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established. Ptolemy does not give them, but in each case when | by Eratosthenes and used by Ilipparchus. This is followed by required applies the theorem of Menelaus for spherics directly. This spherical geometry and trigonometry enough for the determination greatly increases the length of his demonstrations, which the modern of the connexion between the sun's right ascension, declination, reader finds still more cumbrous, inasmuch as in each case it was and longitusle, and for the formation of a table of declinations to necessary to express the relation in terms of chords—the equivalents cach degree of longitude. Delambre says he found both this and of sines—only, cosines and tangents being of later invention. the table of chords very exact.” 3

Such, then, was the trigonometry of the Greeks. Mathe In book ii., after some remarks on the situation of the habitabile matics, indeed, has ever been, as it were, the handmaid parts of the earth, Ptolemy proceeds to inake deductions from the of astronomy, and many important methods of the former principles established in the preceding book, which he does by

means of the theorem of Menelans. The length of the longest arose from the needs of the latter. Moreover, by the found- day being given, he shows how to determine the arcs of the horizon ation of trigonometry, astronomy attained its final general intercepted between the equator and the ecliptic the amplitude constitution, in which calculations took the place of dia- of the eastern point of the celiptie at the solstico--for different grams, as these latter had been at an earlier period sub-hegrees of obliquity of the sphere; hence he finds the height of the stituted for mechanical apparatus in solving the ordinary at what places and times the sun becomes vertical and how to p'roblems. Further, we find in the application of trigon calculate the ratios of gnomons to their equinoctial and solstitial ometry to astronomy frequent examples and even a sys- latter methol is wanting in precision. All these matters he contematic use of the method of approximations,—the basis, siders fully and works out in detail for the parallel of Rhodes. in fact, of all application of mathematics to practical Theon gives us three reasons for the selection of that parallel by questions. There was a disinclination on the part of the Ptolemy: the tirst is that the height of the pole at Rhodes is 360, Greek geometer to be satisfied with a mere approximation,

a whole number, whereas at Alexandria he believed it to be 30' 58': were it ever so close; and the unscientific agrimensor tions; the third is that the climate of Rhodes holds the mean place

the second is that Hipparchus hail maile at Rhodes many obser'Valshirked the labour involved in acquiring the knowledge of the seven climates subsequently described. Delambre suspects which was indispensable for learning trigonometrical cal a fourth reason, which he thinks is the true one, that I'tulemy hai culations. Thus the development of the calculus of taken his examples from the works of Hipparehus, who observed at approximations fell to the lot of the astronomer, who was

Rhodes and had made these calculations for the place where he livedl. both scientific and practical."

In chapter vi. Ptolemy gives an exposition of the most important

properties of each parallel, commencing with the equator, which he We now proceed to notice briefly the contents of the Almagest. considers as the southern limit of the habitable quarter of the It is divided into thirteen books. The first book, which may be earth. For each parallel or climate, which is determined by the rgarlel as introductory to the whole work, opens with a short length of the longest day, he gives the latitude, a principal julance prefare, in which Ptolemy, after some observations on the distine on the parallel, and the lengths of the shadows of the unomon at iion between theory and practice, gives Aristotle's division of the the solstices and cquinox. In the next chapter he enters into par. vienes and remarks on the certainty of mathematical knowledge, ticulars and inquires what are the arts of the equator which cross

inasmuch as the demonstrations in it proceed by the incontrovert the horizon at the same time as given arcs of the ecliptic, or, which ible ways of arithmetic and geoinetry." He concludes his preface comes to the same thing, the time which a given are of the ecliptic with the statement that he will make use of the discoveries of his takes to cross the horizon of a given place. He arrives at a formula porcedecessors, and relate briefly all that has been sufficiently explained for calculating ascensional differeuces and gives tables of ascensions loy the ancients, but that he will treat with more care and develop arranged by 10' of longitude for the clifferent climates from the ment whatever has not been well understood or fully treated. equator to that where the longest day is seventeen hours. He Ptolemy unfortunately loes not always bear this in mind, and it then shows the use of these tables in the investigation of the length is sometimes difficult to distinguish what is due to him from that of the clay for a given climato, of the manner of reuring temporal' which he has borrowed from his predecessors.

to equinoctial hours and tier Irs, and of the nongresimal point Ptolemy then, in the first chapter, presupposing some preliminary and the point of orientation of the ecliptic. In the following notions on the part of the reader, announces that he will treat in chapters of this book he determines the angles formed by the inters onder-what is the relation of the earth to the heavens, what is the sertions of the cliptic first with the meridian, then with the "sition of the oblique circle (the ecliptic), and the situation of the horizon, and lastly with the vertical«ircle- no concludes by giving inlı ahitot parts of the earth; that he will point out the differences tables of the angles and arrs formed by the intersection of these of climates; that he will then pass on to the consideration of the circles, for the seven climates, from the pallel of Merceithirteen motion of the sun and moon, without which one cannot have a hours to that of the mouth of the Borysthenes sistoon hour. just theory of the stars ; lastly, that he will consider the sphere of These tables, le adels, shoulil luo completed by the situation of the thir ficel stars and then the theory of the five stars called "planets." | chief towns in all countries according to their latitudes and longi. All the things-1.6., the phenomena of the heavenly boilies -he tudes; this he promises to do in a separate treatise and has in fait sir he will onleavour to explain in taking for principle that which one in his lineproph!. s priilent, real, and certain, in resting everywhere on the surest Book iii. ileals of the motion of the sun and of the length of the obwrvations and applying geometrical methods. He then enters In order to understand the litt ulties of this question 011. summary oposition of the general principles on which his Ptolemy' says one should mail the books of the ancients, and openi Nyhris is based, and alluce's arguments to show that the heaven ally those of Hipparchus, whom lee paises is a loves of labour in of in polerical form and that it moves after the manner of a anil a lover of truth" laro, qulotóry tr ouni nai qulantei. He pohor, that the earth also is of a form which is sensibly spherical, begins by telling is hos Hippur; dous win bord to divor the pur. that the earth is in the centre of the heavens, that it is but a quint cession of the quinones tel.sters tloobosel Victiulis will in comparison with the distances of the stars, and that it has not Hippurhuus to his rond great discover thuit of the meantiity any motion of translation. With respect to the revolution of the of the solar orbit, and gives the lay will win of the picentrilis

