16 REPORT 1873. 



< Tabulae Logarithmicae, or Two Tables of Logaritlimes ' (1633), the explanatory- 

 portion of whicli was written by Wingate, decimal points are used everywhere ; 

 thus we have (p. 29) " As 1 is to "OZOSTS : so is the square of the circumference to 

 the superficiall Content;" and he takes the case of circumference 88*75, and 

 obtains by multiplication (performed by logarithms) 626-8 for the result. Wingate 

 refers for explanation on the decimal point to his 'Arithmetic ; ' but I have not seen 

 any edition of this work that was published previously to Roe's tables (Watt gives 

 one 1630). In his ' Construction and Use of the Line of Proportion,' 1628, Wingate 

 also uses decimals and decimal points. 



On the whole, therefore, it appears that both Napier and Briggs saw that a 

 mere separator to distinguish integers from decimals was quite sufficient without 

 any exponential marks being attached to the latter— but that Napier used a simple 

 point for the purpose, while Briggs employed a bent or curved line, for which in 

 print he substituted merely a horizontal bar subscript to the decimals — that Gunter 

 and Wingate followed Napier, while Oughtred adopted Briggs's method and made 

 an improvement in the mode of printing it. Napier has left so many instances 

 of the decimal point as to render it pretty certain that he thoroughly appreciated 

 its use ; and there is every reason to believe that Briggs had (in 1619) an equal 

 command over his separator, although there are not enough printed instances of 

 that date to prove it so conclusively as in Napier's case (there is no instance in 

 the * Lucubrationes ' in which a quantity begins with a decimal point ; and there 

 could not well be one). Napier did not use the decimal point in the 'Descriptio' 

 (1614), nor in his book of arithmetic, first printed under the editorship of Mr. 

 Mark Napier in 1839 ; and there is only the single doubtful case in the ' Rabdo- 

 logia,' 1617 ; so that there is reason to believe that he did not regard it as generally 

 applicable in ordinary arithmetic. The only previous publication of Briggs's that 

 I have seen is his ' Chilias,' 1617, which contains no letterpress at all. The 

 fact that Napier and Briggs use different separating notations is an argument 

 against either having been indebted to the other, as whoever adopted the other's 

 views would probably have accepted his separator too. It is doubtful whether, if 

 Napier had written an ordinary arithmetic at the close of his life, he would have 

 used his decimal point. Wingate employed the point with much more boldness, 

 and regarded it much more in the light of a permanent symbol of arithmetic 

 than did (or could) Napier. The Napierian point and the Briggian separator 

 differ but little in writing ; and as far as MS. work is concerned it is quite easy to 

 see why many should have considered the latter preferable ; for it was clear, and 

 interfered with no existing mark. A point is the simplest separator possible ; but 

 it had already another use in language. In all the editions of Oughtred's ' Clavis * 

 (which work held its ground till the beginning of the last century) the rectangular 

 separator was used ; and it is not unlikely that it was ultimately given up, for the 

 same reason as that which I believe will lead to the abandonment of the similar 

 sign now used in certain English books to denote factorials, viz. because it was 

 troublesome to jrrint. But be this as it may, it is not a little remarkable that the 

 first separator used (or, more strictly, one of the first two) should have been that 

 which was finally adopted after a long period of disuse. All through the seven- 

 teenth century exponential marks seem to have been common, on which see the 

 accounts in Sir Jonas Moore's * Moor's Arithmetick,' London, 1660, p. 10, and 

 Samuel Jeake's 'Compleat Body of Arithmetick,' London, 1701 (written in 1674), 

 p. 208, which are unfortunately too long to quote in this abstract. 



In his account Peacock is inaccurate in saying that the ' Logarithmicall Arith- 

 metike' was published by Gellibrand and others, the mistake having arisen no 

 doubt from a confusion with the ' Trigonometria Britannica,' ] 633 ; and in any 

 case the reference is not a good one, as the ' Arithmetike ' of 1631 shows (for 

 reasons which must be passed over here) a less knowledge of decimal arithmetic 

 than do any of the chief logarithmic works of this period. Also Briggs died in 

 1631, not 1630. 



There is no doubt whatever that decimal fractions were fh'st introduced by Ste- 

 vinus in his tract ' La Disme.' De Morgan (' Arithmetical Books,' p. 27) is quite 

 right in his inference that it appeared in French in 1585 attached to the 'Pratique 

 d'Arithm^tique.' A copy of this work (1585) with 'La Disme' appended is now 



