangle contained by A, BD, that contained by A, DE and
that contained by A, EC.

For let BF be drawn from B at right angles to BC; [I. 11] let BG be made equal to A, [I. 3] through G let GH be drawn
parallel to BC, [I. 31] and through D, E, C let DK, EL, CH be drawn parallel to BG.

Then BH is equal to BK, DL, EH.

Now BH is the rectangle A, BC, for it is contained by GB, BC, and BG is equal to A;

BK is the rectangle A, BD, for it is contained by GB, BD, and BG is equal to A;

and DL is the rectangle A, DE, for DK, that is BG [I. 34], is equal to A.

Similarly also EH is the rectangle A, EC.

Therefore the rectangle A, BC is equal to the rectangle A, BD, the rectangle A, DE and the rectangle A, EC.

Therefore etc. Q. E. D. 654


Proposition 2.

If a straight line be cut at random, the rectangle contained by the whole and both of the segments is equal to the square on the whole.

For let the straight line AB be cut at random at the point C; I say that the rectangle contained by AB, BC together with the rectangle contained by BA, AC is equal to the square on AB.

For let the square ADEB be described on AB [I. 46], and let CF be drawn through C parallel to either AD or BE. [I. 31]

Then AE is equal to AF, CE.

Now AE is the square on AB;

AF is the rectangle contained by BA, AC, for it is contained by DA, AC, and AD is equal to AB;

and CE is the rectangle AB, BC, for BE is equal to AB.

Therefore the rectangle BA, AC together with the rectangle AB, BC is equal to the square on AB.

Therefore etc. Q. E. D.


Proposition 3.

If a straight line be cut at random, the rectangle contained by the whole and one of the segments is equal to the rectangle contained by the segments and the square on the aforesaid segment.

For let the straight line AB be cut at random at C; I say that the rectangle contained by AB, BC is equal to the rectangle contained by AC, CB together with the square on BC.

For let the square CDEB be described on CB; [I. 46] let ED be drawn through to F, and through A let AF be drawn parallel to either CD or BE. [I. 31]

Then AE is equal to AD, CE.

Now AE is the rectangle contained by AB, BC, for it is contained by AB, BE, and BE is equal to BC;

AD is the rectangle AC, CB, for DC is equal to CB;

and DB is the square on CB
. Therefore the rectangle contained by AB, BC is equal to the rectangle contained by AC, CB together with the square on BC.

Therefore etc. Q. E. D.


Proposition 4.

If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.

For let the straight line AB be cut at random at C;
I say that the square on AB is equal to the squares on AC, CB and twice the rectangle contained by AC, CB.

For let the square ADEB be described on AB, [I. 46]
let BD be joined; through C let CF be drawn parallel to either AD or EB, and through G let HK be drawn parallel to either AB or DE. [I. 31]

Then, since CF is parallel to AD, and BD has fallen on them, the exterior angle CGB is equal to the interior and opposite angle ADB. [I. 29]

But the angle ADB is equal to the angle ABD,

since the side BA is also equal to AD; [I. 5]
therefore the angle CGB is also equal to the angle GBC, so that the side BC is also equal to the side CG. [I. 6]

But CB is equal to GK, and CG to KB; [I. 34]

therefore GK is also equal to KB; therefore CGKB is equilateral.

I say next that it is also right-angled.

For, since CG is parallel to BK,

the angles KBC, GCB are equal to two right angles. [I. 29]

But the angle KBC is right;

therefore the angle BCG is also right, so that the opposite angles CGK, GKB are also right. [I. 34]

Therefore CGKB is right-angled; and it was also proved equilateral;

therefore it is a square; and it is described on CB.

For the same reason

HF is also a square; and it is described on HG, that is AC. [I. 34]

Therefore the squares HF, KC are the squares on AC, CB.

Now, since AG is equal to GE, and AG is the rectangle AC, CB, for GC is equal to CB,

therefore GE is also equal to the rectangle AC, CB.

Therefore AG, GE are equal to twice the rectangle AC, CB.

But the squares HF, CK are also the squares on AC, CB; therefore the four areas HF, CK, AG, GE are equal to the squares on AC, CB and twice the rectangle contained by AC, CB.

But HF, CK, AG, GE are the whole ADEB,
which is the square on AB.

Therefore the square on AB is equal to the squares on AC, CB and twice the rectangle contained by AC, CB.

Therefore etc. Q. E. D. 655 656 657


Proposition 5.

If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half.

For let a straight line AB be cut into equal segments at C and into unequal segments at D; I say that the rectangle contained by AD, DB together with the square on CD is equal to the square on CB.

For let the square CEFB be described on CB, [I. 46] and let BE be joined; through D let DG be drawn parallel to either CE or BF, through H again let KM be drawn parallel to either AB or EF, and again through A let AK be drawn parallel to either CL or BM. [I. 31]

Then, since the complement CH is equal to the complement HF, [I. 43] let DM be added to each;

therefore the whole CM is equal to the whole DF.

But CM is equal to AL,

since AC is also equal to CB; [I. 36] therefore AL is also equal to DF.
Let CH be added to each;
therefore the whole AH is equal to the gnomon NOP.

But AH is the rectangle AD, DB, for DH is equal to DB,

therefore the gnomon NOP is also equal to the rectangle AD, DB.

Let LG, which is equal to the square on CD, be added to each;

therefore the gnomon NOP and LG are equal to the rectangle contained by AD, DB and the square on CD.

But the gnomon NOP and LG are the whole square CEFB, which is described on CB;

therefore the rectangle contained by AD, DB together with the square on CD is equal to the square on CB.

Therefore etc. Q. E. D. 658


Proposition 6.

If a straight line be bisected and a straight line be added to it in a straight line, the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half is equal to the square on the straight line made up of the half and the added straight line.

For let a straight line AB be bisected at the point C, and let a straight line BD be added to it in a straight line;

I say that the rectangle contained by AD, DB together with the square on CB is equal to the square on CD.

For let the square CEFD be described on CD, [I. 46] and let DE be joined; through the point B let BG be drawn parallel to either EC or DF, through the point H let KM be drawn parallel to either AB or EF, and further through A let AK be drawn parallel to either CL or DM. [I. 31]

Then, since AC is equal to CB, AL is also equal to CH. [I. 36]
But CH is equal to HF. [I. 43]
Therefore AL is also equal to HF.

Let CM be added to each;

therefore the whole AM is equal to the gnomon NOP.

But AM is the rectangle AD, DB,

for DM is equal to DB; therefore the gnomon NOP is also equal to the rectangle AD, DB.

Let LG, which is equal to the square on BC, be added to each;

therefore the rectangle contained by AD, DB together with the square on CB is equal to the gnomon NOP and LG.

But the gnomon NOP and LG are the whole square CEFD, which is described on CD;

therefore the rectangle contained by AD, DB together with the square on CB is equal to the square on CD.

Therefore etc. Q. E. D.


Proposition 7.

If a straight line be cut at random, the square on the whole and that on one of the segments both together are equal to twice the rectangle contained by the whole and the said segment and the square on the remaining segment.

For let a straight line AB be cut at random at the point C;

I say that the squares on AB, BC are equal to twice the rectangle contained by AB, BC and the square on CA.

For let the square ADEB be described on AB, [I. 46] and let the figure be drawn.

Then, since AG is equal to GE, [I. 43] let CF be added to each;

therefore the whole AF is equal to the whole CE.

Therefore AF, CE are double of AF.

But AF, CE are the gnomon KLM and the square CF; therefore the gnomon KLM and the square CF are double of AF.

But twice the rectangle AB, BC is also double of AF; for BF is equal to BC; therefore the gnomon KLM and the square CF are equal to twice the rectangle AB, BC.

Let DG, which is the square on AC, be added to each; therefore the gnomon KLM and the squares BG, GD are equal to twice the rectangle contained by AB, BC and the square on AC.

But the gnomon KLM and the squares BG, GD are the whole ADEB and CF,

which are squares described on AB, BC;
therefore the squares on AB, BC are equal to twice the rectangle contained by AB, BC together with the square on AC.

Therefore etc. Q. E. D.


Proposition 8.

If a straight line be cut at random, four times the rectangle contained by the whole and one of the segments together with the square on the remaining segment is equal to the square described on the whole and the aforesaid segment as on one straight line.

For let a straight line AB be cut at random at the point C;

I say that four times the rectangle contained by AB, BC together with the square on AC is equal to the square described on AB, BC as on one straight line.

For let [the straight line] BD be produced in a straight line [with AB], and let BD be made equal to CB; let the square AEFD be described on AD, and let the figure be drawn double.

Then, since CB is equal to BD, while CB is equal to GK, and BD to KN, therefore GK is also equal to KN.

For the same reason

QR is also equal to RP.

And, since BC is equal to BD, and GK to KN, therefore CK is also equal to KD, and GR to RN. [I. 36]

But CK is equal to RN, for they are complements of the parallelogram CP; [I. 43] therefore KD is also equal to GR; therefore the four areas DK, CK, GR, RN are equal to one another.

Therefore the four are quadruple of CK.

Again, since CB is equal to BD, while BD is equal to BK, that is CG, and CB is equal to GK, that is GQ,

therefore CG is also equal to GQ.

And, since CG is equal to GQ, and QR to RP,

AG is also equal to MQ, and QL to RF. [I. 36]

But MQ is equal to QL, for they are complements of the parallelogram ML; [I. 43]

therefore AG is also equal to RF;
therefore the four areas AG, MQ, QL, RF are equal to one another.
Therefore the four are quadruple of AG.
But the four areas CK, KD, GR, RN were proved to be quadruple of CK;
therefore the eight areas, which contain the gnomon STU, are quadruple of AK.

Now, since AK is the rectangle AB, BD, for BK is equal to BD, therefore four times the rectangle AB, BD is quadruple of AK.

But the gnomon STU was also proved to be quadruple of AK;

therefore four times the rectangle AB, BD is equal to the gnomon STU.

Let OH, which is equal to the square on AC, be added to each;

therefore four times the rectangle AB, BD together with the square on AC is equal to the gnomon STU and OH.

But the gnomon STU and OH are the whole square AEFD,

which is described on AD.
therefore four times the rectangle AB, BD together with the square on AC is equal to the square on AD

But BD is equal to BC; therefore four times the rectangle contained by AB, BC together with the square on AC is equal to the square on AD, that is to the square described on AB and BC as on one straight line.

Therefore etc. Q. E. D.


Proposition 9.

If a straight line be cut into equal and unequal segments, the squares on the unequal segments of the whole are double of the square on the half and of the square on the straight line between the points of section.

For let a straight line AB be cut into equal segments at C, and into unequal segments at D;

I say that the squares on AD, DB are double of the squares on AC, CD.

For let CE be drawn from C at right angles to AB, and let it be made equal to either AC or CB; let EA, EB be joined, let DF be drawn through D parallel to EC, and FG through F parallel to AB, and let AF be joined.

Then, since AC is equal to CE,

the angle EAC is also equal to the angle AEC.

And, since the angle at C is right,

the remaining angles EAC, AEC are equal to one right angle. [I. 32]

And they are equal;

therefore each of the angles CEA, CAE is half a right angle.

For the same reason

each of the angles CEB, EBC is also half a right angle; therefore the whole angle AEB is right.

And, since the angle GEF is half a right angle, and the angle EGF is right, for it is equal to the interior and opposite angle ECB, [I. 29]

the remaining angle EFG is half a right angle; [I. 32] therefore the angle GEF is equal to the angle EFG, so that the side EG is also equal to GF. [I. 6]

Again, since the angle at B is half a right angle, and the angle FDB is right, for it is again equal to the interior and opposite angle ECB, [I. 29]

the remaining angle BFD is half a right angle; [I. 32] therefore the angle at B is equal to the angle DFB, so that the side FD is also equal to the side DB. [I. 6]

Now, since AC is equal to CE,

the square on AC is also equal to the square on CE; therefore the squares on AC, CE are double of the square on AC.

But the square on EA is equal to the squares on AC, CE, for the angle ACE is right; [I. 47]

therefore the square on EA is double of the square on AC.

Again, since EG is equal to GF,

the square on EG is also equal to the square on GF; therefore the squares on EG, GF are double of the square on GF.

But the square on EF is equal to the squares on EG, GF;

therefore the square on EF is double of the square on GF.

But GF is equal to CD; [I. 34] therefore the square on EF is double of the square on CD.

But the square on EA is also double of the square on AC;

therefore the squares on AE, EF are double of the squares on AC, CD.

And the square on AF is equal to the squares on AE, EF, for the angle AEF is right; [I. 47] therefore the square on AF is double of the squares on AC, CD.

But the squares on AD, DF are equal to the square on AF, for the angle at D is right; [I. 47] therefore the squares on AD, DF are double of the squares on AC, CD.

And DF is equal to DB; therefore the squares on AD, DB are double of the squares on AC, CD.

Therefore etc. Q. E. D.


Proposition 10.

If a straight line be bisected, and a straight line be added to it in a straight line, the square on the whole with the added straight line and the square on the added straight line both together are double of the square on the half and of the square described on the straight line made up of the half and the added straight line as on one straight line.

For let a straight line AB be bisected at C, and let a straight line BD be added to it in a straight line;

I say that the squares on AD, DB are double of the squares on AC, CD.

