established. Ptolemy does not give them, but in each case when
required applies the theorem of Menelaus for spherics directly. This
greatly increases the length of his demonstrations, which the modern
reader finds still more cumbrous, inasmuch as in each case it was
necessary to express the relation in terms of chords-the equivalents
of sines-only, cosines and tangents being of later invention.
Such, then, was the trigonometry of the Greeks. Mathe-
matics, indeed, has ever been, as it were, the handmaid
of astronomy, and many important methods of the former
arose from the needs of the latter. Moreover, by the found-
ation of trigonometry, astronomy attained its final general
constitution, in which calculations took the place of dia-
grams, as these latter had been at an earlier period sub-
stituted for mechanical apparatus in solving the ordinary
problems. Further, we find in the application of trigon-
ometry to astronomy frequent examples and even a sys-
tematic use of the method of approximations, the basis,
in fact, of all application of mathematics to practical
questions. There was a disinclination on the part of the
Greek geometer to be satisfied with a mere approximation,
were it ever so close; and the unscientific agrimensor
shirked the labour involved in acquiring the knowledge
which was indispensable for learning trigonometrical cal-
culations. Thus the development of the calculus of
approximations fell to the lot of the astronomer, who was
both scientific and practical.2

by Eratosthenes and used by Hipparchus. This "is followed by spherical geometry and trigonometry enough for the determination of the connexion 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." 3

In book ii., after some remarks on the situation of the habitable

parts of the earth, Ptolemy proceeds to make deductions from the
principles established in the preceding book, which he does by
means of the theorem of Menelaus. The length of the longest
day being given, he shows how to determine the arcs of the horizon
intercepted between the equator and the ecliptic-the amplitude
of the eastern point of the ecliptic at the solstice-for different
degrees of obliquity of the sphere; hence he finds the height of the
pole and reciprocally. From the same data he shows how to find
at what places and times the sun becomes vertical and how to
calculate the ratios of gnomons to their equinoctial and solstitial
shadows at noon and conversely, pointing out, however, that the
siders fully and works out in detail for the parallel of Rhodes.
latter method is wanting in precision. All these matters he con-
Theon gives us three reasons for the selection of that parallel by
Ptolemy: the first is that the height of the pole at Rhodes is 36°,
a whole number, whereas at Alexandria he believed it to be 30° 58';
tions; the third is that the climate of Rhodes holds the mean place
the second is that Hipparchus had made at Rhodes many observa-
of the seven climates subsequently described. Delambre suspects
a fourth reason, which he thinks is the true one, that Ptolemy had
taken his examples from the works of Hipparchus, who observed at
Rhodes and had made these calculations for the place where he lived.
In chapter vi. Ptolemy gives an exposition of the most important
properties of each parallel, commencing with the equator, which he
considers as the southern limit of the habitable quarter of the
earth. For each parallel or climate, which is determined by the
length of the longest day, he gives the latitude, a principal place
on the parallel, and the lengths of the shadows of the gnomon at
the solstices and equinox. In the next chapter he enters into par-
ticulars and inquires what are the arcs of the equator which cross
the horizon at the same time as given arcs of the ecliptic, or, which
comes to the same thing, the time which a given are of the ecliptic
takes to cross the horizon of a given place. He arrives at a formula
for calculating ascensional differences and gives tables of ascensions
arranged by 10° of longitude for the different climates from the
equator to that where the longest day is seventeen hours.
then shows the use of these tables in the investigation of the length
of the day for a given climate, of the manner of reducing temporal
to equinoctial hours and vice versa, and of the nonagesimal point
and the point of orientation of the ecliptic. In the following
chapters of this book he determines the angles formed by the inter-
sections of the ecliptic-first with the meridian, then with the
horizon, and lastly with the vertical circle-and concludes by giving
tables of the angles and arcs formed by the intersection of these
circles, for the seven climates, from the parallel of Meroe (thirteen
hours) to that of the mouth of the Borysthenes (sixteen hours).
These tables, he adds, should be completed by the situation of the
chief towns in all countries according to their latitudes and longi-
tudes; this he promises to do in a separate treatise and has in fact
done in his Geography.

