Denis Roegel
This document is part of the LOCOMAT project: http://locomat.loria.fr
All the work done by Briggs and Vlacq was, as it were, work done for all time.
J.W.L. Glaisher, 1872.
1 From Briggs to Vlacq (1624–1628) In 1628, Adriaan Vlacq published his Arithmetica logarithmica [101], which was the found- ing stone of most later tables of logarithms for 300 years. In order to understand what exactly Vlacq has done, it is necessary to go back a few years.
1.1 Briggs’ Arithmetica logarithmica (1624)
Briggs’ Arithmetica logarithmica [12], the first extensive table of decimal logarithms, was started in 1616 or 1617 by Henry Briggs (1561–1631)1 and published in 1624. Briggs first published a table of the logarithms of the first 1000 integers in 1617 to 14 places and the Arithmetica logarithmica gave the logarithms of all integers from 1 to 20000 and from 90001 to 100000 with 14 decimals. Briggs did not cover the whole range 1–100000 and exhorted others to fill the gap. The chapter 13 of Briggs’ Arithmetica logarithmica gave in particular a method for filling the range 20001–90000 [67].
1.2 The De Decker-Vlacq tables (1626)
The early involvement of Adriaan Vlacq (ca. 1600–1667) in tables of logarithms is some- what obscure. Initially, Vlacq was not a publisher. For our subject, and from the vantage point of publication, things start in 1626, when Ezechiel de Decker (ca. 1603–ca. 1647),2 a Dutch surveyor, published a book containing translations of works by Napier and Stevin [18]. The title of the book was “Eerste deel van de nieuwe telkonst .”3 Contrary to De Decker, Vlacq had money and a command of Latin, and his involvement in this book must have been for the translation of the works from Latin into Dutch. Vlacq certainly also provided the funding, but he was not responsible for the mathematical content of the book. Still in 1626, De Decker alone published a table of the logarithms of the numbers from 1 to 10000, the Nieuwe telkonst [19], copied from Briggs [33], but reduced to 10 places [46, pp. 48–49].4 This book also contained Gunter’s table of logarithms [43]. In
1Briggs was baptized on February 23, 1560 (old style), which is 1561 new style. He died on January 26, 1630 (old style), which is 1631 new style [50].
2A few biographical elements on De Decker are given by Smeur [20, p. 10]. 3A description of this work can be found in Henderson’s index of tables [46, p. 51–52]. According to
Smeur (preface to [20]), only five copies were known by 1964. Smeur’s introduction contains a facsimile of the first page of the Eerste deel .
4The layout of this table was apparently different from that of Briggs’ table, but also from the 1628 Arithmetica logarithmica. According to Bierens de Haan, it had 60 logarithms per page, on two columns [7, p. 205]. According to Glaisher, every copy of Wells’ Sciographia published in 1635 [107] contains the Nieuwe telkonst bound at the end (and was meant to), but almost always without its title page [33, p. 292]. Gunter’s table, which is part of the Nieuwe telkonst , is however not exactly appended to the Sciographia. Instead, an extension of Gunter’s table, with secants (Gunter had only logarithms of sines and tangents) was added [33, p. 296].
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the preface, De Decker explained that he has published this work because the printing of the larger work (Groote Werck) must occupy some time [33, p. 293].
1.3 De Decker’s Tweede deel (1627)
The “larger work” was of course the table of the logarithms from 1 to 100000, filling the gap left by Briggs in his Arithmetica logarithmica and De Decker had planned to publish it in Dutch. The promised work appeared as soon as 1627, with a preface dated 2 October 1627. Only very few copies of it are known [20].5 It is the “Tweede deel van de nieuwe telkonst .”6 The Tweede deel contains 36 pages of introduction, in Dutch, followed by tables, with the Dutch title Tafel der logarithmi voor de ghetallen van 1 af tot 100000 . A partial facsimile (entire text, but only two pages of tables) was published in 1964 [20]. From Decker’s 1626 table from 1 to 10000, and the Tweede deel , it is clear that the choice to reduce the number of places to 10 was made as early as 1626, but it is hard to attribute the choice to Vlacq or De Decker alone.
In the Tweede deel , De Decker wrote that Vlacq had done the largest part of the work, but the fact that De Decker remains sole author certainly confirms that De Decker was at the origin of the project.
At the end of his introduction, De Decker annouced a reissue of the Latin text, as well as French and Dutch translations.
5The first copy which was widely publicized was discovered in 1920 by Marius van Haaften in the library of the Life Insurance Society “De Utrecht” in Utrecht [83, 84]. This finding was also reported in other articles [85, 86, 87, 46, 22, 27]. This insurance company was founded in 1883 and its historic library now (2010) contains about 12000 titles. The Tweede deel is still part of the library which is now situated in the head office of ASR Verzekeringen N.V. (ASR Insurance Group), Archimedeslaan 10, 3584 BA Utrecht, The Netherlands. In 2011, the library will be physically transferred to the NEHA (Dutch Economic Historical Institute). The journey from “De Utrecht” to ASR goes as follows: in 1920 the A.M.E.V. group was founded, an insurance group with the primary company “De Utrecht.” In 1985 it was decided to introduce the name AMEV also in the working companies/products of the group. “De Utrecht” was then called AMEV Life Insurance Company. In 1990 AMEV Group merged with AG Group in Belgium, which resulted in Fortis Group. In 2000 the (still operating by the name) AMEV companies in the Netherlands merged with the Stad Rotterdam companies. The holding company in the Netherlands became ASR Group. In 2005 the working companies introduced the name Fortis e.g. Fortis ASR Life Insurance Company. Due to the disintegration of Fortis in 2008 the insurance activities in the Netherlands are now part of the ASR Insurance Group (100% owned by the Dutch State). Meanwhile the library was managed independently, since about 1975 through a Foundation, whose chairman is now (2010) Hans de Jongh. (Informations on the history of the companies kindly provided by Mr. de Jongh.) Another copy of the Tweede deel —perhaps the one discovered in 1920— had already been seen by
Gieswald around 1856, 64 years before van Haaften’s discovery [31, p. 15]. Gieswald had recognized that it anticipated Vlacq’s table, but his note was hidden in an important article on Bürgi’s tables and may therefore have gone unnoticed. Gieswald had however not noticed that the tables were exactly the same as in the Arithmetica logarithmica. In 1941, Fletcher reported a French edition of Vlacq’s Arithmetica logarithmica bound with pages 1–36 of the Tweede deel and located at the Harald Cohen Library of the University of Liverpool (call number SPEC RYL.P.2.12) [27].
6See Henderson for a description (including a reproduction of the first page) of these tables [46, pp. 52–54].
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1.4 Vlacq’s Arithmetica logarithmica (1628)
Two years ago, more or less, there came into my hand, I know not by what chance, the Arithmetica Logarithmica of Dr. Briggs
Vlacq, 1628 [56, p. 173]
Finally, in 1628, Vlacq published his Arithmetica logarithmica, whose tables are identi- cal to De Decker’s 1627 Tweede deel , but without any reference to him. Vlacq’s Arithme- tica logarithmica was published in two different versions, with Latin [101] and French [100] introductions. According to Briggs, Vlacq had 1000 copies printed, in Latin, French and Dutch, but no Dutch copy is known other than the Tweede deel . Vlacq presented his Arithmetica logarithmica as a second edition of Briggs’ table for obvious commercial rea- sons.7 The original table had been extended from 1 to 100000, but cut to 10 decimals, as De Decker did in 1626 for the table of logarithms from 1 to 10000 [101].8
Vlacq’s introduction is almost the same as Briggs’, except that Vlacq dropped Briggs’ important chapters 12 and 13. Chapter 12 was on the interpolation with second dif- ferences, and chapter 13 on the interpolation using quinquisection. The importance of chapter 13 lies in its potential use to fill the gap from 20001 to 90000, but chapter 12 is of no real use for this task. It is therefore not totally clear why Vlacq dropped these two chapters. One reason is of course that by omitting them, the work to fill the gap left by Briggs seems much more important than it really is. Another reason is that once the gap was filled, these two chapters, and especially chapter 13, were no longer needed. Time pressure would also not account for dropping the chapters, especially in the Latin version where the text does not differ from Briggs’. On the other hand, there are other chapters that Vlacq could have dropped, since they only made sense when building the initial table. It therefore seems that there is no good explanation for this quirk, other than perhaps the desire to conceal the details of the making of the complete table and Vlacq’s wrong assumption that chapter 12 was instrumental for that purpose.