irth round its aris, which he says some have held, I'tolemy, while while he explained the in'nality of ille si!!'* motion. Toleri Imitting that this supposition renders the explanation of the concludes this book hy sienipormition of the citiem pohononiena of the heavens much more simple, vet regards it as namn on which the cution of the top.nl 111 il.is !! dito.meher rili,ulous Lastly, he lays down that there are two redder will find in the artilo.J-11:OVY Wol.in. p.750. I'telu prin ipuil an different motions in the heavens - one loy which all morrover, applies polloman's luy potlesis of the selo 10 expoliai! iler sin ar carried from east to wrst uniformly about the poles of the inequality of the sun's mutoli

, ed shorts that it lolo titl... the Mopfor: than other, which is peculiar to some of the stars, is in Sme risults is tl., lyquelle aim of the nirii Il por foratlar Intrary direction to the former motion and takes plac round luttur hypoi:19 120 mp. r .;'.i1in" on : 11.2 1198 two !1tforenie wolpe. The preliminary notions, which are all older than motions, inil. "ely tie lucription :/:"!,, 1:. In the 1. form the subjects of the sumonil and following chapters. Mondhatter then tot el 1.:. :).il attention Hopi poremented to the construction of his table of Chonts, of noull belirotul. Wifimilir?:? When thuit for the Shih Ho hare given an around, and which is indispensabile to cyanation of pe 1:A ON 301 il...Diep hispatlaj primaisemnomy. The employment of this table prrallapse this is in quiloloofsis,'1-1. po:i.!.!.:!12 it in ......!i viji, iho rulation of the obliquits of the cliptic, the knowledge of by the colon 11.11% 111 pitan! 1007.: T. - : :1. Juli :* who is in loond the foundation of all astronomical science. l'oleur

I MI....! lo... 10 th- nrte ohapter indicates two means of determining this angle!

illi, tig oberration, les ribes the instruments he emplovi-l for that

6.1."1'1111, 11.11'." parpone, ao fin is the same ralue which had already been found • Kann471,,:..! !!. Ten! ".-;,.11: ::

the nell; :!

... - 1.1 ..... ... ... 11 Emile. Suotime de Produtique Powfin, iii. 324.

divisi 1.fi. lio'1.-5.1 SL-' .':,:13 ! . It'App, l'orlesungen der Geschichte der Viithematik, p. 376 Sli.., :1. II., 1.1:.


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which is of universal application, may, we think-regarıl being paid | tudes, arranged according to their constellations; and the eighth to its place in the Almagest—be justly attributed to Hipparchus. book commences with a similar catalogue of the stars in the conIt is the first law of the “philosophia prima” of Cointe. We find in stellations of the southern hemisphere. This catalogue has been the same page another principle, or rather practical injunction, that the subject of keen controversy amongst modern astronomers

. in investigations founded on observations where great delicacy is Some, as Flamsteed and Lalande, maintain that it was the same required we should select those made at considerable intervals of catalogue which Hipparchus had drawn up 265 years before Ptolemy, time in order that the errors arising from the imperfection which is whereas others, of whom Laplace is one, think that it is the work inherent in all observations, even in those made with the greatest of Ptolemy himself. The probability is that in the main the care, may be lessened by being distributed over a large number of catalogue is really that of Hipparchus altered to suit Ptolemy's years. In the same chapter we find also the principle laid down own time, but that in making the changes which were necessary that the object of mathematicians ought to be to represent all the a wrong precession was assumed. This is Delambre's opinion"; celestial phenomena by uniform and circular motions. This prin- he says, "Whoever may have been the true author, the catalogue ciple is stated by Ptolemy in the manner which is unfortunately is unique, and does not suit the age when Ptolemy lived ; by subtoo common with him,—that is to say, he does not give the least tracting 2° 40' from all the longitudes it would suit the age of indication whence he derived it. We know, however, from Sim- Hipparchus; this is all that is certain.”6. It has been remarked plicius, on the authority of Sosigenes,that I'lato is said to have that Ptolemy, living at Alexandria, at which city the altitude of proposed the following problem to astronomers : “What regular the pole is 5o less than at Rhodes, where Hipparchus observed, and determined motions being assumed would fully account for could have seen stars which are not visible at Rhodes; none of the phenomena of the motions of the planetary bodies?” We know, these stars, however, are in Ptolemy's catalogue. The eighth book too, from the same source that Eulemus says in the second book of contains, moreover, a description of the nilky way and the manner his History of Astronomy that “Eudoxus of Cnidus was the first of construeting a celestial globe; it also treats of the configuraof the Greeks to take in hand hypotheses of this kind,"3 that he tion of the stars, first with regard to the sun, moon, and planets, was in fact the first Greek astronomer who proposed a geometrical and then with regard to the horizon, and likewise of the different hypothesis for explaining the periodic motions of the planets—the aspects of the stars and of their rising, culmination, and setting famous system of concentric spheres. It thus appears that the simultaneously with the sun. principle laid down here by Ptolemy can be traced to Enoxus and The remainder of the work is devoted to the planets. The ninth I’lato ; and it is probable that they derived it from the same source, book commences with what concerns them all in general. The namely, Archytas and the Pythagoreans. We have indeed the planets are much nearer to the earth than the fixed stars and direct testimony of Geminus of Rhodes that the Pythagoreans inore distant than the moon. Saturn is the most distant of all, endeavoured to explain the phenomena of the heavens by uniform then Jupiter and then lars. These three planets are at a greater and circular motions. *