He

We now proceed to notice briefly the contents of the Almagest. It is divided into thirteen books. The first book, which may be regarded as introductory to the whole work, opens with a short preface, in which Ptolemy, after some observations on the distinction between theory and practice, gives Aristotle's division of the sciences and remarks on the certainty of mathematical knowledge, "inasmuch as the demonstrations in it proceed by the incontrovertible ways of arithmetic and geometry." He concludes his preface with the statement that he will make use of the discoveries of his predecessors, and relate briefly all that has been sufficiently explained by the ancients, but that he will treat with more care and develop ment whatever has not been well understood or fully treated. Ptolemy unfortunately does not always bear this in mind, and it is sometimes difficult to distinguish what is due to him from that which he has borrowed from his predecessors. Ptolemy then, in the first chapter, presupposing some preliminary notions on the part of the reader, announces that he will treat in order-what is the relation of the earth to the heavens, what is the position of the oblique circle (the ecliptic), and the situation of the inhabited parts of the earth; that he will point out the differences of climates; that he will then pass on to the consideration of the motion of the sun and moon, without which one cannot have a just theory of the stars; lastly, that he will consider the sphere of the fixed stars and then the theory of the five stars called "planets." All these things-i.c., the phenomena of the heavenly bodies-he says he will endeavour to explain in taking for principle that which is evident, real, and certain, in resting everywhere on the surest Book iii. treats of the motion of the sun and of the length of the observations and applying geometrical methods. He then enters year. In order to understand the difficulties of this question on a summary exposition of the general principles on which his Ptolemy says one should read the books of the ancients, and especi Syntaris is based, and adduces arguments to show that the heaven ally those of Hipparchus, whom he praises "as a lover of labour is of a spherical form and that it moves after the manner of a and a lover of truth” (ἀνδρὶ φιλοπόνῳ τε ὁμοῦ καὶ φιλαληθεί). He sphere, that the earth also is of a form which is sensibly spherical, begins by telling us how Hipparchus was led to discover the prethat the earth is in the centre of the heavens, that it is but point cession of the equinoxes; he relates the observations which led in comparison with the distances of the stars, and that it has not any motion of translation. With respect to the revolution of the Hipparchus to his second great discovery, that of the eccentricity earth round its axis, which he says some have held, Ptolemy, while of the solar orbit, and gives the hypothesis of the eccentric by admitting that this supposition renders the explanation of the which he explained the inequality of the sun's motion. Ptolemy phenomena of the heavens much more simple, yet regards it as concludes this book by giving a clear exposition of the circumAll this the altogether ridiculous. Lastly, he lays down that there are two stances on which the equation of time depends. principal and different motions in the heavens-one by which all reader will find in the article ASTRONOMY (vol. ii. p. 750). Ptolemy, moreover, applies Apollonius's hypothesis of the epicycle to explain the equator; the other, which is peculiar to some of the stars, is in the stars are carried from east to west uniformly about the poles of the inequality of the sun's motion, and shows that it leads to the a contrary direction to the former motion and same results as the hypothesis of the eccentric. He prefers the ferent poles. These preliminary notions, which takes place round latter lay both as equally fit to clear up the difficulties. hypothesis as more simple, requiring only one and not two Ptolemy, form the subjects of the second and following chapters. second chapter there are some general remarks to which attention next proceeds to that for the Practire have given an account, and which is indisp choruse of shplanation of phenomena one should adopt the simplest hypothesis practical astronomy. The employment of this table presupposes that it is possible to establish, provided that it is not contradicted

In the

the evaluation of the obliquity of the ecliptic, the knowledge of by the observations in any important respect. This fine principle,

which is indeed the foundation of all astronomical science. Ptolemy
in the next chapter indicates two means of determining this angle
by observation, describes the instruments he employed for that
purpose, and finds the same value which had already been found

Comte, Système de Politique Positive, iii. 324.

Cantor, Vorlesungen über Geschichte der Mathematik, p. 356.

8. v.

3 De Morgan, in Smith's Dictionary of Greek and Roman Biography, "Ptolemæus, Claudius."

Kaipiral, temporal or variable. These hours varied in length with the seasons; they were used in ancient times and arose from the division of the natural day (from sunrise to sunset) into twelve parts. Alm., ed. Halma, i. 159.

XX. 12

book commences with a similar catalogue of the stars in the con-
stellations of the southern hemisphere. This catalogue has been
the subject of keen controversy amongst modern astronomers.
Some, as Flamsteed and Lalande, maintain that it was the same
catalogue which Hipparchus had drawn up 265 years before Ptolemy,
whereas others, of whom Laplace is one, think that it is the work
of Ptolemy himself. The probability is that in the main the
catalogue is really that of Hipparchus altered to suit Ptolemy's
own time, but that in making the changes which were necessary
a wrong precession was assumed. This is Delambre's opinion;
he says,
"Whoever may have been the true author, the catalogue
is unique, and does not suit the age when Ptolemy lived; by sub-
tracting 2° 40' from all the longitudes it would suit the age of
Hipparchus; this is all that is certain." It has been remarked
that Ptolemy, living at Alexandria, at which city the altitude of
the pole is 5° less than at Rhodes, where Hipparchus observed,
could have seen stars which are not visible at Rhodes; none of
these stars, however, are in Ptolemy's catalogue. The eighth book
contains, moreover, a description of the milky way and the manner
of constructing a celestial globe; it also treats of the configura-
tion of the stars, first with regard to the sun, moon, and planets,
and then with regard to the horizon, and likewise of the different
aspects of the stars and of their rising, culmination, and setting
simultaneously with the sun.