In any case, for almost 300 years, until the beginning of the 20th century, Vlacq was considered as the sole author of the tables he published. When Glaisher and Bierens de Haan wrote about De Decker, they either viewed the Arithmetica logarithmica as the announced Groote Werck ,9 or as a different initiative by Vlacq that made De Decker’s obsolete, but it was still felt strange that De Decker’s name was not mentioned in it. In fact, Vlacq’s mathematical fame rests on his tables of logarithms.
Finally, Vlacq was given the honors of the 300th anniversary of his tables in 1933 in Rio de Janeiro [56, pp. 163–164], somewhat paralleling Napier’s tercentenary in 1914.
7The Latin copy from the Complutense University of Madrid is available on google books (id: -XR-2oDEb9oC), but it only contains the Latin introduction, the trigonometric supplement, and the fragment of the logarithms of numbers which was appended to Vlacq’s Trigonometria artificialis. This may be the result of a binding error. The French edition (from the university of Lausanne) is also available on google books (id: vSc_AAAAcAAJ) and contains the complete tables.
8There are numerous descriptions of these tables, going into varying details. A good description can be found in Henderson’s catalogue [46, pp. 55–57]. Our recomputation gives a fairly accurate idea of the tabular part.
9This was also the case of Cantor for whom De Decker’s Nieuwe telkonst and Vlacq’s Arithmetica logarithmica are two volumes of the same work, where the authors split the authorship [17].
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1.5 The case of plagiarism and theft
Vlacq obviously plagiarized part of Henry Briggs’ work and stole most of the work from the real author, De Decker. But why is De Decker not mentioned in the Arithmetica logarithmica, when in fact the tables were published under De Decker’s sole name in 1627?
When Glaisher wrote about it in 1872, De Decker’s Tweede deel had not yet surfaced, and Glaisher considered that Vlacq’s silence of any reference to De Decker’s work pub- lished in 1626 and projects was a proof that Vlacq quarrelled with De Decker, or at least had a set purpose to ignore his book entirely [33, p. 294]. However, at that time, the picture was not yet clear, since Glaisher did not know that the tables had entirely been published one year earlier.
At about the same time, and still before De Decker’s 1627 book was discovered, Bierens de Haan started to write about Vlacq and De Decker, and held that Vlacq was the first to publish decimal logarithms on the continent, but this proved wrong [11].
By the beginning of the 20th century, everybody, including Glaisher and Bierens de Haan, naturally considered that Vlacq was the author of the 1628 Arithmetica logarith- mica.
This perspective changed in 1920 when it appeared that Vlacq took credit for work which was not his. Additional documents were found in the Gouda archives in 1938 [22], so that the picture is now fairly clear. A good summary of the new perspective was given by Leo Miller in his article on Milton and Vlacq [56] and additional elements are given by Smeur [20] and Bruins [15].
The facts, as they are now known, are as follows, according to Miller [56]. In 1625, Vlacq was acquainted with Ezechiel De Decker, who was then a young surveyor and teacher of mathematics. Miller assumes that De Decker was the first in Gouda to become aware of Napier’s and Briggs’ work and had wanted to publish it in Dutch. De Decker’s problem was that he needed to translate Briggs’ work in Dutch and get the work printed, but he knew little Latin and also had little money. Adriaan Vlacq instead was skilled in Latin, and had the money.
The rest of the story unfolds as follows.10 At the end of 1625, we have three contracts related to the upcoming publications:
1. on 24 December 1625, the States General granted to Adriaan Vlacq the copyright of the forthcoming book Nieuwe telkonst for ten years [22, p. 313], [56, p. 165], [15, p. 251];
2. then on 27 December 1625, a contract was settled between Adriaan Vlacq on one side, and Ezechiel De Decker and Lourens Borreman(s) on the other side, both parties agreeing to put out a book giving in particular the logarithms of all prime11 numbers from 20000 to 90000; this was a publishing contract, where Vlacq was paying De Decker a specified amount of money, and where Borreman(s) appears as a reserve force [22, p. 312], [56, p. 165], [15, p. 251];
10Many of the details of the Vlacq-De Decker contracts were only published in Dutch in journals of limited circulation, and many later authors are still unaware of the findings of van Haaften and Endenburg. We have tried to give the references to all Dutch articles at the end of this introduction.
11In his account of the events, Miller did omit the restriction to prime numbers [56, p. 165] .
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3. and finally on 29 December 1625, there was a similar contract between Vlacq and the printer Pieter Rammaseijn.
Initially, only one book was planned. But the task was probably greater than expected and in September 1626, two books were published, both under the name of De Decker: first, the Eerste deel van de nieuwe telkonst [18], then the Nieuwe telkonst [19], containing the logarithms of the numbers 1 to 10000 (copied and shortened from Briggs), and which could be considered a substitute for the upcoming Tweede deel which was going to be delayed. Vlacq and Decker were obviously racing to get the work out before Briggs would publish his.
One year later, on 31 December 1626, probably since the work was going on too slowly for Vlacq, there was a notice of a threatened lawsuit [22, pp. 316–317], [56, p. 170], [15, p. 253]. Vlacq was certainly worried and wanted to alter the contract. First, Vlacq asked De Decker whether it would not be a better idea to compute the logarithms of all numbers between 20000 and 90000, and not only the logarithms of the prime numbers, as initially planned. Second, if De Decker did not agree, Vlacq offered to complete one half of the table before May 1st, 1627, and the other half then had to be completed by De Decker and Borreman(s) at the same time. Third, if De Decker refused these conditions, Vlacq would be satisfied that he left the work, with no salary, so that Vlacq could have the work done by others. From this contract, it is clear that Vlacq wanted to publish his table first, and that commercial interests12 eventually superseded the interests of the initial author, De Decker.
Eventually the computation was done and the Tweede deel was completed. The preface was dated 2 October 1627. In this work, although Vlacq had apparently done part of the work, and actually even more than the half, De Decker acknowledges Vlacq’s computation, but remains the sole author. For Bruins, on the other hand, De Decker did all the work but did not use the planned method.
There is of course still a gap in the story, in that this still does not explain why the Tweede deel has almost totally vanished, and why Vlacq has totally dropped De Decker’s name in the Arithmetica logarithmica (1628). It seems obvious that there was some disagreement between them, perhaps based on De Decker’s wanting to remain the sole author of the tables and Vlacq having only commercial interests. The title page of the Tweede deel may have been designed without Vlacq being aware of it, and Vlacq may then have disagreed with it. And perhaps De Decker then agreed to abandon the authorship against some money, but these are all pure speculations. Perhaps the contract was made void because De Decker did not finish the work before May 1st, 1627, and somehow Vlacq must have taken the opportunity to take possession of the authorship, and ignore De Decker altogether.
