Ans. 195 20 come to, at 16 cents pr. lb. ? 10. *At the rate of 15 deg. pr. hour, how much of the equator will revolve through the meridian in 12 hours 2 min. 26 sec.? Ans. 179 deg. 36 min. 30 sec. 11. When the Sun is on the meridian of London, what o'clock is it at Mexico North America, 100 degrees 5 min. 45 seconds.? Ans. 5 o'clock, 19 m. 37 sec. A. M. 12. What o'clock is it at Moscow 37 deg. 45 min. east long. when it is noon at London ? Ans. 2 o'clock 31m. P. M. meridian of London, 4h. 45m. Meridian of Cambridge, what Ans. 71 deg. 20 m. w. 13 If the Sun comes to the 20 sec. sooner than it does at the is the longitude of Cambridge? 14. suppose a Gentleman has an income of $ 1940 a year, and he spends 3 dols. 46 cents pr. day, how much will he have saved, at the years end? Ans. 683 10 15. Sound uninterrupted, moves about 1142 feet in a second, how long then, after firing a cannon at Springfield before it will be heard at Hartford, it being 26 miles ? Ans. 2 m. sec. 120 16. In a thunder storm it was observed, that it was 6 seconds between the lightning and thunder, at what distance was the explosion? Ans. 6852 ft. 17. Suppose a rocket was seen at the instant of discharge, 12 1 mile. seconds before the report, at what distance was the gun. Ans. 2 miles. 18. If 100 in one year gain $ 6, what will $314 15 cts. gain in the same time? Ans. 18 84c. 9m. 19. If cent ? 212 25 c. gain & 12 374 in one year what is that pr. Ans. 6 20. A owes B & 1736 59 cts. but becoming a bankrupt, he is unable to pay more than 65 cents on the dollar, what does B receive for the debt? Ans. 1128 73c. 3,5m. 21. If a man buy merchandize to the amount of $ 560, and gain by the sale $ 190 40, how much will he gain by laying out 150 at the same rate? Ans. 50 00 22. If 30 men perform a piece of work in 11 days, how many men can accomplish another piece of work 4 times as large in, a fifth part of the time? Ans. 600 23. A wall that is to be built to the height of 27 feet, was raised 9 feet by 12 men in 6 days, how many men must be employed to finish the wall in 4 days, working at the same rate? Ans. 36 24. If a stick 8 feet long, cast a shadow on level ground 12 *Note. The equator may always be supposed to revolve through the meridian, at the rate of 15 degrees in 1 hour of solar time, without any sensible errour; though it is a fraction wide of the truth. feet, what is the width of a river, over which a tower, known to be 180 feet in height casts its shade. Ans. 270 feet. OF THE LEVER OR STEELYARD. It is a principle in Mechanicks, that the power is to the weight, as the velocity of the weight, to the velocity of the power; therefore to find what weight may be raised or balanced by any given power, say; As the distance between the body to be raised, or balanced and the fulcrum, or prop, is to the distance between the prop and the point where the power is applied: so is the power to the weight which it will balance. If a man weighing 160lb. rest on the end of a lever 10 feet long, what weight will he balance on the other end, supposing the prop 1 foot from the weight? The distance between the weight and the prop being 1 foot, the distance from the prop to the power is 10—19 feet, therefore, ft. ft. lb. lb. As 1:9: 160 1440 Ans. If a weight of 1440 be placed 1 foot from the prop, at what distance from the prop must a power of 160lb. be applied to balance it? As 160 1440 : : 19 feet. Ans. At what distance from a weight of 1440lb. must a prop be placed, so that a power of 160lb. applied 9 feet from the prop, may balance it. As 1440 160 91 ft. Ans.' The celebrated Archimedes said he could move the Earth, if he had a place at distance from it to stand upon, to manage his machiebery.. Now suppose the Earth to contain in round numbers 4,000,000,000,000,000,000,00000 lb. or 400000 Trillions of lbs. and that Archimedes was suspended from the end of a lever 12,000,000.000,000,000,000,006,000 miles in length, and the fulcrum, or centre of motion of the lever to be 6000 miles from the Earths centre, how much must Archimedes weigh to balance the Earth? Ans. 200 lb. OF THE WHEEL AND AXLE. The proportion of the wheel and axle, (where the power is applied to the circumference of the wheel, and the weight to be raised is suspended by a cord, which coyls about the axle as the wheel turns round,) is as the diameter of the axle to the diame ter of the wheel, so is the power applied to the wheel, to the weight suspended from the axle. Suppose a windless is constructed in such a manner, that 