Let the numbers be 164 and 165: Then 165-164-1. 164-1= 163 and 165+1=166. Proof, As 1643: 1653 :: 163: 166 nearly. For a quadruplicate proportion subtract, and add once and a half the difference, and so on, for each higher power, increasing the number to be subtracted and added by 5. To reduce a Ratio, consisting of large numbers, to its least terms, and very nearly of the same value. RULE. 1. Divide the greater of the terms by the less, and the least divisor by the remainder, and so on continually, till nothing remain, in the same manner as we get the greatest common measure for reducing a vulgar fraction: This will give a number of ratios, from which we can choose one, that will suit our purpose. 2. Place the first quotient under unit for the first ratio; multiply that by the next quotient, adding nothing to the numerator, and I to the product of the denominator, for a new denominator, and it will give a second ratio, nearer than the first: Then, multiply the last ratio by the next quotient, adding the preceding ratio, and so on, continually till you have gone through. EXAMPLES. 1. Sir Isaac Newton has demonstrated, in his Principia, that the velocity of a comet, moving in a parabola, is to that of a planet, moving in a circular orb, at the same distance from the sun, as 2 to 1. Let this be taken for an example. 2-1-4142; those motions, then, are as 1.4142 to 1; or as 14142 The late Profeffor Winthrop chofe 7 to 5 for a proportion. 2 Geometers 2. Geometers have found the proportion of the circumference of a eircle to its diameter, to be as 3-1416 to 1: Let this ratio be reduced. 3. The area of a circle is to its circmuscribing square, as ⚫7854 to 1, very nearly: Let this be reduced. Therefore, as 14: 11 :: the square of the diameter of a circle to its area. To estimate the Distance of Objects on level ground, or at sea, having only the height given. RULE.-1. To the earth's diameter, (viz. 42056462 feet,) add the height of the eye, and multiply the sum by that height, then the square root of the product is the distance, at which an object on the surface of the earth or water, can be seen by an eye so elevated. 2. As objects are seen in a straight line, and that line is a tangent to the earth's surface; therefore, To find the distance of two elevated objeas, when the right line joining them touches the earth's surface between those objects, (for instance, the line from the eye of the observer to the distance found by the first part of the rule, and from thence to the object ;) work for each object separately, and the sum of the square roots of the products is the distance of the two objects from each other. EXAMPLE. EXAMPLE. How far may a mountain be seen on level ground, or at sea, which is a mile high, supposing the eye of the observer elevated 5 feet above the surface? To estimate the Height of Objects on level ground, or at sea, having only the distance given. RULE.-1. From the given distance, take the distance which the elevation of your eye above the surface will give, found by the last problem. 2. Divide the square of the remainder in feet by 42056462 feet, and the quotient will be the height required. EXAMPLE. Being on my return from a foreign voyage, and finding by my reckoning I was about 54 leagues from Boston light-house, it being in the dusk of the evening, with my telescope I descried the lamp of the light-house in the horizon, at which time, my eye was elevated 6 feet above the surface of the water: Now, supposing my reckoning to be true, what is the height of the light-house above the water? 5 leagues 16.5 miles; then 16·5—42056462+6x6 = 13·943 miles = 73619 feet nearly, and 73619×73619÷42056462 = 129 feet nearly, Ans. MISCELLANEOUS QUESTIONS, WITH THE METHOD OF SOLUTION. 1. What part of 9d. is of 7d ? 2 7 14 -of- -=-) and 5 1 5 9 14 1X14 14 1 5 9x5 45 Ans. 2. What number is that, from which being taken, the remainder will be ? 12 313 3. What number is that, to which if of of 12 be added, the total will be 1 ? 