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. Number of Sides Perimeter of the in- Perimeter of the circum- 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 way 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. area I. plane, when it makes right angles with every straight line II. drawn in one of the planes perpendicular to the common III. contained by that line and another straight line drawn from Suppl. IV. contained by two straight lines drawn from any point V. each other which two other planes have, when the angles VI. not meet the plane, though produced ever so far. VII. meet, though produced ever so far. VIII. two plane angles in one point which are not in the same 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 |