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probably be very unequal, and would correspond to the urgency of the motives which impelled them to prosecute the study and to their views of its practical importance in promoting those arts which ameliorate and embellish social life. There might likewise be found in every nation individuals of superior intelligence and discernment who might attain to greater proficiency in geometry than was usual in the age in which they lived, and by their means the knowledge of its truths, and their application, might to a certain extent be disseminated. Before, however, it could assume a systematic form, it was evidently necessary that the minds of men should, either by natural or providential circumstances, be earnestly and steadily directed to its cultivation. Now, in the early ages of the world, human beings were principally urged to vigorous exertions of body or mind by the desire of increasing their physical enjoyments, or of providing against whatever might have a tendency to endanger either their own security or that of their possessions. It appears therefore by no means improbable that geometry as a science had, according to the commonly received tradition, its origin in Egypt, in consequence of its physical peculiarities. The early civilisation of that country, the peculiar nature of its climate, the fertility and value of its soil, annually enriched by a deposit of mud left on its surface by the inundations of the Nile, which must in the course of years have obliterated the boundaries of property, would gradually impress on the minds of its inhabitants the absolute necessity of discovering some method of accurately ascertaining the dimensions of their respective possessions, that thus they might be enabled to restore the boundaries which had been effaced. Their first efforts to attain their object would no doubt be tentative and very clumsy and inaccurate, but these, by directing their attention to the forms of the different portions of land, and the figures of the parts into which they could most conveniently be divided for the purpose of measurement, would induce them by degrees to investigate the peculiar properties of such figures, and the methods by which their contents could most accurately be determined. On the knowledge thus acquired, a system of rules of measurement would be founded which experience would improve, and to these new truths would be added as they were discovered. They appear also to have been in possession of the method of comparing magnitudes by supraposition, or by mentally applying the one of them to the other, in order to ascertain the conditions of their equality. The particular steps, etc., of their intellectual progress from the rude perceptions of sense to the refined and abstract conceptions which are presented to us in the definitions of geometry, it is now impossible to ascertain, as no record of them has been preserved. The geometrical knowledge of the Egyptians, however, appears to have scarcely extended beyond its elementary truths; but as they have left no writings on the subject, it is impossible to form any just estimate of their proficiency.
From Egypt, geometry was conveyed into Greece, where it was for a long period cultivated with great success. In the earlier ages of the world knowledge could not be acquired from books or written documents, as probably few or none existed. Men of learning and science, in different countries remote from each other, and not books, were then the only depositaries of knowledge. The philosopher, therefore, who was ambitious to extend the sphere of his information before he could gratify so laudable a desire, had, with great danger and inconvenience to himself, to travel to those countries where men eminent for learning and science had their residence. By holding intercourse with these sages, he could both invigorate his own mental powers and enlarge his views, and, by friendly conference and discussion, become master of whatever additional knowledge they had to communicate. With this view, the philosopher Thales is said, even at an advanced period of life, to have travelled to Egypt, and after gaining an acquaintance with learning and philosophy, as then cultivated in that country, he on his return introduced the study of these, and particularly that of geometry, into his native country. This event happened about 640 years before the Christian era. There are no certain traces of geometry in Greece previous to the time of Thales; from him, therefore, the study of the science in that country took its origin. He founded what is called the Ionian school, and his celebrity for learning and science drew to him many disciples. In this school the study of geometry was diligently prosecuted for many years, both by Thales himself during his life, and afterwards by several eminent men who became his successors, and who, by their combined labours, instructions, and example, imparted to their countrymen a taste for abstract speculation. About a century later, his scholar Pythagoras, in imitation of his master, travelled to India, Egypt, and other countries, in quest of knowledge. Having conversed with men eminent for science and philosophy wherever they were to be found, and having improved and enriched his mind by the wisdom of their communications, he, on his return, founded a school at Crotona in Italy, where he acquired great renown both as a philosopher and a geometrician. Thales and Pythagoras are reported to have made many discoveries in the science which cannot now be distinguished, and which have gone to enrich the common stock of geometrical knowledge. Thales, however, is said to have been the first that applied the circle to the measurement of angular magnitude, and he is certainly known to have discovered the important proposition, that all the angles in a semicircle are right angles. Pythagoras also has the singular honour of discovering and demonstrating the still more important theorem, that the square described on the hypothenuse of a right angled triangle is equal to the squares described on the two sides. These discoveries are convincing proofs that the knowledge of geometry which Thales and Pythagoras acquired in Egypt and elsewhere must have been of a very elementary description; for a system of geometry of which these two propositions form no part must be regarded as exceedingly defective. Pythagoras was also the first that observed that relation of lines called their incommensurability as that of the diagonal of a square to its side, and invented the five regular solids afterwards called the Platonic bodies. He appears, indeed, to have been one of the most eminent men of antiquity, possessed both of mental powers of the first order, and also of singular sagacity, who at that early period conceived and firmly maintained that theory of the solar system which has been so satisfactorily established in modern times. His doctrine of metempsychosis, or transmigration of souls, was no doubt extremely absurd, but in no degree more senseless and ridiculous than the modern theories of development, which, all rest upon the groundless assumption that matter is possessed of efficiency, a power of agitating, moulding, and expanding itself, a property of which the inductive philosopher declares it to be totally destitute, because it can be established by experiment that any change, either within or upon it, can be produced only by the operation of some extrinsic agency.
The researches of the earliest geometers were confined to the properties of rectilineal figures, or rather to those of triangles, into which all rectilineal figures may be resolved. The circle probably engaged their attention only as a convenient instrument in the construction of such figures. The properties of this curve soon, however, became objects of inquiry, and a few of them were discovered at an early period in the progress of the science. The straight line and circle, and what could be effected by their combination, terminated the investigations of the first geometers with regard to lines and plane figures.
Their attention was next directed to plane surfaces and their intersections, to solids bounded by planes, and particularly to the three circular solids—the cone, the cylinder, and the sphere. The cone they supposed to be generated by the revolution of a right angled triangle about one of the sides containing the right angle; the cylinder by the revolution of a rectangle about one of its sides; and the sphere by that of a circle about its diameter. The discovery of the superficial and solid contents of these circular bodies were objects to which the ancient philosophers devoted much attention: this formed their sublime geometry, and their most strenuous efforts were put forth in its successful prosecution.
Such were the principal subjects of speculation which en