Elements of algebra, compiled from Garnier's French translation of L. Euler. To which are added, solutions of several miscellaneous problems1824 |
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... Progression XL . On the Summation of Arithmetical Progressions 198 XLI . Of Geometrical Ratio XLII . Of the greatest Common Measure XLIII . Of Geometrical Proportion XLIV . Of Geometrical Progression · 204 · 209 219 225 XLV . Of the ...
... Progression XL . On the Summation of Arithmetical Progressions 198 XLI . Of Geometrical Ratio XLII . Of the greatest Common Measure XLIII . Of Geometrical Proportion XLIV . Of Geometrical Progression · 204 · 209 219 225 XLV . Of the ...
Page 193
... progression may be either increasing or decreasing . The former distinction is applied when the terms go on increasing , that is to say , when the second exceeds the first , the third the second , the fourth the third , & c . by the the ...
... progression may be either increasing or decreasing . The former distinction is applied when the terms go on increasing , that is to say , when the second exceeds the first , the third the second , the fourth the third , & c . by the the ...
Page 194
... progression is that in which the terms go on con- tinually diminishing by the same quantity , as 9 , 7 , 5 , 3 , 1 . 308. Let us suppose the numbers a , b , c , to be in arith- metical progression : then a - b - b - c ; whence it ...
... progression is that in which the terms go on con- tinually diminishing by the same quantity , as 9 , 7 , 5 , 3 , 1 . 308. Let us suppose the numbers a , b , c , to be in arith- metical progression : then a - b - b - c ; whence it ...
Page 195
... progression , because they continually diminish by the same quantity . 311. The number or quantity by which the several terms of an arithmetical progression are increased or diminished , is termed the common difference . So that when ...
... progression , because they continually diminish by the same quantity . 311. The number or quantity by which the several terms of an arithmetical progression are increased or diminished , is termed the common difference . So that when ...
Page 196
... progression ; thus in the 2d there is a + b , 3d 4th a + 2b , • • a + 3b , 5th 6th • a + 1b , · · a + 5b , & c . 4 We know therefore that the 10th term will be a + 9b , and generally that ... progression of 9 196 ARITHMETICAL PROGRESSION .
... progression ; thus in the 2d there is a + b , 3d 4th a + 2b , • • a + 3b , 5th 6th • a + 1b , · · a + 5b , & c . 4 We know therefore that the 10th term will be a + 9b , and generally that ... progression of 9 196 ARITHMETICAL PROGRESSION .
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Common terms and phrases
already seen arithmetic means arithmetic series arithmetical progression assume binomial cent CHAP coefficient common difference Completing the square consequently consider contains cube root decimal determine divided dividend divisible equal equation evident example exponent expressed Extracting the root factors find the greatest Find the sum find the values formula four roots fourth term geometric means geometrical progression given number gives greater number greatest common divisor greatest common measure Hence infinite series infinitum instance integer irrational last term less letters logarithm manner method multiplied negative numbers number of permutations number of terms obtain quadratic surds quotient radical sign ratio reduced remainder represented required to find rule second degree second term square root subtracted suppose third degree three numbers tion transposition unity unknown quantity whence whole number
Popular passages
Page 46 - Now .} of f- is a compound fraction, whose value is found by multiplying the numerators together for a new numerator, and the denominators for a new denominator.
Page 24 - If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts.
Page 228 - There are three numbers in geometrical progression ; the sum of the first and second of which is 9, and the sum of the first and third is 15.
Page 36 - Multiplying or dividing both the numerator and denominator of a fraction by the same number does not change the value of the fraction.
Page 248 - The logarithm of a product is equal to the sum of the logarithms of its factors.
Page 58 - We call this new species of numbers, irrational numbers ; they occur whenever we endeavour to find the square root of a number which is not a square. Thus, 2 not being a perfect square, the square root of 2, or the number which, multiplied by itself, would produce 2, is an irrational quantity. These numbers are also called surd quantities, or incommensurables.
Page 243 - Find two numbers, such, that their sum, their product, and the difference of their squares shall be all equal to each other.
Page 77 - any quantity may be transferred from "one side of the equation to the other, by changing its sign ;" and and it is founded upon the axiom, that " if equals be added to " or subtracted from equals, the sums or remainders will be
Page 113 - Ans. 3 and 7 8. The difference of two numbers is 2, and the difference of their cubes is 98; required the numbers. Ans. 5 and 3 9.
Page 37 - If the numerator and denominator are both, multiplied or both divided by the same number, the value of the fraction will not be altered.