Descartes to Newton

tables could be reduced to the quadrature of hyperbolic spaces. Following up some suggestions of Wallis, William Neil succeeded in rectifying the cubical parabola, and Wren in

rectifying any cycloidal arc.

A prominent English mathematician and contemporary of Wallis was Isaac Barrow (1630-1677). He was professor of

mathematics in London, and then in Cambridge, but in 1669 he resigned his chair to his illustrious pupil, Isaac Newton, and renounced the study of mathematics for that of divinity. As a mathematician, he is most celebrated for his method of tangents. He simplified the method of Fermat by introducing

two infinitesimals instead of one, and approximated to the course of reasoning afterwards followed by Newton in his doctrine on Ultimate Ratios.

He considered the infinitesimal right triangle $AB{B}^{\prime }$ having

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for its sides the difference between two successive ordinates, the distance between them, and the portion of the curve intercepted by them. This triangle is similar to $BPT$, formed by the ordinate, the tangent, and the sub-tangent. Hence, if we know the ratio of $B^{\prime }A$ to $BA$, then we know the ratio of the ordinate and the sub-tangent, and the tangent can be constructed at once. For any curve, say $y^2=px$, the ratio of $B^{\prime }A$ to $BA$ is determined from its equation as follows: If $x$ receives an infinitesimal increment $P{P}^{\prime }=e$, then $y$ receives an increment $B^{\prime }A=a$, and the equation for the ordinate $B^{\prime }{P}^{\prime }$ becomes $y^2+2ay+a^2=px+pe$. Since $y^2=px$, we get $2ay+a^2=pe$; neglecting higher powers of the infinitesimals, we have $2ay=pe$, which gives $a:e=p:2y=p:2\sqrt{px}.$

But $a:e=\text{the ordinate}:\text{the sub-tangent}$; hence $p:2\sqrt{px}=\sqrt{px}:\text{sub-tangent},$ giving $2x$ for the value of the sub-tangent. This method differs from that of the differential calculus only in notation.31