# Remarkable numerical calculations before electronic computers

I know the story that Cole found the factoring of the big number $$2^{67}-1$$.

Is there any other remarkable achievement of hand calculation?

• By a single person or by a group of human computers? – Geremia Mar 23 at 0:47
• Both are okay of course. – wdlang Mar 23 at 7:18
• Between hand calculation and electronic computers you may insert mechanical calculators. – Gerald Edgar Mar 24 at 0:31
• All calculations were done by hand before 20th century, and even some early ones for the space program. There are just too many examples for this question to have a reasonable size answer, please narrow it down. Early calculations of $\pi$, e.g. by Huygens, were painstaking, so was Kepler's fitting of Mars's orbit to various epicyclic theories, or Napier's calculations of logarithmic tables. Euler and Gauss are famous for elaborate number theoretic calculations, Adams and Leverrier for calculating the orbit of Neptune, etc. – Conifold Mar 25 at 6:20

Another example of a "remarkable achievement of hand calculation" was in the field of mathematical astronomy.

During 1758, Alexis Clairaut and his collaborators in Paris worked to refine Edmond Halley's prediction (published in 1705) of a return in about 1758 of the comet that now carries Halley's name. Halley's original prediction had been for 1758, based on his assessment that the comet of 1682 was periodic, with a mean orbital period of about 75.5 years. (Halley later somewhat modified the prediction by an informed guess that a close encounter with Jupiter could delay the event by a few months, perhaps to early 1759.)

Clairaut's calculations refined the prediction, forecasting a return with perihelion passage in mid-April 1759 -- give or take a month. Observations of the comet then showed that indeed it was a return of a comet with closely similar orbital elements as the 1682 comet, verifying the return prediction, and showing that it passed perihelion on 13 March, just within the margin of error that Clairaut had allowed.

The calculation was remarkable both for what was done and for the public reaction to which it contributed. Clairaut, assisted by Jerome Lalande and Mme Lepaute, had calculated by hand a numerical integration of the perturbing effects of Jupiter and Saturn. Lalande wrote that for six months Clairaut and his collaborators in Paris calculated from morning to night, sometimes even continuing at the table at mealtimes. They verified that the perturbing effects acted to delay the return of the comet by some months relative to the original date-estimate of 1758.

Much of the public attention was on the event itself, but the calculations also drew attention and increased Clairaut's fame (not without controversy, spurred partly by rivalries as well as by scientifically-based objections).

References:

-- For the nature of the (hand) calculations : "Clairaut's Calculation of the Eighteenth-century Return of Halley's Comet" (C A Wilson, Journal for the History of Astronomy, v.24 (1993), pp.1-15).

-- For the public reception of the events and calculations in France : "Clairaut et le retour de la « comète de Halley » en 1759" (R Taton, L'Astronomie 100 (1986), 379-408).

-- For a report of other contemporary views of the events : "Comet Halley's First Expected Return" (C B Waff, Journal for the History of Astronomy, v.17 (1986), pp.1-37).

-- For a scientific history of Halley's comet : "The History of Halley's Comet " (D W Hughes et al., Philosophical Transactions of the Royal Society, Series A, 323 (1987), 349-367).

I've always liked logarithms because of their properties, and for some time I wondered who got the idea in the first place and how were the tables computed. It turns out logarithms were developed and computed simultaneously and independently by John Napier and Joost Bürgi. Both of them calculated huge logarithm tables by hand:

Napier computed almost ten million entries from which he selected the appropriate values. Napier himself reckoned that computing this many entries had taken him twenty years, which would put the beginning of his endeavors as far back as 1594.

https://www.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-john-napier-introduces-logarithms

Bürgi computed logarithms for 100000000 to 1000000000. This filled fifty-eight pages of tables for a total of 23,030 entries (23,027, plus an additional 3 entries) to be computed to eight significant digits.

https://www.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-joost-b-rgi-introduces-logarithms

What about Archimedes computing $$\pi$$ with such a precision that he was able to prove that$$3+\frac{10}{71}<\pi<3+\frac17?$$Or the ancient Babylonians being able to compute$$\sqrt2\simeq1+\frac{24}{60}+\frac{51}{60^2}+\frac{10}{60^3}?$$

One famous hand calculation was that by William Shanks, who attempted to calculate the first 707 digits of pi by hand, using Machin's formula and the Maclaurin series for the arctangent function. He did this all by hand. Unfortunately, only the first 527 places were correct. This error was discovered over 70 years later by someone using a desk calculator. The error diminishes the accomplishment, but it is still a "remarkable achievement of hand calculation." It certainly has been remarked on often.

The linked Wikipedia article also states that "Shanks also calculated e and the Euler–Mascheroni constant γ to many decimal places. He published a table of primes up to 60 000 and found the natural logarithms of 2, 3, 5 and 10 to 137 places."

Also in the field of Astronomy:

Urbain Le Verrier's most famous achievement is his prediction of the existence of the then unknown planet Neptune, using only mathematics and astronomical observations of the known planet Uranus.

Quoting this time from the Wiki page on the Discovery of Neptune:

It was a sensational moment of 19th-century science, and dramatic confirmation of Newtonian gravitational theory. In François Arago's apt phrase, Le Verrier had discovered a planet "with the point of his pen".

Having taught the basics of perturbation theory in classical mechanics, I can attest that this kind of work can only be described a monumental, both from a calculation perspective and from mental ability required to understand and accurately perform such calculations.

A few more. "Remarkable approximations for pi are given in Indian texts including 3.1416 of Aryabhata (499 AD), 3.14159265359 of Madhava (14th century AD) and 355/113 of Nilakanta (1500 AD). An anonymous work Karanapaddhati (believed to have been written by Putumana Somayajin in the 15th century AD) gives the value 3.14159265358979324 which is correct up to seventeen decimal places." From VoL7,No.4,pp.ll-13;No.IO,p.6,2002. Mathematics in Ancient India by Amartya Kumar Dutta.

https://www.ias.ac.in/article/fulltext/reso/007/04/0004-0019