Page:The Construction of the Wonderful Canon of Logarithms.djvu/114

90 NOTES. Notwithstanding the simplicity and elegance of the last of these, it was long after Napier’s time—in fact, not till the eighteenth century—that it came into general use.

The subject is referred to by Mark Napier in the ‘Memoirs,’ pp. 451- 455, and by Mr Glaisher in the Report of the 1873 Meeting of the British Association, Transactions of the Sections, p. 16.

It is evident that a mistake must somewhere have occurred in the computation of the Second table, since the last proportional therein is given (sec. 17) as 9995001.222927, whereas on trial it will be found to be 9995001.224804.

This mistake introduced an error into the logarithms of the Radical table, as the logarithm of the first proportional in that table is deduced from the logarithm of the last proportional in the Second table by finding the limits of their difference, But these limits are obtained from the proportionals themselves, and, as shown above, one of these proportionals was incorrect: the limits therefore are incorrect, and consequently the logarithm of the first proportional in the Radical table.

We see the effect of this in the logarithm of the last proportional in the Radical table, which is given (sec. 47) as 6934250.8, whereas it should be 6934253.4, the given logarithm thus being less than the true logarithm by 2.6, or rather more than a three millionth part.

The logarithms as published in the original Canon are affected by the above mistake, and also, as mentioned in sec. 60, by the imperfection of the table of sines. It seems desirable, therefore, to enquire whether in addition any error might have been introduced by the method of computation employed.

Before entering on this enquiry, we should premise that in comparing Napier’s logarithms with those to the base e&minus;1 (which is the base required by his reasoning, though the conception of a base was not formally known to him), it must be kept in view that in making radius 10,000,000