History of the Intellectual Development of Europe, Volume I (of 2) / Revised Edition

by John William Draper

It is said that Euclid opened a geometrical school in Alexandria about B.C. 300. He occupied himself not only with mathematical, but also with physical investigation. Besides many works of the former class supposed to have been written by him, as on Fallacies, Conic Sections, Divisions, Porisms, Data, there are imputed to him treatises on Harmonics, Optics, and Catoptrics, the two latter subjects being discussed, agreeably to the views of those times, on the hypothesis of rays issuing from the eye to the object, instead of passing, as we consider them to do, from the object to the eye. It is, however, on the excellencies of his Elements of Geometry that the durable reputation of Euclid depends; and though the hypercriticism of modern mathematicians has perhaps successfully maintained such objections against them as that they might have been more precise in their axioms, that they sometimes assume what might be proved, that they are occasionally redundant, and their arrangement sometimes imperfect, yet they still maintain their ground as a model of extreme accuracy, of perspicuity, and as a standard of exact demonstration. They were employed universally by the Greeks, and, in subsequent ages, were translated and preserved by the Arabs.

The writings and works of Archimedes.

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.

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.

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.

Astronomy of Eratosthenes.

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. Attempts of Aristarchus to find the distance of the sun. 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.

Biography of the Ptolemies.