Page:A Source Book in Mathematics.djvu/663

 equation for the line sought, from which I look for the tangent to the line according to my method of tangents published in the Acta, October, 1684, which does not preclude transcendentals. Thence, comparing what I discover with the given property of the tangents to the curve, I find, not only the assumptions, the letters a, b, c, etc., but also the special nature of the transcendental.

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Let the ordinate be $$x$$, the abscissa $$y$$, let the interval between the perpendicular and the ordinate...be $$p$$; it is manifest at once by my method that

$pdy=xdx$,

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which differential equation being turned into a summation becomes $\int p\, dy = \int x\, dx$;

But from what I have set forth in the method of tangents, it is manifest that

$d\overline{\frac{1}{2}x}x=xdx$;

therefore, conversely,

$\frac{1}{2}xx=\int x\, dx$

(for as powers and roots in common calculation, so with us sums and differences or ∫ and $$d$$, are reciprocals). Therefore we have

$\int p\, dx=\frac{1}{2}xx$.Q. E. D.

Now I prefer to employ $$dx$$ and similar [symbols], rather than letters for them, because the $$dx$$ is a certain modification of the $$x$$, and so by the aid of this it turns out that, since the work must be done through the letter $$x$$ alone, the calculus obviously proceeds with its own powers and differentials, and the transcendental relations are expressed between $$x$$ and another [quantity]. For which reason, likewise, it is permissible to express transcendental