Method of Indivisibles

Excerpts

• Method of Indivisibles

  A line is considered as the sum of an infinite number of points, a surface of an infinite number of lines, and a solid of an infinite number of surfaces.

• Method of Indivisibles

  If twenty points be taken on a straight line, the sum of the twenty-one lines which lie between point and point is equal to the whole line; which cannot be if the points by themselves constitute any part of the line, however small.

• Method of Indivisibles

  We would therefore recommend to the student not to regard any proposition derived from this method as true on that account; for falsehoods, as well as truths, may be deduced from it.

• Method of Indivisibles

  Though a point is not treated as a length, or as any part of space whatever, it is considered as having weight; and two points are spoken of as having different weights.

• Method of Indivisibles

  A cubic inch of gold weighs more than a cubic inch of water; hence gold is *denser* than water. If the first weighs $19$ times as much as the second, gold is said to be $19$ times more dense than water, or the density of gold is $19$ times that of water.

• Method of Indivisibles

  But if the weight of every two cubic inches is different, we can only find the weight of the whole by the integral calculus.

• Method of Indivisibles

  The integral of $bwx^{2}\, dx$ is $\frac{1}{3}bwx^{3}$, which taken from $x = 0$ to $x = a$ is $\frac{1}{3}bwa^{3}$.

Equations

• Method of Indivisibles

  AB = a

  The whole length of the bar AB is called a.

• Method of Indivisibles

  n\, dx = a

  Dividing the length a into n equal parts, each of width dx, makes the n parts together equal the whole length.

• Method of Indivisibles

  wb\, dx

  The weight of a bulk of water equal to a slice of volume b dx equals w times that volume.

• Method of Indivisibles

  x^{2} × bw\, dx

  If the density at the slice were uniform and equal to x squared, the weight of the slice would be approximately x squared times b w dx; this is only approximately true.

• Method of Indivisibles

  bwx^{2}\, dx + \alpha

  The real weight of the part pq is represented by the term bwx^2 dx plus a correction alpha, which can be made as small a part as we please of the first term.

• Method of Indivisibles

  \frac{1}{3}bwx^{3}

  The integral of b w x^2 dx is one third of b w x cubed.

• Method of Indivisibles

  \frac{1}{3}bwa^{3}

  Taking the integral from x = 0 to x = a gives the weight of the whole bar of length a and section b when the density is x squared.

Problems

No exercises in this chapter.