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To cite this version:
Denis Roegel. A reconstruction of Crelle’s Rechentafeln (1820). [Research Report] 2011.
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Crelle’s Rechentafeln
(1820)
Denis Roegel
21 October 2011
This document is part of the LOCOMAT project:
http://locomat.loria.fr
He is now most famous for having founded the Journal für die reine und angewandte
Mathematik in 1826, but he also published several mathematical treatises and textbooks,
as well as works on railway engineering and railway economics. Among other things, he
also published several mathematical tables, which are the subject of interest here.
Figure 1: August Leopold Crelle (1780–1855) (source: Wikipedia)
1
Multiplication tables before Crelle
Multiplication tables have become so mundane that it is hard to imagine that they too
have a rich history.1
The 10 × 10 tables have been extended early on. A table giving the
products up to 100 × 1000 is for instance cited in the 16th century [17]. In 1610, Herwart
von Hohenburg published his Tabulæ arithmeticæ προσθαφαιρεσεος universales [13, 10, 11]
which gave the products of all numbers from 1 to 1000. It was printed on 999 large
pages [14, volume 1, p. 644], [9, pp. 16–17].
There have been many other tables, but many of the tables were only giving the
product of a one digit number by a larger number, for instance up to 10 × 1000, or up to
10 × 100000.
In 1805, Oyon published a table giving the products n × m for 2 ≤ n ≤ 1000 and
10 ≤m ≤ 509 [15]. But Oyon did not factor the many repetitions.
1
For a good overview of the history of tables of multiplication, see Glaisher’s report [9] and Weiss’
article [17].
Crelle’s Rechentafeln (“computation tables”) published in 1820 [3] were a major step
forward in the development of extended multiplication tables. Apart from Herwart’s
table, this was the first widely distributed table giving products up to 1000 × 1000.
The layout of Crelle’s table appears very close to Herwart’s and it is likely that Crelle
was acquainted with it. But Crelle improved on Herwart’s table, essentially by factoring
the redundancies.
Crelle’s original tables were spanning 1798 pages (not counting intertitles), and were
split in two volumes. In our reconstruction, we have gathered these two volumes in a
unique volume.
Crelle gave the multiples of 1 to 999, skipping the multiples of 10 [9, pp. 19–20]. In
Herwart’s table, each number was covering one page and the products were given in full.2
In Crelle’s table, each number instead spans two pages. For instance, the first page of
the multiples of 39 gives the values of 39 × x with x = 100a + b, and 0 ≤ a ≤ 9 and
1 ≤b ≤ 49. The second page corresponds to 51 ≤ b ≤ 99. Values of b multiple of 10 are
omitted in the table.
Each multiplication is split in two parts, on the one hand the hundreds, and on the
other hand the tens and units. 39 × 51 = 1989, for instance, is split in 19 (hundreds) and
89 (tens and units). 39 × 151, 39 × 251, . . . , 39 × 951, all share the same tens and units,
and this fact is used in the table.
In Herwart’s table, the layout is the same, but all products are given explicitely, and
not in two parts. This results in considerable redundance and an increase in the size of
the table.
2.2
Later editions
There were several further editions, including the 1857 one by Bremiker [6]. It was a
stereotype edition and was reprinted several times afterwards. This edition reformated
the tables and put four former pages on one new page. For instance, the two pages for 2
were put one over each other in the left part of a page, whereas the two pages for 3 were
put one over each other on the right part of the page.
Seeliger’s 1914 edition [7] (and perhaps previous ones) filled the gaps of Crelle’s
orig-inal and added values b which are multiples of 10, although these additions could have
been dispensed of.
Yano also published a reprint of Crelle’s original table in Japan in 1913 [18].
2
We stress that we have not seen Herwart’s table, and our assumptions are based on a sketch of one
page reproduced by Weiss [17]. Assuming this sketch to be faithful to the original table, Herwart’s table
should be very easy to reconstruct. Glaisher also gave a detailed description of the table [9].
the products of all numbers up to 10 millions by 2, 3, . . . , 9. This table could easily be
used to compute the product of two numbers up to 10 millions each.
Crelle also worked on factor tables and completed the fourth, fifth and sixth millions,3
after Burckhardt’s initial work [1]. Crelle’s tables were however discovered to be too
inaccurate to publish. Crelle described methods for computing factor tables in 1853 [5].
