ON MATHEMATICAL TABLES. 15 



Art. 18. Logistic and proportional logarithms. 



19. Tables of Gaussian logaritlims. 



20. Tables to convert Briggian into hyperbolic logarithms, and vice versa, 



21. Interpolation tables. 



22. Mensuration tables. 



23. Dual logarithms. 



24. Mathematical constants. 



25. Miscellaneous tables, figurato numbers, &c. 



Art. 1. MnU'ii^llcation Tables. 



The use of the multiplication table is so essential a part of the history of 

 Numeration and Arithmetic, that for information witli regard to its introduc- 

 tion and application -^e must refer to Peacock's ' History of Arithmetic ' in 

 the ' Encyclopaedia Metropolitana,' to De Morgan's ' Arithmetical Books ' 

 (London, 1847), as well as to Heilbronner, Delambre, &c. (see § 2, art. 3), 

 to Leslie's * Philosophy of Arithmetic,' and perhaps to Barlow's * Theory of 

 jS'umbers' (London, 1811), in most of which references to other works Avill 

 be found. There is abundant evidence that, till comparatively recent times 

 (say the beginning of the eighteenth century), multiplication was regarded 

 as a most laborious operation ; this is testified not only indirectly by the very 

 simple examples given in old arithmetics, but explicitly by Decker in his 

 ' Eerste Deel vande Nieuwe Telkonst ' (see Phil. Mag. Suppl. Number, Dec. 

 1872). The great popularity of Napier's bones, and the eagerness with 

 which they were received all over Europe, show how great an assistance the 

 simplest contrivance for reducing the labour of multiplications was considered 

 to be. It would be interesting to know how- mucli of the multiplication 

 computers were in the habit of committing to memory, as the bones would 

 be no great help to any one who knew it as far as nine times nine. In this 

 Report, however, we are only concerned with extended multiplication tables 

 (viz. such as are to be used as tables, and were not intended to be committed 

 to memory). The earliest printed table of multiplication we have seen re- 

 ferred to is Thomas Finck's ' Tabulae Multiplicationis et Divisionis, seorsim 

 ctiam Moneta3 Danicas accommodatae,' Hafnia;, 1604 (which title De Morgan 

 obtained from Prof. "Werlauff, Royal Librarian at Copenhagen) ; but the 

 work, from its title, must have been rather a ready reckoner than a proper 

 scientific table. The earliest largo table, which, strange to say, is still as exten- 

 sive as aiiy (it has been equalled, but not surpassed, by Ckelie, 1864), is Heeavaet 

 AB Hohenbukg's ' Tabulae Arithmetical 7rpoi7da(j>aipeaeios Universales,' 1010, 

 described at length below. Of double-entry tables, Creli-e's ' Rechentafeln,' 

 1804, is the most useful, and the most used, for general purposes. The other 

 important tables are chiefly for multiplication by a single digit. 



A multiplication table is usually of double entry, the two arguments being 

 the two factors ; and when so arranged, it is frequently called a " Pythagorean 

 Table.'' The great amount of room occupied by Pythagorean tables (no 

 table so arranged could extend to 1000 x 10,000, and be of practicable size) 

 has directed attention to modes of arrangement by which multiplication can 

 be performed by a table of single entry ; the most important of these are 

 tables of quarter-squares, which are described in § 3, art. 3, where are also 

 added some remarks on multiplication tables of single entry. See also Dilling, 

 described below. 



It is almost unnecessary to add that, when not more than seven or ten 

 figures arc required, multiplication can be performed at once by logarithms, 

 which (though not the best method for two factors when either a Pythagorean 