For let CE be drawn from the point C at right angles to AB [I. 11], and let it be made equal to either AC or CB [I. 3]; let EA, EB be joined; through E let EF be drawn parallel to AD, and through D let FD be drawn parallel to CE. [I. 31]

Then, since a straight line EF falls on the parallel straight lines EC, FD,

the angles CEF, EFD are equal to two right angles; [I. 29] therefore the angles FEB, EFD are less than two right angles.

But straight lines produced from angles less than two right angles meet; [I. Post. 5]

therefore EB, FD, if produced in the direction B, D, will meet.

Let them be produced and meet at G, and let AG be joined.

Then, since AC is equal to CE, the angle EAC is also equal to the angle AEC; [I. 5] and the angle at C is right;

therefore each of the angles EAC, AEC is half a right angle. [I. 32]

For the same reason

each of the angles CEB, EBC is also half a right angle; therefore the angle AEB is right.

And, since the angle EBC is half a right angle, the angle DBG is also half a right angle. [I. 15]

But the angle BDG is also right,
for it is equal to the angle DCE, they being alternate; [I. 29]
therefore the remaining angle DGB is half a right angle; [I. 32]
therefore the angle DGB is equal to the angle DBG,
so that the side BD is also equal to the side GD. [I. 6]

Again, since the angle EGF is half a right angle, and the angle at F is right, for it is equal to the opposite angle, the angle at C, [I. 34]

the remaining angle FEG is half a right angle; [I. 32] therefore the angle EGF is equal to the angle FEG, so that the side GF is also equal to the side EF. [I. 6]

Now, since the square on EC is equal to the square on CA, the squares on EC, CA are double of the square on CA.

But the square on EA is equal to the squares on EC, CA; [I. 47] therefore the square on EA is double of the square on AC. [C. N. 1]

Again, since FG is equal to EF, the square on FG is also equal to the square on FE; therefore the squares on GF, FE are double of the square on EF.

But the square on EG is equal to the squares on GF, FE; [I. 47] therefore the square on EG is double of the square on EF.

And EF is equal to CD; [I. 34]

therefore the square on EG is double of the square on CD.
But the square on EA was also proved double of the square on AC; therefore the squares on AE, EG are double of the squares on AC, CD.

And the square on AG is equal to the squares on AE, EG; [I. 47] therefore the square on AG is double of the squares on AC, CD. But the squares on AD, DG are equal to the square on AG; [I. 47] therefore the squares on AD, DG are double of the squares on AC, CD.

And DG is equal to DB; therefore the squares on AD, DB are double of the squares on AC, CD.

Therefore etc. Q. E. D.


Proposition 11.

To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment.

Let AB be the given straight line; thus it is required to cut AB so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment.

For let the square ABDC be described on AB; [I. 46] let AC be bisected at the point E, and let BE be joined; let CA be drawn through to F, and let EF be made equal to BE; let the square FH be described on AF, and let GH be drawn through to K.

I say that AB has been cut at H so as to make the rectangle contained by AB, BH equal to the square on AH.

For, since the straight line AC has been bisected at E, and FA is added to it,

the rectangle contained by CF, FA together with the square on AE is equal to the square on EF. [II. 6]

But EF is equal to EB;

therefore the rectangle CF, FA together with the square on AE is equal to the square on EB.

But the squares on BA, AE are equal to the square on EB, for the angle at A is right; [I. 47]

therefore the rectangle CF, FA together with the square on AE is equal to the squares on BA, AE.

Let the square on AE be subtracted from each;

therefore the rectangle CF, FA which remains is equal to the square on AB.

Now the rectangle CF, FA is FK, for AF is equal to FG; and the square on AB is AD;

therefore FK is equal to AD.

Let AK be subtracted from each;

therefore FH which remains is equal to HD.

And HD is the rectangle AB, BH, for AB is equal to BD; and FH is the square on AH;

therefore the rectangle contained by AB, BH is equal to the square on HA. therefore the given straight line AB has been cut at H so as to make the rectangle contained by AB, BH equal to the square on HA. Q. E. F.


Proposition 12.

In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle.

Let ABC be an obtuse-angled triangle having the angle BAC obtuse, and let BD be drawn from the point B perpendicular to CA produced;

I say that the square on BC is greater than the squares on BA, AC by twice the rectangle contained by CA, AD.

For, since the straight line CD has been cut at random at the point A, the square on DC is equal to the squares on CA, AD and twice the rectangle contained by CA, AD. [II. 4]

Let the square on DB be added to each; therefore the squares on CD, DB are equal to the squares on CA, AD, DB and twice the rectangle CA, AD.

But the square on CB is equal to the squares on CD, DB, for the angle at D is right; [I. 47]

and the square on AB is equal to the squares on AD, DB; [I. 47]
therefore the square on CB is equal to the squares on CA, AB and twice the rectangle contained by CA, AD;
so that the square on CB is greater than the squares on CA, AB by twice the rectangle contained by CA, AD.

Therefore etc. Q. E. D.


Proposition 13.

In acute-angled triangles the square on the side subtending the acute angle is less than the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acutc angle.

Let ABC be an acute-angled triangle having the angle at B acute, and let AD be drawn from the point A perpendicular to BC;

I say that the square on AC is less than the squares on CB, BA by twice the rectangle contained by CB, BD.

For, since the straight line CB has been cut at random at D,

the squares on CB, BD are equal to twice the rectangle contained by CB, BD and the square on DC. [II. 7]

Let the square on DA be added to each; therefore the squares on CB, BD, DA are equal to twice the rectangle contained by CB, BD and the squares on AD, DC.

But the square on AB is equal to the squares on BD, DA, for the angle at D is right; [I. 47] and the square on AC is equal to the squares on AD, DC; therefore the squares on CB, BA are equal to the square on AC and twice the rectangle CB, BD,

so that the square on AC alone is less than the squares on CB, BA by twice the rectangle contained by CB, BD.

Therefore etc. Q. E. D.


Proposition 14.

To construct a square equal to a given rectilineal figure.

Let A be the given rectilineal figure; thus it is required to construct a square equal to the rectilineal figure A.

For let there be constructed the rectangular parallelogram BD equal to the rectilineal figure A. [I. 45]

Then, if BE is equal to ED, that which was enjoined will have been done; for a square BD has been constructed equal to the rectilineal figure A.

But, if not, one of the straight lines BE, ED is greater.

Let BE be greater, and let it be produced to F; let EF be made equal to ED, and let BF be bisected at G.

With centre G and distance one of the straight lines GB, GF let the semicircle BHF be described; let DE be produced
to H, and let GH be joined.

Then, since the straight line BF has been cut into equal segments at G, and into unequal segments at E,

the rectangle contained by BE, EF together with the square on EG is equal to the square on GF. [II. 5]

But GF is equal to GH; therefore the rectangle BE, EF together with the square on GE is equal to the square on GH.

But the squares on HE, EG are equal to the square on GH; [I. 47]
therefore the rectangle BE, EF together with the square on GE is equal to the squares on HE, EG.

Let the square on GE be subtracted from each;

therefore the rectangle contained by BE, EF which remains is equal to the square on EH.

But the rectangle BE, EF is BD, for EF is equal to ED;

therefore the parallelogram BD is equal to the square on HE.

And BD is equal to the rectilineal figure A.

Therefore the rectilineal figure A is also equal to the square
which can be described on EH.

Therefore a square, namely that which can be described on EH, has been constructed equal to the given rectilineal figure A. Q. E. F.

659 660

1 Proclus, ed. Friedlein, p. 68, 6-20.

2 The word γέγονε must apparently mean “flourished,” as Heiberg understands it (Litterargeschichtliche Studien über Euklid, 1882, p. 26), not “was born,” as Hankel took it : otherwise part of Proclus' argument would lose its cogency.

3 So Heiberg understands ἐπιβαλὼν τῷ πρώτῳ (sc. Πτολεμαίῳ). Friedlein's text has καὶ between ἐπιβαλὼν and τῷ πρώτῳ; and it is right to remark that another reading is καὶ ἐν τῷ πρώτῳ (without ἐπιβαλών) which has been translated “in his first book,” by which is understood On the Sphere and Cylinder I., where (1) in Prop. 2 are the words “let BC be made equal to D by the second (proposition) of the first of Euclid's (books),” and (2) in Prop. 6 the words “For these things are handed down in the Elements” (without the name of Euclid). Heiberg thinks the former passage is referred to, and that Proclus must therefore have had before him the words “by the second of the first of Euclid”: a fair proof that they are genuine, though in themselves they would be somewhat suspicious.

4 The same story is told in Stobaeus, Ecl. (II. p. 228, 30, ed. Wachsmuth) about Alexander and Menaechmus. Alexander is represented as having asked Menaechmus to teach him geometry concisely, but he replied : “O king, through the country there are royal roads and roads for common citizens, but in geometry there is one road for all.”

5 Proclus, p. 68, 20, καὶ τῇ προαιρέσει δὲ Πλατωνικός ἐστι καὶ τῇ φιλοσοφίᾳ ταύτῃ οὶκεῖος.

6 ibid. p. 71, 8.

7 ibid. p. 70, 19 sqq.

8 Pappus, VII. p. 678, 10-12, συσχολάσας τοῖς ὑπὸ Εὐκλείδου μαθηταῖς ἐν Ἀλεξανδρεί<*> πλεῖστον χρόνον, ὅθεν ἔσχε καὶ τὴν τοιαύτην ἔξιν οὐκ ἀμαθῆ.

9 Pappus, VII. pp. 676, 25-678, 6. Hultsch, it is true, brackets the whole passage pp. 676, 25-678, 15, but apparently on the ground of the diction only.

10 Stobaeus, l.c.

11 VIII. 12, ext. 1.

12 Letter to Fernandus Acuna, printed in Maurolycus, Historia Siciliae, fol. 21 r. (see Heiberg, Euklid-Studien, pp. 22-3, 25).

13 Preface to translation (Pisauri, 1572).

14 Diog. L. II. 106, p. 58 ed. Cobet.

15 Casiri, Bibliotheca Arabico-Hispana Escurialensis, I. p. 339. Casiri's source is alQifti (d. 1248), the author of the Ta'rīkh al-H<*>ukamā, a collection of biographies of philosophers, mathematicians, astronomers etc.

16 The Fihrist says “son of Naucrates, the son of Berenice (?)” (see Suter's translation in Abhandlungen zur Gesch. d. Math. VI. Heft, 1892, p. 16).

17 The same predilection made the Arabs describe Pythagoras as a pupil of the wise Salomo, Hipparchus as the exponent of Chaldaean philosophy or as the Chaldaean, Archimedes as an Egyptian etc. (H<*>ăjī Khalfa, Lexicon Bibliographicum, and Casiri).

18 The authorities for these statements quoted by Casiri and H<*>ājī Khalfa are al-Kindi's tract de instituto libri Euclidis (al-Kindī died about 873) and a commentary by Qād<*>īzāde ar-Rūmī (d. about 1440) on a book called Ashkāl at-ta' sīs (fundamental propositions) by Ashraf Shamsaddīn as-Samarqandī (c. 1276) consisting of elucidations of 35 propositions selected from the first books of Euclid. Nas<*>īraddīn likewise says that Euclid cut out two of 15 books of elements then existing and published the rest under his own name. According to Qād<*>īzāde the king heard that there was a celebrated geometer named Euclid at Tyre: Nas<*>īraddīn says that he sent for Euclid of T<*>ūs.

19 So says the Fihrist. Suter (op. cit. p. 49) thinks that the author of the Fihrist did not suppose Apollonius of Perga to be the writer of the Elements, as later Arabian authorities did, but that he distinguished another Apollonius whom he calls “a carpenter.” Suter's argument is based on the fact that the Fihrist's article on Apollonius (of Perga) says nothing of the Elements; and that it gives the three great mathematicians, Euclid, Archimedes and Apollonius, in the correct chronological order.

20 Heiberg's Euclid, vol. V. p. 6.

21 See, for example, Loria, Le scienze esatte nell' antica Grecia, 1914, pp. 245-268; 1. L. Heath, History of Greek Mathematics, 1921, I. pp. 421-446. Cf. Heiberg, Litterargeschichtliche Studien über Euklid, pp. 36-153; Euclidis opera omnia, ed. Heiberg and Menge, Vols. VI.—VIII.

22 Proclus, p. 70, 1-18.

23 Heiberg points out that Alexander Aphrodisiensis appears to allude to the work in his commentary on Aristotle's Sophistici Elenchi (fol. 25 b): “Not only those (ἔλεγχοι) which do not start from the principles of the science under which the problem is classed...but also those which do start from the proper principles of the science but in some respect admit a paralogism, e.g. the Pseudographemata of Euclid.” Tannery (Bull. des sciences math. et astr. 2^{e} Série, VI., 1882, I^{e\re} Partie, p. 147) conjectures that it may be from this treatise that the same commentator got his information about the quadratures of the circle by Antiphon and Bryson, to say nothing of the lunules of Hippocrates. I think however that there is an objection to this theory so far as regards Bryson; for Alexander distinctly says that Bryson's quadrature did not start from the proper principles of geometry, but from some principles more general.

