EXAMPLES 1. How many changes may be 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. 2. A 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. 2. 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. 2. Roots... 3. 4. 6. 7. SQUARES. 1. 4. 9. 4. 9. 16. 25. 36. 49. 64. 9. 84. EXAMPLES 3. What is the square root of 119025? 119025(345 64)290 256 685)3425 3425 Ans. 345 2. 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 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 ? Anf. 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+ 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. EXAMPLES. 19. What is the square root of 51 } ? 20. What is the square root of 27 16 ? What is the square root of 545? Anf. 73 21. 22. SURDS. 14 ? Anf. 9,27+ 23. What is the square root of 8ş? Anr. 2,9519+ 24. What is the square root of 6*? 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 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 numbers is the mean proportional fought. EXAMPLES Ans. 3X12=36 then ✓ 36=6 the mean proportional. Anf. 1897,4+ 1. 2. 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 ? Ans. 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. 9. The wall of a town is 25 feet high, which is surrounded by a moat of 30 feet in breadth, I defire 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 hypothenufe 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 hypothenuse and base is the heighth 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 the num. ber of men either in rank or file. An army consisting of 331776 men, I desire to know how many in rank and file ? Aní. 576. II. A certain square pavement contains 48841 square stones, all of the same size, I demand how many are contained in one of the fides. Anf. 221. 10. EXTRACTION OF THE CUBE ROOT. 2. TO extract the Cube Root is to find out a number which being multiplied into itself, and then into that product, produceth the given number. RULE. Point every third figure of the cube given, beginning at the unit's place, seek the greatest cube to the first point and subtract it therefrom, put the root in the quotient, and bring down the figures in the next point to the remainder for a resolvend. Find a divisor by multiplying the square of the quotient by 3. See how often it is contained in the resolvend, rejecting the units and tens, and put the answer in the quotient. 3. To find the fubtrahend. 1. Cube the last figure in the quotient. 2. Multiply all the figures in the quotient by 3, except the laft, and that product by the square of the last. 3. Multiply the divisor by the last figure. Add these products together, gives the subtrahend, which subtract from the resolvend ; to the remainder bring down the next point and proceed as before. Roots. 3. 4. 5. 6. 9. CUBES, 8. 27. 64. 125. 216. 343. 512: 729. 1. 2. 1. |