The cosine, radius, and secant have also been omitted, partly for the same reason, and partly because the radius with which they should both agree will be found carried out in the Second Part of this book; for Case 6 they should agree like the sine and tangent with each other to 72 decimal places, and for Case 7 to 144 decimal places. 1 By Case 6, the tangent is 1972063063734639263984455073299118 880' and the square which forms the unit of comparison is the square of the tangent, which is, 38890327273464518838061949606599216563 67162016449544406781628584372454400' is Again, by Case 6, the rectangle contained by the radius and the sine. 2 2788918330588564181308597538924774401 87651718961354874269698779794778624034. - and the number of sides 07554307875531968564842627379673198284800; therefore there are exactly 24445348571891983269638939752719507554307875531968564842627379673198284800 squares in the given polygon, each of which is expressed by 1 388903272734645188380619496065992165636 7162016449544406781628584372454400° Then, multiplying the number of squares by the value of each square, 244453485718919832696389397527195075543078753196856388903272734645188380619496065992165636716201644954 we have 73198284800 372454400 44 7 e number of sides. 62 area of the inscribed poly ducted from the assumed radius, and twice that amount from the assumed diameter. 62 Therefore, by hypothesis, 1972063063734639263984455073299118880 must be deducted from the circumference to complete the polygon. 1 835002744095575440* By Case 5, the tangent is 1 And, by Case 6, the tangent is 197206306373463926398445507329 9118880 Then, by division and cancellation, we have 2361744410637427202; therefore the tangent No. 5 has been divided into 2361744410637427202 parts, each of which is equal to tangent No. 6. By Case 5, the polygon was carried to 37113126452873856034 sides, each of which contained two tangents of 1 835002744095575440' 905747712062 175303437922709748539397559589557248124 4095575440 1972063063734639263984455073299118880 ; and, 62 by hypothesis, 1972063063734639263984455073299118880 subtracted from the circumference to complete the polygon. must be 175303437922709748539397559589557248124 Then 1972063063734639263984455073299118880 3734639263984455073299118880 589557248062 3299118880 62 197206306 175303437922709748539397559197206306373463926398445507 the circumference of the polygon No. 6. 2788918330588564181308597538924774 By Case 6, secant No. 6, 1972063063734639263984455073299118 Then, by ARTICLES 1 and 2, dividing the assumed circumference by the assumed diameter, we have by cancellation 175303437922709748 197206306373463926 857, or 34, the true ratio between the circumference and diameter of the given circle. By Case 2, the sine of the given arc is is 12. Then, by ARTICLE 7, 1/2 X 1 9912, and the given radius Then, by ARTICLE 7, multiplying the radius by the sine, we have, 2 2788918330588564181308597538924774401 double triangle for half the number of sides. area of the inscribed 2788918330588564181308597538924774401 87651718961354874269698779794778624034 Х 17530343792270974 27889183305885641 3.142857, or 34; which is the true ratio of the circumference to the diameter of the given circle. By Case 2, the difference between the inscribed and circumscribed 1 polygons is 22051 2; and, by Case 3, the difference between the in 1 12. 86440410V 2. Then, dividing the difference of Case 2 by the difference of Case 3, we have 2÷ 1 86440410 1 2 86440410 = 220512 X 1 1/2 = 1 2205 39202, which shows that the difference between the polygons has diminished 39202 times. By Case 3, the tangent is 1 13860 ; and, by Case 4, the tangent is 1 543339720 Then, dividing tangent No. 3 by tangent No. 4, we have 1 1 X 543339720 13860 543339720 = 13860 39202; therefore the tangent No. 3 is 39202 times as large as tangent No. 4. Consequently the difference between the tangents Nos. 3 and 4 is exactly the same as the difference between the differences of the inscribed and circumscribed polygons of Cases 2 and 3. By Case 3, the difference between the inscribed and circumscribed 1 polygons is 864404101 2; and, by Case 4, the difference between the inscribed and circumscribed polygons is 1 1328413456515688201 2. Then, dividing the difference of Case 3 by the difference of Case 4, 1536796802; which shows that the differ ence between the polygons has diminished 1536796802 times. |