A reconstruction of Bauschinger and Peters's eight-place table of logarithms (volume 1, 1910)

(1)

HAL Id: hal-01357798

https://hal.inria.fr/hal-01357798

Submitted on 30 Aug 2016

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Denis Roegel

To cite this version:

Denis Roegel. A reconstruction of Bauschinger and Peters’s eight-place table of logarithms (volume 1, 1910). [Research Report] LORIA, UMR 7503, Université de Lorraine, CNRS, Vandoeuvre-lès-Nancy.

2016. �hal-01357798�

(2)

Bauschinger and Peters’s eight-place table of logarithms

(volume 1, 1910)

Denis Roegel

29 August 2016

This document is part of the LOCOMAT project:

http://locomat.loria.fr

(3)

(4)

117] in the 17th century and remained the standard tables up into the 20th century, only being proofchecked, reorganized, and completed by various table makers. By 1900, very few independent computations had been attempted, except for those of Prony [85] and Sang [28], and their tables had mostly been left unpublished. The largest widely available tables of logarithms gave only the logarithms to 7 places. There were only two tables to 8 places, but these tables each had their own problems. It is in this context that Bauschinger and Peters sought to provide new 8-place tables of logarithms, computed independently, at least to some extent. This project culminated in the publication of two volumes in 1910 and 1911 [17], and the present document explains their methods and gives a reconstruction (recomputation) of the first volume. The second volume is provided in a separate document.

2 Bauschinger and Peters

Julius Bauschinger was a German astronomer [113]. He was born in Fürth (Germany) in 1860, the son of the engineer Johann Bauschinger (1834–1893). In 1896, he became director of the Astronomisches Rechen-Institut in Berlin, and in 1909, director of the Strasbourg observatory, then in Germany. Afterwards, he was director of the Leipzig observatory where he died in 1934.

Among his many publications are tables of theoretical astronomy (1901) [13] and a book on the determination of the orbits of celestial bodies (1906) [15].

Figure 1: Julius Bauschinger (source: Astronomisches Rechen-Institut, Heidelberg) Johann Theodor Peters was born in 1869 in Köln (Germany) [41, 35, 40, 12]. He studied in Bonn and obtained his PhD there. In 1899, he came to the Astronomisches Rechen-Institut in Berlin, whose director was Bauschinger.

3

(5)

undertaking is the present joint work with Julius Bauschinger. Many other tables will follow, in part based on that foundational work, and these tables are described together with their reconstructions.

Peters died in 1941. For details on Peters and his various other tables, see our sur- vey [107].

Figure 2: From left to right, the astronomer Karl Heinrich Willy Kruse (1889–1945), Leslie John Comrie (1893–1950) and Peters, 1930. (from [12])

3 The history of the new tables

In the introduction to their tables, Bauschinger and Peters give a detailed account on their construction which is summarized here.

¹

These tables stemmed from the need to have 8-place tables of logarithms, because the usual 7-place tables were more and more inadequate in astronomy and geodesy as a consequence of the observations’ increased accuracy. When the need for a greater accuracy arose, one usually resorted to Vega’s Thesaurus [115], which gave the logarithms to 10 places, but whose use was inefficient and required interpolations with second differences.

1We are not aware of earlier technical descriptions of the construction of Bauschinger and Peters’s tables, but historical accounts have recently been given by Ulf Hashagen, in particular at the meeting organized for the 400th anniversary of Napier’s logarithms in 2014.

4

(6)

geodesy. Earlier in 1904, Bruns and Bauschinger also mentioned Mendizabal-Tamborrel’s 8-place table [30] which they found impractical.

³

In 1904 Bruns and Bauschinger therefore conceived the plan to construct a new 8- place table of logarithms, which should be as convenient as possible and based on the sexagesimal division of the quadrant. A plan was described early on and gave detailed consideration on the format of the table [23]. This plan also gave an estimate of the cost (50000 Mark), and of the time (three years of work and two years of printing), and considered the use of a calculating machine, namely a Burroughs adding machine with printer [44]. The years 1905 to 1908 were used to obtain the funds, so that work on the tables could start in Spring of 1908.

The first year was devoted to hand calculations for the preparation of interpolations.

This work involved three or four computers

⁴

and was completed in May 1909. At the same time as the hand computations were begun, contact was made with the engineer Christel Hamann (1870–1948) [47] for the construction of a specialized difference engine that could add second differences and print the computed values. Hamann completed the construction of the machine in early 1909 and we describe it briefly in section 7 below.

This machine was then used by two computers to compute the interpolation on 828000 values in the span of one year.

⁵

Almost simultaneously, work was done to prepare and check the shortened 8-place manuscript for the printer.

Printing was started even before the interpolations were completed. It started in May 1909 and was completed in November 1909 for the first volume. The second volume was essentially completed during the year 1910.

Bauschinger and Peters early on decided not to recompute all values ab initio, espe- cially since Briggs’s table [19, 20] gave the logarithms accurately to 12 places (the 14th place in Briggs’s table is essentially random, so that the 13th place gives a good rounding of the 12th). Instead, Bauschinger and Peters set themselves the task to secure the 8th place, and to provide all the functions in such intervals that interpolations could be kept as simple as possible. That led to the decision to provide the logarithms of all numbers from 1 to 200000 and the logarithms of the trigonometric functions at one-second inter- vals, as well as the the auxiliary functions S and T for the first degree of the quadrant.

For these calculations, they decided to use Briggs’s original tables, and not the tables of Vega [115] and Vlacq [116, 117] which had been derived from them.

The first task was therefore to shorten Briggs’s tables to 12 places so that the error on the 12th place should not exceed 0.6 units.

⁶

According to Bauschinger and Peters,

2Things would change in the 1920s and 1930s and Peters then published several decimal tables.

3Apart from the two 1891 8-place tables, there is also John Newton’s 8-place table, published in 1658 [46, 8], and which Bauschinger and Peters did not mention. Newton’s table gave the logarithms of the integers from 1 to 100000 as well as the logarithms of the sines and tangents.

4Bauschinger and Peters’s introduction ends with a list of all the persons which were involved in the projet, and we do not repeat this list here.

5These 828000 values are the number of final values, namely 180000 integers (20000 to 200000), and 45×3600×4 = 648000 trigonometric logarithms, ignoring the values of the functionsS andT.

