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To cite this version:
Denis Roegel. A reconstruction of Peters’s ten-place table of logarithms (volume 1, 1922) . [Re- search Report] LORIA, UMR 7503, Université de Lorraine, CNRS, Vandoeuvre-lès-Nancy. 2016.
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table of logarithms (volume 1, 1922)
Denis Roegel
29 August 2016
This document is part of the LOCOMAT project:
http://locomat.loria.fr
The 8-place table was itself based on a 12-place manuscript table, which was again used by Peters in the preparation of a new 7-place table of logarithms of trigonometrical functions for every second of the quadrant published in 1911 [39].
2 Peters’s table of 10-place logarithms (1922)
In 1922, Peters published the first volume of a table of logarithms to ten places, but the second volume giving the logarithms of trigonometrical functions [45], as well as associated auxiliary tables [44], had actually been published first in 1919. Preliminary work for Peters’s 10-place table had actually been made before WWI, but the completion of the work was postponed because of the war. Towards the end of WWI, the surveying administration conceived the plan of new tables, and Peters’s preliminary work fit into this project. The publication of this first volume was delayed for various reasons. All three volumes were reprinted in 1957 [100], and the reprinted auxiliary volume contains a number of corrections to the original volume 1.
The present document gives a reconstruction of the main tables of the first volume.
The tables of the second volume are described and reconstructed separately [92, 76].
2.1 Table of logarithms
Peters’s table of logarithms contains two main parts. A first table gives the logarithms of the integers from 1 to 1000 to ten places, and a second table gives the logarithms from 10000 to 100000, also to ten places. First differences are given.
These tables are not immediately based on earlier 10 (or more)-place tables, such as those by Briggs [16], Vlacq [102] or Vega [101]. Instead, Peters’s table is based on the unpublished 12-place manuscript table mentioned above and made during the preparation of Bauschinger and Peters’s table of logarithms [15]. This manuscript table gave the logarithms to at most one unit of error on the 12th place.
In most cases, 10-place values correct to half a unit could be derived from the 12-place values, but the 12-place values which were ending with 49, 50 and 51 had to be subjected to a special treatment. This was achieved partly using Steinhauser’s auxiliary tables [98]
and partly though a recomputation using
2log(N + h) = log N + 2M ·
"
h
2N + h + 1 3
h
2N + h
3+ · · ·
#
1
For more information on Peters’s tables, we refer the reader to our summary [93].
2
This formula had already been used during the preparation of Bauschinger and Peters’s 8-place table [15] and its derivation is given in our introduction to these tables [73].
3
log 29888 = 4.47549 68545 49999 33 . . . log 42244 = 4.62576 50339 49999 75 . . . log 49295 = 4.69280 28709 50004 94 . . . log 49692 = 4.69628 64765 50002 89 . . . log 60757 = 4.78359 63215 50007 52 . . . log 71734 = 4.85572 50484 50003 42 . . . log 74281 = 4.87087 77417 50006 78 . . . log 98053 = 4.99146 08857 49995 52 . . .
As a result of these computations, the logarithms of all integers from 1 to 100000 were obtained with an error no larger than half a unit of the tenth place.
A long and tedious process ensured the accuracy of the stereotyped plates, as described in detail in the original introduction to the tables. In the final stage, comparisons were made with the 8-place tables of Bauschinger and Peters [15], with Vlacq’s table [102], and with Duffield’s table [25], whose errors were found to coincide rather too often with those of Vlacq. . .
As explained in the second volume [92], interpolations can be made using an auxiliary table taking into account the second differences. The auxiliary table for the present first volume only comprises two pages and these are not (yet) reproduced here, but the separate volume [76] can also be used for the same purpose. With the auxiliary table published in 1919, the first differences are first corrected, then used for interpolation in the usual way. With the shorter auxiliary table contained in the first volume, interpolation is done first, then the result is corrected using the auxiliary table.
This correction merely corresponds to the application of the interpolation formula log(N + h) = log N + h∆
1+ h(h − 1)
2 ∆
2.
Since ∆
2< 0, we have
log(N + h) = log N + h∆
1+ h(1 − h) 2 |∆
2|
= log N + h∆
1+ 10
−10× h(1 − h)
2 × |10
10∆
2|
= log N + h∆
1+ 10
−10× f(h, 10
10|∆
2|)
where f (x, y) =
x(1−x)2×y. The auxiliary table then merely tabulates f(x, y) as a function of h and 10
10|∆
2|. For instance, taking one of the examples in Peters’s introduction for finding the logarithm of 17273.47319, we have h = 0.47319 and 10
10|∆
2| = 14, hence the
4
which is correct to 10 places.
2.2 Supplementary material
The original volume contains a number of auxiliary tables not reproduced here, as well as a long appendix by Peters and Johannes Stein with methods for the computation of logarithms of trigonometrical functions to 20 and 22 places, tables of constants and their multiples, tables of powers, tables of reciprocals powers, of factorials and their logarithms, of binomial coefficients, of Bernoulli numbers, of primes, of 48-place natural logarithms, etc.
5
[1] ???? On the eight-figure table of Peters and Comrie. Mathematical Tables and other Aids to Computation, 1(2):64–65, 1943. [The title is ours, and there are actually two notices, on the accuracy of the table published in 1939 [58], and its comparison with other tables.]
[2] Marie Henri Andoyer. Nouvelles tables trigonométriques fondamentales contenant les logarithmes des lignes trigonométriques. . . . Paris: Librairie A. Hermann et fils, 1911. [Reconstruction by D. Roegel in 2010 [68].]
[3] Marie Henri Andoyer. Nouvelles tables trigonométriques fondamentales contenant les valeurs naturelles des lignes trigonométriques. . . . Paris: Librairie A. Hermann et fils, 1915–1918. [3 volumes, reconstruction by D. Roegel in 2010 [69].]
[4] Raymond Clare Archibald. J. T. Peters, Achtstellige Tafel der trigonometrischen Funktionen für jede Sexagesimalsekunde des Quadranten. Mathematical Tables and other Aids to Computation, 1(1):11–12, 1943. [review of the edition published in 1939 [58]]
[5] Raymond Clare Archibald. J. T. Peters, Seven-place values of trigonometric functions for every thousandth of a degree. Mathematical Tables and other Aids to Computation, 1(1):12–13, 1943. [review of the edition published in 1942 [43]]
[6] Raymond Clare Archibald. Tables of trigonometric functions in non-sexagesimal arguments. Mathematical Tables and other Aids to Computation, 1(2):33–44, 1943.
[7] Raymond Clare Archibald. J. T. Peters, Eight-place table of trigonometric functions for every sexagesimal second of the quadrant. Achtstellige Tafel der trigonometrischen Funktionen für jede Sexagesimalsekunde des Quadranten.
Mathematical Tables and other Aids to Computation, 1:147–148, 1944. [review of the edition published in 1939 [58]]
[8] Raymond Clare Archibald. J. T. Peters, Siebenstellige Logarithmentafel.
Mathematical Tables and other Aids to Computation, 1:143–146, 1944. [review of the edition published in 1940 [59]]
3