which is of universal application, may, we think-regard being paid | tudes, arranged according to their constellations; and the eighth
to its place in the Almagest-be justly attributed to Hipparchus.
It is the first law of the "philosophia prima" of Comte.1 We find in
the same page another principle, or rather practical injunction, that
in investigations founded on observations where great delicacy is
required we should select those made at considerable intervals of
time in order that the errors arising from the imperfection which is
inherent in all observations, even in those made with the greatest
care, may be lessened by being distributed over a large number of
years. In the same chapter we find also the principle laid down
that the object of mathematicians ought to be to represent all the
celestial phenomena by uniform and circular motions. This prin-
ciple is stated by Ptolemy in the manner which is unfortunately
too common with him, that is to say, he does not give the least
indication whence he derived it. We know, however, from Sim-
plicius, on the authority of Sosigenes,2 that Plato is said to have
proposed the following problem to astronomers: "What regular
and determined motions being assumed would fully account for
the phenomena of the motions of the planetary bodies?" We know,
too, from the same source that Eudemus says in the second book of
his History of Astronomy that "Eudoxus of Cnidus was the first
of the Greeks to take in hand hypotheses of this kind," that he
was in fact the first Greek astronomer who proposed a geometrical
hypothesis for explaining the periodic motions of the planets-the
famous system of concentric spheres. It thus appears that the
principle laid down here by Ptolemy can be traced to Eudoxus and
Plato; and it is probable that they derived it from the same source,
namely, Archytas and the Pythagoreans. We have indeed the
direct testimony of Geminus of Rhodes that the Pythagoreans
endeavoured to explain the phenomena of the heavens by uniform
and circular motions.4

Books iv., v. are devoted to the motions of the moon, which are very complicated; the moon in fact, though the nearest to us of all the heavenly bodies, has always been the one which has given the greatest trouble to astronomers.5 Book iv., in which Ptolemy follows Hipparchus, treats of the first and principal inequality of the moon, which quite corresponds to the inequality of the sun treated of in the third book. As to the observations which should be employed for the investigation of the motion of the moon, Ptolemy tells us that lunar eclipses should be preferred, inasmuch as they give the moon's place without any error on the score of parallax. The first thing to be determined is the time of the moon's revolution; Hipparchus, by comparing the observations of the Chaldeans with his own, discovered that the shortest period in which the lunar eclipses return in the same order was 126,007 days and 1 hour. In this period he finds 4267 lunations, 4573 restitutions of anomaly, and 4612 tropical revolutions of the moon less 7 q.p.; this quantity (73) is also wanting to complete the 345 revolutions which the sun makes in the same time with respect to the fixed stars. He concluded from this that the lunar month contains 29 days and 31′ 50′′ 8" 20""" of a day, very nearly, or 29 days 12 hours 44' 3" 20"". These results are of the highest importance. (See ASTRONOMY.) In order to explain this inequality, or the equation of the centre, Ptolemy makes use of the hypothesis of an epicycle, which he prefers to that of the eccentric. The fifth book commences with the description of the astrolabe of Hipparchus, which Ptolemy made use of in following up the observations of that astronomer, and by means of which he made his most important discovery, that of the second inequality in the moon's motion, now known by the name of the "evection." In order to explain this inequality he supposed the moon to move on an epicycle, which was carried by an eccentric whose centre turned about the earth in a direction contrary to that of the motion of the epicycle. This is the first instance in which we find the two hypotheses of eccentric and epicycle combined. The fifth book treats also of the parallaxes of the sun and moon, and gives a description of an instrument-called later by Theon the "parallactic rods"-devised by Ptolemy for observing meridian altitudes with greater accuracy.

The subject of parallaxes is continued in the sixth book of the Almagest, and the method of calculating eclipses is there given. The author says nothing in it which was not known before his time. Books vii., viii. treat of the fixed stars. Ptolemy verified the fixity of their relative positions and confirmed the observations of Hipparchus with regard to their motion in longitude, or the precession of the equinoxes. (See ASTRONOMY.) The seventh book concludes with the catalogue of the stars of the northern hemisphere, in which are entered their longitudes, latitudes, and magni1 Système de Politique Positive, iv. 173.

This Sosigenes, as Th. H. Martin has shown, was not the astronomer of that name who was a contemporary of Julius Cæsar, but a Peripatetic philosopher who lived at the end of the 2d century A.D.

