).
A few words will be nec ssary on

Of Ptolemy himself we know absolutely nothing but his date, which an astronomer always leaves in his works.
He certainly observed in Alexandrinus. If the canon presently mentioned be genuine (and it is not doubted), he survived Antoninus, and therefore was alive Bibl. Graec. vol. x. p. 411), in the thirteenth century, describes him as of Ptolemais in the Thebaid, called Hermeius. Accordingly, our personal knowledge of one of the most illustrious men that ever lived, both in merits and fame, and who resided and wrote in what might well be called the sister university to Athens, is limited to two accounts of one circumstance, between the uncertainties of which it is impossible to decide, and which give his birth to opposite sides of the Nile. Weidler (Hist. Astron. p. 177) cites some description of his personal appearance from an Arabic writer, who does not state his source of information. Some writers call him king Ptolemy, probably misled by the name, which is nevertheless known to have been borne by private persons, besides the astronomer. On this, and some other gossip not worth citing, because no way Greek, see Halma's preface, p. lxi. Ptolemy is then, to us, the author of certain works; and appears in the character of promulgator of his own researches, and deliverer and extender of those of Hipparchus.
In this last character there is some difficulty about his writings.
It is not easy to distinguish him from his illustrious predecessor.
It is on this account that we have deferred specific mention of HIPPARCHUS, as an astronomer, to the present article.

The writings of Ptolemy (independently of the work on geography, which will be noted apart) are as follows :--

Mathematica Syntaxis, in several places of Schweiger, Hoffmann Syntaxis carried, that it was applied to various astrological works having nothing to do with Ptolemy. Hoffman has two works in his list which he supposes to be English translations of the astrological syntaxis, because they bear as titles "the

On the manuscripts of the Almagest, see Fabricius (Bibl. Graec. vol. v. p. 281) and Halma's preface, p. xlv. &c. Doppelmayer (we copy Halma) says the manuscript used by Grynoeus, the first therefore printed from, was given to the Nuremberg library by Regiomontanus, to whom it was given (probably as a legacy) by Cardinal Bessarion. De Murr could not find this manuscript at Nuremberg, but only that of Theon's commentary, given by Regiomontanus, as described : but Montignot testifies to having caused it to be consulted for his version of the catalogue. Halma somewhat hastily concludes that there are difficulties in the way of supposing this manuscript to have been used : but public libraries do sometimes lose their manuscripts. This Basle edition may count as one manuscript unknown. Halma corrected its text by various others, in the Royal Library at Paris, principally five, as follows :--First, a Paris manuscript (No. 2389) nearly perfect, cited by some who have used it as of the sixth century, but pretty certainly not later than the eighth.
It bears a presentation inscription to John Lascaris, of the imperial family, who is known to have been sent by Lorenzo di Medicis twice to Constantinople, after its occupation by the Turks, to procure manuscripts. Secondly, a Florence manuscript of the twelfth century, marked 2390. Thirdly, a Venice manuscript, marked 313, supposed to be of the eleventh century. Fourthly, two Vatican manuscripts, marked 560 and 184, of about the twelfth century. These Florence, Venice, and Vatican manuscripts were probably returned to their original owners at the peace of 1815.
The seizures made by the French in Italy have procured us the only two editions of Euclid and Ptolemy which give various readings.

seen an earlier edition of the version of Trapezuntius than that of Venice, 1528, folio (with a red lily in the title page); and Hoffman sets down none earlier. Its title (from a copy before us) is "Claudii Ptolemaei Pheludiensis Alexandrini Almagestum.... latina donatum lingua ab Georgio Trapezuntio.... anno salutis MDXXVII. labente."
This version is stated in the preface to have been made from the Greek quem seems to refer to Trapezuntius, who had long been dead : Gauricus explains how he came by a copy. Andrew Trapezuntius, in his preface to his father's work (which follows that of Gauricus), though dedicating to the pope, does not hint at the manuscript from the pope's library, nor at any manuscript in particular.

PAPPUS].Almagest, Tetrabiblon, Centiloquium, and Inerrantium Stellarum Significationes of Ptolemy, and the Hypotyposes of Proclus. Except as containing the first professed collection of the works, it is not of note.
As to its Almagest, it is Trapezuntius as given by Gauricus.
The publisher, H. Petrus, seems to have found reason

From the time of Galileo, at which we are now arrived, we cannot find that any complete version of the Almagest (Greek edition there certainly was none) was published until that of Halma, to which we now come. We shall not attempt to describe the dissertations by Delambre, Ideler, &c., contained in this splendid collection, but shall simply note the contents of the first four volumes : for the rest see THEON. Of the manuscripts we have already spoken.
The descriptions are--Paris, 1813, 1816, 1819, 1820, quarto.
The first two volumes contain the Almagest, in Greek and French, with the various readings.
The third contains the GEMINUS. The fourth contains the

The part of the Almagest which really concerns the modern astronomer, as part of the effective records of his science, is the catalogue of stars in the seventh and eighth books. Of this catalogue there have been several distinct editions.

