by the other, shows them to be Levers of the Third Order. Now I would know what the Length of the Stroke in each of the Barrels will be, if the Crank be made to play just nine Inches round its Centre. (72) With what Force ought that Water-wheel to be driven, which, circumstanced as in the last Question, raises 3 Cubic Feet of Water at every Revolution of the Wheel, each experimentally weighing 624lb. Avoirdupois, the Friction of the Machine rejected? MOTION of BODIES, with their Velocities. 1. If the Quantities of Matter, in any two or more Bodies put in Motion, be equal, the Forces wherewith they are moved will be in Proportion to their Velocities. 2. If the Velocities of these Bodies be equal, their Forces will be directly as the Quantities of Matter contained in them. 3. If both the Quantities of Matter and the Velocities be unequal, the Force with which the Bodies are moved will be in a Proportion compounded of the Quantities of Matter they contain, and of the Velocities wherewith they move. (73) There are two Bodies; the one contains 25 Times the Matter of the other or 25 Times heavier, but the lesser moves with 1000 Times the Swiftness of the greater. In what Proportion are the Forces by which they are moved? (74) There are two Bodies; one of which weighs 100lb. the other 60lb. but the lesser Body is impelled by a Force 8 Times greater than the other. The Proportion of the Velocities wherewith these Bodies move is required. (75) There are two Bodies; the greater contains 8 Times the Quantity of Matter of the less, and is moved with a Force 48 Times greater. The Ratio of the Velocity of these two Bodies is required. 1. In comparing the Motions of Bodies, if their Velocities be equal, the Spaces described by them are in direct Pro portion of the Times in which they are described. 2. If the Times be equal, then the Spaces described will be as their Velocities. 3. If the Times and the Velocities be unequal, the Spaces will be in a Proportion compounded of the Times and Velocities. (76) There are two Bodies, one of which moves forty Times swifter than the other, but the swifter Body has moved but one Minute, whereas the other has been in Motion two Hours. The Ratio of the Spaces described by these two Bodies is required. (77) Suppose one Body to move thirty Times swifter than another; as also the swifter to move 12 Minutes, the other only 1 what Difference will there be between the Spaces by them described, supposing the last has moved 60 Inches? (78) There are two Bodies; one whereof has described fifty Miles, the other only five; but the first hath moved with five Times the Velocity of the second. What is the Ratio then of the Times they have been describ ing those Spaces? XIII. RECIPROCAL PROPORTION, OR The RULE of THREE INVERSE. RECIPROCAL PROPORTION is, when of four Numbers the third beareth the same Ratio to the first as the second doth to the fourth; therefore the less the third Term is in respect to the first, the greater will the fourth Term be in respect to the second. RULE. Multiply the first and second Terms together, and divide their Product by the third Term: the Quotient will be the Answer. EXAMPLES. (1) If 48 Men can perform a Piece of Work in 12 Days, how many Men can do the same in 72 Days? (2) How much in Length, that is four Inches broad, will make a Foot square? (3) Suppose I lend my Friend 500/ for six Months, (allowing the Month to be 30 Days,) afterwards he would requite my Kindness by lending me 2201. Required རྨ the Time I must have it, to requite my former Kind ness. (4) A Garrison being besieged, has three Months' Provisions in it, at the Rate of 14 Ounces per Day each Man: but being informed that it cannot be relieved till the End of eight Months, how many Ounces per Day must each Man have, that the said Provisions may last that Time? (5) If, when the Price of a Bushel of Wheat is 4s 6d the Penny-loaf weighs 12 oz. what must the Penny-loaf weigh when the said Bushel is worth but 3s. ? (6) Suppose 275 Yards of Cloth, which is 5 qrs. wide, make Coats for 130 Men; how many Yards of Shalloon, of 3 qrs. wide, will line the said Coats? (7) In