## A Napierian logarithm before Napier

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In the ordinary histories of mathematics there are very few suggestions about the way in which John Napier conceived the idea of his great discovery, truly one of the most beautiful made by man, not only As supplying a new method for saving time and trouble in tedious calculations, but also as forming one of the most important steps towards the discovery of the infinitesimal calculus.

Generally the only reference made is to ... Archimedes.

I have lately observed that in the *Summa de Arithmetica* of Fra Luca Pacioli, printed in Venice in 1494, there is the following problem:

(Fol. 181, n. 44.) 'A voler sapere ogni quantità a tanto per 100 I'anno, in quanti anni sarà tornata doppia tra utile e capitale, tieni per regola 72, a mente, il quale sempre partirai per l'interesse, e quello che ne viene, in tanti anni sarà raddoppiato. Esempio: Quando l'interesse è a 6 per 100 I'anno, dico che si parta 72 per 6; ne vien 12, e in 12 anni sarà raddoppiato il capitale.'

Luca Pacioli says that the number of years necessary to double a capital placed at compound interest, is the number resulting from the division of the fixed number 72 by the rate of interest per 100.

If we try to explain the mystery of this number 72 (and the reason of this mystery was impenetrable to the succeeding arithmeticians, for instance, Tartaglia), we easily see in modern notation that

(1 +

r/100)^{x}= 2

or, taking Napierian logarithms

xlog(1 +r/100) = log 2

and to a first approximation, if *r* is small:

x= 100 log 2/r

therefore 72 is only a rough calculation of the number 100 log 2.

This problem is to be found, without explanation, in modern treatises, for instance in the introduction to the

*Tables d'intérêt composé*of Pereyre.

Sometimes the number 70 is given instead of 72.

If this problem were known to Napier, might it not have been a suggestion leading to his further discovery? Perhaps a research in his manuscripts can explain this point.

In any case it is curious to note that the Napierian logarithm of 2 was printed before the year 1500, with an approximation of 3 per 100.

JOC/EFR March 2006

The URL of this page is:

http://www-history.mcs.st-andrews.ac.uk/Extras/Pacioli_logarithm.html