Chapter VI: The Greek Age of Reason (2)

Great as is the fame of Euclid, it is eclipsed by that of Archimedes the Syracusan, born B.C. 287, whose connection with Egyptian science is not alone testified by tradition, but also by such facts as his acknowledged friendship with Conon of Alexandria, and his invention of the screw still bearing his name, intended for raising the waters of the Nile. Among his mathematical works, the most interesting, perhaps, in his own estimation, as we may judge from the incident that he directed the diagram thereof to be engraved on his tombstone, was his demonstration that the solid content of a sphere is two-thirds that of its circumscribing cylinder. It was by this mark that Cicero, when Quæstor of Sicily, discovered the tomb of Archimedes grown over with weeds. This theorem was, however, only one of a large number of a like kind, which he treated of in his two books on the sphere and cylinder in an equally masterly manner, and with equal success. His position as a geometer is perhaps better understood from the assertion made respecting him by a modern mathematician, that he came as near to the discovery of the Differential Calculus as can be done without the aid of algebraic transformations. Among the special problems he treated of may be mentioned the quadrature of the circle, his determination of the ratio of the circumference to the diameter being between: 3·1428 and 3·1408, the true value, as is now known, being 3·1416 nearly. He also wrote on Conoids and Spheroids, and upon that spiral still passing under his name, the genesis of which had been suggested to him by Conon. In his work entitled "Psammites" he alludes to the astronomical system subsequently established by Copernicus, whose name has been given to it. He also mentions the attempts which had been made to measure the size of the earth; the chief object of the work being, however, to prove not only that the sands upon the sea-shore can be numbered, but even those required to fill the entire space within the sphere of the fixed stars; the result being, according to our system of arithmetic, a less number than is expressed by unity followed by 63 ciphers. Such a book is the sport of a geometrical giant wantonly amusing himself with his strength. Among his mathematical investigations must not be omitted the quadrature of the parabola. His fame depends, however, not so much on his mathematical triumphs as upon his brilliant discoveries in physics and his mechanical inventions. How he laid the foundation of Hydrostatics is familiar to everyone, through the story of Hiero's crown. A certain artisan having adulterated the gold given him by King Hiero to form a crown, Archimedes discovered while he was accidentally stepping into a bath, that the falsification might be detected, and thereby invented the method for the determination of specific gravity. From these investigations he was naturally led to the consideration of the equilibrium of floating bodies; but his grand achievement in the mechanical direction was his discovery of the true theory of the lever: his surprising merit in these respects is demonstrated by the fact that no advance was made in theoretical mechanics during the eighteen centuries intervening between him and Leonardo da Vinci. Of minor matters not fewer than forty mechanical inventions have been attributed to him. Among these are the endless screw, the screw pump, a hydraulic organ, and burning mirrors. His genius is well indicated by the saying popularly attributed to him, "Give me whereon to stand, and I will move the earth," and by the anecdotes told of his exertions against Marcellus during the siege of Syracuse; his invention of catapults and other engines for throwing projectiles, as darts and heavy stones, claws which, reaching over the walls, lifted up into the air ships and their crews, and then suddenly dropped them into the sea; burning mirrors, by which, at a great distance, the Roman fleet was set on fire. It is related that Marcellus, honouring his intellect, gave the strictest orders that no harm should be done to him at the taking of the town, and that he was killed, unfortunately, by an ignorant soldier--unfortunately, for Europe was not able to produce his equal for nearly two thousand years.

[Sidenote: The writings and works of Eratosthenes.]

