When one speaks today of "computer error," one is most often referring to a flaw in the electronic or mechanical functioning of a programmed machine. An extraordinary case of computer error occurred long before the advent of mechanical and electronic computers, however, as noted by Charles Babbage in an 1827 publication entitled On Errors Common to Many Tables of Logarithms.
Early astronomers spent long periods of time making calculations for tracking and predicting the orbits of comets, planets, and moons, and for determining the occurrence of eclipses. For example, in calculating the orbit of Halley's Comet, the French astronomer Joseph de Lalande (1732-1807) said:
During six months, we calculated from morning till night, sometimes even at meals, the consequence of which was, that I contracted an illness which changed my constitution for the remainder of my life. The assistance rendered by Madame Lepaute was such, that without her, we never could have dared to undertake the enormous labour, in which it was necessary to calculate the distance of each of the two planets, Jupiter and Saturn, from the comet, and their attraction upon that body, separately for every successive degree, and for 150 years.(1)
Charles Fort often ridiculed astronomers for this activity, because they would explain by mathematical calculations what they could not observe through their telescopes, in other words, because they were substituting theory for empirical fact. The profusion of false astronomical predictions and pronouncements that Fort so gleefully pointed out may be largely attributed to the "computer error" that Babbage noted.
According to Florian Cajori, writing in 1894, the main advancements in modern mathematics were "the Arabic notation, decimal fractions, and logarithms."(2) The invention of logarithms has been credited to John Napier, Baron of Merchiston (Scotland), in 1594, when he described his invention to Tycho Brahe. The invention was illustrated with examples and described in Napier's Mirifica logarithmorum canonis descriptio, published in 1614.(3)
Joost Brugi, of Prague, who is credited with an independent invention of logarithms after Napier, published a table of anti-logarithms in 1620, and Edmund Gunter published a table of logarithms to seven places of decimals in the same year. Most significant, however, was Henry Briggs' Logarithmetica Brittanica (1624), which contained the first set of logarithms to 14 places for the numbers from 1 to 20,000 and from 90,000 to 100,000.(4) Adrain Vlacq (also Flack), of Holland, completed Briggs' work with tables published in 1628 at Gouda under the title Arithmetica logarithmica, wherein he presented the first complete set of logarithms by his inclusion of numbers from 20,000 to 90,000.(5)
The great advantage of logarithms is that factors that would be multiplied could simply be summed (added). Cajori states: "It is no exaggeration to say that the invention of logarithms `by shortening the labours doubled the life of the astronomer.'"(6) E.T. Bell writes: "Kepler's laws were the climax of thousands of years of an empirical geometry of the heavens. They were discovered as the result of twenty-two years of incessant calculation, without logarithms.... The contemporaneous invention of logarithms was to reduce all such inhuman labour as Kepler's to more manageable proportions."(7)
When an astronomer is faced with a calculation involving the multiplication of several multi-digit numbers, the advantage of adding numbers from a table and finding the sum in the same table is obvious. One can expect to save time and to avoid errors that might occur in long multiplications. The fact that errors would occur in logarithmic tables should have been expected, however, printers in the 17th and 18th centuries were unlikely to be as careful as mathematicians and astronomers would have wished.
One might wonder how many errors were included in such an effort as Lalande's when one realizes how many calculations may have been made with erroneous logarithms. An historian of mathematics, W.W. Rouse Ball. Writing in 1912, says, "The Arithmetica logarithmica of Briggs and Vlacq are substantially the same as existing tables: parts have at different times been recalculated, but no tables of an equal range and fulness entirely founded on fresh computations have been published since."(8) When Briggs' work was republished by Cambridge University in 1952, it included six folio pages of errors.(9)
In 1785, Charles Hutton published his Mathematical Tables with a preface of scorn, as follows:
The undertaking was occasioned by the great incorrectness of all the editions of Sherwin's and Gardiner's Tables, and more especially by the bad arrangement in the fifth or last edition. Finding, as well from the report of others, as from my own experience, that those editions (to say nothing of the very improper alteration in the form of the table of sines, tangents, and secants in the last of them) were so very incorrectly printed, the errors being multiplied beyond all tolerable bounds, and no dependence to be placed on them for anything of real practice. I was led to undertake the painful office of preparing a correct edition of another similar work.(10)
Hutton's work included an "original history," which cited problems found in other logarithmic tables. For example, in the powers of 2, he writes:
...the third column contains the logarithms of all the numbers in the first column: a property which, if Dr. [John} Newton had been aware of, he could not well have committed such gross mistakes as are found in a table of his similar to that above given, in which most of the numbers in the latter quaternions are totally erroneous; and his confused and imperfect account of his method would induce one to believe that he did not well understand it.(11)
Hutton further points out, "I have a list of several thousand errors which I have corrected in it [the last or fifth edition of Gardiner's Tables], as well as in Gardiner's octavo edition."(12) The lists of errata found in Gardiner's 1742 edition, in the Avignon edition of 1770, and in Callet's editions of 1783 and 1795, which were included in Hutton's 1801 edition, were omitted from his 1830 edition (published after his death in 1823).(13)
The problem of how some errors may have persisted for an extensive period of time was shown by Charles Babbage, who bluntly accused the authors of logarithmic tables as being plagiarists rather than mathematicians:
A few years ago, it was found desirable to compute some very accurate logarithmic tables for the use of the great national survey of Ireland, which was then, still in progress; and on that occasion a careful comparison of various logarithmic tables was made. Six remarkable errors were detected, which were found to be common to several apparently independent sets of tables. This singular coincidence led to an unusually extensive examination of the logarithmic tables published both in England and in other countries; by which it appeared that thirteen sets of tables, published in London between the years 1633 and 1822, all agreed in these six errors. Upon extending the enquery to foreign tables, it appeared that two sets of tables published at Paris, one at Gouda, one at Avignon, one at Berlin, and one at Florence, were infected by exactly the same six errors. The only tables which were found free from them were those of Vega, and the more recent impressions of Callet.[(14)] It happened that the Royal Society possessed a set of tables of logarithms printed in the Chinese character, and on Chinese paper, consisting of two volumes: these volumes contained no indication or acknowledgment of being copied from any other work. They were examined; and the result was the detection in them of the same six errors.
