A Treatise on Algebra: In Practice and Theory, ... with Notes and Illustrations; Containing a Variety of Particulars Relating to the Discoveries and Improvements that Have Been Made in this Branch of Analysis, Volume 2

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J. Johnson and Company, 1813

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Page 307 - When, in any problem, there are two lines, or quantities, alike related to other parts of! the figure, or problem, the best way is not to make use of either of them separately, but to substitute for their sum...
Page 27 - In any series of numbers in arithmetical progression, the sum of the two extremes is equal to the sum of any two terms equally distant from them; as in the latter of the above series 6 + 1=4+3, and =5+2.
Page 331 - Given the hypothenuse (10) of a right angled triangle, and the difference of two lines drawn from its extremities to the centre of the inscribed circle (2), to determine the base and perpendicular. Ans. 8.08004 and 5.87447 PROBLEM XIX.
Page 329 - To find the side of an equilateral triangle, inscribed in a circle whose diameter is d ; and that of another circumscribed about the same circle. Ans. %di/3, and di/3. PROBLEM IV. To find the side of a regular pentagon, inscribed in a circle, whose diameter is d.
Page 31 - V ad j where it is evident, that the product of the extremes of three proportionals is equal to the square of the mean ; or, that the mean is equal to the square root of the product of the two extremes. Also, if each member of the equation ad = be be successively divided by bd, dc, ac, &c. the results will give =/| ', 6 " Or the proportions \b:a::d:c &>'.
Page 307 - ... sum, difference, or rectangle, or the sum of their alternate quotients ; or for some other line or lines in the figure, to which they have both the same relation. 4th. When the area, or the perimeter, of a figure is given, or such parts of it as have only a remote relation to the parts that are to be found, it will sometimes be of use to assume another figure similar to the proposed one, that shall have one of its sides equal to unity, or to some other known quantity ; as the other parts of the...
Page 59 - It is obvious that, by analogous methods, an equation may be transformed into another, the roots of which shall be to those of the proposed equation, in any required ratio. But the subject need not be enlarged on here. The preceding succinct view will suffice for the usual purposes, so far as relates to the nature and chief properties of equations.
Page 19 - In any equation x + ^/y = a + ^b which involves rational quantities and quadratic surds, the rational parts on each side are equal, and also the irrational parts.
Page 11 - The greatest common measure of two quantities is not altered, by multiplying or dividing either of them, by any quantity which is not a divisor of the other, and which contains no factor which is a divisor of the other. The common measure of ab and ac is a.
Page 45 - ... is equal to the coefficient of the second term of that equation, with its sign changed.

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