e; add these five equations together, and the sum is (5sx-y-z-v-w=5s-s=) 4s=a+b+c+d+e; whence s= a+b+c+d+e 4 ; now if this value be substituted for s in the five preceding equations, we shall thence obtain the required numbers, viz. w=32, v=30, z=24, y=21, and x=20, being the ages of E, D, C, B, and A, respectively. 39. To find a point in the straight line which joins two luminaries, or in the line produced, which is equally enlightened by both ". Let a=their distance apart, x=the distance of the least of them from the required point, then a+x=the distance of the other: let the quantity of light emitted by the first in a given time be to that emitted by the second in the same time, as m to n; then be the ratio of the effects they produce, supposing 1 will : X2 1 2 equal: but these effects are by hypothesis equal; whence ··· ma2 +2 amx+mx2 ==nx2, or m—n.x2 +2 amx= will be the ratio, supposing m,and n un m-n m-n' 40. The weight w, and the specific gravity of a mixture, and the specific gravities a and b, of the two simples which compose it, being given, to find the quantity of each " ? u • A luminary, (from the Latin lumen, light,) is a body that gives light, as the sun, moon, a planet, star, &c. The double sign serves both cases, viz. a +x when the point required is beyond the smaller luminary, and a-r when it is between them; also in the answer, the upper sign - applies to the first case, and the lower sign to the second. The gravity of a body, (from the Latin gravis, heavy,) is its weight, Let x=the weight of the simple, whose specific gravity is the greatest, then w―x=the weight of the other. 41. Suppose two bodies, A and B, to move in opposite directions towards the same point with given velocities, the distance of the places from whence they set out, and the difference of the times in which they begin to move, being likewise given, thence to determine the point where they meet? Let d the distance from A to B at the time of setting out, x=A's distance from the point of meeting, then d-x=B's distance from the point of meeting; let t=the difference between the times of their beginning to move, and suppose A moves through the space a in the time n, and B through the space b in the time m, then nx (a: n :: x :) =the time of A's motion, and (b: m :: d-x: a the time of B's motion; whence by the problem, bt+dm =t, :x= .a. b bn+am пх a and the specific gravity is its weight compared with that of a body of equal bulk, but of a different kind: thus, a cubic foot of common water weighs 1000 ounces avoirdupois, and a cubic inch of each of the following substances weighs as follows; viz. fine gold, 19640 oz. fine silver, 11091 oz. cork, 240 oz. new fallen snow, 86 oz. common air, 1.232 oz. &c. &c. these numbers, then, represent the specific gravities of the above-mentioned substances respectively, compared with common water.-Tables of the specific gravity of a great variety of bodies, both solid and fluid, may be found in the writings of Mersenne, Muschenbroeck, Ward, Cotes, Emerson, Hutton, Vyse, Martin, &c. and are useful for computing the weight of such bodies as are too large and unwieldy to be moved; by means of their kind and dimensions, which must be previously known. EXAMPLES.-1. A sets out from London towards Durham distant 257 miles, and travels 11 miles in 4 hours; B sets out from Durhamn 8 hours later, and travels towards London at the rate of 10 miles in 3 hours: whereabouts on the road will they meet? Here d=257, t=8, a=11, n=4, b=10, m=3. Then x= 10 × 4+11x3 17 X11=128 miles from London. 73 2. Supposing Africa to be 20,000 miles round, and a ship to sail from the Isthmus of Suez down the Red Sea, with intent to coast it round that vast continent, sailing on an average 24 miles an hour;-a week after another ship sails from the opposite side of the same Isthmus with the same intent, and passing the Straits of Gibraltar, sails at the rate of 34 miles an hour;near what place on the coast will they meet? 42. If two bodies, A and B, move in the same direction and in the same straight line, their velocities, distance at setting out, and the interval between the times of their beginning to move, being given, thence to determine the point where they will come together. Let A be the farthest from the required point, d=the distance from A to B, x=A's distance from the point, then will x—d=B's distance; also let t=the interval of time between their setting out, and let A move through the space a in the time r, and B through Tx the space b in the time s; then will (a: r :: x :) =the time of a EXAMPLES.-1. A ship sails from the Downs, east, towards Petersburg, at the rate of 54 miles in 23 hours; 24 hours after another ship sails from Lisbon, distant from the Downs 550 miles west, in pursuit of her, and goes at the rate of 8 miles an hour: whereabouts will the latter ship overtake the former? Here d=550, t=24, a=8, r=1, b=54, s=23; and because x8= B sets out first, therefore x= 858.21538, &c. miles from Lisbon, or (858.21538, &c.—550=) 308.21538, &c. miles from the Downs. 2. Suppose the ship from Lisbon sets sail 24 hours before the other ? Then x= 54 × 24-23 x 550 x8=698.7138, &c. miles from Lisbon, or (698.7138, &c.-550=) 148.7138, &c. miles from the Downs. 3. A is 100 miles south of London, and sets out on a journey northward, travelling 37 miles every 24 hours; B from London pursues the same rout, setting out 49 hours after A, and travelling at the rate of 11 miles every 8 hours: where will they be together? 43. Given the forces of several agents separately, to determine their joint force? Let A, B, C, D, &c. be the agents, and suppose Call the given effect 1, and let x=the time in which they can produce it, all operating together: An agent, (in Latin agens, from ayw to drive,) is that by which any thing is done or effected. Philosophers call that the agent, which is the immediate cause of any effect, and that on which the effect is produced they But the sum of these effects is equal to the given effect 1, produced by the joint operation of all the agents, in the time x ; whence ax m bx сх dx α b с d + + + &c. 1, or x. + + + &c.=1,. x= EXAMPLES.-1. A can reap 5 acres of wheat in 8 days, B can reap 4 acres in 7 days, and C 6 acres in 9 days; how long will they require to reap a field of 30 acres, all working together? Here m=8, a=5, n=7, b=4, r=9, c=6. 2. A vessel has three cocks, A, B, and C; A can fill it twice in 3 hours, B 3 times in 4 hours, and C 4 times in 5 hours; in what time will it be filled with the three cocks all open together? 44. If two agents, A and B, can jointly produce an effect in the time m, A and C in the time n, and B and C in the time r ; in what time will each alone produce the same effect? call the patient; the effect, as communicated by the agent, they call an action; but as received by the patient, a passion: a smith striking on an anvil has been frequently proposed as a proper example; thus the smith is the superior agent, the hammer with which he strikes is the inferior agent, the blow he strikes is the action, the anvil is the patient, and the blow it receives, the passion. |