The preceding explanations establish clearly the propriety of the name employed to designate the science which we are considering. This denomination, which has taken to-day so definite a meaning by itself signifies simply _science_ in general. Such a designation, rigorously exact for the Greeks, who had no other real science, could be retained by the moderns only to indicate the mathematics as _the_ science, beyond all others--the science of sciences.
Indeed, every true science has for its object the determination of certain phenomena by means of others, in accordance with the relations which exist between them. Every _science_ consists in the co-ordination of facts; if the different observations were entirely isolated, there would be no science. We may even say, in general terms, that _science_ is essentially destined to dispense, so far as the different phenomena permit it, with all direct observation, by enabling us to deduce from the smallest possible number of immediate data the greatest possible number of results. Is not this the real use, whether in speculation or in action, of the _laws_ which we succeed in discovering among natural phenomena? Mathematical science, in this point of view, merely pushes to the highest possible degree the same kind of researches which are pursued, in degrees more or less inferior, by every real science in its respective sphere.
We have thus far viewed mathematical science only as a whole, without paying any regard to its divisions. We must now, in order to complete this general view, and to form a just idea of the philosophical character of the science, consider its fundamental division. The secondary divisions will be examined in the following chapters.
This principal division, which we are about to investigate, can be truly rational, and derived from the real nature of the subject, only so far as it spontaneously presents itself to us, in making the exact analysis of a complete mathematical question. We will, therefore, having determined above what is the general object of mathematical labours, now characterize with precision the principal different orders of inquiries, of which they are constantly composed.
_Their different Objects._ The complete solution of every mathematical question divides itself necessarily into two parts, of natures essentially distinct, and with relations invariably determinate. We have seen that every mathematical inquiry has for its object to determine unknown magnitudes, according to the relations between them and known magnitudes. Now for this object, it is evidently necessary, in the first place, to ascertain with precision the relations which exist between the quantities which we are considering. This first branch of inquiries constitutes that which I call the _concrete_ part of the solution. When it is finished, the question changes; it is now reduced to a pure question of numbers, consisting simply in determining unknown numbers, when we know what precise relations connect them with known numbers. This second branch of inquiries is what I call the _abstract_ part of the solution. Hence follows the fundamental division of general mathematical science into _two_ great sciences--ABSTRACT MATHEMATICS, and CONCRETE MATHEMATICS.