Let $\theta = \tan^{-1}(t)$. Nowadays it is taught:

1º that $$ \frac{d\theta}{dt} = \frac 1{dt\,/\,d\theta} = \frac 1{1+t^2}, \tag1 $$

2º that, via the fundamental theorem of calculus, this is equivalent to $$ \theta = \int_0^t\frac{du}{1+u^2}, \tag2 $$

3º that, developing the integrand in a geometric series and integrating term by term, this becomes the ~~Nilakantha ~~Madhava-Gregory-Leibniz formula
$$
\theta = t - \frac{t^3}3+\frac{t^5}5-\frac{t^7}7+\dots.
\tag3
$$

Question:Who first proved $(1)$ in print as we do, by deriving an inverse function?

I can't find it in Nilakantha:

According to Ranjan Roy (1990, p.300), Nilakantha first published $(3)$ without proof in his *Tantrasangraha* (1501); a later commentary known as *Yuktibhasa* contains a proof by rectification of an arc of circle, which is beautiful but certainly not quite the same as $(1)$.

I can't find it in Gregory:

According to Dehn & Hellinger (1943, p.149), Gregory communicated $(3)$ in a 1671 letter to Collins, and never published a proof; speculation exists that he found it by deriving $\tan^{-1}$ enough times to figure out its Taylor series at $0$, but in any event, he left no trace of how he may have computed these derivatives.

I can't find it in Leibniz:

According to González-Velasco (2011, p.347), Leibniz communicated $(3)$ in 1674 letters to Oldenburg and Huygens, and later published the case $t=1$ in *Acta Eruditorum* (1682, pp.41-46); his unpublished proof is available (many times over) in the 700+ pages of his Collected Works, Vol. VII,6. There, or in the nice exposition given in Hairer & Wanner (1996, 2nd printing, pp.49-50), one sees that he was squaring the circle in an elaborate way which has nothing to do with $(1)$.

Of course Leibniz must have become aware of $(1)$ and $(2)$ *at some point*, as (later!) inventor of the notation that makes them almost automatic. Unfortunately, I can't find any written evidence of that. Maybe someone else will have better luck!

(The closest I can find is a 1707 letter of Wolff to Leibniz, where the new notation is used to write in effect that $d\theta = dt:(1+t^2)$, and then deduce $(2)$ and $(3)$. The two correspondents may well have had in mind the modern proof $(1)$ of this differential relation, but neither says so.)

I can't find it in Jacob Bernoulli:

With Leibniz notation spreading, one might think that a disciple would write $(1)$ at the first opportunity. But that's not what Jacob does while quickly squaring the circle in *Positionum de Seriebus Infinitis Pars Tertia* (Basel, 1696, Prop. XLV): he ends up integrating $dt:(1+t^2)$, but for him this differential comes not from $(1)$ but from $dx:2\sqrt{2x-x^2}$ via a clever ("diophantine") substitution.

I can't find it in Johann Bernoulli:

When faced with the task of integrating $dt:(1+t^2)$ in his paper on rational integrals (1702), Johann proposes two substitutions:

The first (in Probl. I, Corol.) comes from the partial fraction decomposition $\frac1{1+t^2}=\frac{1/2}{1+it} + \frac{1/2}{1-it}$, and consists in putting $u = \frac{1+it}{1-it}$ so that $ \frac{dt}{1+t^2} = d\left[\frac1{2i}\log\frac{1+it}{1-it}\right] $.

The second (in Probl. II) consists in putting $u=\frac1{1+t^2}$ so that $\frac{dt}{1+t^2} = \frac{-du}{2\sqrt{u-u^2}}$, which differential Bernoulli recognizes (how? he doesn't say) as that "of a circular sector or arc", i.e. our $d\theta$.

Neither of these is the substitution $\theta = \tan^{-1}(t)$, which via $(1)$ would have led directly to $\frac{dt}{1+t^2} = d\theta$. And in later papers (1712, 1719) Bernoulli is content to describe this relation as "well-known".

I can't find it in Euler:

Euler was of course well aware of $(2)$, which appears for instance on line 2 of this paper written in 1739. But when it comes to *proving* $(2)$ or $(3)$, then again he eschews $(1)$:

In his precalculus book (1748, §§139-140), he chooses to first establish Bernoulli's above formula $$ \theta = \frac1{2i}\log\frac{1+it}{1-it} \tag4 $$ (this he does by multiplying numerator and denominator by $\cos\theta$ so they become $e^{\pm i\theta}$), and then to deduce $(3)$ by plugging $it$ in the series for $\log\frac{1+x}{1-x}$. None of this requires $(1)$, $(2)$, or any calculus.

In his differential calculus book (1755, §§194-197), he first differentiates a similar logarithmic formula for $\theta = \sin^{-1}(s)$, namely $\theta = -i\log(\sqrt{1-s^2} + is)$, to obtain $$ d\theta = \frac{ds}{\sqrt{1-s^2}}; \tag5 $$ plugging $s = t/\sqrt{1+t^2}$ into $(5)$ then gives him $(2)$. He might as well have differentiated $(4)$ directly! Either way, $(1)$ is not used, although to be fair, Euler at least gives (§195) an alternative proof of $(5)$ which proceeds for $s$ essentially as $(1)$ does for $t$.

So where can you find it?

It's in Lacroix (1797, pp.113-114) and its progeny. Still I have trouble believing it took over 100 years for $(1)$ to become the standard proof — hence my question.

De Analysi(written c. 1669, published 1711, Opera Omnia I, p. 264) obtains the value (3) of the integral (2) by method 3º. I see no mention of (1), which is what the present question is about. $\endgroup$ – Francois Ziegler Jun 6 '17 at 7:53