24 Pappus, VII. p. 638.

25 Vol. VI. in the Teubner edition of Euclidis opera omnia by Heiberg and Menge. A translation of the Data is also included in Simson's Euclid (though naturally his text left much to be desired).

26 Proclus, p. 69, 4.

27 ibid. 144, 22-26.

28 Steinschneider places him in the 10th c. H. Suter (Bibliotheca Mathematica, IV_{3}, 1903, pp. 24, 27) identifies him with Abū (Bekr) Muh. b. 'Abdalbāqī al-Baġdādī, Qād|ī (Judge) of Māristān (circa 1070-1141), to whom he also attributes the Liber judei (? judicis) super decimum Euclidis translated by Gherard of Cremona.

29 De superficierum divisionibus liber Machometo Bagdadino adscriptus, nunc primum Ioannis Dee Londinensis et Federici Commandini Urbinatis opera in lucem editus, Pisauri, 1570, afterwards included in Gregory's Euclid (Oxford, 1703).

30 R. C. Archibald, Euclid's Book on the Division of Figures with a restoration based on Woepck's text and on the Practica geometriae of Leonardo Pisano, Cambridge, 1915, pp. 4-9.

31 Fournal Asiatique, 1851, p. 233 sqq.

32 L. F. Ofterdinger, Beiträge zur Wiederherstellung der Schrift des Euklides über die Theilung der Figuren, Ulm, 1853.

33 . There is a remarkable similarity between the propositions of Woepcke's text and those of Leonardo, suggesting that Leonardo may have had before him a translation (perhaps by Gherard of Cremona) of the Arabic tract.

34 Heiberg, Euklid-Studien, pp. 56-79, and Loria, op. cit., pp. 253-265.

35 Pappus, ed. Hultsch, VII. pp. 648-660. I put in square brackets the words bracketed by Hultsch.

36 I adopt Heiberg's reading of a comma here instead of a full stop.

37 Heiberg points out that Props. 5-9 of Archimedes' treatise On Spirals are porisms in this sense. To take Prop. 5 as an example, DBF is a tangent to a circle with centre K. It is then possible, says Archimedes, to draw a straight line KHF, meeting the circumference in H and the tangent in F, such that

where c is the circumference of any circle. To prove this he assumes the following construction. E being any straight line greater than c, he says: let KG be parallel to DF, “and let the line GH equal to E be placed verging to the point B.” Archimedes must of course have known how to effect this construction, which requires conics. But that it is possible requires very little argument, for if we draw any straight line BHG meeting the circle in H and KG in G, it is obvious that as G moves away from C, HG becomes greater and greater and may be made as great as we please. The “later writers” would no doubt have contented themselves with this consideration without actually constructing HG.

38 As Heiberg says, this translation is made certain by a preceding passage of Pappus (p. 648, 1-3) where he compares two enunciations, the latter of which “falls short of the former in hypothesis but goes beyond it in requirement.” E.g. the first enunciation requiring us, given three circles, to draw a circle touching all three, the second may require us, given only two circles (one less datum), to draw a circle touching them and of a given size (an extra requirement).

39 I translate Heiberg's reading with a full stop here followed by πρὸς ἀρχῇ δὲ ὅμως [πρὸς ἀρχὴνδεδομένον) Hultsch] τοῦ πρώτου βιβλίου....

40 The four straight lines are described in the text as (the sides) ὑπτίου παρυπτίου, i.e. sides of two sorts of quadrilaterals which Simson tries to explain (see p. 120 of the Index Graecitatis of Hultsch's edition of Pappus).

41 In other words (Chasles, p. 23; Loria, p. 256), if a triangle be so deformed that each of its sides turns about one of three points in a straight line, and two of its vertices lie on two straight lines given in position, the third vertex will also lie on a straight line.

42 Loria (p. 256, n. 3) gives the meaning of this as follows, pointing out that Simson was the discoverer of it: “If a complete n-lateral be deformed so that its sides respectively turn about n points on a straight line, and (n-1) of its nn-1)/2 vertices move on as many straight lines, the other (n-1)(n-2)/2 of its vertices likewise move on as many straight lines: but it is necessary that it should be impossible to form with the (n-1) vertices any triangle having for sides the sides of the polygon.”

43 Reading, with Heiberg, τοῦ βιβλίου [τοῦ ζ Hultsch].

44 Proclus, pp. 212, 14; 301, 22.

45 ibid. p. 212, 12. “The term porism is used of certam problems, like the Porisms written by Euclid.”

46 ibid. pp. 301, 25 sqq.

47 This was thus expressed by Chasles: “Le porisme est une proposition dans laquelle on demande de démontrer qu'une chose ou plusieurs choses sont données, qui, ainsi que l'une quelconque d'une infinité d'autres choses non données, mais dont chacune est avec des choses données dans une même relation, ont une certaine propriété commune, décrite dans la proposition.”

48 Commentary on Apollonius' Conics (vol. II. p. 180, ed. Heiberg).

49 Die Lehre von den Kegelschnitten im Altertum, chapter VIII.

50 Pappus, VII. p. 636.

51 ibid. VII. p. 1004.

52 Bulletin des sciences math. et astron., 2^{o} Série, VI. 149.

53 Further particulars will be found in The Works of Archimedes, pp. lxii—lxiv, and in Zeuthen, Die Lehre von den Kegelschnitten, p. 425 sqq.

54 Pappus, VII. pp. 1006-1014, and Hultsch's Appendix, pp. 1270-3.

55 For further details see The Works of Archimedes, pp. lxiv, lxv, and Zeuthen, l. c.

56 Aperçu historique, pp. 273-4.

57 Pappus, VII. p. 672.

58 For details of these propositions see my Apollonius of Perga, pp. xxxv, xxxvi.

59 Phaenomena, ed. Menge, p. 6: “If a cone or a cylinder be cut by a plane not parallel to the base, the section is a section of an acute-angled cone, which is like a shield (θυρεός).”

60 Euclidis opera omnia, vol. VIII., 1916, pp. 2-156.

61 Heiberg, Euklid-Studien, p. 46; Hultsch, Autolycus, p. XII; A. A. Björnbo, Studien über Menelaos' Sphärik (Abhandlungen zur Geschichte der mathematischen Wissenschaften, XIV. 1902), p. 56sqq.

62 Euclidis opera omnia, vol. VII. (1895).

63 Heiberg, Euclid's Optics, etc. p. l.

64 Proclus, p. 69, 3.

65 Both treatises edited by Jan in Musici Scriptores Graeci, 1895. pp. 113-166, 167-207, and by Menge in Euclidis opera omnia, vol. VIII., 1916, pp. 157-183, 185-223.

66 Comptes rendus de l' Acad. des inscriptions et belles-lettres, Paris, 1904, pp. 439-445. Cf. Bibliotheca Mathematica, vI3, 1905-6, p. 225, note I.

67 Heiberg, Euklid-Studien, pp. 52-55; Jan, Musici Scriptores Graeci, pp. 169-174.

68 H. Suter, Das Mathematiker-Verzeichniss im Fihrist in Abhandlungen zur Geschichte der Mathematik, VI., 1892, pp. 1-87 (see especially p. 17). Cf. Casiri, 1. 339, 340, and Gartz, De interpretibus et explanatoribus Euclidis Arabicis, 1823, pp. 4, 5.

69 Suter, op. cit. pp. 49, 50. Wenrich translated the word as “utilia.” Suter says that the nearest meaning of the Arabic word as of “porism” is use, gain (Nutzen, Gewinn), while a further meaning is explanation, observation, addition: a gain arising out of what has preceded (cf. Proclus' definition of the porism in the sense of a corollary).

70 Heiberg, Euklid-Studien, pp. 9, 10.

71 Suter, op. cit. p. 50.

72 Proclus, p. 84, 8.

73 i.e. we reach a certain height, use the platform so attained as a base on which to build another stage, then use that as a base and so on.

74 Proclus, p. 200, 10.

75 ibid. p. 432, 15.

76 ibid. p. 289, II; p. 328, 16.

77 Proclus, p. 305, 24; p. 346, 13.

78 ibid. p. 41, 10.

79 ibid. p. 196, 16; p. 323, 7; p. 429, 13.

80 Athenaeus, Deipno-Soph. iv., c. 75, p. 174 bc.

81 ibid. XI., c. 97, p. 497 bc.

82 Martin, Recherches sur la vie et les ouvrages d'Héron d'Alexandrie, Paris, 1854, p. 27.

83 Philo, Mechan. Synt., p. 50, 38, ed. Schöne.

84 Heronis Alexandrini opera, vol. v. (Teubner, 1914), p. IX.

85 Fuller details of the various arguments will be found in my History of Greek Mathematics, 1921, vol. II., pp. 298-306.

86 Das Mathematiker- Verzeichniss im Fihrist (tr. Suter), p. 16.

87 ibid. p. 22.

88 Codex Leidensis 399, 1. Euclidis Elementa ex interprctatione al-Hadschdschadschii cum commentariis al-Narizii. Five parts carrying the work to the end of Book iv. were issued in 1893, 1897, 1900, 1905 and 1910 respectively.

89 Anaritii in decem libros priores elementorum Euclidis commentarii ex interpretatione Gherardi Cremonensis...edidit Maximilianus Curtze (Teubner, Leipzig, 1899).

90 Cf. Proclus, 275, 7 εὶ δὲ λέγοι τις τόπον μὴ εὶδέναι..., 289, 18 λέγει οὖν τις ὄτι οὐκ ἔστι τόπος....

91 De Procli fontibus, Lugduni-Batavorum, 1900,

92 Proclus, pp. 297, 1-298, 10.

93 ibid. p. 352, 13, 14 and the pages preceding,

94 Fihrist (tr. Suter), p. 9, 10 and p. 45 (note 5).

95 Van Pesch, De Procli fontibus, pp. 129, 130. Heiberg assigned them as above in his Euklid-Studien (p. 160), but seems to have changed his view later. (See Besthorn-Heiberg. Codex Leidensis, p. 93, note 2.)

96 Van Pesch, op. cit. pp. 130-1.

97 Euclid's Data, ed. Menge, p. 262.

98 Fihrist (tr. Suter), p. 22.

99 Mémoirés présentés à l'académie des sciences, 1856, XIV. pp. 658-719.

100 Woepcke read the name of the author, in the title of the first book, as B. los (the dot representing a missing vowel). He quotes also from other MSS. (e.g. of the Ta)rīkh alHukamā and of the Fihrist) where he reads the name of the commentator as B. lis, B.n.s or B.l.s. Woepcke takes this author to be Valens, and thinks it possible that he may be the same as the astrologer Vettius Valens. This Heiberg (Euklid-Studien, pp. 169, 170) proves to be impossible, because, while one of the mss. quoted by Woepcke says that “B.n.s, le RoÛmi” (late-Greek) was later than Claudius Ptolemy and the Fihrist says “B.l.s, le RoÛmi” wrote a commentary on Ptolemy's Planisphaerium, Vettius Valens seems to have lived under Hadrian, and must therefore have been an eldercontemporary of Ptolemy. But Suter shows (Fihrist, p. 22 and p. 54, note 92) that Banos is only distinguished from Babos by the position of a certain dot, and Balos may also easily have arisen from an original Babos (there is no P in Arabic), so that Pappus must be the person meant. This is further confirmed by the fact that the Fihrist gives this author and Valens as the subjects of two separate paragraphs, attributing to the latter astrological works only.

101 Heiberg, Euklid-Studien, p. 173; Euclid's Data, edd. Menge, pp. 256, lii.

102 Proclus, pp. 189, 190.

103 ibid. p. 197, 6-10.

104 ibid. p. 198, 3-15.

105 Proclus, p. 429, 9-15.

106 Van Pesch, De Procli fontibus, p. 134 sqq.

107 Proclus, p. 272, 10.

108 Pappus, v. pp. 304-350; for Zenodorus' own treatise see Hultsch's Appendix, pp. 1189 —1211.

109 Proclus, pp. 396-8.

110 ibid. pp. 403-4.

111 ibid. pp. 236-7.

112 Pappus, 111. pp. 104-130.

113 Proclus, p. 328, 15.

114 Proclus, p. 165, 24; cf. pp. 328, 329.

115 See Pappus, ed. Hultsch, pp. 1154, 1206.

116 Fihrist (tr. Suter), p. 21.

117 An-Nairĩzĩ, ed. Besthorn-Heiberg, pp. 9-41, 119-133, ed. Curtze, pp. 1-37, 65-73. The Codex Leidensis, from which Besthorn and Heiberg's edition is taken, has unfortunately lost some leaves, so that there is a gap from Def. 1 to Def. 23 (parallels). The loss is, however, made good by Curtze's edition of the translation by Gherard of Cremona.

118 Proclus, p. 177, 21.

119 ibid. p. 176, 7.

