Elementary Illustrations of the Differential and Integral Calculus
On the Ratio of Magnitudes that Vanish Together
Excerpts
On the Ratio of Magnitudes that Vanish Together
while the magnitudes diminish, we may not assume either that their ratio increases, diminishes, or remains the same, for we have shown that a diminution of two magnitudes is consistent with either of these.
On the Ratio of Magnitudes that Vanish Together
Here both $M$ and $N$ decrease at every step, but $M$ loses at each step a larger fraction of itself than $N$, and their ratio continually diminishes.
On the Ratio of Magnitudes that Vanish Together
This is what we mean by saying that $\dfrac{M}{N}$ is an increasing ratio, the limit of which is $2$.
On the Ratio of Magnitudes that Vanish Together
For example, let a point $A$ move on a circle towards a fixed point $B$. The arc $AB$ will then diminish, as also the chord $AB$, and by bringing the point $A$ sufficiently near to $B$, we may obtain an arc and its chord, both of which shall be smaller than a given line, however small this last may be.
On the Ratio of Magnitudes that Vanish Together
But while the magnitudes diminish, we may not assume either that their ratio increases, diminishes, or remains the same, for we have shown that a diminution of two magnitudes is consistent with either of these.
On the Ratio of Magnitudes that Vanish Together
The first possible case is that the ratio of $M$ to $N$ may decrease without limit, that is, $M$ may be a smaller fraction of $N$ after a decrease than it was before, and a still smaller after a further decrease, and so on; in such a way, that there is no fraction so small, to which $\dfrac{M}{N}$ shall not be equal or inferior, if the decrease of $M$ and $N$ be carried sufficiently far.
On the Ratio of Magnitudes that Vanish Together
The second possible case is that in which the ratio of $M$ to $N$, though it increases or decreases, does not increase or decrease without limit, that is, continually approaches to some ratio, which it never will exactly reach, however far the diminution of $M$ and $N$ may be carried.
On the Ratio of Magnitudes that Vanish Together
The difference between this case and the last is, that the ratio of $M$ to $N$, though perpetually increasing, does not increase without limit; it is never so great as $2$, though it may be brought as near to $2$ as we please.
On the Ratio of Magnitudes that Vanish Together
Therefore (1) $\dfrac{M}{N}$ continually increases; (2) may be brought as near to $2$ as we please; (3) can never be greater than $2$. This is what we mean by saying that $\dfrac{M}{N}$ is an increasing ratio, the limit of which is $2$.
On the Ratio of Magnitudes that Vanish Together
In introducing the notion of time, we consult only simplicity. It would do equally well to write any number of successive values of the two quantities, and place them in two columns.
Equations
On the Ratio of Magnitudes that Vanish Together
\dfrac{2}{x(x + 1)}The x-th value of M in the second table is 2 divided by x(x+1), where M and N are the two decreasing quantities in the example.
On the Ratio of Magnitudes that Vanish Together
\dfrac{1}{x^{2}}The x-th value of N in the second table is 1 divided by x squared.
On the Ratio of Magnitudes that Vanish Together
\dfrac{M}{N} = \dfrac{2x^{2}}{x(x + 1)}The x-th value of the ratio M to N equals 2x squared over x(x+1), obtained by dividing the two sequence values above.
On the Ratio of Magnitudes that Vanish Together
\dfrac{2x}{x + 1}The ratio M to N simplifies to 2x divided by (x+1); it is always less than 2 yet approaches 2 as x grows.
On the Ratio of Magnitudes that Vanish Together
1 - \dfrac{1}{x + 1}x/(x+1) equals 1 minus 1/(x+1), so it differs from 1 by 1/(x+1), which can be made as small as we please.
On the Ratio of Magnitudes that Vanish Together
1 + 2 + 3 + \dots + x,\quad\text{or}\quad \frac{x(x + 1)}{2}The denominator of the x-th value of M is the sum of the integers 1 through x, which equals x(x+1)/2.
Problems
No exercises in this chapter.