3 4. What number is that, of which 19, is? 19; then, As: 5. In an orchard of fruit trees, of them bear apples, 1 pears, plums, 60 of them peaches, and 40 cherries: How many trees does the orchard contain? 12 ++;=, and 1-1; therefore, as 12: ::: 1200 Ans. 60+ 40 1 6. A person, who was possessed of of a vessel, sold of his interest for £.375: What was the ship worth at that rate? of. As 4: 375: 6.1500 Ans. 7. If of 3 of 4 of a ship be worth of of of the cargo, val18 ued at 1000 What did both ship and cargo cost? 28 28 39 39 of of, and 3 of 3 of 3 of 499-7990, then, as: '90 :: 28x7000x28 6x39x28 .837 12s. 13 d. the cost of the ship; and £.1000 [Answer. +L.837 125. 138d.=£.1837 12s. 123d. value of the ship and cargo, 8. Two ships, A and B, sailed from a certain port at the same time; A sailed north 8 miles an hour, and B east 6 miles an hour: Required, by an easy method, to find their distance asunder at every hour's end? √ 8x8+6x6=10 miles distant in 1 hour, and 10×2=20 miles in 2 hours, &c. Ans. 9. If a body be weighed in each scale of a balance, whose beam is unequally divided, and those different weights of the body be multiplied together, the square root of the product will be the true weight of that body. Suppose the weight of a bar of silver, in one scale, to be 10oz. and in the other scale 12oz.; required the true weight of the bar? oz. pwt. gr. ᎾᏃ . 12X10 = 10.95445+ = 10 19 2·1384+ Ans. 7 10. A younger brother received D.3125 92c. which was just of his elder brother's fortune; and 5 times the elder's money, was 13 the value of the father's estate: Pray, what was their father worth? As 7: 3125'92 :: 12: 5358-72 the elder brother's fortune; then, 5358-72×51 D.17281 87c. 2m. Ans. 11. A gentleman divided his fortune among his sons, giving A 91. as often as B 51. and to C but 31 as often as to B 71. and yet C's dividend was 153741. What did the whole estate amount to? As 7:53:2; then, as 24: 15371⁄2 :: 9+5+24 : £.11583 8s. 10d. Ans. 12. A gentleman left his son a fortune, of which he spent in 3 months, of of the remainder lasted him 9 months longer, when he had only 5371. left: Pray, what did his father bequeath him? whole legacy, left at three months, then, of of 11 I 165 6-3839 16 1 1 154.537, therefore, as 537 16 384-6 L.2082 185. 2,3 d. Ans. 128 13. A 13. A gay young fellow soon got the better of of his fortune; he then gave £.1500 for a commission, and his profusion continued till he had but .450 left, which he found to be just of his money, after he had purchased his commission: What was his fortune at first? As 6: 450 16: 1200, and 1200+1500-£.2700 of his for tune, and, as 5: 2700 :: 7 : £3780 Ans. 14. A merchant begins the world with D.5000, and finds that by his distillery he clears D.5000 in 6 years; by his navigation D.5000 in 7 years, and that he spends in gaming D.5000 in 3 years: How long will his estate last? 8331/ 666 { 16661/ As 16663-133} + 6661⁄2 : 1 :: 5000: 30 years Ans. 15, A has £.100 of B's money in his hands, for the remittance of which B allows him 9 per cent.: What sum must he remit, to discharge himself of the £.100 ? 16. Said Harry to Edmund, I can place four 1s, so that, when added, they shall make precisely 12: Can you do so too? 17. A and B are on opposite sides of a circular field 268 poles about; they begin to go round it, both the same way, at the same instant of time; A goes 22 rods in 2 minutes, and B 34 rods in 3 minutes: How many times will they go round the field, before the fwifter overtakes the slower? min. po. min. po. 2:22 3:34 } 263 :: 1: A goes in a minute. therefore, B gains 11-11 of a pole of A every minute. And, aspo.: 1 min. :: po. (= half round the field): 402 min. (= the time in which B will overtake A.) Then, 18. If 15 men can perform a piece of work in 11 days, how many men will accomplish another piece of work, four times as large in a fifth part of the time? I work. men. works. men. time men. time. men. As 1 15 4: 60 As ÷ : oo :: 3 : 300 Ans. 19. If A can do a piece of work alone in let them both go about it together: In what time will they finish it 7 days, and B in 12; As |