Elements of Algebra

Leonhard Euler

Longman, Hurst, Rees, Orme, 1822 - Algebra - 593 pages

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Selected pages

SECTION I	1

Explanation of the signs + plus and minus	3

Of the Multiplication of Simple Quantities	6

Of the nature of whole Numbers or Integers with respect to their Factors	10

Of the Division of Simple Quantities	13

Of the properties of Integers with respect to their Divisors	16

Of Fractions in general	20

Of the Properties of Fractions	24

Of the Calculation of Irrational Quantities	104

the demonstration of the preceding Rule	115

Of the Resolution of Negative Powers	123

Of Arithmetical Proportion	129

OfGeometrial Ratio	146

Of Geometrical Proportions	152

Of the Calculation of Interest 177	177

SECTION IV	186

Of the Addition and Subtraction of Fractions	27

Of the Multiplication and Division of Fractions	30

Of Square Numbers L	36

Of Square Roots and of Irrational Numbers re sulting from them	38

Of Impossible or Imaginary Quantities which arise from the same source	42

Of Cubic Numbers	45

Of Cube Roots and of Irrational Numbers re sulting from them	46

Of Powers in general	48

Of the Calculation of Powers	52

Of Roots with relation to Powers in general	54

Of the Method of representing Irrational Num bers by Fractional Exponents	56

Of the different Methods of Calculation and of their Mutual Connexion	60

Of Logarithms in general	63

Of the Logarithmic Tables that are now in use	66

Of the Method of expressing Logarithms	69

SECTION II	76

Of the Subtraction of Compound Quantities	78

Of the Multiplication of Compound Quantities	79

Of the Division of Compound Quantities	84

Of the Resolution of Fractions into Infinite Series	88

Of the Squares of Compound Quantities	97

Of the Solution of Questions relating to the pre	194

Of the Resolution of two or more Equations	206

Of the Resolution of Pure Quadratic Equations	216

Of the Resolution of Mixed Equations of	222

Of the Extraction of the Roots of Polygonal	230

Of the Nature of Equations of the Second	244

Of the Resolution of Complete Equations	253

Of the R ule of Cardan or that of Scipio Ferreo	262

Of the Resolution of Equations of the Fourth	272

PART II	299

Of Compound Indeterminate Equations in which	317

Of the casesin which the Formula a + bx +	335

Of the Cases in Integer Numbers in which	342

Of a particular Method by which the Formula	351

Of the Method of rendering the Irrational Formula	361

Of the Method of rendering rational the incom	368

Of the Method of rendering rational the irrational	379

Of the Resolution of the Formula axxbry+cy	387

Of the Transformation of the Formula ax1+cy	396

Of some Expressions of the Form ax + In	405

Common terms and phrases

according added already answers arithmetical become become a square begin calculation called CHAP common divisor consequently consider contains continued crowns cube denominator determine difference divided divisible easily easy equal equation evident example explained expressed factors farther formula four fourth fraction given gives greater Hence immediately impossible infinite integer numbers kind known last term lastly least less letters likewise logarithm manner means method multiplied namely nature negative observed obtain perceive positive possible preceding present prime principal progression proportion proposed quantity question quotient ratio reason reduced relation remainder remark represented resolve result rule seen shew sides solution sought square root substitute subtract suppose Table taken third tion transform unity whence wherefore whole numbers write

Popular passages

Page 145 - Three, then, is based on the principle, that the product of the extremes is equal to the product of the means...

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Page 197 - A privateer running at the rate of 10 miles an hour discovers a ship 18 miles off making way at the rate of 8 miles an hour : how many miles can the ship run before being overtaken ? 56.

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Page 198 - A hare is 50 leaps before a greyhound, and takes 4 leaps to the greyhound's 3 ; but 2 of the greyhound's leaps are equal to 3 of the hare's ; how many leaps must the greyhound take, to catch the hare?

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Page xxvi - The manner in which we generally calculate a person's property, is a good illustration of what has just been said. We denote what a man really possesses by positive numbers, using, or understanding the sign + ; whereas his debts are represented by negative numbers, or by using the sign — Thus, when it is said of any one that he has 100 crowns, but owes 50, this means that his property really amounts to 100 — 50 ; or, which is the same thing, + 100 — 50, that is to say 50.

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Page 246 - The coefficient pt of the fourth term with its sign changed is equal to the sum of the products of the roots taken three by three ; and so on, the signs of the...

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Page 215 - The general rule, therefore, which we deduce from this, in order to resolve the equation xx = — px + g, is founded on this consideration : That the unknown quantity x is equal to half the coefficient, or multiplier of x on the other side of the equation, plus or minus the square root of the square of this number, and the known quantity which forms the third term of the equation.

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Page 18 - mb 90. In order therefore to reduce a given fraction to its least terms, it is required to find a number by which both the numerator and denominator may be divided. Such a number is called a common divisor, and so long as we can find a common divisor to the numerator and the denominator, it is certain that the fraction may be reduced to a lower form ; but, on the contrary, when we see that except unity no other common divisor can be found, this shews that the fraction is already in the simplest...

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Bibliographic information

Title	Elements of Algebra
Author	Leonhard Euler
Edition	3
Publisher	Longman, Hurst, Rees, Orme, 1822
Original from	Harvard University
Digitized	Sep 29, 2005
Length	593 pages

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