Suppl. AH(AC+CG)=500×1866.02545-933012.73-; therefore P965.92585-, because (965.92585)2 is greater than 933012.73. Hence AC+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) 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, SAH(AC÷+Q)=500 (1991.4449-)-995722.475-, therefore S 997.85895-, because (997.85895) 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) is less than 4282.1; therefore the perimeter of a polygon of ninety-six sides inscribed in the circle (65.4377+) 966282.019+. 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+1 to 1; therefore the circumference of the circle has a greater ratio to the diameter than 3+ 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+1, or as 71 to 223, so the given diameter to a fourth proportional, which will be less than the circumference. and is 79 of the dia COR. 2. Because the difference between the lines found by these proportions differ by meter. Therefore the difference of either of them from the circumference must be less than the 497th part of the dia meter, COR. 3. As 7 is to 22, so is the square of the radius to the area of the circle nearly. Book I. For the diameter of a circle is to its circumference as the square of the radius to the area of the circle; 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. 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 100 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 numbers 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 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 Toooooo 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 R×6.283185, or 2Ř×3.141592, and less than 2Rx3.141593. But these two numbers differ from each other only by a millioneth part of the radius. So also R2x3.141592 is less, and R2×3.141593 greater 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 area 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 exactly determined, though the errors of both may be confined within limits which are less than any that can be assigned. ELEMENTS OF GEOMETRY. SUPPLEMENT. BOOK II. OF THE INTERSECTION OF PLANES. DEFINITIONS. 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. Į 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. / 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. See N. PROP. I. THEOR. 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 plane. Let it be produced to D; then A D |