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 2J. Johnson and Company, 1813 |
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Page 1
... of a ; and consequently , if the quantities 5a and 74 are to be incorporated , or added together , their VOL . II . B sum will be twelve times the thing denoted by a CONTENTS VOL (THE THEORETICAL PART LEADING rules of the science.
... of a ; and consequently , if the quantities 5a and 74 are to be incorporated , or added together , their VOL . II . B sum will be twelve times the thing denoted by a CONTENTS VOL (THE THEORETICAL PART LEADING rules of the science.
Page 2
... denoted by a , or 12a . -- Also , in the other part of this case , which re- lates to the adding of what are usually called ne- gative quantities , as 2a and -3a , it is plain that no other idea can be attached to such expressions ...
... denoted by a , or 12a . -- Also , in the other part of this case , which re- lates to the adding of what are usually called ne- gative quantities , as 2a and -3a , it is plain that no other idea can be attached to such expressions ...
Page 3
... denoted ; so that to subtract one positive quantity from another , of the same kind , is equivalent to adding the same quantity taken negatively ; and to subtract a negative quantity , is equivalent to adding the same quantity taken ...
... denoted ; so that to subtract one positive quantity from another , of the same kind , is equivalent to adding the same quantity taken negatively ; and to subtract a negative quantity , is equivalent to adding the same quantity taken ...
Page 6
... denote the number of accents , that should be placed over the remainder that pre- cedes unity , we shall only have to prove , that I × p ( n ) = p ( n ! × 1 ; which being of itself evident , the proposition is true . 1 Hence , likewise ...
... denote the number of accents , that should be placed over the remainder that pre- cedes unity , we shall only have to prove , that I × p ( n ) = p ( n ! × 1 ; which being of itself evident , the proposition is true . 1 Hence , likewise ...
Page 25
... denoted by the ambiguous sign , as it would have been , if taken without any regard to its origin ; and the same may be shown of several other equations of this kind . dily derived from the examples here given ; and the LEADING RULES OF ...
... denoted by the ambiguous sign , as it would have been , if taken without any regard to its origin ; and the same may be shown of several other equations of this kind . dily derived from the examples here given ; and the LEADING RULES OF ...
Common terms and phrases
abscissa according algebraical arise asymptote axis become binomial binomial theorem bx² centre circle coefficients common consequently continued fraction cubic equation curve cx² cycloid denominator denoted derived determined diameter divided divisor drawn ellipse equa evident farther former formula fraction functions give given equation greater Hence hyperbola imaginary indefinitely infinite kind last equation last expression likewise logarithms manner mentioned method multinomial theorem multiplying negative observed obtained odd number ordinate parabola perpendicular positive roots powers proposed equation quadratic quadratic equation quadratrix quotient readily result right angled right line rule shown side signs similar spiral square subtangent subtracting supposed taken taking tangent theorem three roots tion triangle unknown quantity value be substituted Whence whole number
Popular passages
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.