ANCIENT OR SYNTHETIC GEOMETRY.
The geometrical method of the ancients necessarily constituting a preliminary department in the dogmatical system of geometry, designed to furnish general geometry with indispensable foundations, it is now proper to begin with determining wherein strictly consists this preliminary function of special geometry, thus reduced to the narrowest possible limits.
ITS PROPER EXTENT.
Lines; Polygons; Polyhedrons. In considering it under this point of view, it is easy to recognize that we might restrict it to the study of the right line alone for what concerns the geometry of lines; to the quadrature of rectilinear plane areas; and, lastly, to the cubature of bodies terminated by plane faces. The elementary propositions relating to these three fundamental questions form, in fact, the necessary starting point of all geometrical inquiries; they alone cannot be obtained except by a direct study of the subject; while, on the contrary, the complete theory of all other figures, even that of the circle, and of the surfaces and volumes which are connected with it, may at the present day be completely comprehended in the domain of general or analytical geometry; these primitive elements at once furnishing equations which are sufficient to allow of the application of the calculus to geometrical questions, which would not have been possible without this previous condition.
It results from this consideration that, in common practice, we give to elementary geometry more extent than would be rigorously necessary to it; since, besides the right line, polygons, and polyhedrons, we also include in it the circle and the "round" bodies; the study of which might, however, be as purely analytical as that, for example, of the conic sections. An unreflecting veneration for antiquity contributes to maintain this defect in method; but the best reason which can be given for it is the serious inconvenience for ordinary instruction which there would be in postponing, to so distant an epoch of mathematical education, the solution of several essential questions, which are susceptible of a direct and continual application to a great number of important uses. In fact, to proceed in the most rational manner, we should employ the integral calculus in obtaining the interesting results relating to the length or the area of the circle, or to the quadrature of the sphere, &c., which have been determined by the ancients from extremely simple considerations. This inconvenience would be of little importance with regard to the persons destined to study the whole of mathematical science, and the advantage of proceeding in a perfectly logical order would have a much greater comparative value. But the contrary case being the more frequent, theories so essential have necessarily been retained in elementary geometry. Perhaps the conic sections, the cycloid, &c., might be advantageously added in such cases.
Not to be farther restricted. While this preliminary portion of geometry, which cannot be founded on the application of the calculus, is reduced by its nature to a very limited series of fundamental researches, relating to the right line, polygonal areas, and polyhedrons, it is certain, on the other hand, that we cannot restrict it any more; although, by a veritable abuse of the spirit of analysis, it has been recently attempted to present the establishment of the principal theorems of elementary geometry under an algebraical point of view. Thus