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 bc, 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 ab x=b=c' their denominator becoming 0 when bc we have a b x= y=20 ac We do not easily perceive what may be the quotient of a division when the divisor is zero; we see merely that if we con sider 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 expression of the form the denominator of which is zero, is re m 0' garded 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 xy=a, x y -= which answer to the case where bc, we have by the first, or or lastly 0=0, since a x0 = 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 ab expression removes the absurdity of the result found in art. Ο 67. For this purpose, let the two members of the equation The error here consists in the quantity, by which the sec x ond 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 x-y=0 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 a (a3-b3) This quantity becomes in its present form when a=b; but if we reduce it first to its most simple expression, by suppressing the factor a - b, common to the numerator and denominator, we find * For the sake of conciseness, analysts apply to the same equations the epithet, identical. y y %/= is an identical equation, 5 — Sx = 5— S ≈ is another, and b when two equations express only the same thing, we say that these equations also are identical. a (a + b), which gives 2 a when a = b. It is not the same with the values of x and y, found in the preceding article, for they are not susceptible of being reduced to a more simple expression. It follows, from what I have just said, that when we meet with an expression which becomes, it is proper before pronouncing upon its value, to see if the numerator and denominator have not a common factor, which becoming nothing, renders the two terms at the same time equal to zero, and which being suppressed, the true value of the proposed expression is obtained. There are, notwithstanding, some cases which elude this method, but the limits of this work will only allow me to note the analyt ical fact. It belongs properly to the differential calculus to give the general processes for finding the true value of quantities, which become g. 71. It is very evident, from what has been said, that algebraic solutions either answer perfectly to the conditions of a problem, when it is possible, or they indicate a modification to be made in the enunciation, when the things given imply contradictions that cannot be reconciled; or lastly, they make known an absolute impossibility, when there is no method of resolving with the same things given, a question analogous in a particular sense to the one proposed. 72. It may be remarked, that in the different solutions of the preceding question, the changing of the signs of the unknown quanties x and y corresponds to a change in the direction of the journeys represented by the unknown quantities. When the unknown quantity y was counted from B towards A, it had in the equation x+y=a, the sign+, and it takes the sign for the second case, when the motion is in the opposite direction from B towards C, art. 65, since we had for the first equation |