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elements of algebra 4e: euler leonhard, hewlett john [pdf]

### Elements of Algebra (4E) by Leonhard Euler, John Hewlett

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Elements of Algebra (4E) written by Leonhard Euler, John Hewlett.
The background assumed is that usually obtained in the freshman-sophomore calculus sequence. Linear algebra is not assumed to be known but is developed in the first chapter. Subjects discussed include all the topics usually found in texts on advanced calculus. However. there is more than the usual emphasis on applications and on physical motivation. Vectors are introduced at the outset and serve at many points to indicate geometrical and physical significance of mathematical relations. Numerical methods are touched upon at various points, both because of their practical value and because of the insights they give into the theory. A sound level of rigor is maintained throughout. Definitions are clearly labeled as such and all important results are formulated as theorems. A few of the finer points of real variable theory are treated at the ends of Chapters 2, 4, and 6. A large number of problems (with answers) are distributed throughout the text. These include simple exercises as well as complex ones planned to stimulate critical reading. Some -points of the theory are relegated to the problems. with hints given where appropriate. Generous references to the literature are given, and each chapter concludes with a list of books for supplementary reading. Starred sections are less essential in a first course.

Book Detail .-
Title. Elements of Algebra
Edition. 4th
Author(s). Leonhard Euler, John Hewlett
Publisher. Printed for Logman, Rees Orme and Co. Paternoster-Row
Series.
Year. 1984
Pages. 638
Type. PDF
Language. English
ISBN. 978-1-4613-8513-4,978-1-4613-8511-0
Country. Russia
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Leonhard Euler traslated from Latin Institutiones Calculi Differentialis, Chapter 1 to 9. Leonhard Euler (1707–1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[3] He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. Euler was one of the most eminent mathematicians of the 18th century and is held to be one of the greatest in history. He is also widely considered to be the most prolific mathematician of all time.

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Book Contents .-
Elements of Algebra (4E) written by Leonhard Euler, John Hewlett cover the following topics.
PART I. Containing- the Analysis o/" Determinate Quantities.
SECTION I. Ofthe Different Methods of calculating Simple Quantities.
I. Of Mathematics in general
II. Explanation of the signs + plus and — minus
III. Of the Multiplication of Simple Quantities
IV. Of the nature of whole Numbers, or Integers with respect to their Factors
V. Of the Division of Simple Quantities
VI. Of the properties ofIntegers, with respect to their Divisors
VII. Of Fractions in general
VIII. Of the Properties of Fractions
IX. Of the Addition and Subtraction of Fractions
X. Of the Multiplication and Division of Fractions
XI. Of Square Numbers
XII. Of Square Roots, and of Irrational Numbers resulting from them
XIII. Of Impossible, or Imaginary Quantities, which arise from the same source
XIV. Of Cubic Numbers
XV. Of Cube Roots, and of Irrational Numbers resulting from them
XVI. Of Powers in general
XVII. Of the Calculation of Powers
XVIII. Of Roots with relation to Powers in general
XIX. Of the Method of representing Irrational Numbers by Fractional Exponents
XX. Of the different Methods of Calculation, and of their Mutual Connexion
XXI. Of Logarithms in general
XXII. Of the Logarithmic Tables that are now in use
XXIII. Of the Method of expressing Logarithms
SECTION II. Of the different Methods of calculating Compound Quantities.
I, Of the Addition of Compound Quantities
II. Of the Subtraction of Compound Quantities
III. Of the MuItipHcation of Compound Quantities
IV. Of the Division of Compound Quantities
V. Of the Resolution of Fractions into Infinite Series
VI. Of the Squares of Compound Quantities
VII. Of the Extraction of Roots applied to Compound Quantities
VIII. Of the Calculation of Irrational Quantities
IX. Of Cubes, and of the Extraction of Cube Roots
X. Of the higher Powers of Compound Quantities
XI. Of the Transposition of the Letters, on which the demonstration of the preceding Rule is founded
XII. Of the Expression of Irrational Powers by Infinite Series
XIII. Of the Resolution of Negative Powers
SECTION III. Of Ratios and Proportions.
I. Of Arithmetical Ratio, or the Difference between two numbers