6Although Bauschinger and Peters do not state it explicitely, they seem to have assumed that the

5

(7)

4 The computation of the logarithms of numbers

4.1 Subtabulation

Briggs’s table [19] gives the logarithms of the natural numbers from 1 to 20000 and from 90001 to 100000, all to 14 decimal places. But the new table was to contain the logarithms of the integers from 20000 to 200000.

Therefore the logarithms from 10000 to 20000 were interpolated to give the loga- rithms from 100000 to 200000. The logarithms from 2000 to 10000 were also interpolated and gave the logarithms from 20000 to 100000. Although Briggs’s table contains the logarithms from 90001 to 100000, they were recomputed for purposes of homogeneity.

Similarly, the 8-place logarithms of the integers from 10000 to 119999 from the Service géographique de l’armée [111] were not used, in order to ensure that the new computation was an independent one. In addition, Bauschinger and Peters wanted to secure a 12-place table, at least in manuscript, and this would not have been possible by using the values of the 1891 table which are only to 8 places. This table was only used when reading the proofs, and only one error was found in its logarithms.

In summary, the purpose was to interpolate nine values between the values given by Briggs. Let us first consider an example, here with 9 places and second differences. Given the larger differences

a log(a) ∆

¹

∆

²

999 2.999565488

0.000434512

1000 3.000000000 −0.000000435

0.000434077

1001 3.000434077 −0.000000432

0.000433645 1002 3.000867722

Bauschinger and Peters wanted to obtain the following differences:

rounded 13th place in Briggs’s table is at most wrong by one unit. In that case, it it easy to see that shortening the table to 12 places will ensure that the error is not larger than 0.6 units of the 12th place.

The worst case is to have some value 3.49. . . where the correct value is 3.6, and round it to 3 instead of 4, which results in an error of 0.6.

6

(8)

1000.1 3.000043427 −0.000000004 0.000043423

1000.2 3.000086850 −0.000000004

0.000043419

1000.3 3.000130269 −0.000000005

0.000043414

1000.4 3.000173683 −0.000000004

0.000043410

1000.5 3.000217093 −0.000000004

0.000043406

1000.6 3.000260499 −0.000000005

0.000043401

1000.7 3.000303900 −0.000000004

0.000043397

1000.8 3.000347297 −0.000000005

0.000043392

1000.9 3.000390689 −0.000000004

0.000043388 1001.0 3.000434077

4.2 Computing the initial values

The idea was to compute initial values for the subtabulated differences (the first values of ∆

¹

and ∆

²

), then to add up the differences with an adding machine. The first value of ∆

¹

is added to the first logarithm and yields the second logarithm. Then ∆

²

is added to ∆

¹

and yields the new value of ∆

¹

, which is then added to the second logarithm and returns the third logarithm, and so forth. Approximations of the initial values of the subtabulated differences can be obtained from the larger differences, as we will see shortly.

The first step was for Bauschinger and Peters to have the first, second, third and fourth differences computed by hand to 12 places using Briggs’s table. Hence about 18000 × 4 = 72000 differences had to be computed. Although the third and fourth differences are not used in the new interpolation, they are useful for checking some values, as explained below.

In order to express precisely the relations between the differences, Bauschinger and Peters used Bruns’ notations [22, p. 18]:

7

(9)

(a −

₂

, 1) (a −

₂

, 3)

(a, 0) (a, 2) (a, 4)

(a +

¹₂

, 1) (a +

¹₂

, 3)

(a + 1, 0) (a + 1, 2) (a + 1, 4) . . .

In this scheme, the values of the logarithms are (a, 0), (a + 1, 0), (a + 2, 0), etc., and these are differences of order 0. a is the argument of the logarithms, for instance the integer whose logarithm is taken. If a = 100, we have (a, 0) = 2.00000 . . ., (a + 1, 0) = 2.00432 . . ., (a + 2, 0) = 2.00860 . . ., and so on. We also have (a − 1, 0) = 1.99563 . . . The differences are then located using these arguments and the order of the differences.

For instance, (a +

¹₂

, 1) is the first difference located between the values (a, 0) and (a + 1, 0). With the above example, (a +

¹₂

, 1) = 0.00432 . . . These notations are particularly convenient to express the process of interpolation.

The corresponding notations for the subdivision of the original interval into ten in- tervals are:

value ∆

¹

∆

²

(a, 0)

(a +

₂₀¹

, 1)

(a +

₁₀¹

, 0) (a +

₁₀¹

, 2) (a +

₂₀³

, 1)

(a +

₁₀²

, 0) (a +

₁₀²

, 2) (a +

₂₀⁵

, 1)

. . .

(a +

₁₀⁹

, 0) (a +

₁₀⁹

, 2) (a +

¹⁹₂₀

, 1)

(a + 1, 0)

One should however be careful when using these notations, as for instance (a +

₁₀⁵

, 2) is not equal to (a +

¹₂

, 2). The positional fractions should never be simplified.

Now, in the first example, we have a = 1000 and (a, 0) = 3.000000000, (a + 1, 0) = 3.000434077, (a +

¹₂

, 1) = 0.000434077, (a, 2) = −0.000000434, etc.

And in the subdivided interval, we have (a +

₂₀¹

, 1) = 0.000043427, (a +

₁₀¹

, 2) =

−0.000000004, etc.

There is a relationship between the differences in both cases. We are going to in- vestigate this relationship, and for that purpose, we will use Bessel’s interpolation for- mula. First, we write the differences as follows, where u

i

is the value of the function,

∆u

_i

= u

_i+1

− u

_i

and ∆

^j+1

u

_i

= ∆

^j

u

_i+1

− ∆

^j

u

_i

for j > 0:

8

(10)

∆u

−2

∆ u

−3

u

−1

∆

²

u

−2

∆

⁴

u

−3

∆u

₋₁

∆

³

u

₋₂

u

₀

∆

²

u

−1

∆

⁴

u

−2

∆u

₀

∆

³

u

−1

u

₁

∆

²

u

₀

∆

⁴

u

₋₁

∆u

₁

∆

³

u

₀

u

₂

∆

²

u

₁

∆

⁴

u

₀

∆u

2

∆

³

u

1

u

₃

∆

²

u

₂

∆

⁴

u

₁

Forms of Bessel’s formula are given by many authors, for instance Milne-Thomson [45], but we take Freeman’s notations [33]. With these notations, the value u

_x

is interpolated between u

₀

and u

₁

as follows:

u

_x

= 1

2 (u

₀

+ u

₁

) + (x − 1

2 )∆u

₀

+ x(x − 1) 2!

