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Let the numbers be 164 and 165: Then 165-164-1. 164-1 =163 and 165+1=166.

Proof, As 164 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 ratius, 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 to 10000 ?

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* The late Professor Winthrop chose 7 to 5 for a proportion.

2. Geometers have found the proportion of the circumference of a circle to its diameter, to be as 3-1416 to 1: Let this ratio be

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88

3. The area of a circle is to its circumscribing square, as ⚫7854

to 1, very nearly: Let this be reduced.

7354)10000(1

1

7854

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26 &c.

portion gen

erally used.

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 objects, 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.

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?

√42056462 + 5 × 5=2·746 miles.

√42056462+5280×5280=89.253 miles.

Ans. 91-999 miles.

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 5 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+6×6=13·943 miles=73619 feet nearly, and 73619×73619÷42056462=129 feet nearly, Ans.

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4. What number is that, of which 19,3 is

?

1X 10955

19,3250; then, As: 0 :: 7:2613 Ans.

25
13

5. In an orchard of fruit trees, of them bear apples, pears, plums, 60 of them peaches, and 40 cherries: How many trees does the orchard contain?

+1+1=1, and -; therefore, as: ::: 1200 Ans.

12

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?

7. If of of

valued at £1000:

Ans. £1500.

of a ship be worth of of 3 of the cargo, What did both ship and cargo cost?

£837 12s. 1d. the cost of the ship; and £1837 12s. 135d. value of the ship and cargo, Ans.

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?

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.

OZ.

7

12×10 10-95445+10 19 2-1384+ Ans. 10. A younger brother received $3125 92c. which was just a of his elder brother's fortune; and 5 times the elder's money was 12 the value of their father's estate: Pray, what was their father worth? Ans. $17281 87c. 2m.

11. A gentleman divided his fortune among his sons, giving A £9 as often as B £5, and to C but £3 as often as to B £7, and yet C's dividend was £15378: What did the whole estate amount to? Ans. £11583 8s. 10d.

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 £537 left: Pray, what did his father bequeath him?

16

whole legacy, 18-11 left at three months, then of of 5, and 11-10===£537, therefore, as

16

=

3841

16 16
1651584-

16 384 6144

537: £2082 18s. 2d. Ans.

33

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 £150 left, which he found to be just of his money after he had purchased his commission: What was his fortune at first? Ans. £3780.

14. A merchant begins the world with $5000, and finds that by his distillery he clears $5000 in 6 years: by his navigation $5000 in 7 years, and that he spends in gaming $5000 in 3 years: How long will his estate last?

Ty

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As 16663-8331+6663 : 1 :: 5000 : 30 years, Ang. 15. A bas £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 disAns. 91, charge 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?

Ans. 11.

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 swifter overtakes the slower?

min. po.
2: 22

3:34

}

min. "po.

:: 1:

S11 A goes in a minute.
114 B do.

3

do,

therefore, B gains 11-11 of a pole of A every minute. And, as po.: 1min. :: 23po. (half round the field): 402min. (=the time in which B will overtake A.) Then,

min. po.
511 7

As 1:

And,

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4423
268

min. DO.

:: 402:4122 A travels.

4556 B travels.

16 times round the field, A travels;

and 455617 times round the field, B travels.

268

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? Ans. 300 men.

19. If A can do a piece of work alone in 7 days, and B in 12; let them both go about it together: In what time will they finish it? Days.work.day works. work, work, work, work.day.work.day.

As

(7:11:Then +
12:1:1:S

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20. A and B together can build a boat in 20 days; with the assistance of C they can do it in 12: In what time would C do it by himself?

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W. W. W. W. D. W. D.'

(20:1:1: Then,& as 8: 1 :: 240: 30 Ans. 12:1:: : 12

21. A can do a piece of work alone in 13 days, and A and B together in 3 days: In what time can B do it alone?

Ans. 20 days. 22. A, B, and C can complete a piece of work in 12 days; A can do it alone in 23 days, and B in 37 days: In what time can C do it by himself? Ans. 771 days.

Another question; Four persons can perform a certain work in the following manner, viz. A, B, and C can do it in 6 days; B, C, and D in 7 days; A, C, and D in 8 days, and A, B, and D in 9 days:

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