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INTRODUCTION.

The aim of the following brief notices, is to give some account of the origin and progress of the science of Geometry, together with the names of the men by whom it has been successively advanced. Respecting the history of science it has been remarked that it serves, at least, to commemorate the benefactors of mankind; an object which can scarcely be considered as unworthy or unimportant. It is probable that this science had its origin, like all others, in the necessities of men. The word Geometry (yewuet pía), formed from two Greek words, yn and pet péw, seems to have been originally applied to the measuring of land. The earliest information on this subject is derived from Herodotus (Book 11. 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 is 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 recognized among men. This recognition is found in the oldest historical records known in any language. The narrative in the 23rd chapter of the book of Genesis refers to Palestine, and belongs to a period 1860 years B.c. In Egypt

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also, about 160 years later, as we learn from Gen. xlvii. not only was property in land recognized, 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 arrangement 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. In 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 1. 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, in his Commentary on Euclid's Elements of Geometry (Book 11. c. 4), 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. (Herod. 1. 170. Diog. Laert. 1. 22.) 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. Plutarch relates this story, and adds, that Amasis, who was then king of Egypt, was astonished at the sagacity of Thales. If this tradition, recorded both by Diogenes Laertius and by Plutarch, be deemed worthy of credit, it would appear that the idea suggested by Thales was, to the Egyptians, a new application of a geometrical truth. Whatever mathematical knowledge the Egyptians might possess in the age of Thales, there are no writings, either cotemporary or of later times, which exhibit its extent. 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. The theorems which stand as the 5th, 15th, and 26th Propo

sitions of the first Book of Euclid's Elements of Geometry are attributed to him by Proclus, and Prop. 31. Book 111, also Props. 2, 3, 4, . 5, of Book iv. He is reported by Herodotus (Book 1. 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 365 days. All this implied not only some acquaintance with Geometry, but a considerable knowledge of the motions and periodical revolutions of the heavenly bodies. Anaximander of Miletus and Anaximenes are mentioned as disciples of Thales. The opinions of the latter are discussed by Aristotle immediately after those of Thales. Ameristus, the brother of Stesichorus the poet, is named by Hippias of Elis as a celebrated geometer.

Nearly at the same time with the commencement of speculative philosophy in Ionia, a spirit of enquiry began to shew 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. After having been a disciple of Thales, he is reported to have visited Egypt, where he became the pupil of Oinuphis at Heliopolis, (which in the book of Genesis is called On), once a famous city of Lower Egypt. (Plutarch de Iside et Osiride, s. 10.) After his return from Egypt, he established a school at Crotona, an Achæan colony in Magna Græcia, which became very celebrated, and continued for nineteen generations. According to the account of Proclus (Book 11. c. 4), Pythagoras was the first who gave to Geometry the form of a deductive science, by shewing the connexion of the geometrical truths then known, and their dependence on certain first principles. There are not known to be extant any particular accounts, or even fragments, of the earliest attempts to reduce geometrical truths to a system. It is, however, scarcely possible that any arrangement could have been attempted before a considerable number of geometrical truths had been discovered, and their connexion observed. 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. The passage of Cicero (De Nat. Deor. 111. 36) may be referred to as evidence that later writers were unable to give any precise account of the mathematical discoveries of Pythagoras. To Pythagoras, however, is attributed the discovery of some of the most important elementary properties contained in the first book of Euclid's Elements. The very important truth contained in Prop. 47, Book 1. is also ascribed to Pythagoras. Whether his discovery of this truth resulted from geometrical, or from numerical considerations is not certainly known: Proclus attributes to him the discovery of that right-angled triangle, the three sides of which are respectively 3, 4, and 5 units. To Pythagoras also belongs the discovery, that there are only three kinds of regular polygons which can be placed so as to fill up the space round a point; namely, six equilateral triangles, four squares, and three regular hexagons. Proclus attributes to him the doctrine of incommensurables, and the discovery of the five regular solids, which, if not due to Pythagoras, originated in his school. In 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. It is probable they were composed by some one of his school, and contain the substance of his moral teaching. The speculations of the early philosophers did not end in the investigation of the properties of number and space. The Pythagoreans attempted to find, and dreamed they had found, in the forms of geometrical figures and in certain numbers, the principles of all science and knowledge, whether physical or moral. The figures of Geometry were regarded as having reference to other truths besides the mere abstract properties of space. They regarded the unit, as the point; the duad, as the line; the triad, as the surface; and the tetractys, as the geometrical volume. They assumed the pentad as the physical body with its physical qualities. They seem to have been the first who reckoned the elements to be five in number, on the supposition of their derivation from the five regular solids. They made the cube, earth; the pyramid, fire; the octohedron, air ; the icosahedron, water; and the dodecahedron, æther. The analogy of the five senses and the five elements was another favourite notion of the Pythagoreans.