Eventually, De Decker continued to publish other works on his own, in particular Stevin’s Thiende in 1630. This work was not in Vlacq’s copyright, and De Decker was
12As van Haaften wrote in 1941, by publishing his table in 1628, Vlacq obtained two advantages: first, by taking almost the same title as Briggs’, his book became attractive to readers, and second, it prevented Briggs from publishing a second edition on his side, and therefore thwarted competition [89, pp. 63–64]. This is in contrast with earlier opinions, especially by Bierens de Haan, who considered that Vlacq was not boasting, and in fact a “very modest man.” [8, p. 374]. For Bruins, Vlacq was the one who had the idea of raiding Briggs [15]. This is certainly true, but I would then also add that Vlacq certainly had also the idea of raiding De Decker.
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therefore not bound by it [56, p. 168]. But Vlacq never mentionned De Decker’s participation and several times claimed that
he had the idea to make the tables and that he alone computed them [56].
1.6 The fate of De Decker’s tables
Although the 36 pages-introduction of De Decker’s Tweede deel has almost completely vanished, the tables themselves were obviously not lost.
A mere examination of the printer’s signatures confirms that the 1627 and 1628 tables are identical. It is therefore most likely that the 1627 tables were reused by Vlacq, although they might have been reprinted (but not reset).13 The same printer plates were used for both tables.
An additional problem is the delay between the Tweede deel and the Arithmetica lo- garithmica. This delay can easily be explained by the preparation of the Latin (almost identical to Briggs’) and French introductions, and by the computation of the trigono- metric parts.
Moreover, in 1631 George Miller published another table of logarithms of the num- bers 1 to 100000 to 10 places, a table that Miller asserted to be a reprint of Vlacq’s 1628 book [102].14 However, all seems to indicate that the tables in Vlacq’s Arithmetica loga- rithmica and Miller’s book are actually De Decker’s 1627 tables, merely stripped of every mention of De Decker. Miller’s table actually uses the Dutch titles (the pages introducing the tables),15 and these tables are therefore certainly all or part of those projected by Vlacq for the Dutch translation, but which was apparently not published. According to Leo Miller [56, p. 157], George Miller’s editions vary widely.
1.7 Vlacq’s later career
In 1633, Vlacq moved to London, perhaps in order to exploit the publication of his tables in England. He lived in London from 1633 to 1643. He then settled in Paris, where he published the second edition of Grotius’ De imperio in 1648 [42, p. 87]. In 1652, he took his residence in The Hague where he published in particular works of Huygens, Gassendi, and Boyle.16
13This, however, is extremely unlikely, because it would have meant for the printer to keep the plates, which take a lot of space, are very heavy, and it would have prevented the metal of being used for other books.
14See Glaisher [34, pp. 256–257] and Henderson [46, pp. 58–59] for a description of these tables. 15Moreover, Miller’s book uses the Dutch title from the Tweede deel , namely Tafel der logarithmi voor
de ghetallen van 1 af tot 100000 , and not the translation from titles used in the Latin and French editions of the Arithmetica logarithmica. There is therefore no doubt that Miller’s edition of the tables is De Decker’s, and that there was more than one copy of the Tweede deel .
16A somewhat edulcorated autobiography by Vlacq is reproduced in the 2001 edition of Grotius’ De imperio and this autobiography has a paragraph on Vlacq’s involvement in tables of logarithms [42, pp. 995–1000]. See also [33, pp. 299–300].
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2 Briggs’ plans Henry Briggs initially had in mind the completion of his table to 14 decimals [46, p. 54], and he was certainly unhappy by Vlacq’s publication, which destroyed his efforts.17 It is possible that Briggs’ had actually almost finished the computations of his complete table, but there was no point going to the printer anymore. Briggs must then have switched to the work on the logarithms of trigonometric functions, the Trigonometria britannica, which was published posthumously in 1633.
Glaisher [36, pp. 380–381] cites an excerpt from a letter dated October 25th, 1628 from Briggs to Pell in which Briggs mentions his projects and how they were altered by Vlacq:18
My desire was to have those Chiliades that are wantinge betwixt 20 and 90 cal- culated and printed, and I had done them all almost by my selfe, and by some frendes whom my rules had sufficiently informed, and by agreement the busi- ness was conveniently parted amongst us; but I am eased of that charge and care by one Adrian Vlacque, an Hollander, who hathe done all the whole hun- dred chiliades and printed them in Latin, Dutche, and Frenche, 1000 bookes in these 3 languages, and hathe sould them almost all. But he hathe cutt of 4 of my figures throughout; and hathe left out my Dedication, and to the reader, and two chapters the 12 and 13, in the rest he hathe not varied from me at all.
In 1631, Richard Norwood wrote, referring to Miller’s 1631 edition of Vlacq’s tables:
And here I have just occasion to blame the ill dealings of these men, not only towards my selfe, as I have before noted in the Preface to the Reader, but likewise towards Mr. Briggs, both in the matter before mentioned, and in printing a second edition of his Arithmetica Logarithmica in Latine, whilest he lived, against his minde and liking: and brought them over to sell, when the first were unsolde; so frustrating those additions which Mr. Briggs intended in his second Edition, and moreover leaving out some things, that were in the first edition, of special moment. A practice of very ill consequence, and tending to the great discouragement of such as take paines in this kind.19
According to Hallowes, Briggs must have felt that he could not complete both the tables of the logarithms of numbers and the logarithms of trigonometric functions, that the completion of the Arithmetica logarithmica to 14 places had been reduced to a certain routine, and he must then have turned to the trigonometric functions exclusively [45, p. 85].
Briggs obviously was not deterred by the reduction of accuracy introduced by Vlacq, and must still have believed that 14 place logarithms were useful, as testified by the Trigonometria britannica.
17On the other hand, if Briggs’ complete table had not been interrupted, would the Trigonometria britannica have been completed?
18The full letter is reproduced in Halliwell’s Rara mathematica [44, pp. 55–57]. 19[57], cited in [56, p. 172]
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3 De Decker and Vlacq’s computation methods Neither De Decker, nor Vlacq have given details about the actual methods used to com- pute their table, but we know from the notarial contracts that De Decker started to compute the logarithms of primes. This computation was probably far from finished by the end of 1626, and this caused a change in the methods employed for the calculation. The cause of this change was obviously the race between Vlacq or De Decker and Briggs, and the only way by which they could outwin Briggs was not only to reduce the accuracy of the tables, but also to use only Briggs’ values as a basis.
Before considering the computation of the logarithms of primes, we can review some methods that might have been employed by De Decker.
3.1 Interpolation with Briggs’ second differences method
We consider first the accuracy of finding the logarithms of 20001–20005 and 89996–89999 (the extreme values to be computed) using the values of the logarithms of 20000, 20005, ..., 89995, 90000, ..., which are themselves easily obtained from Briggs’ table. The problem is then reduced to the interpolation of four values between log 4000 and log 4001, and between log 17999 and log 18000, and to the addition of log 5. Briggs’ second differences method is akin to the use of Newton’s forward difference formula
zx = A+ x1 + x · x− 1
2 2 + x · x− 1
2 · x− 2
3 3 + · · · ,
the differences n being forward differences. If the second differences are constant, this expression reduces to z = A + x1 + x · x−1
2 2. We can take x = 0, 1
5 , 2
5 1
In chapter 12 of his Arithmetica logarithmica, Briggs gave examples of interpolating nine values between log(2n) and log(2n+ 10) and by taking every other value, we readily can interpolate four values between log(2n) and log(2n+10), and therefore between log(n) and log(n+ 5). All the necessary material is given in Arithmetica logarithmica.