The remainder of the work is devoted to the planets. The ninth book commences with what concerns them all in general. The planets are much nearer to the earth than the fixed stars and more distant than the moon. Saturn is the most distant of all, then, Jupiter and then Mars. These three planets are at a greater distance from the earth than the sun. So far all astronomers are agreed. This is not the case, he says, with respect to the two remaining planets, Mercury and Venus, which the old astronomers placed between the sun and earth, whereas more recent writers have placed them beyond the sun, because they were never seen on the sun. He shows that this reasoning is not sound, for they might be nearer to us than the sun and not in the same plane, and consequently never seen on the sun. He decides in favour of the former opinion, which was indeed that of most mathematicians. The ground of the arrangement of the planets in order of distance was the relative length of their periodic times; the greater the circle, the greater, it was thought, would be the time required for its description. Hence we see the origin of the difficulty and the difference of opinion as to the arrangement of the sun, Mercury, and Venus, since the times in which, as seen from the earth, they appear to complete the circuit of the zodiac are nearly the samea year. 10 Delambre thinks it strange that Ptolemy did not see that these contrary opinions could be reconciled by supposing that the two planets moved in epicycles about the sun; this would be stranger still, he adds, if it is true that this idea, which is older than Ptolemy, since it is referred to by Cicero," had been that of the Egyptians. 12 It may be added, as strangest of all, that this doctrine was held by Theon of Smyrna, 13 who was a contemporary of Ptolemy or somewhat senior to him. From this system to that of Tycho Brahe there is, as Delambre observes, only a single step.

3 Brandis, Schol. in Aristot. edidit Acad. Reg. Borussica (Berlin, 1836), p. 498. 4 Eloaywyn els тà pairóμeva, c. i. in Halma's edition of the works of Ptolemy, vol. iii. ("Introduction aux Phénomènes Célestes, traduite du Grec de Géminus," p. 9), Paris, 1819.

5 This has been noticed by Pliny, who says, "Multiformi hæc (luna) ambage torsit ingenia contemplantium, et proximum ignorari maxime sidus indignantium" (N. H., ii. 9).

We have seen that the problem which presented itself to the astronomers of the Alexandrian epoch was the following: it was required to find such a system of equable circular motions as would represent the inequalities in the apparent motions of the sun, the moon, and the planets. Ptolemy now takes up this question for the planets; he says that "this perfection is of the essence of celestial things, which admit of neither disorder nor inequality," that this planetary theory is one of extreme difficulty, and that no one had yet completely succeeded in it. He adds that it was owing to these difficulties that Hipparchus-who loved truth above all things, and who, moreover, had not received from his predecessors observations either so numerous or so precise as those that he has left-had succeeded, as far as possible, in representing the motions of the sun and moon by circles, but had not even commenced the theory of the five planets. He was content, Ptolemy 6 Delambre, Histoire de l'Astronomie Ancienne, ii. 264.

7 This is true of their mean distances; but we know that Mars at opposition is nearer to us than the sun.

8 Eratosthenes, for example, as we learn from Theon of Smyrna.

9 Transits of Mercury and Venus over the sun's disk, therefore, had not beea

observed.

10 This was known to Eudoxus. Sir George Cornewall Lewis (An Historical Survey of the Astronomy of the Ancients, p. 155), confusing the geocentric revolu 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 "the error with respect to Mercury and Venus is considerable"; this, however, is an error not of Eudoxus but of Cornewall Lewis, as Schiaparelli has remarked. 11 "Hunc [solem] ut comites consequuntur Veneris alter, alter Mercurii cursus (Somnium Scipionis, De Rep., vi. 17). This hypothesis is alluded to by Pliny, N. H., ii. 17, and is more explicitly stated by Vitruvius, Arch., ix. 4. 12 Macrobius, Commentarius ex Cicerone in Somnium Scipionis, i. 19. 13 Theon (Smyrnans Platonicus), Liber de Astronomia, ed. Th. H. Martin (Paris, 1849), pp. 174, 294, 296. Martin thinks that Theon, the mathematician, four of whose observations are used by Ptolemy (Alm., ii. 176, 193, 194, 195, 196, ed. Halma), is not the same as Theon of Smyrna, on the ground chiefly that the

latter was not an observer.

Ptolemy is hardly less celebrated as a geographer than as an astronomer, and his great work on geography exercised as great an influence on the progress of that science as did his Almagest on that of astronomy. It became indeed the paramount authority on all geographical questions for a period of many centuries, and was only gradually superseded by the progress of maritime discovery in the 15th and 16th centuries. This exceptional position was due in a great measure to its scientific form, which rendered it very convenient and easy of reference; but,

Ptolemy concludes his great work by saying that he has included in it everything of practical utility which in his judgment should find a place in a treatise on astronomy at the time it was written, with relation as well to discoveries as to methods. His work was justly called by him MaoŋμаTIKỲ Zúvrais, for it was in fact the mathematical form of the work which caused it to be preferred to all others which treated of the same science, but not by the sure methods of geometry and calculation." Accordingly, it soon spread from Alexandria to all places where astronomy was cultivated; numerous copies were made of it, and it became the object of serious study on the part of both teachers and pupils. Amongst its numerous commentators may be mentioned Pappus and Theon of Alexandria in the 4th century and Proclus in the 5th. It was translated into Latin by Boetius, but this translation has not come down to us. The Syntaris was translated into Arabic at Baghdad by order of the lightened caliph Al-Mamún, who was himself an astronomer, about 524 apart from this consideration, it was really the first attempt