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. Ænopides of Chios was somewhat junior to Anaxagoras, both of whom are mentioned by Plato in his Rivals. Proclus ascribes to Ænopides the discovery of the truths which form the 12th and 23d Props. of the first Book of Euclid. Briso and Antipho are mentioned by Aristotle, as distinguished geometers, but no records are known to be extant of their writings or their discoveries.

About 450 B.C., Hippocrates of Chios, their cotemporary, 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. He discovered the quadrature of the lunes which bear his name, by describing, in the same direction, semicircles on the three sides of an isosceles right-angled triangle, and observing that the sum of the two lunes, between the two quadrants of the larger semicircle and the two smaller semicircles, was equal to the area of the triangle. By means of the lunes he attempted the quadrature of the circle, but without success. The account of his attempts will be found in the Commentary of Simplicius, on the first book of Aristotle's Physics. Hippocrates also solved the problem of the duplication of the cube, which is, to find the length of the edge of a cube which shall be twice as great as a given cube. This problem, at that time, and for ages afterwards, excited very great attention among philosophers. He shewed that the solution depended on finding two geometric mean proportionals between two given lines.

About 100 years after Pythagoras, Zenodorus wrote a tract, preserved in Theon's commentary on the Almagest of Ptolemy, in which he shews that plane figures having equal perimeters have not equal

Democritus, a native of Abdera, about the 80th Olympiad was cele

areas.

brated 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 instructors 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, Ojdeis åyewuét pntos cioitw: “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. Plato both cultivated and advanced the science of Geometry, as we learn from the testimony of Proclus and Pappus. The character of his writings, though not confined to discourses on Mathematics, affords incontestable evidence how great an admirer he was of Geometry, and how zealously he cultivated that science. 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. Hippocrates had, before his time, reduced the problem of the duplication of the cube to that of finding two mean proportionals between the edge of the given cube, and a line double that length; Plato attempted a solution of the problem in this form, by means of the right line and circle only. In this attempt, however, he failed, but effected a solution by means of two rulers, which could not be admitted as purely geometrical, since it involved other considerations besides those of the straight line and circle. It is uncertain whether the restriction of constructions in Geometry to the right line and circle, originated in the School of Plato, or at an earlier period. Plato is said to have discovered the Conic Sections,—curves which result from planes intersecting the surface of a cone,--and some of their more remarkable properties. Great attention was given to this subject, both during his lifetime, and after his death, by his cotemporaries and their successors. The Conic Sections and their properties were considered a distinct branch of the science, and called the Higher Geometry.” The trisection of an angle, or an arc of a circle, was another famous problem, which engaged the attention of the School of Plato. This, as well as the duplication of the cube, was, for many ages, believed possible, by means of the right line and circle, and has been repeatedly attempted from the earliest times, without success. It baffled the genius of Archimedes and others, who were not aware of the impossibility of its solution by the method they applied. The problem of the trisection of an angle or arc is reducible to the following problem: To draw a right line from a given point, cutting the semi-circumference of a circle in two points, and the diameter produced, so that the chord intercepted between the two points of intersection may be equal to the radius. This condition leads to the algebraical equation of one of the conic sections, whose properties are not the same as those of the right line an circle; and hence its impossibility is inferred.

From the Academy of Plato, proceeded many who successfully cultivated, and very considerably extended, the bounds of Geometrical

Proclus names thirteen of the disciples and friends of Plato

science.

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