1 2 (∆

²

u

₋₁

+ ∆

²

u

₀

) + (x −

¹₂

)x(x − 1)

3! ∆

³

u

−1

+ · · ·

(1)

with 0 ≤ x ≤ 1.

Using Bruns’ notations with (a, 0) = u

₀

and (a + 1, 0) = u

₁

, as well as setting (a +

1

2

, 0) =

¹₂

[(a, 0) + (a + 1, 0)], (a +

¹₂

, 2) =

¹₂

[(a, 2) + (a + 1, 2)], and so on, this becomes (a + x, 0) = (a + 1

2 , 0) + (x − 1

2 )(a + 1

2 , 1) + x(x − 1) 2! (a + 1

2 , 2) + (x −

¹₂

)x(x − 1)

3! (a + 1

2 , 3) + · · ·

(2)

And therefore:

(a + 2p + 1

20 , 1) = (a + p + 1

10 , 0) − (a + p 10 , 0)

= 1

10 (a + 1

2 , 1) + ( p

100 − 0.045)(a + 1

2 , 2) + · · ·

(3)

Assuming all second differences to be constant, we have

9

(11)

20 2 2 (a + 5

20 , 1) = 0.1 × (a + 1

2 , 1) − 0.025 × (a + 1

2 , 2) (6)

. . . . (7)

(a + 19

20 , 1) = 0.1 × (a + 1

2 , 1) + 0.045 × (a + 1

2 , 2) (8)

and these equations are suitable for tabulating the new tables using the differences com- puted from Briggs’s table.

All that is left to know are the second differences (a +

₁₀¹

, 2). We can write

(a + 1 10 , 2) =

(a + 2

10 , 0) − (a + 1 10 , 0)

−

(a + 1

10 , 0) − (a+, 0)

= (a + 2

10 , 0) − 2(a + 1

10 , 0) + (a, 0)

(9)

Using Bessel’s expression for (a + x, 0), it is straightforward to obtain (a + 1

10 , 2) ≈ 0.01(a + 1

2 , 2) (10)

this expression becoming an equality when the third differences are equal to zero.

4.3 Checking the interpolations

Once the first difference (a +

₂₀¹

, 1) and second difference (a +

₁₀¹

, 2) had been computed, the interpolation was done with four additional digits, so that the computation was done on sixteen places. This was done in order to avoid rounding errors.

⁷

Each interpolation was checked by comparing the last value of one interpolation with the first one of the next interpolation.

According to Bauschinger and Peters, the last first difference (a+

¹⁹₂₀

, 1) of each interval was also computed, so that the following formula could be checked:

(a + 1

20 , 1) − (a − 1

20 , 1) = 0.01(a, 2) − 0.0225(a, 4)

However. . . this formula turns out to be incorrect! The correct formula is (a + 1

20 , 1) − (a − 1

20 , 1) = 0.01(a, 2) − 0.000825(a, 4)

and it is therefore not clear whether this formula was really used to check the interpo- lations. In addition, even using the correct formula may be useless if the subtabulated differences are computed using only the first and second original differences.

7These 16 places must have been chosen because the values of (a+ ¹₂,2) are on 13 places, and multiplying them by 0.045 yields values on 16 places.

10

(12)

As seen above, the contribution of the third difference

₂

in Bessel’s interpo- lation formula is

(t −

¹₂

)t(t − 1) 3! (a + 1

2 , 3)

where t is the phase varying from 0 to 10. Bauschinger and Peters gave a small table facilitating the computation of the influence of that term.

Each logarithm whose value was not certain after rounding the result of the interpo- lation to 8-places and taking into account the maximal error produced by ignoring the third differences, was recomputed directly using the series:

log(x + h) = log(x) + 2M

( h

2x + h + 1 3

h 2x + h

3

+ · · · )

where M = 1/ ln 10. This formula follows from ln(1+b)−ln(1−b) = 2 h

b +

^b₃³

+

^b₅⁵

+ · · · i , replacing b with

_2x+h^h

:

ln(1 + h x ) = 2

( h

2x + h + 1 3

h 2x + h

3

+ · · · )

5 The computation of the logarithms of trigonometric functions

5.1 The auxiliary functions

The Briggs-Gellibrand table gives the values of log sin every 36

⁰⁰

, that is every 100th of a degree, to 14 places and from 0

^◦

to 45

^◦

. The values of log tan are only given to 10 places.

These values, together with the logarithms of numbers, were used to compute the values of the auxiliary functions S and T :

S(α) = log sin α − log(A) + 10 (11)

T (α) = log tan α − log(A) + 10 = S(α) − log cos α (12) where A is the angle α expressed in sexagesimal seconds.

⁸

These values were then interpolated for every sexagesimal second using the method described in the next section. The interpolated values were then added to the logarithms of the seconds in order to obtain the values of log sin and log tan from 0

^◦

to 5

^◦

. The values of log cos were obtained with log cos = log sin − log tan.

8One should be aware that the definitions ofS andT are not universal, and that different tables may use different definitions. For instance, in Peters’s 10-place table of logarithms [56, 55], A is the angle expressed in degrees, and not in sexagesimal seconds.

11

(13)

The computed values were checked with the 14-place values computed by Bruhns for his tables published in 1870 [21].

The values of log cot were only included to 8-places in the printer’s manuscript.

5.2 The main trigonometric table

According to Bauschinger and Peters, the values of log sin and log cos were directly taken to 12 places from Briggs and Gellibrand at intervals of 36

⁰⁰

, but this is only possible from 0

^◦

to 45

^◦

for log sin and from 45

^◦

to 90

^◦

for log cos, and log tan was computed by subtraction. Could it be that Bauschinger and Peters used Bruhns’ 14-place manuscript table for log cos?

The first, second, third and fourth differences were then computed and the authors concluded that the values of the logarithms were wrong by at most 0.6 units of the 12th place.

