The first term, ratio, and number of terms given, to find the fum of all the terms. RULE. Find the laft term as before, then fubtract the first from it, and divide the remainder by the ratio lefs one, to the product of which add the greater, and it gives the fum required. EXAMPLES. I. A fervant skilled in numbers agreed with a gentleman to ferve him 12 months, provided he would give him a farthing for his first month's fervice, a penny for the fecond, and 4d. for the third, &c.what did his wages amount to? O. I. 2. 3. 4. 256X256 65536, then 65536×64=4194304 1. 4. 16. 64. 256. (4+4+3=11 No. of terms lefs 1.) 2. 4194304-1 1398101; then 4-I 1398101+4194304=5592405 farthings. Anf. £.5825 85. 51d. A man bought a horse, and by agreement was to give a farthing for the first nail, three for the fecond, &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? Anf. £.204 155. 4. A laceman well verfed 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 fecond, &c. in triple proportion. I defire to know what he fold the lace for, if the pins were valued at 100 for a farthing; alfo, what the laceman got or loft by the fale thereof, fuppofing the lace ftood him in £.7 pounds per yard. Anf. The lace fold for £.326886 os. gd. PERMUTATION IS the changing or varying of the order of things. RULE. Multiply all the given terms, one into another, and the laft product will be the number of changes required. Cc EXAMPLES. I. How many changes may be rung upon 12 bells, and how long would they be ringing but once over, fuppofing 10 changes might be rung in one minute, and the year to contain 365 days 6 hours? 1X2X3X4X5X6X7X8X9X10X 11 X 12=479001600 changes, which 1047900160 minutes, and if reduced is = 91 years 3 weeks 5 days and 6 hours. 2. A young fcholar 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 fhould ftay, afked him what he must give him for fo long as he could place his family (confifting of fix perfons befides himfelf) in different pofitions, every day, at dinner; the gentleman, thinking it could not be long, tells him .5, to which the fcholar agrees; what time did the fcholar ftay with the gentle man? Anf. 5040 days. EXTRACTION OF THE SQUARE ROOT. EXTRACTING the Square Root is to find out fuch a number as being multiplied into itfelf, 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 fo upon every fecond figure throughout. 2. Seek the greateft fquare number in the first point, towards the left hand, placing the fquare number under the first point, and the root thereof in the quotient; fubtract the fquare number from the first point, and to the remainder bring down the next point, and call that the refolvend. 3. Double the quotient, and place it for a divifor on the left hand of the refolvend; seek how often the divifor is contained in the refolvend (referving always the unit's place) and put the answer in the quotient, and alfo on the right hand fide of the divifor; then multiply by the figure laft put in the quotient, and subtract the product from the refolvend; bring down the next point to the remainder (if there be any more) and proceed as before. When the given number confifts of a whole number and decimals together, make the number of decimals even, by adding cyphers to them, fo that there may be a point fall on the unit's place of the whole number. 7. 9. 10. 4,372594 What is the square root of 3271,4007? 11. Anf. 57,19+ ? ? Anf. 1.50701+ Ans. 1,1269+ To extract the fquare root of a vulgar fraction. RULE. Reduce the fraction to its lowest terms, then extract the fquare root of the numerator for a new numerator, and the fquare root of the denominator for a new denominator. If the fraction be a surd, (i. e.) a number whofe root can never be exactly found, reduce it to a decimal, and extract the root from it. To extract the fquare 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 furd, reduce the fractional part to a decimal, annex it to the whole number, and extract the square root therefrom. I. The APPLICATION. Anf. 7. Anf. 9,27+ Anf. 2,9519+ Anf. 2,5298+ There is an army confifting of a certain number of men, who are placed rank and file, that is, in the form of a square, each fide having 576 men, I defire to know how many the whole square conAnf. 331776. tains. 2. A certain pavement is made exactly square, each fide 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 fquare root of the product of the given numbers is the mean proportional fought. 1. 2. EXAMPLES. What is the mean proportional between 3 and 12 ? Anf. 3X 12-36 then √ 366 the mean proportional. What is the mean proportional between 4276, and 842 ? Anf. 1897,4+ To find the fide of a fquare equal in area to any given fuperficies. RULE. The fquare root of the content of any given superficies, is the fquare equal fought. EXAMPLES. 3. If the content of a given circle be 160, what is the fide of the fquare equal ? Anf. 12,64911. 4. If the area of a circle is 750, what is the fide of the fquare Anf. 27,38612 equal? The area of a circle given to find the diameter. RULE. As 355 452, or as I : 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. 5. 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 grafs, and no more, fuppofing 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 I : 12,56637: the area to the fquare of the periphery, or multiply the square 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? Anf. 44,84. Any two fides of a right angled triangle given to find the third fide. 1. The base and perpendicular given to find the hypothenuse. RULE. The fquare root of the fum of the squares of the base and perpendicular is the length of the hypothenuse. EXAMPLES. 8. The top of a caftle from the ground is 45 yards high, and is furrounded 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. |