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solved without employing any instruments beside the rule and compasses: for, by the ancients, no operations were considered as geometrical, unless performed by means of these two instruments alone; those that required others being called mechanical.
According to an old tradition generally spread through Greece, a public calamity, in which religion was concerned, gave rise to this research. It was said, that, Apollo having afflicted the athenians with a dreadful pestilence, to revenge an affront he had received from them, the oracle of the temple of Delos, being consulted on the means of appeasing his wrath, answered: Double the altar. The altar of Apollo at Athens, to which the oracle alluded, was a perfect cube; and the problem was immediately proposed to all the geometricians of Greece. The priests, who never forget their own interests, added a condition, which they represented as a religious duty, but which happily did not increase the diffi-. culty of the problem: they required the material of the new altar to be gold. The question at first sight appeared easy; but this mistake was soon corrected, and all the sagacity of the geometricians of Greece. was baffled by it.
On turning the problem every way, it was perceived, and the discovery is attributed to Hippocrates of Chios, that if two geometrical mean proportional... lines could be inserted between the side of the cube given and the double of this side, the first of these two lines would be the side of the cube sought. This: new point of view revived for a moment the hope of accomplishing the solution by means of the rule and
compasses but the difficulty was only disguised; it had merely changed it's form: thus it was still insurmountable, and the geometricians, tired with the labour they had already exerted on the problem, let it sleep for a time.
Still, however, geometry advanced. A. c. 390, Plato cultivated it with care, and acquired profound knowledge in the science. It is true we have no work of his written expressly on the subject: but we see by various passages in his other works, that he was master of it; and the ancient historians have transmitted to us the results of several discoveries, with which he enriched it. He placed it in the first rank of mental acquirements, and made it the principal object of the instructions he gave his scholars. He had written over the door of his school: 'Let no one enter here, who is ignorant of geometry.' The problem of doubling the cube could not fail of engaging his attention. Having attempted in vain to solve it with the rule and compasses, he invented, for the purpose of finding the two mean proportionals, an instrument composed of two rules, one of which moved in the grooves of two arms at right angles with the other, so as always to continue parallel with it. But this solution was of the mechanical kind, and did not satisfy the wish of geometricians.
He was more fortunate in another speculation of a kind entirely new. Before his time the circle was the sole curve considered in geometry: he introduced into it the theory of the conic sections, or those celebrated curves, which are formed on the surface of a
cone, when cut by a plane in different directions. On attentively examining the generation of these curves, he discovered several properties of them, These first notions, being spread through his school, germinated there rapidly. His principal scholars or friends, Aristeus, Eudoxus, Menechmus, Dinostratus, &c. penetrated deeply into this branch of geome try. In a short time it was so extended as to form a distinct part of the science, of a more exalted order than the common geometry, whence it derived the name of the higher or sublime geometry. Some other ancient curves, which I shall have occasion to notice, were afterward comprised under the same des nomination.
Aristeus composed five books on conic sections, of which the ancients have spoken with the greatest eulogies, but unfortunately they have not reached us. A. C. 380. Of Menechmus we have two learned applications of the same theory to the problem of doubling the cube. The properties of the conic sections, and those of geometrical progressions, led him to remark, that, on constructing, conformably to the conditions of the problem, two conic sections, which should intersect each other, the two ordinates corresponding to the point of intersection might represent the two mean proportionals. Hence he framed two solutions: in the first he constructed two parabolas, having one common summit, with their axes perpendicular to each other, and for their respective parameters the side of the given cube and the double of that side: then the two ordinates, drawn to the point of intersection of the two curves, are the two
mean proportionals sought. The second solution proceeds by the intersection of a parabola and an equilateral hyperbola: the parabola has for it's parameter the side of the given cube, or the double of this side; it's summit is the centre, and it's axis one of the assymptotes of the equilateral hyperbola; and the power of the hyperbola is the product of the side of the given cube by the double of that side. Lastly the ordinates of the two curves, drawn to the point of intersection, are the two mean proportionals required. The reader who is tolerably versed in geometry will make out the demonstrations of these theorems without difficulty.
Thus it appears, that, if we possessed the means of describing conic sections with one continued motion, and in as simple a manner as we trace a circle. with the compasses, the solutions of Menechmus would have all the advantages of geometrical constructions, in the sense which the ancients applied to the term. But there exists no instrument for describing the conic sections in this manner. These solutions, therefore, do not fulfil the desired purpose in practice, though they are perfect in theory, and must be considered as an effort of inventive genius. It was afterward found, that the same end might be attained by the intersection of a circle and a parabola; an easy simplification of the problem, which detracts nothing from the honour due to Menechmus.
This discovery is so much the more remarkable, as it has been the source of the celebrated theory of loci geometrici, of which ancient and modern geometricians have made so many important applica
tions. Let us add, that the method of Menechmus includes likewise the germe of geometrical analysis, or of that art, by which, comsidering a problem as solved, and treating the unknown quantities as known, we proceed from reasoning to reasoning,: from consequence to consequence, till we obtain an expreffion, which we may call the geometrical translation of all the conditions of the problem. This art is not algebra; but algebra lends it important assistance and in this respect the moderns have a great, advantage over the ancients, though these became versed in geometrical analysis after the solutions given by Menechmus.
The problem of the trisection of an angle, which is of the same nature with that of doubling the cube, was equally agitated in the school of Plato. Without attaining it's solution by means of the rule and compasses, it was reduced at least to a very simple and very curious proposition. This consists in drawing a right line from a given point to the semiperiphery of a circle, which line shall cut this periphery, and the prolongation of the diameter that forms it's base, so that the part of the line comprised between the two points of intersection shall be equal to the radius: a result which gives rise to several easy con-structions. The intersection of the conic sections is applied to this problem likewise, as it was by Menech mus to the duplication of the cube.
According to the modern methods, each of these two problems leads to an equation of the third order, with this difference, that the equation relative to the duplication