The claims of Byrgius are discussed in Kästner’s Geschichte der Mathematik, ii. 375, and iii. 14; Montucla’s Histoire des mathématiques, ii. 10; Delambre’s Histoire de l’astronomie moderne, i. 560; de Morgan’s article on “Tables” in the English Cyclopaedia; Mark Napier’s Memoirs of John Napier of Merchiston (1834), p. 392, and Cantor’s Geschichte der Mathematik, ii. (1892), 662. See also Gieswald, Justus Byrg als Mathematiker und dessen Einleitung in seine Logarithmen (Danzig, 1856).
[5] See Mark Napier’s Memoirs of John Napier of Merchiston (1834), p. 362.
[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 ... CIɔ IɔC 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.