(4.) Extract the square root of 3272869681.

(5.) Extract the square root of 15241578750190521. (6.) Required the square root of 57132. (7.) What is the square root of 75.347? (8.) Required the square root of 1788-57'. (9.) What is the square root of ⚫4325? (10.) Required the square root of 5'3'.

Examples to Prop. 2.

(11.) What is the square root of 2915?

2025×2116 4284900, the square root of which is 2070; then 2025-18, or 2078-8, the root required.

2070

(12.) Extract the square root of 3.

7x9-63, the square root of which is 7.9372532332; this root divided by 9, the denominator, gives 8819171036, &c. for the square root of 3.

(13.) What is the square root of $?
(14.) Required the square root of 33
(15.) Required the square root of
(16.) What is the square root of 15g?
(17.) Required the square root of 29.

Examples to Prop. 3.

(18.) Find a mean proportional between 3 and 27.

√3x27=81=9. Answer.

For 39: 9:27.

(19.) Of three numbers in geometrical progression, the first is 18, and the third 32, what is the middle one?

(20.) In a pair of scales, a body weighed 90lb. in one scale, and only 40lb. in the other scale: required the true weight, and the ratio of the lengths of the two arms of the balance on each side of the point of suspension?

Examples to Prop. 4.

(21.) The area of a fish-pond is 9 acres, 2 roods, 15 perches; how many yards are contained in the side of a square of equal superficies?

T

a. r. p.

9. 2. 15-1535 square perches. 5×5-30 square yards in 1 perch.

√1535x301=√46433·75=215-485 yards, nearly. Answer. (22.) An army of 56169 men is to be formed into a square, how many men will the front contain?

(23.) If the area of a circular piece of ground be 231.2575 acres, how many yards will the side of a square be that will contain the same number of acres?

Examples to Prop. 5.

(24.) A circular fish-pond is to be dug in a garden that shall take up just an acre, what must be the length of the cord which describes the circle?

Then

An acre 4840 square yards.

48407854=√6162·465—78.514 yards the diameter of the circle, hence the length of the cord, or the radius, 39.257 yards.

(25.) The area of one end of a circular piece of timber is 4356'6 square inches, what is the diameter?

(26.) In a field adjoining my house I wish to plant four acres of wood in the form of a circle, and to have a gravel walk round it of six feet wide; what must the lengths of the cords be which describe each of the circles?

Examples to Prop. 6,

(27.) The base of a right-angled triangle is 24 feet, the perpendicular 18 feet; what is the length of the hypothenuse?

24×24576 the square of the base.

18x18-324 the square of the perpendicular.

Sum 900, the square-root of which is 30 the hypothenuse,

(28.) The wall of a fort standing on the brink of a river is 42.426 feet high, the breadth of the river is 23 yards; what length must a cord be to reach from the top of the fort across the river?

(29.) Two ships sail from the same port, the one due

east 50 miles, the other due south 84 miles; how far are they asunder?

Examples to Prop 7.

(30.) The hypothenuse of a right-angled triangle is 30, and the base 24; what is the length of the perpendicular?

30+2454 sum of the given sides.

30-24 6 difference of the given sides.

54x6 324=18. Answer.

(31.) A line 27 yards long will reach from the top of a fort on the opposite bank of a river to the water edge on this side of the river; what is the height of the fort, the river being 24 yards across?

(32.) A ladder of 100 feet in length was placed against a building of 100 feet high, in such a manner that the top of it reached the top of the building within 6 inches; what was the distance of the foot of the ladder from the base of the edifice?

CLASS II.

(33.) A gentleman hired a number of labourers, at a shilling per day each, to dig a fish-pond. When they had finished their work, their wages amounted together to 120l. 1s. What were the wages of one man, each man worked as many days as there were men in company?

(34.) A number of men, drinking porter in London, spent, at a reckoning, half a crown and a farthing; when they came to pay the landlord, they found that each man had as many farthings to pay as there were men in company. Pray how many men were there?

(35.) The wall of a town, which is surrounded by a moat 24 feet wide, is 18 feet high; what length must a ladder be made to reach from the outer edge of the moat to the top of the wall?

(36.) A ladder, 50 feet long, will reach to a window 30 feet from the ground on one side of the street; and, without moving the foot, will reach a window 40 feet high on the other side. The breadth of the street is required.

(37.) The longer diameter of an ellipsis is 81 inches,

and the shorter diameter 64 inches; what is the diameter of a circle of equal superficies?

(38.) If a ladder 50 feet in length will exactly reach the coping of a house when the foot is 10 feet from the upright of the building, how long must a ladder be to reach the bottom of the second-floor window, which is 17.9897 feet from the coping, the foot of this ladder standing 6 feet from the upright of the building; and what is the height of the wall of the house?

(39.) A society collected among themselves, for charitable purposes, the sum of 301. 9s. 24d., each member contributed as many farthings as there were members in the whole society. What did each contribute?

(40.) A line of 380 feet will reach from the top of a precipice that stands close by the side of a brook, to the opposite bank; the precipice is 128 feet high, how broad is the brook?

(41.) There are five numbers in geometrical progression, the first is 5, and the fifth is 1280, what are all the rest?

(42.) An irregular piece of ground, consisting of 420 acres, 3 roods, 14 perches, is to be exchanged for a square piece of the same surface; what will be the length of one of its sides? This square is likewise to be divided into 40 equal squares, what will be the extent of a side of each?

(43.) There are three towers, A, B, and C, standing in a direct line, the heights whereof are 64, 90·249, and 50, feet respectively. The distance between the top of the tower A and that of B is 97 feet; and the distance between the bottom of the tower B and that of C is 76 feet. By these data it is required to find the distances the tops and bottoms of the towers are from each other?

(44.) A gentleman has a garden in the form of an equilateral triangle, the sides whereof are each 50 feet: at each corner of the garden stands a tower;-the height of A is 30 feet, that of B 34 feet, and that of C 28 feet. At what distance from the bottom of each of these towers must a ladder be placed that it may just reach the top of each tower, and what will be the length of the ladder, the ground of the garden being horizontal?

Prop. 1.

CUBE ROOT.

To extract the cube root of any number.

RULE I.

1. If there be decimals in the given number, make them to consist of three, six, or nine, &c. places, by annexing ciphers to the right-hand; then, separate the whole into periods of three figures each, beginning at the right-hand. The left-hand period may consist of one, two, or three figures.

2. Find the nearest less cube to the left-hand period, and subtract it therefrom; put the root in the quotient, and bring down the figures in the next period for a dividend.

3. Find a divisor by multiplying the square of the quotient by 300, seek how often it is contained in the dividend, and put the answer in the quotient.

4. Multiply the last figure in the quotient by the preceding figure (or figures,) and that product by 30; add the result, together with the square of the last quotient figure, to the divisor; this sum, when multiplied by the last quotient figure, will give the subtrahend.

5. Take the subtrahend from the dividend, and bring down the next period for a new dividend. Then find a divisor as above, and repeat the operation.

For the proof. Cube the root found, and to the product add the remainder, if any, and that sum will be the same as the number given to be extracted.

Note. The above rule is the same in principle as those usually given by other authors; but, when the number to be extracted is large, or has not a rational root, and is required to be extracted to several figures, the operation by this rule is very tedious: the following rules will be found preferable in those cases.

RULE II.

1. Find the root to three places of figures, by Rule I., and call it the assumed root. Then,

2. As the sum of the given number and double the cube of the assumed root, is to the sum of double the given number and the cube of the assumed root, so is the assumed root to the root required, nearly.