The Notion of Infinitely Small Quantities

Excerpts

• The Notion of Infinitely Small Quantities

  Since $(a + h)^{2} = a^{2} + 2ah + h^{2}$, it appears that if $a$ be increased by $h$, $a^{2}$ is increased by $2ah + h^{2}$.

• The Notion of Infinitely Small Quantities

  The smaller $h$ is made, the more near does this proportion diminish towards that of $2a$ to $1$, to which it may be made to approach within any quantity, if it be allowable to take $h$ as small as we please.

• The Notion of Infinitely Small Quantities

  The proposition, therefore, that $h$ can be taken so small that $2ah + h^{2}$ and $2ah$ are rigorously equal, though not true, and therefore entailing error upon all its subsequent consequences, yet is of this character, that, by taking $h$ sufficiently small, all errors may be made as small as we please.

• The Notion of Infinitely Small Quantities

  Again, it would be said (1) that if $AB$ be infinitely small, $MA$ is infinitely less than $BM$.

• The Notion of Infinitely Small Quantities

  In this case he used the phrase that $n$ is *infinitely* small with respect to $m$.

• The Notion of Infinitely Small Quantities

  In this reasoning there is evidently an absolute error; for it is impossible that $h$ can be so small, that $2ah + h^{2}$ and $2ah$ shall be the same.

• The Notion of Infinitely Small Quantities

  Nothing is either small or great in itself, these terms only implying a relation to some other magnitude of the same kind, and even then varying their meaning with the subject in talking of which the magnitude occurs, so that both terms may be applied to the same magnitude: thus a large field is a very small part of the earth.

• The Notion of Infinitely Small Quantities

  Here all dispute about a standard of smallness is avoided, because, be the standard whatever it may, the proportion of $h^{2}$ to $h$ may be brought under it.

• The Notion of Infinitely Small Quantities

  The desire of combining simplicity with the appearance of rigorous demonstration, probably introduced the notion of infinitely small quantities; which was further established by observing that their careful use never led to any error.

• The Notion of Infinitely Small Quantities

  If $a$ be increased by $h$, $a^{2}$ is increased by $2ah + h^{2}$, which, whatever may be the value of $h$, is to $h$ in the proportion of $2a + h$ to $1$. The smaller $h$ is made, the more near does this proportion diminish towards that of $2a$ to $1$, to which it may be made to approach within any quantity, if it be allowable to take $h$ as small as we please. Hence the ratio, $\emph{increment of } a^{2} ÷ \emph{increment of } a$, is a decreasing ratio, whose limit is $2a$.

• The Notion of Infinitely Small Quantities

  This should be considered as an abbreviation of the proposition proved (10), and of the following: If a polygon be inscribed in a circle, the greater the number of its sides, and the smaller their lengths, the more nearly will the perimeters of the polygon and circle be equal to one another; and further, if any straight line be given, however small, the difference between the perimeters of the polygon and circle may be made less than that line, by sufficient increase of the number of sides and diminution of their lengths.

Equations

• The Notion of Infinitely Small Quantities

  (a + h)^{2} = a^{2} + 2ah + h^{2}

  Squaring a sum a + h expands to a^2 + 2ah + h^2, so increasing a by h increases a^2 by 2ah + h^2.

• The Notion of Infinitely Small Quantities

  1:h :: h:h^{2}

  As 1 contains h, so many times, h contains h^2; the four quantities 1, h, h, h^2 are in proportion.

Problems

No exercises in this chapter.