Logarithmic tables

History of Mathematics video from 1980s, discussing Paris, French Revolution, and math: https://www.youtube.com/watch?v=upu1YZ-M0vI&list=PL1BAF8C53CA724006&index=9

M. Bradley, A career biography of Gaspard-Clair-François-Marie Riche de Prony: bridge-builder, educator and scientist (1998)

Graham Jagger, "The making of logarithm tables", in The History of Mathematical Tables: From Sumer to Spreadsheets, ed. Martin Campbell-Kelly

Henry Briggs A table to find the height of the pole, the magnetical declination being given (Blundeville, 1602)

Wright and Briggs, Certaine errors in navigation (London, 1610)

John Napier, Mirifici Logarithmorium canonis descriptio (Edinburgh: Printed by Andrew Hart, 1614)

• 90 pages of tables

John Napier, A description of the admirable table of logarithms (trans. E. Wright) (London, 1616)

• used by navigators for exploration, East India Company

John Napier, Rabdologiae (1617)

• described calculating devices including numbering rods (Napier's Bones)

John Napier, Mirifici logarithmorum canonis constructio (1619)

• posthumously published by his son Robert, explaining tables in Descriptio

Henry Briggs, Logarithmorum chilias prima (privately printed, 1617)

• https://archive.org/details/bim_early-english-books-1475-1640_logarithmorum-chilias-pr_briggs-henry_1617
• extremely rare, 1 pages, earliest printed table to base 10
• Translation of preface in Jagger:
  • "Here is the first thousand logarithms which the author has had printed, not with the intention of becoming public property, but partly to satisfy on a private basis the wish of some of his own intimate friends: partly so that with its help he might more conveniently solve not only several following thousands but also the integral table of Logarithms used for the calculation of all triangles. For he has the Table of Sines, accurately drawn up from first elements by himself ten years earlier, through algebraic equations and differences, proportional to the actual sines instead of individual degrees and hundredths of degrees: this he hopes to publish, God willing, together with the appended logarithms, as soon as conveniently may be."

John Speidell, New logarithmes (1619)

• "

a book of tables derived directly from those in the Descriptio. This work, despite containing no explanation of the tables nor any worked examples, had gone through nine editions by 1627" (Jagger 69)

Edmund Gunter, Canon triangulorum sive tabulœ sinuum et tangentium arti�cialium ad radium 10000.0000 & ad scrupulum prima quadrantis (London: William Jones, 1620)

• "Gunter's table, like those of Napier, presents the logarithms of sines and tangents for every minute of the quadrant, semi-quadrantally arranged. These tables are to eight �gures, split into two groups of four by a vertical bar. Leading digits are omitted where these do not change from the line above. This may have been a deliberate device to render the page more readable, but it is not impossible that it is the result of a desire to economize on the use of type." (Jagger 62-3)

"In the years immediately following the publication of the Descriptio, these two works, Gunter's Canon triangulorum and Briggs's Logarithmorum chilias prima, established the techniques and conventions that were to underpin the two great canons of the next decade, the Arithmetica logarithmica and the Trigonometria Britannica." (Jagger 65)

Henry Briggs, Arithmetica Logarithmica (London, 1624)

• http://locomat.loria.fr contains a reconstruction of this table
• logarithms of 30k natural numbers to 14 decimal places

Adrian Vlacque, Arithmetica logarithmica" (1628)

• published the completed canon to 10 digits

Briggs (posthumous), Trigonometria Britannica (Gouda, 1633)

• Gellibrand succeeded Briggs as professor of astronomy at Gresham College; published tables Briggs worked out
• http://locomat.loria.fr contains a reconstruction of this table

"Briggs's two great works, the Arithmetica logarithmica and the Trigonometria Britannica, formed the basis of all logarithm tables published for the next 200 years or so. It was not until the end of the eighteenth century that any further calculation of logarithms took place." (Jagger 66)

"But logarithms to base 10, as set forth by Briggs in the Arithmetica logarithmica and Trigonometrica Britannica, were not at once readily accessible to the student and practitioner. These two works were large, relatively expensive and written in Latin, and were, perhaps, destined more for the coffee table than the working library. There was a need for smaller and cheaper volumes, written in English. One of the earliest writers who sought to �ll this gap was John Newton (1621–1678). During the Interregnum Newton, a staunch Royalist, taught mathematics in Oxford and in the years 1654 to 1660 wrote eight books, all of them in English, on arithmetic, geometry, trigonometry, and astronomy. Five of these volumes have a signi�cant tabular content: tables of logarithms of the natural numbers and of angular functions and their logarithms, all of them explicitly derived from those of Briggs." (Jagger 69-70)

Oughtred, Trigonometria (London: Joseph Moxon, 1657)

