arcs, and thereby to increase the number of sides of the inscribed polygons, the ares subtended by these sides become nearer and nearer to the sides themselves, and finally the difference between them will become imperceptible.

Q. And what conclusion can you now draw respecting the whole circumference of a circle?

A. That the circumference of a circle differs very little from the sum of all the sides of a regular inscribed polygon of a great number of sides; therefore if the number of sides of the inscribed polygon is very great, (several thousand for instance) the polygon will differ so little from the circle itself, that without perceptible error the one may be taken for the other.

QUERY XXIII.

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It has been shown in the last query, that a circle be considered as a regular polygon of a very great number of sides; what inferences can you now draw with regard to the circumferences and areas of circles?

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A

A. 1. The circumferences of two circles are in proportion to the radii of these circles; that is, a straight line as long as the circumference of the first circle, is as many times greater than

a straight line, as long as the circumference of the second circle; as the radius AB of the first circle, is greater than the radius ab of the second circle. For if in each of the two circles a regular polygon of a very great number of sides is inscribed, the sums of all the sides of the two polygons are to each other in proportion to the radii, AB, ab, of the circumscribed circles (page 157, 1st); and as the difference between the circumference of a circle and the sum of all the sides of an inscribed polygon of a great number of sides is imperceptible (last Query), we may say that the circumferences themselves are in the same ratio as the radii AB, a b.*

2. The areas of two circles are in proportion to the squares constructed upon their radii; that is, the area of the greater circle is as many times greater than the area of the smaller circle, as the area of the square upon the radius of the greater circle, is greater than the area of the square constructed upon the radius of the smaller circle. For if in each of these circles a regular polygon of a great

* The teacher may give an ocular demonstration of this principle by taking two circles, cut out of pasteboard or wood; and measuring their circumferences by passing a string around them. The measure of the one will be as many times greater than the measure of the other, as the radius of the first circle is greater than the radius of the second circle.

number of sides is inscribed, the difference between the areas of the polygons and the areas of the circles themselves will be imperceptible; and because the areas of two regular polygons of the same number of sides are in the same ratio, as the areas of the squares upon the radii of the circles in which they are inscribed, (page 158, 2dly) the areas of these circles will themselves be in the ratio of the squares upon their radii.

3. The area of a circle is found by multiplying the circumference of the circle, given in rods, feet, inches, &c. by half the radius, given in units of the same kind. Because a circle differs so little from a regular inscribed polygon of a great number of sides, that the area of the polygon may, without perceptible error, be taken for the area of the circle. Now, the area of regular polygon inscribed in a circle, is found by multiplying the sum of all the sides by the radius of the inscribed circle (page 159) and dividing the product by 2; therefore the area of the circle itself is found by multiplying the circumference (instead of the sums of all the sides of the inscribed polygon) by the radius, and dividing the product by 2. For it has been shown in the last Query, that the sides of a regular inscribed polygon grow nearer and nearer the circumference of the circumscribed circle, in proportion

as these sides increase in number; consequently the circumference of a circle, inscribed in a regular polygon of a great number of sides, will also grow nearer and nearer the circumference of the circumscribed circle; until finally the two circumferences will differ so little from each other, that the radius of the one may, without perceptible error, be taken for the radius of the other.

Remark. Finding the area of a circle is sometimes called squaring the circle. The problem, to construct a rectilinear figure, for instance a rectangle, whose area shall exactly equal the area of a given circle, is that which is meant by finding the quadrature of the circle. For the area of any geometrical figure, terminated by straight lines only, can easily be found by the rule given in Query 5, Sect. III; or, in other words, we can always construct a square, which shall measure exactly as many square rods, feet, inches, &c. as a rectilinear figure of any number of sides.

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M

Now it is easy to show, that there is nothing absurd in the idea of constructing a rectilinear figure, for instance a rectangle, whose area shall be equal to the area of a given circle. For let us take a semicircle ABM, and let us for a moment imagine the diameter G AB to move parallel to itself be- E tween the two perpendiculars AI, BK. It is evident that when the A diameter AB is very near its original

C

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F

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B

position, for instance in CD, the area of the rectangle ABCD is smaller than the area of the semicircle ABM; but the diameter continuing to move parallel to itself in the direction from A to I, there will be a point in the line AI, where the area of the rectangle ABIK is greater than the area of the semicircle ABM. Now as there is a point in the line AI, below the point I, in which the area of the rectangle ABCD is smaller than the area of the

semicircle ABM, and as the diameter by continuing to move in the same direction makes in different points C, E, G, &c. of that same line, the rectangles ABCD, ABEF, ABGH, &c. whose areas become greater and greater, until finally they become greater than the area of the semicircle itself; there must evidently be a point in the line AI, in which a line drawn parallel to the diame ter AB makes with it and the perpendiculars AI, BK, a rectangle, which, in area, is equal to the semicircle ABM; and as there is a rectangle which, in area, is equal to the semicircle ABM, by doubling it, we shall have a rectangle which, in area, is equal to the whole circle.

Neither is it difficult to find the area of a circle mechanically. For the area of a circle being found by multiplying the circumference by the length of the radius, and dividing the product by 2 (page 165, 3dly), we need only pass a string around the circumference of a circle, and then multiply the length of that string by the length of the radius; the product divided by 2 will be the area of the circle. Having thus found the comparative length of the radius and circumference of one circle, we might determine the circumference, and thereby the area of any other circle, when knowing its radius. For the circumferences of two circles being in proportion to the radii of the two circles, we should have three terms of a geometrical proportion given; viz. the radii of the two circles, and the circumference of the one; from which we might easily find the fourth term (theory of Proportions, page 75, 6thly), which would be the circumference of the other circle.

But the expressions of the circumference and area of a circle, thus obtained by measurement, are never so correct as is required for very nice and accurate mathematical calculations; we must therefore resort to other means such as geometry itself furnishes, to calculate the ratio of the radius or diameter to the circumference of the circle; and herein consists the difficulty of the quadrature of the circle. For if the ratio of the radius to the circumference is once determined, we can easily find the circumference of any circle, when its radius is given; and knowing the circumference and the radius, we can find the area of the circle. (page 165, 3dly).