what Time will 600/ gain 50/ Interest, when 80/ does it in 15 Years? (8) Suppose 200/ would defray the Expences of ten Men for 43 Weeks and 5 Days, how long would six Men be in spending the same Sum ? (9) How many Yards of Paper, that is 3 qrs. wide, will hang a Room that is 30 Feet long, and 24 Feet wide? (10) A Garrison consisting of 1500 Men, being besieged, have Provisions only for three Months; but it being necessary they should hold out five Months, how many Men must depart that the said Provisions may serve that Time? (11) There is a Vessel, having a Cock, which will empty it in six Hours. I demand how many Cocks of the same Capacity there must be to empty the said Vessel in fifteen Minutes? A LEVER of the FIRST ORDER. A Lever of the First Order hath the Power at one of its Ends; the Weight to be raised is put at the other, and the Fulcrum or Prop somewhere between them. In this Order, the Power applied at one End will be recprocally proportional to the Distances of those Ends from the Fulcrum, or Point supported; or in the SteelYards, as the Distance of the Weight from the Point of Suspension. QUESTIONS for Exercise at Leisure Hours. (12) What Weight will a Person be able to raise, who presses with the Force of a Hundred and Half on the End of an equipoised Hand-spike 100 Inches long, which is to meet with a convenient Prop exactly 7 Inches above the other End of the Machine? (13) What Weight, hung at 70 Inches' Distance from the Fulcrum of a Steel-yard, will equipoise a hhd. of Tobacco, weighing 94 cwt. freely suspended at 2 Inches' Distance on the contrary Side? MOTION of BODIES, with their Velocities. In comparing the Motion of Bodies, the Ratio or Proportion between their Velocities will be compounded of the direct Ratio of the Forces wherewith they are moved, and the reciprocal of the Quantities of Matter they contain.. (14) The Battering Ram of Vespasian weighed, suppose 100000lb. and was moved, let us admit, with such a Velocity, by Strength of Hands, as to pass through 20 Feet in one Second of Time; and this was found sufficient to demolish the Walls of Jerusalem. what Velocity must a Bullet that weighs but 30lb. be moved, in order to do the same Execution? With (15) A Body weighing 200lb. is impelled by such a Force, as to send it 100 Feet in a Second. With what Velocity would a Body of 8lb. move, if it were impelled by the same Force? XIV. CONTRACTIONS in the RULE of THREE. THIS being considered, you may oftentimes perform the Work much shorter than by the general Rule. I. Divide the Third Term by the First, multiply that Quotient by the Second, and their Product will be the An swer. EXAMPLES. (1) If four Yards of Broad-cloth cost 31 17s 6d. what will a Piece, containing 28 Yards, come to, at the same Rate? (2) If three Chests of Tea, each 3 cwt. cost 11/ 13s what must I give for 72 cwt. of the same? 2 Divide the Second Term by the First, multiply that Quotient by the Third, and their Product will be the Answer. EXAMPLES. (3) Suppose I give 787. for 26 cwt. of Cheese, what must be given 156 cwt of the same? (4) If for 3lb. of Tea I give 18s. what is the Value of 1 cwt. of the same? 3 Divide the First Term by the Second, and divide the Third by that Quotient; which will give the answer. EXAMPLES. (5) Suppose I give for 6 Gowns, each containing 8 Yards of Stuff, 67 what must be given for 64 Yards at the same Rate? (6) If I give 12s for 24lb. of Sugar, what must be given for 14 cwt. of the same? 4 Divid ethe First Term by the Third, divide the Second by that Quotient, and the last Quotient will be the Answer. EXAMPLES. (7) Suppose I give for 3 Pieces of Broad Cloth, each 24 Yards, 647 14s what must be given for 8 Yards of the same? (8) If for 6 Parcels of Tea, each 3lb. I gave 12/ 19s what must I give for 6lb. of the same at that Rate? 5 Divide the first Term by the Third, multiply the Second by that Quotient, and the Product will be the An swer. EXAMPLES. (9) How much in Length, that is 3 Inches broad, will make a Foot square? (10) If for 48s I have 225cwt. carried 512 Miles, how many Hundred can I have carried 64 Miles for the same Money? H |
