[842] It must, apparently, have been through his knowledge of this property of the coefficient of the second term, that Cardan recognised the existence of equal roots, even when affected by the same sign (Cossali, ii. 362); which, considered in relation to the numerical problems then in use, would seem a kind of absurdity.
[843] Kästner, p. 161. In one place Cossali shows, that Cardan had transported all the quantities of an equation on one side, making the whole equal to zero; which Wallis has ascribed to Harriott, as his leading discovery, p. 324. Yet in another passage we find Cossali saying: una somma di quantità uguale al zero avea un’aria mostruosa, e non sapeasi di equazion si fatta concepire idea, p. 159.
Imperfections of algebraic language. 6. These anticipations of Cardan are the more truly wonderful, when we consider that the symbolical language of algebra, that powerful instrument not only in expediting the processes of thought, but in suggesting general truths to the mind, was nearly unknown in his age. Diophantus, Fra Luca, and Cardan make use occasionally of letters to express indefinite quantities, besides the res or cosa, sometimes written shortly, for the assumed unknown number, of an equation. But letters were not yet substituted for known quantities; and it has been seen in a note, that Tartaglia first discovered, and that by a geometrical construction, what appears so very simple as the equation between the cube of a line and that of any two parts into which it may be divided. Michael Stifel, in his Arithmetica Integra, Nuremberg, 1544, is said to have first used the signs + and -, and numeral exponents of powers.[844] It is very singular that discoveries of the greatest convenience, and not above the ingenuity of a parish schoolmaster, should have been overlooked by men of extraordinary acuteness, like Tartaglia, Cardan, and Ferrari, and hardly less so, that by dint of this acuteness, they dispensed with the aid of these contrivances in which we almost fancy the utility of algebraic expression consists.
[844] Hutton, Kästner.
Copernicus. 7. But the great boast of science during this period is the treatise of Copernicus on the revolutions of the heavenly bodies, in six books, published at Nuremberg, in 1543.[845] This founder of modern astronomy was born at Thorn, of a good family, in 1473; and after receiving the best education his country furnished, spent some years in Italy, rendering himself master of all the mathematical and astronomical science at that time attainable. He became possessed afterwards of an ecclesiastical benefice in his own country. It appears to have been about 1507, that after meditating on various schemes besides the Ptolemaic, he began to adopt and confirm in writing that of Pythagoras, as alone capable of explaining the planetary motions with that simplicity which gives a presumption of truth in the works of nature.[846] Many years of exact observation confirmed his mind in the persuasion that he had solved the grandest problem which can occupy the astronomer. He seems to have completed his treatise about 1530; but perhaps dreaded the bigoted prejudices which afterwards oppressed Galileo. Hence he is careful to propound his theory as an hypothesis; though it is sufficiently manifest that he did not doubt of its truth. It was first publicly announced by his disciple Joachim Rhœticus, already mentioned for his trigonometry, in the Narratio de Revolutionibus Copernici, printed at Dantzic, in 1540. The treatise of Copernicus himself, three years afterwards, is dedicated to the pope, Paul III., as if to shield himself under that sacred mantle. But he was better protected by the common safeguard against oppression. The book reached him on the day of his death; and he just touched with his hands the great legacy he was to bequeath to mankind. But many years were to elapse before they availed themselves of the wisdom of Copernicus. The progress of his system, even among astronomers, as we shall hereafter see, was exceedingly slow.[847] We may just mention here, that no kind of progress was made in mechanical or optical science during the first part of the sixteenth century.
[845] The title-page and advertisement of so famous a work, and which so few of my readers will have seen, are worth copying from Kästner, ii. 595. Nicolai Copernici Torinensis, de Revolutionibus Orbium Cœlestium, libri vi.
Habes in hoc opere jam recens nato et edito, studiose lector, motus stellarum tam fixarum quam erraticarum, cum ex veteribus tum etiam ex recentibus observationibus restitutos; et novis insuper ac admirabilibus hypothesibus ornatos. Habes etiam tabulas expeditissimas, ex quibus eosdem ad quodvis tempus quam facillime calculare poteris. Igitur eme lege, fruere. Αγεωμετρητος ουδεις εισιτω Noribergæ, apud Joh. Petreium, anno MDxliii.
[846] This is the proper statement of the Copernican argument, as it then stood; it rested on what we may call a metaphysical probability, founded upon its beauty and simplicity; for it is to be remembered that the Ptolemaic hypothesis explained all the phenomena then known. Those which are only to be solved by the supposition of the earth’s motion were discovered long afterwards. This excuses the slow reception of the new system, interfering as it did with so many prejudices, and incapable of that kind of proof which mankind generally demand.
[847] Gassendi, Vita Copernici. Biogr. Univ. Montucla. Kästner. Playfair. Gassendi, p. 14-22, gives a short analysis of the great work of Copernicus, de orbium Cœlestium Revolutionibus, p. 22. The hypothesis is generally laid down in the first of the six books. One of the most remarkable passages in Copernicus is his conjecture that gravitation was not a central tendency, as had been supposed, but an attraction common to matter, and probably extending to the heavenly bodies, though it does not appear that he surmised their mutual influences in virtue of it: gravitatem esse affectionem non terræ totius, sed partium ejus propriam, qualem soli etiam et lunæ cæterisque astris convenire credibile est. These are the words of Copernicus himself, quoted by Gassendi, p. 19.
Sect. II.