More precisely, Briggs gave the values of z0.1 − z0.0, z0.2 − z0.1, etc., as functions of the first difference 1 and the second difference 2. De Decker might immediately have computed z0.2, z0.4, etc., using Briggs’s expressions, probably simplifying them. Finally, he would have added log 5.
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log 4000 = 3.60205 99913 2796
log 4001 = 3.60216 85513 7900
log 4002 = 3.60227 70843 0019
log 17999 = 4.25524 83769 6186
log 18000 = 4.25527 25051 0331
log 18001 = 4.25529 66319 0434
log 5 = 0.69897 00043 3602
the values being here correctly rounded. Briggs’ values may slightly differ from these values, of course.
De Decker may then have computed (using the above rounded values)
log 20000 = log 4000 + log 5 = 4.30102 99956 6398
log 20005 = log 4001 + log 5 = 4.30113 85557 1502
log 20010 = log 4002 + log 5 = 4.30124 70886 3621
log 89995 = log 17999 + log 5 = 4.95421 83812 9788
log 90000 = log 18000 + log 5 = 4.95424 25094 3933
log 90005 = log 18001 + log 5 = 4.95426 66362 4036
The first three values happen to be correctly rounded, but the last one if off by one unit of the last decimal place.
For the 20000–20005 interval, the differences are (in units of the 14th place and using the above rounded values)20
1 1 = 10 85600 5104
1 2 = 10 85329 2119
2 = −271 2985
z0,4 = 4.30107 34229 3998
z0,6 = 4.30109 51349 5019
z0,8 = 4.30111 68458 7520
20An earlier version of this introduction gave the wrong sign for 2, resulting in less accurate inter- polations, which led us to conclude erroneously that the use of second differences was not sufficient to achieve the accuracy of Vlacq’s tables. We thank Peter Eberhard, from Zittau, Germany, for noticing our error.
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log 20001 = 4.30105 17098 4523
log 20002 = 4.30107 34229 4084
log 20003 = 4.30109 51349 5094
log 20004 = 4.30111 68458 7563
The error is at most about 8 · 10−13. This shows that the interpolation with second differences is sufficient to obtain 10 correct decimals on that interval, but it is not for filling Briggs’ Arithmetica logarithmica to 14 places.
For the 89995–90000 interval, the differences are (in units of the 14th place and using the above rounded values)
1 1 = 2 41281 4145
1 2 = 2 41268 0103
2 = −13 4042
log 89996 = 4.95422 32070 3341
log 89997 = 4.95422 80327 1531
log 89998 = 4.95423 28583 4360
log 89999 = 4.95423 76839 1827
We now have about 14 correct places. In both cases, the interpolation could have been done either on the final interval (e.g. 20000–20005) or first on the reduced interval (in that case 4000–4001). The result would have been the same.
3.2 Interpolation with third or fourth differences
The accuracy would have been even better if the interpolations were using third differ- ences (which was not described by Briggs), or Briggs’s quinquisection, which is actually equivalent to the use of an interpolation with fourth differences. But the use of second differences is more than enough, and was clearly described by Briggs, and there is no reason to consider that De Decker and Vlacq used another method.
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3.3 Logarithms of primes
Another approach is to compute the logarithms of primes on the interval 20000–90000, the logarithms of all other values being then obtained from the logarithms of prime numbers. We know from De Decker and Vlacq’s contract that De Decker was supposed to compute the logarithms of the primes, but that this was later changed.
There are 2262 primes smaller than 20000 and 8713 primes smaller than 90000 which makes 6451 primes between 20000 and 90000. De Decker might have computed the logarithms of all these prime numbers, but this would have been a great amount of work.
Henderson remarks that De Decker and Vlacq did a lot less work than Briggs [46, p. 54]. According to him, the primes between 22500 and 50000 could be found using log 2, log 3, log 4 and the logarithms of the numbers 90000–100000 found in Briggs’ tables. However, Henderson must have made a mistake, since only the logarithms of the primes in the intervals 22500–25000, 30000–33333, and 45000–50000 can be obtained easily that way. For instance, a prime number around 40000 cannot have any of its multiples in the interval 90000–100000. Nevertheless, this would have saved the computation of the logarithms of a number of primes.
So, eventually, De Decker would have had to compute the logarithms of the primes in the intervals 20000–22500, 25000–30000, 33333–45000, and 50000–90000, that is a total of 3593 primes. We can assume that De Decker did compute some of them, perhaps the half, but that Vlacq then wanted to change the contract and impose delays.
Bruins [15] suggests that under Vlacq’s pressure, De Decker only computed the log- arithms of odd numbers by interpolation. Indeed, it is easy to see that we have (with M = 1
ln 10 )
2 =
1
2(n2 + 2n)
This expression gives the difference between the logarithm of an odd number and the arithmetic mean of the logarithms of the surrounding even numbers, and log 2n would have been obtained from log 2 and log n. Such a computation would in fact only have been required for the 6451 prime numbers, since the logarithms of odd numbers which are not prime can be obtained from other primes.
It is however far from certain that De Decker has reached this formula, especially since our equations are anachronistic.
As an illustration of this method, we can consider two examples, say the computation of the first and the last prime numbers of the 20000–90000 interval. The first is 20011 and the last is 89989. For 20011, De Decker may have computed log 20010 = log 2 + log 10005 and log 20012 = log 2 + log 10006 using Briggs’ 14 place values. For 89989, he may have computed log 89988 using log 44994 computed earlier, or even using a further decomposition to rely only on Briggs’ values. The same applies for log 89990. Let us
13
assume that De Decker went back to Briggs’ values in both cases. Assuming correct 14 place values, he would have found:
log 2 ≈ 0.30102 99956 6398
log 10005 ≈ 4.00021 70929 7223
log 10006 ≈ 4.00026 04985 4739
log 20010 ≈ 4.30124 70886 3621
log 20012 ≈ 4.30129 04942 1137
log 7499 ≈ 3.87500 33536 0004
log 12 ≈ 1.07918 12460 4762
log 89988 ≈ 4.95418 45996 4766
log 8999 ≈ 3.95419 42518 1586
log 10 = 1.00000 00000 0000
log 89990 ≈ 4.95419 42518 1586
Then:
δ(89989) ≈ M
From this it comes (rounded to 10 places)
log 20011 ≈ log 20010 + log 20012
2 + 5.4 · 10−10 ≈ 4.30126 87920
log 89989 ≈ log 89988 + log 89990
2 + 0.27 · 10−10 ≈ 4.95418 94258
and these two values are the exact values. In general, however, this process will introduce errors, and it would have been safer for De Decker to keep the original 14-place accuracy and to round the values only at the end. On the other hand, if 14-places had been used throughout following Briggs’ scheme, De Decker and Vlacq could certainly have filled the canon to Briggs’ accuracy. One must therefore assume that De Decker and Vlacq’s computations were not accurate enough to secure 14 places.
According to Bruins, the above computations could have been done in 100 hours by Decker. In any case, even if it took longer, the computations were not very difficult if done that way. Henderson considers that a good computer could have done the computation of the logarithms of primes in six months, and in fact it probably took less than a year.
3.4 Settling the matter
In order to solve the question of the method used by De Decker and Vlacq, one should first check if the logarithms of numbers 5n between 20001 and 90000 have been obtained
14
using log n and log 5, or some other similar relation. It is of course possible that the underlying structure of the tables varies and that the same methods have not been used throughout. The accuracy of log n where n is prime should be checked.
