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other as the radius CA to OB; (8.5;) that is,

per. MNP, &c. per. EFG, &c. :: CA: OB.

And if the arcs which subtend the sides of the polygons, are continually bisected, the number of sides of the polygons will become indefinitely increased, and their perimeters will be equal to the circumferences of the circles: (9. 5. Cor. 1:) hence we shall have, (11. 3,)

circ. CA: circ. OB:: CA: OB.

Again, the areas of inscribed polygons are to each other as the of the radii of the circumscribed circles; that is,

squares

area MNP, &c. : area EFG, &c. :: CA': OB3.

But if the number of sides of the polygons be indefinitely increased, their areas will be respectively equal to the areas of the circles. (9. 5. Cor. 2.)

Hence,

area CA area OB:: CA2: OB2.

Therefore, The circumferences of circles, &c.

Cor. The similar arcs

AB, DE, are to each other as their radii AC, DO; and the similar sectors, ACB, DOE are to each other as the squares of those radii.

A

B

D

E

For since the arcs are similar, the angle C, (Def. 4.4,) is equal to the angle O; but C is to four right-angles as the arc AB is to the whole circumference described with the radius AC; (12.2;) and O is to four right-angles, as the arc DE is to the circumference described with the radius OD: hence the arcs AB, DE, are to each other as the circumferences of which they form part: but these circumferences are to each other as the radii AC, DO; hence,

arc. AB arc. DE :: AC : DO.

For a like reason, the sectors ACB, DOE, are to eac

other as the whole circles; which are as the squares of their radii; therefore, sect. ACB sect. DOE :: AC2: DO2.

[blocks in formation]

The area of a circle is equal to the product of its circumference by half the radius.

Let the area of the circle whose radius is OT, be designated by area ON; then will area ON-OT x circ. TNP. For the area of the polygon I GHIK, described about the circle TNP, is equal to its perimeter multiplied by half of the radius OT. (7. 5.)

HT

G

N

P

K

M

Now if the number of sides of the circumscribed polygon be indefinitely increased, its perimeter will coincide with the circumference of the circle, and its area will become equal to the area of a circle. (9. 5. Cor. 2.) Hence the area of the polygon will be equal to the circumference TNP × 10T; and the area of the circle is equal to the same; that is, area ON=OT × circ. TNP.

Hence, The area of a circle, &c.

Cor. 1. The surface of a sector is equal to the arc of that sector multiplied by half its radius.

For, the sector ACB, is to the whole circle as the arc AMB is to the whole cir

cumference ABD, (12. 2. Sch. 2,) or as AMB X AC is to ABD XAC.

But the

M

B

whole circle is equal to ABDxAC; hence the sector

ACB is measured by AMB

AC.

Cor. 2. Let the circumference of the circle whose diam

eter is unity be denoted by: then, because circumferences are to each other as their radii or diameters, we shall have the diameter 1 to its circumference as the diameter 2CA is to the circumference whose radius is CA; that is,

1: : : 2CA: circ. CA, therefore circ. CA=2π × CA.

Multiply both terms by CA; we have CA x circ. CA=π × CA2, or surf. CA=π× CA2: hence the surface of a circle is equal to the product of the square of its radius by the constant number, which represents the circumference whose diameter is 1, or the ratio of the circumference to the diameter.

In like manner the surface of the circle, whose radius is OB, will be equal to XOB2; but XCA2 : π × OB2 : : CA': OB2; hence the surfaces of circles are to each other as the squares of their radii, which agrees with the preceding theorem.

Scholium. It has been already observed, that the problem of the quadrature of the circle consists in finding a square equal in surface to a circle, the radius of which is known. Now it has just been proved, that a circle is equivalent to the rectangle contained by its circumference and half its radius; and this rectangle may be changed into a square, by finding a mean proportional between its length and its breadth. (Prob. 3. 4.) To square the circle, therefore, is to find the circumference when the radius is given; and for effecting this, it is enough to know the ratio of the circumference to its radius, or its diameter.

Hitherto, the ratio in question has never been determined except approximately; but the approximation has been carried so far, that a knowledge of the exact ratio would afford no real advantage whatever beyond that of the approximate ratio. Accordingly, this problem, which engaged geometers so deeply, when their methods of approximation were less

perfect, may now be regarded as an idle question unworthy of the scholar's attention.

Archimedes showed that the ratio of the circumference to the diameter is included between 30 and 31; hence 34 or affords at once a pretty accurate approximation to the number above designated by ; and the simplicity of this first approximation has brought it into very general use. Metius, for the same number, found the much more accurate value. At last the value of π, developed to a certain order of decimals, was found by other calculators to be 3.1415926535897932, &c.; and some have had patience enough to continue these decimals to the hundred and twentyseventh, or even to the hundred and fortieth place. Such an approximation is evidently equivalent to perfect correctness: the root of an imperfect power is in no case more accurately known.

BOOK VI.

PLANES AND SOLID ANGLES.

DEFINITIONS.

1. A straight line is perpendicular to a plane, when it is perpendicular to all the straight lines which pass through its foot in the plane. Conversely, the plane is perpendicular to the line.

The foot of the perpendicular is the point at which that line meets the plane.

2. A line is parallel to a plane, when it cannot meet that plane, to whatever distance both be produced. Conversely, the plane is parallel to the line.

3. Two planes are parallel to each other, when they cannot meet, to whatever distance both be produced.

4. The angle or mutual inclination of two planes which meet or intersect one another, is the opening between them. This quantity is measured by the angle contained between two perpendiculars drawn in these planes, at the same point of their common intersection.

This angle may be acute, or right, or obtuse.

5. If it is a right-angle, the two planes are perpendicular to each other.

6. A solid angle is the angular space included between several planes which meet at the same point.

Thus, the solid angle S, is formed by the union of the planes ASB, BSC, CSD, DSA.

Three planes at least, are requisite to form a solid angle.

S

B

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