Mr. Baily remarks, as to the catalogue, and the same appears as to other parts of the Almagest, that Halma often gives in the text he has chosen readings different from those of all his principal subjects of collation.
This means that he has, in a considerable number of cases, either amended his text conjecturally, or preferred the reading of some minor manuscript, without particular mention.
This is no great harm, since, as the readings of all his great sources are always given, it amounts to having one more choice from an unnamed quarter.
But it is important that the critical reader of the edition have notice of it; and the more so, inasmuch as the readings are at the end of each volume, without

On the preceding summary of the bibliographical history of the Almagest, we shall remark that the reader is not to measure the currency of it by the number of its editions. It was the gold which lay in the Bank, while paper circulated on its authority. All the European books on astronomy were fashioned upon it, and it was only the more learned astronomers who went to the common original. Euclid was actually read, and accordingly, as we have seen, the presses were crowded with editions of the Elements. But Ptolemy, in his own words, was better known by his astrology than by his astronomy. We now come to his other writings, on which we have less to say.

Tetrabiblon, or Quadripartitum de Apotelesmatibtus et Judiciis Astrorum. With this goes another small work, called Fructus Librorum Suorum, often called Centiloquium, from its containing a hundred aphorisms. Both of these works are astrological, and it has been doubted by some whether they be genuine.
But the doubt merely arises from the feeling that the contents are unworthy of Ptolemy. The Tetrabiblon itself is, like the Almagest and other writings, dedicated to his brother Syrus : it refers, in the introduction, to another work on the mathematical theory. Both works If editors will put the various readings at the end of their volumes, instead of at the bottom of the pages, we should wish, when there are more volumes than one, that the readings for one volume should be inserted at the end of another.
It would then be practicable to have the text and its variations before the reader at one and the same moment, when two or three instances come close together, is very desirable.

They have been twice printed in Greek, and together ; first,

The

This is a catalogue of Assyrian, Persian, Greek, and Roman sovereigns, with the length of their reigns, several times referred to by Syncellus, and found, with continuation, in Theon. It is considered an undoubted work of Ptolemy. It is a scrap which has been printed by Scaliger, Calvisius (who valued it highly), Petavius and Dodwell; but most formally by Bainbridge (in the work presently cited), and by Halma, as above noticed.

`φάσεις ἀπλανῶν ἀστέρων καὶ συναγωγὴ ἐπισημασειῶν`

(This is an annual list of sidereal phaenomena.

It has been printed three times in Greek :

These works are obtained from the Arabic. Fabricius, who had not seen them, conjectures that they are the same, which is not correct. The Analemma is a collection of graphical processes for facilitating the construction of sun-dials, grounded on what we now call the orthographic projection of the sphere, a perspective in which, mathematically speaking, the eye is at an infinite distance. The

De Analemmate, edited by Commandine, Rome, 1562, 4to.

This is a brief statement of the principal hypotheses employed in the Almagest (to which it refers in a preliminary address to Syrus) for the explanation of the heavenly motions. Simplicius refers to two books of hypotheses, of which we may suppose this is one.

This is a treatise on the theory of the musical scale.

A metaphysical work, attributed to Ptolemy.

In Lalande we find attributed to Ptolemy, "Regulae Artis Mathematicae" (Gr. Lat.),--1569, 8vo., with explanations by Erasmus Reinhold.

The collection made by Fabricius of the lost works of Ptolemy is as follows :--From Simplicius, two books of hypotheses. From Suidas, three books

It must rest an unsettled question whether the work written by Ptolemy on optics be lost or not. The matter now stands thus: Alhazen, the principal Arab writer on optics, does not mention Ptolemy, nor indeed, any one else. Some passagesfrom Roger Bacon, taken to be opinions passed on a manuscript purporting to be that of Ptolemy, led Montucla to speak highly of Ptolemy as an optical writer. This mention probably led Laplace to examine a Latin version from the Arabic, existing in the Royal Library at Paris, and purporting to be Ptolemy's treatise. The consequence was Laplace's assertion that Ptolemy had given a detailed account of the phenomenon of astronomical refraction. This remark of Laplace led Humboldt to examine the manuscript, and to call the attention of Delambre to it. Delambre accordingly gave a full account of the work in his

We now come to speak of Ptolemy as an astronomer, and of the contents of the Almagest. And with his name we must couple that of his great predecessor, Hipparchus.
The latter was alive at