In order to use the machine for adding up the differences, the first and second differ- ences for the 1

⁰⁰

intervals had to be computed. This was accomplished using the following formulæ which can easily be derived from Bessel’s interpolation formula when the third differences are equal to zero:

a + 1

72 , 1

= 1 36

a + 1

2 , 1

− 35 2 · 36

²

a + 1

2 , 2

(13)

a + 1 36 , 2

to

a + 35

36 , 2

= 1 36

²

a + 1

2 , 2

(14) When these values were computed, four places were added, although this could not prevent small rounding errors.

At the same time, the last first difference was computed with

a + 71 72 , 1

= 1 36

a + 1

2 , 1

+ 35 2 · 36

²

a + 1

2 , 2

(15) and according to Bauschinger and Peters, similarly as in the case of the logarithms of numbers, and ignoring differences beyond the fourth differences, these values could be checked with

a + 1

72 , 1

−

a − 1 72 , 1

= 1

36

²

(a, 2) − 35

4 · 36

²

(a, 4). (16) But here again, this formula turns out to be incorrect! The correct formula is

a + 1

72 , 1

−

a − 1 72 , 1

= 1

36

²

(a, 2) − 36

²

− 1

12 · 36

⁴

(a, 4). (17) In general, when subdividing into n intervals, we have

a + 1

2n , 1

−

a − 1 2n , 1

= 1

n

²

(a, 2) − n

²

− 1

12 · n

⁴

(a, 4). (18)

12

(14)

ferences.

Once the first differences had been computed, they were then used as inputs for the machine calculation.

The values of log cot were computed independently from those of log tan, and not by the use of log cot(x) = − log tan(x), in part because Bauschinger and Peters wanted to produce a complete 12-place manuscript of the logarithms of all four functions sin, cos, tan and cot.

The influence of the neglected third differences was again estimated using the expres- sion

(t −

¹₂

)t(t − 1) 6 (a + 1

2 , 3)

where t is the phase ranging from

₃₆⁰

to

³⁶₃₆

. Bauschinger and Peters here too gave a table facilitating the computation of the maximal error.

In those cases where the 12th place of the logarithms were uncertain, the sines and cosines were recomputed using the formulæ

sin x = x − x

³

3! + x

⁵

5! ∓ · · · (19)

cos x = 1 − x

²

2! + x

⁴

4! ∓ · · · (20)

and then their logarithms were computed.

The logarithms of the tangents could be recomputed using

log tan x = log sin x − log cos x (21)

6 Independent calculations

The computation of the logarithms of numbers and trigonometric functions was performed twice and independently, using different methods. In the first case, the computations were done as described by one of the authors (presumably Peters), and in the other case by Dr. Witt, probably the astronomer Carl Gustav Witt (1866–1946). For the trigonometric computations, Peters could use his auxiliary 21-place table of sines and cosines [49].

¹⁰

Both computations agreed at least on 16 places.

9The two incorrect expressions have remained until the last edition published in 1970. Incidentally, in the introduction to the 10-place tables published in 1922 [59], Peters gives the incorrect general formula

a+ 1

2n,1

−

a− 1 2n,1

= 1

n²(a,2)−n−1 4·n²(a,4).

10We assume that this is the table referred to, but Bauschinger and Peters cite a 21-place table of logarithms of sines and cosines.

13

(15)

12th place. Although Bauschinger and Peters did not print this table, they considered it as a worthwhile investment, and made it practical to use and in particular organized it in small units stored in a closet and kept at the Astronomisches Recheninstitut in Berlin.

In retrospect, Fletcher wrote in 1962 that Bauschinger and Peters’s table is extremely accurate and that no error was known to exist in the first volume, and that it was thought that there was none. One typo was found in the first edition of the second volume, but that error was corrected in the second edition [32, p. 787]. This error was taken over to the 7-place table published in 1911 [50].

7 Hamann’s difference engine

Hamann’s machine is described in the introduction to Bauschinger and Peters’s table and we give here only a small sketch of its principles.

¹¹

It was made of two independent calculating machines performing their calculations on 16 places, and of a printer (figure 3).

The purpose of the machine was to add up first and second differences on intervals where the second difference remains constant.

Each machine contains a result value (Zählwerk ). Machine 2 contains the value of the function, and machine 1 the first difference, the latter being at the same time stored in the adding work 2 (Schaltwerk ). The second difference is introduced in the first adding work, and through the rotation of a crank a it is added to the first difference in the result work, as well as to the second adding work. A second crank b adds the first difference to the value of the function, which is stored in the second result work. The printer is also operated through cranks. A specimen of printout is shown in the introduction of the original tables.

It is thought that a skilled operator could compute 36 table values in five minutes.

8 Printing

The output stripes of the printer were corrected and then used to set the pages and produce stereotypes. These stereotypes were again proofread. According to Bauschinger and Peters, during the last step of checking, an average of no more than one error per page was found. As claimed by the authors, “the tables are as correct as human work can be.”

During the correction process, comparisons were made with a number of other tables and errors were found in Bruhns’ 7-place tables [21], in Bremiker’s 6-place tables [18], in Vega’s Thesaurus [115], in Callet’s tables [24], in Briggs’s Arithmetica logarithmica [19], and in Briggs/Gellibrand’s Trigonometria britannica [20], of which detailed lists have been given.

11We might describe this machine in more detail in a future article. Among earlier and shorter descriptions, we direct the reader to Galle’s overview [34, pp. 44–48] and Weiss’ recent articles [118, 119].

14

(16)

15

(17)

an accurate 8-place table, their plan was also to produce a practical table. They did in particular follow the layout used in Bremiker’s tables [18], and they decided to avoid altogether the use of interpolation tables in the trigonometric part for matters of space.

Instead, they thought that the use of a table of multiplication, or of a small table of logarithms would be easier than that of an interpolation table. For that reason, con- secutive values were put over each other, so that the differences could be reckoned with ease. Bauschinger and Peters also explained their choice for the symbol ‘*’ representing a prefix change.

The first volume contains the main tables of logarithms on pages 2–363, and these are supplemented by four tables which have not (yet) been reproduced here, namely a small table giving the values of log sin, log tan, S and T for every 10

⁰⁰

from 0

^◦

to 20

^◦

(page 364), a table of the multiples of M = 1/ ln 10 and of ln 10 (page 365), a table for the conversion of angles into time (with 360

^◦

= 24 hours, page 366), and a table for the conversion of angles into seconds (page 367). Finally, page 368 contains a list of twelve mathematical constants useful in calculations. In the first volume (logarithms of numbers), it should be observed that the differences given in column “d.” are between the last value of the line and the first value of the next line.