120 Bibliotheca Mathematica, 113, 1900, pp. 9-11.

121 Zeitschrift für Math. u. Physik, XLIV., hist.-litt. Abth. p. 61.

122 The above argument seems to me quite insuperable. The other arguments of Tannery do not, however, carry conviction to my mind. I do not follow the reasoning based on Aganis' definition of an angle. It appears to me a pure assumption that Geminus would have seen that Posidonius' definition of parallels was not admissible. Nor does it seem to me to count for much that Proclus, while telling us that Geminus held that the postulate ought to be proved and warned the unwary against hastily concluding that two straight lines approaching one another must necessarily meet (cf. a curve and its asymptote), gives no hint that Geminus did try to prove the postulate. It may well be that Proclus omitted Geminus' “proof” (if he wrote one) because he preferred Ptolemy's attempt which he gives (pp. 365-7).

123 Proclus, p. 361, 21.

124 My task in this chapter is made easy by the appearance, in the nick of time, of the dissertation De Procli fontibus by J. G. van Pesch (Lugduni-Batavorum, Apud L. van Nifterik, mdcccc). The chapters dealing directly with the subject show a thorough acquaintance on the part of the author with all the literature bearing on it; he covers the whole field and he exercises a sound and sober judgment in forming his conclusions. The same cannot always be said of his only predecessor in the same inquiry, Tannery (in La Géométrie grecque, 1887), who often robs his speculations of much of their value through his proneness to run away with an idea; he does so in this case, basing most of his conclusions on an arbitrary and unwarranted assumption as to the significance of the words οἱ περί τινα (e.g. Η῞ρωνα, Ποσειδώνιον etc.) as used in Proclus.

125 Simplicius on Aristotle's Physics, ed. Diels, pp. 54-69.

126 ibid. p. 68, 32.

127 Cf. Martin, Recherches sur la vie et les ouvrages d'Héron d'Alexandrie, pp. 240-2.

128 Zeller calls him “Der Gelehrte, dem kein Feld damaligen Wissens verschlossen ist.”

129 Van Pesch observes that in his commentaries on the Tïmaeus (pp. 671-2) he speaks as no real mathematician could have spoken. In the passage referred to the question is whether the sun occupies a middle place among the planets. Proclus rejects the view of Hipparchus and Ptolemy because “ θεουργός” (sc. the Chaldean, says Zeller) thinks otherwise, “whom it is not lawful to disbelieve.” Martin says rather neatly, “Pour Proclus, les Éléments d'Euclide ont l'heureuse chance de n'e=tre contredits ni par les Oracles chaldaïques, ni par les spéculations des pythagoriciens anciens et nouveaux......”

130 Proclus, p. 84, 13.

131 ibid. p. 429, 12.

132 ibid. p. 31, 20.

133 ibid. p. 32, 2.

134 ibid. p. 27, 27 to 28. 7; cf. also p. 21, 25, pp. 46, 47.

135 Proclus, p. 210, 18.

136 ibid. p. 384, 2.

137 ibid. p. 272, 12.

138 ibid. p. 238, 12.

139 ibid. p. 298, 14.

140 Cf. p. 224, 15 (on 1. 2).

141 ibid. p. 84, 9.

142 ibid. p. 105.

143 ibid. p. 113.

144 ibid. p. 375, 8.

145 ibid. pp. 368-373.

146 Proclus, p. 280, 9; p. 282, 20; pp. 335, 336.

147 ibid. p. 272, 14.

148 ibid. p. 398, 18.

149 ibid. p. 423, 6.

150 Heiberg, Euklid-Studien, p. 165, note.

151 Proclus, p. 432, 9.

152 The words in the Greek are: εἰ μὲν δυνηθείημεν καὶ τοῖς λοιποῖς τὸν αὐτὸν τρόπον ἐξελθεῖν. For ἐξελθεῖν Heiberg would read ἐπεξλθεῖν

153 True, a Vatican Ms. has a collection of scholia on Books I. (extracts from the extant commentary of Proclus), II., V., VI., X. headed Εἰς τὰ Εὐκλείδου στοιχεῖα προλαμβανόμενα ἐκ τῶν Πρόκλου .σποράδην καὶ κατ̓ ἐπιτομήν. Heiberg holds that this title itself suggests that the authorship of Proclus was limited to the scholia on Book I.; for προλαμβανόμενα ἐκ τῶν Πρόκλου suits extracts from Proclus' prologues, but hardly scholia to later Books. Again, a certain scholium (Heiberg in Hermes XXXVIII., 1903, p. 341, No. 17) purports to quote words from the end of “a scholium of Proclus” on X. 9. The words quoted are from the scholium X. No. 62, one of the Scholia Vaticana. But none of the other, older, sources connect Proclus' name with X. No. 62; it is probable therefore that a Byzantine, who had in his Ms. of Euclid the collection of Schol. Vat. and knew that those on Book I. came from Proclus, himself attached Proclus' name to the others.

154 While one class of scnona (Schol. Vat.) have some better readings than our MSS. of Proclus have, and partly fill up the gaps at 1. 36, 37 and 1. 41-43, the other class (Schol. Vind.) derive from an inferior Proclus MS. which also had the same lacunae.

155 Knoche, Untersuchungen über des Proklus Diadochus Commentar zu Euklid's Elementen (1862), p. 11.

156 Proclus, p. 200, 10-13.

157 Instances of the application of this criterion will be found in the discussion of Proclus' indebtedness to the commentaries of Heron, Porphyry and Pappus.

158 Van Pesch attributes this converse and proof to Pappus, arguing from the fact that the proof is followed by a passage which, on comparison with Pappus' note on the postulate that all right angles are equal, he feels justified in assigning to Pappus. I doubt if the evidence is sufficient.

159 Van Pesch illustrates this by an objection refuted in the note on 1. 9, p. 273, 11 sqq. After using the above expression to introduce the objection, Proclus uses further on (p. 273, 25) the term “they say” (φασίν).

160 Proclus, p. 22, 15.

161 ibid. p. 56, 25.

162 ibid. p. 71, 18.

163 ibid. p. 105, 5.

164 ibid. p. 125, 8.

165 ibid. p. 200, 2.

166 ibid. p. 241, 19.

167 ibid. p. 352, 15.

168 ibid. p. 362, 15.

169 See the passages referred to by van Pesch (p. 70). The most glaring case is a passage (p. 21, 19) where he quotes Plotinus, using the expression “Plotinus says......” Comparison with Plotinus, Ennead. 1. 3, 3, shows that very few words are those of Plotinus himself; the rest represent Plotinus' views in Proclus' own language.

170 Proclus, P. 330, 19 sqq

171 See pp. 20 to 27 above.

172 Proclus, p. 68, 4-7.

173 De Procli fontibus, pp. 73-75.

174 See above, p. 29.

175 Simplicius, loc. cit., ed. Diels, p. 60, 27.

176 Simplicius, loc. cit., ed. Diels, p. 60, 27.

177 Archimedes, ed. Heiberg, vol. III. p. 228.

178 Proclus, p. 352, 14-18.

179 ibid. p. 299, 3.

180 ibid. p. 333, 5.

181 ibid. p. 379, 1-16.

182 ibid. p. 419, 15-18.

183 ibid. p. 157, 10, 11.

184 Cantor (Gesch. d. Math. 13, p. 221) points out the connexion between the reductio ad absurdum and the analytical method said to have been discovered by Plato. Proclus gives the proof by reductio ad absurdum to meet an imaginary critic who desires a mathematical proof; possibly Thales may have been satisfied with the argument in the same sentence which mentions Thales, “the cause of the bisection being the unswerving course of the straight line through the centre.”

185 Proclus, p. 211, 19-23.

186 ibid. p. 250, 20.

187 ibid. p. 283, 7-10.

188 ibid. pp. 304, 11-305, 3.

189 ibid. pp. 428, 7-429, 9.

190 ibid. p. 426, 6-9.

191 Diogenes Laertius, I. 24, p. 6, ed. Cobet.

192 Plutarch, non posse suaviter vivi secundum Epicurum, 11; Symp. VIII, 2.

193 Diog. Laert. VIII. 12, p. 207, ed. Cobet: Ἡνίκα Πυθαγόρης τὸ περικλεὲς εὕρετο γράμμα, κεῖν̓ ἐφ̓ ὄῳ κλεινὴν ἤγαγε βουθυσίην. See on this subject Tannery, La Géoinétrie grecque, p. 105.

194 Proclus, pp. 64-70.

195 The plural is well explained by Tannery, La Géoinétrie grecque, pp. 73, 74. No doubt the author of the summary tried to supplement Eudemus by means of any other histories which threw light on the subject. Thus e.g. the allusion (p. 64, 21) to the Nile recalls Herodotus. Cf. the expression in Proclus, p. 64, 19, παρὰ τῶν πολλῶν ίστόρηται.

196 Tannery, La Géométrie grecque, p. 75.

197 ibid. pp. 66-75.

198 ibid. p. 19.

199 Diog. Laertius, IX. 40, p. 237, ed. Cobet.

200 Proclus, p. 64, 8.

201 Proclus, p. 211, 19 sqq.; the passage is quoted above, p. 36.

202 ibid. p. 125, 8.

203 Tannery, La Géométrie grecque, p. 26.

204 Simplicius, ed. Diels, p. 69, 23.

205 ibid. p. 60, 29.

206 Cf. Simplicius, p. 63, 19 sqq.; p. 64, 25 sqq.; also Usener's note “de supplendis Hippocratis quas omisit Eudemus constructionibus” added to Diels' preface, pp. XXIII—XXVI.

207 Manitius, Gemini elementa astronomiae (Teubner, 1898), pp. 237-252; Tittel, art. “Geminos” in Pauly-Wissowa's Real-Encyclopädie der classischen Altertumswissenschaft, vol. VII.. 1910.

208 Pappus, ed. Hultsch, p. 1026, 9.

209 Apollonius, ed. Heiberg, vol. II. p. 170.

210 Tannery, La Géométrie grecque, pp. 18, 19.

211 Proclus, pp. 38, 1-42, 8.

212 Heron, ed. Hultsch, pp. 246, 16-249, 12.

213 Van Pesch, De Procli fontibus, pp. 97-113. The dissertation of Tittel is entitled De Gemini Stoici studiis mathematicis (1895).

214 Proclus, pp. 38, 1-42, 8, except the allusion in p. 41, 8-10, to Ctesibius and Heron and their pneumatic devices (θανματοποιϊκή), as regards which Proclus' authority may be Pappus (VIII. p. 1024, 24-27) who uses very similar expressions. Heron, even if not later than Geminus, could hardly have been included in a historical work by him. Perhaps Geminus may have referred to Ctesibius only, and Proclus may have inserted “and Heron” himself.

215 Proclus, pp. 103, 21-107, 10; pp. 111, 1-113, 3.

216 ibid. pp. 117, 14-120, 12, where perhaps in the passage pp. 117, 22-118, 23 we may have Geminus' own words.

217 ibid. pp. 176, 18-177, 25; perhaps also p. 175. The note ends with the words “These things too we have selected from Geminus' Φιλοκαλία for the elucidation of the matters in question.” Tannery (p. 27) takes these words coming at the end of the commentary on the definitions as referring to the whole of the portion of the commentary dealing with the definitions. Van Pesch properly regards them as only applying to the note on parallels. This seems to me clear from the use of the word too (τοσαῦτα καί).

218 Proclus, p. 176, 5-17.

219 ibid. pp. 178-182, 4; pp. 183, 14-184, 10; cf. p. 188, 3-11.

220 ibid. p. 185, 6-25.

221 ibid. p. 192, 5-29.

222 ibid. pp. 200, 21-202, 25.

223 Proclus, p. 251, 2-11.

224 ibid. pp. 277, 25-279, 11.

225 ibid. pp. 394, 11-395, 2 and p. 395, 13-21.

226 ibid. p. 395, 8-12.

227 ibid. pp. 33, 21-34, 1.

228 ibid. pp. 57, 9-58, 3.

229 ibid. pp. 72, 3-75, 4.

230 ibid. p. 77, 3-6.

231 ibid. pp. 77, 7-78, 13, and 79, 3-81, 4.

232 ibid. pp. 78, 13-79, 2.

233 ibid. pp. 102, 22-103, 18.

234 ibid. pp. 126, 7-127, 16.

235 ibid. pp. 159, 12-160, 9.

236 ibid. pp. 162, 27-164, 6.

237 ibid. p. 143, 5-11.

238 ibid. p. 168, 4-12.

239 ibid. p. 187, 19-27.

240 ibid. pp. 220, 7-222, 14; also p. 330, 6-9.

241 ibid. pp. 244, 14-246, 12.

242 ibid. pp. 252, 5-254, 20.

243 ibid. pp. 301, 21-302, 13.

244 ibid. pp. 322, 4-323, 3.

245 Próclus, pp. 355, 20-356, 16.

246 ibid. p. 364, 9-12; pp. 364, 20-365, 4.

247 ibid. p. 71, 19; p. 356, 8, 6.

248 ibid. p. 74, 23, 24.

249 ibid. pp. 105, 5, 6, 14, 15.

250 ibid. p. 100, 5-19.

251 ibid. p. 114, 20-25.

252 ibid. p. 123, 15-19 (cf. p. 124, 17, p. 125, 17).

253 ibid. p. 183, 13, 14.

254 ibid. pp. 194, 25-195, 5.

255 ibid. pp. 279, 16-280, 4.

256 ibid. p. 282, 8-19.

257 ibid. pp. 335, 16-336, 5.