Eratosthenes was contemporary with Archimedes. He was born at Cyrene, B.C. 276. The care of the library appears to have been committed to him by Euergetes; but his attention was more specially directed to mathematical, astronomical, geographical, and historical pursuits. The work entitled "Catasterisms," doubtfully imputed to him, is a catalogue of 475 of the principal stars; but it was probably intended for nothing more than a manual. He also is said to have written a poem upon terrestrial zones. Among his important geographical labours may be mentioned his determination of the interval between the tropics. He found it to be eleven eighty-thirds of the circumference. He also attempted the measurement of the size of the earth by ascertaining the distance between Alexandria and Syene, the difference of latitude between which he had found to be one-fiftieth of the earth's circumference. It was his object to free geography from the legends with which the superstition of ages had adorned and oppressed it. In effecting this he well deserves the tribute paid to him by Humboldt, the modern who of all others could best appreciate his labours. He considered the articulation and expansion of continents; the position of mountain chains; the action of clouds; the geological submersion of lands; the elevation of ancient sea-beds; the opening of the Dardanelles and of the Straits of Gibraltar; the relations of the Euxine Sea; the problem of the equal level of the circumfluous ocean; and the necessary existence of a mountain chain running through Asia in the diaphragm of Dicæarchus. What an advance is all this beyond the meditations of Thales! Herein we see the practical tendencies of the Macedonian wars. In his astronomical observations he had the advantage of using the armils and other instruments in the Observatory. He ascertained that the direction of terrestrial gravity is not constant, but that the verticals converge. He composed a complete systematic description of the earth in three books--physical, mathematical, historical--accompanied by a map of all the parts then known. Of his skill as a geometer, his solution of the problem of two mean proportionals, still extant, offers ample evidence; and it is only of late years that the fragments remaining of his Chronicles of the Theban Kings have been properly appreciated. He hoped to free history as well as geography from the myths that deform it, a task which the prejudices and interests of man will never permit to be accomplished. Some amusing anecdotes of his opinions in these respects have descended to us. He ventured to doubt the historical truth of the Homeric legends. "I will believe in it when I have been shown the currier who made the wind-bags which Ulysses on his homeward voyage received from Æolus." It is said that, having attained the age of eighty years, he became weary of life, and put an end to himself by voluntary starvation.

[Sidenote: Chronology of Eratosthenes.]

I shall here pause to make a few remarks suggested by the chronological and astronomical works of Eratosthenes. Our current chronology was the offspring of erroneous theological considerations, the nature of which required not only a short historical term for the various nations of antiquity, but even for the existence of man upon the globe. This necessity appears to have been chiefly experienced in the attempt to exalt certain facts in the history of the Hebrews from their subordinate position in human affairs, and, indeed, to give the whole of that history an exaggerated value. This was done in a double way: by elevating Hebrew history from its true grade, and depreciating or falsifying that of other nations. Among those who have been guilty of this literary offence, the name of the celebrated Eusebius, the Bishop of Cæsarea in the time of Constantine, should be designated, since in his chronography and synchronal tables he purposely "perverted chronology for the sake of making synchronisms" (Bunsen). It is true, as Niebuhr asserts, "He is a very dishonest writer." To a great extent, the superseding of the Egyptian annals was brought about by his influence. It was forgotten, however, that of all things chronology is the least suited to be an object of inspiration; and that, though men may be wholly indifferent to truth for its own sake, and consider it not improper to wrest it unscrupulously to what they may suppose to be a just purpose, yet that it will vindicate itself at last. It is impossible to succeed completely in perverting the history of a nation which has left numerous enduring records. Egypt offers us testimonials reaching over five thousand years. As Bunsen remarks, from the known portion of the curve of history we may determine the whole. The Egyptians, old as they are, belong to the middle ages of mankind, for there is a period antecedent to monumental history, or indeed, to history of any kind, during which language and mythology are formed, for these must exist prior to all political institutions, all art, all science. Even at the first moment that we gain a glimpse of the state of Egypt she had attained a high intellectual condition, as is proved by the fact that her system of hieroglyphics was perfected before the fourth dynasty. It continued unchanged until the time of Psammetichus. A stationary condition of language and writing for thousands of years necessarily implies a long and very remote period of active improvement and advance. It was doubtless such a general consideration, rather than a positive knowledge of the fact, which led the Greeks to assert that the introduction of geometry into Egypt must be attributed to kings before the times of Menes. Not alone do her artificial monuments attest for that country an extreme antiquity; she is herself her own witness; for, though the Nile raises its bed only four feet in a thousand years, all the alluvial portion of Egypt has been deposited from the waters of that river. A natural register thus re-enforces the written records, and both together compose a body of evidence not to be gainsaid. Thus the depth of muddy silt accumulated round the pedestals of monuments is an irreproachable index of their age. In the eminent position he occupied, Eusebius might succeed in perverting the received book-chronology; but he had no power to make the endless trade-wind that sweeps over the tropical Pacific blow a day more or a day less; none to change the weight of water precipitated from it by the African mountains; none to arrest the annual mass of mud brought down by the river. It is by collating such different orders of evidence together--the natural and the monumental, the latter gaining strength every year from the cultivation of hieroglyphic studies--that we begin to discern the true Egyptian chronology, and to put confidence in the fragments that remain of Eratosthenes and Manetho.

