Page:The Construction of the Wonderful Canon of Logarithms.djvu/121

Notes. 97

The first method of construction, described on pages 48-50, involves the extraction of fifth roots, from which we may infer that Napier was acquainted with a process by which this could be done. The inference is confirmed by an examination of his ‘Ars Logistica,’ at p. 49 of which (Lib. II., ‘Logistica Arithmetica,’ cap. vii.) he indicates a method by which roots of all degrees may be computed. This method of extraction is referred to by Mark Napier in the ‘Memoirs,’ p. 479 seg., and a translation is there given of the greater part of the chapter above referred to. A method based on the same principles is given by Mr Sang in the chapter “On roots and fractional powers” in his ‘Higher Arithmetic,’ and these principles are also made use of by Mr Sang in his tract on the ‘Solution of Algebraic Equations of all Orders,’ published in 1829.

No general method of extracting roots was known at the time, and it does not appear that Napier had communicated his method to Briggs. At any rate, Briggs did not employ the first method described in computing the logarithms for his canon.

The second method, described on page 51, is a method suitable for finding the logarithms of prime numbers when the logarithms of any two other numbers as 1 and 10 are given. This is done by inserting geometrical means between the numbers, and arithmetical means between their logarithms. The example given is to find the logarithm of 5, but as the example terminates abruptly after the second operation, I append the following table from the article on Logarithms in the ‘Edinburgh Encyclopedia’ (1830), which will sufficiently exhibit the method of working out the example, though it is not carried to the same number of places as that in the text.

THE TABLE.