The second original volume (logarithms of trigonometric functions) contains 950 pages of tables, on pages 2–951, and these tables have all been reconstructed. In that volume, the lines “d.”, at the top and bottom of each page, give the first and last differences in each column, in units of the last place. Asterisks indicate changes in prefixes. However, the asterisk that should have been given on the first page, for the cosines (between 00000 and 99999), was not given in the original table.

The layout of several later tables was directly based on the present tables. This is the case for Peters’s 7-place table of logarithms of trigonometric functions [50], published in 1911, which can be seen as an abridged version of the second volume of Bauschinger and Peters’s table. The table of trigonometric functions to 8-places published by Peters in 1939 [69] also follows this layout.

16

(18)

17

(19)

18

(20)

19

(21)

Peters’s table. 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] ???? 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 [69], 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 [79].]

[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 [80].]

[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 [69]]

[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.

[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.

[8] Raymond Clare Archibald. J. T. Peters, Siebenstellige Logarithmentafel.

Mathematical Tables and other Aids to Computation, 1:143–146, 1944.

12Note 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.

The reader should keep in mind that this list is not meant as a facsimile of the original works. The original style information could no doubt have been added as a note, but we have not done it here.

20

(22)

Funktionen von Tausendstel zu Tausendstel des Neugrades. Mathematical Tables and other Aids to Computation, 2(19):298–299, 1947.

[review of 9th edition of [68]

published in 1944]

[11] Raymond Clare Archibald. J. T. Peters, Siebenstellige Werte der

trigonometrischen Funktionen von Tausendstel zu Tausendstel des Neugrades.

Mathematical Tables and other Aids to Computation, 2(19):299, 1947.

[review of the 1941 edition [71]]

[12] Raymond Clare Archibald. Mathematical table makers. Portraits, paintings, busts, monument. Bio-bibliographical notes. New York: Scripta Mathematica, 1948.

[contains a photograph of Peters]

[13] Julius Bauschinger. Tafeln zur theoretischen Astronomie. Leipzig: Wilhelm Engelmann, 1901.

[14] Julius Bauschinger. Interpolation. In Wilhelm Franz Meyer, editor, Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, volume 1(2), pages 799–820. Leipzig: B. G. Teubner, 1904.

[a French translation appeared in [110]]

[15] Julius Bauschinger. Die Bahnbestimmung der Himmelskörper. Leipzig: Wilhelm Engelmann, 1906.

[16] Julius Bauschinger and Johann Theodor Peters. Logarithmic-trigonometrical tables with eight decimal places etc. Leipzig: Wilhelm Engelmann, 1910–1911.

[2 volumes, English introduction. See [17] for the German edition.]

[17] Julius Bauschinger and Johann Theodor Peters. Logarithmisch-trigonometrische Tafeln mit acht Dezimalstellen etc. Leipzig: Wilhelm Engelmann, 1910–1911.

[2 volumes, German introduction. See [16] for the English edition; these volumes have been reprinted in 1936, 1958 and 1970, but the introductions vary. In particular, details of the construction of Hamann’s machine were dropped in the last editions. Volume 2 is reconstructed in [87].]

[18] Carl Bremiker. Logarithmisch-trigonometrische Tafeln mit sechs Decimalstellen.

Berlin: Nicolaische Verlagsbuchhandlung, 1869.

[first stereotype edition, earlier editions were published in 1852 and 1860]

[19] Henry Briggs. Arithmetica logarithmica. London: William Jones, 1624.

[The tables were reconstructed by D. Roegel in 2010. [83]]

[20] Henry Briggs and Henry Gellibrand. Trigonometria Britannica. Gouda: Pieter Rammazeyn, 1633.

21

(23)

Teubner, 1903.

[23] Heinrich Bruns and Julius Bauschinger. Denkschrift über neue achtstellige Logarithmentafeln für den astronomischen Gebrauch. Vierteljahrsschrift der Astronomischen Gesellschaft, 39:158, 232–240, 1904.

[24] Jean-François Callet. Tables portatives de logarithmes, contenant les logarithmes des nombres, depuis 1 jusqu’à 108000 ; etc. Paris: Firmin Didot, 1795.

[There have been numerous later printings of these tables.]

[25] Martin Campbell-Kelly, Mary Croarken, Raymond Flood, and Eleanor Robson, editors. The history of mathematical tables: from Sumer to spreadsheets. Oxford:

Oxford University Press, 2003.

[brief description of Hamann’s machine on pp. 135–136]

[26] Leslie John Comrie. Logarithmic and trigonometrical tables. Monthly Notices of the Royal Astronomical Society, 85(4):386–388, 1925.

[mentions several of Peters’s tables]

[27] Leslie John Comrie. J. T. Peters, Sechsstellige Tafel der trigonometrischen Funktionen,. . . , Berlin, 1929. Mathematical Tables and other Aids to

Computation, 1(5):162, 1944.

[Attributes errors in the first edition of [60] to one of the proofreaders of the table.]

[28] Alex D. D. Craik. The logarithmic tables of Edward Sang and his daughters.

Historia Mathematica, 30(1):47–84, February 2003.

[29] Harold Thayer Davis, editor. Tables of the higher mathematical functions.

Bloomington, In.: The principia press, Inc., 1933, 1935.

[2 volumes]

[30] Joaquín de Mendizábal-Tamborrel. Tables des Logarithmes à huit décimales des nombres de 1 à 125000, et des fonctions goniométriques sinus, tangente, cosinus et cotangente de centimiligone en centimiligone et de microgone en microgone pour les 25000 premiers microgones, et avec sept décimales pour tous les autres microgones. Paris: Hermann, 1891.

[A sketch of this table was reconstructed by D. Roegel [84].]

[31] Alan Fletcher, Jeffery Charles Percy Miller, and Louis Rosenhead. An index of mathematical tables. London: Scientific computing service limited, 1946.

[32] Alan Fletcher, Jeffery Charles Percy Miller, Louis Rosenhead, and Leslie John Comrie. An index of mathematical tables (second edition). Reading, Ma.:

Addison-Wesley publishing company, 1962.