Finally, if it is deemed that interpolation was used, the errors in the interpolations might help understanding which kind of interpolation was used.
Such a study will hopefully be possible soon, using the digital versions of Vlacq’s Arithmetica logarithmica which are now widely available.
4 Vlacq-based tables Almost all the tables of logarithms published after Vlacq’s Arithmetica logarithmica and Trigonometria artificialis until the 20th century were based on them, with various cor- rections, extensions, or simplifications.
As a most curious example, the 1628 Arithmetica logarithmica and Vlacq’s 1633 trigonometric table were even used as a basis for the Chinese tables of logarithms pub- lished around 1720 [71].
Probably the most popular such tables were those of Vega [4, 26], which went into many editions. Vega’s great Thesaurus logarithmorum completus [99] published in 1794 gave the logarithms to 10 places, borrowing them from Vlacq from 1 to 100000, and perhaps from Briggs from 100000 to 101000. The logarithms of trigonometric functions were taken from Vlacq’s Trigonometria artificialis , only extending them for the first degree of the trigonometric functions.
5 The tables’ errors Vlacq’s table, like that of Briggs, contained numerous errors, and the original table was supplemented with an errata. Several later authors also added their errata, so that there are many overlapping erratas. An errata was also given by Lefort in 1858 [53, pp. 148– 150] and Lefort used Prony’s Tables du cadastre as a means of comparison [72]. For errors in Vlacq’s tables, see also [34], who gives in particular an errata of Lefort’s errata table (!). [32] is an addition to the previous article. According to Glaisher, the logarithms of integers from 1 to 100000 contain about 300 errors, exclusive of errors of one unit in the last figure [28, p. 925]. In 1910, before the publication of his own tables, Henri Andoyer wrote that the last place of Vlacq’ tables was usually wrong and that the error could reach six units [1].
Many of Vlacq’s errors are also found in Vega’s table [99] and in other tables based on Vlacq’s.21 Andoyer writes for instance that the last place of Vega’s tables is usually wrong [1]. An overview of the erratas in Vega’s and Vlacq’s tables is given by Fletcher et al. [28, pp. 919–925].
Using his own table of logarithms as a comparison, and given that his computations had reached 300000 by 1874, Sang looked for errors in Vlacq’s table in the interval 20000– 30000. Every tenth value in Sang’s table had to agree with a value in Vlacq’s table, but
21On the propagation and persistence of certain errors in Vlacq’s and other tables, see in particular Babbage [5, 65].
15
Sang found 42 errors [74]. Almost all of these errors are unit errors on the 10th place. However, as observed in an article in Nature [3], most of the errors found by Sang had already been found by Lefort, but this was not known by Sang at that time.
Apart from the Briggs-Vlacq tables, there are very few extensive independent cal- culations of logarithms. The only works of some extent are those of Prony (Tables du cadastre) [72] and by Sang [63]. One might consider using one table to check another one. However, Edward Sang was very critical towards the usefulness of the Tables du cadastre for checking the tables of Briggs or Vlacq [76]. Assuming that the errors in the logarithms of the Tables du cadastre are smaller than 5·10−13, the 13th and 14th decimals are therefore uncertain and the values of the Tables du cadastre cannot be used reliably (except in certain cases) to check these decimals in other tables with certainty.
Sang gives seven examples where the values given by the Tables du cadastre lead to an incorrect rounding to 10 places, and therefore cannot always be used with Vlacq’s ta- bles. For instance, using the Tables du cadastre, Lefort concluded that Vlacq’s value for log 26188 should have been 4.41810 23323, because the Tables du cadastre have 4.41810 23322 50 . . .. But in fact Vlacq’s initial value (and also Vega’s) 4.41810 23322 is the correct one, because the exact value is log 26188 = 4.41810 23322 49959 . . . [53]. There are six other similar examples where the Tables du cadastre cannot establish the certainty of the 10th place.
These examples are the following:22
Number Logarithm (20th place rounded)
26188 4.41810 23322 49959 00920 29163 4.46483 21978 49968 31667 30499 4.48428 55999 50010 73882 31735 4.50153 85026 49975 27403 34162 4.53354 32883 50038 92375 34358 4.53602 78753 50011 99957 60096 4.77884 55662 49998 09339
Sang correctly gave the 20 first rounded decimals, using his 28-place table.
6 Extended trigonometric tables
6.1 The appended trigonometric tables
At the end of Vlacq’s Arithmetica logarithmica, there is a short canon of logarithms of trigonometric functions.23 This canon gives the logarithms of the six trigonometric functions to 10 places for every minute of the quadrant. This table was certainly computed by Vlacq who knew apparently enough to be able to compute such tables without De Decker. Like Gunter’s table, this table actually gives the values of 10+log sin, 10+log cos, 10 + log tan, and 10 + log cot. It does not, however, use a radius of 1011 as claimed by Vlacq and copied by others [56, p. 156].
22Sang’s article mistakenly gives the number 34182 instead of 34162. 23This table was not part of the only known copy of the Tweede deel [56, p. 172].
16
This table extends Gunter’s table which gave the logarithms of sines and tangents to 7 places for every minute [43] and it prefigures Vlacq’s 1633 Trigonometria artificialis [103] which gives these logarithms to 10 places every 10 seconds of the quadrant.24 The small table published in 1628 has been reproduced, as soon as 1631 [25].25
This table was presumably computed by Vlacq who only needed to compute the logarithms of sines from 0 to 45. Vlacq explained his method of construction in the preface of the Arithmetica logarithmica. He actually based himself on Pitiscus’ Thesaurus mathematicus [59], computing logarithms of known sines using the radix method.26
If log sinα has been computed for α ≥ 45, then we can use log sinα = log sin 2α + log sin 30 − log sin(90 − α) for α < 45 and the logarithms of tangents and secants can be obtained with
log tanα = log sinα− log sin(90 − α)
log secα = − log sin(90 − α)
Such a method does however lead to an accumulation of errors. Delambre writes that sometimes eight logarithms were used to compute a new one, which causes the last decimal to be sometimes incorrect [21, vol. 2, p. 421].
6.2 Vlacq’s small tables
Vlacq has also published smaller tables of logarithms of trigonometric functions which remained in print well into the nineteenth century [42, p. 88]. They were probably Vlacq’s most popupar works. These tables gave the sines, tangents, secants, log. sines, log. tangents with seven places for every minute of the quadrant and were merely derived from the 1628 tables. These tables also gave the logarithms of numbers from 1 to 10000 with seven places [104]. Such tables were printed in Latin, Dutch, German or French, at least in 1651, 1661, 1665, 1670, 1689, 1699, 1726, 1742, 1748, 1757, 1763, 1775, 1808, and 1821 (all versions on google books), and perhaps also in 1636, 1642, 1663, 1666, 1667, 1673, 1683, 1706, 1721, 1732, 1738, 1767, 1768, 1778, 1790 (all unverified) or other years. The 1808 edition also contained the log. sin and log. tan for every second of the first degree, to seven places [105]. According to Miller [56, pp. 160–161], the first edition of these tables was in 1651 and not 1636, contrary to many accounts.
7 Structure of the tables and recomputation Vlacq’s Arithmetica logarithmica contains an introduction of 80 pages, followed by two tables. The first table gives the logarithms of all integers from 1 to 100000 to 10 places. It is followed by a table of logarithms of the six trigonometric functions for every minute of the quadrant, also to 10 places. This trigonometric table anticipates the table in Vlacq’s Trigonometria artificialis [103].
24The Trigonometria artificialis also contains the logarithms of numbers 1–20000 taken from Vlacq’s Arithmetica logarithmica.
