SIM SO N.

THE wonderful progress that has been made in the pure mathematics since the application of algebra to geometry, begun by Vieta in the sixteenth, completed by Des Cartes in the seventeenth century, and especially the still more marvellous extension of analytical science by Newton and his followers, since the invention of the Calculus, has, for the last hundred years and more, cast into the shade the methods of investigation which preceded those now in such general use, and so well adapted to afford facilities unknown while mathematicians only possessed a less perfect instrument of investigation. It is nevertheless to be observed that the older method possessed qualities of extraordinary value. It enabled us to investigate some kinds of propositions to which algebraic reasoning is little applicable; it always had an elegance peculiarly its own; it exhibited at each step the course which the reasoning followed, instead of concealing that course till the result came out; it exercised the faculties more severely, because it was less mechanical than the operations of the analyst. That it afforded evidence of a higher character, more rigorous in its nature than that on which algebraic reasoning rests,

cannot with any correctness be affirmed; both are equally strict; indeed if each be mathematical in its nature, and consist of a series of identical propositions arising one out of another, neither can be less perfect than the other, for of certainty there can be no degrees. Nevertheless it must be a matter of regret— and here the great master and author of modern mathematics has joined in expressing it-that so much less attention is now paid to the Ancient Geometry than its beauty and clearness deserve; and if he could justly make this complaint a century and a half ago, when the old method had but recently, and only in part, fallen into neglect and disuse, how much more are such regrets natural in our day, when the very name of the Ancient Analysis has almost ceased to be known, and the beauties of the Greek Geometry are entirely veiled from the mathematician's eyes! It becomes, for this reason, necessary that the life of Simson, the great restorer of that geometry, should be prefaced by some remarks upon the nature of the science, in order that, in giving an account of his works, we may say his discoveries, it may not appear that we are recording the services of a great man to some science different from the mathematical.

The analysis of the Greek geometers was a method of investigation of peculiar elegance, and of no inconsiderable power. It consisted in supposing the thing as already done, the problem solved, or the truth of the theorem established; and from thence it reasoned until something was found, some point reached, by pursuing steps each one of which led to the next, and by only assuming things which were already known

being ascertained by former discoveries. The thing thus found, the point reached, was the discovery of something which could by known methods be performed, or of something which, if not self-evident, was already by former discovery proved to be true; and in the one case a construction was thus found by which the problem was solved, in the other a proof was obtained that the theorem was true, because in both cases the ultimate point had been reached by strictly legitimate reasoning, from the assumption that the problem had been solved, or the assumption that the theorem was true. Thus, if it were required from a given point in a straight line given by position, to draw a straight line which should be cut by a given circle in segments, whose rectangle was equal to that of the segments of the diameter perpendicular to the given line-the thing is supposed to be done; and the equality of the rectangles gives a proportion between the segments of the two lines, such that, joining the point supposed to be found, but not found, with the extremity of the diameter, the angle of that line with the line sought but not found, is shown by similar triangles to be a right angle, i. e., the angle in a semicircle. Therefore the point through which the line must be drawn is the point at which the perpendicular cuts the given circle. Then, suppose the point given through which the line is to be drawn, if we find that the curve in which the other points are situate is a circle, we have a local theorem, affirming that, if lines be drawn through any point to a line perpendicular to the diameter, the rectangle made by the segments of all the lines cutting the perpendicular is constant; and this theorem would be

demonstrated by supposing the thing true, and thus reasoning till we find that the angle in a semicircle is a right angle, a known truth. Lastly, suppose we change the hypothesis, and leave out the position of the point as given, and inquire after the point in the given straight line from which a line being drawn through a point to be found in the circle, the segments will contain a rectangle equal to the rectangle under the perpendicular segments-we find that one point answers this condition, but also that the problem becomes indeterminate; for every line drawn through that point to every point in the given straight line has segments, whose rectangle is equal to that under the segments of the perpendicular. The enunciation of this truth, of this possibility of finding such a point in the circle, is a Porism. The Greek geometers of the more modern school, or lower age, defined a Porism to be a proposition differing from a local theorem by a defect or defalcation in the hypothesis; and accordingly we find that this porism is derived from the local theorem formerly given, by leaving out part of the hypothesis. But we shall afterwards have occasion to observe that this is an illogical and imperfect definition, not coextensive with the thing defined; the above proposition, however, answers every definition of a Porism.

The demonstration of the theorem or of the construction obtained by investigation in this manner of proceeding, is called synthesis, or composition, in opposition to the analysis, or the process of investigation; and it is frequently said that Plato imported the whole system in the visits which he made, like Thales of Miletus and Pythagoras, to study under the Egyptian

geometers, and afterwards to converse with Theodorus at Cyrene, and the Pythagorean School in Italy. But it can hardly be supposed that all the preceding geometers had worked their problems and theorems at random; that Thales and Pythagoras with their disciples, a century and a half before Plato, and Hippocrates, half a century before his time, had no knowledge of the analytical method, and pursued no systematic plan in their researches, devoted as their age was to geometrical studies. Plato may have improved and further systematized the method, as he was no doubt deeply impressed with the paramount importance of geometry, and even inscribed upon the gates of the Lyceum a prohibition against any one entering who was ignorant of it. The same spirit of exaggeration which ascribes to him the analytical method, has also given rise to the notion that he was the discoverer of the Conic Sections; a notion which is without any truth and without the least probability.

Of the works written by the Greek geometers some have come down to us; some of the most valuable, as theElements' and 'Data' of Euclid, and the 'Conics' of Apollonius. Others are lost; but, happily, Pappus, a mathematician of some merit, who flourished in the Alexandrian school about the end of the fourth century, has left a valuable account of the geometrical writings of the elder Greeks. His work is of a miscellaneous nature, as its name, Mathematical Collections,' implies; and excepting a few passages, it has never been published in the original Greek. Commandini, of Urbino, made a translation of the whole six books then discovered; the first has never been.

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