This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1810 Excerpt: ...unity. After seeding therefore for the approximate integer1 value of that root, and calling that value ft we shall then make y = /3-J--, which will give an. "" ' equation in s, having likewise a root greater than unity, whose approximate integer value we must next seek, and so on. In 'this manner the root required will ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1810 Excerpt: ...unity. After seeding therefore for the approximate integer1 value of that root, and calling that value ft we shall then make y = /3-J--, which will give an. "" ' equation in s, having likewise a root greater than unity, whose approximate integer value we must next seek, and so on. In 'this manner the root required will be found expressed by the continued fraction-, &c. which will be terminated if the root is commensu-rable; but will necessarily go on ad infinitum, if it be incommensurable. In the Memoirs before quoted there will be found all the principles and details necessary to render this method and its application easy, and even different means of abridging many of the operations which it requires; I believe that I have scarcely left, any thing farther to be said on this important subject. With regard to the roots of equations of the second degree, we shall afterwards give (Art. 33 et seq.) a particular and very simple method of changing them into continued fractions. 10. After having thus explained the genesis of continued fractions, we shall proceed to show their application and their principal properties. It is evident, that the more terms we take in a continued fraction, the nearer we approximate the true value of the quantity which we have expressed by that fraction f so that if we successively stop at each term of the fraction, we shall have a series of quantities converging towards the given quantity. Thus, having reduced the value of a to the continued fraction, which approach nearer and nearer to the value of a. In order to judge better of the law and of the convergence of these quantities, it must be remarked, that by the formulae of Articles, we have o=a-, b=P+, c=y+ &c. oca Whence we immediately perceive, that a. is the fi...
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