A Napierian logarithm before Napier

At its meeting of 15 July 1912, the Council of the Royal Society of Edinburgh resolved to commemorate the tercentenary of the publication in 1614 of Napier's Mirifici Logarithmorum Canonis Descriptio. From Saturday 25 July 1914 to Monday 27 July 1914 the Royal Society of Edinburgh held a Congress in Edinburgh to honour the Tercentenary. A fine volume was published in the following year C G Knott (ed.), Napier Memorial Volume (Royal Society of Edinburgh, London, 1915). Knott writes in the Preface:-

As regards the Congress itself it is pleasant to recall the goodwill and friendliness which characterised its meetings, attended though these were by men and women whose nationalities were fated to be in the grip of war before a week had passed.

Our library in St Andrews contains at least two copies of this Napier Memorial Volume , one of which still retains many uncut pages.

A number of interesting articles in this volume are difficult to obtain elsewhere. We produce below one by Giovanni Vacca. When he wrote the article Vacca was Professore Incarito of Chinese in the Royal University of Rome. The article makes an interesting comment on the appearance of a logarithm in Pacioli's Summa de Arithmetica.

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.'

Here is a rough translation of the Italian: You want to know for every percentage interest per year, how many years will be required to return double the original capital, you hold the rule 72 in mind, which always you will divide by the interest, and the result will determine in how many years it will be doubled. Example: When the interest is 6 per 100 per year, I say that you divide 72 by 6; that is 12, so in 12 years the capital will be doubled.

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

x log(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.

The correct result is 100 log 2 = 69.31471806. However, the result of the approximation made above actually often makes 72 a better value than 70. For example for Pacioli's own example of r = 6, the correct answer is 11.89566105. We see here that 72/6 = 12 is a better approximation than 70/6 = 11.66666667

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.

Note that 70 is a much better approximation to 100 log 2 = 69.31471806 than 72 but as we noted above that often 72 gives more accurate answers to the problem

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

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