Astra 1461) and completed by his pupil and successor in the professorship of
the Almagest in print. The first complete edition of the Almagest is that of P.
the At epitome is that of Venice, 1496, and this was the first appear edition
Liechtenstein (Venice, 1515),a Latin version from the Arabic.
translation of George of Trebizond was first printed in 1528, at Venice. The
Greek text, which was not known in Europe until the 15th century, was first
published in the 16th by Simon Grynæus, who was also the first editor of the
reek text of Euclid, at Basel, 1538. This edition was from a manuscript in
the library of Nuremberg-where it is no longer to be found-which had been
presented by Regiomontanus, to whom it was given by Cardinal Bessarion.
The last edition of the Almagest is that of Halma, Greek with French translation,
in two vols., Paris, 1813-16. On the manuscripts of the Almagest and its biblio-
graphical history, see Fabricius, Bibliotheca Graca, ed. Harles, vol. v. p. 280,
and Halma's preface. An excellent summary of the bibliographical history is
given by De Morgan in his article on Ptolemy already quoted.

Other works of Ptolemy, which we now proceed to notice very briefly, are
as follows. (1) Φάσεις ἀπλανῶν ἀστέρων καὶ συναγωγὴ ἐπισημασιών, οι
the Apparitions of the Fired Stars and a Collection of Prognostics. It is a calendar
of a kind common amongst the Greeks under the name of apámŋyμa, or a
collection of the risings and settings of the stars in the morning or evening
twilight, which were so many visible signs of the seasons, with prognostics of
the principal changes of temperature with relation to each climate, after the
hrvations of the best meteorologists, as, for example, Meton, Democritus,
Eudoxus Hipparchus, &c. Ptolemy, in order to make his Parapegma useful
to all the Greeks scattered over the enlightened world of his time, gives the
apparitions of the stars not for one parallel only but for each of the five parallels
Delambre compares these mean motions with those of our modern tables
and fines them tolerably correct. By "motion in longitude" must be under-
to the motion of the centre of the epicycle about the eccentric, and by
"anomaly the motion of the star on its epicycle.

In this elition the Greek text and the French translation are given in parallel columns; the latter, however, should not be read without reference to

XV.

means of determination are not available. The greater part
of the treatise of Marinus was occupied with the discussion of
these authorities, and it is impossible for us, in the absence
of the original work, to determine how far he had succeeded
in giving a scientific form to the results of his labours;
but we are told by Ptolemy himself that he considered
them, on the whole, so satisfactory that he had made the
work of his predecessor the basis of his own in regard to
all the countries bordering on the Mediterranean, a term
which would comprise to the ancient geographer almost
all those regions of which he had really any definite know-
ledge. With respect to the more remote regions of the
world, Ptolemy availed himself of the information imparted
by Marinus, but not without reserve, and has himself ex-
plained to us the reasons that induced him in some instances
to depart from the conclusions of his predecessor. It is
very unjust to term Ptolemy a plagiarist from Marinus, as
has been done by some modern authors, as he himself
acknowledges in the fullest manner his obligations to
that writer, from whom he derived the whole mass of his
materials, which he undertook to arrange and present to
his readers in a scientific form. It is this form and ar-
rangement that constitute the great merit of Ptolemy's
work and that have stamped it with a character wholly
distinct from all previous treatises on geography. But
at the same time it possesses much interest, as showing
the greatly increased knowledge of the more remote por-
tions of Asia and Africa which had been acquired by geo-
graphers since the time of Strabo and Pliny.

It will be convenient to consider separately the two
different branches of the subject,—(1) the mathematical
portion, which constitutes his geographical system, properly
so termed; and (2) his contributions to the progress of
positive knowledge with respect to the Inhabited World.
See Plate 1. Mathematical Geography. As a great astronomer, Ptolemy
VII., vol. was of course infinitely better qualified to comprehend and explain
the mathematical conditions of the earth and its relations to the
celestial bodies that surround it than any preceding writers on
the special subject of geography. But his general views, except
on a few points, did not differ from those of his most eminent
precursors Eratosthenes and Strabo. In common with them, he
assumed that the earth was a globe, the surface of which was
divided by certain great circles-the equator and the tropics—
parallel to one another, and dividing the earth into five great
zones, the relations of which with astronomical phenomena were
of course clear to his mind as a matter of theory, though in regard
to the regions bordering on the equator, as well as to those ad-
joining the polar circle, he could have had no confirmation of his
conclusions from actual observation. He adopted also from Hip-
parchus the division of the equator and other great circles into
360 parts or "degrees" (as they were subsequently called, though
the word does not occur in this sense in Ptolemy), and supposed
other circles to be drawn through these, from the equator to the
pole, to which he gave the name of "meridians." He thus conceived
the whole surface of the earth (as is done by modern geographers)
to be covered with a complete network of "parallels of latitude" and
"meridians of longitude," terms which he himself was the first ex-
tant writer to employ in this technical sense. Within the network
thus constructed it was the task of the scientific geographer to
place the outline of the world, so far as it was then known by
experience and observation.