[Sidenote: Astronomy of Eratosthenes.]

[Sidenote: Attempts of Aristarchus to find the distance of the sun.]

At the time of which we are speaking--the time of Eratosthenes--general ideas had been attained to respecting the doctrine of the sphere, its poles, axis, the equator, arctic and antarctic circles, equinoctial points, solstices, colures, horizon, etc. No one competent to form an opinion any longer entertained a doubt respecting the globular form of the earth, the arguments adduced in support of that fact being such as are still popularly resorted to--the different positions of the horizon at different places, the changes in elevation of the pole, the phenomena of eclipses, and the gradual disappearance of ships as they sail from us. As to eclipses, once looked upon with superstitious awe, their true causes had not only been assigned, but their periodicities so well ascertained that predictions of their occurrence could be made. The Babylonians had thus long known that after a cycle of 223 lunations the eclipses of the moon return. The mechanism of the phases of that satellite was clearly understood. Indeed, Aristarchus of Samos attempted to ascertain the distance of the sun from the earth on the principle of observing the moon when she is dichotomized, a method quite significant of the knowledge of the time, though in practice untrustworthy; Aristarchus thus finding that the sun's distance is eighteen times that of the moon, whereas it is in reality 400. In like manner, in a general way, pretty clear notions were entertained of the climatic distribution of heat upon the earth, exaggerated, however, in this respect, that the torrid zone was believed to be too hot for human life, and the frigid too cold. Observations, as good as could be made by simple instruments, had not only demonstrated in a general manner the progressions, retrogradations and stations of the planets, but attempts had been made to account for, or rather to represent them, by the aid of epicycles.

[Sidenote: Biography of the Ptolemies.]

It was thus in Alexandria, under the Ptolemies, that modern astronomy arose. Ptolemy Soter, the founder of this line of kings, was not only a patron of science, but likewise an author. He composed a history of the campaigns of Alexander. Under him the collection of the library was commenced, probably soon after the defeat of Antigonus at the battle of Ipsus, B.C. 301. The museum is due to his son Ptolemy Philadelphus, who not only patronized learning in his own dominions, but likewise endeavoured to extend the boundaries of human knowledge in other quarters. Thus he sent an expedition under his admiral Timosthenes as far as Madagascar. Of the succeeding Ptolemies, Euergetes and Philopator were both very able men, though the later was a bad one; he murdered his father, and perpetrated many horrors in Alexandria. Epiphanes, succeeding his father when only five years old, was placed by his guardians under the protection of Rome, thus furnishing to the ambitious republic a pretence for interfering in the affairs of Egypt. The same policy was continued during the reign of his son Philometor, who, upon the whole, was an able and good king. Even Physcon, who succeeded in B.C. 146, and who is described as sensual, corpulent, and cruel--cruel, for he cut off the head, hands, and feet of his son, and sent them to Cleopatra his wife--could not resist the inspirations to which the policy of his ancestors, continued for nearly two centuries, had given birth, but was an effective promoter of literature and the arts, and himself the author of an historical work. A like inclination was displayed by his successors, Lathyrus and Auletes, the name of the latter indicating his proficiency in music. The surnames under which all these Ptolemies pass were nicknames, or titles of derision imposed upon them by their giddy and satirical Alexandrian subjects. The political state of Alexandria was significantly said to be a tyranny tempered by ridicule. The dynasty ended in the person of the celebrated Cleopatra, who, after the battle of Actium, caused herself, as is related in the legends, to be bitten by an asp. She took poison that she might not fall captive to Octavianus, and be led in his triumph through the streets of Rome.

If we possessed a complete and unbiased history of these Greek kings, it would doubtless uphold their title to be regarded as the most illustrious of all ancient sovereigns. Even after their political power had passed into the hands of the Romans--a nation who had no regard to truth and to right--and philosophy, in its old age, had become extinguished or eclipsed by the faith of the later Cæsars, enforced by an unscrupulous use of their power, so strong was the vitality of the intellectual germ they had fostered, that, though compelled to lie dormant for centuries, it shot up vigorously on the first occasion that favouring circumstances allowed.

[Sidenote: They patronize literature as well as science.]

