Baron De Prony's Description of the Construction of Mathematical Tables



The original of this document probably copied at Charles Babbage's request by an amanuensis was written in a fine copperplate hand.


By M. de PRONY (Translated from the French by C.J.D.lRoberts)

[This notice was read at the public session of the French Royal Academy of Sciences on June 7th 1824]

[This extract was made from the 'Notice sur les Grandes Tables Logarithmique &c. Par M. Prony printed in a quarto pamphlet 'Recueil des Discourses lus dans la seance publique de L'Academie Royale Des Sciences, 7 Juin 1824.' It commences on page 33. The extract presented here begins on page 36.]

The work which I am about to have the honour to present to the Academy, is one of the many which Baron Fourier, one of its permanent secretaries, mentioned in his historical summary on the progress of the Sciences, read during our last public session. This work has resulted in a monumental set of logarithmic and trigonometric tables, the most vast and most complete which has ever existed, executed by new procedures, by which means scientific operations have been transformed into operations which one could call manufacturing, entrusted to men for whom, to be employed on this project, it sufficed merely for them to know the first two rules of arithmetic and be able to write figures legibly.

Negotiations have been opened for the publication of these great tables, so calculated, between the French and British governments. It was the latter which has taken the intiative by offering to pay for half the printing expenses.

In the present state of affairs it seemed fit to present, at this year's public session, some details serving to supplement and develop the mention made of my tables made last year by M. le Baron Fourier. In discharging this task I shall economise with the Assembly's time, and only give those parts of my report necessary to render it intelligible.

The new French metric system only allows, for the sub-divisions of the units of measure, of whatever type, decimal fractions; the quadrant or quarter of a circle, and angular measures were also made to conform to the common rule; from then on all existing trigonometric tables, whether presented in natural or logarithmic form, adapted to the needs of Astronomy and Geodesy became useless; and it was found necessary to calculate new ones. At the time I was director general of the Cadastre de la France [Ordnance Survey of France], to which I had been appointed in 1791, and it so befell me that I was asked to produce these decimalised tables. But as one wished to give everything that was connected with the French metric system, an air of grandeur to excite the attention, even the admiration, and a superiority over anything which had previously been produced, with which to inspire confidence, "I was engaged expressly not only to compile tables which left nothing to be desired about their accuracy, but also to make of them "a monument to calculation the greatest and the most impressive that had ever been executed or even conceived". These are the exact expressions that were used in the brief that I was given; not only did this impose on me the condition of having to calculate the values of the rows in the tables of the trigonometric functions and their logarithms, to a large number of figures and in a succession of very close divisions, but it was also necessary, in addition, to redo the tables of logarithms of the natural numbers, extending them to twice the length of the greatest known tables, that is to carry them forward from the number 100,000 to 200,000 to 14 or 15 decimal places instead of 8 or 10.

The burning desire I had to find myself exclusively and irrevocably associated with this work of science and art, made me determined to accept unconditionally the proposals that had been put to me; but it was evident to me that by using the methods employed by Rheticus, Oshon, Pitiscus and Briggs, even if I improved on them to make them more effective, I could not hope to live long enough to finish the project; it was thus I found myself in such a embarassment more arduous than I could hide: a circumstance happily as it had happened unexpectedly helped me out of this embarassment. Having one day noticed, in the shop of a seller of old books a copy of the first English edition 1776, of Smith's "Treatise on the Wealth of Nations", I decided to acquire it, and on opening the book at random, I came across the chapter where the author had written about the division of labour; citing, as an example of the great advantages of this method, the manufacture of pins. I conceived all of a sudden the idea of applying the same method to the immense job with which I had been burdened, to manufacture my logarithms as one manufactures pins. I have reasons to believe that, without realising it, I had already been prepared for this realisation from certain parts of mathematical anaylsis, on which I had then been giving tuition at the Ecole Polytechnique.

I decided to spend a few days in the country in order to establish the basis for my new manufacture, which did not delay in becoming active. I formed my band of fellow-workmen into three sections amongst whom were to be apportioned the successive operations of my system of the division of labour.

The first was composed of four or five geometricians of very high merit; this group occupied itself purely with the analytical part of the work, and the calculation of some fundamental numbers [for the tables].

The second group contained seven or eight calculators, who possessed a knowledge of analysis, and had considerable experience in converting formulae into numbers: their duty was to deduce from the first group's general calculations numbers serving as starting points, with which to form the top most rows of each sheet in the grand folio volumes, on each of which had been drawn one hundred lines; the ninety-nine remaining lines were to be filled in by the workers of the third group.

This third group comprised no less than seventy or eighty individuals; but it was the easiest to form, because, as I had foreseen, they did not need, in order to be admitted [to this group] any preliminary instruction; the one essential condition, for their admission [to the group], was for them to know the first two rules of arithmetic: so it was thus we were presented with a singular gathering of persons who had lived in very different worlds.

I remember with satisfaction that many among them came to seek and find, in this special workshop, a safeguard, a refuge which, happily, was not violated, and that the political circumstances of that time rendered these fully necessary; they were able, thanks to the system of the division of labour, without being scholars to live in safety, under the shield of science.

