which bodies acquire in falling through them, are proportional to their lengths; and the times of describing these chords are all equal to one another, and are severally equal to the time of describing the diameter. When a body moves over a system of planes, the velocity lost in passing from one plane to the succeeding one, is to the velocity it had acquired at the bottom of the former plane, as the versed sine of the angle formed by the planes to radius. The times of descending through similar curves similarly situated with respect to the horizon, are as the square roots of the lengths of those

curves.

THE PENDUlum. A pendulum is a body suspended by a right line from any point, moving freely about that point as a center. A cycloid is the curve described by a point in the circumference of a circle rolling in a straight line on a plane. The times of vibration of pendulums of different lengths acted upon by different accelerative forces, will vary as the square roots of the lengths directly, and the square roots of the forces inversely. The times of vibrations of pendulums of the same length, vary inversely as the square roots of the accelerative forces. The lengths of pendulums vibrating in the same time vary as the forces which accelerate them. The number of vibrations performed in a given time by pendulums of different lengths acted upon by differ. ent accelerative forces, are directly as the square roots of the forces, and inversely as the square roots of the lengths.

SOLUTION OF PROBLEMS.

ARITHMETIC.

I. A privateer having taken a prize worth $6850, it is divided into 100 shares, of which the captain is to have 11; 2 lieutenants, each 5; 12 midshipmen, each 2; and the remainder is to be divided equally among the sailors, who are 105 in number. What is each person's share? Daboll, p. 63. If $6850 be divided by 100, the number of shares, each share will $68,50.

$68,50

11

$68,50
5

$68,50
2

$753,50=Capt's. $342,50=each Lt's$137,00=mid's. All the officers had 45 shares, which=$3082,50, which subtracted from $6850=$3767,50, which divided by 105= $35,88, each sailor's part.

II. There is a room 6 yards long and 5 wide; how many yards of carpeting, a yard wide, will be sufficient to cover the floor, if the hearth and fire-place occupy 3 square yards? Adams, p. 55.

6×5=30 yards, and 30-3=27 yards, the answer.

III. A man married at the age of 23; he lived with his wife 14 years; she then died, leaving him a daughter, 12 years of age; 8 years after, the daughter was married to a man 5 years older than herself, who was 4 years of age when the father died. How old was the father at his death? Adams, p. 56.

23+14+8=45=the age of the father at the marriage of his daughter, who was herself then 20 years old, and her husband was 25 years old. Fifteen years after the father died, which added to 45-60, the answer.

IV. The forward wheels of a wagon are 14 feet in cir

cumference, and the hind wheels 15 feet 9 inches; how ma ny more times would the forward wheels turn round than the hind wheels, in running from Boston to New-York, it being 248 miles? Daboll, p. 73.

248

320 rods 1 mile.

4960

744

79360

5 yards 1 rod.

396800

39680

436480

3 feet=1 yard.

14 feet 174 inches

1309440

12 inches 1 foot.

15713280 in. from B. to N.Y.

V. A owes B 3475 pounds; but B compounds with him for 13 shillings and 4 pence on the pound; what must B receive for his debt? Daboll, p. 102.

This question might be expressed by saying, If A is to pay 13 shillings and 4 pence on 1 pound, what must he pay on 3475 pounds? Then we have the proportion,

55600-12=46333 shillings 4 pence.

46333-20=2316, 13, 4, the answer.

VI. A and B depart from the same place and travel the same road; but A goes 5 days before B, at the rate of 15 miles a day; B follows at the rate of 20 miles a day; what distance must he travel to overtake A? Daboll, p. 107.

B has to gain 75 miles before he overtakes A. Then, if 5 miles gain cost 20 miles travel, what travel will 75 miles gain cost? 20:5::75: 300, the answer.

VII. 650 men are in garrison, with provisions enough for 2 months. How many must leave the garrison, that the same provisions may last those who remain 5 months? Barnard, p. 244.

=

2: 650:5: 260 the number who can be sustained for 5 months with their present provisions. 650-260=390=the number which must leave.

VIII. What length of board 7 inches wide, will make a square foot? 12: 12::7: 19}, the answer.

IX. Suppose a certain body, put in motion, should move the length of one barley corn the first second of time, and so continue to increase its motion in triple proportion geometri. cal; how many yards would the said body move in the term of half a minute?

205891132094648-2-102945566047324÷8= 12)34315188682441 and 1 remainder.