Page:The Construction of the Wonderful Canon of Logarithms.djvu/109

NOTES, 85

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.

Logarithmus, corresponding to which Numerus Vulgaris is used, is however employed in the title-page and headings of the Constructio, and in the Appendix and following papers. It is also used throughout the Descriptio published in 1614; and as the word was not invented till several years after the completion of the Constructio (see the second page of the Preface, line 12), the latter must have been written some years prior to 1614.

For shortness, Napier sometimes uses the expression logarithm of an arc for the logarithm of the sine of an arc.

The Antilogarithm of an arc, meaning log. sine complement of arc, and the Differential of an arc, meaning log. tangent. of arc (see Descriptio, Bk. I., chap. iii.), are terms used in the original, but as they have a different signification in modern mathematics, we do not use them in the translation.

Prosthapharesis was a term in common use at the beginning of the seventeenth century, and is twice employed by Napier in the Spherical Trigonometry of the Constructio as well as in the Descriptio. The following short extract from Mr Glaisher’s article on Napier, in the ‘Encyclopedia Britannica,’ indicates the nature of this method of calculation.

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 ), which had its origin in the solution of spherical triangles. The method consists in the use of the formula sin a sin b = ½ {cos (a &minus; b) &minus; 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; and as such products occur in the solution of