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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, expanding 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in 1674 letters to Oldenburg and Huygens, and later published the case $t=1$ in Acta Eruditorum (1682, pp.41-46); his unpublished proof is available (many times over) in the 700+ pages of his Collected Works, Vol.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 B. does to rectify a unit circle in Positionum de Seriebus Infinitis Pars Tertia (Basel, 1696, Prop. XLV): instead, he parametrizes one with $(x,y)=$ $\bigl(x,\sqrt{2x-x^2}\bigr)$ and then expresses the resulting arc length differential — also seen in Leibniz (1686) — as $$ d\theta =\sqrt{\smash{dx^2+dy^2}\vphantom{a^2}} =\frac{dx}{\sqrt{2x-x^2}} =\frac{2d\mathsf t}{1+\mathsf t^2} \tag{$*$} $$ $(=\mathrm{LH}$ on his Fig. 3$)$ by introducing a “diophantine” (a.k.a. Weierstraß) substitution $\smash{\mathsf t=\frac xy}$ $=\smash{\tan\frac\theta2}$ $(=\mathrm{BI}$ on the figure, as he notes; so his $\mathsf t=\tan\mathrm{BAI}$ is not our $t=\tan\mathrm{BAH})$. While this still proves $(2)$ and $(3)$ for the halved angle and its tangent, the argument definitely isn’t $(1)$.

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I can’t find it in Johann Bernoulli:

When faced with the task of integrating $\smash{\frac{dt}{1+t^2}}$ in his paper on rational integrals (1702), Johann B. 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}=\frac{du}{2iu}=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 $\frac{dt}{1+t^2} = \frac{-du}{\sqrt{4(u-u^2)}}$, which he knows (perhaps by recognizing half $(*)$ with $x=2(1-u)$?) is a “circular sector or arc differential”.

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 de Moivre:

Schneider (1968, footnotes 248 & 250) seems to claim that a 1708 letter of de Moivre to Bernoulli has the proof $(1)$ and also the “Euler” formula $\theta=\smash{\frac1i}\log(\cos\theta+i\sin\theta)$. But this is not borne out by the letter’s text in (1931, pp.241-257): there, as also in his paper (1703, p.1124) and book (1730, p.44), de Moivre simply states $\smash{d\theta=\frac{dt}{1+t^2}}$ without proving it anew.

I can’t find it in Euler:

Euler was of course well aware of $(2)$, which appears for instance in his fifth paper (1729, pp.93, 95) and in his later E60, 65, 66, 125, 129, 130, 162, 217, 391, 475, 482, etc. But when it comes to proving $(2)$ or $(3)$, then again he seems to eschew $(1)$:

  • 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 $\smash{\log\frac{1+x}{1-x}}$. 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 $(5)$ using the argument $(1)$, but applied to $\smash{\sin^{-1}}$ instead of $\smash{\tan^{-1}}$.

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 proof $(1)$ should have become standard before Lacroix) relied on the misconception that tangent was easier to differentiate than arctangent. In fact the calculus of inverse trigonometric functions took off earlier, as has been explained by G. Eneström (1905), C. Boyer (1947), or V. J. Katz (1987):

it was quite common [at first] to deal with what we call the arcsine function rather than the sine.

C. Wilson (2001, 2007) concurs and stresses that our trig functions with periodic graphs weren’t much seen or differentiated until Euler “found” them to solve 2nd order linear differential equations (1741); so much so that he still wrote in (1749, p.15):

as this way of operating is not yet commonplace, it will be apropos to warn that the differentials of the formulas $\sin.𝜑$ : $\cos.𝜑$ : $\mathrm{tang}.𝜑$ : $\cot.𝜑$ are $d𝜑\,\cos.𝜑$ : $-d𝜑\,\sin.𝜑$ : $\smash[b]{\frac{d𝜑}{\cos.𝜑\,^2}}$ & $\smash[b]{-\frac{d𝜑}{\sin.𝜑\,^2}}$

— and e.g. in (1796, p.163) L’Huilier still computed $\tan'$ from $\arctan'$ rather than vice versa.

In this vein, $d𝜃 = \frac{dt}{1+t^2}$ was not proved like $(1)$, but by a differential triangle argument similar to $(*)$ but simpler and attributed to Cotes (Aestimatio errorum, 1722): in modern notation, parametrize the unit circle with $(x,y)=\frac{(1,\,t)}{\sqrt{1+t^2}}$ and obtain $$ d\theta =\sqrt{\smash{dx^2+dy^2}\vphantom{a^2}} =\frac{dt}{1+t^2} \tag6 $$ ($=\mathrm{CE}$ in Cotes’ figure, which became standard in many books even before his own — the list could almost be described as “everyone but Euler”):

1708 Charles-René Reyneau §590 fig. 41   1718 John Craig pp.52–54   1722 Roger Cotes (posthumous) Lemma II   1730 Edmund Stone p.63 fig. 13   1736 James Hodgson p.230   1736 John Muller §247 fig. 153  1737 Thomas Simpson §143   1742 Colin MacLaurin §195 fig. 52   1743 William Emerson pp.171–172 fig. 76   1748 Maria Agnesi p.639 fig. 4   1749 Charles Walmesley (credits Cotes) pp.3,53 fig. 10   1749 William Emerson p.29 fig. 6   1750 Thomas Simpson §142   1754 Louis-Antoine de Bougainville p.24 fig. 9   1761 Abraham Kästner  §299 fig. 18   1765 Jean Le Rond D’Alembert p.640 fig. 25   1767 Étienne Bézout p.146 fig. 46   1768 Thomas Le Seur & François Jacquier p.63 fig. 7   1774 Jean Saury pp.25,63 fig. 3   1779 Samuel Horsley pp.298–299   1786 Simon L’Huilier pp.103–104 fig. 20   1795 Simon L’Huilier §76 fig. 17 

As to our usual proof $(1)$, it appears before Lacroix in 18-year-old Legendre’s Theses mathematicæ (1770), then in a book by their common teacher J.-F. Marie and several others:

1770 Adrien-Marie Legendre pp.10,16   1772 Joseph-François Marie §904   1777 Jacques-Antoine Joseph Cousin p.81   1781 Claude Bertrand p.140   1781 Louis Lefèvre-Gineau p.31   1795 Simon L’Huilier §76   1797 Sylvestre-François Lacroix pp.113–114   1801 Joseph-Louis Lagrange p.81   1810 Sylvestre-François Lacroix pp.lii,203–204  

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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 calling it Madhava’s instead (e.g. also in Resonance 2 (1997), nº 11, p. 112). Regarding whether Madhava’s proof was “basically” $(1)$ I agree with Pingree’s opinion. $\endgroup$ – Francois Ziegler Dec 23 '19 at 8:05

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