258 Philologus, vol. XLIII. p. 489.

259 See above, pp. 2, 3.

260 Marinus in Euclidis Data, ed. Menge, p. 234, 16.

261 Fihrist, tr. Suter, p. 19.

262 Proclus, p. 191, 23.

263 ibid. pp. 362, 14-363, 18; pp. 365, 7-367, 27.

264 ibid. p. 200, 1-3.

265 ibid. pp. 199, 11-200, 1.

266 ibid. pp. 214, 18-215, 13; pp. 216, 10-218, 11.

267 ibid. p. 216, 21.

268 ibid. p. 218, 1.

269 ibid. pp. 241, 19-243, 11.

270 ibid. pp. 242, 22-243, 11.

271 Proclus, p. 241, 21, 22.

272 ibid. p. 123, 19.

273 Tannery, La Géométrie grecque, pp. 25, 26.

274 Van Pesch, De Procli fontibus, p. 139.

275 ibid. p. 155.

276 Proclus, p. 84, 13, p. 432, 14, 15.

277 Cf. ibid. p. 289, 11-15; p. 432, 15-17.

278 ibid. p. 432, 17.

279 ibid. p. 202, 9-25.

280 See Proclus, p. 270, 5-24 (I. 8); pp. 309, 3-310, 8 (I. 16); pp. 310, 19-311, 23 (I. 17); pp. 316, 14-318, 2 (I. 18); p. 384, 13-21 (I. 32); pp. 426, 22-427, 8 (I. 47).

281 Proclus, p. 198, 5-15.

282 ibid. p. 250, 12-19.

283 ibid. p. 190, 9-23.

284 ibid. p. 243, 12-29.

285 ibid. pp. 405, 6-406, 9.

286 ibid. p. 368, 1-23.

287 ibid. pp. 371, 11-373, 2.

288 Aristotle, de caelo, I. 5 (271 b 28-30).

289 Proclus, p. 371, 10.

290 Berlin Aristotle, vol. IV. p. 214 a 9-12.

291 The material for the whole of this chapter is taken from Heiberg's edition of the Elements, introduction to vol. v., and from the same scholar's Litterargeschichtliche Studien über Euklid, p. 174 sqq. and Paralipomena zu Euklid in Hermes, XXXVIII., 1903.

292 I. p. 201 ed. Halma=p. 50 ed. Basel.

293 See Pauly-Wissowa, Real-Encyclopädie der class. Altertumswissenschaft, vol. II., 1896, p. 675.

294 Heiberg, vol. v. pp. xxix—xxxiii.

295 Zeitschrift fiir Math. u. Physik, XXIX., hist.-litt. Abtheilung, p. 13.

296 [εἰς τ]ὰ τοῦ Εὐκλείδου στοιχεῖα προλαμβανόμενα ἐκ τῶν Πρόκλου σποράδην καὶ κατ̓ ἐπιτομήν. Cf. p. 32, note 8, above.

297 Heiberg, Paralipomena zu Euklid in Hermes, XXXVIII., 1903, pp. 46-74, 161-201, 321-356.

298 Described by Heiberg in Oversigt over det kngl. danske Videnskabernes Selskabs Forhandlinger, 1900, p. 161.

299 Exhaustive details under all the different heads are given by Heiberg (Vol. v. pp. lii—lxxv).

300 Euclides ab omni naevo vindicatus, Mediolani, 1733.

301 Cf. especially the assumption, without proof or definition, of the criterion for equal solid angles, and the incomplete proof of XII. 17.

302 An-Nairīzī, ed. Curtze, p. 121.

303 Proclus, pp. 194, 10 sqq.

304 Heiberg, Om Scholierne til Euklids Elementer, Kjøbenhavn, 1888. The tract is written in Danish, but, fortunately for those who do not read Danish easily, the author has appended (pp. 70-78) a résumé in French.

305 Om Scholierne til Euklids Elementer, pp. 11, 12: cf. Euklid-Stulien, pp. 170, 171; Woepcke, Mémoires présent. à l' Acad. des Sciences, 1856, XIV. p. 658 sqq.

306 Archimedes, ed. Heiberg, III. P. 28, 19-22.

307 Woepcke, op. cit. p. 702.

308 Art. “Eukleides” in Pauly-Wissowa's Real-Encyclopädie.

309 Hultsch has written upon these in Bibliotheca Mathematica, V_{3}, 1904, pp. 225-233.

310 Thus \sgrt{(27)} is given (allowing for a slight correction by means of the context) as 5 II' 46'' 10''', which gives for \sgrt{3} the value 1 43' 55'' 23''', being the same value as that given by Hipparchus in his Table of Chords, and correct to the seventh decimal place. Similarly \sgrt{8} is given as 2 49' 42'' 20''' 10'''', which is equivalent to\sgrt{2}=1.41421335. Hultsch gives instances of the various operations, addition, subtraction, etc., carried out in these fractions, and shows how the extraction of the square root was effected. Cf. T. L. Heath, Históry of Greek Mathematics, 1., pp. 59-63.

311 Lexico[ndot ] bibliogr. et encyclop. ed. Flügel, III. pp. 91, 92.

312 Klamroth, Zeitschrift der Deutschen Morgenländischen Gesellschaft, XXXV. p. 303.

313 Fihrist (tr. Suter), p. 16.

314 Codex Leidensis 399, 1. Euclidis Elementa ex interpretatione al-Hadschdschadschii cum commentariis al-Narizii, Hauniae, part 1. i. 1893, part I. ii. 1897, part II. i. 1900, part II. ii. 1905, part III. i. 1910.

315 Steinschneider, Zeitschrift für Math. u. Physik, XXXI., hist.-litt. Abtheilung, pp. 85, 86, 99.

316 Klamroth, p. 279.

317 Steinschneider, p. 88.

318 Klamroth, p. 306.

319 These MSS. are described by Nicoll and Pusey, Catalogus cod. mss. orient. bibl. Bodleianae, pt. II. 1835 (pp. 257-262).

320 Klamroth, pp. 306-8.

321 Steinschneider, p. 98. Heiberg has quoted the whole of this preface in the Zeitschrift fűr Math. u. Physik, XXIX., hist.-litt. Abth. p. 16.

322 This seems to include a rearrangement of the contents of Books XIV., XV. added to the Elements.

323 Suter, Die Mathematiker und Astronomen der Araber, p. 151. The Latin title is Euclidis elementorum geometricorum libri tredecim. Ex traditione doctissimi Nasiridini Tusini nunc primum arabice impressi. Romae in typographia Medicea MDXCIV. Cum licentia superiorum.

324 Kästner, Geschichte der Mathematik, I. p. 367.

325 Suter has a note that this MS. is very old, having been copied from the original in the author's lifetime.

326 Suter, p. 151.

327 Described by Kästner, Geschichte der Mathematik, I. pp. 289-299, and by Weissenborn, Die Ubersetzungen des Euklid durch Campano und Zamberti, Halle a. S., 1882, pp. 1-7. See also infra, Chapter VIII, p. 97.

328 Sonderabdruck des Jahresberichtes über die Fortschritte der klassischen Alterthumswissenschaft vom Okt. 1879-1882, Berlin, 1884.

329 Klamroth, p. 271.

330 Curtze, op. cit. p. 20; Heiberg, Euklid-Studien, p. 178.

331 Heiberg's Euclid, vol. v. p. ci.

332 Klamroth, pp. 273-4.

333 Heiberg, Zeitschrift fi<*>r Matk. u. Physik, XXIX., hist.-litt. Abtheilung, p. 21.

334 Klamroth, p. 274; Steinschneider, Zeitschrift für Math. u. Physik, XXXI., hist.-litt. Abth. p. 98.

335 Besthorn-Heiberg read “11?” as the number, Klamroth had read it as 21 (p. 273).

336 See Klamroth, pp. 275-6, 280, 282-4, 314-15, 326;Heiberg, vol. v. pp. xcvi, xcvii.

337 Heiberg in Zeitschrift für Math. u. Physik, XXIX., hist.-litt. Abth. p. 3 sqq.

338 Apollonius, ed. Heiberg, vol. 11. p. 218, 3-5.

339 Pappus, V. p. 436, 5.

340 Proclus, pp. 303-4.

341 Zeitschrift für Math. u. Physik, XXIX., hist.-litt. Abth. p. 6 sqq.

342 Klamroth, p. 310; Steinschneider, pp. 85-6.

343 Klamroth, p. 290, illustrates is[hnull ]āq's method by his way of distinguishing ἐφαρμόζειν (to be congruent with) and ἐφαρμόζεσθαι (to be applied to), the confusion of which by translators was animadverted on by Savile. Is[hnull ]āq avoided the confusion by using two entirely different words.

344 Klamroth, pp. 310-1.

345 ibid. p. 287.

346 ibid. pp. 304-5.

347 ibid. p. 274.

348 [hnull ]ājī Khalfa, I. p. 383.

349 Heiberg, Zeitschrift für Math. u. Physik, XXIX., hist.-litt. Abth. pp. 2, 3.

350 Steinschneider, Zeitschrift für Math. u. Physik, XXXI., hist.-litt. Abth. pp. 86 sqq.

351 Steinschneider, p. 86, Fihrist (tr. Suter), pp. 16, 67; Suter, Die Mathematiker und Astronomen der Araber (1900), p. 45.

352 Supplementum ad Euclidis opera omnia, ed. Heiberg and Menge, Leipzig, 1899.

353 Fihrist, pp. 16, 38; Steinschneider, p. 87; Suter, p. 65.

354 Fihrist, pp. 16, 25; Steinschneider, p. 88; Suter, p. 12.

355 Fihrist, pp. 16, 25, 58.

356 Suter, p. 26, note, quotes the Paris MS. 2467, 16^{o} containing the work “on proportion” as the authority for this conjecture.

357 MS. 2457, 39^{o} (cf. Woepcke in Me/m. pre/s. agrave; l'acad. des sciences, XIV., 1856, p. 669).

358 Fihrist, p. 17.

359 Suter, p. 58, note b.

360 Steinschneider, p. 89.

361 Fihrist, p. 17.

362 Woepcke, Fournal Asiatique, Sér. v. T. v. pp. 218-256 and 309-359.

363 Gesch. d. Math. vol. 13, pp. 743-6.

364 Fihrist, p. 39; Suter, p. 71.

365 Fihrist, p. 17; Suter, p. 17.

366 Fihrist, p. 17.

367 Suter, p. 64.

368 Fihrist, p. 17, 29; Suter, pp. 13, 14.

369 Fihrist, p. 17.

370 Fihrist, p. 17; Suter, p. 66.

371 Fihrist, p. 17, 10-15.

372 The mere catalogue of al-Kindī's works on the various branches of science takes up four octavo pages (11-15) of Suter's translation of the Fihrist.

373 Fihrist, pp. 16, 17.

374 Woepcke, Mém. prés. à l'acad. des sciences, XIV. pp. 666, 668.

375 Suter, p. 68.

376 Fihrist, p. 38; Suter, p. 60.

377 Fihrist, p. 26, and Suter's note, p. 60.

378 Suter, p. 211, note 23.

379 Fihrist, p. 43.

380 Fihrist, p. 43; Suter, p. 41.

381 Suter, pp. 34-8.

382 Fihrist (ed. Suter), p. 59, note 132; Suter, p. 52, note b.

383 See Suter in Bibliotheca Mathematica, IV_{3}, 1903-4, pp. 296-7, review of Julius Lippert's Ibn al-Qiftĩ. Ta'rĩch al-hukamã, Leipzig, 1903.

384 Fihrist, p. 40.

385 Suter, p. 75.

386 Suter, p. 55.

387 Steinschneider, p. 92.

388 Steinschneider, pp. 92-3.

389 Steinschneider, p. 92; Suter, p. 89.

390 Suter, p. 57.

391 Suter, pp. 150-1.

392 Suter, p. 157.

393 ibid. p. 175.

394 ibid. p. 56.

395 ibid. pp. 153-4, 227.

396 Gartz, p. 14; Steinschneider, pp. 94-5.

397 Suter in Bibliotheca Mathematica, IV3, 1903, pp. 25, 295; Suter has also an article on its contents, Bibliotheca Mathematica, VII3, 1906-7, pp. 234-251.

398 Steinschneider, p. 94; Suter, p. 146.

399 Suter, Nachträge und Berichtigungen, in Abhandlungen zur Gesch. der math. Wissenschaften, XIV., 1902, p. 170.

400 . Suter, p. 81, and Nachträge, p. 172.

401 De oratore III. 132.

402 Tusc. 1. 5.

403 Gromatici veteres, I. 97 sq. (ed. F. Blume, K. Lachmann and A. Rudorff. Berlin, 1848, 1852).

404 Censorinus, ed. Hultsch, pp. 60-3.

405 Martianus Capella, VI. 724.

406 ibid. VI. 708 sq.

407 Cf. Cantor, I_{3}, p. 565.

408 The fragment was deciphered by W. Studemund, who communicated his results to Cantor.

409 Cassiodorus, Variae, I. 45, p. 40, 12 ed. Mommsen.

410 See especially Weissenborn in Abhandlungen zur Gesch. d. Math. II. p. 18_{5} sq.; Heiberg in Philologus, XLIII. p. 507 sq.; Cantor, 1_{3}, p. 580 sq.

