setting out from the point A, and proceeding only 10 miles in an hour, should ever be able to overtake the courier setting out from the point B, and travelling 20 miles per hour, and who is in advance of the first. 66. Nevertheless, these same values resolve the question in a certain sense; for, by substituting them in the equations equations which are satisfied; since, by making the reductions, the first member becomes equal to the second; and if we attend to the signs of the terms, which compose the first, we shall see how it is necessary to modify the enunciation of the question, in order to do away the absurdity. Indeed, it is the distance a corresponding to x, and passed over by the first courier, which is in reality subtracted from the distance 2 a, corresponding to y, and passed over by the second courier; it is then just as if we had changed y into x, and x into y, and had supposed that the courier starting from the point B, had run after the other. This change in the enunciation, produces also a change in the direction of the routes of the couriers; they are no longer travelling toward the point C, but in an opposite manner toward the point C', as represented in the figure below; and their coming together takes place in R'. The result from this is positive values, which resolve the question in the precise sense in which it is enunciated. 67. The question we have been considering presents a case, in which it is in every sense absurd. This occurs when we suppose the two couriers to travel equally fast. It is evident, that in whatever direction we suppose them to move, they can never come together, since they preserve constantly the interval of their points of departure. This absurdity, which no modification in the enunciation can remove, is very conspicuous in the equations. We have now b = c, since the couriers, travelling equally fast, pass over the same space in an hour; the equation a result sufficiently absurd, since it supposes a quantity a, the magnitude of which is given, to be nothing. 68. This absurdity shews itself in a manner very singular in the values of the unknown quantities their denominator becoming 0 when b = c, we have We do not easily perceive what may be the quotient of a division when the divisor is zero; we see merely, that if we consider b as nearly equal to c, the values of x and y become very great. To be convinced of this, we need only take and it is manifest, that as the divisor diminishes in proportion to the smallness of the assumed difference of the numbers b and c, we obtain values more and more increased in magnitude. But as a quantity, however minute, can never be taken for zero, it follows, that however small we make the difference of the numbers represented by the letters b and c, and however great may be the consequent values of x and y, we never attain to those which answer to the case where b = c. Since these last cannot be represented by any number, however great we suppose it, they are said to be infinite; and every m expression of the form the denominator of which is zero, is 0' regarded as the symbol of infinity. This example shows that mathematical infinity is a negative idea, since we at length get it only by the impossibility of assigning a quantity that can resolve the question. We may ask here, how the values satisfy the equations proposed; for it is an essential characteristic of algebra, that the symbols of the values of unknown quantities, whatever they may be, being subjected to the operations indicated upon these quantities, shall satisfy the equations of the problem. By substituting them in the equations which answer to the case where b = c, we have by the first, or or lastly, 0 = 0, since a x 0 = 0. The second equation gives, under the same condition, the two members of each equation becoming equal, the equations are satisfied. It remains still to explain how the notion indicated by the a b expression , removes the absurdity of the result found in art. 0 67. For this purpose, let the two members of the equation The error here consists in the quantity, by which the second member exceeds the first; but this error becomes smaller and smaller, in proportion to the assumed magnitude of x. It is then with reason, that algebra gives for x an expression, which cannot be represented by any number, however great, but which, as it proceeds in the order of numbers becoming greater and greater, points out in what manner we may reduce more and more the error of the supposition. 69. If the couriers travelling equally fast, and in the same direction, had set out from the same point, their coming together could not be said to take place at any particular point, since they would be together through the whole extent of their route. It may be worth while to see how this circumstance is represented by the values, which the unknown quantities x and y assume in this case. The points A and B being coincident, we have on this supposition a = 0, and constantly b= c; it follows then, that In order to interpret these values, that indicate a division, in which the dividend and divisor are each nothing, I go back to the equations of the question. The first becoming xy0, gives x = y; and substituting this value in the second equation, which is The last equation having its two members identical, that is to say, composed of the same terms with the same sign, is verified, whatever value is assigned to y, and this unknown quantity can never be determined. Besides, it is evident that the equation. and consequently can express nothing more than the first.* The only result, both from the one and from the other, is, that the two couriers are always together, since the distances x and y from the point A are equal; their value in other respects remains indeterminate. The expression then, is here a symbol of an indeterminate quantity. I say here, for there are cases where it is not; but the expression has not then the same origin as the preceding. 70. To give an example, let there be This quantity becomes b (a - b) ' in its present form, when a = b; but if most simple expression, by suppress– b, common to the numerator and denominator, we reduce it first to its *For the sake of conciseness, analysts apply to the same equations the epithet, identical. y У = is an identical equation, 5 3x5- 3x is another, b b - and when two equations express only the same thing, we say that these equations also are identical. |