3
At this point, it is not clear if Crelle’s manuscript tables do still exist. These tables were deposited
in the Archives of the Academy of sciences in Berlin. We have contacted the Academy of sciences, but
their archives are closed until the beginning of 2012. It should therefore be investigated further whether
Crelle’s manuscript do still exist, and whether they were really as inaccurate as has been written.
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] Johann Karl Burckhardt. Table des diviseurs pour tous les nombres des 1er
, 2e
et 3e
million, etc. Paris: Vve Courcier, 1817.
[2] Moritz Cantor. Crelle, August Leopold. In Historische Kommission bei der
Bayerischen Akademie der Wissenschaften, editor, Allgemeine Deutsche Biographie,
volume 4, pages 589–590. Leipzig: Duncker & Humblot, 1876.
[3] August Leopold Crelle. Rechentafeln, welche alles Multipliciren und Dividiren mit
Zahlen unter Tausend ganz ersparen, bei grösseren Zahlen aber die Rechnung
erleichtern und sicherer machen. Berlin: Maurerschen Buchhandlung, 1820. [2
volumes]
[4] August Leopold Crelle. Erleichterungs-Tafel für jeden, der zu rechnen hat :
enthaltend die 2, 3, 4, 5, 6, 7, 8, und 9fachen aller Zahlen von 1 bis 10 Millionen.
Berlin, 1836. [not seen, tentative reconstruction in [16]]
[5] August Leopold Crelle. Wie eine Tafel der untheilbaren Factoren der Zahlen bis zu
beliebiger Höhe möglichst leicht und sicher aufzustellen sei. Journal für die reine
und angewandte Mathematik, 51(1):61–99, 1856.
[6] August Leopold Crelle. Dr. A. L. Crelle’s Rechentafeln, welche alles Multipliciren
und Dividiren mit Zahlen unter Tausend ganz ersparen, bei grösseren Zahlen aber
die Rechnung erleichtern und sicherer machen. Berlin: Georg Reimer, 1857. [one
volume, with preface by C. Bremiker, second edition in 1864, seventh in 1895]
[7] August Leopold Crelle. Dr. A. L. Crelle’s Rechentafeln, welche alles Multipliciren
und Dividiren mit Zahlen unter Tausend ganz ersparen, bei grösseren Zahlen aber
die Rechnung erleichtern und sicherer machen. Berlin: Georg Reimer, 1914. [edited
by Oskar Seeliger, not seen]
[8] Review of Dr. A. L. Crelle’s Rechentafeln, 1857. The mechanic’s magazine,
68(1795):9–10, 1858. [review of [6]]
[9] 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
4Note 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.
[10] James Whitbread Lee Glaisher. On Herwart ab Hohenburg’s Tabulæ Arithmeticæ
προσθαφαιρεσέος universales, Munich, 1610. Proceedings of the Cambridge
Philosophical Society, 2:386–392, 1876.
[11] James Whitbread Lee Glaisher. On multiplication by a table of single entry.
Philosophical magazine, 6(38):331–347, 1878.
[12] James Whitbread Lee Glaisher. Table, mathematical. In Hugh Chisholm, editor,
The Encyclopædia Britannica, 11th edition, volume 26, pages 325–336. Cambridge,
England: at the University Press, 1911.
[13] Hans Georg Herwart von Hohenburg. Tabulæ arithmeticæ προσθαφαιρεσεος
universales. Munich: Nicolai Henrici, 1610. [not seen]
[14] Charles Hutton. A philosophical and mathematical dictionary. London, 1815.
[15] Jean-Baptiste Oyon. Tables de multiplication, etc. Paris: Hte
Pérez et Cie, 1824.
[3rd edition. First edition in 1805, second edition in 1812, and 4th edition in 1864.]
[16] Denis Roegel. A reconstruction of Crelle’s Erleichterungstafel (1836). Technical
report, LORIA, 2011. [This is a reconstruction of [4].]
[17] Stephan Weiss. Die Multipliziertafeln: ihre Ausgestaltung und Verwendung, 2003.
[available at http://www.mechrech.info/publikat/MTafel1.pdf]
[18] Yano Tsuneta (矢野恒太). 乘除表: 三位三位 (Calculating tables). 東京: 博文館
(Tokyo: Hakubunkan), 1913. [reprint of Crelle’s 1820 table, not seen]