[2 volumes]

[33] Harry Freeman. An elementary treatise on actuarial mathematics. Cambridge:

University Press, 1932.

22

(24)

Reichsvermessungsdienst. Mitteilungen des Reichsamts für Landesaufnahme, 17:346–350, 1941.

[36] James Henderson. Bibliotheca tabularum mathematicarum, being a descriptive catalogue of mathematical tables. Part I: Logarithmic tables (A. Logarithms of numbers), volume XIII of Tracts for computers. London: Cambridge University Press, 1926.

[37] Samuel Herrick, Jr. Natural-value trigonometric tables. Publications of the Astronomical Society of the Pacific, 50(296):234–237, 1938.

[38] Peter Holland. Biographical notes on Johann Theodor Peters, 2011.

www.rechnerlexikon.de/en/artikel/Johann_Theodor_Peters

[39] Wilhelm Rudolf Alfred Klose. Prof. Dr. Jean Peters gestorben. Zeitschrift für Angewandte Mathematik und Mechanik, 22(2):120, 1942.

[obituary notice]

[40] Otto Kohl. Jean Peters. Vierteljahresschrift der Astronomischen Gesellschaft, 77:16–20, 1942.

[includes one photograph]

[41] August Kopff. Jean Peters †. Astronomische Nachrichten, 272(1):47–48, 1941.

[42] Christine Krause. Das Positive von Differenzen : Die Rechenmaschinen von Müller, Babbage, Scheutz, Wiberg, . . . , 2007.

[43] A. V. Lebedev and R. M. Fedorova. A guide to mathematical tables. Oxford:

Pergamon Press, 1960.

[44] Ernst Martin. Die Rechenmaschinen und ihre Entwicklungsgeschichte.

Pappenheim: Johannes Meyer, 1925.

[English translation as “The calculating machines,”

published in 1992]

[45] Louis Melville Milne-Thomson. The calculus of finite differences. London:

Macmillan and Co., 1933.

[46] John Newton. Trigonometria Britanica, or, the doctrine of triangles. London:

R. & W. Leybourn, 1658.

[47] Heinz Nix. Hamann, Christel Bernhard Julius. In Neue Deutsche Biographie, volume 7, page 573. Berlin: Duncker & Humblot, 1966.

[48] Johann Theodor Peters. Neue Rechentafeln für Multiplikation und Division mit allen ein- bis vierstelligen Zahlen. Berlin: G. Reimer, 1909.

[also published in 1919 and 1924 by Walter de Gruyter & Co.; the library of the Paris observatory also has a variant of the 1909 edition with the French title “Nouvelles tables de calcul pour la multiplication et la division de tous les nombres de un à quatre chiffres” (as well as a French introduction), which

23

(25)

[49] Johann Theodor Peters. Einundzwanzigstellige Werte der Funktionen Sinus und Cosinus : zur genauen Berechnung von zwanzigstelligen Werten sämtlicher trigonometrischen Funktionen eines beliebigen Arguments sowie ihrer

Logarithmen. Berlin: Verlag der Königl. Akademie der Wissenschaften, 1911.

[54 pages, Appendix 1 to the “Abhandlungen der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse.”, not seen, but reprinted at the end of the English edition of [69]]

[50] Johann Theodor Peters. Siebenstellige Logarithmentafel der trigonometrischen Funktionen für jede Bogensekunde des Quadranten. Leipzig: Wilhelm Engelmann, 1911.

[reconstructed in [93]]

[51] Johann Theodor Peters. Fünfstellige Logarithmentafel der trigonometrischen Funktionen für jede Zeitsekunde des Quadranten. Berlin: Reimer, 1912.

[52] Johann Theodor Peters. Tafeln zur Berechnung der Mittelpunktsgleichung und des Radiusvektors in elliptischen Bahnen für Excentrizitätswinkel von 0

^◦

bis 24

^◦

. Berlin: Ferd. Dümmler, 1912.

[second edition in 1933]

[53] Johann Theodor Peters. Dreistellige Tafeln für logarithmisches und numerisches Rechnen. Berlin: P. Stankiewicz, 1913.

[not seen, second edition in 1948 (seen),

reconstructed in [88]]

[54] Johann Theodor Peters. Siebenstellige Werte der trigonometrischen Funktionen von Tausendstel zu Tausendstel des Grades. Berlin-Friedenau: Verlag der Optischen Anst. Goerz, 1918.

[Reprinted in 1938 and 1941, as well as in 1942 in English with the title “Seven-place Values of trigonometric functions for every thousandth of a degree.”, all four editions seen. Reconstructed in [94].]

[55] Johann Theodor Peters. Zehnstellige Logarithmentafel : Hilfstafeln zur

zehnstelligen Logarithmentafel. Berlin: Preuß. Landesaufnahme, 1919.

[not seen, second edition in 1957 (seen), reconstructed in [89]]

[56] Johann Theodor Peters. Zehnstellige Logarithmentafel, volume 2 : Zehnstellige Logarithmen der trigonometrischen Funktionen von 0

^◦

bis 90

^◦

für jedes

Tausendstel des Grades. Berlin: Reichsamt f. Landesaufnahme, 1919.

[not seen, second edition in 1957 (seen); also Russian editions in 1964 and 1975; reconstructed in [106]]

[57] Johann Theodor Peters. Sechsstellige Logarithmen der trigonometrischen Funktionen von 0

^◦

bis 90

^◦

für jedes Tausendstel des Grades. Berlin: Verlag der preussischen Landesaufnahme, 1921.

[58] Johann Theodor Peters. Siebenstellige Logarithmen der trigonometrischen Funktionen von 0

^◦

bis 90

^◦

für jedes Tausendstel des Grades. Berlin: Verlag der preussischen Landesaufnahme, 1921.

24

(26)

mathematical tables are by Peters, J. Stein and G. Witt]

[60] Johann Theodor Peters. Sechsstellige Tafel der trigonometrischen Funktionen : enthaltend die Werte der sechs trigonometrischen Funktionen von zehn zu zehn Bogensekunden des in 90

^◦

geteilten Quadranten u. d. Werte d. Kotangente u.

Kosekante f. jede Bogensekunde von 0

^◦

0

⁰

bis 1

^◦

20

⁰

. Berlin: Ferd. Dümmler, 1929.

