tain spectra, or simulacra, thrown off from the surfaces of bodies, and received into the eye.

Aristotle speculated on the subject in a metaphysical point of view, and called in question this doctrine of Empedocles; contending that light is not a material substance. He supported this by arguments drawn from its velocity, which he supposed infinite. He seems to have regarded it as something like an impulse propagated through a medium. But his whole doctrine was so mixed up with verbal mysticism, that it is difficult to determine whether this idea was really a sort of conjectural anticipation of the theory of undulations.

Mathematical Sciences.

In the review already made of the earlier astronomical and physical discoveries of the ancients, we have found scarcely any instances of sound investigation, without an accompanying admixture of extravagant and gratuitous hypothesis. In turning, however, to their inventions in pure mathematics, we find a pleasing contrast. In this department we recognise the sure, though very gradual, developement of the most important elementary truths, established with a perfection of logical accuracy in the reasoning which is, to this day, the theme of general approbation and the model of universal imitation. And, unless, indeed, the obscure mystical properties ascribed to numbers by Pythagoras be deemed an exception, we do not perceive these investigations ever degraded by a mixture with frivolous conceits or visionary speculations.

Those elementary truths, which necessarily form the foundation of the whole science of quantity in all its species and dimensions, we have already seen, have been claimed as the inventions of several early nations. It is highly probable that the discovery of them took place among different people without any communication. They have been handed down as the speculations of the earliest age; and such primary and simple relations

are among those first subjects which would naturally exercise the skill of a contemplative mind devoting itself to the consideration of the different combinations to which geometrical figures and magnitudes can be subjected. It was, doubtless, long before the scattered truths were collected and arranged in any systematic form.

Pythagoras, no doubt, directed his attention to these subjects; and the story related of his sacrificing a hecatomb for joy on the discovery of the fundamental theorem of geometry, would hardly have been invented if he had not at least enjoyed the reputation of originality in the discovery.

The early Greek philosophers employed themselves in speculating upon all the relations they could discover in the simple geometrical figures, and in those constructions which involved only the use of the straight line and the circle by degrees they extended their enquiries to the properties of planes and solids, especially the regular solids contained by plane sides, and those generated by the revolution of a circle, a triangle, or a rectangle, the sphere, the cone, and cylinder.

It was in the Platonic school, and, as some contend, by Plato himself, that some of the most valuable accessions to geometry were made. The science had as yet taken cognisance of no other curves than the circle. Plato perceived, that if a cone be cut by a plane in certain positions, the intersection of the surface of the cone with the plane will neither be a portion of a circle, nor rectilinear, but will take the form of certain peculiar curves. It was soon found that only three distinct species of these curves could be formed; they were termed accordingly the conic sections, and named (according to another analogy arising out of one of their properties) the parabola, the ellipse, and hyperbola. The` manner in which the formation of the curves was first conceived, affords a striking instance of the slow progress of discovery among a class of truths as yet new to the apprehension. A plane was conceived touching a cone along one of its sides: another plane, perpendicular to

this, of course cut the cone, and gave rise to the curve. If the cone had a right angle at its summit, the curve was a parabola: if the vertical angle was less than a right angle, an ellipse: if greater, an hyperbola. A different species of cone was thus supposed necessary to give each different curve. A century elapsed before it was seen that they might all be obtained from one and the same cone of any species, by merely altering the inclination of the cutting plane. Of the high importance of these curves in the researches of modern physical science, this is not the place to speak. We must here confine ourselves to noticing that the ancients happily attached due value to them, and continued with unremitting diligence to investigate their various properties. In these researches, none was more eminent than Menechme, the friend and disciple of Plato.

But the Platonic school was scarcely less distinguished for originating other important branches of mathematical speculation. Among these, the most remarkable, perhaps, was the geometrical analysis. This invention is expressly ascribed to Plato himself, by Proclus. Any geometrical question, whether problem or theorem, being submitted to analysis, is assumed as solved, or as From this assumption, a chain of consequences is drawn, which, by the ingenuity of the geometer, is continued, until he arrives at some proposition known to be true or false, possible or impossible. The final consequence points out whether the question be true or possible; and by retracing the steps, a synthetic proof or solution may be found.

true.

Another class of speculations commenced and pursued in this school, was the geometric loci. These, in their simplest form, arose out of problems where it was required to find a point determined, for instance, by the intersection of two lines under certain given conditions, and where it was found that there were an infinite number of points fulfilling the requisition, but each restricted to a certain position: so that if they were all assigned, they would all lie in a certain line or locus,

straight or curved. For example, the locus of the vertices of triangles of equal area on the same base, is a straight line parallel to the base; and the locus of the vertices of right-angled triangles on the same hypothenuse, is a semicircle. These loci were chiefly employed as affording the means of solving other problems of the determinate kind.

Of these, one which largely occupied the attention of the ancient geometers, and received its solution in the Platonic school, was a response of the oracle of Delos, requiring, that of the altar in that temple, which was an exact cube, the exact double in solid content should be made also in the form of a cube. This was done at first mechanically; but Menechme applied to the solution the resources of the method of loci, and produced a geometrical construction. The trisection of a circular arc was another problem of some celebrity, which, in like manner, was made to yield to the growing powers of geometry, though it had resisted all attempts by means of the elementary methods.

To the principles developed in these several discoveries, we shall, at a future period, have occasion to We will merely remark, in this place, that they unquestionably contain the germs, as it were, of the most valuable inventions of modern times.

recur.

Upon the whole, the geometry of the ancient schools is that portion of their speculations to which we look back with by far the greatest interest and satisfaction ; and to which we are really indebted for the removal of the difficulties which we should otherwise have to contend with, in limine, in all our enquiries. What were to them matters of high and original discovery, now form the necessary elements of all well-conducted education: and even though modern science had given us more direct and easy methods of arriving at those results which we want for their actual applications, yet, as a subject of abstract study, for the refined elegance of their train of deduction, for the exact taste of their style, and the fastidious precision of their reasoning,

the ancient geometers will retain their pie-eminence as models to succeeding ages.

The invention of mechanical modes of construction, by which certain curves could be traced out and some problems solved, and which began to prevail with Eudoxus, Archytas, and their followers, was much censured by Plato, who considered such methods derogatory to the abstract philosophic dignity of geometry, and destructive to its purely intellectual character. This sentiment, to a certain degree perfectly just, was extensively adopted by the philosophers of the Platonic school, and produced the effect of alienating mechanical invention from mathematical speculation: a result highly injurious to the former science, and repressive of much of the spirit of invention and improvement in the latter. Indeed, it was, probably, to the wide distinction maintained in the ideas of many of the ancient philosophers, between the respective characters of geometrical and of physical investigation, that we may attribute much of the neglect of the latter, and the slowness of its progress, from the want not only of the powerful aid it might have derived from the former, but even of recognition as a legitimate branch of philosophy. Nothing, indeed, can be more striking, than the contrast afforded by the manner in which these two classes of enquiry were respectively carried on: and we can hardly help feeling astonished, that the same philosophical genius which was so rigorously precise in its demands for the perfect demonstration of the most primary notions on which mathematical truth was to rest, should have been ready to satisfy itself with the most flimsy conjectures or unsubstantial analogies, in matters of physical speculation.