What Hipparchus did is to be collected mostly from the writings of Ptolemy himself, who has evidently intended that his predecessor should lose no fame in his hands.
The historian who has taken most pains to discriminate, and to separate what be held rather too partial to the predecessor of Ptolemy, those who think so will be obliged to admit that he gives his verdict upon the evidence, and not upon any prepossession gained before trial.
He is too much given, it may be, to try an old astronomer by what he has done for us, but this does not often disturb his estimate of the relative merit of the ancients. And it is no small testimony that an historian so deeply versed in modern practice, so conversant with ancient writings, so niggard of his praise, and so apt to deny it altogether to any which has since been surpassed, cannot get through his task without making it evident that Ilipparchus has become a chief favourite.
The summing up on the merits of the trite father of astronomy, as the historian calls him, is the best enumeration of his services which we can make, and will save the citation of authorities.
The following is translated from the preliminary discourse (which, it is important to remember, means the last part written) of the

"Let no one be astonished at the errors of half a degree with which we charge Hipparchus, perhaps with an air of reproach. We must bear in mind that his astrolabe was only an armillary sphere ; that its diameter was but moderate, the subdivisions of a degree hardly sensible; and that he had either telescope, vernier, nor micrometer. What could we do even now, if we were deprived of these helps, if we were ignorant of refraction and of the true altitude of the pole, as to which, even at Alexandria, and in spite of armillary circles of every kind, an error of a quarter of a degree was committed.
In our day we dispute about the fraction of a second; in that of Hipparchus they could not answer for the fraction of a degree; they might mistake nearest degree, which leaves them open to any error under half a degree, which is about the diameter of the sun or moon.il a entrevu) of the second inequality; he made all the observations necessary for a discovery the honour of which was reserved for Ptolemy; a discovery which perhaps he had not time to finish, but for which he had prepared every thing.
He showed that all the hypotheses of his predecessors were insufficient to explain the double inequality of the planets; he predicted that nothing would do except the combination of the two hypotheses of the excentric and epicycle. Observations were wanting to him, because these demand intervals of time exceeding the duration of the longest life : he prepared them for his successors. We owe to his catalogue the important knowledge of the retrograde motion of the equinoctial points. We could, it is true, obtain this knowledge from much better observations, made during the last hundred years : but such observations would not give proof that the motion is sensibly uniform for a long succession of centuries ; and the observations of Hipparchus, by their number and their antiquity, in spite of the errors which we cannot help finding in them, give us this important confirmation of one of the fundamental points of Astronomy.
He was here the first discoverer.
He invented the planisphere, or the mode of representing the starry heavens upon a plane, and of producing the solutions of problems of spherical astronomy, in a manner often as exact as, and more commodious than, the use of the globe itself.
He is also the father of true geography, by his happy idea of marking the position of spots on

We shall now proceed to give a short synopsis of the subjects treated in the Almagest : the reader will find a longer and better one in the second volume of the work of Delambre just cited.

The first book opens with some remarks on theory and practice, on the division of the sciences, and the certainty of mathematical knowledge : this preamble concludes with an announcement of the author's intention to avail himself of his predecessors, to run over all that has been sufficiently explained, and to dwell upon what has not been done completely and well.
It then describes as the intention of the work to treat in order:--the relations of the earth and heaven; the effect of position upon the earth; the theory of the sun and moon, without which that of the stars cannot be undertaken; the sphere of the fixed stars, and those of the five stars called planets. Arguments are then produced for the spherical form and motion of the heavens, for the sensibly spherical form of the earth, for the earth being in the centre of the heavens, for its being but a point in comparison with the distances of the stars, and its having no motion of translation. Some, it is said, admitting these reasons, nevertheless think that the earth may have a motion of rotation, which causes the (then) only apparent motion of the heavens. Admiring the simplicity of this solution, Ptolemy then gives his reasons why it cannot be.
With these, as well as his preceding arguments, our readers are familiar. Two circular celestial motions are then admitted: one which all the stars have in common, another which several of them have of their own. From several expressions here used, various writers have imagined that Ptolemy held the opinion maintained by many of his followers, namely, that the celestial spheres are solid. Delambre inclines to the contrary, and we follow him.
It seems to us that, though, as was natural, Ptolemy was led into the phraseology of the solid-orb system, it is only in the convenient mode which is common enough in all systems. When a modern astronomer speaks of the variation of the eccentricity of the moon's orbit as producing a certain effect upon, say her longitude, any one might suppose that this orbit was a solid transparent tube, within which the moon is materially restrained to move. Had it not been for the notion of his successors, no one would have attributed the same to Ptolemy: and if the literal meaning of phrases have weight, Copernicus is at least as much open to a like conclusion as Ptolemy.

