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Denis Roegel
To cite this version:
Denis Roegel. A reconstruction of the tables of Thompson’s Logarithmetica Britannica (1952).
[Re-search Report] 2011. �hal-00654453�
of the tables of Thompson’s
Logarithmetica Britannica
(1952)
Denis Roegel
20 December 2011
joined the statistical staff of the General Register Office, Somerset House, London. He
retired in 1945 from the General Register Office. We do not Thompson’s exact death,
but we know that he was a member of the BAASMTC
1from 1929 and until 1965. He
possibly died in 1968 or 1973.
2
Thompson’s table of logarithms (1924–1952)
In 1922 Thompson started the project of a new table of logarithms, celebrating the 300th
birthday of Briggs’ Arithmetica logarithmica [2]. Briggs had computed the logarithms of
the numbers from 1 to 20000 and from 90001 to 100000 to 14 places. In 1628, Vlacq filled
the gap from 20001 to 90000, but only to 10 places [21]. Thompson, instead, decided to
compute the logarithms of all integers from 1 to 100000 to 20 places.
Thompson’s table was published in nine parts from 1924 to 1952. Each part covered
a range of 10000 integers, from 10000 to 100000. The first published part was part IX,
covering the range 90000 to 100000, and published in 1924. Then came part VIII (1927),
part IV (1928), part V (1931), part VI (1933), part I (1934), part VII (1935), part III
(1937), and finally part II (1952). Thompson’s table was actually completed in 1939,
but the war broke out and delayed the writing of the introduction. The whole table, as
well as several auxiliary tables and introductions, was published in two volumes in 1952.
It was the last large table of logarithms not computed by computer. In 1961 and 1972,
Thompson’s table was also translated in Russian [17, 18, 19, 20].
Thompson used various methods for the construction of his table. He made in
par-ticular heavy use of mechanical calculating machines, as explained in the introduction of
the table. For an introduction to Thompson’s methods, we refer to Weiss’ article [22].
3
Everett’s interpolation formula
Thompson’s table gives the values of the logarithms, and the second (δ
2) and fourth (δ
4)
differences, which can be used for interpolation with Everett’s central-difference
interpo-lation formula [5]. Indeed, in 1921 Thompson had published a table of the coefficients
of Everett’s formula which facilitate the interpolation [14]. Similar tables were later
published by Chappell [3] and Dijkstra and Wijngaarden [4].
u
0δ
2u
0δ
4u
0δ
1u
0δ
3u
0u
1δ
2u
1δ
4u
1δ
1u
1δ
3u
1u
2δ
2u
2δ
4u
2δ
1u
2δ
3u
2u
3δ
2u
3δ
4u
3δ
1u
3δ
3u
3u
0, u
1, etc., are the values of the function, δ
1are the first differences, δ
2the second
differences, and so on.
In order to interpolate between u
0and u
1, only the boxed values are used and we have
used the names given by Thompson.
More exactly, Everett’s interpolation formula is the following, also in Thompson’s
notations:
u
θ= φu
0+ θu
1+
φ(φ
2− 1)
3!
δ
2u
0+
θ(θ
2− 1)
3!
δ
2u
1+
φ(φ
2− 1)(φ
2− 4)
5!
δ
4u
0+
θ(θ
2− 1)(θ
4− 1)
5!
δ
4u
1+ · · · (1)
where
u
θis the value interpolated between u
0and u
1;
φ and θ = 1 − φ are the two fractions dividing the interval from u
0to u
1; θ = 0
corresponds to u
0and θ = 1 to u
1;
δ
2u
0
, δ
2u
1are the central differences of the second order;
δ
4u
0
, δ
4u
1are the central differences of the fourth order.
Thompson uses the following abbreviations:
2(θ) =
θ(θ
2− 1)
3!
2(φ) =
φ(φ
2− 1)
3!
(2)
4(θ) =
θ(θ
2− 1)(θ
2− 4)
5!
4(φ) =
φ(φ
2− 1)(φ
2− 4)
5!
(3)
. . .
Everett’s formula then becomes
u
θ= φu
0+ θu
1+
2(φ)δ
2u
0+
2(θ)δ
2u
1+
4(φ)δ
4u
0+
4(θ)δ
4u
1+ · · ·
(4)
Thompson illustrates this formula by the interpolation of log 2.7182818 using the
values of log 27182 and log 27183. Thompson continues by showing how the logarithm of
places. It is prefixed by a table of the logarithms of the first 1000 integers to 21 places.
Both tables were reconstructed.
For every value ending with 0 or 5, Thompson adds if the real value is smaller (−)
or larger (+). For instance, log 3 is given as 0.47712 12547 19662 43729 5+, because
the real value is 0.47712 12547 19662 43729502 . . . On the other hand, log 16 is given as
1.20411 99826 55924 78085 5− because the real value is 1.20411 99826 55924 78085 495 . . .
We have added these marks whenever they were needed.
All the tables were recomputed using the GNU mpfr multiple-precision floating-point
library developed at INRIA [8].
Not all items of this list are mentioned in the text, and the sources which have not been
seen are marked so. We have added notes about the contents of the articles in certain
cases.
[1] Raymond Clare Archibald. Mathematical table makers: Portraits, paintings, busts,
monuments; bio-bibliographical notes. New York: Scripta Mathematica, 1948.
