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

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

EXAMPLES, A servant skilled in numbers agreed with a gentleman to serve him 12 months, provided he would give him a farthing for his first 6 month's service, a penny for the second, and 4d. for the third, &c.what did his wages amount to ?

256 X 256365536, then 65536 x 64=4194304 3. 4

41943041 1. 4. 16. 64. 256.

1398101 ; then (4+4+3=11 No. of terms lefs 1.) 4-I

1398101 +4194304=5592405 farthings.

Anf. £.5825 8s. 510. 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 ?

Anf. £.965114681693 135. 4d. 3. A certain person married his daughter on new-year's day, and gave her husband one fhilling towards her portion, promising to double it on the first day of every month for one year ; what was her portion ?

Ans. £ 204 1550 4. A laceman well versed in numbers agreed with a gentleman to fell him 22 yards of rich gold brocaded lace, for 2 pins the first yard, 6 pins the second, &c. in triple proportion. I desire to know what he sold the lace for, if the pins were valued at 100 for a farthing ; also, what the laceman got or loft by the sale thereof, fuppofing the lace stood him in £ .7 pounds per yard.

Ans. The lace sold for £ -326886

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



How many changes may


rung upon 12 bells, and how long would they be iinging but once over, supposing to changes might be rung in one minute, and the year to contain 365 days 6 hours ? 1 X2 X3 X4 X5 X6X7 X 8 X 9 X 10X11 X 12=479001600 chan. ges, which • 10=47900160 minutes, and if reduced is = 91 years 3 weeks 5 days and 6 hours.



young scholar coming into a town for the conveniency of a 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, aiked him what he must give him for so long as he could place his family (consisting of six perfons besides himself) in different positions, every day, at dinner ; the gentleman, thinking it could not be long, tells him £ 5, to which the icholar agrees; what time did the (cholar' stay with the gentle. man ?

Aní. 5040 days.


EXTRACTION OF THE SQUARE ROOT. ExtrACTING 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 hundred's, and so upon every second figure throughout.

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 ; fubtract 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 ihe resolvend ; bring down the next point to the remainder (if there be any more and proceed as before.


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3. 4.


7. SQUARES. 1. 4. 9.

4. 9. 16. 25. 36. 49. 64.


EXAMPLES 3. What is the square root of 119025?






Ans. 345


Ans. 327

What is the square root of 106929 ? 3. What is the square root of 2268741 ? Ans. 1506,237 4. What is the square root of 7596796 ? Ans. 2756,228 +5. What is the square root of 36372961 ?

Anf. 6031
What is the square root of 22071204 ?

Ans. 4698 When the given number consists of a whole number and decimals together, make the number of decimals even, by adding cyphers 10 them, so that there may be a point fall on the unit's place of the whole number. 7. What is the square root of 3271,4007?

Ans. 57,19+ 8. What is the square root of 4795,25731 ? Anf: 69.247+ 9. What is the square root of 4,372594 ?

Anf. 2,091 + What is the square root of 2,2710957?

Anf. 1,50701+ What is the square root of,00032754


Anf. ,01809+ 12. What is the square root of 1,270054?

Ans. 1,1269+

10. 11.

To extract the square root of a vulgar fra£tion. RULE. Reduce the fraction to its lowest terms, then extract the {quare 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.

EXAMPLES. 13. What is the square root of 784 ? 14. What is the square root of 27.1? 15. What is the square root of 12 ?


SURDS 16. What is the square root of ili ? 17. What is the square root of 17 18. What is the square root of

Anf. ,89802+
Anf. ,86602 +
Anl. ,93308-40

To extract the square root of a mixed number. RULE. 1. Reduce the fractional part of the 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 number, and extract the square root therefrom.

Anf. 75.
Anf. 54
Anf. 31.






EXAMPLES. 19. What is the square root of 51 if ?

What is the square root of ?
What is the square root of 5


What is the square root of 85 If ?

Anf. 9,27+ 23. What is the square root of 83?

Anf. 2,9519+ 24 What is the square root of 6ş?

Anf. 2,5298+ The APPLICATION. 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.

Anf. 331776. 2. A certain pavement is made exa&tly square, each side of which contains 97 feet, I demand how many square feet are contained therein ?

Anf. 9409.

To find a mean proportional between any two given numbers. Rule. The square root of the product of the given numbers is the mean proportional fought.

What is the mean proportional between 3 and 12 ?

Anf. 3 X 12=36 then v 36=6 the mean proportional.
What is the mean proportional between 4276, and 842 ?

Anf. 1897,4+



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

Rule. The square root of the content of any given superficies, is the square equal fought.

EXAMPLES. 3. If the content of a given circle be 160, what is the side of the square equal ?

Anf. 12,64911.

4. If the area of a circle is 750, what is the side of the square

Anf. 27,38612

equal ?

The area of a circle given to fine the diameter. Rule. As 355 : 452, or as 1 : 1,273239 :: fo 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 diameter.

EXAMPLE. 5. 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 ?

Anf. 6,136 perches.

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

Rule. As 113 : 1420, or as 1 : 12,56637 :: the area : to the {quare of the periphery, or multiply the iquare root of the area by 3,5449, and the product is the circumference.

EXAMPLES. 6. When the area is 12, what is the circumference ?

Anf. 12,2798. 7. When the area is 160, what is the periphery ?

Ans. 44,84.

Any two sides of a right angled triangle given to find the third fide.

1. The base and perpendicular given to find the hypothenuse.

Rule. The square root of the sum of the squares of the base and perpendicular is the length of the hypothenuse.

EXAMPLES. 8. The top of a castle from the ground is 45 yards high, and is surrounded with a ditch 60 yards broad, what length must a ladder be to reach from the outside of the ditch to the top of the castle ?

Anf. 75 yards.

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