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
- 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
- Advertisement
- Chap. I Of Continued Fractions
- Chap. II Solution of some New and Curious Arithmetical Problems
- Chap. III Of the Resolution in Integer Numbers of Equations of the First Degree containing two Unknown Quantities
- Chap. IV General Method for resolving in, Integer Equations of two Unknown Quantities, one of which does not exceed the First Degree
- Chap. V A direct and general Method for finding the values of x, that will render Rational Quantities of the form √(a + bx + cx2), and for resolving, in Rational Numbers, the indeterminate Equations of the second Degree, which have two Unknown Quantities, when they admit of Solutions of this kind
- Chap. VI Of Double and Triple Equalities
- Chap. VII A direct and general Method for finding all the values of y expressed in Integer Numbers, by which we may render Quantities of the form √(Ay2 + B), rational, A and B being given Integer Numbers; and also for finding all the possible Solutions, in Integer Numbers, of indeterminate Quadratic Equations of two unknown Quantities
- Chap. VIII Remarks on Equations of the Form p2=Aq2 + 1, and on the common Method of resolving them in whole Numbers
Chap. IX - Of the Manner of finding Algebraic Functions of all Degrees, which, when multiplied together, may always produce similar Functions
Published online by Cambridge University Press: 05 July 2011
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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
- 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
- Advertisement
- Chap. I Of Continued Fractions
- Chap. II Solution of some New and Curious Arithmetical Problems
- Chap. III Of the Resolution in Integer Numbers of Equations of the First Degree containing two Unknown Quantities
- Chap. IV General Method for resolving in, Integer Equations of two Unknown Quantities, one of which does not exceed the First Degree
- Chap. V A direct and general Method for finding the values of x, that will render Rational Quantities of the form √(a + bx + cx2), and for resolving, in Rational Numbers, the indeterminate Equations of the second Degree, which have two Unknown Quantities, when they admit of Solutions of this kind
- Chap. VI Of Double and Triple Equalities
- Chap. VII A direct and general Method for finding all the values of y expressed in Integer Numbers, by which we may render Quantities of the form √(Ay2 + B), rational, A and B being given Integer Numbers; and also for finding all the possible Solutions, in Integer Numbers, of indeterminate Quadratic Equations of two unknown Quantities
- Chap. VIII Remarks on Equations of the Form p2=Aq2 + 1, and on the common Method of resolving them in whole Numbers
Summary
[APPENDIX TO CHAP. XI. AND XII.]
88. I believe I had, at the same time with M. Euler, the idea of employing the irrational, and even imaginary factors of formulæ of the second degree, in finding the conditions, which render those formulæ equal to squares, or to any powers. On this subject, I read a Memoir to the academy in 1768, which has not been printed; but of which I have given a summary at the end of my researches on Indeterminate Problems, which are to be found in the volume for the year 1767, printed in 1769, before even the German translation of M. Euler's Algebra.
In the place now quoted, I have shewn how the same method may be extended to formulæ of higher dimensions than the second; and I have by these means given the solution of some equations, which it would perhaps have been extremely difficult to resolve in any other way. It is here intended to generalise this method still more, as it seems to deserve the attention of mathematicians, from its novelty and singularity.
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- Elements of Algebra , pp. 583 - 593Publisher: Cambridge University PressPrint publication year: 2009First published in: 1822
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