Lectures on Elementary Mathematics by Joseph Louis Lagrange

Author:Joseph Louis Lagrange
Language: eng
Format: epub, mobi, pdf
Publisher: Dover Publications

From the first equation we obtain , which substituted in the second gives

Removing the fractions and arranging the terms, we get finally

an equation of the fourth degree with the second term missing.

Ferrari (1522–1565). Bombelli.

According to Bombelli, of whom we shall speak again, Louis Ferrari of Bologna resolved the problem by a highly ingenious method, which consists in dividing the equation into two parts both of which permit of the extraction of the square root. To do this it is necessary to add to the two numbers quantities whose determination depends on an equation of the third degree, so that the resolution of equations of the fourth degree depends upon the resolution of equations of the third and is therefore subject to the same drawbacks of the irreducible case.

The Algebra of Bombelli was printed in Bologna in 1579* in the Italian language. It contains not only the discovery of Ferrari but also divers other important remarks on equations of the second and third degree and particularly on the theory of radicals by means of which the author succeeded in several cases in extracting the imaginary cube roots of the two binomials of the formula of the third degree in the irreducible case, so finding a perfectly real result and furnishing thus the most direct proof possible of the reality of this species of expressions.

Such is a succinct history of the first progress of algebra in Italy. The solution of equations of the third and fourth degree was quickly accomplished. But the successive efforts of mathematicians for over two centuries have not succeeded in surmounting the difficulties of the equation of the fifth degree.

Yet these efforts are far from having been in vain. They have given rise to the many beautiful theorems which we possess on the formation of equations, on the character and signs of the roots, on the transformation of a given equation into others of which the roots may be formed at pleasure from the roots of the given equation, and finally, to the beautiful considerations concerning the metaphysics of the resolution of equations from which the most direct method of arriving at their solution, when possible, has resulted. All this has been presented to you in previous lectures and would leave nothing to be desired if it were but applicable to the resolution of equations of higher degree.

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