The Lumber Room

"Consign them to dust and damp by way of preserving them"

The same in every country

(TODO: Learn and elaborate more on their respective histories and goals.)

The formula
$\frac{\pi}{4} = 1 - \frac13 + \frac15 - \frac17 + \frac19 - \frac1{11} + \dots$
(reminded via this post), a special case at $x=1$ of
$\arctan x = x - \frac{x}3 + \frac{x}5 - \frac{x}7 + \dots,$

was found by Leibniz in 1673, while he was trying to find the area (“quadrature”) of a circle, and he had as prior work the ideas of Pascal on infinitesimal triangles, and that of Mercator on the area of the hyperbola $y(1+x) = 1$ with its infinite series for $\log(1+x)$. This was Leibniz’s first big mathematical work, before his more general ideas on calculus.

Leibniz did not know that this series had already been discovered earlier in 1671 by the short-lived mathematician James Gregory in Scotland. Gregory too had encountered Mercator’s infinite series $\log(1+x) = x - x^2/2 + x^3/3 + \dots$, and was working on different goals: he was trying to invert logarithmic and trigonometric functions.

Neither of them knew that the series had already been found two centuries earlier by Mādhava (1340–1425) in India (as known through the quotations of Nīlakaṇṭha c.1500), working in a completely different mathematical culture whose goals and practices were very different. The logarithm function doesn’t seem to have been known, let alone an infinite series for it, though a calculus of finite differences for interpolation for trigonometric functions seems to have been ahead of Europe by centuries (starting all the way back with Āryabhaṭa in c. 500 and more clearly stated by Bhāskara II in 1150). Using a different approach (based on the arc of a circle) and geometric series and sums-of-powers, Mādhava (or the mathematicians of the Kerala tradition) arrived at the same formula.

[The above is based on The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha by Ranjay Roy (1991).]

This startling universality of mathematics across different cultures is what David Mumford remarks on, in Why I am a Platonist:

As Littlewood said to Hardy, the Greek mathematicians spoke a language modern mathematicians can understand, they were not clever schoolboys but were “fellows of a different college”. They were working and thinking the same way as Hardy and Littlewood. There is nothing whatsoever that needs to be adjusted to compensate for their living in a different time and place, in a different culture, with a different language and education from us. We are all understanding the same abstract mathematical set of ideas and seeing the same relationships.

The same thought was also expressed by Mean Girls:

Written by S

Tue, 2016-03-15 at 13:53:32

Posted in history, mathematics

3 Responses

1. Testing math:

$x$

$x$

S

Thu, 2022-05-12 at 06:53:21

2. ${f(x + \epsilon) = f(x) + o_{\epsilon \rightarrow 0; f,x}(1)}$

S

Thu, 2022-05-12 at 06:58:25

3. Thanks for this formalization: many people who have only occasional encounters with asymptotic notation get bothered by it and propose using set notation everywhere instead of “abusing” the equals sign; this finally is something to point to about how to use and interpret asymptotic notation in its usual form.

Donald Knuth once mentioned an unfulfilled dream (see here) of writing a book called “O Calculus”, doing everything with O notation: for example, he wrote, a function $f$ is continuous at a point $x$ if $f(x + \epsilon) = f(x) + o(1)$. In the language of this post, I guess that the more precise version would be: ${f(x + \epsilon) = f(x) + o_{\epsilon \rightarrow 0; f,x}(1)}$ for all $x$. Anyway, looking at this post, it seems that writing such a text would be both a bit more nontrivial than expected, but also actually possible and likely instructive. Interesting to think about…

S

Thu, 2022-05-12 at 07:03:57

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