(4) What is the area of the sector, when the chord of half its arc is 14 feet 2 inches, and the versed sine 6 feet 8 inches ?*

Problem 11. To find the area of a circular Segment.

RULE 1. Find the area of the sector; and also the area of the triangle formed by the chord and the two radii of the sector: their difference, when the segment is less than a semicircle, or their sum, when it is greater, will be the area of the segment.

RULE 2. Divide the height of the segment by the diameter, and find the quotient in the column of heights in the following table. Multiply the corresponding area by the square of the diameter, for the area of the segment.t

(1) What is the area of a segment, when the chord of the whole arc is 60, and the chord of half the arc 371⁄2?

* When the half chord (see AE, Fig. 7) of the arc is found by the properties of a right angled triangle, then AE the versed sine (DE) X the remaining part of the diameter; whence the diameter (and consequently the radius) will be known.

+ When there is a remainder (or fraction) after the second quotient figure, in dividing the height by the diameter; having taken out the area answering to the two figures, add to it such fractional part of the difference between that and the next succeeding area, for the sake of greater accuracy.

(2) What is the area of a segment whose height is 18, and the diameter of the circle 48 ?

(3) Required the area of a circular segment whose height is 2, and chord 20.

(4) What is the area of the segment of a circle whose radius is 24, the chord of the whole arc 20, and the chord of half the arc 10-2 ?

(5) If the radius of a circle is 10 feet, what is the area of the segment whose chord is 12 feet?

Problem 12. To find the circumference of an Ellipse, the transverse and conjugate diameters being given.

RULE. Multiply the square

root of half the sum of the squares of the two diameters by 31416, and the product. will be the circumference nearly.*

(1) What is the circumference of an ellipse whose transverse diameter is 24, and conjugate 18?

Fig. 10.

C

B

(2) The two axes of an ellipse are 60 and 45 yards respectively what is the circumference?

Problem 13. To find the area of an Ellipse.

RULE. Multiply the product of the axes by 7854, for the area.

(1) Required the area of an ellipse whose axes are 35 and 25. (2) What will be the expense of trenching an elliptic garden, whose axes are 70 and 50 feet, at 33d. per square yard? (3) Required the area of the ellipse in Grosvenor Square, London; the transverse diameter being 8.40 chains, and the conjugate 6.12 chains.

Problem 14. To find the area of an Elliptic Segment, the base being parallel to either axis.

RULE. Divide the height of the segment by that axe of which it is a part, and find, in the Table of Circular Segments a versed sine equal to the quotient.

* If the half sum of the two diameters be multiplied by 3-1416, the product will give the circumference sufficiently near for most practical purposes.

Multiply the corresponding tabular area and the two axes continually together, and the product will be the area required.

(1) What is the area of an elliptic segment cut off by a line (called a double ordinate) parallel to the conjugate diameter, at the distance of 36 yards from the centre; the axes being 120 and 40 yards respectively?

(2) Required the number of square yards in the segment of an ellipse, cut off by an ordinate parallel to the transverse diameter; the height being 5 feet, and the two axes 35 and 25 feet respectively.

A COLLECTION OF QUESTIONS.

(1) WHAT is the value of 14 barrels of soap, at 44d. per lb. each barrel containing 254 lb.? Ans. £66..13..6. (2) A and B joined in partnership; A put into the joint stock £320. for 5 months, and B £460. for 3 months: they gained £100. What is each man's share of the gain?

Ans. A's £53..13..938. and B's £46..6..228. (3) How many yards of cloth, at 17s. 6d. per yard, can I have for 13 cwt. 2 qrs. of wool, at 14d. per lb.? Ans. 100 yards, 33 qrs. (4) If I buy 1000 ells of Flemish linen for £90. at what price must I sell it per English ell, to gain £10. by the whole? Ans. 3s. 4d. per ell.

(5) A has 648 yards of cloth, at 14s. per yard, ready money, but in barter will have 16s. B has wine at £42. per tun, ready money: what must he charge it per tun in barter, and what quantity must be given in exchange for the cloth? Ans. £48. per tun, and the quantity, 10 tuns, 3 hhds. 123 gals. (6) A jeweller sold jewels to the value of £1200. for which he has received in part 876 French pistoles, at 16s. 6d. each. How much more is due to him? Ans. £477..6.

(7) An oilman bought 417 cwt. 1 qr. 15 lb. gross weight of train oil, tare 20 lb. per cwt. how many neat gallons were there, allowing 74 lb. to a gallon? Ans. 5120 gallons.