was determined by actual observation; but we learn from Ptolemy
himself that this was not the case, and that such observations for
latitude were very few in number, while the means of determining
longitudes were almost wholly wanting.1 Hence the positions laid
down by him were really, with very few exceptions, the result of
computations of distances from itineraries and the statements of
travellers, estimates which were liable to much greater error in
ancient times than at the present day, from the want of any accurate
mode of observing bearings, or portable instruments for the measure-
ment of time, while they had no means at all of determining dis-
tances at sea, except by the rough estimate of the time employed
in sailing from point to point. The use of the log, simple as it
appears to us, was unknown to the ancients. But, great as would
naturally be the errors resulting from such imperfect means of cal-
culation, they were in most cases increased by the permanent error
arising from the erroneous system of graduation adopted by Ptolemy
in laying them down upon his map. Thus, if he had arrived at
the conclusion from itineraries that two places were 5000 stadia
from one another, he would place them at a distance of 10° apart,
and thus in fact separate them by an interval of 6000 stadia.

Unfortunately at the very outset of his attempt to realize this conception he fell into an error which had the effect of vitiating all his subsequent conclusions. Eratosthenes was the first writer who had attempted in a scientific manner to determine the circumference of the earth, and the result at which he arrived, that it amounted to 250,000 stadia or 25,000 geographical miles, was generally adopted by subsequent geographers, including Strabo. Posidonius, however, who wrote about a century after Eratosthenes, had made an independent calculation, by which he reduced the circumference of the globe to 180,000 stadia, or less than threefourths of the result obtained by Eratosthenes, and this computation, on what grounds we know not, was unfortunately adopted by Marinus Tyrius, and from him by Ptolemy. The consequence of this error was of course to make every degree of latitude or longitude (measured at the equator) equal to only 500 stadia (50 geographical miles), instead of its true equivalent of 600 stadia. effects would indeed have been in some measure neutralized had there existed a sufficient number of points of which the position

Its

Another source of permanent error (though one of much less im-
portance) which affected all his longitudes arose from the errone-
ous assumption of his prime meridian. In this respect also he
followed Marinus, who, having arrived at the conclusion that the
Fortunate Islands (the Canaries) were situated farther west than
any part of the continent of Europe, had taken the meridian
through the outermost of this group as his prime meridian, from
whence he calculated all his longitudes eastwards to the Indian
Ocean. But, as both Marinus and Ptolemy were very imperfectly
acquainted with the position and arrangements of the islands in
question, the line thus assumed was in reality a purely imaginary
one, being drawn through the supposed position of the outer island,
Vincent), which was regarded by Marinus and Ptolemy, as it had
which they placed 24° west of the Sacred Promontory (Cape St
been by all previous geographers, as the westernmost point of the
continent of Europe,-while the real difference between the two is
not less than 9° 20. Hence all Ptolemy's longitudes, reckoned east-
wards from this assumed line, were in fact about 7° less than they
would have been if really measured from the meridian of Ferro,
which continued so long in use among geographers in modern
times. The error in this instance was the more unfortunate as the
longitude could not of course be really measured, or even calculated,
from this imaginary line, but was in reality calculated in both
directions from Alexandria, westwards as well as eastwards (as
Ptolemy himself has done in his eighth book) and afterwards re-
versed, so as to suit the supposed method of computation.
It must be observed also that the equator was in like manner
placed by Ptolemy at a considerable distance from its true geo-
graphical position. The place of the equinoctial line on the sur-
face of the globe was of course well known to him as a matter of
theory, but as no observations could have been made in those
remote regions he could only calculate its place from that of the
tropic, which he supposed to pass through Syene. And as he here,
as elsewhere, reckoned a degree of latitude as equivalent to 500
stadia, he inevitably made the interval between the tropic and the
equator too small by one-sixth; and the place of the former on the
surface of the earth being fixed by observation he necessarily carried
up the supposed place of the equator too high by more than 230
geographical miles. But as he had practically no geographical
acquaintance with the equinoctial regions of the earth this error
was of little importance.

