Memoirs of John Napier of Merdmton. 283 



particular geometrical progression which he adopted in the 

 Arenarius. 



Such is the invention of Napier. He has rendered continu- 

 ous and universal, throughout the whole system of numbers, 

 those advantages which Archimedes had only obtained intermit- 

 tingly, and for particular numbers. If it be asked why Archi- 

 medes did not make this second step, which now appears to us 

 so little removed from the first, a plausible reason, in our opinion, 

 may be found in the nature of the literal symbols employed in 

 his time to designate numbers. For the signification of these 

 characters being absolute, numbers only differing slightly from 

 each other were often expressed by characters having no appa- 

 rent mutual relation ; or, if their expressions had any elements in 

 common, the ratio of their magnitudes to quantities of a different 

 kind was not manifested by the numerical expression itself; 

 whereas, in our actual method of writing the numbers, those two 

 kinds of evidence exist, and, as it were, appeal to the eye, es- 

 pecially when, generalizing the idea which attaches a value of 

 position to the numerals, we extend it inversely to the subdivi- 

 sions of unity by means of decimals. We have here, then, 

 another of those examples of the influence of symbols over the 

 extension of our ideas, in which the history of mathematics 

 abounds. And, in reference to this subject, be it remarked, 

 that Napier was the first in Europe who employed that genera- 

 lization, so simple in the method of noting the decimal sub- 

 divisions, which was indispensable to effect those successive 

 subtractions, and confine them betwixt the limits assigned 

 to his error. To convince ourselves that that idea was not so 

 easy a discovery as we are apt to believe, now that use has 

 familiarized it, we have only to look at the complicated and 

 nearly impracticable methods by which Stevin, an experienced 

 and ingenious geometrician, essayed to write the decimals, very 

 shortly before. Pitiscus, indeed, substituted the present notation 

 in the second edition of his trigonometry, 1612; and the Canon 

 mirifictis, in which Napier employs that notation, was not 

 published until the year 1614, which leaves with Pitiscus 

 the merit so far as priority of pubhcation is concerned. But 

 that Napier, who constantly employed it in his tables, must 

 have conceived it independently of Pitiscus, appears incon- 