25See Henderson [46, pp. 57–58] and Miller [56, p. 162]. 26This was also observed by Delambre [21, vol. 2, p. 421], Glaisher [37, pp. 442–443] and Wolf [109,
vol. 1, pp. 169–175].
17
These two tables have been recomputed using the GNU mpfrmultiple-precision floating- point library developed at INRIA [29], and the exact values are given. The comparison of our table and Vlacq’s will therefore immediately show where Vlacq’s tables contain errors.
We have tried to be as faithful as possible to the original layout, but apart from the accuracy of the values, there are some minor differences. Vlacq’s first table does for instance not give the difference between log 2 and log 1, and we have filled this gap which was certainly an oversight of the printer. Another small difference is located on the last page of the logarithms of numbers. Vlacq omits the horizontal division line after 99965, but this too seems a printing error and we have left this line. In the second table, Vlacq had log sin 0 = log tan 0 = 0 and we have put Infinitum instead, as well as in the next difference line which had no values where the differences were truly infinite. The first difference of the logarithms of secants was also not given, and we have added it.
We have also added the original signatures of the pages (for both tables), and these signatures make it obvious that the second part of the Trigonometria artificialis is a leftover from the 1628 printing, and in fact even from De Decker’s tables. These signatures can be compared with those of the 1627 tables.
18
References The following list covers the most important references27 related to Vlacq’s tables. 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] Marie Henri Andoyer. Nouvelles Tables trigonométriques fondamentales. Comptes rendus hebdomadaires des séances de l’Académie des sciences, 150:591–592, janvier-juin 1910.
[2] Anonymous. Contributions to scientific bibliography. The Edinburgh journal of science (New series), 5(11):219–222, October 1831.
[3] Anonymous. Note about Edward Sang’s project of computing a nine-figure table of logarithms. Nature, 10:471, 1874. [Issue of 8 October 1874. This note was reproduced in [75].]
[4] Raymond Clare Archibald. (Vega’s tables). Mathematical Tables and other Aids to Computation, 2(16):161–165, 1946.
[5] Charles Babbage. Notice respecting some errors common to many tables of logarithms. Memoirs of the astronomical society of London, 3(1):65–67, 1827.
[6] Charles Babbage. Tables of logarithms of the natural numbers from 1 to 108,000. London: Mawman, 1827. [A complete reconstruction of the tables was made by D. Roegel in 2010 [65].]
[7] David Bierens de Haan. Notice sur des tables logarithmiques hollandaises. Bulletino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche, 6:203–238, May 1873.
[8] David Bierens de Haan. On certain early logarithmic tables. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, Series 4, 45(301):371–376, 1873.
[9] David Bierens de Haan. Adriaan Vlack en Ezechiel De Decker. Verslagen en mededeelingen der Koninklyke Akademie van wetenschappen, Afd. Natuurkunde, 8 (2nd ser.):57–99, 1874.
[10] David Bierens de Haan. Adriaan Vlack en zijne logarithmentafels. Verslagen en mededeelingen der Koninklyke Akademie van wetenschappen, Afd. Natuurkunde, 8 (2nd ser.):163–199, 1874.
27Note 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.
19
[11] David Bierens de Haan. Bouwstoffen voor de geschiedenis der wis- en natuurkundige wetenschappen in de Nederlanden. Amsterdam, 1878. [Available on http://www.dbnl.be/tekst/bier009bouw01_01. Parts 1 and 3 discuss the tables of Vlacq and De Decker.]
[12] Henry Briggs. Arithmetica logarithmica. London: William Jones, 1624. [The tables were reconstructed by D. Roegel in 2010. [67]]
[13] Ian Bruce. The agony and the ecstasy — the development of logarithms by Henry Briggs. The Mathematical Gazette, 86(506):216–227, July 2002.
[14] Evert Marie Bruins. Computation of logarithms by Huygens. Janus, 65:97–104, 1978.
[15] Evert Marie Bruins. On the history of logarithms: Bürgi, Napier, Briggs, de Decker, Vlacq, Huygens. Janus, 67(4):241–260, 1980.
[16] Moritz Cantor. Adriaen Vlacq. In Historische Kommission bei der Bayerischen Akademie der Wissenschaften, editor, Allgemeine Deutsche Biographie, volume 40, page 86. Leipzig: Duncker & Humblot, 1896.
[17] Moritz Cantor. Vorlesungen über Geschichte der Mathematik. Leipzig: B. G. Teubner, 1900. [volume 2, pp. 743–747 on Vlacq]
[18] Ezechiel de Decker. Eerste deel van de nieuwe telkonst, inhoudende verscheyde manieren van rekenen, waerdoor seer licht konnen volbracht worden de geometrische ende arithmetische questien. Gouda: Pieter Rammazeyn, 1626. [not seen]
[19] Ezechiel de Decker. Nieuwe telkonst inhoudende de logarithmi voor de ghetallen beginnende van 1 tot 10000, ghemaeckt van Henrico Briggio Professor van de Geometrie tot Ocxfort. Gouda: Pieter Rammazeyn, 1626. [not seen]
[20] Ezechiel de Decker. Tweede deel van de nieuwe tel-konst ofte wonderliicke konstighe tafel, inhoudende de logarithmi, voor de getallen van 1 tot 100000 toe. Gouda: Pieter Rammazeyn, 1627. [500 partial facsimile copies were printed in 1964 by Nieuwkoop & B. de Graaf with an introduction by Alphonse Johannes Emile Marie Smeur.]
[21] Jean-Baptiste Joseph Delambre. Histoire de l’astronomie moderne. Paris: Veuve Courcier, 1821. [2 volumes, see vol. 1, p. 545, vol. 2, pp. 420–432]
[22] Pieter Johannes Teunus Endenburg. De oudste Nederlandsche logarithmen-tafels en hun makers. Het Boek, 25:311–320, 1939.
[23] Jean-Marie Farey et Patrick Perrin. Les logarithmes de Briggs (1624). In La mémoire des nombres, pages 319–341. IREM de Basse Normandie, 1997. [The same article was also published separately in 1995 [24].]
[24] Jean-Marie Farey and Patrick Perrin. Les logarithmes de Briggs. Repères-IREM, 21:61–77, October 1995. [This is a separate publication of [23].]
20
[26] Gerlinde Faustmann. Jurij Vega — the most internationally distributed logarithm tables. [8 pages, no date, no later than 2004, http://www.rechenschieber.org/vega.pdf]
[27] Alan Fletcher. Early logarithmic works. Nature, 148(3763):728, 1941.
[28] 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]
[29] 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.
[30] David Gibb. A course in interpolation and numerical integration for the mathematical laboratory, volume 2 of Edinburgh Mathematical Tracts. London: G. Bell & sons, Ltd., 1915.
[31] Hermann Robert Gieswald. Justus Byrg als Mathematiker und dessen Einleitung in seine Logarithmen. In Programm Danzig Realgymnasium St. Johannis, pages 1–36. Danzig: Realgymnasium St. Johannis, 1856. [This article contains on page 15 the first known mention of De Decker’s Tweede deel.]
[32] James Whitbread Lee Glaisher. Addition to a paper on errors in Vlacq’s ten-figure logarithms, published in the last number of the Monthly Notice. Monthly Notices of the Royal Astronomical Society, 32(8):288–290, June 1872.
[33] James Whitbread Lee Glaisher. Notice respecting some new facts in the early history of logarithmic tables. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, Series 4, 44(293):291–303, 1872.
