When the indices begin with a cypher, the sum of the indices made choice of must he always one less than the number of terms given in the question, for 1 in the indices is over the second term, and 2 over the third, &c. Add any two of the indices together, and that sum will agree with the product of their respective terms. As in the first table of indices 2+ 5= 7 Then in the second .... 4 X 32 128 2+4= 6 In any Geometrical Progression proceeding from unity, the ratio being known, to find any remote term, without producing all the intermediate terms. RULE. Find what figures of the indices added together would give the exponent of the term wanted, then multiply the numkers standing under such exponent into each other, and it will give the term required. NOTE. When the exponent 1 stands over the second term, the number of exponents must be 1 less than the number of terins. EXAMPLES. 1. A man agrees for 12 peaches, to pay only the price of the last, reckoning a farthing for the first, a half-penny for the se cond, &c. doubling the price to the last; what must he give for them? 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 a piece; the gentleman bids him £.15 a piece, 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? 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 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. 1. A servant skilled in numbers agreed with a gentleman to serve him 12 months, provided he would give him a farthing U 2 for his first month's service, a penny for the second, and 4d. for the third, &c.-what did his wages amount to ? 256x256 65536, then 65536 x 64-4194304 0. 1. 2. 3. 4. 1.4. 16. 64. 256. (4+4+3=11. No. of terms less 1.) 4194304-1 1398101; then 4 1 13981014194304-5592455 farthings. Ans. £.5925 8s. 54d. 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 oneyear; what was her portion ? Ans. £.204 158. 4. A laceman well versed in numbers, agreed with a gentleman to sell 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 lost by the sale thereof, supposing the lace stood him in £.7 per yard. Ans. The lace sold for £.326886 Os. 9d. Gain £.326732 Os. 9d. PERMUTATION 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. 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 1047900160 minutes, and if reduced is91 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. 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. 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. Ans. 327 2. What is the square root of 106929 ? 3. What is the square root of 2268741 ? Ans. 1506,23 + 4. What is the square root of 7596796? Ans. 2756,228+ 5. What is the square root of 36372961? Ans. 6031 6. 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 to 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? 8. What is the square root of 4795,25731 ? 9. What is the square root of 4,372594? 10. What is the square root of 2,2710957 ? 11. What is the square root of,00032754 ? 12. What is the square root of 1,270054 ? Ans. 57,19+ 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. |