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Suppl. AH(AC+CG)=500X1866.02545-933012.73—; therefore

P=965.92585—, because (965.92585)2 is greater than 933012.73. Hence AČ+P=1965.92585

As Q is the perpendicular drawn from the centre on the chord of one-twenty-fourth of the circumference, Q2= AH(AC+P)=500x1965.92585—=982962.93—; therefore Q=991.44495-because (991.44495)2 is greater than 982962.93. Hence AC+Q=1991.44495

Because S is the perpendicular from C on the chord of oneforty-eighth of the circumference, Sø=-AH(AC+Q)=500 (1991.4449–)=995722.475—therefore S997.85895-because (997.85895)2 is greater than 995722.475.

But the square of the chord of the ninety-sixth part of the circumference=AB(AC-S)=2000(2.14105+) = 4282.1-+; and the chord=65.4377+, because (65.4377)2 is less than 4282.1; therefore the perimeter of a polygon of ninety-six sides inscribed in the circle = (65.4377+) 96=6282.019+. But the circumference of the circle is greater than the perimeter of the inscribed polygon ; therefore the circumference is greater than 6282.019 of the parts of which the radius contains 1000, or than 3141.009 of the parts of which the radius contains 500, or the diameter contains 1000. Now 3141.009 has to 1000 a greater ratio than 3+4 to 1; therefore the circumference of the circle has a greater ratio to the diameter than 3+41 has to 1; that is, the excess of the circumference above three times the diameter is greater than ten of the parts of which the diameter contains 71; and it has already been shown to be less than ten of the parts of which the diameter contains 70, Therefore, the circumference, &c. Q. E. D,

COR. 1. Hence, the diameter of a circle being given, the circumference may be found nearly by this proportion, as 7 to 22, so the given diameter to a fourth proportional, which will be greater than the circumference ; or by this proportion, as 1 to 3+41, or as 71 to 223, so the given diameter to a fourth proportional, which will be less than the circumference.

Cor. 2. Because the difference between 1 and 1 is 1979 the lines found by these proportions differ by an of the diameter. Therefore the difference of either of them from the circumference must be less than the 497th part of the diameter,

COR. 3. As 7 is to 22, so is the square of the radius to Book I. the area of the circle nearly.

For the diameter of a circle is to its circumference as the square of the radius to the area of the circles; and the dia- k Cor. meter is to the circumference nearly as 7 to 22; therefore 5.1. Sup. the square

of the radius is to the area of the circle nearly as 7 to 22.

SCHOLIUM.

It is evident that the method employed in this proposition for finding the limits of the ratio of the circumference to the diameter may be carried to a greater degree of exactness, by finding the perimeters of an inscribed polygon and of a circumscribed polygon of a greater number of sides than 96. The manner in which the perimeters of such polygons approach nearer to each other, as the number of their sides increases, may be seen from the following Table, which is constructed on the principles explained in the foregoing proposition, and in which the radius is supposed=1.

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Number of Sides Perimeter of the in- Perimeter of the circum-
of the Polygon. scribed Polygon. scribed Polygon.

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The part by which the numbers in the second column are less than the entire perimeter of any of the inscribed polygons is less than unit in the sixth decimal place; and the part by which the numbers in the last column exceed the perimeter

Suppl. of any of the circumscribed polygons is less than unit in the

sixth decimal place, that is, than toodoor of the radius.

Because the numbers in the second column are less than the perimeters of the inscribed polygons, each of them is less than the circumference of the circle; and because the num, bers in the third column exceed the perimeters of the circumscribed polygons, each of them is greater than the circumference of the circle. But when the arch of į of the circumference is bisected ten times, the number of sides of the polygon is 6144, and the numbers in the table differ from one another only by Toodoo part of the radius, and therefore the perimeters of the polygons differ by less than that quantity; consequently the circumference of the circle, which is greater than the least and less than the greatest of these numbers, is determined within less than the millioneth part of the radius.

Hence, if R be the radius of any circle, the circumference
is greater than Rx6.283185, or 2RX3.141592, and less than
2RX3.141593. But these two numbers differ from each other
only by a millioneth part of the radius. So also RPx3.141592
is less, and Rox3.141593 graater than the area of the circle ;
and these numbers differ from each other only by a millioneth
part of the square of the radius.
In this

way
the circumference and the of the circle

may be found still nearer to the truth; but neither by this, nor by any other method yet known to geometers, can they be ex. actly determined, though the errors of both may be confined within limits which are less than any that can be assigned.

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I.
A STRAIGHT line is perpendicular, or at right angles to a Book II.

plane, when it makes right angles with every straight line
which it meets in the plane. ).

II.
A plane is perpendicular to a plane, when the straight lines

drawn in one of the planes perpendicular to the common
section of the two planes are perpendicular to the other
plane.

III.
The inclination of a straight line to a plane is the acute angle

contained by that line and another straight line drawn from
the point in which the first line meets the plane to the
point in which a perpendicular drawn from any point in
the first line to the plane meets the plane.

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Suppl.

IV.
The angle made by two planes which cut each other is the
angle

contained by two straight lines drawn from any point
in the line of their common section, at right angles to that
line, one line in one plane, and the other line in the
other plane. Of the two adjacent angles made by two lines
drawn in this manner, that which is acute is also called the
inclination of the planes to each other.)

V.
Two planes are said to have the same, or a like inclination to

each other which two other planes have, when the angles
of inclination above defined are equal to each other.

VI.
A straight line is said to be parallel to a plane, when it does

not meet the plane, though produced ever so far.

VII.
| Planes are said to be parallel to one another, which do not

meet, though produced ever so far.

VIII.
A solid angle is an angle made by the meeting of more than

two plane angles in one point which are not in the same
plane.

PROP. I. THEOR.

See N.

ONE part of a straight line cannot be in a plane, and another part above it.

If it be possible, let AB, part of the straight line ABC, be in the plane, and the part BC above it. Since the straight

line AB is in the plane, it can a 2 Post. 1. be produced in that planea.

Let it be produced to D; then

A

D

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