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duplication of the cube has but one real root, while that relating to the trisection of the angle has three.
Most of the ancient geometricians were so possessed with the hope of resolving these problems by means of the rule and compasses, that they could not bring themselves to give it up. They made many fruitless attempts; and this eagerness became a kind of epidemic disease, which has been transmitted from age to age down to the present day. But it must find an end; and in fact it was relinquished by those who kept pace with the progress of mathematics, when, in modern days, algebra began to be applied to geometry. At present, the disease is incurable in those, who attack these questions with the weapons of the ancients; for, as they are ignorant of the present state of the sciences, there are no means of curing them.
Though the ancient geometricians, of whom I have just spoken, did not attain their principal object, their researches were useful in other respects: geometry is indebted to them for several new theories, and some ingenious instruments for solving the two problems in question, so as to approximate the truth, and more than sufficient for practical purposes. Most of these methods are lost: but we have those of four illustrious geometricians, Dinostratus, Nicomedes, Pappus, and Diocles, who deserve to be mentioned with honour. The first was of the school of Plato, and contemporary with Menechmus, of whom he is even supposed to have been the brother: the other three flourished in the school of Alexandria.
Dinostratus invented a curve, which would have possessed the double advantage of giving the trisection or multiplication of an angle, and the quadrature of the circle, whence it derived the appellation of quadratrix, if it could have been described with one continued motion by means of the rule and compasses. It is formed by the intersection of the radii of a quadrant with a rule, which is made to move uniformly and parallel to one of the extreme radii of the quadrant: but it is of the number of mechanical curves, and does not rigorously fulfil either of the objects for which it was designed.
The conchoid of Nicomedes is a geometrical curve, which applies equally to both problems. A. c. 280. It is generally constructed by fixing a rule on a table, and revolving round one of it's extremities another rule furnished with two points, which are kept constantly equidistant from each other: one of these points traverses the fixed rule, and the other describes the curve. This mechanism is susceptible of several variations. The position of the polar axis, and the distance of the two movable points, are determined by the conditions of whichever of the two problems is to be solved. Newton, in an appendix to his Arithmetic, passes the highest encomium on the invention of Nicomedes; he prefers it for the geometrical construction of determinate equations of the third or fourth order to the methods derived from the intersections of the conic sections.
Pappus, in his Mathematical Collections, proposes an ingenious method for finding the two mean proportionals in the problem of doubling the cube, or mul
tiplying it in general.
A. c. 450.
Of the two extreme lines he forms the two sides of a rectangular triangle; and from the summit of the right angle, taking the longest side as the radius, he describes a semicircle, the diameter of which is consequently double that side: then from the two extremities of the diameter he draws two indefinite right lines, one of which has the same direction as the hypothenuse, the other cuts this produced, as also the shortest side of the triangle produced, and the semiperiphery: and he orders it so, that the middlemost of these three points of intersection is equidistant from the other two. The distance from this middle point to the centre will then be the greater of the two mean proportionals required.
This method, it is obvious, requires a proceeding by supposition, which is liable to some uncertainty. A. c. 460, Diocles improved it by means of the curve, called the cissoid, which bears his name. This curve is constructed by describing a semicircle on the double of the greatest extreme line as a diameter; raising on one of the extremities of the diameter an indefinite perpendicular which serves as a directrix; drawing from the other extremity an infinite number of transverse lines cutting the semiperiphery and the directrix; and taking on each transverse line a point, the distance of which from the commencement of the line is equal to the portion comprised between the directrix and the semiperiphery. This series of points forms the cissoid. The rectangled triangle of Pappus is then constructed, and the cissoid cuts the line produced from the hypothenuse in a point, through which is to pass the transverse line, that determines,
termines, on the line produced from the shortest side of the triangle, the middle point of Pappus.
I must now turn back, and resume the history of geometry from a little after the time of Plato.
In proportion as this science was extended, particular treatises appeared from time to time, in which all the known propositions were collected and arranged in systematic order. Such was the object proposed by Euclid, a geometrician of the alexandrian school, in his celebrated Elements. A. C. 300. This work is divided into fifteen books, eleyen of which belong to pure geometry: the other four treat of proportions in general, and of the principal properties of commensurate and incommensurate numbers. Though the theory of the conic sections was considerably advanced at the time when Euclid wrote, he has not mentioned them, as his object then was simply elementary geometry: but it appears by his data, and by some fragments of other works, that he was well versed in their theory.
No book of science ever met with success.comparable with that of Euclid's Elements. They have been taught exclusively for several centuries in every mathematical school, and translated and com◄ mented upon in all languages: a certain proof of their excellence.
The ancient geometricians sought the utmost strictness in their demonstrations. From a small number of axioms, or selfevident propositions, they deduced in an incontestable manner the truth of the secondary propositions, which they aimed to establish, without indulging themselves in any of those suppo
suppositions, sometimes savouring of boldness, which the moderns occasionally employ to simplify their reasoning and the consequences deduced from them. One of their grand principles was the reductio ad absurdum: they concluded that two ratios must be equal, when they had proved, that, on the supposition of their being unequal, one must be at the same time both greater and less than the other, which implies a contradiction. For instance, were it required to demonstrate, that the circumferences of two circles are as their diameters; they would have imagined, that they were offending against the strictness of geometry, if, after having proved, that the perimeters of two regular and similar polygons, inscribed in two circles, were always as the diameters, whatever might be the number of the sides of the polygons, they had finished with confounding the peripheries of the circles and the perimeters of the polygons, and consequently the two ratios, by multiplying the number of sides of the polygons to infinity. Their mode was less diffuse. They began by establishing, that if we continue to subdivide into two equal parts each of the arcs subtended by the sides of the polygons, the perimeters of the new polygons, still proportional to the diameters, would continually approach the peripheries of the circles, so as ultimately not to differ from them by any assignable quantity. They then showed, that we could not suppose the ratio of the two peripheries to be greater or less than that of the perimeters of the two last rectilinear polygons, or of the diameters, without an absurdity: