Elementary Illustrations of the Differential and Integral Calculus
On the Ratios of Continuously Increasing or Decreasing Quantities
Excerpts
On the Ratios of Continuously Increasing or Decreasing Quantities
assign a line and an angle, however small, $B$ can be placed so near to $A$ that the lines and angles above alluded to shall be severally less than the assigned line and angle.
On the Ratios of Continuously Increasing or Decreasing Quantities
To illustrate this result from the trigonometrical tables, observe that if the radius $OA$ be the linear unit, and $\angle BOA = \theta$, $BM$ and $BA$ are respectively $\sin\theta$ and $2\sin\frac{1}{2}\theta$.
On the Ratios of Continuously Increasing or Decreasing Quantities
In geometry and mechanics, it is necessary to consider quantities as increasing or decreasing *continuously*; that is, a magnitude does not pass from one value to another without passing through every intermediate value. Thus if one point move towards another on a circle, both the arc and its chord decrease continuously.
On the Ratios of Continuously Increasing or Decreasing Quantities
Again, $OT$ diminishes and $OM$ increases, but neither without limit, for the first is never less, nor the second greater, than the radius.
On the Ratios of Continuously Increasing or Decreasing Quantities
Let $\theta = 1°$; then $\sin\theta = .0174524$ and $2\sin\frac{1}{2}\theta = .0174530$; whence $2\sin\frac{1}{2}\theta ÷ \sin\theta = 1.00003$ very nearly, so that $BM$ differs from $BA$ by less than four of its own hundred-thousandth parts.
On the Ratios of Continuously Increasing or Decreasing Quantities
Thus if $\angle BOA = 1°$, $BM ÷ MA = 114.589$ and $BA ÷ MA = 114.593$ very nearly; that is, $BM$ and $BA$ both contain $MA$ more than $114$ times.
On the Ratios of Continuously Increasing or Decreasing Quantities
The arc $BA$ always lies between $BA$ and $BN + NA$, or $BM + MA$; hence $\dfrac{\arc BA}{\chord BA}$ lies between $1$ and $\dfrac{BM}{BA} + \dfrac{MA}{BA}$. But $\dfrac{BM}{BA}$ has been shown to approach continually towards $1$, and $\dfrac{MA}{BA}$ to decrease without limit; hence $\dfrac{\arc BA}{\chord BA}$ continually approaches towards $1$.
Equations
On the Ratios of Continuously Increasing or Decreasing Quantities
OD ÷ OA = BM ÷ BAThe ratio of OD to OA equals the ratio of BM to BA, because the right triangles ODA and BMA are similar.
On the Ratios of Continuously Increasing or Decreasing Quantities
\sin\theta = .0174524Table value: the sine of one degree is about 0.0174524.
On the Ratios of Continuously Increasing or Decreasing Quantities
2\sin\frac{1}{2}\theta ÷ \sin\theta = 1.00003For an angle of one degree, twice the sine of half the angle divided by the sine of the angle is very nearly 1.00003, so the chord BA differs from BM by less than four hundred-thousandths of itself.
On the Ratios of Continuously Increasing or Decreasing Quantities
BM ÷ MA = 114.589For an angle BOA of one degree, the ratio of BM to MA is very nearly 114.589, so BM contains MA more than 114 times.
On the Ratios of Continuously Increasing or Decreasing Quantities
\angle BOA = \thetaThe angle at the centre O subtended by the arc AB is denoted theta.
Problems
No exercises in this chapter.