EXAMPLES. 1. If the diameter of a circle be 17, what is the circumference? Here 3.1416x 17-53.4072-circumference. 2. If the circumference of a circle be 354, what is the diameter ? 3. What is the circumference of a circle whose diameter is 40 feet? Ans. 125.6640. 4. What is the circumference of a circle whose diameter is 12 feet? Ans. 37.6992. 5. If the circumference of the earth be 25000 miles, what is its diameter ? Ans. 7958 nearly. 6. The base of a cone is a circle; what is its diameter when the circumference is 54 feet? Ans. 20.3718. troublesome and laborious that it must have cost him incredible pains. It is said to have been thought so curious a performance, that the numbers were cut on his tomb-stone in St. Peter's Church-yard, at Leyden. This last number has since been confirmed and extended to double the number of places, by the late ingenious Mr. Abraham Sharp, of Little Horton, near Bedford, in Yorkshire. But since the invention of Fluxions, and the Summation of Infinite Series, there have been several methods discovered for doing the same thing with much more ease and expedition. The late Mr. John Machin, Professor of Astronomy in Gresham College, has by these means given a quadrature of the circle which is true to 100 places of decimals; and M. de Lagny, M. Euler, &c. have carried it still further. All of which proportions are so extremely near the truth, that, except the ratio could be completely obtained, we need not wish for a greater degree of accuracy. PROBLEM X. To find the length of any arc of a circle. RULE.* 1. When the chord of the arc and the versed sine of half the arc are given. To 15 times the square of the chord, add 33 times the square of the versed sine,† and reserve the number. To the square of the chord, add 4 times the square of the versed sine, and the square root of the sum will be twice the chord of half the arc. Multiply twice the chord of half the arc by 10 times the * Demon. Put c= the chord of the arc, and v=the versed sine of half the arc, then the rule may be expressed thus: ✓(4c2+v2).10v2 60c2 +33v2 · = 2 √ (c2 + v2)·(1+ —270) = 2 √ do (1 2 √ dv (1+60d—27v do(1 i 3v2 523 Now 2√dv.(1+6d+40d2 + + &c.) is known 112d3 to be the length of an arc whose diameter is d, and the versed sine of half the arc ; and this differs from the Here, as in many other places in the following part of the work, the term versed sine is used instead of versed sine of half the arc, but in all cases of the kind, it is the versed sine of half the arc that is to be understood. square of the versed sine, divide the product by the reserved number, and add the quotient to twice the chord of half the arc: the sum will be the length of the arc very nearly. When the chord of the arc, and the chord of half the arc are given.-From the square of the chord of half the arc subtract the square of half the chord of the arc, the remainder will be the square of the versed sine: then proceed as above. 2. When the diameter and the versed sine of half the arc are given. From 60 times the diameter subtract 27 times the versed sine, and reserve the number. Multiply the diameter by the versed sine, and the square root of the product will be the chord of half the arc. Multiply twice the chord of half the arc by 10 times the versed sine, divide the product by the reserved number, and add the quotient to twice the chord of half the arc; the sum will be the length of the arc very nearly. Note 1.-When the diameter and chord of the arc are given, the versed sine may be found thus: From the square of the diameter subtract the square of the chord, and extract the square root of the remainder. Subtract this root from the diameter, and half the remainder will give the versed sine of half the arc. 2. The square of the chord of half the arc being divided by the diameter will give the versed sine, or being divided by the versed sine will give the diameter. 3. The length of the arc may also be found by multiplying together the number of degrees it contains, the radius and the number .01745329. Or, as 180 is to the number of degrees in the arc, so is 3.1416 times the radius, to the length of the arc. Or, as 3 is to the number of degrees in the arc, so is .05236 times the radius, to the length of the arc. * EXAMPLES. 1. If the chord DE be 48, and the versed sine CB 18 what is the length of the arc? Ans. 64.2959. 45252 reserved number. 482=2304 the square of the chord. 182 × 4=1296=4 times the square of the versed sine. √3600=60=twice the chord of half the arc ACB. 60 × 182 x 10 194400 Now =4.2959, which added to twice the chord of half the arc gives 64.2959=the length of the arc. 2. Given the diameter CE 50, and the versed sine CD 18, what is the length of the arc? Ans. 64.2959. * When very great accuracy is required, the following theorem may be used. Let d denote the diameter of the circle, and v the versed sine of half the arc, then the arc 2 √ dv × (1+ v 3v2 + 5v3 350* 63v5 6d 40d2 112d3 1152d4 2816d5 50 x 60=3000 18x27 486 2514 reserved number. AC=√50 × 18=30=the chord of half the arc. 30 x 2 x 18 x 10 2514 10800 =4.2959, which added to twic the chord of half the arc gives 64.2959=the length of the arc ACB. 3. The chord of the whole arc is 7, and the versed sine 2, what is the length of the arc? Ans. 8.4343. 4. The chord of the whole arc is 40, and the versed sine 15, what is the length of the arc ? Ans. 53.5800. 5. The chord of the whole arc is 50, and the chord of half the arc 27, required the length of the arc. Ans. 55.3720. 6. Given the diameter of the circle 100, and the versed sine 9, required the length of the arc. Ans. 60.9380. 7. Given the chord of the whole arc 16, and the diameter of the circle 20, required the length of the arc. Ans. 18.5439. 8. The diameter of the circle is 50, and the chord of half the arc 30, what is the length of the arc? Ans. 64.2959. Ans. 53.5800. 9. The chord of half the arc is 25, and the versed sine 15, required the length of the arc. PROBLEM XI. To find the area of a circle. RULE I.* Multiply half the circumference by half the diameter, and the product will be the area. * Demon. A circle may be considered as a regular polygon of an infinite number of sides, the circumference being equal to the perimeter, and the radius to the perpendicular. But the area of a regular polygon is equal to half the perimeter multiplied by the perpendicular, |