[pp. 78–79 on Thompson]
[2] Henry Briggs. Arithmetica logarithmica. London: William Jones, 1624.
[The tables
were reconstructed by D. Roegel in 2010. [13]]
[3] Edwin Chappell. A table of coefficients to facilitate interpolation by means of the
formulæ of Gauss, Bessel and Everett. London: published by the author, 1929.
[4] Edsger Wybe Dijkstra and Adriaan van Wijngaarden. Table of Everett’s
interpolation coefficients. The Hague: Excelsior’s Photo-Offset, 1955.
[Report R294 of
the computation department of the mathematical centre, Amsterdam]
[5] Joseph David Everett. On a new interpolation formula. Journal of the Institute of
Actuaries, 35:452–458, 1901.
[6] Alan Fletcher. Review of “Logarithmetica Britannica, Part II” by A. J. Thompson.
The Mathematical Gazette, 37(321):221–223, September 1953.
[7] Alan Fletcher, Jeffery Charles Percy Miller, Louis Rosenhead, and Leslie John
Comrie. An index of mathematical tables. Oxford: Blackwell scientific publications
Ltd., 1962.
[2nd edition (1st in 1946), 2 volumes]
[8] Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier, and Paul
Zimmermann. MPFR: A multiple-precision binary floating-point library with
correct rounding. ACM Transactions on Mathematical Software, 33(2), 2007.
[9] James Whitbread Lee Glaisher. Report of the committee on mathematical tables.
London: Taylor and Francis, 1873.
[Also published as part of the “Report of the forty-third
meeting of the British Association for the advancement of science,” London: John Murray, 1874.
A review by R. Radau was published in the Bulletin des sciences mathématiques et
astronomiques, volume 11, 1876, pp. 7–27]
[10] Herman Heine Goldstine. A history of numerical analysis from the 16th through the
19th century. New York: Springer, 1977.
2
Note on the titles of the works: Original titles come with many idiosyncrasies and features (line
splitting, size, fonts, etc.) which can often not be reproduced in a list of references. It has therefore
seemed pointless to capitalize works according to conventions which not only have no relation with the
original work, but also do not restore the title entirely. In the following list of references, most title
words (except in German) will therefore be left uncapitalized. The names of the authors have also been
homogenized and initials expanded, as much as possible.
Press, 1926.
[12] Denis Roegel. A reconstruction of De Decker-Vlacq’s tables in the Arithmetica
logarithmica (1628). Technical report, LORIA, Nancy, 2010.
[This is a recalculation of
the tables of [21].]
[13] Denis Roegel. A reconstruction of the tables of Briggs’ Arithmetica logarithmica
(1624). Technical report, LORIA, Nancy, 2010.
[This is a recalculation of the tables of
[2].]
[14] Alexander John Thompson. Table of the coefficients of Everett’s central-difference
interpolation formula, volume V of Tracts for computers. London: Cambridge
University press, 1921.
[second edition in 1943]
[15] Alexander John Thompson. Henry Briggs and his work on logarithms. The
American Mathematical Monthly, 32(3):129–131, March 1925.
[16] Alexander John Thompson. Logarithmetica Britannica, being a standard table of
logarithms to twenty decimal places of the numbers 10,000 to 100,000. Cambridge:
University press, 1952.
[2 volumes]
[17] Александр (Alexander) Джон (John) Томпсон (Thompson). Таблицы
десятичных логарифмов чисел. Т.1: Логарифмы чисел от 10000 до 55000.
Москва: Вычислительный центр Академия наук СССР, 1961.
[not seen]
[18] Александр (Alexander) Джон (John) Томпсон (Thompson). Таблицы
десятичных логарифмов чисел. Т.2: Логарифмы чисел от 55000 до 100000.
Москва: Вычислительный центр Академия наук СССР, 1961.
[not seen]
[19] Александр (Alexander) Джон (John) Томпсон (Thompson). Таблицы
десятичных логарифмов чисел. Т.1: Логарифмы чисел от 10000 до 55000.
Москва: Вычислительный центр Академия наук СССР, 1972.
[not seen]
[20] Александр (Alexander) Джон (John) Томпсон (Thompson). Таблицы
десятичных логарифмов чисел. Т.2: Логарифмы чисел от 55000 до 100000.
Москва: Вычислительный центр Академия наук СССР, 1972.
[not seen]
[21] Adriaan Vlacq. Arithmetica logarithmica. Gouda: Pieter Rammazeyn, 1628.
[The
introduction was reprinted in 1976 by Olms and the tables were reconstructed by D. Roegel in
2010. [12]]
[22] Stephan Weiss. Triumphator hoch vier, oder Die Differenzmaschine von
A. J. Thompson und die Logarithmetica Britannica, 2007.
[available on
http://www.mechrech.info/publikat/DMThomp.pdf]
N, –
N
log N
N
log N
N
log N
N
log N
+ + + − + − + + + − + − + + + − + + − − + + + + + + −
log N, –
N, –
N
log N
N
log N
N
log N
N
log N
+ + − − + + − + + + + + + + + + − + + + − + + + +
log N, –
N, –
N
log N
N
log N
N
log N
N
log N
− − + + − + − + + + + + + + − + − − + + + − − − − + + −
log N, –
N, –
N
log N
N
log N
N
log N
N
log N
+ − + − + + − − − + − − − + + + + + + − − −