Then follows the geometrical exposition of the mode of obtaining a table of chords, and the table itself to half degrees for the whole of the semicircle, with differences for minutes, after the manner of recent modern tables. This morsel of geometry is one of the most beautiful in the Greek writers: some propositions from it are added to many editions of Euclid. Delambre, who thinks as meanly as he can of Ptolemy on all occasions, mentions it with a doubt as to whether it is his own, or collected from his predecessors. In this, as in many other instances, he shows no attempt to judge a mathematical argument by ally thing except its result : had it been otherwise, the unity and power of this chapter would have established a strong presumption in favour of its originality. Though Hipparchus constructed chords, it is to be remembered we know nothing of his manner as a mathematician; nothing, indeed, except some results. The next chapter is on the obliquity of the ecliptic as determined by observation. It is followed by spherical geometry and trigonometry enough for the determination of the connection between the sun's right ascension, declination, and longitude, and for the formation of a table of declinations to each degree of longitude. Delambre says he found both this and the table of chords very exact.

The second book is one of deduction from the general doctrine of the sphere, on the effect of position on the earth, the longest days, the determination of latitude, the points at which the sun is vertical, the equinoctial and solsticial shadows of the gnomon, and other things which change with the spectator's position. Also on the arcs of the ecliptic and equator which pass the horizon simultaneously, with tables for different climates, or parallels of latitude having longest days of given durations.
This is followed by the consideration of oblique spherical problems, for the purpose of calculating angles made by the ecliptic with the vertical, of which he gives tables.

The third book is on the length of the year, and on the theory of the solar motion. Ptolemy informs us of the manner in which Hipparchus made the discovery of the precession of the equinoxes, by observation of the revolution from one equinox to the same again being somewhat shorter than the actual revolution in the heavens.
He discusses the reasons which induced his predecessor to think there was a small inequality in the length of the year, decides that he was wrong, and produces the comparison of his own observations with those of Hipparchus, to show that the latter had the true and constant value (one three-hundredth of a day less than 365 1/4 days).
As this is more than six minutes too great, and as the error, in the whole interval between the two, amounted to more than a day and a quarter, Delambre is surprised, and with reason, that Ptolemy should not have detected it.
He hints that Ptolemy's observations may have been calculated from their required result; on which we shall presently speak.
It must be remembered that Delambre watches every process of Ptolemy with the eye of a lynx, to claim it for Hipparchus, if he can; and when it is certain that the latter did not attain it, then he might have attained it, or would if he had lived, or at the least it is to be matter of astonishment that he did not.

Ptolemy then begins to explain his mode of applying the celebrated theory of excentrics, or revolutions in a circle which has the spectator out of its centre; of epicycles, or circles, the centres of which revolve on other circles, &c.
As we cannot here give mathematical explanations, we shall refer the reader to the general notion which he probably has on this subject, to Narrien's History of Astronomy, or to Delambre himself.
As to the solar theory, it may be sufficient to say that Ptolemy explains the one inequality then known, as Hipparchus did before him, by the supposition that the circle of the sun is an excentric; and that he does not

On this theory of epicycles, we may say a word once for all.
The commbn notion is that it was a cumbrous and useless apparatus, thrown away by the moderns, and originating in the Ptolemaic, or rather Platonic, notion, that all celestial motions must either be circular and uniform motions, or compounded of them.
But on the contrary, it was an elegant and most efficient mathematical instrument, which enabled Hipparchus and Ptolemy to represent and predict much better than their predecessors had done; and it was probably at least as good a theory as their instruments and capabilities of observation required or deserved. And many readers will be surprised to hear that the modern astronomer to this day resolves the same motions into epicyclic ones. When the latter expresses a result by series of sines and cosines (especially when the angle is a mean motion or a multiple of it) he uses epicycles; and for one which Ptolemy scribbled on the heavens, to use Milton's phrase, he scribbles twenty.
The difference is, that the ancient believed in the necessity of these instruments, the modern only in their convenience; the former used those which do not sufficiently represent actual phenomena, the latter knows how to choose better; the former taking the instruments to be the actual contrivances of nature, was obliged to make one set explain every thing, the latter will adapt one set to latitude, another to longitude, another to distance. Difference enough, no doubt; but not the sort of difference which the common notion supposes.

The fourth and fifth books are on the theory of the moon, and the sixth is on eclipses.
As to the moon, Ptolemy explains the first inequality of the moon's motion, which answers to that of the sun, and by virtue of which (to use a mode of expression very common in astronomy, by which a word properly representative of a phenomenon is put for its cause) the motions of the sun and moon are below the average at their greatest distances from the earth, and above it at their least.
This inequality was well known, and also the motion of the lunar apogee, as it is called; that is, the gradual change of the position of the point in the heavens at which the moon appears when her distance is greatest. Ptolemy, probably more assisted by records of the observations of Hipparchus than by his own, detected that the single inequality above mentioned was not sufficient, but that the lunar motions, as then known, could not be explained without supposition of another inequality, which has since been named the evection. Its effect, at the new and full moon, is to make the effect of the preceding inequality appear different at different times; and it depends not only on the position of the sun and moon, but on that of the moon's apogee.
The disentanglement of this inequality, the magnitude of which depends upon three angles, and the adaptation of an epicyclic hypothesis to its explanation, is the greatest triumph of ancient astronomy.