distance from the earth than the sun.? So far all astronomers are Books iv., v. are devoted to the motions of the moon, which are agreed. This is not the case, he says, with respect to the two very complicated; the moon in fact, though the nearest to us of remaining planets, Mercury and Venus, which the old astronomers all the heavenly bodies, lias always been the one which has given placed between the sun and earth, whereas more recent writers the greatest trouble to astronomers.5 Book iv., in which Ptolemy have placed them beyond the sun, because they were never seen on follows Hipparchus, treats of the first and principal inequality of the sun. He shows that this reasoning is not sound, for they the inoon, which quite corresponds to the inequality of the sun might be nearer to us than the sun and not in the same plane, and treated of in the third book. As to the observations which shoull | consequently never seen on the sun. He decides in favour of the be employed for the investigation of the motion of the moon, former opinion, which was indeed that of most mathematicians. Ptolemy tells us that lunar eclipses should be preferred, inasmuch The ground of the arrangement of the planets in order of distance as they give the moon's place without any error on the score of was the relative length of their periodic times; the greater the parallax. The first thing to be determined is the time of the circle, the greater, it was thought, would be the time required for moon's revolution ; Hipparchus, by comparing the observations of its description. llence we see the origin of the difficulty and the the Chaldæans with his own, discovered that the shortest perioil ditference of opinion as to the arrangement of the sun, Mercury, in which the lunar eclipses return in the same order was 126,007 and Venus, since the times in which, as seen from the earth, they days and 1 hour. In this period he finds 4267 lunations, 4573 appear to complete the circuit of the zodiac are nearly the samerestitutions of anomaly, and 4612 tropical revolutions of the moon a year. 10 Delambre thinks it strange that l’tolemy did not see that less 7° 9.1). ; this quantity (73%) is also wanting to complete the these contrary opinions could be reconciled by supposing that the 345 revolutions which the sun makes in the same time with respect two planets moved in epicycles about the sun; this would be to the fixed stars. He concludel from this that the lunar month stranger still, le ailds, if it is true that this idea, which is older contains 29 days and 31' 50' 8'' 20'''' of a day, very nearly, or 29 than Ptolemy, since it is referred to by Cicero, 11 had been that of the days 12 hours 14' 3" 20". These results are of the highest import- Egyptians. 1. It may be allded, as strangest of all, that this doctrine

(See ASTRONOMY.) In oriler to explain this inequality, or was held by Theon of Smyrna, 13 who was a contemporary of Ptolemy
the equation of the centre, Ptolemy makes use of the hypothesis of or somewliat senior to him. From this system to that of Tycho
an epicycle, which he prefers to that of the eccentric. The fifth Brahe there is, as Delambre observes, only a single step.
book commences with the description of the astrolabe of Ilip We have seen that the problem which presented itself to the
parchus, which Ptolemy made use of in following up the observa- astronomers of the Alexanılrian epoch was the following: it was
tions of that astronomer, and by means of which he made his most required to find such a system of equable circular motions as would
important discovery, that of the second inequality in the moon's represent the inequalities in the apparent motions of the sun, the
motion, now known by the name of the evection." In order moon, and the planets. Ptolemy now takes up this question for
to explain this inequality he supposed the moon to move on an the planets; he says that “this perfection is of the essence of
epicycle, which was carrieıl by an eccentric whose centre turned celestial things, which admit of neither disorder nor inequality,”
about the earth in a direction contrary to that of the motion of that this planetary theory is one of extreme dilliculty, and that
the epicycle. This is the first instance in which we find the two no one had yet completely succeeded in it. He adds that it was
hypotheses of eccentric and epicycle combined. The fifth book owing to these difficulties that Ilipparchus—who loved truth abovo
treats also of the parallaxes of the sun and moon, and gives a all things, and who, moreover, had not received from his prede-
description of an instrument-called later by Theon the "parallactic c'essors observations either so numerous or so precise as those that
rods ”—Levised by Ptolemy for observing meridian altitudes with he has left-had succeeded, as far as possible, in representing tho
greater accuracy.

motions of the sun and moon by circles, but had not even com-
The subject of parallaxes is continued in the sixth book of the menced the theory of the five planets. He was content, Ptolemy
Almagest, and the method of calculating eclipses is there given.
The author says nothing in it which was not known before his time.

6 Delambre, Histoire de l'Istronomie Ancienne, ii. 204.

istances; but we know that Mars at opposition Books vii., viii. treat of the fixed stars. Ptolemy verified the fixity of their relative positions and confirmed the observations of

8 Eratosthenes, for example, as we learn from Theon of Smyrna. Hipparchus with regard to their motion in longitude, or the pre 9 Transits of Mercury and Venus over the sun's disk, therefore, had not been cession of the cquinoxes. (See ASTRONOMY.) The seventh book

10 This was known to Eudoxus. Sir George Cornewall Lewis (An Historical concludes with the catalogue of the stars of the northern hemi Surrey of the Astronomy of the Incients, 1. 155), confusing the geocentric revolusphere, in which are entered their longitudes, latitudes, and magni tions assigned by Eudoxus to these two planets with the heliocentric revolu

tions in the Copernican system, which are of course quite different, says that 1 Système dle Politique Positive, iv. 173.

“the error with respect to Mercury and Venus is considerable"; this, however, 2 This Sosigenes, as Th. H. Martin has shown, was not the astronomer of that

is an error not of Eudoxus but of Cornewall Lewis, as Schiaparelli has remarked. name who was a contemporary of Julius Cæsar, but a Peripatetic philosopher who lived at the end of the 2d century A.D.

11 "Hunc solem) ut comites consequuntur Veneris alter, alter Mercurii cursus 3 Brandis, Schol. in Aristot. eiidit Acad. Reg. Borussica (Berlin, 1836), p. 498.