The folio-sheets that were distributed amongst them were each drawn with and divided into 100 intervals, 50 on the recto side and the remainder on the verso. The top line of numbers, on one side of the folio sheets only, was as I have mentioned previously, provided by the workers of the second section. The ninety-nine remaining lines were then filled in by means of purely mechanical operations carried out by the 3rd section, each of whom was performing 900 to 1000 additions or subtractions per day, nearly all of whom not having the least theoretical notion on the work which they were doing.

All the calculations were done twice: expedite means of verification, but very rigorous, were prepared in advance; and I noticed that the sheets the most exempt from error were in particular, furnished by those who had the most limited intelligence, an existence, so to speak, 'automatique' [of routine].

The set of tables thus produced, within the space of less than two years, contained 10000 Natural Sines, calculated to 25 decimals with 7 or 8 columns of auxiliary numbers known by the name "Differences", and which are very useful to calculators; 200000 Logarithms of Sines and Tangents calculated to 14 decimal places with 4 columns of differences; 10000 Logarithms relating to the tables of Sines and Tangents of the Arc, to facilitate interpolation in calculations using small angles, calculated to 14 decimal places with three columns of differences; Logarithms of the first 10000 numbers calculated to 19 decimals, and finally Logarithms of the following numbers, from 10000 as far as 200000, calculated to 14 decimal places with 5 columns of differences; the result about 2,300,000 numbers, of which 4 or 500,000 consisted of 14 to 25 digits; about 1 per cent of these numbers, at the most, had been calculated from analytical formulae, and the remaining 99 per cent deduced from the 1 per cent by means of a manufacturing procedure.

A very extensive introduction [to the work] holds the complete explanation of the methods used, tables of the fundamental numbers, etc.

I shall avoid entering into the technical details relative to the great superiority of the work I have just given a glimpse of, on the most considerable works of the same kind, executed since Copernicus up to the present [1824]; one can on this point consult many of the notices published in Volume Five of the memoirs of the Physical and Mathematical Sciences Division of the Institute and particularly that report prepared by Messrs. Lagrange, Laplace and Delambre, where the following remark was made: "For the first time the work of preparing trigometric tables has come to be executed using all the resources that analysis can furnish, and the greatest monument of its kind will be that which will have really given the least difficulty and have taken the least time."

The late M. Delambre, editor of this report, had had the great tables at his disposal for several months and had submitted them to both numerous and rigourous tests of verification; here is how he expressed himself when he gave an account of these verifications: "It is after a careful and deep examination of these tables [that I am able to say], that their high value is as much due to their accuracy as to their extent." Such a judgement made by such a man as Delambre must surely inspire confidence.

To give an example of the speed of work performed by the sections of calculators which I had formed, and whose dissolution I regretted, I shall cite the following facts: the minister asked me to produce an abridged transcription of the tables of logarithms, sines and tangents, to put them in a portable edition; I reflected that the manuscript so formed would necessarily conceal copyist errors which upon inspection would lead to embarassment and an increase in time. I therefore took it upon myself to construct afresh, by my manufacturing process, the portable tables, without having recourse to the grand tables, and in the space of nine days these portable tables were calculated in duplicate [for the checking procedure] and without errors, for the 10000 partitions of the quarter of a circle, to 9 decimal places and 2 or 3 columns of differences; if one had made extracts from the Grand Tables, the work would have subject to unavoidable errors, and would have taken many months.

The seventeen grand in-folio volumes which enclose this immense collection, have been for many years deposited at the [Paris] Observatory, and have already been of considerable use to the authors and editors of diverse logarithmic and trigonometric tables published both here in France and outside of France, for about 30 years, and even to those authors of works of Pure Analysis, that is even the late Callet, has taken all that was related to the centesimal partitioning of the quadrant into his two volumes of tables: Delambre, editor of the Tables of Borda, checked all of them against mine. A similar verification was done for Messrs. Hobert and Idler's tables published in Berlin. In Volume 5 of the Memoirs of the Institute, cited above, I have already given a table of errors to be found in the Opus Palatinum by Oshon, and of the 14th and 15th digits of the Thesaurus Matematicus; in the 3rd volume of 'Des Exercises de Calcul Integral' by my colleague M. Legendre, the 19 figure logarithms which he extracted from, etc. etc.

Page 41 to end (page 42)

Messrs. Lagrange, Laplace and Delambre have in the above report already expressed that it is their formal wish the printing of my tables should be continued with, which at the time of preparing this report for the printers, had been commenced by M. Firmin Didot, forming a folio volume containing one hundred rows on each page. Four hundred printing plates using movable type to which were given when they had been assembled the properties of stereotype, had already been composed, when various causes due partly to the conditions of the contract between M. Didot and the [French] Government, and the collapse in value of paper money [Assignats], forced the project to be suspended.

I have said that the British Government had subsequently made the 9 proposals to the French Government relative to this printing; a letter from Lord Castlereagh addressed in Paris, at the end of 1819, to Sir Charles Stuart, informed him that Sir Charles Blagden was to act as the British commissioner in France for this important matter (these are the expressions used in the letter) of the publication of the Grand French Tables. The death of the honorable Blagden has caused negotiations to be halted, which meanwhile do not appear to have been abandoned judging by a letter recently written by His Excellency the Minister for the Interior to the Secretary of the Bureau of Longitude; Europe's scholars are impatiently waiting the outcome of these negotiations; Europe, worried, sees a monument, the greatest of its kind and whose loss will never probably be made good, existing only as a manuscript and finding itself exposed to the risks of destruction which could cause everlasting regrets to the friends of Science.