TRANSACTIONS Of THE SECTIONS. 15 



quot sunt fiourae post peiiodum. Ut 1000000004 valet idem, quod 10000000^^. 

 Item 2o-803, idem quod 25^%Vo- Item 9999998-0005021, idem valet quod 

 9999998i^f fifJ^^, & sic de ceteris." On p. 8 we have 10-502 multiplied by 3-216, 

 and tlie result found to be 33-774432 ; and on pp. 23 and 24 occur decimals not 

 attached to integers, viz. -4999712 and -0004950. These show that Napier was 

 in possession of all the conventions and attributes that enable the decimal point 

 to complete so symmetrically our system of notation, viz. (1) he saw that a point 

 or separatrix was quite enough to separate integers from decimals, and that no 

 signs to indicate primes, seconds, &c. were required ; (2) he used ciphers after the 

 decimal point and preceding- the first significant figure ; and (3) he had no objec- 

 tion to a decimal standing by itself without any integer. Napier thus had com- 

 plete command over decimal fractions, and understood perfectly the nature of the 

 decimal point ; and I believe (except, perhaps, Briggs) he is the first person of 

 whom this can be said. When I first read the ' Constructio ' I felt some doubt 

 as to whether Napier really appreciated the value of the decimal point in all its 

 bearings, as he seemed to have regarded it to some extent as a mark to separate 

 figures that were to be rejected from those that were to be retained ; but a careful 

 examination has led me to believe that his views on the subject were pretty nearly 

 identical with those of a modem arithmetician. There are perhaps 200 decimal 

 points in the book, afibrding abundant evidence on the subject. 



The claim of Napier to the invention of the decimal point is not here noticed 

 for the first time, as both Delambre (' Hist, de I'Astron. mod.,' t. i. p. 497) and 

 Hutton allude to the decimal fractions in the ' Constructio ' (though the latter 

 claims priority for Pitiscus), and Mr. Mark Napier (' Memoirs of John Napier,' 

 p. 454) devotes a good deal of space to it. 



Briggs also used decimals, but in a form not quite so convenient as Napier. 

 Thus he writes 63-0957379 as 63 0957379 , viz. he prints a bar under the decimals : 

 this notation first appears, without any explanation, in his ' Lucubrationes,' ap- 

 pended to the * Constructio " *. Briggs used this notation all his life (he died in 

 1631), and he explains it in the ' Arithmetica Logarithmica ' of 1624. Oughtred's 

 symbol, first used (as far as I know) in his ' Arithmeticse in numeris . . . Clavis,' 

 1631, dittered only from Briggs 's in the insertion of a vertical bar to separate the 

 decimals from the integers more completely — thus, 63j 0957379. Oughtred's and 

 Briggs's notations are essentially the same, the improvement of the former being 

 no doubt due to the uncertainty that sometimes might be felt as to which was 

 the first figui-e above Briggs's line. From an inspection of MSS. of Briggs and 

 Oughtred (the Birch MSS. contain abetter of Briggs to Pell ; and the Royal Society 

 has a Peter Ramus, with many of his MS. notes, while the Cambridge University 

 copy of the ' Constructio ' is annotated in MS. by Oughtred) it is apparent that, 

 in writing, Briggs and Oughtred both made the separating rectangle in exactlv 

 the same way ; viz. they wrote it 6t: [ 095 7379, the upright mark usually being just 

 high enough to fix distinctly what two figures it was intended to separate, and they 

 rarely took the trouble to continue the horizontal line to the end of the decimals 

 if there were many. Thus Oughtred was a follower of Briggs, and only made an 

 improvement in the printed notation. It is clear that, in writing, Briggs's rect- 

 angle was pretty nearly as convenient as Napier's point ; and there is every proba- 

 bility that Briggs appreciated all the properties of the " separatrix " as clearly as 

 Napier ; but in his 8 pp. of ' Lucubrationes ' he has left much less to judge by than 

 has Napier. In 1624, as we can see from his ' Arithmetica Logarithmica,' he had 

 full command over decimal arithmetic in its present form (except that he used 

 the rectangular " separatrix " instead of the point. Gunter was a follower of 

 Napier, and employed the point (but see De Morgan). In his 'Description and 

 use of the Sector' (1623) he uses the point throughout pretty much as we do at 

 present (e. g. p. 40 of the ' First Booke of the Crosse-Staffe,' "As 4-50 unto 1-00 : so 

 1-000 unto 0-222 "), except that he called the decimals parts in the text. In Roe's 



* A curious blunder is made in Bartholomew Vincent's reprint of the ' Constructio,' 

 Lyons, 1620 (of which there is a copy in the Royal Society's Library). The printer, un- 

 aware that the position of Briggs's subscript rules had any meaning, has disposed them 

 symmetrically under all the figures. 



