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 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 65536×64-4194304 4194304-1 0. 1. 2. 3. 4. 1. 4. 16. 64. 256. (4+4+3=11. No. of terms less 1.) 4-1 =1398101; then 1398101+4194304-5592405 farthings. Ans. £5325 88. 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? 1X2×3×4X5×6×7×8×9X10X11X12-479001600 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 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? Anş. 5040 days. EXTRACTION OF THE SQUARE ROOT. EXTRACTION OF THE SQUARE Roor 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. 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 ract 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 number, 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 36=6 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? Any two sides of a right angled triangle given, to find the third side. The base and perpendicular given, to find the hypothe nuse. RULE. The square root of the sum of the squares of the base and perpendicular is the length of the hypothenuse. EXAMPLES. 1. 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? Ans. 75 yards. 2. The wall of a town is 25 feet high, which is surrounded by a moat of 30 feet in breadth, I desire to know the length of a ladder that will reach from the outside of the moat to the top of the wall. Ans. 39,05 feet. The hypothenuse and perpendicular given, to find the base. RULE. The square root of the difference of the squares of the hypothenuse and perpendicular is the length of the base. The base and hypothenuse given, to find the perpendicular. RULE. The square root of the difference of the squares of the hypothenuse and base is the height of the perpendicular. N. B. The two last questions may be varied for examples to the two last propositions. Any number of men being given, to form them into a square battle, or to find the number of ranks and files. RULE. The square root of the number of men given, is Me number of men either in rank or file. EXAMPLES. 1. An army consists of 331776 men, I desire to know how many in rank and file? Ans. 576. 2. A certain square pavement contains 48841 square stones, all of the same size, I demand how many are contained in one of the sides? Ans. 221. To find the area of a piece of land in form of a triangle. RULE. Add together the three sides, from half their sum subtract each side, and note the remainder, then multiply the half sum by one of those remainders, and that product by another remainder; the square root of the last product will be the area. |