A Theoretical and Practical Treatise on Algebra: In which the Excellences of the Demonstrative Methods of the French are Combined with the More Practical Operations of the English : and Concise Solutions Pointed Out and Particularly Inculcated

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J. Ernst, 1850 - Algebra - 328 pages
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Page 20 - ... the product of the two, plus the square of the second. In the third case, we have (a + b) (a — 6) = a2 — b2. (3) That is, the product of the sum and difference of two quantities is equal to the difference of their squares.
Page 30 - To reduce a mixed number to an improper fraction, Multiply the whole number by the denominator of the fraction, and to the product add the numerator; under this sum write the denominator.
Page 25 - Divide the first term of the dividend by the first term of the divisor, and write the result as the first term of the quotient. Multiply the whole divisor by the first term of the quotient, and subtract the product from the dividend.
Page 193 - Three quantities are said to be in harmonical proportion, when the first is to the third, as the difference between the first and second is to the difference between the second and third.
Page 84 - It is required to divide the number 24 into two such parts, that the quotient of the greater part divided by the less, may be to the quotient of the less part divided by the greater, as 4 to 1.
Page 198 - There are four numbers in geometrical progression, the second of which is less than the fourth by 24 ; and the sum of the extremes is to the sum of the means, as 7 to 3. What are the numbers ? Ans.
Page 193 - Three lines are in harmonical proportion, when the first is to the third, as the difference between the first and second, is to the difference between the second and third ; and the second is called a harmonic mean between the first and third. The expression 'harmonical proportion...
Page 200 - If four magnitudes are in proportion, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference.
Page vii - Algebraic operations are based upon definitions and the following axioms : — 1. If the same quantity, or equal quantities, be added to equal quantities, the sums will be equal. 2. If the same quantity, or equal quantities, be subtracted from equal quantities, the remainders will be equal. 3. If equal quantities be multiplied by the same quantity, or equal quantities, the products will be equal. 4. If equal quantities be divided by the same quantity, or equal quantities, the quotients will be equal....
Page 57 - For, if we have given ab' = a'b, then, dividing by bb', we obtain Corollary. The terms of a proportion may be written In any order which will make the product of the extremes equal to the product of the means.

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