(Somnium Scipionis, De Rep., vi. 17). This hypothesis is alluded to by Pliny,

V. II., ii, 17, and is more explicitly stated by Vitruvius, Arch., ix. t. 4 Eibaywyn eis palvbueva, c. i. in Halina's edition of the works of 19 Macrobius, Commentariuser Cicerone in Somnium Scipionis, i. 19. Ptolemy, vol. iii. ("Introduction aus Phénomènes Célestes, traduite du Grec 13 Theon (Smyrna'us Platonicus), Liber de Astronomii, ed. Th. H. Martin (Paris, de Géminus," p. 9), Paris, 1819.

1819), PP. 174, 294, 290. Martin thinks that Theon, the mathematician, four of 5 This has been noticel by Pliny, who says, "Multiformi hæc (luna) ambaye whose observations are used by Ptolemy (11 m., ii. 176, 193, 194, 195, 196, el. torsit ingenia contemplantiuin, et proximum ignorari maxime sidus indignan- Halma), is not the same as Theon of Smyrnu, on the ground chietly that the tiun" (N. II., ii. 9).


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7 This is true of their mean is nearer to us than the sun.


latter was not an observer.

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continues, to arrange the observations which had been made on in which the length of the longest lay varies from 13 hours to 15} hours,them in a methodic order and to show thence that the phenomena that is, from the latitude of Syene to that of the middle of the Euxine. This did not agree with the hypotheses of mathematicians at that time.

work has been printeil by Petavius in his l'rmologium, Paris, 1050, and by

IIalma in his cilition of the works of Ptolemy, vol. iii., Paris, 1919. (-) He showed that in fact each planet had two inequalities, which

Υποθέσεις των πλανωμένων και των ουρανίων κύκλων κινήσεις, (On the are different for cach, that the retrogradations are also different, I'lanetary II ypothesis. This is a summary of a portion of the 11 mungest, and conwhilst other astronomers admitted only a single inequality and tains a brief statement of the principal hypotheses for the explanation of the the same retrogradation ; he showed further that their motions motions of the heavenly boilies. It was first publishel (Gr., Lat.) by Bainbridlye,

the Savilian professor of astronomy at Oxford, with the Sphere of Proclus and cannot be explained by eccentrics nor by epicycles carried along

the Kavwv faollec@v, London, 1020, anul afterwards by Halma, vol. ir., Paris, concentrics, but that it was necessary to combine both hypotheses.

1820. (3) Kavwv Baoilecwv, 1 Table of livigns. This is a chronological table After these preliminary notions he gives from Hipparchus the of Assyrian, Persian, Greek, and Roinan soverrigns, with the length of their periodic motions of the five planets, together with the shortest rrigns, from Nabonasar to Antoninus Pius. This talle (comp. G. Synellus, times of restitutions, in which, moreover, he has maile some slight Chronogr., el. Din., i. 388 sq.) has been printed by Scaliger, Calvisius, Patavius, corrections. He then gives tables of the mean motions in longitude

Bainbridge (as above noted), and by Ilalia, vol. iii., Paris, 1819. (4) 'Apuoviand of anomaly of each of the five planets,' and shows how the

KWv Biblía Ý. This Treatise on Jusic was published in Greek and Latin by

Wallis at Oxforil, 1682. It was afterwards reprinted with Porphyry's Collmotions in longitude of the planets can be represented in a general manner by means of the hypothesis of the eccentric combined with

mentary in the thiril volume of Wallis's works, Oxford, 16:9. (5) Tetpá, 313\os that of the epicycle. He next applies his theory to each planet

ouvražis, Tetrabiblon or Qualripartitum. This work is astrological, as is also

the small collection of aphorisms, caller Kaptós or Centiloquium, by which it and concludes the ninth book by the explanation of the various

is followedl. It is doubtful whether these works are genuine, but the doubt phenomena of the planet Mercury. In the tenth and eleventh merely arises from the feeling that they are unworthy of Ptolemy. They were books he treats, in like manner, of the various phenomena of the both published in Greek and Latin by Camerarius, Nurembery, 153.), and by planets Venus, Mars, Jupiter, and Saturn.

Melanchthon, Basel, 15. (6) Do nolemmute. The original of this work of

Ptolemy is lost. It was translated from the Arabic and published by comBook xii. treats of the stationary and retrograde appearances of maudine, Rome, 1562. The 1 nolemme is the lescription of the sphere on a each of the planets and of the greatest elongations of Mercury and plane. We find in it the sections of the different circles, as the diurnal parallels, Venus. The author tells us that some mathematicians, and amongst

and verything which can facilitate the intelligence of gnomonics. I his dis them Apollonius of Perga, employed the hypothesis of the epicycle

scription is made by perpendiculars let fall on the plane; whener it has burn

called by the moderns "orthographic projection." (7) I’lanispliipii 11. T? to explain the stations and retrogradations of the planets. Ptolemy Planisphone. The Greek text of this work also is lost, and we have only a goes into this theory, but does not change in the least the theorems Latin translation of it froin the Arabic. The "planisphere" is a projection of Apollonius; he only promises simpler and clearer demonstra

of the sphere on the equator, the eye being at the pole, in fact what is 101:

called "storeographic" projection. The best edition of this work is that of tions of them. Delambre remarks that those of Apollonius must

Commandine, Venice, 15:58. (5) Optics. This work is known to is only loy have been very obscure, since, in order to make the demonstrations imperfect manuscripts in Paris and Oxforil, which are Latin translation from in the lingest intelligible, he (Delambre) was obliged to recast

the Arabic; some extracts from them have been recently published. The polics

consists of live books, of which the fifth presents most interest: i: trents of them. This statement of l'toleiny is important, as it shows that

the refraction of luminous rays in their passare through inedia of different the mathematical theory of the planetary motions was in a toler densities, and also of astronomical refractions, on which subject tlir throry is hly forwarı state long before his time. Finally, book xiii. treats more complete than that of any astronomer before the time of Cassini. of the motions of the planets in latitude, also of the inclinations of allusion to the Almugst or to the subjent of refraction in the 1st it-elf:

Morgan loubts whether this work is genuine on account of the alience of their orbits and of the magnitule of these inclinations.

but his chief reason for doubting its authenticity is that the author of the pitics Thuy who wish to go into details and learn the mathematical explanation of was a poor geometer.