This Egyptian dynasty extended its protection and patronage to literature as well as to science. Thus Philadelphus did not consider it beneath him to count among his personal friends the poet Callimachus, who had written a treatise on birds, and honourably maintained himself by keeping a school in Alexandria. The court of that sovereign was, moreover, adorned by a constellation of seven poets, to which the gay Alexandrians gave the nickname of the Pleiades. They are said to have been Lycophron, Theocritus, Callimachus, Aratus, Apollonius Rhodius, Nicander, and Homer the son of Macro. Among them may be distinguished Lycophron, whose work, entitled Cassandra, still remains; and Theocritus, whose exquisite bucolics prove how sweet a poet he was.

[Sidenote: The writings of Apollonius.]

To return to the scientific movement. The school of Euclid was worthily represented in the time of Euergetes by Apollonius Pergæus, forty years later than Archimedes. He excelled both in the mathematical and physical department. His chief work was a treatise on Conic Sections. It is said that he was the first to introduce the words ellipse and hyperbola. So late as the eleventh century his complete works were extant in Arabic. Modern geometers describe him as handling his subjects with less power than his great predecessor Archimedes, but nevertheless displaying extreme precision and beauty in his methods. His fifth book, on Maxima and Minima, is to be regarded as one of the highest efforts of Greek geometry. As an example of his physical inquiries may be mentioned his invention of a clock.

[Sidenote: The writings of Hipparchus.]

[Sidenote: The theory of epicycles and eccentrics.]

Fifty years after Apollonius, B.C. 160-125, we meet with the great astronomer Hipparchus. He does not appear to have made observations himself in Alexandria, but he uses those of Aristyllus and Timochares of that place. Indeed, his great discovery of the precession of the equinoxes was essentially founded on the discussion of the Alexandrian observations on Spica Virginis made by Timochares. In pure mathematics he gave methods for solving all triangles plane and spherical: he also constructed a table of chords. In astronomy, besides his capital discovery of the precession of the equinoxes just mentioned, he also determined the first inequality of the moon, the equation of the centre, and all but anticipated Ptolemy in the discovery of the evection. To him also must be attributed the establishment of the theory of epicycles and eccentrics, a geometrical conception for the purpose of resolving the apparent motions of the heavenly bodies, on the principle of circular movement. In the case of the sun and moon, Hipparchus succeeded in the application of that theory, and indicated that it might be adapted to the planets. Though never intended as a representation of the actual motions of the heavenly bodies, it maintained its ground until the era of Kepler and Newton, when the heliocentric doctrine, and that of elliptic motions, were incontestably established. Even Newton himself, in the 37th proposition of the third book of the "Principia," availed himself of its aid. Hipparchus also undertook to make a register of the stars by the method of alineations--that is, by indicating those which were in the same apparent straight line. The number of stars catalogued by him was 1,080. If he thus depicted the aspect of the sky for his times, he also endeavoured to do the same for the surface of the earth by marking the position of towns and other places by lines of latitude and longitude.

[Sidenote: The writings of Ptolemy.]

[Sidenote: His great work: the mechanical construction of the heavens.]

Subsequently to Hipparchus, we find the astronomers Geminus and Cleomedes; their fame, however, is totally eclipsed by that of Ptolemy, A.D. 138, the author of the great work "Syntaxis," or the mathematical construction of the heavens--a work fully deserving the epithet which has been bestowed upon it, "a noble exposition of the mathematical theory of epicycles and eccentrics." It was translated by the Arabians after the Mohammedan conquest of Egypt; and, under the title of Almagest, was received by them as the highest authority on the mechanism and phenomena of the universe. It maintained its ground in Europe in the same eminent position for nearly fifteen hundred years, justifying the encomium of Synesius on the institution which gave it birth, "the divine school of Alexandria." The Almagest commences with the doctrine that the earth is globular and fixed in space; it describes the construction of a table of chords and instruments for observing the solstices, and deduces the obliquity of the ecliptic. It finds terrestrial latitudes by the gnomon; describes climates; shows how ordinary may be converted into sidereal time; gives reasons for preferring the tropical to the sidereal year; furnishes the solar theory on the principle of the sun's orbit being a simple eccentric; explains the equation of time; advances to the discussion of the motions of the moon; treats of the first inequality, of her eclipses, and the motion of the node. It then gives Ptolemy's own great discovery--that which has made his name immortal--the discovery of the moon's evection or second inequality, reducing it to the epicyclic theory. It attempts the determination of the distances of the sun and moon from the earth, with, however, only partial success, since it makes the sun's distance but one-twentieth of the real amount. It considers the precession of the equinoxes, the discovery of Hipparchus, the full period for which is twenty-five thousand years. It gives a catalogue of 1,022 stars; treats of the nature of the Milky Way; and discusses, in the most masterly manner, the motions of the planets. This point constitutes Ptolemy's second claim to scientific fame. His determination of the planetary orbits was accomplished by comparing his own observations with those of former astronomers, especially with those of Timochares on Venus.

