[6] In the _Rabdologia_ (1617) he speaks of the canon of logarithms as "a me longo tempore elaboratum." [7] A careful examination of the history of the method is given by Scheibel in his _Einleitung zur mathematischen Bücherkenntniss_, Stück vii. (Breslau, 1775), pp. 13-20; and there is also an account in Kästner's _Geschichte der Mathematik_, i. 566-569 (1796); in Montucla's _Histoire des mathématiques_, i. 583-585 and 617-619; and in Klügel's _Wörterbuch_ (1808), article "Prosthaphaeresis." [8] Besides his connexion with logarithms and improvements in the method of prosthaphaeresis, Byrgius has a share in the invention of decimal fractions. See Cantor, _Geschichte_, ii. 567. Cantor attributes to him (in the use of his prosthaphaeresis) the first introduction of a subsidiary angle into trigonometry (vol. ii. 590). [9] The title of this work is--_Benjaminis Ursini_ ... _cursus mathematici practici volumen primum continens illustr. & generosi Dn. Dn. Johannis Neperi Baronis Merchistonij &c. Scoti trigonometriam logarithmicam usibus discentium accommodatam_ ... _Coloniae_ ... _CI[~C] I[~C]C XIX_. At the end, Napier's table is reprinted, but to two figures less. This work forms the earliest publication of logarithms on the continent. [10] The title is _Logarithmorum canonis descriptio, seu arithmeticarum supputationum mirabilis abbreviatio_. _Ejusque usus in utraque trigonometria ut etiam in omni logistica mathematica, amplissimi, facillimi & expeditissimi explicatio. Authore ac inventore Ioanne Nepero, Barone Merchistonii, &c. Scoto. Lugduni_.... It will be seen that this title is different from that of Napier's work of 1614; many writers have, however, erroneously given it as the title of the latter. [11] In describing the contents of the works referred to, the language and notation of the present day have been adopted, so that for example a table to radius 10,000,000 is described as a table to 7 places, and so on. Also, although logarithms have been spoken of as to the base e, &c., it is to be noticed that neither Napier nor Briggs, nor any of their successors till long afterwards, had any idea of connecting logarithms with exponents. [12] The smallest number of entries which are necessary in a table of logarithms in order that the intermediate logarithms may be calculable by proportional parts has been investigated by J. E. A. Steggall in the _Proc. Edin. Math. Soc._, 1892, 10, p. 35. This number is 1700 in the case of a seven-figure table extending to 100,000. [13] Accounts of Sang's calculations are given in the _Trans. Roy. Soc. Edin._, 1872, 26, p. 521, and in subsequent papers in the _Proceedings_ of the same society.