The seventh and eighth books are devoted to the stars.
The celebrated catalogue (of which we have before spoken) gives the longitudes and latitudes of 1022 stars, described by their positions in the constellations.
It seems not unlikely that in the main this catalogue is really that of Hipparchus, altered to Ptolemy's own time by assuming, the value of the precession of the equinoxes given by IIipparchus as the least which could be ; some changes having also been made by Ptoiemy's own observations.
This catalogue is pretty well shown by Delambre (who is mostly successful when he attacks Ptolemy as an observer) to represent the heaven of Hipparchus, altered by a wrong precession, better than the heaven of the time at which the catalogue was made. And it is observed that though Ptolemy observed at Alexandria, where certain stars are visible which are not visible at Rhodes (where Hipparchus observed), none of those stars are in Ptolemy's catalogue.
But it may also be noticed, on the other hand, that one original mistake (in the equinox) would have the effect of making all the longitudes wrong by the same quantity; and this one mistake might have occurred, whether from observation or calculation, or both, in such a manner as to give the suspicious appearances.

The remainder of the thirteen books are devoted to the planets, on which Hipparchus could do little, except observe, for want of long series of observations. Whatever we may gather from scattered hints, as to something having been done by Hipparchus himself, by Apollonius, or by any others, towards an explanation of the great features of planetary motion, there can be no doubt that the theory presented by Ptolemy is his own.

These are the main points of the Almagest, so far as they are of general interest. Ptolemy appears in it a splendid mathematician, and an (at least) indifferent observer.
It seems to us most likely that he knew his own deficiency, and that, as has often happened in similar cases, there was on his mind a consciousness of the superiority of Hipparchus which biassed him to interpret all his own results of observation into agreement with the predecessor from whom he feared, perhaps a great deal more than he knew of, to differ.
But nothing can prevent his being placed as a fourth geometer with Euclid, Apollonius, and Archimedes. Delambre has used him, perhaps, harshly; being, certainly in one sense, perhaps in two, an indifferent judge of the higher kinds of mathematical merit.

As a literary work, the Almagest is entitled to a praise which is rarely given; and its author has shown abundant proofs of his conscientious fairness and nice sense of honour.
It is pretty clear that the writings of Hipparchus had never been public property : the astronomical works which intervene between Hipparchus and Ptolemy are so poor as to make it evident that the spirit of the former had not infused itself into such a number of men as would justify us in saying astronomy had a scientific school of followers. Under these circumstances, it was open to Ptolemy, had it pleased him, most materially to underrate, if not entirely to suppress, the labours of Hipparchus; and without the fear of detection. Instead of this, it is from the former alone that we now chiefly know the latter, who is constantly cited as the authority, and spoken of as the master. Such a spirit, shown by Ptolemy, entitles us to infer that had he really used the catalogue of Hipparchus in the manner hinted at by Delambre, he would have avowed what he had done; still, under the circumstances of agreement noted above, we are not at liberty to reject the suspicion. We imagine, then, tnat Ptolemy was strongly biassed towards those methods both of observation and interpretation, which app. citt. ; Weidler, Hist. Astron. ; Lalande, Bibliogr. Astron. ; Hoffman, Lexic. Bibliogr. ; the editions named, except when otherwise stated ; Fabric. Bibl. Graec., &c.) [A. De M.]

The

The treatise of Ptolemy was based on an earlier work by Marinus of Tyre, of which we derive almost our whole knowledge from Ptolemy himself (1.6, &c.). He tells us that Marinus was a diligent inquirer, and well acquainted with all the facts of the science, which had been collected before his time; but that his system required correction, both as to the method of delineating the sphere on a plane surface, and as to the computation of distances : he also informs us that the data followed by Marinus had been, in many cases, superseded by the more accurate accounts of recent travellers. It is, in fact, as the corrector of those points in the work of Marinus which were erroneous or defective, that Ptolemy introduces himself to his readers; and his discussion of the necessary corrections occupies fifteen chapters of his first book (cc. 6-20). The most important of the errors which he ascribes to Marinus, is that he assigned to the known part of the world too small a length from east to west, and too small a breadth from north to south. He himself has fallen into the opposite error.

Before giving an account of the system of Ptolemy, it is necessary to notice the theory of Brehmer, in his

In examining the geographical system of Ptolemy, it is convenient to speak separately of its mathematical and historical portions; that is, of his notions respecting the figure of the earth, and the mode of determining positions on its surface, and his knowledge, derived from positive information, of the form and extent of the different countries, and the actual positions and distances of the various places in the then known world.

1. The Mathematical Geography of Ptolemy.-- Firstly, as to the figure of the earth. Ptolemy assumes, what in his mathematical works he undertakes to prove, that the earth is neither a plane surface, nor fan-shaped, nor quadrangular, nor pyramidal, but spherical.
It does not belong to the present subject to follow him through the detail of his proofs.