(G. J. 1.) this celebrate system of "cocentries" and "epicycles" are referred to the Ilmoipeat itself, which can be most conveniently studied in Halma's edition, to

Geography. Illamhuris lliboire de l'Astronomia Ancieme, the second volume of which is for Ptolemy is harılly less celebrated as a geographer than the most part le vote to the .Ilmuge:1, 3 or to Sarrien's History of Istronomy, in which tlie subjert is treated with reat clearness.

as an astronomer, and his great work on geography exerPolens conclu les his great work by saying that he has included in it everyThin: of practical utility which in his judgment should find a place in a

cised as great an intuence on the progress of that science frater on astronomy nt the time it was written, with relation as well to dis as did his mongest on that of astronomy. It became Misirled as to Diethos. His work was justly callel by him Madnuatiņ

indeed the paramount authority on all geographical quesSivrafis, for it was in fact the mathematical form of the work which caused

forms preferred to all others which trratrol of the same science, but not loy tions for a period of many centuries, and was only craduthe core method of yourfry and calculation." Accordingly, it soon spreni

ally superscled by the progress of maritime discovery in inum Ilocan.lria to all places where astronomy was cultivated; numerous copies Top 100 le of it, and it wcame the object of serious stuly on the part of lioth the 15th and 16th centuries. This exceptional position ochen and pupils. Amongst its numerous commentators may be mentioned l'appus and ton of Alexawrin in the 4th century and Proclus in the 5th. was due in a great measure to its scientific form, which 1. au translate into latin lay Bevetius, but this translation has not come rendered it very convenient and easy of reference; but, l.wn to ne. The incis was translated into Arabic at Baghdad by order of The enlightenel caliph 11. Jamin, who was himself an astronomer, about $27 apart from this consiileration, it was really the first attempt A N., anl the Arabic translation was revised in the following century by ever madle to place the study of wograpy on a truly Thali ilin Kurta. The emperor Frederick II. caused the .171.".post to ln trans. Ly'n frun the Arabic into Latin at Naples about 10:0. In the 19th century it scientific basis. The great astronomer Hipparchus had #translate from a linurk manuscript in the l'atican by Geory of Trebizond.

indeed pointed out, three centuries before the time of In ihr ume rentnry an aputon of the Ilmu Wis commenced loy Purlach

1wl) anel cumplrtal by his jopil and 110Cdor in the professorship of Ptolemy, that the only way to construct a really trustArmy in the univerity of Vienna, Riomontanus. The marliest clition If threrpitone is that of Venice, 10., an'l this was the first apfwarance of

worthy map of the Inhabited World would be by observa... Live in print. The first Complete elit fun of the low. w is that of l. tions of the latitude and longitude of all the principal T: hnetrin (Venire, 1315), a latin version from the Irable. The laten innslatin flor of Treluzeud was first printed in 1.399, at Venic The points on its surface, and laying down a map in accorilance lintut, which we not known in Europe until the 15th century, was first

with the positions thus determinel. But the material: :chal in the loth Ls Simon Gryn-14, who was also the first editor of the 1. text of Euclil, at Basel, 1:33 This clition was from a manuscript in for such a course of proceeding We're almost wholly wanting, :-!mry or Suremlwo-where it is no longer to le found which had been ritol Of Rosemontanna, to whom it was given by Carlinal Bessarion.

and, though Hipparchus made some approach to a corrert Thala alitun nofthem.Jes" is that of Halme, Cinek with French translation, division of the known worlil into zones of latitude, os ?!, the manuscripts the its

al history, or Falricing, E.Winthern cron, c. Harles, cul. I p. 2-0, "climata," as he termeil them, trustworthy observations Ploua e por fame. In excellent samary of the bibliograplıical history is :5-7 lw Woman in his article on Ptolemy already quotel.

eren of this character were in his time very few in unler, 7: Top of Ptolerus, which we nu piered to notive tisy loriofly, ama while the means of determinin:- longituiles coulil Hardly 14 ( 1:.·(1Ράσεις απλανών αστέρων και συναγωγή επισημασιαν, be said to exist. IIence prolalily it arose that no attempt * "iiv the Fired.-4.4 1.1 Cilation of Ing. It is a culplar irl cummin among the Gnrks nn less the name of Tamánua, sa

was made liy terecling geographers to follow the im9. The ruins anl -finanf the star in the mornin: os prin: portant

sio of lliurhus. Marina of Tire, who 4, which were many visible signs of the season with pries ini hanno fun with plation for climate, after the liveil shortly before the time of liolems, and winne work

1. Tnf the best mrtas.zista, 44, fris tampile, Veton, Democritus! is known to 115 only throuh that writer, aprire to love F.:' 174 Hiperbuk kr. Molemy, in or les to make his 'JU.7./11.ful

1.linke til tre the rnlightennl world of his fine, Vee the been the first to rime the poroololem tlu piroquel, and .-.ngerif the car net forp oogie prall.l.nly fort for each of the firporallel

lay down the map of the known work in according with It'ua. P.111 thrus mean moting with those of our molen tales the precejot u of Hirsjure1:17. His materials for the enfor?? 1a le the paralls miret. By "motion in lunsitule on lisP9f the cont of the picrel alwut the enemntrie, ant ly

tion of sich all-241 1. inde el misrally ina...11.13 a'r 'the motion of the star in ite pierria. : 17 th: 1din the correk fort anl the Frnch translation are girn in

and he was forced to content him-eli for the one posit 1-2-comie : the laster, bwver, sbr 1700€ lurral without re-forence : with determinations derisul net iruna-inomioisi prisus hom hepire le analşale of the Image of th="Trades

vations but from the calls of distance from itin tarina 1-rom,!e colans la plate mathematiquel Poolmnur."

and other such Inritules ha- still continue to live on• TuTon. 4. Aus dinius the tryis in prg/4-1 -1..

