664 



SCIENCE 



[N. S. Vol. XLV. No. 1174 



tion by "Wright of Napier's work, and while it 

 did receive examination by Napier before pub- 

 lication, yet the probability is that the in- 

 novation of the decimal point was introduced 

 by Wright. To prove even independent de- 

 velopment of the decimal point one would 

 have to show that neither Wright nor Napier 

 had the 1608 or 1912 edition of Pitiscus or 

 any knowledge of the work ; the earlier edition 

 of Pitiscus is cited by Napier. 



Cantor does refer^ to the use in manuscript, 

 as he carefully mentions, of the decimal point 

 by Biirgi " wahrscheinlieh kurz nach . . . 

 August 1592 " ; even of this use in MS. Cantor 

 says^ " scheint zuerst," whereas of the use by 

 Pitiscus, he says " angefiihrte Thatsache, dass 

 Pitiscus in Tabellenanhange seiner Trigo- 

 nometrie von 1608 sowie von 1612 . . . das 

 Decimalstellen abtrennende Piinktchen be- 

 nutzt hat." My own feeling is that Cantor 

 mentions Biirgi's use of the point in MS. here 

 to strengthen Cantor's similar claim on MS. 

 evidence for priority for Biirgi in the inven- 

 tion of logarithms.-"* 



So far as the complete explanation of 

 decimal fractions is concerned that appears 

 in the work of Stevin, " La Disme," of 1585, 

 which work in Flemish and in French evi- 

 dently attained rapidly wide circulation. In 

 1603 Johann Beyer in his " Logistica deci- 

 malis " gave the explanation of operations 

 with decimal fractions, using a period in com- 

 bination with Stevin's method of designating 

 the last order of the last place by a super- 



V 



imposed numeral; thus, 8.798 for 8.00798. 

 Even in 1616 Kepler in his " Auszug auss der 

 \u'alten Messe-Kunst Archimedis " used a 

 comma (reversed) quite as we do a decimal 

 point, and gives a suiBciently complete exposi- 

 tion of the use of decimal fractions. 



It seems to me to be distinctly unfortunate 

 that Gravelaar should have given, without any 

 ground as I believe I have shown, credit to 

 Napier for a somewhat important contribu- 



8 ' ' Vorlesungeu, ' ' Vol. II., second edition, p. 

 618. 



dLoc. cit., pp. 617, 619. 



10 See Cajori, "Napier Terc. Volume," pp. 

 101-102. 



tion to the theory of notation of decimal 

 fractions; certain writers have been misled 

 into according to Napier honor in this field, 

 where no credit is due. 



Except for the fact that a question of 

 nationality is introduced we might say that 

 neither Pitiscus nor Napier is worthy of more 

 than a passing note in the history of the de- 

 velopment of decimal fractions. Decimal 

 fractions and a more or less convenient nota- 

 tion for the same were historically inevitable, 

 just as logarithms, the analytic geometry, the 

 graphical representation of complex numbers, 

 the calculus, and many other developments 

 of mathematics were inevitable. The earlier 

 steps leading to these processes can now be 

 traced and it is increasingly evident that a 

 succession of thinkers made possible these at- 

 tainments. Similarly the preparatory con- 

 tributions of many minds have made possible 

 the simultaneous discovery, frequently, of ap- 

 parently new mathematical theories. 



In the pamphlet by Professor Smith, which 

 I have cited, there appears the reproduction 

 of a page in ChristofF Rudolfi's arithmetic of 

 1526 in which there is the computation of 

 interest at 5 per cent., involving funda- 

 mentally decimal fractions, using a bar as a 

 separatrix. Concerning this Enestrom says :^^ 



In the year 1492 Francesco PeUizzati (or 

 Pellos) published at Turin an arithmetic in 

 which, in one example, division by 100 of 

 6976587 is given by 69765.87; the point is 

 also used in dividing by 30, 400, 3000, and 

 the like, pointing oil one, two, or three places 

 and afterwards writing the remainder as an 

 ordinary fraction. PeUizzati can, however, 

 not be credited with any real comprehension 

 of decimal fractions. The page in question 



u Bibliotheca Mathematica, Vol. X., p. 243, m 

 his article, ' ' Tiber das angebliche Dezimalbnich- 

 zeichen einiger der altesten gedruckten Eechen- 

 biicher. ' ' 



Hier hat EudolfE wirklich mit Dezimalbriichen 

 gerechnet; wenn man ferner bemerkt, dass der 

 Strich bei Eudolff gewissermassen ein Komma ist 

 ... so kaHn man sagen, dass sowohl die moderne 

 Anwendung wie die moderne Bezeichnung bei 

 Eudolff vorkommt . . . ohne dass Eudolff die Trag- 

 weite seines Verfahrens verstanden hat. 



