Book contents
- Frontmatter
- ADVERTISEMENT
- MEMOIR OF THE LIFE AND CHARACTER OF EULER, BY THE LATE FRANCIS HORNER, ESQ., M. P.
- ADVERTISEMENT BY THE EDITORS OF THE ORIGINAL, IN GERMAN
- ADVERTISEMENT BY M. BERNOULLI, THE FRENCH TRANSLATOR
- Contents
- PART I Containing the Analysis of Determinate Quantities
- SECTION I Of the Different Methods of calculating Simple Quantities
- SECTION II Of the different Methods of calculating Compound Quantities
- Chap. I Of the Addition of Compound Quantities
- Chap. II Of the Subtraction of Compound Quantities
- Chap. III Of the Multiplication of Compound Quantities
- Chap. IV Of the Division of Compound Quantities
- Chap. V Of the Resolution of Fractions into Infinite Series
- Chap. VI Of the Squares of Compound Quantities
- Chap. VII Of the Extraction of Roots applied to Compound Quantities
- Chap. VIII Of the Calculation of Irrational Quantities
- Chap. IX Of Cubes, and of the Extraction of Cube Roots
- Chap. X Of the higher Powers of Compound Quantities
- Chap. XI Of the Transposition of the Letters, on which the demonstration of the preceding Rule is founded
- Chap. XII Of the Expression of Irrational Powers by Infinite Series
- Chap. XIII Of the Resolution of Negative Powers
- SECTION III Of Ratios and Proportions
- SECTION IV Of Algebraic Equations, and of the Resolution of those Equations
- PART II Containing the Analysis of Indeterminate Quantities
- ADDITIONS BY M. DE LA GRANGE
Chap. VII - Of the Extraction of Roots applied to Compound Quantities
Published online by Cambridge University Press: 05 July 2011
- Frontmatter
- ADVERTISEMENT
- MEMOIR OF THE LIFE AND CHARACTER OF EULER, BY THE LATE FRANCIS HORNER, ESQ., M. P.
- ADVERTISEMENT BY THE EDITORS OF THE ORIGINAL, IN GERMAN
- ADVERTISEMENT BY M. BERNOULLI, THE FRENCH TRANSLATOR
- Contents
- PART I Containing the Analysis of Determinate Quantities
- SECTION I Of the Different Methods of calculating Simple Quantities
- SECTION II Of the different Methods of calculating Compound Quantities
- Chap. I Of the Addition of Compound Quantities
- Chap. II Of the Subtraction of Compound Quantities
- Chap. III Of the Multiplication of Compound Quantities
- Chap. IV Of the Division of Compound Quantities
- Chap. V Of the Resolution of Fractions into Infinite Series
- Chap. VI Of the Squares of Compound Quantities
- Chap. VII Of the Extraction of Roots applied to Compound Quantities
- Chap. VIII Of the Calculation of Irrational Quantities
- Chap. IX Of Cubes, and of the Extraction of Cube Roots
- Chap. X Of the higher Powers of Compound Quantities
- Chap. XI Of the Transposition of the Letters, on which the demonstration of the preceding Rule is founded
- Chap. XII Of the Expression of Irrational Powers by Infinite Series
- Chap. XIII Of the Resolution of Negative Powers
- SECTION III Of Ratios and Proportions
- SECTION IV Of Algebraic Equations, and of the Resolution of those Equations
- PART II Containing the Analysis of Indeterminate Quantities
- ADDITIONS BY M. DE LA GRANGE
Summary
317. In order to give a certain rule for this operation, we must consider attentively the square of the root a + b, which is a2 + 2ab + b2, in order that we may reciprocally find the root of a given square.
318. We must consider therefore, first, that as the square, a2 + 2ab + b2, is composed of several terms, it is certain that the root also will comprise more than one term; and that if we write the terms of the square in such a manner, that the powers of one of the letters, as a, may go on continually diminishing, the first term will be the square of the first term of the root; and since, in the present case, the first term of the square is a2, it is certain that the first term of the root is a.
319. Having therefore found the first term of the root, that is to say, a, we must consider the rest of the square, namely, 2ab + b2, to see if we can derive from it the second part of the root, which is b. Now, this remainder, 2ab + b2, may be represented by the product, (2a + b)b; wherefore the remainder having two factors, (2a + b), and b, it is evident that we shall find the latter, b, which is the second part of the root, by dividing the remainder, 2ab + b2, by 2a + b.
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- Information
- Elements of Algebra , pp. 100 - 104Publisher: Cambridge University PressPrint publication year: 2009First published in: 1822