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that this science had its origin, like all others, in the necessities of men. The word Geometry seems to have been originally applied to the measuring of land. The earliest information on this subject is derived from Herodotus, (book ii., c. 109,) where he describes the customs of the Egyptians in the age of Sesostris, who reigned in Egypt from about 1416 to 1357, B.C. The account of Herodotus to this effect: “I was informed by the Priests at Thebes, that King Sesostris made a distribution of the territory of Egypt among all his subjects, assigning to each an equal portion of land in the form of a quadrangle, and that from these allotments he used to derive his revenue by exacting every year a certain tax. In cases, however, where a part of the land was washed away by the annual inundations of the Nile, the proprietor was permitted to present himself before the King, and signify what had happened. The King then used to send proper officers to examine and ascertain, by admeasurement, how much of the land had been washed away, in order that the amount of tax to be paid for the future might be proportional to the land which remained. From this circumstance I am of opinion, that Geometry derived its origin; and from hence it was transmitted into Greece.” The natural features and character of the land of Egypt, where rain is unknown, are such as to give credibility, at least, to the tradition recorded by Herodotus. In the earliest records of history, the population of Egypt is represented as numerous; and in the valley of the Nile, the extent of cultivated land is comparatively small. Its extreme fertility is also placed in close contrast with the barrenness of the districts beyond the limits of the inundations of the Nile, by which the boundaries of the land on its margin are annually liable to alteration. There appear, therefore, some grounds for the belief that the geometrical allotment of land had its origin on the banks of the Nile. But independently of the tradition of Herodotus, it seems reasonable to suppose, that the science of Plane Geometry may have originated in the necessity of measuring and dividing lands, which must have arisen as soon as property in land came to be recognised among men. This recognition is found in the oldest historical records known in any language. The narrative in the 23d chapter of the Book of Genesis refers to Palestine, and belongs to a period 1860 years B.C. In Egypt also, about one hundred and sixty years later, as we learn from Gen. xlvii., not only was property in land recognised, but taxes were raised from the possessors and cultivators of the soil. This necessarily implies that there existed some method of estimating and dividing land, rude, probably, and inaccurate at first; but as society advanced and its wants increased, gradually becoming more exact.
The existence of the pyramids, the ruins of temples, and other architectural remains, supply evidence of some knowledge of Geometry ; although it is possible that the geometrical properties of figures, necessary for such works, might have been known only in the form of practical rules, without any scientific arrangements of geometrical truths.
The word Geometry is used in a more extensive sense, as the science of Space; or that science which discusses and investigates the properties and relations existing between definite portions of space, under the fourfold division of lines, angles, surfaces, and volumes, without regard to any properties they may have of a physical nature. Of the origin and progress of Geometry, in this sense, it is proposed here to give some short account, as far as can be ascertained.
Whatever geometrical or astronomical science may have been possessed
by the earlier Chaldeans and Egyptians, there are not known to be any historical records, which supply definite views of its limits or extent. În the most ancient Jewish writings, there is not the least allusion from which to infer that scientific Geometry was known and cultivated by that people. The traditions recorded by Josephus on this subject (book i., c. 3, 9) can scarcely be considered worthy of being received as historical truth, since the subsequent history of the Jews does not inform us that they were, at any period, a scientific people.
Other ancient writers also confirm the tradition of Herodotus, that from Egypt the knowledge of Geometry passed over into Greece, where it attained a high degree of cultivation. Proclus attributes to Thales the merit of having first conveyed the knowledge of Geometry from Egypt to Greece. Thales was a native of Miletus, at that time the most flourishing of the Greek colonies of Ionia in Asia Minor. He was born about 640 B.C., and was descended from one of the most distinguished families, originally of Phænicia. Thales, from a desire of knowledge, is reported by Diogenes Laertius to have travelled into Egypt, and to have held a friendly intercourse with the Priests of that country ; thus obtaining an acquaintance with the science of the Egyptians. The same writer also adds, that he learned the art of Geometry among the Egyptians, and suggested a method of ascertaining the altitude of the pyramids by the length of their shadows. Thales is also said to have been the discoverer of some geometrical theorems, and to have left to his successors the principles of many others. He is reported by Herodotus (book i., c. 74) to have foretold the year in which an eclipse of the sun would happen. He also designated the seasons, and found the year to consist of three hundred and sixty-five days.
Nearly at the same time with the commencement of speculative philosophy in Ionia, a spirit of inquiry began to show itself in some of the Achæan and Dorian colonies in Magna Græcia. The most distinguished man of these times was Pythagoras. He was born at Samos, about 568 B.C. ; and his descent is referred by Diogenes Laertius to the Tyrrhenian Pelasgi. According to the account of Proclus, Pythagoras was the first who gave to Geometry the form of a deductive science, by showing the connexion of the geometrical truths then known, and their dependence on certain first principles. The traditionary account, that Pythagoras was the founder of scientific mathematics, is, in some degree, supported by the statement of Diogenes Laertius, that he was chiefly occupied with the consideration of the properties of number, weight, and extension, besides music and astronomy. He is reputed to have held, that the sun is the centre of the system, and that the planets revolve round it. This has been called, from his name, the Pythagorean System, which was revived by Copernicus, A.D. 1541, and proved by Newton. As a moral philosopher, many of his precepts relating to the conduct of life will be found in the verses which bear the name of the Golden Verses of Pythagoras.
