274 M. Biot's Abstract of Mr Napier's 



to discuss unknown pretensions not brought forward, and 

 which no one, in the present day, can either discover or appre- 

 ciate. At the epoch of Napier's conception of the logarithms, 

 all mathematicians, all astronomers, who by that time were 

 in great numbers, felt every moment the necessity of some 

 invention which would simplify the frightful numerical cal- 

 culations to which they were incessantly constrained to de- 

 vote themselves for the solution of astronomical triangles, the 

 sole application of mathematics at that time known. Various 

 passages of the scientific history of the period bear witness to 

 the attempts made with this object in view, as well by Byrge 

 as doubtless by many others, among whom Kepler reckons him- 

 self. And, truly, when we consider what it must have been 

 to calculate numerically tables of sines and natural tangents, for 

 a radius expressed by a million, or even ten millions of parts, of 

 which it was then composed, — when we reflect that all this de- 

 manded continual divisions and multiplications which required 



applied to the ranges of sexagesimal derivations then in use, the observations 

 of Archimedes upon the geometrical and arithmetical progressions, considered 

 in relation to each other ; he may perhaps have even perceived the simpli- 

 fications that might be deduced when it was necessary to multiply or di- 

 vide such progressions by each other. But, in order to bring within such a 

 progression all possible numbers, a mighty step was to be made, and therein 

 consists the idea peculiar to Napier. Moreover, if Byrge had obtained a 

 glimpse of that idea, he had neither followed it out nor published it ; who, 

 then, can assign a value to it in the present day ? Lastly, the proof that 

 Kepler did not attach to this remark of Byrge any right, more or less ap- 

 proximating to the invention of logarithms, is, that in those same Rudolphine 

 Tables, he says expressly that Napier is the inventor — M. Biot. 



All that is urged in this note by M. Biot wiU be found to be comprehended 

 in Mr Napier's investigations of the matter. (Memoirs, c. X.) Nor can we agree 

 with M. Biot in his somewhat impatient remark, " Mais qu'est-il besoin de 

 discuter des titres inconnus, que Ton ne produit pas, et que personne au- 

 jourd'huis ne peut voir, ni apprecier ?" He appears to have overlooked Mr 

 Napier's exposition of the very great injustice done to Napier, in this matter, 

 by Dr Charles Hutton ; and Montucla and others had attached too much im- 

 portance to the claim made for Byrge. It was the duty of Napier's biogra- 

 pher to clear up completely, as he has done, and, as we believe, required to 

 be done, the sole and unquestionable right of Napier to the invention of lo- 

 garithms, both in their original form, which may be termed the parent loga- 

 rithms, and in their modified form, called common logarithms, and sometimes 

 attributed solely to Briggs. Mr Napier's investigation of the matter has left 

 no dubiety, and has, moreover, elicited this additional and highly valuable 

 testimony and tribute from M. Biot. — Translator. 



