being 65.4386-, the perimeter of that polygon will be (65.4386 -) 96=6282.1056—. Let the perimeter of the circumscribed polygon of the same number of sides, he M, then (2. Cor. 2.1. Sup.)T: AC:: 6282.1056: M, that is, (since T-999.46458+, as already shewn), 999.46458 + 1000 :: 6282.1056—: M; if then N be such, that 999.46458 1000 :: 6282.1056-: N; ex æquo perturb. 999.46458+: 999.46458 :: N: M; and, since the first is greater than the second, the third is greater than the fourth, or Nis greater than M. Now, if a fourth proportional be found to 999.46458, 1000 and 6282.1056 viz. 6285.461—, then, because, 999.46458 : 1000 :: 6282.1056: 6285.461-, and as before, 999.4645S: 1000 :: 6282.1056-: N; therefore, 6282.1056 : 6282.1056-:: 6285.461-N, and as the first of these proportionals is greater than the second, the third, viz. 6285.461 is greater than N, the fourth. But N was proved to be greater than M; much more, therefore, is 6285.461 greater than M, the perimeter of a polygon of ninety-six sides circumscribed about the circle; that is, the perimeter of that polygon is less than 6285. 461; now, the circumference of the circle is less than the perimeter of the polygon; much more, therefore, is it less than 6285.461; wherefore the circumference of a circle is less than 6285.461 of those parts of which the radius contains 1000. The circumference, therefore, has to the diameter a less ratio (8.5.) than 6285.461 has to 2000, or than 3142.7305 has to 1000: but the ratio of 22 to 7 is greater than the ratio of 3142.7305 to 1000, therefore the circumference has a less ratio to the diameter than 22 has to 7, or the circumference is less than 22 of the parts of which the diameter contains 7. It remains to demonstrate, that the part by which the circumference exceeds the diameter is greater than. of the diameter. 10 71 It was before shewn, that CG2 750000; wherefore CG-S66. 02545-, because (866.02545) is greater than 750000; therefore AC+CG 1866.02545-. Now, P being, as before, the perpendicular from the centre on the chord of one twelfth of the circumference, P= AH (AC+CG) =500×(1866.02545) - 933012.73-; and P=965.925S5-, because (965.92585) is greater than 933012.73. Hence also, AC+P - Next, as Q the perpendicular drawn from the centre on the chord of one twenty-fourth of the circumference, Q' AH (AC+ P)=500× (1965.92585-)=982962.93—; and Q=991.44495 because (991.44495) is greater than 982962.93. Hence also, AC+ Q=1991.44495—. 2 In like manner, as S is the perpendicular from C on the chord of one forty-eighth of the circumference,S=AH(AC+Q)=500(1991 .44495) 995722.475-, and S (997.85895-) because (997. $5895) 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 itself: = 65.4377+ because (65. 4377) is less than 4282.1: Now the chord of one ninety-sixth part of the circumference being =65.4377+, 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 those 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+ to 1; 10 71 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 10, 71 above three times the diameter is greater than ten of those parts of which the diameter contains 71; and it has already been shown to be less than ten of those of which the diameter contains 70. Therefore, &c. Q. E. D. 10 COR. 1. Hence the diameter of a circle being given, the circumference may be found nearly, by making as 7 to 22, so the given diameter to a fourth proportional, which will be greater than the circumference. And if as 1 to 3+ or as 71 to 223, so the given di71' ameter to a fourth proportional, this will be nearly equal to the circumference, but will be less than it. Cor. 2. Because the difference between and 10 1 497 the lines found by these proportions differ by 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 to 22, so the square of the radius to the area of the circle nearly. For it has been shewn, that (1. Cor. 5. 1. Sup.) the diameter of a circle is to its circumference as the square of the radius to the area of the circle; but the diameter is to the circumference nearly as 7 to 22, therefore the square of the radius is to the area of the circle nearly in that same ratio. SCHOLIUM. It is evident that the method employed in this proposition, for finding the limits of the ratio of the circumference of the diameter, may be carried to a greater degree of exactness, by finding the perimeter of an inscribed 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 one another, 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 that is wanting in the numbers of the second column, to make up the entire perimeter of any of the inscribed polygons, is less than unit in the sixth decimal place; and in like manner, the part by which the numbers in the last column exceed the perimeter of any of the circumscribed polygons is less than a unit in the sixth t decimal place, that is than Too of the radius. Also, as the numbers in the second column are less than the perimeters of the inscribed polygons, they are each of them less than the circumference, of the circle; and for the same reason, each of those in the third column is greater than the circumference. But when the arch of of the circumference is bisected ten times, the number of sides in the polygon is 6144, and the numbers in the Table differ from one another only by Too part of the radius, and therefore the perimeters of the polygons differ by less than that quantity; and 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 millionth part of the radius. 1 Hence also, if R be the radius of any circle, the circumference is greater than Rx6.283185, or than 2Rx3.141592, but less than 2R x3.141593; and these numbers differ from one another only by a . millionth part of the radius. So also R+3.141592 is less, and R' x3.141593 greater than the area of the circle; and these numbers differ from one another only by a millionth part of the square of the radius. In this way, also, 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 reduced to a less quantity than any that can be assigned. ELEMENTS 1 OF GEOMETRY. SUPPLEMENT. BOOK II. OF THE INTERSECTION OF PLANES. DEFINITIONS. 1. A STRAIGHT line is perpendicular or at right angles to a plane, when it makes right angles with every straight line which it meets in that 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 straight line, and another drawn from the point in which the first line meets the plane, to the point in which a perpendicular to the plane, drawn from any point of the first line, meets the same plane. IV. The angle made by two planes which cut one another, is the angle contained by two straight lines drawn from any, the same point in the line of their common section, at right angles to that line, the one, in the one plane, and the other, in the other. 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 one another. |