CHAP. II. Origin and Progress of Geometry. DIFFERENT origins, more or less ancient, are ascribed to geometry. Most authors assign Egypt for it's birthplace; as Herodotus for example, the first who began to write history in prose, A, C. 450.; for in the remotest antiquity, the memory of past events was preserved only in a feeble and mutilated state in some rude songs; and afterward it was confounded with fiction in the poems of Hesiod and Homer, where every thing else was sacrificed to the embellishment of the subject. We will recite the account, which Herodotus gives of what he himself learned respecting it at Thebes and at Memphis. "I was told," says he, that Sesostris had divided Egypt among all his subjects, and that he had given each an equal quantity of land, on condition of paying annually a proportionate tribute. If the allotment of any one were diminished by the river, he repaired to the king, and related what had befallen his land. The king then sent to the spot, and caused his land to be measured, that he might know what diminution it had undergone, and require a tribute only in proportion to what remained. I believe,' adds Herodotus, that here geometry took it's birth, and that hence it was transmitted to the greeks.' In this passage we perceive two distinct things: the account of a verification depending on geometry, and and the private opinion of Herodotus respecting the origin of this science. If, as many chronologists suppose, Sesostris be the same with Shishak, who made war on Rehoboam, the son of Solomon, it would follow from the opinion of Herodotus, that the origin of geometry preceded the christian era about a thousand years only: but it may be carried much higher; for the measuring of the fields, directed by Sesostris, is not only far from fixing precisely the origin of geometry, but even appears to indicate, that the science must have made some progress at that time. If we were inclined to indulge in frivolous conjectures, we should carry back the origin of geometry to the invention of the rule, square, and compasses, since it makes the greatest use of these in-* struments in practice: but the same argument of their use should rather lead to the supposition, that they were invented at the commencement of society, and that the invention was prompted by simple ne cessity, without the aid of any theory, for the pur pose of constructing huts or houses. If we confine ourselves, in beginning this abstract of the history of geometry, to the time when it assumes, at least to us, the character of a real science, we shall at once transport ourselves to Greece, and the age of Thales. Whether this philosopher taught the egyptians, or learned from them, the method of measuring the height of the pyramids of Memphis by the extent of their shadows, we discover, that he was versed in the theory and practice of geometry. A. c. 640. All the ancient writers indeed speak of him as a very learned geometrician. To him is ascribed the first employment of the circumference of a circle for the measure of angles. No doubt he made other discoveries in geometry, now lost, or confounded among those, which have been collected and transmitted to posterity by the authors of elementary works. He possessed an ample share of knowledge in every branch of mathematics and physics, as we have already observed. In astronomy he will appear again before us with distinction. The name of Pythagoras is rendered immortal in the annals of geometry, by the discovery which he made, that the square of the hypothenuse of a rectangled triangle is equal to the sum of the squares of the other two sides. A. c. 590. Some authors relate, that he was so transported with joy and gratitude to the gods for having inspired him with it, that he sacrificed to them a hundred oxen: but we can hardly reconcile this hecatomb with the moderate fortune the philosopher possessed, still less with his religious opinions concerning the transmigration of souls. But be this as it may, never had enthusiasm a better foundation. This problem of Pythagoras ranks in the first class of geometrical truths, both from the singularity of it's result, and the number and importance of the cases to which it is applicable in every branch of mathematics. The author himself derived from it this consequence, that the diagonal of a square is incommensurate to it's side it led also to the discovery of several ge neral neral properties of other incommensurate lines or -numbers, In the long series of grecian philosophers, which extends from Thales and Pythagoras to the destruction of the Alexandrian school, there is scarcely one by whom the mathematics were neglected. Astronomy is in general the science on which they were chiefly occupied; but the most celebrated of them applied themselves to geometry as the principal, without which all the rest must remain lifeless and inactive. The propositions, which constitute the bulk of what we now call elementary geometry, were almost all invented by the philosophers of Greece. One of the most ancient of these geometricians, mentioned after Thales and Pythagoras, is Enopidus of Chios, the author of some very simple problems, as of letting fall a perpendicular upon a right line from a given point, making an angle equal to another angle, dividing an angle into two equal parts, &c. A. c. 480. Zenodorus, his contemporary, and the first of the ancients of whose geometrical writings any have reached us, one being preserved by Theon in his Commentary on Ptolemy, went farther. He showed the falsity of the opinions then entertained, that figures with equal circumferences must have equal areas. The demonstration of this was not easy to discover, and proves that geometry had then made considerable progress. The ingenious theory of the regular bodies originated about the same time in the pythagorean school. Hippocrates Hippocrates of Chios distinguished himself by the quadrature of the famous lunula of the circle, which bear his name. · A. c. 450. Having described three semicircles on the three sides of a rightangled isos-· celes triangle as diameters, that on the hypothenuse being in the same direction as the others, he found, that the sum of the areas of the two equal lunes, comprised between the two quadrants answering to the hypothenuse and the semicircles answering to the other two sides, was equal to the area of the triangle. This is the first instance of a curvilinear space being found equal to a rectilinear, and it has been extended to other more abstruse and curious quadratures, in proportion as geometry has improved. The attainments of Hippocrates of Chios in geometry were very extensive. He wrote Elements of Geometry, much esteemed in his time, though other works of the same kind, particularly Euclid's, have occasioned his to be lost and forgotten. He appeared with honour in the list of geometricians, who attempted to solve the celebrated problem of the duplication of the cube, which at that period began to be pursued with ardour. The object of this problem was to construct a cube, that should be double a given cube; not in respect to it's sides, about which there could be no question; or even it's superficies, for this could have been easily accomplished by the geometry of that time; but in it's solidity, so that it's weight should be double that of the other, supposing them both to be made of one homogeneal substance. And it was to be re |