2. A country gentleman, going to a fair to buy some oxen, meets with a person who had 23, he demanding the price of them, was answered £16 apiece; the gentleman bids him £15 apiece, and he would buy all; the other tells him it would not be taken, but if he would give what the last ox would come to, at a farthing for the first, and doubling it to the last, he should have all. What was the price of the oxen? Ans. £4369 1s. 4d.

In any Geometrical Progression, not proceeding from unity, the ratio being given, to find any remote term, without producing all the intermediate terms.

RULE. Proceed as in the last, only observe that every product must be divided by the first term.

EXAMPLES.

1. A sum of money is to be divided among eight persons, the first to have £20, the second £60, and so on in triple proportion, what will the last have?

540x540

14580X60

0. 1. 2. 3.

14580 then

=43740

20. 60. 180. 540. 20

20 Ans. £43740.

3+3+1=7 one less than the number of terms. 2. A gentleman dying, left 9 sons, to whom and to his executor, he bequeathed his estate in manner following: To his executor £50; his youngest son was to have as much more as the executor, and each son to exceed the next younger by as much more; what was the eldest son's portion? Ans. £25600.

The first term, ratio, and number of terms given, to find the sum of all the terms.

RULE. Find the last ferm as before, then subtract the first from it, and divide the remainder by the ratio less one, to the quotient of which add the greater, and it gives the sum required.

EXAMPLES.

1. A servant skilled in numbers agreed with a gentleman to serve him 12 months, provided he would give him a farthing for his first month's services, a penny for the second, and 4d. for the third, &c.-what did his wages amount to? 256X256 65536, then 65536X64-4194304

0. 1. 2. 3. 4. 1. 4. 16. 64. 256.

4194304-1

(4+4+3=11. No. of terms less 1.) 4—1

1398101; then

13981014194304-5592405 farthings. Ans. £5325 8s. 51d.

2. A man bought a horse, and by agreement was to give a farthing for the first nail, three for the second, &c.; there were 4 shoes, and in each shoe 8 nails: what was the worth of the horse? Ans. £965114681693 13s. 4d.

3. A certain person married his daughter on new-year's day, and gave her husband one shilling towards her portion, promising to double it on the first day of every month for one year; what was her portion? Ans. £204 15s.

PERMUTATION

Is the changing or varying of the order of things. RULE. Multiply all the given terms into one another, and the last product will be the number of changes required.

EXAMPLES.

1. How many changes may be rung upon 12 bells, and how long would they be ringing but once over, supposing 10 changes might be rung in one minute, and the year to contain 365 days 6 hours?

1×2×3×4×5×6×7×8×9×10×11×12=479001600

changes, which÷10-47900160 minutes, and if reduced is =91 years, 3 weeks, 5 days and 6 hours.

2. A young scholar coming into a town for the conveniency of a good library, demands of a gentleman with whom he

lodged, what his diet would cost for a year, who told him £10; but the scholar not being certain what time he should stay, asked him what he must give him for so long as he could place his family (consisting of 6 persons besides himself) in different positions, every day at dinner; the gentleman, thinking it could not be long, tells him £5, to which the scholar agrees: what time did the scholar stay with the gentleman? Ans. 5040 days.

EXTRACTION OF THE SQUARE ROOT.

EXTRACTION OF THE SQUARE ROOT is to find out such a number as being multiplied into itself, the product will be equal to the given number.

RULE. 1. Point the given number, beginning at the unit's place, then to the hundreds', and so upon every second figure throughout.

2. Seek the greatest square number in the first point, towards the left hand, placing the square number under the first point, and the root thereof in the quotient; subtract the square number from the first point, and to the remainder bring down the next point, and call that the resolvend.

3. Double the quotient, and place it for a divisor on the left hand of the resolvend; seek how often the divisor is contained in the resolvend (reserving always the unit's place) and put the answer in the quotient, and also on the right hand side of the divisor; then multiply by the figure last put in the quotient, and subtract the product from the resolvend; bring down the next point to the remainder (if there be any more) and proceed as before.

Roots. 1. 2. 3. 4 5. 6. 7. 8. 9.
Squares. 1. 4. 9. 16 25. 36. 49. 64. 81.
EXAMPLES.

1 What is the square root of 119025?

2. What is the square root of 106929?

Ans. 327. 3. What is the square root of 2268741? Ans. 1506,23+ When the given number consists of a whole number and decimals together, make the number of decimals even, by adding ciphers to them, so that there may be a point fall on the unit's place of the whole number.

4. What is the square root of 3271,4007? Ans. 57,19+ 5. What is the square root of 4795,2571? Ans. 69,247+

To extract the square root of a vulgar fraction.

RULE. Reduce the fraction to its lowest terms, then extract the square root of the numerator for a new numerator, and the square root of the denominator for a new denominator.

If the fraction be a surd, (i. e.) a number whose root can never be exactly found, reduce it to a decimal, and extract the root from it.

To extract the square root of a mixed number.

RULE. 1. Reduce the fractional part of a mixed number to its lowest term, and then the mixed number to an improper fraction.

2. Extract the roots of the numerator and denominator for a new numerator and denominator.

If the mixed number given be a surd, reduce the fractional part to a decimal, annex it to the whole uumber, and extract the square root therefrom.

EXAMPLES.

1. There is an army consisting of a certain number of men, who are placed rank and file, that is, in the form of a square, each side having 576 men, I desire to know how many the whole square contains? Ans. 331776

2. A certain pavement is made exactly square, each side of which contains 97 feet, I demand how many square feet are contained therein? Ans. 9409.

To find a mean proportional between any two given numbers. RULE. The square root of the product of the given number is the mean proportional sought.

1. What is the mean proportional between 3 and 12? Ans. 3X12-36 then/366 the mean proportional. 2. What is the mean proportional between 4276 and 842? Ans. 1897,4+

To find the side of a square equal in area to any given super

fices.

RULE. The square root of the content of any given superfices, is the equal square sought.

EXAMPLES.

1. If the content of a given circle be 160, what is the side of the square equal? Ans. 12,64911.

2. If the area of a circle is 750, what is the side of the square equal?

Ans. 27,38612.

The area of a circle given, to find the diameter. RULE. AS 355: 452, or as 1 : 1,273239 :: so is the area: to the square of the diameter;—or, multiply the square root of the area by 1,12837, and the product will be the diam

eter.

EXAMPLE.

What length of cord will fit to tie to a cow's tail, the other end fixed in the ground, to let her have liberty of eating an acre of grass, and no more, supposing the cow and tail to be 5 yards and a half? Ans. 6,136 perches.

The area of a circle given, to find the periphery or circumfe

rence.

RULE. AS 113: 1420, or as 1 : 12,56637 :: the area: to the square of the periphery, or multiply the square root of the area by 3,5449, and the product is the circumference.

EXAMPLES.

1. When the area is 12, what is the circumference?

Ans. 12,2798.

Ans. 44,84.

2. When the area is 160, what is the periphery?