(20) Suppose, with Dr. Keil, the distance of the sun to be from us 115 of his diameters; how much hotter is it then at the surface of the sun than under our equator? (21) A ball, descending by the force of gravity from the top of a tower, was observed to fall half the way in the last second of time. Required the tower's height, and the whole time of descent. The less porous a body is, the greater is its density. (22) The compactness or density of the moon is to that of the earth, as 132 is to 100. What proportion then is there between the quantity of matter in the earth and that in the moon, since the earth's diameter is 7970 miles, and that of the moon 2170? (23) There is a vast country in Ethiopia Superior, to whose inhabitants the moon always appears to be most enlightened when she is least enlightened, and to be least when most, according to Gordon's Geographical Grammar. Admitting the mean distance of the earth and moon's centres 240,000 miles, in what proportion is this illumination? Velocities acquired by heavy Bodies falling. The velocity acquired by heavy bodies falling near the surface of the earth is 16 feet in the first second: and as 16 feet are to the square of one second, or 1, so is the given distance to the square of the seconds required: or, on the contrary, to determine what space a heavy body has passed in any time given, is, By multiplying 16, the descent of a heavy body in one second of time, by as many of the odd numbers, beginning from unity, as there are seconds in the given time, viz. by I for the first, 3 for the second, 5 for the third, 7 for the fourth, &c.: the sum total will give the space it has passed. (24) Suppose a stone let go into an abyss should be stopped at the end of the eleventh second after its delivery, what space would it have gone through? (25) What is the difference between the depth of two wells, into each of which should a stone be dropped at the same instant, one will meet with the bottom conds, the other at 10? at 6 se (26) If a stone be 19 seconds in descending from the top of a precipice to the bottom, what is the height of the same? (27) In what time would a musket ball, dropped from the top of Salisbury steeple, said to be 400 feet high, be at the bottom? (28) If a hole could be bored through the centre of the earth, in what time after the delivery of a heavy body on its surface would it arrive at its centre? LIX. The DOUBLE RULE of THREE in DECIMALS. REDUCE the fractional parts to decimals, and then proceed as in whole numbers. EXAMPLES. (1) If 17. 2s. worth of wine will suffice a club of 12 persons when the wine is sold at the rate of 251. 4s. per hhd. how many persons will 17. 12s. worth serve, when the wine is sold after the rate of 18 guineas per hhd. (2) If 6 lb. of pepper be worth 13 lb. of ginger, and 19 lb. of this be worth 44b. of cloves, and 10 lb. be equivalent to 63 lb. of sugar at 5d. per lb. what is the value of 1 cwt. of pepper ? (3) What money, at 3 per cent. will clear 387. 10s. in a year and a quarter's time? QUESTIONS for Exercise at leisure Hours. (4) A. lent his good friend B. fourscore and eleven guineas, from the 11th of Dec. to the 12th of May following; B. on another occasion, let A. have 100 marks, from September 3 to Christmas following. Quere, how long ought the person obliged to let his friend use 40l. fully to retaliate the favour? (5) A. B. and C. will trench a field in 12 days; B. C. and D. in 14; C. D. and A. will do it in 15, and D. A. and B. in 18. In what time will it be done by all of them together, and by each of them singly? (6) A young hare starts 5 rods before a greyhound, and is not perceived by him till she has been up 34 seconds; she scuds away at the rate of 12 miles an hour; and the dog, on view, makes after her at the rate of 20. How long will the course hold, and what ground will be run, beginning with the outsetting of the dog? VIBRATION of PENDULUMS. IT has been found by experiment, that a pendulum 39,2 inches long, in our latitude, vibrates 60 times in one minute; and that the lengths of the pendulums are to one another reciprocally as the square of the number of their vibrations made in the same space of time. (7) What difference is there between the length of a pendulum that vibrates half a second, or 120 times in a minute, and another that swings double seconds, or 30 times in a minute? (8) What difference will there be in the number of vibrations made by a pendulum of 6 inches long, and another of 12 inches long, in an hour's time? (9) What difference is there in the length of two pendulums, the one swinging 30 times, the other 100 times in an hour? (10) Give the length of a pendulum that will swing once in a third, ditto in a second, ditto in a minute, ditto in an hour, ditto in a day. (11) Observed, that while a stone was descending to measure the depth of a well, a string and plummet, that from the point of suspension, or the place where it was held, to the centre of oscillation, or that part of the bob, which, being divided by the circular line, struck from the above centre, would divide it into two parts of equal weight, measured just 18 inches, had made 8 vibrations. Pray what was the depth, allowing 1150 feet per second for the return of sound to the ear? LX. FELLOWSHIP. How to perform fellowship, either single or double, without that tedious and laborious task of making so many different statings as there are persons concerned. RULE. 1. DIVIDE the whole gain or loss by the whole stock. 2. Multiply the quotient by each person's particular stock; and the several products will be the respective gain or loss of each. Note. This rule is best adapted for decimals. EXAMPLES. (1) Three persons making a joint stock; A. puts in 750l. B. 450l. C. 300l. with which they trade a certain time; and, when they balance accounts, find that they have gained 300l. What is the share of each? (2) Three merchants, A, B, and C. traded together; A. put in 1201. for 8 months; B. 2501, for 4 months; and C. 1001. for 5 months: they gained 1847. 10s. What is each man's share of the gain? (3) Once as I walk'd upon the banks of the Wye, I, in the meads, three beauteous nymphs did spy, Anne takes four and three-sevenths her part to make; For more examples, see Sections XXV. and XXVI. Of Simple Interest, Annuities, or Pensions, &c. LXI. SIMPLE INTEREST. Here are five letters to be observed, viz. Pany principal or sum put to interest. I=the interest. T=the time of the principal's continuance at interest. Note.-The ratio is the simple interest of 11. for one year, at any given rate; and is thus found, Viz. 100: 5 1,05 the ratio at 5 per cent. per ann. Case 1. When the principal, time, and rate per cent, are given, to find the interest. RULE. Multiply the principal, rate, and time, continually into one another, the product is the interest sought. Or, if p the principal, t=the time, r=the rate, and I= the interest, then the theorem will be as follows: THEOREM 1. p tr r=1. EXAMPLES. (1) What is the interest of 2607. 17s. 6d. for 5 years at 4 per cent. per annum ? |