II. Of Arithmetical Proportion
III. Of Arithmetical Progressions
IV. Of the Summation of Arithmetical Progressions
V. Of Figurate, or Polygonal Numbers
VI. Of Geometrical Ratio
VII. Of the greatest Common Divisor of two given Numbers
Vlll. Of Geometrical Proportions
IX. Observations on the Rules of Proportion and their Utility
X, Of Compound Relations
XI. Of Geometrical Progressions
XII. Of Infinite Decimal Fractions
XIII. Of the Calculation of Interest
SECTION IV. Of Algebraic Equations, and of the Resolution of those Equations.
I. Of the Solution of Problems in General
II. Of the Resolution of Simple Equations, or Equations of the First Degree
III. Of the Solution of Questions relating to the preceding Chapter
IV. Of the Resolution of two or more Equations of the First Degree
V. Of the Resolution of Pure Quadratic Equations
VI. Of the Resolution of Mixed Equations of the Second Degree
VJI. Of the Extraction of the Roots of Polygonal Numbers
VIII. Of the Extraction of Square Roots of Binomials
IX. Of the Nature of Equations of the Second Degree
X. Of Pure Equations of the Third Degree
XI. Of the Resolution of Complete Equations of the Third Degree
XII. Of the Rule of Cardan, or that o^ Scipio Ferreo
XIII. Of the Resolution of Equations of the Fourth Degree
XIV. Of the Rule of Bomhelli, for reducing the Resolution of Equations of the Fourth Degree to that of Equations of the Third Degree
XV. Of a new Method of resolving Equations of the Fourth Degree
XVI. Of the Resolution of Equations by Approximation

PART II. Containing the hna\ys,\% o/" Indeterminate Quantities.
I. Of the Resolution of Equations of the First Degree, which contain more than one unknown Quantity
II. Of the Rule which is called Regu/a Cceci, for determining, by means of two Equations, three or more Unknown Quantities
III. Of Compound Indeterminate Equations, in which one of the Unknown Quantities does not exceed the First Degree
IV. Of the Method of rendering Surd Quantities, of the form (^/a + ax + c/t^"-). Rational
V. Of the Cases in which the Formula a -f- b.v -\- c.%^ can never become a Square
VI. Of the Cases in Integer Numbers, in which the Formula ax~ -\- b becomes a Square
VII. Of a particular Method, by which the Formula an^ -\- 1 becomes a Square in Integers
VIII. Of the Method ofrendering the Irrational Formula (v/a + bx -f- cx^ -h dx^) Rational
IX. Of the Method of rendering rational the incommensurable Formula {\/ x-\- hx \-cx"-\-dji^-\- ex*
X. Of the Method of rendering rational the irrational Formula (Va -|- bx +cx^ + da,^)
XI. Of the Resolution of the Formula o^^-f hxy + cyinto its Factors
XII. Of the Transformation of the Formula ax- -j- c^- into Squares and higher Powers
Xlll. Of some Expressions of the Form r/a* + by*^ which are not reducible to Squares
XIV. Solution of some Questions that belong to this Part of Algebra
XV. Solutions of some Questions in which Cubes are required

I. Of Continued Fractions
II. Solution of some New and Curious Arithmetical Problems
III. Of the Resolution in Integer Numbers of Equations of the First Degree containing two Unknown Quantities
IV. General Method for resolving in Integer Equations of two Unknown Quantities, one of which does not exceed the First Degree
V. A direct and general Method for finding the values of x, that will render Quantities of the form »y{a-\- bx +cx^) Rational, 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 Resolution of the Equation Ap^ +
sq^ — z^ in Integer Numbers
VI. Of Double and Triple Equalities - - 547
VII. A direct and general Method for finding all the values of 2/ expressed in Integer Numbers, by which we may rejider Quantities of the form ^/ {A.y^ + b), rational; a and b being given Integer Numbers; and also for finding all the possible Solutions, in Integer Numbers, of indeterminaie Quadratic Equations of two unknown Quantities Resolution of the Equation Cj/^— 2«_y:2 + nz^=. 1 in Integer Numbers. First Method, Second Method. Of the Manner of finding all the possible Solutions of the Equations cy- — 2nyz + Bz^ = 1, when we know only one of them. Of the Manner of finding all the possible Solutions, in whole Numbers, of Indeterminate Quadratic Equations of two Unknov*n Quantities
VIII. Remarks on Equations of the Form j3*= Aq--{- \, and on the common Method of resolving them in whole Numbers
IX. Of the Manner of finding Algebraic Functions of all Degrees, which, when multiplied together, may always produce similar Functions

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