[seen, reprinted in 1939, 1946, 1953, 1962, 1968 and 1971; in Russian in 1975, and perhaps already in 1937 and 1938; reconstructed in [95]]

[61] Johann Theodor Peters. Tafeln zur Verwandlung von rechtwinkligen

Platten-Koordinaten und sphärischen Koordinaten ineinander. Berlin: Ferd.

Dümmler, 1929.

[Veröffentlichungen des Astronomischen Rechen-Instituts zu Berlin-Dahlem, number 47]

[62] Johann Theodor Peters. Multiplikations- und Interpolationstafeln für alle ein- bis dreistelligen Zahlen. Berlin: Wichmann, 1930.

[reprinted from [63]; reconstructed in [92]]

[63] Johann Theodor Peters. Sechsstellige trigonometrische Tafel für neue Teilung.

Berlin: Wichmann, 1930.

[seen, third edition in 1939 and fourth in 1942; an excerpt was reprinted as [62]; reconstructed in [96]]

[64] Johann Theodor Peters. Präzessionstafeln für das Äquinoktium 1950.0. Berlin:

Ferd. Dümmler, 1934.

[65] Johann Theodor Peters. Tafeln zur Berechnung der jährlichen Präzession in Rektaszension für das Äquinoktium 1950.0. Berlin: Ferd. Dümmler, 1934.

[66] Johann Theodor Peters. Hilfstafeln zur Verwandlung von Tangentialkoordinaten in Rektaszension und Deklination. Berlin: Ferd. Dümmler, 1936.

[67] Johann Theodor Peters. Sechsstellige Werte der Kreis- und

Evolventen-Funktionen von Hundertstel zu Hundertstel des Grades nebst einigen Hilfstafeln für die Zahnradtechnik. Berlin: Ferd. Dümmler, 1937.

[not seen, reprinted in 1951 and 1963 (seen); reconstructed in [101]]

[68] Johann Theodor Peters. Sechsstellige Werte der trigonometrischen Funktionen von Tausendstel zu Tausendstel des Neugrades. Berlin: Wichmann, 1938.

[seen, 3rd edition in 1940, 5th and 6th in 1942, 7th in 1943, 9th in 1944, 10th in 1953, 12th in 1959, 14th in 1970, and other editions in 1973 and other years; reconstructed in [97]]

[69] Johann Theodor Peters. Achtstellige Tafel der trigonometrischen Funktionen für jede Sexagesimalsekunde des Quadranten. Berlin: Verlag des Reichsamts für

25

(27)

was reconstructed in [90]]

[70] Johann Theodor Peters. Siebenstellige Logarithmentafel. Berlin: Verlag des Reichsamts für Landesaufnahme, 1940.

[2 volumes, 1: Logarithmen der Zahlen,

Antilogarithmen, etc., 2: Logarithmen der trigonometrischen Funktionen für jede 10. Sekunde d.

Neugrades, etc.; reconstructed in [98] and [99]]

[71] Johann Theodor Peters. Siebenstellige Werte der trigonometrischen Funktionen von Tausendstel zu Tausendstel des Neugrades. Berlin: Verlag des Reichsamts für Landesaufnahme, 1941.

[reprinted in 1952, 1956 and 1967; reconstructed in [100]]

[72] Johann Theodor Peters, Alfred Lodge, Elsie Jane Ternouth, and Emma Gifford.

Factor table giving the complete decomposition of all numbers less than 100,000.

London: Office of the British Association, 1935.

[introduction by Leslie J. Comrie, and bibliography of tables by James Henderson, reprinted in 1963] [reconstructed in [86]]

[73] Johann Theodor Peters and Karl Pilowski. Tafeln zur Berechnung der Präzessionen zwischen den Äquinoktien 1875.0 und 1950.0. Berlin: Ferd.

Dümmler, 1930.

[74] Johann Theodor Peters and Johannes Stein. Zweiundfünfzigstellige Logarithmen.

Berlin: Ferd. Dümmler, 1919.

[Veröffentlichungen des Astronomischen Rechen-Instituts zu Berlin, number 43]

[75] Johann Theodor Peters, Walter Storck, and F. Ludloff. Hütte Hilfstafeln : zur I. Verwandlung von echten Brüchen in Dezimalbrüche ; II. Zerlegung der Zahlen bis 10000 in Primfaktoren ; ein Hilfsbuch zur Ermittelung geeigneter Zähnezahlen für Räderübersetzungen. Berlin: Wilhelm Ernst & Sohn, 1922.

[3rd edition]

[76] Johann Theodor Peters and Gustav Stracke. Tafeln zur Berechnung der Mittelpunktsgleichung und des Radiusvektors in elliptischen Bahnen für Exzentrizitätswinkel von 0

^◦

bis 26

^◦

. Berlin: Ferd. Dümmler, 1933.

[Veröffentlichungen des Astronomischen Rechen-Instituts zu Berlin-Dahlem, number 41; second edition, first edition in 1912]

[77] Denis Roegel. A reconstruction of Adriaan Vlacq’s tables in the Trigonometria artificialis (1633). Technical report, LORIA, Nancy, 2010.

[This is a recalculation of the tables of [117].]

[78] Denis Roegel. A reconstruction of De Decker-Vlacq’s tables in the Arithmetica logarithmica (1628). Technical report, LORIA, Nancy, 2010.

26

(28)

(1915–1918). Technical report, LORIA, Nancy, 2010.

[This is a reconstruction of [3].]

[81] Denis Roegel. A reconstruction of the “Tables des logarithmes à huit décimales”

from the French “Service géographique de l’armée” (1891). Technical report, LORIA, Nancy, 2010.

[82] Denis Roegel. A reconstruction of the tables of Briggs and Gellibrand’s

Trigonometria Britannica (1633). Technical report, LORIA, Nancy, 2010.

[83] Denis Roegel. A reconstruction of the tables of Briggs’ Arithmetica logarithmica (1624). Technical report, LORIA, Nancy, 2010.

[84] Denis Roegel. A sketch of Mendizábal y Tamborrel’s table of logarithms (1891).

Technical report, LORIA, Nancy, 2010.

[This is a sketch of Mendizábal’s table [30].]

[85] Denis Roegel. The great logarithmic and trigonometric tables of the French Cadastre: a preliminary investigation. Technical report, LORIA, Nancy, 2010.