411 Cantor, Gesch. d. Math. I3, p. 906.

412 Boncompagni, Della vita e delle opere di Gherardo Cremonese, Rome, 1851, p. 5.

413 Described in an appendix to Studien über Menelaos' Sphärik (Abhandlungen zur Gesckichse der mathematischen Wissenschaften, XIV., 1902).

414 4 See Bibliotheca Mathematica, VI3, 1905-6, PP. 242-8.

415 Anaritii in decem libros priores Elementorum Euclidis Commentarii ex interpretatione Gherardi Cremonensis in codice Cracoviensi 569 servata edidit Maximilianus Curtze, Leipzig (Teubner), 1899.

416 Cantor, II_{1}, p. 88.

417 Tiraboschi, Storia della letteratura italiana, IV. 145mdash;160.

418 H. Weissenborn in Zeitschrift für Math. u. Physik, XXV., Supplement, pp. 143mdash;166, and in his monograph, Die Übersetzungen des Euklid durch Campano und Zamberti (1882); Max. Curtze in Philologische Rundschau (1881), 1. pp. 943-950, and in Fahresbericht über die Fortschritte der classischen Alterthumswissenschaft, XL. (188_{4}, III.) pp. 19mdash;22; Heiberg in Zeitschrift für Math. u. Physik, XXXV., hist.-litt. Abth., pp. 48mdash;58 and pp. 81mdash;6.

419 Quoted by Halliwell in Rara Mathematica (p. 56 note) from MS. Bib. Reg. Mus. Brit. 17 A. 1. f. 2^{4}mdash;3.

420 Saggio di una Bibliografia Euclidea, memoria del Prof. Pietro Riccardi (Bologna, 1887, 1888, 1890, 1893).

421 I. p. 404.

422 Published in C. T. de Murr's Memorabilia Bibliothecarum Norimbergensium, Part I. p. 190 sqq.

423 Curtze (An-Nairīzī, p. xiii) reproduces the heading of the first page of the text as follows (there is no title-page): Preclariffimũ opus elemento<*> Euclidis megarēfis [vmacr ]na cū cōmentis Campani pfpicaciffimi in artē geometriā incipit felicit', after which the definitions begin at once. Other copies have the shorter heading: Preclarissimus liber elementorum Euclidis perspicacissimi: in artem Geometrie incipit quam foelicissime. At the end stands the following: <*> Opus elementorū euclidis megarenfis in geometriā artē Jnid quoq<*> Campani pfpicaciffimi Cōmentationes finiũt. Erhardus ratdolt Augustensis impreffor folertiffimus. venetijs impreffit . Anno falutis . M.cccc.lxxxij . Octauis . Cale[ntilde] . Ju[ntilde] . Lector . Vale.

424 Kästner, Geschichte der Mathematik, I. p. 289 sqq. See also Weissenborn, Die Übersetzungen des Euklid durch Campano und Zamberti, pp. 1-7.

425 “Mea industria non sine maximo labore effeci vt qua facilitate litterarum elementa imprimuntur ea etiam geometrice figure conficerentur.”

426 Curtze in Zeitschrift für Math. u. Physik, XX., hist.-litt. Abth. p. 58.

427 The title begins thus: “Euclidis megaresis philosophi platonicj mathematicarum disciplinarum Janitoris: Habent in hoc volumine quicunque ad mathematicam substantiam aspirant: elementorum libros xiij cum expositione Theonis insignis mathematici. quibus multa quae deerant ex lectione graeca sumpta addita sunt nec non plurima peruersa et praepostere: voluta in Campani interpretatione: ordinata digesta et castigata sunt etc.” For a description of the book see Weissenborn, p. 12 sqq.

428 See Weissenborn, p. 30 sqq.

429 “Atque utinam et alii cognoscere vellent non ostentare aut ea quae nesciunt veluti fumum venditare non conarentur.”

430 Campanus' translation in Ratdolt's edition is as follows: “Quantitates quae dicuntur continuam habere proportionalitatem, sunt, quarum equè multiplicia aut equa sunt aut equè sibi sine interruptione addunt aut minuunt” (!), to which Campanus adds the note: “Continuè proportionalia sunt quorum omnia multiplicia equalia sunt continuè proportionalia. Sed noluit ipsam diffinitionem proponere sub hac forma, quia tunc diffiniret idem per idem, aperte (? a parte) tamen rei est istud cum sua diffinitione convertibile.”

431 “Euclidis Megarensis Geometricorum Elementorum Libri XV. Campani Galli transalpini in eosdem commentariorum libri XV. Theonis Alexandrini Bartholomaeo. Zamberto Veneto interprete, in tredecim priores, commentationum libri XIII. Hypsiclis Alexandrini in duos posteriores, eodem Bartholomaeo Zamberto Veneto interprete, commentariorum libri II.” On the last page (261) is a similar statement of content, but with the difference that the expression “ex Campani...deinde Theonis...et Hypsiclis...traditionibus.” For description see Weissenborn, p. 56 sqq.

432 De arté supputandi libri quatuor.

433 Kästner, I. p. 260.

434 Heiberg, vol. V. p. cvii.

435 Kästner, I. p. 359.

436 ΕΥΚΛΕΙΔΟΥ ΤΑ ΣΩΖΟΜΕΝΑ. Euclidis quae supersunt omnia. Ex recensione Davidis Gregorii M.D. Astronomiae Professoris Saviliani et R.S.S. Oxoniae, e Theatro Sheldoniano, An. Dom. MDCCIII.

437 Euclidis quae supersunt. Les Œuvres d'Euclide, en Grec, en Latin et en Français d'après un manuscrit très-ancien, qui était resté inconnu jusqu'à nos jours Par F. Peyrard. Ouvra<*>e approuvé par l'Institut de France (Paris, chez M. Patris).

438 Euclidis elcmentorum libri sex priores graece et latine commentario e scriptis veterum ac recentiarum mathematicorum et Pfleidereri maxime illustrati (Berolini, sumptibus G. Reimeri). Tom. I. 1824; tom. II. 1825.

439 Described by Boncompagni, Bullettino, II. p. 389.

440 Euclidis elemcntorum librit XV. Accessit XVI. de solidorum regularium comparatione. Omnes perspicuis demonstrationibus, accuratisque scholiis illustrati. Auctore Christophoro Clavio (Romae, apud Vincentium Accoltum), 2 vols.

441 The title-page of the edition of 1565 is as follows: Euclide Megarense philosopho, solo introduttore delle scientie mathematice, diligentemente rassettato, et alla integrità ridotto, per il degno professore di tal scientie Nicolo Tartalea Brisciano. secondo le due tradottioni. con una ampla espositione dello istesso tradottore di nuouo aggiunta. talmente chiara, che ogni mediocre ingegno, sensa la notitia, ouer suffragio di alcun' altra scientia con facilità serà capace a poterlo intendere. In Venetia, Appresso Curtio Troiano, 1565.

442 Das sibend acht und neunt buch des hochberümbten Mathematici Euclidis Megarensis... durch Magistrum Fohann Scheybl, der löblichen universitet zu Tübingen, des Euclidis und Arithmetic Ordinarien, auss dem latein ins teutsch gebracht....

443 .Die sechs erste b'ücher Euclidis vom anfang oder grund der Geometrj...Auss Griechischer sprach in die Teütsch gebracht aigentlich erklärt...Demassen vormals in Teütscher sprach nie gesehen worden...Durch Wilhelm Holtzman genant Xylander von Augspurg. Getruckht zu Basel.

444 Proclus, Comm. on Eucl. I., ed. Friedlein, pp. 72 sqq.

445 τὸ πλῆθος τῶν ἐντὸς ὀρθαῖς ἴσων. If the text is right, we must apparently take it as “the number of the angles equal to right angles that there are inside,” i.e. that are made up by the internal angles

446 τῶν ἀρχικῶν σχημάτων, by which Proclus probably means the regular polyhedra (Tannery, P. 143 n.).

447 We have no more than the most obscure indications of the character of this work in an Arabic MS. analysed by Woepcke, Essai d'une restitution de travaux perdus d'Apollonius sur les quantités irrationelles d'après des indications tirées d'un manuscrit arabe in Mémoires présentés à l'académie des sciences, XIV. 658-720, Paris, 1856. Cf. Cantor, Gesch. d. Math. I_{3}, pp. 348-9: details are also given in my notes to Book X.

448 Proclus, pp. 70, 19-71, 21.

449 Topics VIII. 14, 163 b 23.

450 Topics VIII. 3, 158 b 35.

451 Metaph. 998 a 25

452 Metaph. 1014 a 35-b 5.

453 Proclus, p. 66, 20 ὤστε τὸν Λέοντα καὶ τὰ στοιχεῖα συνθεῖναι τῷ τε πλἡθει καὶ τῆ χρείᾳ τῶν δεικνυμένων ἐπιμελέστερον.

454 Proclus, p. 67, 14 καὶ γὰρ τὰ στοιχεῖα καλῶς συνέταξεν καὶ πολλὰ τῶν μερικῶν [ὁρικῶν (?) Friedlein] καθολικώτερα ἐποίησεν.

455 Proclus, p. 67, 22 τῶν στοιχείων πολλὰ ἀνεῦρε.

456 Mathematisches zu Aristoteles in Abhandlungen zur Gesch. d. math. Wissenschaften, XVIII. Heft (1904), pp. 1-49.

457 Anal. post. 1. 6, 74 b 5.

458 ibid. 1. 10, 76 a 31-77 a 4.

459 Anal. post. 1. 2, 72 a 14-24.

460 Metaph. 1061 b 19-24.

461 Anal. post. 1. 11, 77 a 30.

462 Metaph. 996 b 26-30.

463 Metaph. 997 a 20-22.

464 Proclus, p. 194, 8.

465 Metaph. 997 a 10.

466 ibid. 996 b 26.

467 ibid. 1005 a 21—b 11.

468 ibid. 997 a 5-8.

469 ibid. 1005 b 11-17.

470 ibid. 1006 a 5.

471 ibid. 1006 a 17.

472 ibid. 1006 a 10.

473 ibid. 1006 a 11-15.

474 ibid. 1006 a 18 sqq.

475 Proclus, pp. 75, 10-77, 2.

476 Republic, VI. 510 c. Cf. Aristotle, Nic. Eth. 1151 a 17.

477 H. Jackson, Journal of Philology, vol. x. p. 144.

478 Proclus, pp. 178, 12-179, 8. In illustration Proclus contrasts the drawing of a straight line or a circle with the drawing of a “single-turn spiral” or of an equilateral triangle, the spiral requiring more complex machinery and even the equilateral triangle needing a certain method. “For the geometrical intelligence will say that by conceiving a straight line fixed at one end but, as regards the other end, moving round the fixed end, and a point moving along the straight line from the fixed end, I have described the single-turn spiral; for the end of the straight line describing a circle, and the point moving on the straight line simultaneously, when they arrive and meet at the same point, complete such a spiral. And again, if I draw equal circles, join their common point to the centres of the circles and draw a straight line from one of the centres to the other, I shall have the equilateral triangle. These things then are far from being completed by means of a single act or of a moment's thought” (p. 180, 8-21).

479 Proclus, p. 181,4-11.

480 It is necessary to coin a word to render ἀνισορροπιῶν, which is moreover in the plural. The title of the treatise as we have it is Equilibria of planes or cenires of gravity of planes in Book I and Equilibria of planes in Book II.

481 Proclus, p. 181, 16-23.

482 ibid. p. 182, 6-14.

483 Pp. 118, 119.

484 Proclus, pp. 182, 21-183, 13.

485 Anal. post. 1. 25, 86 a 33-35.

486 Cf. Lardner's Euclid: also Todhunter.

487 Proclus, p. 77, 7-12.

488 ibid. pp. 77, 15-78, 8.

489 ibid. pp. 78, 8-79, 2.

490 Proclus, pp. 79, 11-80, 5.

491 In the text we have τὸ δὲ πρόβλημα answering to τὸ μὲν without substantive: πρόβλημα was obviously inserted in error.

492 Proclus, pp. 80, 15-81, 4.

493 Proclus, p. 81, 5-22.

494 τὸ πέμπτον. This should apparently be the fourth because in the next words it is implied that none of the first three propositions are required in proving it.

495 Proclus, pp. 241, 19-243, 11.

496 ibid. pp. 233, 21-234, 6.

497 Proclus, p. 234, 21.

498 ibid. p. 243, 12-25.

499 ibid. p. 220, 7-12.

500 ibid. pp. 220, 16-221, 6.

501 Proclus, p. 221, 7-11.

502 ibid. pp. 221, 13-222,14.

503 ibid. pp. 203, 1-204, 13; 204, 23-205, 8.

504 Proclus, p. 208, 21.

505 ibid. p. 202, 3.

506 La Géométrie grecque, p. 149 note. Where δεῖ δὴ introduces the closer description of the problem we may translate, “it is then required” or “thus it is required” (to constructetc.): when it introduces the condition of possibility we may translate “thus it is necessary etc.” Heiberg originally wrote δεῖ δὲ in the latter sense in 1. 22 on the authority of Proclus and Eutocius, and against that of the MSS. Later, on the occasion of XI. 23, he observed that he should have followed the MSS. and written δεῖ δὴ which he found to be, after all, the right reading in Eutocius (Apollonius, ed. Heiberg, II. p. 178). δεῖ δὴ is also the expression used by Diophantus for introducing conditions of possibility.

