June 29, 1917] 



SCIENCE 



663 



" National Research Council." Membership 

 of committees. Edwin Bidwell Wilson 



Massachusetts Institute of Technology, 

 Cambridge, Mass. 



THE DECIMAL POINT 



Students of the history of science are con- 

 tinually impressed by the fact that we are 

 truly in scientific achievements the heirs of 

 all the ages with progenitors not Hmited by 

 any boimds of time or place. The historian 

 John Fiske in one of his essays^ pertinently 

 remarks : 



The thinker who elaborates a new system of phi- 

 losophy, deeper and more comprehensive than any 

 yet known to mankind, though he may work in 

 solitude, nevertheless does not work alone. The 

 very fact which makes his great scheme of thought 

 a success, and not a failure, is the fact that it puts 

 into definite and coherent shape the ideas which 

 many people are more or less vaguely entertaining, 

 and that it carries to a grand and triumphant con- 

 clusion processes of reasoning in which many per- 

 sons have already begun taking the earlier steps. 



The late and lamented Pierre Duhem in that 

 wonderful work, " Le Systeme du Monde," 

 with the fifth and concluding volume fortu- 

 nately completed in manuscript before his 

 recent death, opens his work with a statement 

 to the effect that to trace the origin and 

 genesis of great scientific ideas one is gradu- 

 ally led back to the point where history 

 ceases. 



The development of decimal fractions fur- 

 nishes an excellent illustration of the process 

 mentioned. This story was well told in the 

 Teachers College Bulletin of 1910 by David 

 Eugene Smith, in an article entitled, " The 

 invention of the demical fraction." I propose 

 in the present paper to discuss briefly one 

 point of the development, namely the appear- 

 ance of the decimal point itself; I am adding 

 also an early approach to decimal fractions 

 not known to writers on the subject, based 

 upon a study of a Vienna manuscript of the 

 fifteenth century.^ 



1 "A century of science, and other essays," New 

 York, 1899. 



2 Codex Vindobonensis 4770. 



]Sr. L. W. A. Gravelaar, in discussing 

 Ifapier's works' ascribes to Napier priority in 

 the use of the decimal point. In a further 

 article, " De notatie der decimale breuken," 

 Gravelaar* purports to show that Napier was 

 not familiar in 1616 and 1617, with the edi- 

 tions of the " Trigonometry " of Pitiscus 

 which appeared in 1608 and 1612, containing 

 the first appearance in print of the decimal 

 point after Stevin's systematic exposition of 

 the subject of decimal fractions in 1585. To 

 me the whole procedure of Gravelaar borders 

 so closely on the absurd that it would not 

 merit discussion, if it had not been accepted 

 somewhat seriously by other writers.^ 



Enestrom, the editor of the Bibliotheca 

 Mathematica, refers"* to Gravelaar's work, as 

 follows : " Nach Herrn Gravelaar ist Neper 

 der erste, bei dem das Komma (Ehabdologia, 

 1617) und das Piinktchen (Constructio, 1616) 

 als wirkliche Dezimalzeichen vorkommen; 

 Pitiscus hatte zwar schon ein Piinktchen 

 angewendet, aber dies ist nur als ein Scheide- 

 zeichen auzusehen." That the point used 

 with decimal significance in Pitiscus is used 

 with full appreciation by Pitiscus should be 

 evident simply from the fact that at this time 

 the work of Stevin on decimal fractions was 

 widely known; further Pitiscus who uses a 

 bar in the text of his " Trigonometry " of 

 1608 and of 1612, 13/00024 where the point is 

 used in the Tables, explicitly says of 13/00024, 

 " fractione scilicet 24/100000 neglecto," mean- 

 ing that in place of 13.00024 he uses 13 as an 

 approximation. Among others von Braun- 

 mjihl, in his " Vorlesungen iiber Geschichte 

 der Trigonometric,"' mentions the use of the 

 point in decimal fractions by Pitiscus. 



The further fact should be noted that the 

 Constructio of 1616 is the English transla- 



3 Verh. d. Koningl. Akad. v. Wetenschappen, 

 Amsterdam, 1899, Deel. VI., No. 6. 



* Nieuw Archif voor Wislcunde, Amsterdam, 

 1900. 



6 Notably by Glaisher and others in the ' ' Napier 

 Tercentenary Volume. ' ' 



« Bibliotheca Mathematioa, Vol. VI., third series, 

 pp. 108-109. 



' Vol. I., p. 225. 