Pythagoras was followed by Anaxagoras of Clazomene. Aristotle states that he wrote on Geometry ; and Diogenes Laertius reports that he maintained the sun to be larger than the Peloponnesus.
About 450 B.C., Hippocrates of Chios was the most eminent geometer of his time, and is reported to have written a treatise on the Elements of Geometry; no fragments of which, however, are known to be in existence.
Democritus, a native of Abdera, about the 80th Olympiad, was celebrated
for his knowledge both of philosophy and the mathematics. He is stated to have spent his large patrimony in travelling in distant countries.
Theodorus of Cyrene was eminent for his knowledge of Geometry, and is reported to have been one of the instructers of Plato.
We now come to the time of Plato, one of the most distinguished philosophers that ever lived, whose writings are still read, and regarded as of inestimable value. Plato visited Egypt, and, on his return, founded his school at Athens, about 390 B.C. Over the entrance he placed the inscription, Ουδείς αγεωμέτρητος εισίτω" “Let no one ignorant of Geometry enter here.” This is a plain declaration of Plato's opinion respecting Geometry. He considered Geometry as the first of the sciences, and as introductory and preparatory to the pursuit of the higher subjects of human knowledge. To Plato is attributed the discovery of the method of the Geometrical Analysis ; but by what means he was led to it, or to the invention of geometrical loci, is not known. He is said to have discovered the Conic Sections. The trisection of an angle, or an arc of a circle, was another famous problemi, which engaged the attention of the school of Plato.
From the academy of Plato proceeded many who successfully cultivated, and very considerably extended, the bounds of geometrical science. Proclus names thirteen of the disciples and friends of Plato who improved and made additions to the science. Diogenes Laertius reports of Archytas, that he was the first who brought Mechanics into method by the use of mechanical principles, and the first who applied organic motions to Geometrical figures, and found out the duplication of the cube in Geometry. Eudoxus, à native of Cnidus, a town of Caria in Asia Minor, was one of the most intimate of the friends of Plato. He is reported to have written on the Elements, and to have generalised many results which had originated in the school of Plato, and to have advanced the science of Geometry by many important discoveries. Aristotle, though originally a disciple of Plato, and attached to his school for a period of twenty years, became the founder of a new sect of philosophers,—the Peripatetics, and opened a school, B.C. 341, at the Lyceum, on the banks of the Ilissus. There he continued twelve years, till the false accusation of Eurymedon obliged him to flee to Chalcis, where he died at the age of sixty-three.
On this division of the Platonic school, the two sects—both the Academics and the Peripatetics-continued to hold the same opinion on the utility of Geometry, as the necessary introductory knowledge for all who were desirous of proceeding with the study of philosophy. Thus the science of Geometry continued to be cultivated, and to make advancement.
the Peripatetic school are especially celebrated, Theophrastus and Eudemus, who devoted themselves chiefly to mathematical studies.
The birth-place and even the country of Euclid are unknown, and he has been very frequently confounded with another philosopher of the same name, who was a native of Megara. He studied at Athens, and became a disciple of the Platonic school. He flourished in the time of Ptolemy Lagus, (B.C. 323 to 284,) to whom he made the celebrated reply, that “there is no royal road to Geometry.” He is said to have successfully cultivated and taught Geometry and the mathematical sciences at Alexandria, shortly after the school of Philosophy was founded in that city. The school at Alexandria became most distinguished for the eminent mathematicians it produced, both in the lifetime of Euclid and afterwards, until the destruction of the great library at Alexandria, and the subjugation of Egypt by the Arabians. Euclid has become celebrated chiefly by his work on the Elements of Geometry, for which his name has become a synonym.
It has been a question whether Euclid was the author or the compiler of the Elements of Geometry, which bear his name. If Euclid were the discoverer of the propositions contained in the thirteen books of the Elements, and the author of the demonstrations, he would be a phenomenon in the history of science. It is by far more probable that he collected and arranged the books on Geometry in the order in which they have come down to us, and made a more scientific classification of the geometrical truths which were known in his time. Euclid may also have been the discoverer of some new propositions, and may have amended and rendered more conclusive the demonstrations of others. From the slow advances of the human mind in making discoveries, and the general history of the progress of the sciences, it would seem unreasonable to assign to Euclid a higher place than that of the compiler and improver of the Elements of Geometry. These, thus arranged and improved by Euclid, were acknowledged so far superior in completeness and accuracy to the elementary treatises then existing, that they entirely superseded them, and in course of time all have disappeared. The book of Euclid's Elements is therefore the most ancient work on Geometry known to be extant. The Greek arithmetical notation, employed in the arithmetical portion, has yielded to the more perfect system of the Indian ; but the geometrical portion, from the time it was first put forth till the present day, a period of upwards of two thousand years, has maintained its high character as an elementary treatise, in all nations wherever the sciences have been cultivated.