With Marinus and Ptolemy, as with all preceding Greek geo-
graphers, the most important line on the surface of the globe for
all practical purposes was the parallel of 36° of latitude, which
passes through the Straits of Gibraltar at one end of the Mediter-
ranean, and through the Island of Rhodes and the Gulf of Issus at
the other. It was thus regarded by Dicearchus and almost all his
successors as dividing the regions around the inland sea into two
portions, and as being continued in theory along the chain of Mount
Taurus till it joined the great mountain range north of India; and
from thence to the Eastern Ocean it was regarded as constituting
the dividing line of the Inhabited World, along which its length
must be measured. But it sufficiently shows how inaccurate were
the observations and how imperfect the materials at his command,
even in regard to the best known portions of the earth, that Ptolemy,
following Marinus, describes this parallel as passing through Caralis
in Sardinia and Lilybum in Sicily, the one being really in 39°
12' lat., the other in 37° 50'. It is still more strange that he places
so important and well known a city as Carthago 1° 20' south of the
dividing parallel, while it really lies nearly 1° to the north of it.

1 Hipparchus had indeed pointed out long before the mode of determining longitudes by observations of eclipses, but the instance to which he referred of the celebrated eclipse before the battle of Arbela, which was seen also at Carthage, was a mere matter of popular observation, of no scientific value. Yet Ptolemy seems to have known of

no other.

The

The great problem that had attracted the attention and exercised the ingenuity of all geographers from the time of Dicæarchus to that of Ptolemy was to determine the length and breadth of the Inhabited World, which they justly regarded as the chief subject of the geographer's consideration. This question had been very fully discussed by Marinus, who had arrived at conclusions widely dif ferent from those of his predecessors. Towards the north indeed there was no great difference of opinion, the latitude of Thule being generally recognized as that of the highest northern land, and this was placed both by Marinus and Ptolemy in 63° lat., not very far beyond the true position of the Shetland Islands, which had come in their time to be generally identified with the mysterious Thule of Pytheas. The western extremity, as already mentioned, had been in like manner determined by the prime meridian drawn through the supposed position of the Fortunate Islands. But towards the south and east Marinus gave an enormous extension to the continents of Africa and Asia, beyond what had been known to or suspected by the earlier geographers, and, though Ptolemy greatly reduced his calculations, he still retained a very exaggerated estimate of their results.

The additions thus made to the estimated dimensions of the known world were indeed in both directions based upon a real extension of knowledge, derived from recent information; but unfortunately the original statements were so perverted by misinterpretation in applying them to the construction of a map as to give results differing widely from the truth. The southern limit of the world as known to Eratosthenes, and even to Strabo (who had in this respect no further knowledge than his predecessor more than two centuries before), had been fixed by them at the parallel which passed through the eastern extremity of Africa (Cape Guardafui), or the Land of Cinnamon as they termed it, and that of the Sembrite (corresponding to Sennaar) in the interior of the same continent. This parallel, which would correspond nearly to that of 10° of true latitude, they supposed to be situated at a distance of 3400 stadia (340 geographical miles) from that of Meroe (the position of which was accurately known), and 13,400 to the south of Alexandria ; while they conceived it as passing, when prolonged to the eastward, through the island of Taprobane (Ceylon), which was universally recognized as the southernmost land of Asia. Both these geographers were wholly ignorant of the vast extension of Africa to the south of this line and even of the equator, and conceived it as trending away to the west from the Land of Cinnamon and then to the north-west to the Straits of Gibraltar. Marinus had, however, learned from itineraries both by land and sca the fact of this great extension, of which he had indeed conceived so exaggerated an idea that even after Ptolemy had reduced it by more than a half it was still materially in excess of the truth. The castern coast of Africa was indeed tolerably well known, being frequented by Greek and Roman traders, as far as a place called Rhapta, opposite to Zanzibar, and this is placed by Ptolemy not far from its true position in 7° S. lat. But he added to this a bay of great extent as far as a promontory called Prasum (perhaps Cape Delgado), which he placed in 15° S. lat. At the same time he assumed the position in about the same parallel of a region called Agisymba, which was supposed to have been discovered by a Roman general, whose itinerary was employed by Marinus. Taking, therefore, this parallel as the limit of knowledge to the south, while he retained that of Thule to the north, he assigned to the inhabited world a breadth of nearly 80°, instead of less than 60°, which it had occupied on the maps of Eratosthenes and Strabo.

It had been a fixed belief with all the Greek geographers from the earliest attempts at scientific geography not only that the length of the Inhabited World greatly exceeded its breadth, but that it was more than twice as great,-a wholly unfounded assumption, but to which their successors seem to have felt themselves bound to conform. Thus Marinus, while giving an undue extension to Africa towards the south, fell into a similar error, but to a far greater degree, in regard to the extension of Asia towards the east. Here also he really possessed a great advance in knowledge over all his predecessors, the increased trade with China for silk having led to an acquaintance, though of course of a very vague and general kind, with the vast regions in Central Asia that lay to the cast of the Pamir range, which had formed the limit of the Asiatic nations previously known to the Greeks. But Marinus had learned that traders proceeding eastward from the Stone Tower-a station at the foot of this range-to Sera, the capital city of the Seres, occupied seven months on the journey, and from thence he arrived at the enormous result that the distance between the two points was not less than 36,200 stadia, or 3620 geographical miles. Ptolemy, while he justly points out the absurdity of this conclusion and the erroneons mode of computation on which it was founded, had no means of correcting it by any real anthority, and hence reduced it summarily by one half. The effect of this was to place Sera, the easternmost point on his map of Asia, at a distance of 45° from the Stone Tower, which again he fixed, on the authority of itineraries cited by Marinus, at 24,000 stadia or 60° of longitude from the Euphrates, reckoning in both cases a degree of longitude as equivalent to 400