(8) If I buy cloth at 14s. 6d. per yard, and sell it at 16s. 9d. what is the gain per cent. ? Ans. £15..10..4.. (9) Bought 27 bags of ginger, each weighing gross 843 lb. tare 18 lb. per bag, tret as usual, what is the value at 84d. per lb. ? Ans. £76..13..2.

(10) If oz. costs. what will & lb. cost? Ans. 17s. 6d. (11) If of a gallon cost of a £. what will § of a tun cost? Ans. £105. (12) A gentleman who spends one day with another £1..7..10. lays up at the year's end £340. What is his annual income? Ans. £848..14..4. (13) What is the difference in ounces, between 13 fothers of lead, and 39 boxes of tin, each box weighing 388 lb. ? Ans. 212160 ounces.

(14) A captain, commanding a crew of 160 mariners, captured a prize worth £1360. The captain was allowed onefifth, and the rest was equally divided among the sailors. What was each man's share?

Ans. The captain had £272. and each sailor £6..16. (15) At what rate per cent. will £956. amount to £1314.. 10. in 7 years, at simple interest? Ans. £5. per cent.

(16) A has 24 cows worth £3..12. each, and B 7 horses worth £13. each. How much will make good the difference, in case they interchange their droves of cattle?

Ans. £4..12. (17) A man left £120. to be given to three persons, A, B, and C; B to have twice as much as A, and C as much as A and B; what was the share of each?

Ans. A £20. B £40. and C £60. (18) £1000. is to be divided among three men, in such a manner, that if A has £3. B shall have £5. and C £s. How much will each man have?

Ans. A £187..10. B £312..10. and C £500. (19) A piece of wainscot is 8 feet 6 inches long, and 2 feet 9 inches broad: what is the superficial content?

Ans. 24 feet 0' 3" 4" 6"". (20) A garrison of 360 men, who had originally six months' provisions, having endured a siege of 5 months, without obtaining any relief or fresh supply, wish to know how many men must depart, that the provisions may suffice for the residue 5 months longer? Ans. 288 men.

(21) The less of two numbers is 187; the difference 34 The square of their product is required. Ans. 1707920929. (22) A butcher sent his man with £216. to a fair to buy cattle; he bought oxen at £11. cows at 40s. colts at £1..5. and hogs at £1..15. each, and of each a like number. What was the number of each? Ans. 13 of each sort, and £8. over. (23) What number added to 11 will produce 363 ?

Ans. 24.

(24) What number multiplied by will produce 1197 ? Ans. 26. (25) What is the value of 179 hogsheads of tobacco, each weighing 13 cwt. at £2..7..1. per cwt.? Ans. £5478..2..11. (26) My factor informs me that he has bought goods on my account, of the value of £500..13..6. What will his commission come to at £34. per cent.? Ans. £17..10..5..217 qrs. (27) If of 6 were three, what would of 20 be?

144

Ans. 7.

(28) Reduce 3 qrs. 14 lb. to the decimal of a cnt.

Ans. 875 cwt. (29) How many lb. of sugar, at 44d. per lb. must be given in barter for 60 gross of inkle, at 8s. 8d. per gross?

Ans. 13863 lb. (30) If I buy yarn for 9d. per lb. and sell it again for 13 d. per lb. what is the gain per cent.? Ans. £50. (31) A tobacconist mixes 20 lb. of tobacco at 9d. per lb. with 60 lb. at 12d. per lb. 40 lb. at 18d. per lb. and 12 lb. at 2s. per lb. what is a pound of the mixture worth?

Ans. 1s. 24d. I· (32) What is the difference between twice eight and twenty, and twice twenty-eight; also between twice five and fifty, and twice fifty-five? Ans. 20, and 50. (33) Whereas a noble and a mark just 15 yards did buy ; How many ells of the same cloth for £50. had I? Ans. 600 ells. (34) A broker bought for his principal, in the year 1720, £400. South Sea stock, at £650. per cent. and sold it again when it was worth but £130. per cent. What was the whole loss? Ans. £2080.

(35) C has candles at 6s. per dozen ready money, but in barter will have 6s. 6d. per dozen; D has cotton at 9d. per lb. ready money: what price must the cotton be charged in barter; and how much must be exchanged for 100 dozen of candles ? Ans. The cotton at 9åd. per lb. and the quantity, 7 cnt. O qrs. 16 lb.

(36) If a clerk's salary is £73. a year, what is that per day?

Ans. 4s.

(37) B has an estate of £53. per annum, and pays 5s. 10d. to the subsidy: what must C pay, whose estate is worth £100. per annum? Ans. 11s. Od. (38) If I buy 100 yards of riband, at 3 yards for a shilling, and 100 yards more at 2 yards for a shilling, and sell the