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means of determination are not available. The greater part was determined by actual observation ; but we learn from Ptolemy of the treatise of Marinus was occupied with the discussion of himself that this was not the case, and that such observations for these authorities, and it is impossible for us, in the absence latitude were very few in number, while the means of determining of the original work, to determine how far he had succeeded down by him were really, with very few exceptions, the result of

longitudes were almost wholly wanting. Hence the positions laid in giving a scientific form to the results of his labours ; computations of distances from itineraries and the statements of but we are told by Ptolemy himself that he considered travellers

, estimates which were liable to much greater error in them, on the whole, so satisfactory that he had made the ancient times than at the present day, from the want of any accurate work of his predecessor the basis of his own in regard to

mode of observing bearings, or portable instruments for the measure

ment of time, while they had no means at all of determining disall the countries bordering on the Mediterranean, a term tances at sea, except by the rough estimate of the time employed which would comprise to the ancient geographer almost in sailing from point to point. The use of the log, simple as it all those regions of which he had really any definite know- appears to us, was unknown to the ancients. But, great as would ledge. With respect to the more remote regions of the culation, they were in most cases increased by the permanent error

naturally be the errors resulting from such imperfect means of cal. world, Ptolemy availed himself of the information imparted | arising from the erroneous system of graduation adopted by Ptolemy by Varinus, but not without reserve, and has himself ex in laying them down upon his map. Thus, if he had arrived at l»lained to us the reasons that induced him in some instances the conclusion from itineraries that two places were 5000 stadis

from one another, he would place them at a distance of 10° apart, to depart from the conclusions of his predecessor. It is

and thus in fact separate them by an interval of 6000 stadia. very unjust to term Ptolemy a plagiarist from Marinus, as Another source of permanent error (though one of much less imhas been done by some modern authors, as he himself portance) which affected all his longitudes arose from the erroneacknowledges in the fullest manner his obligations to

ous assumption of his prime meridian. In this respect also he that writer, from whom he derived the whole mass of his Fortunate Islands (the Canaries) were situated farther west than

followed Marinus, who, having arrived at the conclusion that the materials, which he undertook to arrange and present to any part of the continent of Europe, had taken the meridian his readers in a scientific form. It is this form and ar through the outermost of this group as his prime meridian, from rangement that constitute the great merit of Ptolemy's whence he calculated all his longitudes eastwards to the Indian work and that have stamped it with a character wholly acquainted with the position and arrangements of the islands in

Ocean. But, as both Marinus and Ptolemy were very imperfectly distinct from all previous treatises on geography. But question, the line thus assumed was in reality a purely imaginary at the same time it possesses much interest, as showing one, being drawn through the supposed position of the outer island, the greatly increased knowledge of the more remote por which they placed 23° west of the Sacred Promontory (Cape St tions of Asia and Africa which had been acquired by geo

Vincent), which was regardeil by Marinus and Ptolemy, as it had

been by all previous geographers, as the westernmost point of the graphers since the time of Strabo and Pliny,

continent of Europe, while the real difference between the two is It will be convenient to consider separately the two not less than 9° 20'. Hence all Ptolemy's longitudes

, reckoned eastdifferent branches of the subject,—(1) the mathematical wards from this assumed line, were in fact about 70 less than they portion, which constitutes his geographical system, properly which continued so long in use among geographers in modern

would have been if really measured from the meridian of Ferro, so termed; and (2) his contributions to the progress of times. The error in this instance was the more unfortunate as the

positive knowledge with respect to the Inhabited World. longitude could not of course be really measured, or even calculated, See Plate 1. Mathematical Geography. As a great astronomer, Ptolemy from this imaginary line, but was in reality calculated in both VII., vol. was of course infinitely better qualified to comprehend and explain directions from Alexandria, westwards as well as eastwards (as

the mathematical conditions of the carth and its relations to the l’tolemy himself has done in his eighth book) and afterwards re-
celestial bodies that surround it than any preceding writers on verseil, so as to suit the supposed method of computation.
the special subject of geography. But his general views, except

It must be observed also that the equator was in like manner on a few points, did not differ from those of his most eminent placed by Ptolemy at a considerable distance from its true gcoprecursors Eratosthenes and Strabo. In common with them, he graphical position. The place of the equinoctial line on the surassumed that the carth was a globe, the surface of which was face of the globe was of course well known to him as a matter of (lividel by certain great circles-the equator and the tropies- theory, but as no observations could have been made in those parallel to one another

, and dividling the carth into five great remote regions he could only calenlate its place from that of the zones, the relations of which with astronomical phenomena were tropic, which he supposed to pass through Syene. And as he here, of course clear to his mind as a matter of theory, though in regarul as elsewhere, reckoned a degree of latitude as equivalent to 500 to the regions bordering on the equator, as well as to those ail- stadia, he inevitably made the interval between the tropic and the joining the polar circle, he could have had no confirmation of his equator too small by one-sixth; and the place of the former on the conclusions from actual observation. Ile adopted also from Ilip- surface of the earth being fixed by observation he necessarily carried parchus the division of the equator and other great circles into up the supposed place of the equator too high by more than 230 360 parts or “degrees” (as they were subsequently called, though geographical miles. But as he hal practically no geographical the word does not occur in this sense in Ptolemy), and supposed acquaintance with the equinoctial regions of the earth this error other circles to be drawn through these, from the equator to the

was of little importance. pole, to which he gave the name of “meridians.” He thus conceived With Marinus and Ptolemy, as with all preceding Greek geothe whole surface of the earth (as is done by modlern geographers) graphers, the most important line on the surface of the globe for to be covered with a complete network of “parallels of latitudo” and all practical purposes was the parallel of 36° of latitude, which "meridians of longitude,” terms which he himself was the first ex passes through the Straits of Gibraltar at one end of the Meditertant writer to employ in this technical sense. Within the network ranean, and through the Island of Rhodes and the Gulf of Issus at thus constructed it was the task of the scientific geographer to the other. It was thus regarded by Dicæarchus and almost all his place the outline of the world, so far as it was then known by successors as dividing the regions around the inland sea into two experience and observation.

