# Was Newton aware of a nascent inverse function theorem?

More specifically, was Newton aware that given an inverse pair of functions $$f$$ and $$h$$ such that

$$f(h(x)) = x = h(f(x))$$ about the origin that, for

$$(x,y)=(h(y),f(x)),$$

the derivatives satisfy

$$f^{'}(x) = 1/h^{'}(y)$$

or

$$dy/dx = 1/(dx/dy)$$

near the origin?

Heuristically, this follows symbolically from

$$dy = f^{'}(x)dx = f^{'}(x)h^{'}(y)dy.$$

And it follows geometrically for a function whose graph lies in the first quadrant by reflection through the bisector of the first quadrant, the line $$y=x$$. Clearly, the slope for any tangent line is inverted by the reflection just as displacements along the $$x-$$axis and the $$y-$$axis are interchanged. d. In fact, it follows directly from the tangent line perspective since $$y = m \; x + b$$ and $$y = \frac{1}{m}(x-b)$$ describe an inverse pair.

Surely, with Newton's mastery of geometric calculus, he was aware of these relationships. Is there evidence of this in Newton's work?

Related MO-Q by Ziegler.

Cross-posted from this MO-Q and an identical question on MSE.

Edit 6/12/17:

An example of a calculation incorporating the IFT that would have been obvious to Newton and plausible for him to have performed if only as a simple check of his general formulas:

It was known well before Newton that

$$\frac{d\tan(x)}{dx} = 1+ \tan^2(x),$$

or, with $$y = \tan(x)$$,

$$\frac{dy}{dx} = 1+ y^2.$$

In terms of fluxions and fluents, this could be put in the form of Newton's implicit function

$$g(x,y,\dot{x},\dot{y})=\dot{y}-(1+y^2)\dot{x}=0.$$

Then

$$\frac{\dot{x}}{\dot{y}}= \frac{1}{1+y^2}=\frac{dx}{dy},$$

and application of the binomial theorem and integration would give the series

$$\arctan(y) = x = y - \frac{y^3}3+\frac{y^5}5-\frac{y^7}7+\dots. \tag3$$

Newton could then have derived a series expression for $$\tan(x)$$ using his series reversion formula (see Ferraro) for finding the series for the compositional inverse of a function from its power series. In fact, the same procedure is applied to finding a series for $$\sin(x)$$ in Ferraro on pages 76-78 following an alleged reconstruction by Horsley of Newton's derivation of the series.

• The comments to the identical MSE and MO questions provide related references. – Tom Copeland Jun 12 '17 at 23:52