The Doctrine and Application of Fluxions: Containing (Besides What Is Common to the Subject) a Number of New Improvements in the Theory, and the Solution of a Variety of New and Very Interesting Problems in Different Branches of Mathematics, Volume 2
The Doctrine and Application of Fluxions: Containing (Besides What Is Common to the Subject) a Number of New Improvements in the Theory, and the Solution of a Variety of New and Very Interesting Problems in Different Branches of Mathematics, Volume 2
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. 1823 edition. Excerpt: ...when both its numerator and denominator become equal to nothing, or vanish, at the same time. Which value (it follows from above) will be found by dividing the fluxion of the numerator by that of the denominator. For, since the value of any fraction, in that circumstance, is to be looked on as the limiting ...
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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. 1823 edition. Excerpt: ...when both its numerator and denominator become equal to nothing, or vanish, at the same time. Which value (it follows from above) will be found by dividing the fluxion of the numerator by that of the denominator. For, since the value of any fraction, in that circumstance, is to be looked on as the limiting ratio towards which its, two terms converge, before they vanish, and seeing the fluxions are always expressed by that ratio, the trulh of the rule, or position, is manifest. An example, however, may not be improper: x---a Let, therefore, the fraction be propounded, to x--a find the value thereof when x=a. In which case, the true value sought, or the fluxion of the numerator divided by that of the denominator, is =--. x = 2x=2a. And that this is the true value, may be confirmed by common division, whereby the fraction proposed is reduced to x + a; whose value, when x=a, is therefore =2a, the very same as before. SECTION VIII. The Use of Fluxions in the Rectification, or finding the Lengths, of Curves. CASE I. 135.-LiET A C G be a Curve of any kind whose Ordinates are parallel to themselves, and perpendicular to the Axis A Q. If the fluxion of the abscissa A M be denoted b M m; or by C n (equal and parallel to M m) and n S equal and parallel to C r, be taken to represent the corresponding fluxion of the ordinate M C; then will the Art. 48 diagonal C S (touching the curve in C) be the line & 49. which the generating point (p) would describe, was its motion to become uniform at C ( Vide Art. 48 and 49), which line, therefore, truly expresses the fluxion of the + Art. 2. space A C gone over, according to the definition, -f Hence, putting A M = x, C M=zy, and A C=z, we have z (= C S = /C + Sn2) = V& + f; from which, and the equation of the...
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All Editions of The Doctrine and Application of Fluxions: Containing (Besides What Is Common to the Subject) a Number of New Improvements in the Theory, and the Solution of a Variety of New and Very Interesting Problems in Different Branches of Mathematics, Volume 2
