[39] A prolonged but indecisive controversy has been carried on, chiefly by French scholars, with regard to the relations of Ptolemy, Abul Wafa, and Tycho in this matter.
[40] For example, the practice of treating the trigonometrical functions as algebraic quantities to be manipulated by formulæ, not merely as geometrical lines.
[41] Any one who has not realised this may do so by performing with Roman numerals the simple operation of multiplying by itself a number such as MDCCCXCVIII.
[42] On trigonometry. He reintroduced the sine, which had been forgotten; and made some use of the tangent, but like Albategnius ([§ 59n.]) did not realise its importance, and thus remained behind Ibn Yunos and Abul Wafa. An important contribution to mathematics was a table of sines calculated for every minute from 0° to 90°.
[43] That of “lunar distances.”
[44] He did not invent the measuring instrument called the vernier, often attributed to him, but something quite different and of very inferior value.
[45] The name is spelled in a large number of different ways both by Coppernicus and by his contemporaries. He himself usually wrote his name Coppernic, and in learned productions commonly used the Latin form Coppernicus. The spelling Copernicus is so much less commonly used by him that I have thought it better to discard it, even at the risk of appearing pedantic.
[46] Nullo demum loco ineptior est quam ... ubi nim’s pueriliter hallucinatur: Nowhere is he more foolish than ... where he suffers from delusions of too childish a character.
[47] His real name was Georg Joachim, that by which he is known having been made up by himself from the Latin name of the district where he was born (Rhætia).
[48] The Commentariolus and the Prima Narratio give most readers a better idea of what Coppernicus did than his larger book, in which it is comparatively difficult to disentangle his leading ideas from the mass of calculations based on them.