The mode of laying down positions on the sur face of this sphere, by imagining great circles passing through the poles, and called meridians, because it is mid-day at the same time to all places through which each of them passes; and other circles, one of which was the great circle equidistant from the poles (the equinoctial line or the equator), and the other small circles parallel to that one; and the method of fixing the positions of these several circles, by dividing each great circle of the sphere into 360 equal parts (now called degrees, but by the Greeks "parts of a great circle"), and imagining a meridian to be drawn through each division of the equator, and a parallel through each division of any meridian ;--all this had been settled from the time of Eratosthenes. What we owe to Ptolemy or to Marinus (for it cannot be said with certainty to which) is the introduction of the terms longitude (latilude (length of the known world, that is, its distance, in degrees, from a fixed meridian, measured along its own parallel; and the latter to describe the position of a place with reference to the breadth of the known world, that is, its distance, in degrees, from the equator, measured along its own meridian. Having introduced these terms, Marinus and Ptolemy designated the positions of the places they mentioned, by stating the numbers which repesent the longitudes and latitudes of each.
The subdivision of the degree adopted by Ptolemy is into twelfths.

Connected with these fixed lines, is the subject of climates, by which the ancients understood belts of the earth's surface, divided by lines parallel to the equator, those lines being determined according to the different lengths of the day (the longest day was the standard) at different places, or, which is the same thing, by the different lengths, at different Clima, 2nd ed.

Next, as to the size of the earth. Various attempts had been made, long before the time of Ptolemy, to calculate the circumference of a great circle of the earth by measuring the length of an arc of a meridian, containing a known number of degrees. Thus Eratosthenes, who was the first to attempt any complete computation of this sort from his own observations, assuming Syene and Alexandria to lie under the same meridian fucts of geography, but only with the opinions of the ancient geographers, we do not stay to correct the errors in the data of these computations.Cyc. Theor. 1.8; Ukert, Geogr. d. Griech. u. Römer, vol. i. pt. 2, pp. 42-45).
The most important of the other computations of this sort were those of Poseidonius, (for he made two,) which were founded on different estimates of the distance between Rhodes and Alexandria : the one gave, like the computation of Eratosthenes, 252,000 stadia for the circumference of a great circle, and 700 stadia for the length of a degree; and the other gave 180,000 stadia for the circumference of a great circle, and 500 stadia for the length of a degree (Cleomed. 1.10; Strab. ii. pp. 86, 93, 95, 125 ; Ukert, l.c. p. 48).
The truth lies just between the two; for, taking the Roman mile of 8 stadia as 1-75th of a degree, we have (75 x 8 =) 600 stadia for the length of a degree. the Dictionary of Antiquities, art. Stadium

Ptolemy followed the second computation of Poseidonius, namely, that which made the earth 180,000 stadia in circumference, and the degree 500 stadia in length; but it should be observed that he, as well as all the ancient geographers, speaks of his computation as confessedly only an approximation to the truth. He describes, in bk. 1.100.3, the method of finding, from the direct distance in stadia of two places, even though they be not under the same meridian, the circumference of the whole earth, and conversely. There having been found, by means of an astronomical instrument, two fixed stars distant one degree from each other, the places on the earth were sought to which those stars were in the zenith, and the distance between those places being ascertained, this distance was, of course (excluding errors), the length of a degree of the great circle passing through those places, whether that circle were a meridian or not.

The next point to be determined was the mode of representing the surface of the earth with its meridians of longitude and parallels of latitude, on a sphere, and on a plane surface.
This subject is discussed by Ptolemy in the last seven chapters of his first book (18-24), in which he points out the imperfections of the system of delineation adopted by Marinus, and expounds his own. Of the two kinds of delineation, he observes, that on a sphere is the easier to make, as it involves no method of projection, but is a direct representation; but, on the other hand, it is inconvenient to use, as only a small portion of the surface can be seen at once : while the converse is true of a map on a plane surface.
The earliest geographers had no guide for their maps but reported distances and general notions of the figures of the masses of land and water. Eratosthenes was the first who called in the aid of astronomy, but he did not attempt any complete projection of the sphere (see ERATOSTHENES, and Ukert, vol. i. pt. 2, pp. 192, 193, and plate ii., in which Ukert attempts a restoration of the map of Eratosthenes). Hipparchus, in his work against Eratosthenes, insisted much more fully on the necessary connection between geography and astronomy, and was the first who attempted to lay down the exact positions of places according to their latitudes and longitudes.
In the science of projection, however, he went no further than the method of representing the meridians and parallels by parallel straight lines, the one set intersecting the other at right angles. Other systems of projection were attempted, so that at the time of Marinus there were several methods in use, all of which he rejected, and devised a new system, which is described in the following manner by Ptolemy (

