14 REPORT 1873. 



conformity with the practice of Stevinus. The same form is adopted in an ex- 

 ample of abbreviated multiplication which subsequently occurs .... The preceding 

 statement will sufficiently explain the reason why no notice is taken of decimals 

 in the elaborate explanations which are given by Napier, Biiggs, and Kepler, of 

 the theory and construction of logarithms; and indeed we tiud no mention oi 

 them in any English author between 1(319 and 16.31. In that year the 'Loga- 

 rithmicall Arithmetike ' was published by Gellibrand and other friends of Briggs 

 (who died the year before), with a much more detailed and popular explanation 

 of the doctrine of logarithms thau was to be found in the ' Aritlimetica Loga- 

 rithmica.'. . . .From this period we may consider the decimal Arithmetic as fully 

 established, inasmuch as the explanation of it began to form an essential part of 

 all books of practical arithmetic. The simple method of marking the separation 

 of the decimals and integers by a comma, of which Napier has given a solitary 

 example, was not, however, generally adopted." 



De Morgan ('Arithmetical Hooks,' 1847, p. xxiii) writes : — " Dr. Peacock mentions 



is the person who first made this distinction a permanent language, not using it 

 merely as a rest in a process, to be useful in pointing out afterwards how another 

 process is to come on or language is to be applied, but making it his final and 

 permanent indication as well of the way of pointing out where the integers end 

 and the fractions begin, as of the manner in which that distinction modifies opera- 

 tions. Now, first, I must submit that Napier did not do this ; secondly, that if 

 he did do it, Richard Witt did it before him." 



De Morgan then states that he has not seen Wright's translation of 1616 ; but he 

 proceeds to examine Napier's claim as resting on the two examples in the ' Rab- 

 dologia,' in the first of which a comma is used, but only in one place. After 

 this examination he proceeds : — " I cannot trace the decimal point in this ; but if 

 required to do so, I can see it more distinctly in Witt, who published four years 

 before Napier. But I can hardly admit him to have arrived at the notation of 

 the decihial point. ..." * 



I agree with De Morgan in all that he has stated in the above extracts, and 

 do not think that the single instance of the comma used in the course of work, 

 and replaced immediately afterwards by exponential marks, is a sutlicieut ground 

 for assigning to Napier the invention of the decimal point, or even afibrds a pre- 

 sumption that he made use of it at all in the expression of results. 



StiU one of the objects of this paper is to claim (provisionally, of course, till 

 evidence of any earlier use is produced, if such there be) the invention of the 

 decimal point for Napier, but not on account of any thing contained in the ' Rab- 

 dologia.' The mathematical works published by Napier in his lifetime (he died 

 in 1617) were his 'Mirifici Logarithmorum Canonis Descriptio,' 1614, containing 

 the first announcement of the invention of logarithms, and the ' Rahdologia,' 

 1617, giving an accoimt of his almost equally remarkable (as it was thought at 

 the time) invention of numbering rods or "bones." In 1619, two years after his 

 death, the ' Mirifici Logarithmorum Canonis Constructio,' containing the method of 

 construction of the canon of logarithms, was published, edited by his son ; and in 

 this work the decimal point is systematically used in a manner identical with 

 that in which we employ it at the present day. I can find no traces of the decimal 

 point in Wright's [translation of the ' Descriptio,' 1616 ; and, as De Morgan says, 

 the use of the decimal separator is not apparent in Witt. The earliest work, 

 therefore, in which a decimal separator was employed seems to be Napier's 

 posthumous work the ' Constructio' (1619), where the following definition of the 

 point occurs on p. 6: — "In numeris periodo sic in se distinctis, quicquid post 

 periodum notatur fractio est, cujus denominator est unitas cum tot cyphrispost se, 



* In an essay "On some points in the history of Arithmetic" (Companion to the 

 Almanac for 1851), De Morgan lias fiu-ther discussed the invention of the decimal point, 

 but in the same spirit as regards Napier. He seems never to have seen Napier's ' Con- 

 structio' of 1619 ; and the work is very rare. The only copy I have been able to see is 

 that in the Cambridge University Library. 



