Elementary Illustrations of the Differential and Integral Calculus

A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines

Excerpts

A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
As $A'B'$ moves towards $AB$, $da$ and $db$ are diminished without limit, $a$ and $b$ remaining the same; hence the limit of the ratio $\dfrac{db}{da}$ is $\dfrac{2a}{2b}$ or $\dfrac{a}{b}$.
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
But here it is necessary to remark that $AB$ is itself one of the positions intermediate between $A'B'$ and $A''B''$, and when two lines are, by the motion of one of them, brought into one and the same straight line, they intersect one another (if this phrase can be here applied at all) in every point, and all idea of one distinct point of intersection is lost.
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
Let $P$ be the point of separation; then every point of $P'P''$, except $P$, is a real point of intersection of $AB$, with one of the positions of $A''B''$, and when $A''B''$ has moved very near to $AB$, the point $P''$ will be very near to $P$; and there is no point so near to $P$, that it may not be made the intersection of $A''B''$ and $AB$, by bringing the former sufficiently near to the latter.
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
The same result may be more simply obtained, by diminishing $da$ and $db$ in equation (5), before obtaining the values of $y$ and $x$.
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
This limit of the intersections is different for every different position of the line $AB$, but may be determined, in every case, by the following simple construction.
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
Hence $BP = AQ$ and $AP = BQ$, or the point $P$ is as far from either extremity of $AB$ as $Q$ is from the other.

Equations

A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
a^{2} + b^{2} = l^{2}
The squares of the two axial segments OA and OB add up to the square of the length of the sliding line AB.
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
(a + da)^{2} + (b - db)^{2} = l^{2}
In the displaced position the line still has length l, with its ends at a + da and b - db along the axes.
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
2a\, da + (da)^{2} - 2b\, db + (db)^{2} = 0\Add{,}
Subtracting the first length equation from the displaced one gives a relation between the increments da and db.
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
\frac{db}{da} = \frac{2a + da}{2b - db}\Add{.}
The ratio of the increment db to the increment da equals (2a + da)/(2b - db); this is equation (1).
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
ay + bx = ab\Add{.}
Every point (x, y) of the line AB satisfies this equation in the coordinate axes; this is equation (2).
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
(a + da)y + (b - db)x = (a + da)(b - db)\Add{;}
The line of the displaced position, which cuts off a + da and b - db from the axes, has this equation; this is equation (3).
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
y\, da - x\, db = b\, da - a\, db - da\, db\Add{.}
Subtracting equation (2) from equation (3) gives this relation for the point P' of intersection; this is equation (4).
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
y - x\, \frac{2a + da}{2b - db} = b - a\, \frac{2a + da}{2b - db} - db\Add{.}
Dividing equation (4) by da and substituting db/da from equation (1) gives this relation; this is equation (5).
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
y - \frac{a}{b}\, x = b - \frac{a^{2}}{b}
Equation (5) with da and db diminished without limit gives this line through the limit point P; the same line in cleared form is equation (6).
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
by - ax = b^{2} - a^{2}\Add{.}
Cleared of fractions, the limit of the intersections satisfies this equation; this is equation (6).
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
x = OM = \frac{a^{3}}{a^{2} + b^{2}} = \frac{a^{3}}{l^{2}}
Solving equations (6) and (2) gives the co-ordinate x of the limit point P as a^3 over l^2.
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
y = MP = \frac{b^{3}}{a^{2} + b^{2}} = \frac{b^{3}}{l^{2}}
Solving equations (6) and (2) gives the co-ordinate y of the limit point P as b^3 over l^2.
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
BP = OM\, \dfrac{BA}{AO} = \dfrac{a^{3}}{l^{2}}\, \dfrac{l}{a} = \dfrac{a^{2}}{l}
By similar triangles the distance BP from B to the limit point equals a^2 over l.
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
PA = \dfrac{b^{2}}{l}
Similarly, the distance PA from A to the limit point equals b^2 over l.
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
AQ = \dfrac{a^{2}}{l}
Since OA is a mean proportional between AQ and AB (OQ perpendicular to BA), AQ equals a^2 over l.
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
BQ = \dfrac{b^{2}}{l}
Similarly, BQ equals b^2 over l, the distance from B to the foot Q.
A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines
BP = AQ
The limit point P lies as far from B as Q lies from A, so the limit of the intersections is found by this construction.

Problems

No exercises in this chapter.