507 . See the passage of Eutocius referred to in last note.

508 Proclus, p. 207, 4-25.

509 Proclus, p. 205, 13-15.

510 Proclus, pp. 211, 1-212, 4.

511 It would appear, says Tannery (p. 151 n.), that Geminus understood a lemma as being simply λαμβανόμενον, something assumed (cf. the passage of Proclus, p. 73, 4, relating to Menaechmus' view of elements): hence we cannot consider ourselves authorised in attributing to Geminus the more technical definition of the term here given by Proclus, according to which it is only used of propositions not proved beforehand. This view of a lemma must be considered as relatively modern. It seems to have had its origin in an imperfection of method. In the course of a demonstration it was necessary to assume a proposition which required proof, but the proof of which would, if inserted in the particular place, break the thread of the demonstration: hence it was necessary either to prove it beforehand as a preliminary proposition or to postpone it to be proved afterwards (ὡς ἑξῆς δειχθήσεται). When, after the time of Geminus, the progress of original discovery in geometry was arrested, geometers occupied themselves with the study and elucidation of the works of the great mathematicians who had preceded them. This involved the investigation of propositions explicitly quoted or tacitly assumed in the great classical treatises; and naturally it was found that several such remained to be demonstrated, either because the authors had omitted them as being easy enough to be left to the reader himself to prove, or because books in which they were proved had been lost in the meantime. Hence arose a class of complementary or auxiliary propositions which were called lemmas. Thus Pappus gives in his Book VII a collection of lemmas in elucidation of the treatises of Euclid and Apollonius included in the so-called “Treasury of Analysis” (τόπος ἀναλνόμενος). When Proclus goes on to distinguish three methods of discovering lemmas, analysis, division, and reductio ad absurdum, he seems to imply that the principal business of contemporary geometers was the investigation of these auxiliary propositions.

512 This passage and another from Diogenes Laertius (III. 24, p. 74 ed. Cobet) to the effect that “He [Plato] explained (εἰσηγήσατο) to Leodamas of Thasos the method of inquiry by analysis” have been commonly understood as ascribing to Plato the invention of the method of analysis; but Tannery points out forcibly (pp. 112, 113) how difficult it is to explain in what Plato's discovery could have consisted if analysis be taken in the sense attributed to it in Pappus, where we can see no more than a series of successive, reductions of a problem until it is finally reduced to a known problem. On the other hand, Proclus' words about carrying up the thing sought to “an acknowledged principle” suggest that what he had in mind was the process described at the end of Book VI of the Republic by which the dialectician (unlike the mathematician) uses hypotheses as stepping-stones up to a principle which is not hypothetical, and then is able to descend step by step verifying every one of the hypotheses by which he ascended. This description does not of course refer to mathematical analysis, but it may have given rise to the idea that analysis was Plato's discovery, since analysis and synthesis following each other are related in the same way as the upward and the downward progression in the dialectician's intellectual method. And it may be that Plato's achievement was to observe the importance, from the point of view of logical rigour, of the confirmatory synthesis following analysis, and to regulariśe in this way and elevate into a completely irrefragable method the partial and uncertain analysis upon which the works of his predecessors depended.

513 Here again the successive bipartitions of genera into species such as we find in the Sophist and Republic have very little to say to geometry, and the very fact that they are here mentioned side by side with analysis suggests that Proclus confused the latter with the philosophical method of Rep. VI.

514 Tannery rightly remarks (p. 152) that the subdivision of a theorem or problem into several cases is foreign to the really classic form; the ancients preferred, where necessary, to multiply enunciations. As, however, some omissions necessarily occurred, the writers of lemmas naturally added separate cases, which in some instances found their way into the text. A good example is Euclid 1. 7, the second case of which, as it appears in our text-books, was interpolated. On the commentary of Proclus on this proposition Th. Taylor rightly remarks that “Euclid everywhere avoids a multitude of cases.”

515 Proclus, p. 212, 5-11.

516 Tannery notes however that, so far from distinguishing his corollaries from the conclusions of his propositions, Euclid inserts them before the closing words “(being) what it was required to do” or “to prove.” In fact the porism-corollary is with Euclid rather a modified form of the regular conclusion than a separate proposition.

517 Proclus, p. 212, 18-23.

518 Anal. prior. II. 26, 69 a 37.

519 ibid. II. 25, 69 a 20.

520 Proclus, pp. 212, 24-213, 11. This passage has frequently been taken as crediting Hippocrates with the discovery of the method of geometrical reduction: cf. Taylor (Translation of Proclus, II. p. 26), Allman (p. 41n., 59), Gow (pp. 169, 170). As Tannery remarks (p. 110), if the particular reduction of the duplication problem to that of the two means is the first noted in history, it is difficult to suppose that it was really the first; for Hippocrates must have found instances of it in the Pythagorean geometry. Bretschneider, I think, comes nearer the truth when he boldly (p. 99) translates: “This reduction of the aforesaid construction is said to have been first given by Hippocrates.” The words are πρῶτον δέ φασι τῶν ἀπορουμένων διαγραμμάτων τὴν ἀπαγωγὴν ποιήσασθαι, which must, literally, be translated as in the text above; but, when Proclus speaks vaguely of “difficult constructions,” he probably means to say simply that “this first recorded instance of a reduction of a difficult construction is attributed to Hippocrates.”

521 Aristotle, Anal. prior. I. 7, 29 b 5; 1. 44, 50 a 30.

522 ibid. I. 21, 39 b 32; I. 29, 45 a 35.

523 Anal. post. I. 24, 85 a 16 etc.

524 Anal. prior. I. 23, 40 b 25.

525 Anal. prior. I. 23, 41 a 24.

526 ibid. II. 14, 62 b 29.

527 Proclus, p. 254, 22-27.

528 Proclus, p. 255, 8-26.

529 Pappus, v. p. 410 sqq.

530 ibid. pp. 410, 27-412, 2.

531 Bretschneider, p. 168. See however Heiberg's recent suggestion (Paralipomena zu Euklid in Hermes, XXXVIII., 1903) that the author was Heron. The suggestion is based on a comparison with the remarks on analysis and synthesis quoted from Heron by an-Nairĩzĩ (ed. Curtze, p. 89) at the beginning of his commentary on Eucl. Book II. On the whole, this suggestion commends itself to me more than that of Bretschneider.

532 Proclus, p. 67, 6.

533 Cantor, Gesch. d. Math. I_{3}, p. 241.

534 Proclus, p. 60, 7-9.

535 ibid. p. 60, 16-19.

536 Pappus, VII. pp. 634-6.

537 Hankel, Zur Geschichte der Mathematik in Alterthum und Mittelalter, 1874, pp. 137-150; Duhamel, Des'méthodes dans les sciences de raisonnement, Part I., 3 ed., Paris, 1885, pp. 39-68; Zeuthen, Geschichte der Mathematik im Altertnm und Mittelalter, 1896, pp. 92-104; Ofterdinger, Beiträge zur Geschichte der griechischen Mathematik, Ulm, 1860; Cantor, Geschichte der Mathematik, I_{3}, pp. 220-2.

538 Hankel, p. 139.

539 Zeuthen, p. 103.

540 Proclus, p. 242, 16, 17.

541 Zeuthen, p. 93.

542 Hankel, p. 141.

543 Pappus, VII. pp. 830-2.

544 Anal. post. II. 13, 96 b 15.

545 Cf. De anima, I. 2, 404 2 9, where “breathing” is spoken of as the ὄρος of “life,” and the many passages in the Politics where the word is used to denote that which gives its special character to the several forms of government (virtue being the ὅρος of aristocracy, wealth of oligarchy, liberty of democracy, 1294 a 10); Plato, Republic, VIII. 551 c.

546 Anal. past. I. 10, 76 a 31 sqq.

547 ibid. II. 7, 92 b 10.

548 ibid. 92 b 19.

549 ibid. 92 b 12 sqq.

550 Anal. post. I. 1, 71 a II sqq.

551 Mill's System of Logic, Bk. I. ch. Viii.

552 It is true that it was in opposition to “the ideas of most of the Aristotelian logicians” (rather than of Aristotle himself) that Mill laid such stress on his point of view. Cf. his observation: “We have already made, and shall often have to repeat, the remark, that the philosophers who overthrew Realism by no means got rid of the consequences of Realism, but retained long afterwards, in their own philosophy, numerous propositions which could only have a rational meaning as part of a Realistic system. It had been handed down from Aristotle, and probably from earlier times, as an obvious truth, that the science of geometry is deduced from definitions. This, so long as a definition was considered to be a proposition ’unfolding the nature of the thing,’ did well enough. But Hobbes followed and rejected utterly the notion that a definition declares the nature of the thing, or does anything but state the meaning of a name; yet he continued to affirm as broadly as any of his predecessors that the ἀρχαί, principia, or original premisses of mathematics, and even of all science, are definitions; producing the singular paradox that systems of scientific truth, nay, all truths whatever at which we arrive by reasoning, are deduced from the arbitrary conventions of mankind concerning the signification of words.” Aristotle was guilty of no such paradox; on the contrary, he exposed it as plainly as did Mill.

553 This has been fully brought out in two papers by G. Vailati, La teoria Aristotelica della definizione (Riuista di Filosofia e scienze affini, 1903), and Di un' opera dimenticata del P. Gerolamo Saccheri (“Logica Demonstrativa,” 1697) (in Rivista Filosofica, 1903).

554 “Definitio quid nominis nata est evadere definitio quid rei per postulatum vel dum venitur ad quaestionem an est et respondetur affirmative.”

555 Opuscules et fragments inédits de Leibniz, Paris, Alcan, 1903, p. 431. Quoted by Vailati.

556 Trendelenburg, Elementa Logices Aristoteleae, § 50.

557 Trendelenburg, Erläuterungen zu den Elementen der aristotelischen Logik, 3 ed. p. 107. On construction as proof of existence in ancient geometry cf. H. G. Zeuthen, Die geometrische Construction als “Existenzbeweis” in der antiken Geometrie (in Mathematische Annalen, 47. Band).

558 Anal. post. II. 13, 96 a 33—b 1.

559 Trendelenburg, Erläuterungen, p. 108.

560 Topics VI. 4, 141 a 26 sqq.

561 Topics VI. 4, 141 b 25-34.

562 Topics VI. 4, 142 a 22-31.

563 ibid. 142 a 34—b 6.

564 Trendelenburg, Erläuterungen, p. 115.

565 Topics VI. 4, 142 b 7-10.

566 Anal. post. II. 2, 90 a 31.

567 Anal. post. II. 2, 90 a 15-21.

568 ibid. II. 8, 93 a 17.

569 ibid. 93 a 20.

570 Trendelenburg, Erläuterungen, p. 110.

571 De anima II. 2, 413 a 13-20,

572 Anal. post. II. 2, 90 a 6,

573 ibid. II. 11, 94 a 28.

574 Other passages in Aristotle may be quoted to the like effect: e.g. Anal. post. I. 2, 71 b 9 “We consider that we know a particular thing in the absolute sense, as distinct from the sophistical and incidental sense, when we consider that we know the cause on account of which the thing is, in the sense of knowing that it is the cause of that thing and that it cannot be otherwise,” ibid. I. 13, 79 a 2 “For here to know the fact is the function of those who are concerned with sensible things, to know the cause is the function of the mathematician; it is he who possesses the proofs of the causes, and often he does not know the fact.” In view of such passages it is difficult to see how Proclus came to write (p. 202, 11) that Aristotle was the originator (Ἀριστοτέλους κατάρξαντος) of the idea of Amphinomus and others that geometry does not investigate the cause and the why (τὸ διὰ τί). To this Geminus replied that the investigation of the cause does, on the contrary, appear in geometry. “For how can it be maintained that it is not the business of the geometer to inquire for what reason, on the one hand, an infinite number of equilateral polygons are inscribed in a circle, but, on the other hand, it is not possible to inscribe in a sphere an infinite number of polyhedral figures, equilateral, equiangular, and made up of similar plane figures? Whose business is it to ask this question and find the answer to it if it is not that of the geometer? Now when geometers reason per impossibile they are content to discover the property, but when they argue by direct proof, if such proof be only partial (ἐπὶ μέρους), this does not suffice for showing the cause; if however it is general and applies to all like cases, the why (τὸ διὰ τί) is at once and concurrently made evident.”

575 Heron, Def. 12 (vol. IV. Heib. pp. 22-24).

576 e.g. in III. 20 and in Data 89.

577 Cf. the chapter on νεύσεις in The Works of Archimedes, pp. c—cxxii.

578 Zeuthen, Die Lehre von den Kegelschnitten im Altertum, ch. 12, p. 262.

579 Heiberg, Mathematisches zu Aristoteles, p. 16.

580 Pappus VII. pp. 670-2.

581 The Greek usage differs from ours in that the definite article is employed in such a phrase as this where we have the indefinite. ἐπὶ τῆς δοθείσης εὐθείας πεπερασμένης, “on the given finite straight line,” i.e. the finite straight line which we choose to take.

