Memoirs of John Napier of' Merchiston, 279 



kind of progression called, progression by equidifference, or arith- 

 metical progression. Archimedes detected and demonstrated all 

 the relations, of which we have given the exposition above, 

 betwixt these two kinds of progressions, when their terms are 

 thus brought into correspondence. And in order to shew that 

 these properties were applicable to any terms of the two series, 

 he conceived the idea of representing generally these terms by 

 means of letters employed solely as signs of quantities, without 

 any particular numerical value ; thus affording the first example 

 of reasoning applied to figurative symbols representing abstrac- 

 tions, in which, properly, algebra consists, — that powerful intel- 

 lectual instrument for the discovery of the general relations of 

 magnitudes.* 



From this to the logarithms there is but one step, and the 

 logarithms themselves are just indices used, according to the 

 Archimedean principle, to express the rank of each number in 

 an indefinite geometrical progression comprehending all num- 



• Mr Napier particularly observes this fact, and also shews that Archimedes, 

 in lug very curious work Arenaritcs, obtained a glimpse of those three great 

 features of the modem science of numbers, Arabic notation, the logarithms, 

 and the language of algebra, unconscious, however, of the mighty mysteries 

 he had touched, and which remained to be successively unfolded in after ages, 

 (pp. 343, 344, 348.) It is complimentary to Mr Napier's treatment of his 

 subject, that M. Biot, in this paper, has followed the same plan of explaining 

 and illustrating the logarithms, by giving a history of the first appearance of 

 the logarithmic principle in the Arenarius of Archimedes, and then drawing 

 (as Doctor Hutton, and others, had failed to do,) the essential distinction 

 betwixt that glimpse of the principle, and the great invention of logarithms. 

 And it is but justice to Mr Napier's work to observe, that it follows out that 

 historical and numerical exposition, much more fully than M. Biot gives it. 

 Mr Napier also makes this observation, " That the Arabic system itself is 

 essentially logarithmic, and that the properties of the Archimedian theorem 

 may present themselves to a very ordinary calculator upon a consideration of 

 the simple notation 1000.'' He illustrates this by shewing, that the cyphers 

 serve to number the steps which the figure has taken in the decuple ascending 

 scale of progression ; and consequently, that in this view they are actually 

 indices of the value of the particular term. So " that the mere addition of the 

 cyphers in the Arabic scale will afford the same result as the multiplication of 

 the terms,*' &c. (p. 437.) M. Biot's immediate adoption, in his exposition of 

 the logarithmic principle, of the Cartesian exponential artifice, which abbre- 

 viates the written expression, tends to obscure this fact, which is certainly 

 interesting in a history of the gradual development of that mighty power of 



numbers, the logarithms {See the Memoirs^ p. 435, et infra.) — Translator. 



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