[86] Denis Roegel. A reconstruction of the table of factors of Peters, Lodge, Ternouth, and Gifford (1935). Technical report, LORIA, Nancy, 2011.

[87] Denis Roegel. A reconstruction of Bauschinger and Peters’s eight-place table of logarithms (volume 2, 1911). Technical report, LORIA, Nancy, 2016.

[88] Denis Roegel. A reconstruction of Peters’s 3-place tables (1913). Technical report, LORIA, Nancy, 2016.

[89] Denis Roegel. A reconstruction of Peters’s auxiliary tables to his ten-place logarithms (1919). Technical report, LORIA, Nancy, 2016.

[90] Denis Roegel. A reconstruction of Peters’s eight-place table of trigonometric functions (1939). Technical report, LORIA, Nancy, 2016.

[91] Denis Roegel. A reconstruction of Peters’s five-place table of logarithms of trigonometric functions (1912). Technical report, LORIA, Nancy, 2016.

[92] Denis Roegel. A reconstruction of Peters’s multiplication and interpolation tables (1930). Technical report, LORIA, Nancy, 2016.

27

(29)

[94] Denis Roegel. A reconstruction of Peters’s seven-place table of trigonometric functions (1918). Technical report, LORIA, Nancy, 2016.

[95] Denis Roegel. A reconstruction of Peters’s six-place table of trigonometric functions (1929). Technical report, LORIA, Nancy, 2016.

[96] Denis Roegel. A reconstruction of Peters’s six-place table of trigonometric functions for the new division (1930). Technical report, LORIA, Nancy, 2016.

[97] Denis Roegel. A reconstruction of Peters’s six-place table of trigonometric functions for the new division (1938). Technical report, LORIA, Nancy, 2016.

[98] Denis Roegel. A reconstruction of Peters’s table of 7-place logarithms (volume 1, 1940). Technical report, LORIA, Nancy, 2016.

[99] Denis Roegel. A reconstruction of Peters’s table of 7-place logarithms (volume 2, 1940). Technical report, LORIA, Nancy, 2016.

[100] Denis Roegel. A reconstruction of Peters’s table of 7-place trigonometrical values for the new division (1941). Technical report, LORIA, Nancy, 2016.

[101] Denis Roegel. A reconstruction of Peters’s table of involutes (1937). Technical report, LORIA, Nancy, 2016.

[102] Denis Roegel. A reconstruction of Peters’s table of logarithms to 6 places (1921).

Technical report, LORIA, Nancy, 2016.

[103] Denis Roegel. A reconstruction of Peters’s table of logarithms to 7 places (1921).

Technical report, LORIA, Nancy, 2016.

[104] Denis Roegel. A reconstruction of Peters’s table of products (1909). Technical report, LORIA, Nancy, 2016.

[105] Denis Roegel. A reconstruction of Peters’s ten-place table of logarithms (volume 1, 1922). Technical report, LORIA, Nancy, 2016.

[106] Denis Roegel. A reconstruction of Peters’s ten-place table of logarithms (volume 2, 1919). Technical report, LORIA, Nancy, 2016.

[107] Denis Roegel. The genealogy of Johann Theodor Peters’s great mathematical tables. Technical report, LORIA, Nancy, 2016.

28

(30)

[109] Karl Schütte. Index mathematischer Tafelwerke und Tabellen aus allen Gebieten der Naturwissenschaften. München: R. Oldenbourg, 1955.

[110] Dmitri˘ı Selivanov, Julius Bauschinger, and Marie Henri Andoyer. Le calcul des différences et interpolation. In Jules Molk, editor, Encyclopédie des sciences mathématiques pures et appliquées, volume 1(4) (fasc. 1), pages 47–160. Paris:

Gauthier-Villars, 1906.

[includes a French edition of [14]]

[111] Service géographique de l’Armée. Tables des logarithmes à huit décimales des nombres entiers de 1 à 120000, et des sinus et tangentes de dix secondes en dix secondes d’arc dans le système de la division centésimale du quadrant publiée par ordre du Ministre de la guerre. Paris: Imprimerie nationale, 1891.

[Reprinted by the Institut Géographique National in 1944 and 1964, and reconstruction by D. Roegel, 2010.[81]]

[112] Daniel Shanks. Jean Peters, Eight-place tables of trigonometric functions for every second of arc. Mathematics of Computation, 18(87):509, 1964.

[113] Gustav Stracke. Julius Bauschinger. Monthly Notices of the Royal Astronomical Society, 95(4):336–337, 1935.

[114] John Todd. J. Peters, Ten-place logarithm table. Mathematical Tables and other Aids to Computation, 12:61–63, 1958.

[review of the 2nd edition published in 1957 [59, 56]]

[115] Georg Vega. Thesaurus logarithmorum completus. Leipzig: Weidmann, 1794.

[116] 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. [78]]

[117] Adriaan Vlacq. Trigonometria artificialis. Gouda: Pieter Rammazeyn, 1633.

[118] Stephan Weiss. Die Differenzmaschine von Hamann und die Berechnung der Logarithmen, 2006.

www.mechrech.info/publikat/HamDiffM.pdf

[119] Stephan Weiss. Difference engines in the 20

^th

century. In Proceedings 16th International Meeting of Collectors of Historical Calculating Instruments, September 2010, Leiden, pages 157–164, 2010.

[120] Roland Wielen and Ute Wielen. Die Reglements und Statuten des Astronomischen Rechen-Instituts und zugehörige Schriftstücke im Archiv des Instituts. Edition der Dokumente. Heidelberg: Astronomisches Rechen-Institut, 2011.

[pp. 255–258 on some archives on Peters]

29

(31)

Astronomisches Rechen-Institut, 2012.

photographs]

[122] Roland Wielen, Ute Wielen, Herbert Hefele, and Inge Heinrich. Die Geschichte der Bibliothek des Astronomischen Rechen-Instituts. Heidelberg: Astronomisches Rechen-Institut, 2014.

[various information on Peters]

[123] Roland Wielen, Ute Wielen, Herbert Hefele, and Inge Heinrich. Supplement zur Geschichte der Bibliothek des Astronomischen Rechen-Instituts. Heidelberg:

Astronomisches Rechen-Institut, 2014.

[lists several of Peters’s tables]

30

(32)