[Sidenote: His geography.]

To Ptolemy we are also indebted for a work on Geography used in European schools as late as the fifteenth century. The known world to him was from the Canary Islands eastward to China, and from the equator northward to Caledonia. His maps, however, are very erroneous; for, in the attempt to make them correspond to the spherical figure of the earth, the longitudes are too much to the east; the Mediterranean Sea is twenty degrees too long. Ptolemy's determinations are, therefore, inferior in accuracy to those of his illustrious predecessor Eratosthenes, who made the distance from the sacred promontory in Spain to the eastern mouth of the Ganges to be seventy thousand stadia. Ptolemy also wrote on Optics, the Planisphere, and Astrology. It is not often given to an author to endure for so many ages; perhaps, indeed, few deserve it. The mechanism of the heavens, from his point of view, has however, been greatly misunderstood. Neither he nor Hipparchus ever intended that theory as anything more than a geometrical fiction. It is not to be regarded as a representation of the actual celestial motions. And, as might be expected, for such is the destiny of all unreal abstractions, the theory kept advancing in complexity as facts accumulated, and was on the point of becoming altogether unmanageable, when it was supplanted by the theory of universal gravitation, which has ever exhibited the inalienable attribute of a true theory--affording an explanation of every new fact as soon as it was discovered, without requiring to be burdened with new provisions, and prophetically foretelling phenomena which had not as yet been observed.

[Sidenote: The later Alexandrian geometers.]

[Sidenote: Decline of the Greek age of Reason.]

From the time of the Ptolemies the scientific spirit of the Alexandrian school declined; for though such mathematicians as Theodosius, whose work on Spherical Geometry was greatly valued by the Arab geometers; and Pappus, whose mathematical collections, in eight books, still for the most part remain; and Theon, doubly celebrated for his geometrical attainments, and as being the father of the unfortunate Hypatia, A.D. 415, lived in the next three centuries, they were not men like their great predecessors. That mental strength which gives birth to original discovery had passed away. The commentator had succeeded to the philosopher. No new development illustrated the physical sciences; they were destined long to remain stationary. Mechanics could boast of no trophy like the proposition of Archimedes on the equilibrium of the lever; no new and exact ideas like those of the same great man on statical and hydrostatical pressure; no novel and clear views like those developed in his treatise on floating bodies; no mechanical invention like the first of all steam-engines--that of Hero. Natural Philosophy had come to a stop. Its great, and hitherto successfully cultivated department, Astronomy, exhibited no farther advance. Men were content with what had been done, and continued to amuse themselves with reconciling the celestial phenomena to a combination of equable circular motions. To what are we to attribute this pause? Something had occurred to enervate the spirit of science. A gloom had settled on the Museum.

[Sidenote: Causes of that decline.]

There is no difficulty in giving an explanation of this unfortunate condition. Greek intellectual life had passed the period of its maturity, and was entering on old age. Moreover, the talent which might have been devoted to the service of science was in part allured to another pursuit, and in part repressed. Alexandria had sapped Athens, and in her turn Alexandria was sapped by Rome. From metropolitan pre-eminence she had sunk to be a mere provincial town. The great prizes of life were not so likely to be met with in such a declining city as in Italy or, subsequently, in Constantinople. Whatever affected these chief centres of Roman activity, necessarily influenced her; but, such is the fate of the conquered, she must await their decisions. In the very institutions by which she had once been glorified, success could only be attained by a conformity to the manner of thinking fashionable in the imperial metropolis, and the best that could be done was to seek distinction in the path so marked out. Yet even with all this restraint Alexandria asserted her intellectual power, leaving an indelible impress on the new theology of her conquerors. During three centuries the intellectual atmosphere of the Roman empire had been changing. Men were unable to resist the steadily increasing pressure. Tranquillity could only be secured by passiveness. Things had come to such a state that the thinking of men was to be done for them by others, or, if they thought at all, it must be in accordance with a prescribed formula or rule. Greek intellect was passing into decrepitude, and the moral condition of the European world was in antagonism to scientific progress.