The Principles of Algebra

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1796 Excerpt: ... while 3 xx is less than b, or xx is less than the increment of x3 will be less than the contempo 3 rary increment of bx; but, when %xx is greater than b, or xx is greater than----, (though less than b,) the increment of x3 will be greater than the contemporary increment of bx. But, while x increases from x to x + x, or receives the increment, and bx receives the increment bx, and 3 receives the increment 3 + yc1 + x3, or (neglecting the two latter quantities,) 3«% the compound quantity bx--x3 will become equal to bx + bx--'--3--3#x2--x3, or (neglecting the two terms xx1 and x) bx + bx---x1--3x1x, or bx--x3 + bx--3, that is, it will be increased by bx, or the increment of bx, and diminilhed by y?x, or the increment of x3. Therefore, so long as hi, or the increment of bx, continues to be greater than 3% or the contemporary increment of the compound quantity bx--x3 will increase; but asterwards, when bx, or the increment of bx, is less than 3.V2, or the contemporary increment of x3, it will decrease. And therefore the greatest magnitude of the compound quantity bx--3 will be that which it has when b, or the increment of bx, is exactly equal to Ja-1, or the contemporary increment of x3, that is, when i is = 3, or x is =----3 T,,. b. /b.,. bJb But, when xx is =--, x is, and x3 is =----, and bx is =. and consequently A---is = V3--bj/b_-jbjb--_ dWb V3 J71 lV3 3V3' 3«/i, Therefore the greatest possible magnitude of the compound quantity bx--x3 is £y£ ani consequently the greatest possible magnitude of c, or the absolute term of the equation bx--xs = c, Is--QE. I.-irf. LXV. It follows from the foregoing investigation of the greatest possible magnitude of the compound quantity bx--x3, that, if c3 or the absolute term of the equation bx--x3--c, is...