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it as its object the indirect measurement of magnitudes, and by saying it constantly proposes to determine certain magnitudes from others by means of the precise relations existing between them.
This enunciation, instead of giving the idea of only an art, as do all the ordinary definitions, characterizes immediately a true science, and shows it at once to be composed of an immense chain of intellectual operations, which may evidently become very complicated, because of the series of intermediate links which it will be necessary to establish between the unknown quantities and those which admit of a direct measurement; of the number of variables coexistent in the proposed question; and of the nature of the relations between all these different magnitudes furnished by the phenomena under consideration. According to such a definition, the spirit of mathematics consists in always regarding all the quantities which any phenomenon can present, as connected and interwoven with one another, with the view of deducing them from one another. Now there is evidently no phenomenon which cannot give rise to considerations of this kind; whence results the naturally indefinite extent and even the rigorous logical universality of mathematical science. We shall seek farther on to circumscribe as exactly as possible its real extension.
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.
ITS TWO FUNDAMENTAL DIVISIONS.
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.