|

stadia, in accordance with his uniform system of allowing 500 stadia to 1° of latitude. Both distances were greatly in excess of the truth, independently of the error arising from this mistaken system of graduation. The distances west of the Euphrates were of course comparatively well known, nor did Ptolemy's calculation of the length of the Mediterranean differ very materially from those of previous Greek geographers, though still greatly exceeding the truth, after allowing for the permanent error of graduation. The effect of this last cause, it must be remembered, would unfortunately be cumulative, in consequence of the longitudes being computed from a fixed point in the west, instead of being reckoned east and west from Alexandria, which was undoubtedly the mode in which they were really calculated. The result of these combined causes of error was to lead him to assign no less than 180°, or 12 hours, of longitude to the interval between the meridian of the Fortunate Islands and that of Sera, which really amounts to about 130°.

All

But in thus estimating the length and breadth of the known world Ptolemy attached a very different sense to these terms from that which they had generally borne with preceding writers. former Greek geographers, with the single exception of Hipparchus, had agreed in supposing the Inhabited World to be surrounded on all sides by sea, and to form in fact a vast island in the midst of a circumfluous ocean. This notion, which was probably derived originally from the Homeric fiction of an ocean stream, and was certainly not based upon direct observation, was nevertheless of course in accordance with the truth, great as was the misconception it involved of the extent and magnitude of the continents included within this assumed boundary. Hence it was unfortunate that Ptolemy should in this respect have gone back to the views of Hipparchus, and have assumed that the land extended indefinitely to the north in the case of Europe and Scythia, to the cast in that of Asia, and to the south in that of Africa. His boundary-line was in each of these cases an arbitrary limit, beyond which lay the But in the last case he was not Unknown Land, as he calls it. content with giving to Africa an indefinite extension to the south: he assumed the existence of a vast prolongation of the land to the cast from its southernmost known point, so as to form a connexion with the south-eastern extremity of Asia, of the extent and position of which he had a wholly erroneous idea.

In this last case Marinus had derived from the voyages of recent navigators in the Indian Seas a knowledge of the fact that there lay in that direction extensive lands which had been totally unknown to previous geographers, and Ptolemy had acquired still more extensive information in this quarter. But unfortunately he had formed a totally false conception of the bearings of the coasts thus made known, and consequently of the position of the lands to which they belonged, and, instead of carrying the line of coast northwards from the Golden Chersonese (the Malay Peninsula) to China or the land of the Sina, he brought it down again towards the south after forming a great bay, so that he placed Cattigarathe principal emporium in this part of Asia, and the farthest point known to him-on a supposed line of coast, of unknown extent, but with a direction from north to south. The hypothesis that this land was continuous with the most southern part of Africa, so that the two enclosed one vast gulf, though a mere assumption, is stated by him as definitely as if it was based upon positive information; and it was long received by medieval geographers as an unquestioned fact. This circumstance undoubtedly contributed to perpetuate the error of supposing that Africa could not be circumnavigated, in opposition to the more correct views of Strabo and other earlier geographers. On the other hand, there can be no doubt that the undue extension of Asia towards the east, so as to diminish by 50° of longitude the interval between that continent and the western coasts of Europe, had a material influence in fostering the belief of Columbus and others that it was possible to reach the Land of Spices (as the East Indian islands were then called) by direct navigation towards the west.

It is not surprising that Ptolemy should have fallen into considerable errors respecting the more distant quarters of the world; but even in regard to the Mediterranean and its dependencies, as well as the regions that surrounded them, with which he was in a certain sense well acquainted, the imperfection of his geographical Here he had indeed some knowledge is strikingly apparent. well-established data for his guidance, as far as latitudes were concerned. That of Massilia had been determined many years before by Pytheas within a few miles of its true position, and the latitude of Rome, as might be expected, was known with approximate Those of Alexandria and Rhodes also were well known, having been the place of observation of distinguished astronomers, and the fortunate accident that the Island of Rhodes lay on the same parallel of latitude with the Straits of Gibraltar at the other end of the sea enabled him to connect the two by drawing the parallel direct from the one to the other. The importance attached to this line (36' N. lat.) by all preceding geographers has been already mentioned. Unfortunately Ptolemy, like his predecessors, supposed its course to lie almost uniformly through the open sea, wholly ignoring the great projection of the African coast towards

accuracy.