HKB
very elementary, to be placed at once in the rank of the most difficult questions, whose complete and rigorous solution surpasses as yet the utmost power of the human intellect.
Inaccessible Distances. Let us take a second example from geometrical phenomena. Let it be proposed to determine a distance which is not susceptible of direct measurement; it will be generally conceived as making part of a figure, or certain system of lines, chosen in such a way that all its other parts may be observed directly; thus, in the case which is most simple, and to which all the others may be finally reduced, the proposed distance will be considered as belonging to a triangle, in which we can determine directly either another side and two angles, or two sides and one angle. Thence-forward, the knowledge of the desired distance, instead of being obtained directly, will be the result of a mathematical calculation, which will consist in deducing it from the observed elements by means of the relation which connects it with them. This calculation will become successively more and more complicated, if the parts which we have supposed to be known cannot themselves be determined (as is most frequently the case) except in an indirect manner, by the aid of new auxiliary systems, the number of which, in great operations of this kind, finally becomes very considerable. The distance being once determined, the knowledge of it will frequently be sufficient for obtaining new quantities, which will become the subject of new mathematical questions. Thus, when we know at what distance any object is situated, the simple observation of its apparent diameter will evidently permit us to determine indirectly its real dimensions, however inaccessible it may be, and, by a series of analogous investigations, its surface, its volume, even its weight, and a number of other properties, a knowledge of which seemed forbidden to us.
Astronomical Facts. It is by such calculations that man has been able to ascertain, not only the distances from the planets to the earth, and, consequently, from each other, but their actual magnitude, their true figure, even to the inequalities of their surface; and, what seemed still more completely hidden from us, their respective masses, their mean densities, the principal circumstances of the fall of heavy bodies on the surface of each of them, &c.
By the power of mathematical theories, all these different results, and many others relative to the different classes of mathematical phenomena, have required no other direct measurements than those of a very small number of straight lines, suitably chosen, and of a greater number of angles. We may even say, with perfect truth, so as to indicate in a word the general range of the science, that if we did not fear to multiply calculations unnecessarily, and if we had not, in consequence, to reserve them for the determination of the quantities which could not be measured directly, the determination of all the magnitudes susceptible of precise estimation, which the various orders of phenomena can offer us, could be finally reduced to the direct measurement of a single straight line and of a suitable number of angles.
TRUE DEFINITION OF MATHEMATICS.
We are now able to define mathematical science with precision, by assigning to 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.