14lb. applied to the wheel will raise 224lb. suspended from the axle, which is 6 inches in diameter, what is the diameter of the wheel? Ans. 8 feet. Inversely, As 224: 6:14:968 feet. Suppose the diameter of the wheel to be 8 feet,required the diameter of the axle, so that 14lb. suspended from the wheel, may balance 224lb. on the axle. Inversely, As 14: 96: 224 : 6 diameter required. Suppose the diameter of the wheel 96 in. and that of the axle 6 in. what weight suspended from tbe axle will balance 14lb. upon the wheel? Inversely, As 96: 14 :: 6 : 224 weight required. OF LOGARITHMS. THE operations of Multiplication and Division, when they are to be often repeated, and the extracting of Roots, especially if they be from the higher powers, become so tedious, that it is an object which has long employed the skill and talents of the most profound mathematicians, to substitute in their place more expeditious, and easier methods of calculation. To effect this, CERTAIN NUMBERS have been so contrived, and adapted to other numbers, that the addition and subtraction of the former, have been made to perform the office of multiplication and division in the latter, with imcomparable facility and expedition. The invention of Logarithms is by some ascribed to Baron Napier. But the kind of Logarithms now in use, was invented by Mr. Henry Briggs, Professor of Geometry in Gresham College, London. LOGARITHMS (from logos, ratio and arithmos, number) are the indices of the ratios of numbers to one another; being a series of numbers in arithmetical progression, corresponding to others in geometrical progression. 50 1 2 5 indices or Logarithms. 21, 10, 100, 1000, 10000, 100000, Thus 3 4 This is the most convenient series of numbers, to which most of the modern TABLES OF LOGARITHMS are calculated. In which it is apparent that if any two. indices, or Logarithms, be added together, their sum will be the index, or logarithm, of that number, which is equal to the product of the two terms, in the geometrick progression, to which those indices, or logarithms belong. Thus, the logarithms 2 and 3, being added together,make 5, corresponding to 100000, the product of 100, into 1000, and the logarithms 1, and 4, being added together, make 5, the logarithm corresponding to 100000, the product of 10 into 10000. Whence it is evident that rowERS of the same ROOT may be multiplied, by adding their exponents, or logarithms. In like manner, if any one index, or logarithm, be subtracted from another, the difference will be the logarithm of that number, which is equal to the quotient of the two terms, to which those logarithms belong. Thus; if from 5, (the logarithm of 100000) be subtracted 2, (the logarithm of 100) the difference 3, will correspond to 1000, the quotient of 100000 divided by 100. Again; if from 5, (the logarithm 100000) be subtracted 3, (the logarithm 1000) the difference is 2, answering to 100, the quotient arising from 100000 divided by 1000. Hence it is manifest, that. A POWER may be divided by another power of the same root, by subtracting the logarithm of the divisor, from the logarithm of the dividend. So also if the logarithm of any number be multiplied by the index of its power, the product will be equal to the logarithm of that power. Thus if 2, (the logarithm 100) be multiplied by 3, the product will be 6, equal to the logarithm of 1000000, or the 3d power of 100. Again, if the logarithm of any number be divided by the index of its root, the quotient will be the logarithm of its root. Thus, the index, or logarithm of 1000000, is 6, and if this number be divided by 3, the quotient will be 2, which is the logarithm of 100, or the cube root of 1000000. In the following series, to wit. 102 10 10° 10-' 10-2 100 10 1 10 -100 Whose Logarithms are 104 103 10000 1000 10-3 10-4 1000 10000 It will be seen, that the logarithms of all the numbers between 1 and 10, are greater than 0, but less than 1; since by the series, it may be seen, that the logarithms of 1 and of 10, are 0, and 1. Each number therefore between 1 and 10, has 0 for its index, with a decimal annexed. For the same reason, if the given number be between the log. and 2 1the decimal part will be 2 and 3 i. c. 2+ the decimal part 100 and 1000 Thus the logarithm of the natural number of 35 is 1. of 175 is of 8795 is 2. 2430380 3. 9442358 Whence we derive this general truth. The index of the logarithm, is always 1 less than the number of integral figures in the natural number, whose logarithm is required; or the in |