INTRODUCTION.

of view, gives rise then to a mathematical question, which consists in substituting for the direct measurement of one of these two magnitudes, when it is impossible, the measurement of the other. It is thus, for example, that we may determine indirectly the depth of a precipice, by merely measuring the time that a heavy body would occupy in falling to its bottom, and by suitable procedures this inaccessible depth will be known with as much precision as if it was a horizontal line placed in the most favourable circumstances for easy and exact measurement. On other occasions it is the height from which a body has fallen which it will be easy to ascertain, while the time of the fall could not be observed directly; then the same phenomenon would give rise to the inverse question, namely, to determine the time from the height; as, for example, if we wished to ascertain what would be the duration of the vertical fall of a body falling from the moon to the earth.

In this example the mathematical question is very simple, at least when we do not pay attention to the variation in the intensity of gravity, or the resistance of the fluid which the body passes through in its fall. But, to extend the question, we have only to consider the same phenomenon in its greatest generality, in supposing the fall oblique, and in taking into the account all the principal circumstances. Then, instead of offering simply two variable quantities connected with each other by a relation easy to follow, the phenomenon will present a much greater number; namely, the space traversed, whether in a vertical or horizontal direction; the time employed in traversing it; the velocity of the body at each point of its course; even the intensity and the direction of its primitive impulse, which may also be viewed as variables; and finally, in certain cases (to take every thing into the account), the resistance of the medium and the intensity of gravity. All these different quantities will be connected with one another, in such a way that each in its turn may be indirectly determined by means of the others; and this will present as many distinct mathematical questions as there may be co-existing magnitudes in the phenomenon under consideration. Such a very slight change in the physical conditions of a problem may cause (as in the above example) a mathematical research, at first 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.

2. 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.