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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in 1674 letters to Oldenburg and Huygens, and later published the case $t=1$ in Acta Eruditorum (1682, pp.41-46); his unpublished proof is available (many times over) in the 700+ pages of his Collected Works, Vol.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.

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    $\begingroup$ suddenly I thought that the first place where it appeared could be… exactly this question $\endgroup$ – Pietro Majer Mar 1 '15 at 2:00
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    $\begingroup$ @PietroMajer Glad to have created some suspense! But nah, it's in just about any textbook $\endgroup$ – Francois Ziegler Mar 1 '15 at 2:15
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    $\begingroup$ Newton devised a formula for series reversion (comp inversion), same as oeis.org/A133437, and he tested it on the asin (see around pg. 75 of the Ferraro ref in the OEIS entry). You would think he would have done the same for tan and atan. Not exactly the same as the symbolic manipulation, but he had geometric arguments related to fluxions. $\endgroup$ – Tom Copeland Jun 6 '17 at 5:15
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    $\begingroup$ In fact, Newton seems to have dealt with basically the first three equations, according to Ferraro, as an example of more general propositions. $\endgroup$ – Tom Copeland Jun 6 '17 at 6:50
  • $\begingroup$ @TomCopeland Ferraro (pp. 69-70) clearly says that Newton in 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
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I now believe that my question (and suggestion that the proof $(1)$ should have become standard before Lacroix) relied on the misconception that tangent is easier to differentiate than arctangent. In fact, as V. J. Katz (1987) explains, the calculus of inverse trig functions took off earlier and “it was quite common [at first] to deal with what we call the arcsine function rather than the sine”.

In this vein, $d\theta:dt= 1:(1+t^2)$ was not proved like $(1)$, nor like Johann Bernoulli, but by a geometrical argument — similarity of differential triangles — apparently due to Roger Cotes in his posthumous Aestimatio errorum in mixta mathesi (1722, Lemma II). This, then, became standard in many textbooks: Reyneau (1708, §590, fig.41), Craig (1718, p.54), Muller (1736, p.122, fig.153), Hodgson (1736, p.230), Maclaurin (1742, §195, fig.52), Agnesi (1748, p.639, fig.4), Walmesley (1749, p.53, fig.10), Emerson (1749, p.29, fig.6; 1757, p.239), Simpson (1750, p.165), Bougainville (1754, p.24), Kästner (1761, §299), Bézout (1767, p.146, fig.46), Le Seur–Jacquier (1768, p.63, fig.7), Horsley (1779, pp.298-299).

Edit (to address the question): Lacroix may have learned the proof $(1)$ from his teacher Joseph-François Marie, whose revision of La Caille (1772, p.389) has perhaps the earliest occurrence. Before Lacroix the argument also appears in Cousin (1777, p.81), Marie (1793, p.56), L’Huilier (1795, p.112; 1796, p.163), and in lectures of Lagrange (see 1801, p.81) which Lacroix credits in his second edition (1810, pp.lii, 203).

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The Madhava–Gregory series, by R. C. Gupta, attributes (3) to Indian mathematician-astronomer Madhava of Sangamagrama (circa 1350–1425). He also writes that a geometric derivation which is basically equivalent to (1) can be found in the book Yuktibhāṣā written by Indian astronomer Jyesthadeva (circa 1500-1601) of the Kerala school of mathematics in about 1530.

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  • $\begingroup$ Thanks Zurab. I had followed Ranjan Roy (1990) in calling $(3)$ the Nilakantha(-Gregory-Leibniz) series, but indeed I see arguments for attributing it to Madhava instead (e.g. also in Resonance 2 (1997), nº 11, p. 112). $\endgroup$ – Francois Ziegler Feb 15 '15 at 19:47

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