Ptolemy then proceeds to describe his own method, which does not admit of an abridged statement, and cannot be understood without a figure.
The reader is therefore referred for it to Ptolemy's own work (1.24), and to the accounts given by Ukert (l.c. pp. 195, &c.), Mannert (vol. i. pp. 127, &c.), and other geographers. All that can be said of it here is that Ptolemy represents the parallels of latitude as arcs of concentric circles (their centre representing the North Pole), the chief of which are those passing through Thule, Rhodes, and Meroe, the Equator, and the one through Prasum.
The meridians of longitude are represented by

2. The Historical or Positive Geography of Ptolemy.-- The limits just mentioned, as those within which Ptolemy's projection of the sphere was contained, were also those which he assigned to the known world. His own account of its extent and divisions is given in the fifth chapter of his seventh book.
The boundaries which he there mentions are, on the east, the unknown land adjacent to the eastern nations of Asia, namely, the Sinae and the people of Serica; on the south, the unknown land which encloses the Indian Sea, and that adjacent to the district of Aethiopia called Agisymba, on the south of Libya; on the west, the unknown land which surrounds the Aethiopic gulf of Libya, and the Western Ocean; and on the north, the continuation of the ocean, which surrounds the British islands and the northern parts of Europe, and the unknown land adjacent to the northern regions of Asia, namely Sarmatia, Scythia, and Serica.

He also defines the boundaries by meridians and parallels, as follows. The southern limit is the parallel of 16 3/12° S. lat., which passes through a point as far south of the equator, as Meroe is north of it, and which he elsewhere describes as the parallel through Prasum, a promontory of Aethiopia : and the northern limit is the parallel of 63° N. lat., which passes through the island of Thule : so that the whole extent from north to south is 79 3/12°, or in round numbers, 80°; that is, as nearly as possible, 40,000 stadia. The eastern limit is the meridian which passes through the metropolis of the Sinae, which is 119 1/2 ° east of Alexandria, or just about eight hours : and the western limit is the meridian drawn through the Insulae Fortunatae (the Canaries) which is 60 1/2 °, or four hours, west of Alexandria, and therefore 180°, or twelve hours, west of the easternmost meridian.
The various lengths of the earth, in itinerary measure, he reckons at 90,000 stadia along the equator (500 stadia to a degree), 40,000 stadia along the northernmost parallel (222 2/9 stadia to a degree), and 72,000 stadia along the parallel through Rhodes (400 stadia to a degree), along which parallel most of the measurements had been reckoned.

In comparing these computations with the actual distances, it is not necessary to determine the true position of such doubtful localities as Thule and the metropolis of the Sinae; for there are many other indications in Ptolemy's work, from which we can ascertain nearly enough what limits he intends. We cannot be far wrong in placing his northern boundary at about the parallel of the Zetland Isles, and his eastern boundary at about the eastern coast of Cochin China, in fact just at the meridian of 110° E. long. fromm Greenwich), or perhaps at the opposite side of the Chinese Sea, namely, at the Philippine Islands at the meridian of 120°. It will then be seen that he is not far wrong in his dimensions from north to south; a circumstance natural enough, since the methods of taking latitudes with tolerable precision had long been known, and he was very careful to avail himself of every recorded observation which he could discover. But his longitudes are very wide of the truth, his length of the known world, from east to west, being much too great. The westernmost of the Canaries is in a little more than 18° W. long., so that Ptolemy's easternmost meridian (which, as just stated, is in 110° or 120° E. long.) ought to have been that of 128 or 138°, or in round numbers 130° or 140°, instead of 180° ; a difference of 50° or 40°, that is, from 1-7th to 1-9th of the earth's circumference.

It is well worthy, however, of remark in passing, that the modern world owes much to this error ; for it tended to encourage that belief in the practicability of a western passage to the Indies, which occasioned the discovery of America by Columbus.

There has been much speculation and discussion as to the cause of Ptolemy's great error in this matter; but, after making due allowance for the uncertainties attending the computations of distance on which he proceeded, it seems to us that the chief cause of the error is to be found in the fact already stated, that he took the length of a degree exactly one sixth too small, namely, 500 stadia instead of 600. As we have already stated, on his own authority, he was extremely careful to make use of every trustworthy observation of latitude and longitude which he could find; but he himself complains of the paucity of such observations ; and it is manifest that those of longitude must have been fewer and less accurate than those of latitude, both for other reasons, and chiefly on account of the greater difficulty of taking them. He had, therefore, to depend for his longitudes chiefly on the process of turning into degrees the distances computed in stadia; and hence, supposing the distances to be tolerably correct, his error as to the longitudes followed inevitably from the error in his scale. Taking Ptolemy's own computation in stadia, and turning it into degrees of 600 stadia each, we get the following results. The length of the known world, measured along the equator, is 90,000 stadia; and hence its length in degrees is 90,000/600 = 150°; the error being thus reduced from 50° or 40° to 20° 10°. But a still fairer method is to take the measurement along the parallel of Rhodes, namely 72,000 stadia. Now the true length of a degree of latitude in that parallel is about 47' = 47/60 of a degree of a great circle = 47/60 x 600 stadia = 470 stadia, instead of 400; and the 72,000 stadia give a little over 153 degrees, a result lamost identical with the former. The remaining error of 20° at the most, or 10° at the least, is, we think, sufficiently accounted for by the errors in the itinerary measures, which experience shows to be almost always on the side of making distances too great, and which, in this case, would of course go on increasing, the further the process was continued eastward. Of this source of error Ptolemy was himself aware; and accordingly he tells us that, among the various computations of a distance, he always chose then least; but, for the reason just stated, that least one was probably still too great.

