Lanterna Rally

Author: Joseph Louis Lagrange

Publisher: Rarebooksclub.com

Published: 2013-09

Total Pages: 38

ISBN-13: 9781230100364

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1901 edition. Excerpt: ...equations. in existence, and we owe to these passages at arms many important discoveries in analysis. Such was the resolution of equations of the fourth degree, which was propounded in the following problem. To find three numbers in continued proportion of which the sum is 10, and the product of the first two 6. Generalising and calling the sum of the three numbers a, the product of the first two b, and the first two numbers themselves x, y, we shall have, first, xy = b. Owing to the continued proportion, the third number v2 will then be expressed by--, so that the remaining condition will give y.x+y+-=a. From the first equation we obtain x=, which sub y stituted in the second gives b f Removing the fractions and arranging the terms, we get finally y + bf--a ly + P--, an equation of the fourth degree with the second term missing. According to Bombelli, of whom we shall speak Ferrari (1522-1565). Bombelli. 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...