It is quite apparent that this remarkable coincidence of error must have arisen from the various tables being copied successively one from another. The earliest work in which they appeared was Vlacq's Logarithms, (folio, Gouda, 1628); and from it, doubtless, those which immediately succeeded it in point of time were copied; from which the same errors were subsequently transcribed into all the other, including the Chinese logarithms.(15)
The Chinese volume was undoubtedly the second part of a work entitled Su-li Ching-yuen, published in 1713 and comprising 53 books. In 1913, Yoshio Mikami identified its contents as including: "...common logarithms, the latter being given for 11 decimal places. These logarithmic tables are said to be the same as those published by Adrian Vlacq in 1628 in Holland."(16) Mikami also states:
The Chinese mathematicians widely employed these tables and became perfectly convinced of the convenience of them. But there are no treatises in China, in which the construction of such tables was explained. The publication of algebraical treatises containing the theory of logarithms is only of a recent date.(17)
When Charles Babbage commenced work on his own set of logarithmic tables, published in 1827 as Table of Logarithms of the Natural Numbers from 1 to 108,000, he utilized Callet's tables and compared them with Hutton's, Vega's, Briggs', Gardiner's, and Taylor's. Even after these comparisons revealed many errors which were then recalculated, the process of re-reading the tables would reveal another thirty-two errors and then eight more errors when reading the proofs, which were corrected on the printing plates.(18)
Babbage noted that sometimes the errors in tables of logarithms were the result of calculating new figures using Vlacq's erroneous tables, "in which nevertheless the erroneous figures in Vlacq are omitted."(19) This could be considered an early instance of a "computer virus," where data is rendered useless from utilizing infected software. The problems encountered by the use of such tables are shown in two examples given by Babbage:
Mr. Baily states that he has himself detected in the solar and lunar tables, from which our Nautical almanac was for a long period computed, more than five hundred errors. In the multiplication table already mentioned, computed by Dr. Hutton for the Board of Longitude, a single page was examined and recomputed: it was found to contain about forty errors.(20)
What Charles Babbage concluded was that the errors encountered in logarithmic
tables could not be avoided until such time as a "calculating engine"
might be employed to recalculate each of the logarithmic figures, which
had not been done since the time of Briggs and Vlacq, and the figures then
published without typographical error. Babbage's invention was not built
for this purpose; but, a century later, modern electronic computers would
produce new logarithmic tables, as Babbage had envisioned. Yet, ironically,
the development of compact and economical computers has now rendered the
need for logarithmic tables and mechanical instruments, such as the slide
rule, obsolete. We may continue to encounter "computer error,"
but this is certainly nothing new.
1. "The approaching comet," Edinburgh Review, 61 (April 1835): 99-100; not volume 66, as given by Fort in chapter 3 of New Lands.
2. Florian Cajori. A History of Mathematics, (New York: Macmillan, 1894), 161.
3. W.W. Rouse Ball. A Short Account of the History of Mathematics, (London: Macmillan, 5th ed., 1912), 195.
4. Ibid, 196; Cajori, op. cit., 164.
5. Cajori, op. cit., 164-5.
6. Ibid, 161.
7. E.T. Bell. The Development of Mathematics, (New York: McGraw-Hill, 1940), 145.
8. Ball, op. cit., 196.
9. Henry Briggs. Arithmetica logarithmica (Cambridge: University Press, 1952): vol. 1, lxxviii-lxxxiii.
10. Charles Hutton. Mathematical Tables, (London, 7th ed., 1830): v.
11. Charles Hutton. Mathematical Tables (London, 3rd ed., 1801): 64.
12. Ibid, 40.
13. Ibid, 342-4.
14. George Freiherr von Vega. Logarithmische, Trigonometrishe..., (Vienna, 1783), or Thesaurus logarithmorum. Jean Francois Callet. Tables portatives de Logarithmes, (Paris, 1783).
15. "Babbage's calculating engine," Edinburgh Review, 59 (July 1834): 277-8.
16. Yoshio Mikami. The Development of Mathematics in China and Japan, (New York: Chelsea, n.d.; reprint of 1913 ed.): 117.
17. Ibid, 127.
18. "Babbage's calculating engine," op. cit., 276.
19. Ibid, 279.
20. Ibid, 276.
"Computer error" was published in the INFO Journal, (no. 78; Autumn 1997): 21-2, 36.
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