• "Undoubtedly the typesetting of the tabular portion of the Trigonometria was Moxon's pièce de résistance, but something had gone badly wrong in the process of its production, and a second edition of the book, this time in English, was rushed out within months of the �rst." (Jagger 71)
• "It is clear that it was Oughtred's intention that the tables of logarithms of sines and tangents should consist of seven �gures after the decimal point: in fact only six were printed. Because of this error the rules given in the appendix for using the tables would not work." (Jagger 71)

"Within just a few decades of their invention, logarithms had become ubiquitous among the mathematical community: tables were easily and cheaply available and many texts describing their use had been printed. What is striking is that their development from Napier's initial formulation was the work not of a few isolated individuals but rather the result of an almost explicit collaborative effort by the members of a closely connected

group, a group centred on Gresham College with Henry Briggs as its driving force." (Jagger 72)

Ivor Grattan-Guinness, "The computation factory: de Prony's project for making tables in the 1790s," in The History of Mathematical Tables

" early on in the new decade de Prony set up a Bureau de Cadastre in Paris, to prepare a detailed map of France to facilitate the accurate measurement of property as a basis of taxation. In connection with this plan, it was decided that a very large set of logarithmic and trigonometric tables would be prepared." (Grattan-Guinness 107)

"de Prony devised the project explicitly following the principles of the division of labour which had been laid down by Adam Smith in A treatise on the wealth of nations of 1776." (108) -- inspired by chapter on pins

divided into three sections:

• mathematicians, choosing the formulas to be used
• calculators, who determined values and differences of various orders
• assistants to calculators, 60-80 many probably hairdressers who lost work after the revolution; "Thus these artists were converted into elementary arithmeticians, executing only additions and subtractions" (109)

when a page was completed, it was returned to the second section to check, using formulas devised by first section

by 1794, 700 results produced each day

project completed in 1801

"The publishers Firmin Didot were charged with the task of printing, and estimated that a volume of 1200 large pages would result, together with the introduction, at a cost of 80 000 francs; 12 apparently about 500 pages were in proof by 1802. But �nancial crises in France began to intervene, and printing stopped." (111)

"Both sets of the manuscript tables are still preserved. One is held in the library of the Paris Observatoire. de Prony himself retained the other set; but his niece and heir did not include it in his manuscripts that she sent to the École des Ponts et Chaussées. It was found only in 1858, in the possession of the niece's daughter and son-in-law, by Pierre Alexandre Francisque Lefort(-Latour) (1809–1878), in the course of making an extensive study of the other set, in the Observatoire." (112)

19 volumes, last one contains an account of methods of calculation; others are large-format, 251 folios each (containing the tables for ar un of 25100 values of the numbers or angles). Vols. 1-8: logs of numbers from 1 to 100000 calculated to 14 decimal places; 1 vol of sines of parts of the radius; 3 vols of sines and of tangents; 1 vol each of the ratios of arcs to sines and of arcs to tanges, and the product of sines with cosines (112-3)

"When de Prony's set had been reported to the Académie des Sciences in 1858, hopes had again been expressed that printing could be achieved. The collation of the two sets was completed in 1862 by of�cers of the Dépôt Général de la Guerre (the last volume of the set is so annotated and signed). In his discussion of the tables, Lefort was doubtful of the value of publication, but thought that a rounded-off version to eight places was desirable." (117)

Philippe Minard, "Agitation in the Work Force," in Revolution in Print: The Press in France 1775-1800, eds. Robert Darnton and Daniel Roche (University of California Press, 1989): 107-23.

on innovations in printing in Revolutionary France:

"The earliest innovations had to do with composition. Printers wanted to be able to pull new impressions without resetting type. At that time the only way to do this was to keep the original formes, thus immobilizing thousands of expensive characters. Because the procedure was costly, the technique of taking an impression of the finished page to form a matrix was developed. this matrix the whole page could be recast at any time in a single block of metal, or plate, bearing the text in relief and serving as a substitute for the forme of movable type. Stereotyped printing was introduced in Paris in the 178os by François-Ignace-Joseph Hoffmann, who after demonstrating his process before the Académie Royale des Sciences was licensed to set up a printing shop and to publish his Journal polytype des sciences et des arts. But his political activities and his printing of clandestine pamphlets hostile to the Crown caused his shop to be closed down in November 1787.23 The benefit of polytyping is that it allows composition to be done with less movable type; once a page has been set, it can be cast, and the type can be reused. If there is no reason to have thousands of characters in stock, the cost of setting up a clandestine printing press is that much lower." Apart from the "polytyping" of bank notes, bills of the Caisse Patrio-tique, and later lottery tickets, between 1790 and the Year IV (1796) we find no books printed from plates earlier than the Year III (1794-95). In 1795 Firmin Didot printed Callot's Table of Logarithms, and in 1798, with Louis-Etienne Hehran, he published stereotyped editions of literary classics. The process was still controversial. The bookseller Stoupe, in the Year VII (1799), reiterated Lottin's argument of 178g that the discovery was a step backward because it took printing back three centuries to a point before the invention of movable type. In fact, the stereotyping system was neither perfected nor made economical before 1799." (115)