of quantity, and, by thus forming the scattered elements of specious analysis into a system, has been justly reckoned the founder of a science which, from its extensive application, has made the old problems of mere numerical algebra appear elementary and almost trifling. "Algebra," says Kästner, "from furnishing amusing enigmas to the Cossists," as he calls the first teachers of the art, "became the logic of geometrical invention." It would appear a natural conjecture, that the improvement, towards which so many steps had been taken by others, might occur to the mind of Vieta simply as a means of saving the trouble of arithmetical operations in working out a problem. But those who refer to his treatise entitled De Arte Analytica Isagoge, or even the first page of it, will, I conceive, give credit to the author for a more scientific view of his own invention. He calls it logistice speciosa, as opposed to the logistice numerosa of the older analysis; his theorems are all general, the given quantities being considered as indefinite, nor does it appear that he substituted letters for the known quantities in the investigation of particular problems. Whatever may have suggested this great invention to the mind of Vieta, it has altogether changed the character of his science. 5. Secondly, Vieta understood the transformation of equations, so as to clear them from co-efficients or surd. roots, or to eliminate the second term. This, however, is partly claimed by Cossali for Cardan. Yet it seems that the process employed by Cardan was much less neat and by all that these algebraists have written on the subject, that they had the clearest conviction they were dealing with continuous, or geometrical, not merely with discrete, or arithmetical, quantity. This gave them an insight into the fundamental truth, which is unintelligible so long as algebra passes for a specious arithmetic, that every value which the conditions of the problem admit may be assigned to unknown quantities, without distinction of rationality and irrationality. To abstract number itself irrationality is inapplicable. Geschichte der Mathematik, i. 63. Forma autem Zetesin ineundi ex arte propria est, non jam in numeris suam logicam exercente, quæ fuit osci tantia veterum analystarum, sed per logisticen sub specie noviter inducendam, feliciorem multo et potiorem numerosa, ad comparandum inter se magnitudines, proposita primum homogeniorum lege, &c. p. i. edit. 1646. A profound writer on algebra, Mr. Peacock, has lately defined it, "the science of general reasoning by symbolical language." In this sense there was very little algebra before Vieta, and it would be improper to talk of its being known to the Greeks, Arabs, or Hindoos. The definition would also include the formulæ of logic. The original definition of algebra seems to be the science of finding an equation between known and unknown quantities, per oppositionem et restaurationem. short than that of Vieta, which is still in use. 3. He obtained a solution of cubic equations in a different method from that of Tartaglia. 4. "He shows," says Montucla, "that when the unknown quantity of any equation may have several positive values, for it must be admitted that it is only these that he considers, the second term has for its co-efficient the sum of these values with the sign-; the third has the sum of the products of these values multiplied in pairs; the fourth the sum of such products multiplied in threes, and so forth; finally, that the absolute term is the product of all the values. Here is the discovery of Harriott pretty nearly made." It is at least no small advance towards it. Cardan is said to have gone some way towards this theory, but not with much clearness, nor extending it to equations above the third degree. 5. He devised a method of solving equations by approximation, analogous to the process of extracting roots, which has been superseded by the invention of more compendious rules. 6. He has been regarded by some as the true author of the application of algebra to geometry, giving copious examples of the solution of problems by this method, though all belonging to straight lines. It looks like a sign of the geometrical relation under which he contemplated his own science, that he uniformly denominates the first power of the unknown quantity latus. But this will be found in older writers." m It is certain that Vieta perfectly knew the relation of algebra to magnitude as well as number, as the first pages of his In Artem Analyticam Isagoge fully show. But it is equally certain, as has been observed before, that Tartaglia and Cardan, and much older writers, Oriental as well as European, knew the same; it was by help of geometry, which Cardan calls via regia, that the former made his great discovery of the solution of cubic equations. Cossali, ii. 147. Cardan, Ars Magna, ch. xi. Latus and radix are used indifferently for the first power of the unknown quantity in the Ars Magna. Cossali contends that Fra Luca had applied algebra to geometry. Vieta, however, it is said, was the first who taught how to construct geometrical figures by 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."" It has been observed above, that this requires a slight limitation as to the solution of problems. means of algebra. Montucla, p. 604. But compare Cossali, p. 427. A writer lately quoted, and to whose knowledge and talents I bow with deference, seems, as I would venture to suggest, to have over-rated 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'algebraistes modernes ont employée avant le géomètre Français. Car outre Leonard de 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. n 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. 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 Equatio, or rather Equetur, 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. = of this 8. Amidst the great progress of algebra through the sixteenth century, the geometers, content with what Geometers the ancients had left them, seem to have had period. 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 commenta tors 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, "Cossali. Hutton. P 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 VOL. II. another, that several modern writers say Vieta's word is æquatio. I have always found it æquetur; a difference not material in itself. Q 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. Joachim 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.' 10. It can excite no wonder that the system of CoperCopernican nicus, simple and beautiful as it is, met with little theory. encouragement for a long time after its promulgation, when we reflect upon the natural obstacles to its reception. Mankind can in general take these theories of the celestial movements only upon trust from philosophers; and in this instance it required a very general concurrence of competent judges to overcome the repugnance of what called itself common sense, and was in fact a prejudice as natural, as universal, and as irresistible as could influence human belief. With this was united another, derived from the language of Scripture; and though it might have been sufficient to answer, that phrases implying the rest of the earth and motion of the sun are merely popular, and such as those who are best convinced of the opposite doctrine must employ in ordinary language, this was neither satisfactory to the vulgar, nor recognized by the church. Nor were the astronomers in general much more favourable to the new theory than either the clergy or the multitude. They had taken pains to familiarise their understandings with the Ptolemaic hypothesis; and it may be often observed that those who have once mastered a complex theory are better pleased 9 Montucla. Kästner. Hutton. Biog. Univ. г Montucla, p. 581. |