A writer lately quoted, and to whose knowledge and talents I bow with deference, seems, as I would venture to suggest, to have overrated the importance of that employment of letters to signify quantities, known or unknown, which he has found in Aristotle, and in several of the moderns, and in consequence to have depreciated the real merit of Vieta. Leonard of Pisa, it seems, whose algebra this writer has for the first time published, to his own honour and the advantage of scientific history, makes use of letters as well as lines, to represent quantities. Quelquefois il emploie des lettres pour exprimer des quantités indéterminées, connues ou inconnues, sans les représenter par des lignes. On voit ici comment les modernes ont été amenés à se servir des lettres d’alphabet (même pour exprimer des quantités connues) long temps avant Viète, à qui on a attribué à tort une notation qu’il faudrait peut-être faire remonter jusqu’à Aristote, et que tant d’algébraistes modernes ont employée avant le géomètre Français. Car outre Leonard di Pise, Paciolo et d’autres géomètres Italiens firent usage des lettres pour indiquer les quantités connues, et c’est d’eux plutôt que d’Aristote que les modernes ont appris cette notation. Libri, vol. ii. p. 34. But there is surely a wide interval between the use of a short symbolic expression for particular quantities, as M. Libri has remarked in Aristotle, or even the partial employment of letters to designate known quantities, as in the Italian algebraists, and the method of stating general relations by the exclusive use of letters, which Vieta first introduced. That Tartaglia and Cardan, and even, as it now appears, Leonard of Pisa went a certain way towards the invention of Vieta, cannot much diminish his glory; especially when we find that he entirely apprehended the importance of his own logistice speciosa in science. I have mentioned above, that, as far as my observation has gone, Vieta does not work particular problems by the specious algebra.
6. “Algebra,” says a philosopher of the present day, “was still only an ingenious art, limited to the investigation of numbers; Vieta displayed all its extent, and instituted general expressions for particular results. Having profoundly meditated on the nature of algebra, he perceived that the chief characteristic of the science is to express relations. Newton with the same idea defined algebra an universal arithmetic. The first consequences of this general principle of Vieta were his own application of his specious analysis to geometry, and the theory of curve lines, which is due to Descartes; a fruitful idea, from which the analysis of functions, and the most sublime discoveries, have been deduced. It has led to the notion that Descartes is the first who applied algebra to geometry; but this invention is really due to Vieta; for he resolved geometrical problems by algebraic analysis, and constructed figures by means of these solutions. These investigations led him to the theory of angular sections, and to the general equations which express the values of chords.”[1356] It will be seen in the notes that some of this language requires a slight limitation.
[1356] M. Fourier, quoted in Biographie Universelle.
7. The Algebra of Bombelli, published in 1589, is the only other treatise of the kind during this period that seems worthy of much notice. Bombelli saw better than Cardan the nature of what is called the irreducible case in cubic equations. But Vieta, whether after Bombelli or not, is not certain, had the same merit.[1357] It is remarkable that Vieta seems to have paid little regard to the discoveries of his predecessors. Ignorant, probably, of the writings of Record, and perhaps even of those of Stifelius, he neither uses the sign = of equality, employing instead the clumsy word Æquatio, or rather Æquetur,[1358] nor numeral exponents; and Hutton observes that Vieta’s algebra has, in consequence, the appearance of being older than it is. He mentions, however, the signs + and -, as usual in his own time.
[1357] Cossali. Hutton.
[1358] Vieta uses =, but it is to denote that the proposition is true both of + and -; where we put ±. It is almost a presumption of copying one from another, that several modern writers say Vieta’s word is æquatio. I have always found it æquetur; a difference not material in itself.
Geometers of this period. 8. Amidst the great progress of algebra through the sixteenth century, the geometers, content with what the ancients had left them, seem to have had little care but to elucidate their remains. Euclid was the object of their idolatry; no fault could be acknowledged in his elements, and to write a verbose commentary upon a few propositions was enough to make the reputation of a geometer. Among the almost innumerable editions of Euclid that appeared, those of Commandin and Clavius, both of them in the first rank of mathematicians for that age, may be distinguished. Commandin, especially, was much in request in England, where he was frequently reprinted, and Montucla calls him the model of commentators for the pertinence and sufficiency of his notes. The commentary of Clavius, though a little prolix, acquired a still higher reputation. We owe to Commandin editions of the more difficult geometers, Archimedes, Pappus, and Apollonius; but he attempted little, and that without success, beyond the province of a translator and a commentator. Maurolycus of Messina had no superior among contemporary geometers. Besides his edition of Archimedes, and other labours on the ancient mathematicians, he struck out the elegant theory, in which others have followed him, of deducing the properties of the conic sections from those of the cone itself. But we must refer the reader to Montucla, and other historical and biographical works, for the less distinguished writers of the sixteenth age.[1359]
[1359] Montucla. Kästner. Hutton. Biogr. Univ.
Joachim Rhæticus. 9. The extraordinary labour of Joachim Rhæticus in his trigonometrical calculations, has been mentioned in our first volume. His Opus Palatinum de Triangulis was published from his manuscript by Valentine Otho, in 1594. But the work was left incomplete, and the editor did not accomplish what Joachim had designed. In his tables the sines, tangents, and secants are only calculated to ten, instead of fifteen places of decimals. Pitiscus, in 1613, not only completed Joachim’s intention, but carried the minuteness of calculation a good deal farther.[1360]
[1360] Montucla, p. 581.