[34] James Whitbread Lee Glaisher. On errors in Vlacq’s (often called Brigg’s or Neper’s) tables of ten-figure logarithms of numbers. Monthly Notices of the Royal Astronomical Society, 32(7):255–262, May 1872.
[35] James Whitbread Lee Glaisher. Supplementary remarks on some early logarithmic tables. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, Series 4, 44(296):500–506, 1872.
[36] James Whitbread Lee Glaisher. On early logarithmic tables, and their calculators. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, Series 4, 45(301):376–382, May 1873.
[37] James Whitbread Lee Glaisher. On logarithmic tables. Monthly notices of the Royal Astronomical Society, 33(7):440–458, 1873.
[38] James Whitbread Lee Glaisher. On the progress to accuracy of logarithmic tables. Monthly Notices of the Royal Astronomical Society, 33:330–345, 1873.
21
[39] 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.]
[40] Herman Heine Goldstine. A history of numerical analysis from the 16th through the 19th century. New York: Springer, 1977.
[41] G. Govi. Rapport sur l’utilité des tables de logarithmes à plus de sept décimales ; à propos d’un projet publié par M. Sang. Atti della Reale Accademia di Scienze di Torino, 8:157–170, 1873. [Reprinted in [73].]
[42] Hugo Grotius. De imperio summarum potestatum circa sacra. Leiden: Brill, 2001. [Introduction, English translation and commentary by Harm-Jan Van Dam, two volumes]
[43] Edmund Gunter. Canon triangulorum, sive tabulæ sinuum et tangentium artificialium ad radium 10000,0000.& ad scrupula prima quadrantis. London: William Jones, 1620. [The tables were reconstructed by D. Roegel in 2010. [66]]
[44] James Orchard Halliwell, editor. Rara mathematica; or, a collection of treatises on the mathematics and subjects connected with them, from ancient inedited manuscripts. London: John William Parker, 1839.
[45] D. M. Hallowes. Henry Briggs, mathematician. Transactions of the Halifax Antiquarian Society, pages 79–92, 1962.
[46] 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.
[47] Charles Hutton. Mathematical tables: containing common, hyperbolic, and logistic logarithms, also sines, tangents, secants, and versed-sines, etc. London: G. G. J., J. Robinson, and R. Baldwin, 1785.
[48] G. Huxley. Briggs, Henry. In Charles Coulston Gillispie, editor, Dictionary of Scientific Biography, volume 2, pages 461–463. New York: Charles Scribner’s Sons.
[49] Graham Jagger. The making of logarithm tables. In Martin Campbell-Kelly, Mary Croarken, Raymond Flood, and Eleanor Robson, editors, The history of mathematical tables: from Sumer to spreadsheets, pages 48–77. Oxford: Oxford University Press, 2003.
[50] Graham Jagger. The will of Henry Briggs. BSHM Bulletin: Journal of the British Society for the History of Mathematics, 21(2):127–131, July 2006.
[51] Johannes Kepler, John Napier, and Henry Briggs. Les milles logarithmes ; etc. Bordeaux: Jean Peyroux, 1993. [French translation of Kepler’s tables and Neper’s descriptio by Jean Peyroux.]
22
[52] E. F. Kossmann. De boekhandel te ’s-Gravenhage tot het eind van de 18de eeuw. Biographisch woordenboek van Boekverkopers, Uitgevers, Boekdrukkers, Boekbinders enz. Met vermelding van hun uitgaven en de veilingen door hen gehouden. ’s-Gravenhage, 1937. [not seen]
[53] Pierre Alexandre Francisque Lefort. Description des grandes tables logarithmiques et trigonométriques, calculées au bureau du cadastre, sous la direction de Prony, et exposition des méthodes et procédés mis en usage pour leur construction. Annales de l’Observatoire impérial de Paris, 4 (supplément):123–150, 1858.
[54] Frédéric Maurice. Mémoire sur les interpolations, contenant surtout, avec une exposition fort simple de leur théorie, dans ce qu’elle a de plus utile pour les applications, la démonstration générale et complète de la méthode de quinti-section de Briggs et de celle de Mouton, quand les indices sont équidifférents, et du procédé exposé par Newton, dans ses Principes, quand les indices sont quelconques. In Additions à la Connaissance des temps ou des mouvements célestes, à l’usage des astronomes et des navigateurs, pour l’an 1847, pages 181–222. Paris: Bachelier, 1844. [A summary is given in the Comptes rendus hebdomadaires des séances de l’Académie des sciences, 19(2), 8 July 1844, pp. 81–85, and the entire article is translated in the Journal of the Institute of Actuaries and Assurance Magazine, volume 14, 1869, pp. 1–36.]
[55] Erik Meijering. A chronology of interpolation: from ancient astronomy to modern signal and image processing. Proceedings of the IEEE, 90(3):319–342, March 2002.
[56] Leo Miller. Milton and Vlacq. Papers of the Bibliographical Society of America, 73:145–208, 1979.
[57] Richard Norwood. Trigonometrie. Or, the doctrine of triangles. London: William Jones, 1631. [Reprinted in 1971 by Theatrum Orbis Terratum (Amsterdam). Not seen.]
[58] James Mills Peirce. The Elements of Logarithms. Boston: Ginn brothers, 1873.
[59] Bartholomaeus Pitiscus. Thesaurus mathematicus sive canon sinuum ad radium 1. 00000. 00000. 00000. et ad dena quæque scrupula secunda quadrantis : una cum sinibus primi et postremi gradus, ad eundem radium, et ad singula scrupula secunda quadrantis : adiunctis ubique differentiis primis et secundis; atque, ubi res tulit, etiam tertijs. Frankfurt: Nicolaus Hoffmann, 1613. [The tables were reconstructed by D. Roegel in 2010. [68]]
[60] Jean-Charles Rodolphe Radau. Études sur les formules d’interpolation. Bulletin Astronomique, Série I, 8:273–294, 1891.
[61] Georg Joachim Rheticus and Valentinus Otho. Opus palatinum de triangulis. Neustadt: Matthaeus Harnisch, 1596. [This table was recomputed in 2010 by D. Roegel [69].]
[62] Donald A. Roberts. Vlacq, Adrian. In William Bridges Hunter, Jr., editor, A Milton encyclopedia, volume 8, pages 143–144. London: Associated University Presses, Inc., 1980.
23
[63] Denis Roegel. A construction of Edward Sang’s projected table of nine-place logarithms to one million (1872). Technical report, LORIA, Nancy, 2010. [This construction is based on the specimen pages [73].]
[64] 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 [103].]
[65] Denis Roegel. A reconstruction of Charles Babbage’s table of logarithms (1827). Technical report, LORIA, Nancy, 2010. [This is a reconstruction of [6].]
[66] Denis Roegel. A reconstruction of Gunter’s Canon triangulorum (1620). Technical report, LORIA, Nancy, 2010. [This is a recalculation of the tables of [43].]
[67] 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 [12].]
[68] Denis Roegel. A reconstruction of the tables of Pitiscus’ Thesaurus Mathematicus (1613). Technical report, LORIA, Nancy, 2010. [This is a recalculation of the tables of [59].]
[69] Denis Roegel. A reconstruction of the tables of Rheticus’s Opus Palatinum (1596). Technical report, LORIA, Nancy, 2010. [This is a recalculation of the tables of [61].]
[70] Denis Roegel. A reconstruction of the tables of Thompson’s Logarithmetica Britannica (1952). Technical report, LORIA, Nancy, 2010. [This is a unpublished reconstruction of the tables in [81], not available for copyright reasons.]
