[1351] Forma autem Zetesin ineundi ex arte propria est, non jam in numeris suam logicam exercente, quæ fuit oscitantia 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 formulas 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.
5. Secondly, Vieta understood the transformation of equations, so as to clear them from coefficients 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 short than that of Vieta, which is still in use.[1352] 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 coefficient 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.[1353] 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He devised a method of solving equations by approximation, analogous to the process of extracting roots, which has been superseded by the invention of more compendious rules.[1354] 6.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.[1355]
[1352] It is fully explained in his work De Recognitione Æquationum, cap. 7.
[1353] Some theorems given by Vieta very shortly and without demonstration, show his knowledge of the structure of equations. I transcribe from Maseres, who has expressed them in the usual algebraic language. Si a + b × x - x2 æquetur ab, x explicabilis est de qualibet illarum duarum a vel b. The second theorem is:—
| a⎫ | ab⎫ | ||
| Si x3 - b⎬ | x2 + | ac⎬ | x |
| c⎭ | bc⎭ | ||
æquetur abc, x explicabilis est de qualibet illarum trium a, b, vel c. The third and fourth theorems extend this to higher equations.
[1354] Montucla, i. 600. Hutton’s Mathematical Dictionary. Biog. Univers. art. Viète.
[1355] 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 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 means of algebra, Montucla, p. 604. But compare Cossali, p. 427.