16 KEPOiiT— 1873. 



or quarter-square table of suitable extent is at hand) have the advantage 

 that by their means any number of factors can be multiplied together at 

 once. 



Geuson's table, 1798,13 for multiplications of a somewhat different kind from 

 the rest. 



Crelle, in the introduction to his ' Itcchentafeln ' (1820), mentions a 

 work, ' Tables do Multiplication, i\ I'usagc do MM. les gcomL'trcs, de Mm. les 

 ingenieurs verificateurs du Cadastre, etc' sec. edit. Paris, Chez Valacc, 1812, 

 which he says extends to 500 x 500, and occupies 500 quarto pages ; while, 

 he adds, his own work, -which is four times the extent, occupies only 1800 

 octavo pages. For the full titles of Picarte's ' Tables de MultipHcation ' and 

 ' Tableau Pithagorique,' see under Picaete (1861), in § 3, art. 7. 



Closely connected with multiplication tables ai-e so-called ProportionaJ-parts 

 tables (described in the next article) ; and very frequently in the latter the 

 last figure is not contracted, so that by a mere chauge of the position of the 

 decimal point they become tables of multiples. 



Herwart ab Hohenburg, 1610. Multiplication table, from 2x 1 to 

 1000 X 1000. The thousand multiples of any one of the numbers are con- 

 tained on the same page, so that (as the number 1 is omitted) there are 999 

 pages of tables. By a strange oversight, the numbering begins with 1 on 

 the first page of the table instead of 2, so that the multiples of n are found 

 on page n — \ : this is inconvenient, as the number of the page alone appears 

 on it, so that (say) to find a multiple of 898 we seek the page headed 897. 

 Each page contains 100 lines, numbered in the left-hand column 1, 2, 3, ... ; 

 and besides this column of arguments there are ten columns headed 0, 100, 

 . . . 900. The first figure of the multiplier is therefore found at the top of 

 the column, and the last two in the left-hand column (on p. 3 it will be 

 noticed 200 and 300 are interchanged at the top of the columns). There 

 being more than 1000 pages of thick paper, the book, as De Morgan Jias 

 observed, forms a folio of almost unique thickness. Also, as the pages con- 

 tain 100 lines, pretty Avell leaded, the size of the book is very large ; so that 

 Leslie (Philosophy of Arithmetic, 2nd edit. 1820, p. 246) was quite right in 

 calling it " a very ponderous folio." De Morgan saj^s"the book is exces- 

 sively rare ; a copy sold by auction a few years ago was the onlj' one we 

 «ver saw." 



Ktistner (' Geschichte,' t. iii. p. 8) quotes the remark of Heilbronner (who 

 gives the title of the work, ' Hist. Math.' p. 801), " Docet in his tabulis sine 

 abaco mulliplicationcm atque divisionem perficere," &c., and adds that Heil- 

 bronner could not have seen the work, or he would have described it ; he 

 remembers to have read that it was like a great multiplication table. The 

 title is given by Murhard, and marked with an asterisk to show that he had 

 seen a copy. Hogg gives the title very imperfectly ; and it is clear the work 

 has not been in his hands. There is a complete copy in the Britisli Museum, 

 and a copy in the Graves Library ; but the latter is imperfect, the pages 

 12-25, 120-145, and 468-517 having been lost, and their places supplied 

 with blank paper. On account of the rarity of the work, and the great in- 

 terest attaching to it from the time when it was published, we have thought 

 it worth while to give tlie title in full in § 5. The clearness of the type 

 and the extent of the table (which has not been surpassed, and only equalled 

 by Crelle, 1864), taken in connexion with its early date (fou]- years before 

 Napier's ' Canon Mirificus '), give the work a peculiar interest. De Morgan 

 writes : — " it is truly remarkable that while the difficulties of trigonometrical 



