All geometrical ideas necessarily relate to the three universal categories; magnitude, form, position. Magnitude already belongs to the domain of quantity. Form can be reduced to position, since every form can be considered as the result of the advance of a point, that is to say of its successive positions. The problem is therefore to bring all ideas of situation whatever back to ideas of magnitude. How did Descartes solve it? By generalising a process which we may say is natural to the human mind, since it comes spontaneously into being under the stress of necessity. Indeed, if we must indicate the situation of an object without showing it immediately, do we not refer it to others which are known, by stating the magnitude of geometrical elements by which we conceive the object to be connected with them? Geographers act in the same way in their science to determine the longitude and latitude of a place, and astronomers to determine the right ascension and the declination of a star. These geographical and astronomical co-ordinates fulfil the same office as the Cartesian co-ordinates. The only difference, but it is a capital one, consists in the fact that Descartes carried this method to the highest degree of abstract generality thus giving it its maximum of fertility and power. Although general geometry is infinitely superior to special geometry it cannot, nevertheless, altogether dispense with the latter. As the ancients did, so it will always be necessary to begin with special geometry. For general geometry rests upon the use of calculation. But if, as Comte has said, geometry is truly a science of facts calculation will evidently never be able to supply us with the first knowledge of these facts. In order to lay the foundations of a natural science simple mathematical analysis would never suffice, nor could it give a fresh demonstration of it, when these foundations have already been laid. Before all things a direct study of the subject is necessary, until the precise relations are discovered. “The application of mathematical analysis can never begin any science whatever, since it could never take place except when the science has been sufficiently elaborated to establish, in relation to the phenomena under consideration, some equations which might serve as a starting-point for analytical work.”[112] In a word, the creation of analytical geometry does not prevent geometry from remaining a natural science. Even when it has become as purely rational as possible, it none the less remains rooted in experience. IV.