# Was a nascent inverse function theorem known to Newton?

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. 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.

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.

• This question probably belongs to the HSM site. – Alexandre Eremenko Jun 7 '17 at 7:25
• @AlexandreEremenko, then so should the linked question and the one you recently responded to mathoverflow.net/questions/270930/…, where no such comments were made. – Tom Copeland Jun 7 '17 at 7:36
• @Alex: I can see no reason why questions on history of mathematics should be moved to a site where it is perfectly legitimate to ask whether Newton had three balls. – Franz Lemmermeyer Jun 7 '17 at 8:13
• I suggest revising the question to provide evidence for positive answers to similar questions, and to say what would count as evidence for a negative answer. Key texts which would provide evidence one way or the other, e.g. the Treatise on Fluxions, are available online. – Matt F. Jun 7 '17 at 9:40
• I was trying to make a point. Last week I deleted my account there because the site is so embarrassing. But this is strictly my opinion - feel free to think of hsm whatever you want to, – Franz Lemmermeyer Dec 2 '17 at 9:41