which shows that the segments, sectors, and circles are proportional. Further, denoting the arcs of the sectors by u, u,, and .. the circumferences by nu, nu,, we have

An Incremental Vanishing Arc of any continuous (403) curve, is to be regarded as a straight line.

Let z be such an arc, c its chord, and t, t2, tan

gents at its extremities, P, P2, that is, coinciding P in direction with z at P, P, and terminating in their point of intersection.

We have (387)

C=
c = tcos(t,c)+t2cos(t2,c);

C

Fig. 67.

PP

but it is obvious from the implied condition of continuity, that as the arc z decreases to its vanishing state and the points P, P2, approach to coincidence, that the tangents t, t, changing their directions by insensible degrees, will come to form one and the same straight line, and the angles, (t,c), (t,,c), decreasing to zero, their cosines become, [cos(t,c)] = 1, [cos(t2,c)] = 1;

Fig. 68.

and (113),

[c] = [t] + [t2];

<t+tz.

> c

Q. E. D.

Scholium. It is obvious that if the curve were other than continuous, no such conclusion as the above would result.

Fig. 69.

PROPOSITION IV.

To find the Derivative of the Segmental Area of any continuous curve referred to rectangular coördinates.

Let Y be segmental area in question, and K its increment; we have (403), (146),

The derivative of the segmental area of any continu- (404) ous curve is equal to the ordinate of that curve regarded as a function of the abscissa. [Y' = y = fx.]

PROPOSITION V.

To find the area of the Ellipse.

Let Y be an elliptical segment embraced by the semiminor axis and the abscissa x; we have (404), (203),

the area sought, no constant being added, since Y2 = 0.

(405)

+

( − 1 ) ( − 2)

2.3

Making x = a and multiplying by 4, we find

Ellipse Circumscribing Circle :: ba: 2b: 2a

:: Minor Axis Major Axis.

(410)

The Parabola is two-thirds the circumscribing rect- (413)

angle.

Scholium. It is obvious that the quadrature of the circle and ellipse can only be obtained approximately, while that of the parabola is exact.

PROPOSITION VII.
ROPOSITION

To find the proximate area of any continuous curve.

Suppose the equation of the curve, referred to rectangular coordinates, to be

then

...

=

(414)

Y = Ax+A1x2 +‡Ã ̧x3 +‡A2+...+constant, but, if the area Y become O when x = 0, which condition is always admissible, since the origin may be taken at pleasure, we have

3

....

Y = А ̧x+‡Ã ̧x2 +‡Ã1⁄2ï3 +‡A ̧x1 + ... But, since three points, Po, P1, P2, determine with considerable accuracy a curve of moderate extent, we will take the foot of the first ordinate, yo, as the origin, and the abscissas, x1 = h, x2 = 2h, so that the corresponding ordinates, yo, Y, Y2 shall be equally distant from each other; .. making x = 0, h, 2h, we have (414)

Yo = Ao,

and

Y2-y1 = Ah+ A2 · 3h2;

Y2-2y+yo= A2 • 2h2;

but when x = 2h, we have (415)

Y = A, 2h+A, • 4h2 + ‡A1⁄2 • 8h3 = 2h(A + A2h + 1‡A2 • h2);

or

•

Y=2h[y1+ (Y2 - 2y1+Yo)],
Y = fh(tyo+2y, +12).

If we continue the same method

of admeasurement and notation, we

(416)

12-2

.. Y2+Y4+Y6+...+Y2n = 3h(zyo +2y1 + Y2+QY3+Y4

+2y5+...+Y2n−2+2Y2n−1 + 1Y2n).

(417)

This beautiful and useful theorem is due to Simpson. The student should enunciate it in common language.

EXERCISES.

1o. Required the diameter of a circle having ten linear chains in circumference to every square chain in area.

2o. A square plate of silver, 3 inches, on the side, is worth $4. What is the value of the greatest circle that can be cut from it? 3°. Had the plate in 2° been an equilateral triangle, what would have been its value?

4°. The two sides including the right angle of a right angled triangle, are three and four rods; what is the area of the circumscribing circle?

5°. Determine a circle circumscribing an isosceles triangle, the two equal sides of which, including an angle of 36°, are 15'15 chs. each?

6°. The equal sides of an isosceles triangle embrace an angle of 473°, and the area of the inscribed circle is one acre. Determine the triangle.

7. A circular plate of brass, 20 inches in diameter, is worth $375. What is the value of the three greatest and equal circles that can be cut from it?

8°. Required the area of a circular segment embraced by an arc and its chord, the length of which is 5'87 chs., and the breadth 1'35? 9°. The dimensions of an elliptical fish-pond are 10 and 15 rods. What is its area?

10°. The ordinate of a parabolic segment is 3 chains, and the corresponding abscissa 7 chains. Required the area.

11°. Required the area of an elliptical segment embraced between the semiminor axis and an ordinate 5657 chains, and having a breadth of one chain.

=

12°. Wishing to ascertain the cross section of a river 100 yards from water's edge to water's edge, I take soundings every 10 yards, and find them to be in yards:

Yo = 0, y1 = 12, y2 = 2061, y;= 25'3, y1 = 28'4, y5 = 2969, y = 29'3, y1 = 26'1, y1 = 20‘9, y, = 12′8, Y10 = 0.