John Napier

“ From the above example we see that the error in the logarithm found by Napier’s method amounts only to unity in the 15th place, so that his method of computation clearly gives accurate results far in excess of his requirements, But it is easy to show that Napier’s method may be adapted to meet any requirements of accuracy. In sec. 60, Napier, in suggesting the construction of a table of logarithms to a greater number of places, proposes to take 100000000 as radius. ”

“ Napier’s Canon or Table of Logarithms does not contain the logarithms of equidifferent numbers, but of sines of equidifferent arcs for every minute in the quadrant. A specimen page of the Table is given in the Catalogue under the 1614 edition of the Descriptio.
The sine of the Quadrant or Radius, which he calls Sinus Totus, was assumed to have the value 100000000.
Numerus Artificialis, or simply Artificialis, is used in the body of the Constructio for Logarithm, the number corresponding to the logarithm being called Numerus Naturalis. ”

“ In the description of the third method, on pages 53-54, it is explained that when log. 1 = 0 and log. 10 is assumed equal to unit with a number of cyphers annexed, a close approximation to the logarithm of any given number may be obtained by finding the number of places in the result produced by raising the given number to a power equal to the assumed logarithm of 10, As an example, Napier mentions that, assuming log. 10 = 1000000000, the number of places, less one, in the result produced by raising 2 to the 1000000000th power will be 301029995. ”

“ The “new invention in Denmark” to which Anthony Wood refers as having given the hint to Napier was probably the method of calculation called prosthapharesis (often written in Greek letters προσθαφαιρεσις) , which had its origin in the solution of spherical triangles. The method consists in the use of the formula sin a sin b = 1⁄2 {cos (a − b) − cos (a + b) }, by means of which the multiplication of two sines is reduced to the addition or subtraction of two tabular results taken from a table of sines ”

“ The error, however, although continually increasing, yet retains always the same ratio to the logarithm, except for a very small disturbing element to be afterwards referred to, so that the true logarithm will always be very nearly equal to the logarithm found by Napier’s method of computation less a six hundred billionth part. ”

“ The Tract on Decimal Arithmetic, which introduced the idea of decimal fractions and a notation for them, was published in 1585 in Dutch, under the title of ‘De Thiende,’ and in the same year in French, under the title of ‘La Disme.’
We find Briggs, in his ‘Remarks on the Appendix,’ while sometimes employing the point, also using the notation 25118865 for , distinguishing the fractional part by retaining the line separating the numerator and denominator, but omitting the latter. The form 2|5118865 has also been used. ”

“ On the Occurrence of a Mistake in the Computation of the Second Table; with an Enquiry into the Accuracy of Napier’s Method of Computing his Logarithms. 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. ”

“ The effect of this would be to throw still further back the error involved in taking the arithmetical mean of the limits for the true logarithm. Thus, using the formula given, substituting 100000000 for n, and multiplying the result by that amount as already explained, we should have for the true logarithm of 99999999, the first proportional after radius in the new First table,
1.000000005000000033333 etc.
If we take the arithmetical mean of the limits, we have
1.000000005000900050000 etc. ”

“ The mistake in the Second table may possibly not have been observed by him, as the preparatory tables for the Canons were different.
The mistake was observed by Mr Edward Sang in 1865, when recomputing in full the preparatory tables of Napier’s Canon to 15 places.
It had been previously pointed out by M. Biot, in his articles on Napier in the ‘Journal de Savants’ for 1835, p. 255. The following translation of the passage is given in the ‘Edinburgh New Philosophical Journal’ for April 1836, p. ”