[71] Denis Roegel. Introduction to Chinese and Japanese tables of logarithms, with a review of secondary sources. Technical report, LORIA, Nancy, 2010. [This introduction is supplemented by three volumes of reconstructed tables patterned after Vlacq’s tables.]
[72] Denis Roegel. The great logarithmic and trigonometric tables of the French Cadastre: a preliminary investigation. Technical report, LORIA, Nancy, 2010.
[73] Edward Sang. Specimen pages of a table of the logarithms of all numbers up to one million...: shortened to nine figures from original calculations to fifteen places of decimals, 1872. [These specimen pages were reprinted in 1874 in a booklet which contained also a reprint of Govi’s report [41], a reprint of Sang’s article on Vlacq’s errors [74], and several other letters by eminent scientists supporting the publication of Sang’s table. The specimen pages were used to construct [63].]
[74] Edward Sang. On last-place errors in Vlacq’s table of logarithms. Proceedings of the Royal Society of Edinburgh, 8:371–376, 1875. [First printed in [73].]
[75] Edward Sang. Remarks on the great logarithmic and trigonometrical tables computed by the Bureau du Cadastre under the direction of M. Prony. Proceedings of the Royal Society of Edinburgh, Session 1874–1875, 8:421–436, 1875. [This article reproduces [3].]
24
[76] Edward Sang. On last-place errors in Vlacq. Nature, 42(1094):593, 1890.
[77] D. J. E. Schrek. David Bierens de Haan. Scripta Mathematica, 21:31–41, 1955.
[78] Dirk Jan Struik. Vlacq, Adriaan. In Charles Coulston Gillispie, editor, Dictionary of Scientific Biography, volume 14, pages 51–52. New York: Charles Scribner’s Sons.
[79] Dirk Jan Struik. The land of Stevin and Huygens; a sketch of science and technology in the Dutch Republic during the Golden Century, volume 7 of Studies in the history of modern science. Dordrecht: D. Reidel publishing company, 1981.
[80] Alexander John Thompson. Henry Briggs and his work on logarithms. The American Mathematical Monthly, 32(3):129–131, March 1925.
[81] 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, unpublished reconstruction by D. Roegel in 2010 [70].]
[82] Johannes Tropfke. Geschichte der Elementar-Mathematik in systematischer Darstellung, volume 2. Leipzig: Veit, 1903.
[83] Marius van Haaften. De Decker’s logarithmentafel “Tweede deel vande nieuwe Telkonst” gevonden. De Verzekeringsbode, 39(49):383–386, 1920. [4 September 1920]
[84] Marius van Haaften. De Deckers Eerste Deel vande Nieuwe Telkonst. De Verzekeringsbode, 39(52):406–410, 1920. [25 September 1920]
[85] Marius van Haaften. De Decker als schrijver over Intrestrekening. De Verzekeringsbode, 40(4):25–28, 1921. [23 October 1920]
[86] Marius van Haaften. De Decker als schrijver over Intrestrekening. De Verzekeringsbode, 40(5):37–39, 1921. [30 October 1920]
[87] Marius van Haaften. De kleine logarithmentafel van De Decker. De Verzekeringsbode, 40(19):146–149, 1921. [5 February 1921]
[88] Marius van Haaften. Ce n’est pas Vlack, en 1628, mais De Decker, en 1627, qui a publié le premier, une table de logarithmes étendue et complète. Nieuw Archief voor Wiskunde, 2/15:49–54, 1928.
[89] Marius van Haaften. Quelques nouvelles données concernant l’histoire des anciennes tables néerlandaises de logarithmes. Nieuw Archief voor Wiskunde, 2/21:59–64, 1941.
[90] Marius van Haaften. De Decker, Vlacq en de oudste Nederlandsche logarithmentafels. I. De Verzekeringsbode, 60(1):2–3, 1941.
[91] Marius van Haaften. De Decker, Vlacq en de oudste Nederlandsche logarithmentafels. II. De Verzekeringsbode, 60(2):7–8, 1941.
25
[92] Marius van Haaften. Endenburg’s nieuwe gegevens omtrent de oudste Nederlandsche logarithmentafels. I. De Verzekeringsbode, 60(8):29–30, 1941.
[93] Marius van Haaften. Endenburg’s nieuwe gegevens omtrent de oudste Nederlandsche logarithmentafels. II. De Verzekeringsbode, 60(9):33–34, 1941.
[94] Marius van Haaften. De rekenmeester Ezechiël de Decker (1603–1647) en zijn geschriften. I. De Verzekeringsbode, 60(12):45–46, 1941.
[95] Marius van Haaften. De rekenmeester Ezechiël de Decker (1603–1647) en zijn geschriften. II. De Verzekeringsbode, 60(13):50–51, 1941.
[96] Marius van Haaften. De eerste volledige grote logarithmentafel van Ezechiël de Decker uit 1627. Nieuw tydschrift voor wiskunde, 31:137–144, 1943–1944.
[97] Kasper van Ommen and Henk Bos. De Bibliotheca mathematica van Bierens de Haan. Nieuw Archief voor Wiskunde, 5/4(3):242–251, September 2003.
[98] Otto E. van Poelje. Adriaen Vlacq and Ezechiel de Decker: Dutch contributors to the early tables of Briggsian logarithms. Journal of the Oughtred Society, 14(1):30–40, 2005.
[99] Georg Vega. Thesaurus logarithmorum completus. Leipzig: Weidmann, 1794.
[100] Adriaan Vlacq. Arithmetique logarithmetique. Gouda: Pieter Rammazeyn, 1628. [This is the Arithmetica logarithmica [101] with a French introduction.]
[101] Adriaan Vlacq. Arithmetica logarithmica. Gouda: Pieter Rammazeyn, 1628. [The introduction was reprinted in 1976 by Olms. A French edition [100] was published at the same time.]
[102] Adriaan Vlacq. Logarithmicall arithmetike. London: George Miller, 1631. [The table of logarithms of numbers contained in this volume must in fact be those of De Decker printed in 1627.] [not seen]
[103] Adriaan Vlacq. Trigonometria artificialis. Gouda: Pieter Rammazeyn, 1633. [The tables were reconstructed by D. Roegel in 2010. [64]]
[104] Adriaan Vlacq. Tabulae sinuum, tangentium et secantium et logarithmi sinuum, tangentium & numerorum ab unitate ad 10000, etc. Lyon: Jean Thioly, 1670.
[105] Adriaan Vlacq. Tabulae sinuum, tangentium et secantium et logarithmorum sinuum, tangentium et numerorum ab 1 ad 10000, etc. Leipzig: Io. Benj. Georg. Fleischer, 1808. [New edition by Johann Jacob Ebert.]
[106] Anton von Braunmühl. Vorlesungen über Geschichte der Trigonometrie. Leipzig: B. G. Teubner, 1900, 1903. [2 volumes]
[107] John Wells. Sciographia, or the art of shadovves... All performed, by the doctrine of triangles... and by helpe of... logarithmes. London: T. Harper, 1635.
26
[108] J. Hill Williams. Briggs’s method of interpolation; being a translation of the 13th chapter and part of the 12th of the preface to the “Arithmetica Logarithmica”. Journal of the Institute of Actuaries and Assurance Magazine, 14:73–88, 1869.
[109] Rudolf Wolf. Handbuch der Astronomie : ihrer Geschichte und Litteratur. Zürich: Friedrich Schulthess, 1890. [2 volumes]
[110] Z. Adrien Vlacq. In Louis Gabriel Michaud, editor, Biographie universelle, ancienne et moderne, nouvelle édition, volume 44, page 1. Paris: Ch. Delagrave et Cie, (no year, ca. 1865). [contains some inaccuracies]
27
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