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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 follows

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 lived

flourished,

as Heiberg understands it (was born,

as Hankel took it : otherwise part of Proclus' argument would lose its cogency.in his first

by which is understood let

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.O king, through the country there are royal roads and roads for common citizens, but in geometry there is one road for all.

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

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

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

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

to make perfect the understanding of the learner in regard to the whole of geometryTo get out of the difficulty he says. ibid . p. 71, 8.

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

One thing is however certain, namely that Euclid taught, and founded a school, at Alexandria. This is clear from the remark of Pappus about Apolloniushe 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 three- and four-line locus,

which in fact was not possible without some theorems first discovered by himself. Pappus says on thisNow 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 Stobaeussome 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 Euclid the geometer.

There is no doubt about the reading, although an early commentator on Valerius Maximus wanted to correct Eucliden

into

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 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 he was different from him of Megara of whom Laertius wrote, and who wrote dialogues

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' descriptionof Megara, or, according to some, of Gela, as Alexander says in the

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

We readEuclid, son of Naucrates, grandson of Zenarchus

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 son of Naucrates, the son of Berenice (?)

(see Suter's translation in

Thusinusalso

let no one unversed in geometry enter my doors; the Arab turned geometry into

their Academies.

Equally remarkable are the Arabian accounts of the relation of Euclid and Apollonius

a carpenter.Suter's argument is based on the fact that the

discoveredthe 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

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

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 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

instrumentum musicum pneumaticum,and who explained the name of Euclid, which they variously pronounced as

Lastly the alternative version, given in brackets above, which says that Euclid made the

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 solidsin the Arabic must be the

Comparison of the five figuresin 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

In giving a list of the Euclidean treatises other than the

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

I. The

I mention this first because Proclus refers to it in the general remarks in praise of the

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

Not only those (Tannery (e)/legxoi ) 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. thePseudographemata of Euclid.

2. The

The

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 accessible

3. The book

This work (For the circle is divisible into parts unlike in definition or notion (

Like

and unlike

here mean, not similar

and dissimilar

in the technical sense, but like

or unlike

(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

the actual, very ancient, copy from which I(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 awrote out ...

liber divisionumoccurs. 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 1851unlike

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

to apply to a straight line a rectangle equal to the rectangle contained byMoreover the treatise is no fragment, but finishes with the wordsAB, AC and deficient by a square .

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 Ofterdinger

4.The

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.

“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 porisms

“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