portions, and as being continued in theory along the chain of Mount Unfortunately at the very outset of his attempt to realize this Taurus till it joined the great mountain range north of India; and conception he fell into an error which had the effect of vitiating from thence to the Eastern Ocean it was regardled as constituting all his subsequent conclusions. Eratosthenes was the first writer the diviling line of the Inhabited World, along which its length who had attempted in a scientific manner to cletermine the cir must be measured. But it sulliciently shows how inaccurate were cumference of the carth, and the result at which he arrived, that the observations and low imperfect the materials at his command, it amounted to 250,000 stadia or 25,000 geographical miles, was adopted by subsequent geographers, including

Strabo following

Marinus, describes this parallel as passing through Caralis Posidonius, however, who wrote about a century after Eratosthenes, in Sardinia and Lilybæum in Sicily, the one being really in 39° had made an independent calculation, by which he reduced the 12'lat., the other in 37° 50'. It is still more strange that he places circumference of the globe to 180,000 stadia, or less than three so important and well known a city as Carthage 1° 20' south of the fourths of the result obtained by Eratosthenes, and this computa- dividing parallel, while it really liés nearly 1° to the north of it. tion, on what grounds we know not, was unfortunately adopted by Marinus Tyrius, and from him by Ptolemy. The consequence of 1 Hipparchus had indeed pointed out long before the mode of dethis error was of course to make every degree of latitude or longi- termining longitudes by observations of eclipses, but the instance to tude (measured at the equator) equal to only 500 stadia (50 geo which he referred of the celebrated eclipse before the battle of Arbela, graphical miles), instead of its true equivalent of 600 stadia. Its which was seen also at Carthage, was a mere matter of popular obsereffects would indeed have been in some measure neutralized hai vation, of no scientific value. Yet Ptolemy seems to have known of there existed a suflicient number of points of which the position no other.


The great problem that had attracted the attention and exercised | stadia, in accordance with his uniform system of allowing 500 the ingenuity of all geographers from the time of Dicæarchus to stadia to 1° of latitude. Both distances were greatly in excess of that of Ptolemy was to determine the length and breadth of the the truth, independently of the error arising from this mistaken Inhabited World, which they justly regarded as the chief subject of system of graduation. The distances west of the Euphrates were the geographer's consideration. This question had been very fully of course comparatively well known, nor did Ptolemy's calculation liscussed by Marinus, who had arrived at conclusions widely dif- of the length of the Mediterrancan diller very materially from those ferent from those of his predecessors. Towards the north indeed of previous Greek geographers, though still greatly exceeding the there was no great difference of opinion, the latitude of Thule being truth, after allowing for the permanent error of graduation. The generally recognized as that of the highest northern lanıl, and this effect of this last cause, it must be remembered, would unfortunately was placed both by Marinus and Ptolemy in 630 lat., not very far be cumulative, in consequence of the longitudes being computer beyond the true position of the Shetland Islands, which had come from a fixed point in the west, instead of being reckoned east and in their time to be generally i lentified with the mysterious Thulo west from Alexanılria, which was undoubteilly the mode in which of Pytheas. The western extremity, as already mentioned, had been they were really calculated. The result of these combined causes in like manner determined by the prime meridian drawn through of error was to lead him to assign no less than 180°, or 12 hours, of the supposeil position of the Fortunate Islands. But towards the longitude to the interval between the meridian of the Fortunato south and east Marinus gave an enormous extension to the con Islands and that of Sera, which really amounts to about 1:30'. tinents of Africa and Asia, beyond what had been known to or But in thus estimating the length and breadth of the known suspected by the earlier geographers, and, though Ptolemy greatly world Ptolemy attached a very ditlerent sense to these terms from reduced his calculations, he still retained a very exaggerated esti that which they had generally borne with preceding writers. All mate of their results.

