The Project Gutenberg eBook #36640: Lectures on Elementary Mathematics

LECTURE V.

ON THE EMPLOYMENT OF CURVES IN THE SOLUTION OF PROBLEMS.

As long as algebra and geometry travelled separate paths their advance was slow and their applications limited. But when these two sciences joined company, they drew from each other

fresh vitality and thenceforward marched on at a rapid pace towards perfection. It is to Descartes that we owe the applica- tion of algebra to geometry,—an application which has furnished the key to the greatest discoveries in all branches of mathemat- ics. The method which I last expounded to you for finding and demonstrating divers general properties of equations by consid- ering the curves which represent them, is, properly speaking, a species of application of geometry to algebra, and since this method has extended applications, and is capable of readily solv- ing problems whose direct solution would be extremely difficult or even impossible, I deem it proper to engage your attention in this lecture with a further view of this subject,—especially since it is not ordinarily found in elementary works on algebra.

You have seen how an equation of any degree whatsoever can be resolved by means of a curve, of which the abscissæ represent the unknown quantity of the equation, and the ordinates the values which the left-hand member assumes for every value of the unknown quantity. It is clear that this method can be applied generally to all equations, whatever their form, and that it only requires them to be developed and arranged according to the different powers of the unknown quantity. It is simply necessary to bring all the terms of the equation to one side, so that the
other side shall be equal to zero. Then taking the unknown quantity for the abscissa x, and the function of the unknown quantity, or the quantity compounded of that quantity and the known quantities, which forms one side of the equation, for the ordinate y, the curve described by these co-ordinates x and y will give by its intersections with the axis those values of x which are the required roots of the equation. And since most frequently it is not necessary to know all possible values of the unknown quantity but only such as solve the problem in hand, it will be sufficient to describe that portion of the curve which corresponds to these roots, thus saving much unnecessary calculation. We can even determine in this manner, from the shape of the curve itself, whether the problem has possible solutions satisfying the proposed conditions.

Suppose, for instance, that it is required to find on the line joining two luminous points of given intensity, the point which receives a given quantity of light,—the law of physics being that the intensity of light decreases with the square of the distance. Let a be the distance between the two lights and x the dis- tance between the point sought and one of the lights, the inten- sity’ of which at unit distance is M , the intensity of the other

at that distance being N . The expressions ^M

_x2

and ^N

(a − x)²

, ac-

cordingly, give the intensity of the two lights at the point in

question, so that, designating the total given effect by A, we have the equation