Whatever we learn from Herodotus respecting Geometry in Egypt, is referred to the age of Ses ris, and does not carry us beyond such processes as might exist without any attempts at science. The Geometry which was brought from Egypt to Greece appears to have been in its infancy; and all that Pythagoras and others borrowed from the Egyptians could not have exceeded some practical rules and their applications. The general tenor of all the traditional and probable evidence tends to show, that the scientific form of the Elements of Geometry is due to the acute intellect of the Greeks. And this presumption is reduced almost to historical certainty by the existing remains of the Greek Geometry.
Archimedes was born at Syracuse, B.C. 287, about the period of the death of Euclid, and became the most eminent of all the Greek mathematicians. His discoveries in Geometry, Mechanics, and Hydrostatics, form a distinguished epoch in the history of mathematical science; and his remaining writings on the pure Mathematics are the most valuable portion of the ancient Geometry. The story of the crown of King Hiero, to whom Archimedes was related, is briefly this. Hiero had delivered to a goldsmith a certain weight of gold to be converted into a votive crown. The King suspected that the crown he received from the smith was not of pure gold, though of the proper weight, but that it was alloyed with silver. He applied to Archimedes to ascertain, without melting the crown, whether it contained alloy. It was observed by Archimedes, on going into a bath full of water, that when his body was immersed in the bath, a quantity of water equal to the bulk of his body flowed over the edge of the bath. It occurred at once to him, that if a weight of pure gold equal to the weight of the crown were immersed in a vessel full of water, and the quantity of water left in the vessel measured, on the gold being taken out, by doing the same with the
crown in the same vessel, he would be able to ascertain whether the bulk of the crown were greater than the bulk of an equal weight of pure gold. For any weight of silver is larger in bulk than an equal weight of pure gold. According to Vitruvius, as soon as he had discovered the method of solution, he leaped out of the bath, and ran hastily through the streets to his own house, shouting, Eūpnka, cüpnka! He was also the inventor of a machine for raising water from lower to higher levels, which was called the screw of Archimedes. An important application of the principle of this screw has lately been made in the propulsion of ships by means of steam power. When Syracuse was besieged by a land and naval armament under Appius and Marcellus, the besieged held out a successful resistance for three years, chiefly by the aid of the machines invented by Archimedes. The city was at length surprised and taken, B.c. 212. Archimedes, when seventy-five years of age, was slain by a soldier, while intent on the solution of a problem. He is reported to have expressed a desire that a sphere inscribed on a cylinder might be engraved on his tomb, to record his discovery of the relation between the volumes and the surfaces respectively of these two solids. About two hundred years after his death, his tomb was discovered near Syracuse by Cicero, while Quæstor in Sicily. It was nearly overgrown with bushes and brambles, which he caused to be cleared away. The tomb was identified as the tomb of Archimedes by the sphere and cylinder engraved upon the stone with the inscription, the latter part of which was completely effaced. (Cic. Tusc. Quæst., lib. iv.)
Conon was the friend and cotemporary of Archimedes, and is celebrated by Virgil in his third Eclogue. In the treatise on the quadrature of the parabola, speaking of his genius, Archimedes exclaims, “How many theorems in Geometry, which to others have appeared impossible, would Conon have brought to perfection, if he had lived !”
Cotemporary with Archimedes was Eratosthenes, a distinguished geometrician and astronomer. He was the first who attempted to measure the circumference of the earth by means of observations of the sun at two different places, near the same meridian, at the time of the solstice. Though he did not completely succeed, on account of the inaccuracy of his data, he pointed out the method. None of his works have descended to modern times except a few fragments, and a list of the names of forty-four constellations, and the principal stars in each constellation.
Apollonius of Perga, a city of Pamphylia, lived about the same time, and stands next in fame to Archimedes. He was born at the time when Ptolemy Euergetes was King of Egypt: he studied the mathematical sciences at Alexandria in the school which Euclid's disciples had founded, and passed there the greater part of his life. He was the author of several works on Geoinetry, and became so eminent in that science, that he was called by his cotemporaries the Great Geometer.
Hipparchus, a native of Nice, though not a writer on the Elements of Geometry, is regarded as the first who reduced Astronomy to a science, and either devised or greatly improved the methods of calculation in Trigonometry, which form the basis of the science of Astronomy. He was the first who discovered the precession of the equinoxes, and taught how to foretell eclipses, and form tables of them. The catalogue of stars which he observed and registered between the years B.c. 160 and 135, is preserved in Ptolemy's Almagest. They are arranged according to their longitudes and their apparent magnitudes. He was also the first who suggested the idea of fixing the position of places on the earth, as he did in the heavens, by