The method pursued by Ptolemy in laying down the actual positions of places has already been incidentally mentioned in the foregoing discussion.
He fixed as many positions as possible by their

His general ideas of the form of the known world were in some paints more correct, in others less so, than those of Strabo.
The elongation of the whole of course led to a corresponding distortion of the shapes of the several countries.
He knew the southern part of the Baltic, but was not aware of its being an inland sea.
He makes the Palus Maeotis far too large and extends it far too much to the north. The Caspian he correctly makes an inland sea (instead of a gulf of the Northern Ocean), but he errs greatly as to its size and form, making its length from E. to W. more than twice that from N. to S.
In the southern and south-eastern parts of Asia, he altogether fails to represent the projection of Hindostan, while, on the other hand, he gives to Ceylon (Taprobane) more than four times its proper dimensions, probably through confounding it with the mainland of India itself, and brings down the southern part of it below the equator.
He shows an acquaintance with the Malay peninsula (his Aurea Chersonesus) and the coast of Cochin China; but, probably through mistaking the eastern Archipelago for continuous land, he brings round the land which encloses his Sinus Magnus and the gulf of the Sinae (probably either the gulf of Siam and the Chinese Sea, or both confounded together) so as to make it enclose the whole of the Indian Ocean on the south.
At the opposite extremity of the known world, his idea of the western coast of Africa is very erroneous.
He makes it trend almost due south from the pillars of Hercules to the Hespera Keras in 85/12 N. lat., where a slight bend to the eastward indicates the Gulf of Guinea; but almost immediately afterwards the coast turns again to the S. S. W.; and from the expression already quoted, which Ptolemy uses to describe the boundary of the known world on this side, it would seem as if he believed that the land of Africa extended here considerably to the west. Concerning the interior of Africa he knew considerably more than his predecessors. Several modern geographers have drawn maps to represent the views of Ptolemy; one of the latest and best of which is that of Ukert (Geogr. d. Griech. u. Römer, vol. i. pl. 3).

Such are the principal features of Ptolemy's geographical system. It only remains to give a brief outline of the contents of his work, and to mention the principal editions of it. Enough has already been said respecting the first, or introductory book. The next six books and a half (ii.--7.4) are occupied with the description of the known world, beginning with the West of Europe, the description of which is contained in book ii.; next comes the East of Europe, in book iii.; then Africa, in book iv.; then Western or Lesser Asia, in book v.; then the Greater Asia, in book vi.; then India, the Chersonesus Aurea, Serica, the Sinae, and Taprobane, in book vii. cc. 1-4. The form in which the description is given is that of lists of places with their longitudes and latitudes, arranged under the heads, first, of the three continents, and then of the several countries and tribes. Prefixed to each section is a brief general description of the boundaries and divisions of the part about to be described; and remarks of a imiscellaueous character are interspersed among the lists, to which, however, they bear but a small proportion.

The remaining part of the seventh, and the whole of the eighth book, are occupied with a description of a set of maps of the known world, which is introduced by a remark at the end of the 4th chapter of the 7th book, which clearly proves that Ptolemy's work had originally a set of maps appended to it. In 100.5 he describes the general map of the world.
In cc 6, 7, he takes up the subject of spherical delineation, and describes the armillary sphere, and its connection with the sphere of the earth.
In the first two chapters of book viii., he explains the method of dividing the world into maps, and the mode of constructing each map and he then proceeds (cc. 3-28) to the description of the maps themselves, in number twenty-six, namely, ten of Europe, four of Libya, and twelve of Asia. The 29th chapter contains a list of the maps, and the countries represented in each; and the 30th an account of the lengths and breadths of the portions of the earth contained in the respective maps.
These maps are still extant, and an account of them is given under AGATHODAEMON, who was either the original designer of them, under Ptolemy's direction, or the constructor of a new edition of them.

Enough has been already said to show the great value of Ptolemy's work, but its perfect integrity is another question.
It is impossible but that a work, which was for twelve or thirteen centuries the text-book in geography, should have suffered corruptions and interpolations; and one writer has contended that the changes made in it during the middle ages were so great, that we can no longer recognise in it the work of Ptolemy (Schlözer, Nord, Gesch. in the

Lex. Bibliog. Script. Graec.