former Greek geographers, with the single exception of llipparchus, The additions thus made to the estimated dimensions of the had agreed in supposing the Inhabited World to be surrounded on known world were indeed in both directions based upon a real exten all sides by sea, and to form in fact a vast island in the midst of sion of knowledge, derival from recent information ; but unfortu a circumfluous ocean. This notion, which was probably deriveil nately the original statements were so perverted by misinterpretation originally from the Homeric fiction of an ocean stream, and was in applying them to the construction of a map as to give results certainly not based upon direct observation, was nevertheless of will ring widely from the truth. The southern limit of the world course in accordance with the truth, great as was the misconception as known to Eratosthenes, and even to Strabo (who had in this it involved of the extent and magnitude of the continents included respect no further knowledge than his predecessor more than two within this assumed boundary. Hence it was unfortunate that renturies before), had been fixed by them at the parallel which | Ptolemy shoull in this respect have gone back to the views of possed through the castern extremity of Africa (Cape Guarılafui), or Ilipparchus, and have assumed that the land extended indefinitely the Land of Cinnamon as they termed it, anıl that of the Sembritir to the north in the case of Europe and Seythia, to the cast in that corresponding to Sennaar) in the interior of the same continent. of Asia, and to the south in that of Africa. His boundary-line was This parallel, which would correspond nearly to that of 10° of true in each of these cases an arbitrary limit, beyond which lay the latituile, they supposed to be situated at a distance of 3400 stadia | Unknown Land, as he calls it. But in the last case he was not (340 geographical miles) from that of Meroe (the position of which content with giving to Africa an indefinite extension to the south: was neurately known), and 13,100 to the south of Alexandria ; | he assumel tlic existence of a rast prolongation of the land to the while they conceived it as passing, when prolonged to the cast waril, cast from its southernmost known point, so as to form a connesion through the island of Taprobano (Ceylon), which was universally with the south-eastern extremity of Asia, of the extent and position ricognized as the southernmost land of Asia. Both these geo- of which he hail a wholly erroneous islea. graphers were wholly ignorant of the vast extension of Africa to In this last case Marimus la derived from the voyages of recent the south of this line and even of the equator, and conceived it as navigators in the Indian Seas a knowledge of the fact that there trending away to the west from the Land of Cinnamon and then lay in that direction extensive lands which had been totally unto the north-rest to the Straits of Gibraltar. Marinus had, how known to previous geographers, and I'tolemy had acquireil still Fer, learnel from itineraries both by land and sea the fart of this more estensive information in this quarter. But unfortunately he niat extension, of which he haul inleed conecivel so exaggeratel hail formed a totally false conception of the bearings of the coasts in iilea that even after Ptolemy hail reduced it by more than a half thus madle known, and consequently of the position of the laus to it was still materially in excess of the truth. The eastern coast of which they belonged, and, instead of carrying the line of coast Africa was inderd tolerally well known, being frequented boy (ireek northwards from the Golden Chersonese ithe Malay Peninsula) to anul Roman traders, as far as a placo called Rhapta, opposite to China or the land of the Sina', he brought it down again towards Zanzibar, and this is placed by Ptolemy not far from its tre posi- the south after forming a great bay, so that he placu (attigaration in i 8. lat. But he adılel to this a bay of great extent as far the principal emporium in this part of lsin, and the farthest point as a promontory callel l'rasum (perhaps (apie Delgado), which he known to him-on a supposed line of cost, of unknown extent, plan in 159° s. lat. At the same time he assumed the position but with a direction from north to south. The liypothesis that in alwut the same juarallel of a region called Ayisymba, which was this land was continuous with the most southern part of Africa, suppl to have been discovered by a Roman general, whose so that the two enclosed one vast gull, though a mere assumption, Iturrary was employed by Marinus. Taking, therefore, this parallel is statel liy him as dutinitely as if it was based upon positive inin the limit of knowledge to the south, while he retained that of formation; and it was long received by medieval geographers as an Thule to the north, he assigned to the inhabited world a breadth of unquestioned fart. This circumstance undoubtedly contributed nearly 50', instead of less than 60', which it had occupied on the to perpetuate the error of -11pposing that Afrira could not lcirmap of Eratosthenes and Strabo.

cumnavigated, opposition to the more correct virws of Strabo It hallucina tiver blief with all the Greek geographers from all other carlier inophers: On the other hand, there can be no the carliest attempts at scientific grovtaplıy not only that the loubit that the undue estension of Ixit towards the ra-t, so as to 1. n.3th of the Inhibited Worlel greatly exceeded its breaslılı

, but diminishi by 30' of longitude the interval let ween that continent that it was more than twice as great, a wholly unfounded assump an the Western costs of Europe, dal a material influence in fostertron, but to whih their steerssors seem to have felt themselves ing the belief of Columbus amiothers that it was possible to trarlı lmunud to conform. Thus Marimus, while giving an unelme extension the Land of Spees lits the East Iwan island were then called to Ifrica fowanis the south, full into a similar error, but to a lar los cirert navigation towarıls the wrist. Tintor degre, in toril to the extension of Asia towards the cast. It is not surprising that livlomy shoull law fullen into conIl palco le really messed a giat advance in knowledge over all silerable errors","ting the movili-fille quilteps of 1. World; hie porader win the inreare traile with China for silk having leidt men in to the Mediterum and its dependenteis, as for an aquaintance, though of course of a very vague anılmal well as the risions that Suod tllli, : h uluit lie was in a hind, with the riut regions in Central Asia that lay to the plant of 17.11n Willained, the imperion of his motripolijal

L'amir in which hail formeel the limit of the Asiatie nations knowledge is 11111:), ap!«1:41:. Her for livet internt sollir firmely known to tfr Cirocks. But Marinus haud learned that Woll-establiebolat to dismin', ils fur in l.11:11.le's War con Prulure proveling eastward from the Stone Tower-a station at the Limnel. Thit op Mondialulunghe tormime 10:01 9 pils ...fore fows of this rang--to Srl, the capital city of the Sures, rupil blytheas wiihin on hit ila 77!!" | Moll, mil litule win months on the journey, and from thience he arrived at the of Roma', as it loys he will approximate 1779 enle that the distanıt }wtween the two points w.1s not 115.41These immanel R.........

!!! syni, Low shin:3:00 tardin, or 3020 geographical miles. I'rolemy, while luving loent!, profondo...:::bund:.-1031007/s liv jietl joints out the almanlity of this conclusion and the propeanl the foreli!!!...!! tha: :i.: loiro Bl..jolly od 1 doo 1?128 para of computation on which it was founulel, hal no 111 of SHIM?! 12:1**, with the solicitar a: ilmother

fing it by any mal anthority, anıl hence mluere it immriisland of the win!:, ! 1.1.1 : ti tulio win the Isona hair. The ettert of this was to polare Sir, the ratenment persile: : : 170 : ...11: 1.1. 11. ???!!!!!! rint on hie map of Axia, at a distance of 45} from the Storm to this link. p... o-rarbeta 1.2 1... ?) Toerr, -hiih apain he fixel, on the authority of itineraries citudley also. 1111! . U..!:11:1::3 I'!..,,!!.9. IN 11

Warings at 24.001) stalia or 6) of longitude from the Euphratis !!!!!! ;-) ; 71.::: t.; mbwaing in both cases a degree of longitude as equivalent to 100: Weil sit:vtillitill.nl Guwar's

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