7.7K
66 THE NATURE OF MATHEMATICS subject in the third chapter: here we shall illustrate the considerations of Fermat (1601-1665) and Barrow (1630- 1677) the intellectual descendants of Kepler by a simple example. Let it be proposed to draw a tangent at a given point P in the circumference of a circle of centre and equation x 2 + y z = 1. Let us take the circle to be a polygon of a great number of sides ; let PQ be one of these sides, and produce it to meet the -axis at T. Then PT will be the tangent in question. Let the co-ordinates of P be X and Y ; those of Q will be X + e and Y -f a, where e and a are infinitely small increments, positive or negative. From a figure in which the ordinates and abscissae of P and Q are drawn, so that the ordinate of P is PR, we can see, by a well-known property of triangles, that TR is to RP (or F) as e is to a. Now, X and Y are related by the equation X 2 + F 2 = l, and, since Q is also on the locus aj2 + 2/ 2 = l, we have (X + e) 2 +(F + a) 2 = 1. From the two equations in which X and Y occur, we conclude that 0, and hence o e TR - But - = : hence TR = - * '. Now, a and e may a Y X + \ be neglected in comparison with X and F, and thus we can F 2 say that, at any rate very nearly, we have TR = - But -A. this is exactly right, for, since TP is at right angles to OP, we know that OR is to RP as PR is to RT. Here X and F are constant, but we can say that the abscissa of the point where the tangent at any point (say y) of the circle cuts the a;-axis v 2 is given by adding - - to x. x Thus, we can find tangents by considering the ratios of infinitesimals to one another. The method obviously applies to other curves besides circles ; and Barrow's method and nomenclature leads us straight to the notation and nomenclature of Leibniz. Barrow called the triangle PQS, where S is where a parallel to the #-axis through Q meets PR, the " differential triangle," and Leibniz denoted Barrow's a and e by dy and dx (short for "the differential ofy" and "the
Was this article helpful?
Yes0No0