582 To be strictly literal we should have to translate in the reverse order “let the given finite straight line be the (straight line) AB” ; but this order is inconvenient in other cases where there is more than one datum, e.g. in the setting-out of I. 2, “let the given point be A, and the given straight line BC,” the awkwardness arising from the omission of the verb in the second clause. Hence I have, for clearness' sake, adopted the other order throughout the book.

583 Two things are here to be noted, (1) the elegant and practically universal use of the perfect passive imperative in constructions, γεγράφθω meaning of course “let it have been described” or “suppose it described,” (2) the impossibility of expressing shortly in a translation the force of the words in their original order. κύκλος γεγράφθω ΒΓΔ means literally “let a circle have been described, the (circle, namely, which I denote by) BCD.” Similarly we have lower down “let straight lines, (namely) the (straight lines) CA, CB, be joined,” ἐπεζεύχθωσαν εὐθεῖαι αί ΓΑ, ΓΒ. There seems to be no practicable alternative, in English, but to translate as I have done in the text.

584 Euclid is careful to adhere to the phraseology of Postulate 1 except that he speaks of “joining” (ἐπεζεύχθωσαν) instead of “drawing” (γράφειν). He does not allow himself to use the shortened expression “let the straight line FC be joined” (without mention of the points F, C) until I. 5.

585 , ἑκατέρα τῶν ΓΑ, ΓΒ and 24. the three straight lines CA, AB, BC, αἱ τρεῖς αἱ ΓΑ, ΑΒ, ΒΓ. I have, here and in all similar expressions, inserted the words “straight lines” which are not in the Greek. The possession of the inflected definite article enables the Greek to omit the words, but this is not possible in English, and it would scarcely be English to write “each of CA, CB” or “the three CA, AB, BC.”

586 I have inserted these words because “to place a straight line at a given point” (πρὸς τῷ δοθέντι σημείῳ) is not quite clear enough, at least in English.

587 It will be observed that in this first application of Postulate 2, and again in I. 5, Euclid speaks of the continuation of the straight line as that which is produced in such cases, ἐκβεβλήσθωσαν and προσεκβεβλήσθωσαν meaning little more than drawing straight lines “in a straight line with” the given straight lines. The first place in which Euclid uses phraseology exactly corresponding to ours when speaking of a straight line being produced is in I. 16: “let one side of it, BC, be produced to D” (προσεκβεβλήσθω αὐτοῦ μία πλευρὰ ΒΓ ἐπὶ τὸ Δ).

588 The Greek expressions are λοιπὴ ΑΛ and λοιπῇ τῇ BH, and the literal translation would be “AL (or BG) remaining,” but the shade of meaning conveyed by the position of the definite article can hardly be expressed in English.

589 It is a fact that Euclid's enunciations not infrequently leave something to be desired in point of clearness and precision. Here he speaks of the triangles having “the angle equal to the angle, namely the angle contained by the equal straight lines” (τὴν γωνίαν τῇ γωνίᾳ ἴσην ἔχῃ τὴν ὑπὸ τῶν ἴσων εὐθειῶν περιεχομένην), only one of the two angles being described in the latter expression (in the accusative), and a similar expression in the dative being left to be understood of the other angle. It is curious too that, after mentioning two “sides,” he speaks of the angles contained by the equal “straight lines,” not “sides.” It may be that he wished to adhere scrupulously, at the outset, to the phraseology of the definitions, where the angle is the inclination to one another of two lines or straight lines. Similarly in the enunciation of I. 5 he speaks of producing the equal “straight lines” as if to keep strictly to the wording of Postulate 2.

590 I agree with Mr H. M. Taylor (Euclid, p. ix) that it is best to abandon the traditional translation of “each to each,” which would naturally seem to imply that all the four magnitudes are equal rather than (as the Greek ὲκατέρα ὲκατέρᾳ does) that one is equal to one and the other to the other.

591 Here we have the word base used for the first time in the Elements. Proclus explains it (p. 236, 12-15) as meaning (1), when no side of a triangle has been mentioned before, the side “which is on a level with the sight” (τὴν πρὸς τῇ ὄψει κειμένην), and (2), when two sides have already been mentioned, the third side. Proclus thus avoids the mistake made by some modern editors who explain the term exclusively with reference to the case where two sides have been mentioned before. That this is an error is proved (1) by the occurrence of the term in the enunciations of I. 37 etc. about triangles on the same base and equal bases, (2) by the application of the same term to the bases of parallelograms in I. 35 etc. The truth is that the use of the term must have been suggested by the practice of drawing the particular side horizontally, as it were, and the rest of the figure above it. The base of a figure was therefore spoken of, primarily, in the same sense as the base of anything else, e.g. of a pedestal or column; but when, as in I. 5, two triangles were compared occupying other than the normal positions which gave rise to the name, and when two sides had been previously mentioned, the base was, as Proclus says, necessarily the third side.

592 ὑποτείνειν ὑπό, “to stretch under,” with accusative.

593 The full Greek expression would be ὑπὸ τῶν ΒΑ, ΑΓ περιεχομένη γωνία, “the angle contained by the (straight lines) BA, AC.” But it was a common practice of Greek geometers, e.g. of Archimedes and Apollonius (and Euclid too in Books X.—XIII.), to use the abbreviation αἱ ΒΑΓ for αἱ ΒΑ, ΑΓ, “the (straight lines) BA, AC.” Thus, on περιεχομένη being dropped, the expression would become first ὑπὸ τῶν ΒΑΓ γωνία, then ὑπὸ ΒΑΓ γωνία, and finally ὑπὸ ΒΑΓ, without γωνία, as we regularly find it in Euclid.

594 The difference between the technical use of the passive ἐφαρμόζεσθαι “to be applied (to),” and of the active ἐφαρμόζειν “to coincide (with)” has been noticed above (note on Common Notion 4, pp. 224-5).

595 Heiberg (Paralipomena su Euklid in Hermes, XXXVIII., 1903, p. 56) has pointed out, as a conclusive reason for regarding these words as an early interpolation, that the text of an-Nairīzī (Codex Leidensis 399, 1, ed. Besthorn-Heiberg, p. 55) does not give the words in this place but after the conclusion Q.E.D., which shows that they constitute a scholium only. They were doubtless added by some commentator who thought it necessary to explain the immediate inference that, since B coincides with E and C with F, the straight line BC coincides with the straight line EF, an inference which really follows from the definition of a straight line and Post. 1; and no doubt the Postulate that “Two straight lines cannot enclose a space” (afterwards placed among the Common Notions) was interpolated at the same time.

596 Where (as here) Euclid's conclusion merely repeats the enunciation word for word, I shall avoid the repetition and write “Therefore etc.” simply.

597 (meaning the equal sides). Cf. note on the similar expression in Prop. 4, lines 2, 3.

598 , εἰλήφθω ἐπὶ τῆς ΒΔ τυχὸν σημεῖον τὸ Ζ, where τυχὸν σημεῖονmeans “a chance point.”

599 δύο αἱ ΖΑ, ΑΓ δυσὶ ταῖς ΗΑ, ΑΒ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ. Here, and in numberless later passages, I have inserted the word “sides” for the reason given in the note on I. 1, line 20. It would have been permissible to supply either “straight lines” or “sides” ; but on the whole “sides” seems to be more in accordance with the phraseology of I. 4.

600 i.e., apparently, common to the angles, as the αὐτῶν in βάσις αὐτῶν κοινὴ can only refer to γωνία and γωνίᾳ preceding. Simson wrote “and the base BC is common to the two triangles BFC, CGB” ; Todhunter left out these words as being of no use and tending to perplex a beginner. But Euclid evidently chose to quote the conclusion of I. 4 exactly; the first phrase of that conclusion is that the bases (of the two triangles) are equal, and, as the equal bases are here the same base, Euclid naturally substitutes the word “common” for “equal.”

601 As “(Being) what it was required to prove” (or “do” ) is somewhat long, I shall henceforth write the time-honoured “Q. E. D.” and “Q. E. F.” for ὅπερ ἔδει δεῖξαι and ὅπερ ἔδει ποιῆσαι.

602 In an English translation of the enunciation of this proposition it is absolutely necessary, in order to make it intelligible, to insert some words which are not in the Greek. The reason is partly that the Greek enunciation is itself very elliptical, and partly that some words used in it conveyed more meaning than the corresponding words in English do. Particularly is this the case with οὐ συσταθήσονται ἐπί “there shall not be constructed upon,” since συνίστασθαι is the regular word for constructing a triangle in particular. Thus a Greek would easily understahd συσταθήσονται ἐπί as meaning the construction of two lines forming a triangle on a given straight line as base; whereas to “construct two straight lines on a straight line” is not in English sufficiently definite unless we explain that they are drawn from the ends of the straight line to meet at a point. I have had the less hesitation in putting in the words “from its extremities” because they are actually used by Euclid in the somewhat similar enunciation of I. 21.

How impossible a literal translation into English is, if it is to convey the meaning of the enunciation intelligibly, will be clear from the following attempt to render literally: “On the same straight line there shall not be constructed two other straight lines equal, each to each, to the same two straight lines, (terminating) at different points on the same side, having the same extremities as the original straight lines” (ἐπὶ τῆς αὐτῆς εὐθείας δύο ταῖς αὐταῖς εὐθείαις ἄλλαι δύο εὐθεῖαι ἴσαι ἑκατὲρα ἑκατέρα οὐ συσταθήσονται πρὸς ἄλλῳ καἰ ἄλλῳ σημείῳ ἐπἰ τὰ αὐτὰ μέρη τὰ αὐτὰ πέρατα ἔχουσαι ταῖς ἐξ ἀρχῆς εὐθείαις).

The reason why Euclid allowed himself to use, in this enunciation, language apparently so obscure is no doubt that the phraseology was traditional and therefore, vague as it was, had a conventional meaning which the contemporary geometer well understood. This is proved, I think, by the occurrence in Aristotle (Meteorologica III. 5, 376 a 2 sqq.) of the very same, evidently technical, expressions. Aristotle is there alluding to the theorem given by Eutocius from Apollonius' Plane Loci to the effect that, if H, K be two fixed points and M such a variable point that the ratio of MH to MK is a given ratio (not one of equality), the locus of M is a circle. (For an account of this theorem see note on VI. 3 below.) Now Aristotle says “The lines drawn up from H, K in this ratio cannot be constructed to two different points of the semicircle A” (αἰ οὖν ἀπὸ τῶν ΗΚ ἀναγόμεναι γραμμαὶ ἐν τούτῳ τῷ λόγῳ οὐ συσταθήσουται τοῦ ἐφ̓ Α ἡμικυκλίου πρὸς ἄλλο καὶ ἄλλο σημεῖον).

If a paraphrase is allowed instead of a translation adhering as closely as possible to the original, Simson's is the best that could be found, since the fact that the straight lines form triangles on the same base is really conveyed in the Greek. Simson's enunciation is, Upon the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise those which are terminated at the other extremity. Th. Taylor (the translator of Proclus) attacks Simson's alteration as “indiscreet” and as detracting from the beauty and accuracy of Euclid's enunciation which are enlarged upon by Proclus in his commentary. Yet, when Taylor says “Whatever difficulty learners may find in conceiving this proposition abstractedly is easily removed by its exposition in the figure,” he really gives his case away. The fact is that Taylor, always enthusiastic over his author, was nettled by Simson's slighting remarks on Proclus' comments on the proposition. Simson had said, with reference to Proclus' explanation of the bearing of the second part of I. 5 on I. 7, that it was not “worth while to relate his trifles at full length,” to which Taylor retorts “But Mr Simson was no philosopher; and therefore the greatest part of these Commentaries must be considered by him as trifles, from the want of a philosophic genius to comprehend their meaning, and a taste superior to that of a mere mathematician, to discover their beauty and elegance.”

603 It would be natural to insert here the step “but the angle ACD is greater than the angle BCD. [C.N. 5].”

604 much greater, literally “greater by much” (πολλῷ μεἰζων). Simson and those who follow him translate: “much more then is the angle BDC greater than the angle BCD,” but the Greek for this would have to be πολλῷ (or πολὺ[ρπαρ ] μᾶλλόν ἐστι...μείζων. πολλῷ μᾶλλον, however, though used by Apollonius, is not, apparently, found in Euclid or Archimedes.

605 The text has here “BA, CA.”

606 fall beside them. The Greek has the future, παραλλάξουσι. παραλλάττω means “to pass by without touching,” “to miss” or “to deviate.”

607 The verb is κείσθω which, as well as the other parts of κεῖμαι, is constantly used for the passive of τίθημι “to place” ; and the latter word is constantly used in the sense of making, e.g., one straight line equal to another straight line.

608 , κάθετον εὐθεῖαν γραμμἡν. This is the full expression for a perpendicular, κάθετος meaning let fall or let down, so that the expression corresponds to our plumb-line. κάθετος is however constantly used alone for a perpendicular, γραμμἡ being understood.

609 , literally “towards the other parts of the straight line AB,” ἐπὶ τὰ ἕτερα μέρη τῆς AB. Cf. “on the same side” (ἐπὶ τὰ αὐτὰ μέρη) in Post. 5 and “in both directions” (ἐφ̓ ἑκάτερα τὰ