On the Ratio or Proportion of Two Magnitudes

Excerpts

• On the Ratio or Proportion of Two Magnitudes

  Let the given ratio be that of the numbers $m$ and $n$. Then, $P$ being a line, $mP$ and $nP$ are in the proportion of $m$ to $n$; and it is evident, that let $m$, $n$, and $A$ be what they may, $P$ can be so taken that $mP$ shall be less than $A$.

• On the Ratio or Proportion of Two Magnitudes

  Thus, the ratio of the diagonal of a square to its side is that of $\sqrt{2}$ to $1$, which is very nearly that of $14142$ to $10000$, and is certainly between this and that of $14143$ to $10000$.

• On the Ratio or Proportion of Two Magnitudes

  We are not, therefore, entitled to say that because two magnitudes are diminished, their ratio is diminished; it is possible that $B$, which we will suppose to be at first a hundredth part of $C$, may, after a diminution of both, be its tenth or thousandth, or may still remain its hundredth, as the following example will show:

• On the Ratio or Proportion of Two Magnitudes

  In estimating the approach to, or departure from equality, which two magnitudes undergo in consequence of a change in their values, we must not look at their differences, but at the proportions which those differences bear to the whole magnitudes.

• On the Ratio or Proportion of Two Magnitudes

  For example, if a geometrical figure, two of whose sides are $3$ and $4$ inches now, be altered in dimensions, so that the corresponding sides are $100$ and $101$ inches, they are nearer to equality in the second case than in the first; because, though the difference is the same in both, namely one inch, it is one third of the least side in the first case, and only one hundredth in the second.

• On the Ratio or Proportion of Two Magnitudes

  Thus, twenty miles would be a material error in talking of a day’s journey, but would not be considered worth mentioning in one of three months, and would be called totally insensible in stating the distance between the earth and sun.

• On the Ratio or Proportion of Two Magnitudes

  In future, when we talk of an approach towards equality, we mean that the ratio is made more nearly equal to unity, not that the difference is more nearly equal to nothing. The second may follow from the first, but not necessarily; still less does the first follow from the second.

• On the Ratio or Proportion of Two Magnitudes

  This is only saying that $P$ can be taken less than the $m$th part of $A$, which is obvious, since $A$, however small it may be, has its tenth, its hundredth, its thousandth part, etc., as certainly as if it were larger.

Equations

• On the Ratio or Proportion of Two Magnitudes

  \dfrac{x + a}{x} = 1 + \dfrac{a}{x}

  The ratio of the two quantities x + a and x equals 1 plus a/x, which expresses how far that ratio is from unity.

• On the Ratio or Proportion of Two Magnitudes

  \dfrac{x + m + a}{x + m} = 1 + \dfrac{a}{x + m}

  After an increase m is given to x, the ratio of x + m + a to x + m equals 1 plus a/(x + m), which lies nearer to unity than before because the same difference a is divided by a larger quantity.

Problems

No exercises in this chapter.