by the feet, and take parts for the inches, feconds, &c. of the multiplier out of the multiplicand; and parts for the like denominations of the multiplicand out of the feet of the multiplier only, and the fum of all the lines will be the anfwer. PR ROO O F. Questions in this rule may be proved by decimals, by practice or by vulgar fractions. EXAMPLE S. 1. Multiply 10 feet 9 ins. 6 pts. by 8 ft. 7 ins. 9 pts. By Practice. 2. Multiply 4 feet 6 ins. by 14 feet 9 ins. 3. Multiply 11 ft. 8 ins. 4. Multiply 4 ft. 7 ins. Anf. 66 ft. 4 ins. 6 pts. by 9 ft. 7 ins. Anf. 111 ft. 9 ins. 8 pts. 8 pts. by 9 ft. 6 ins. Anf. 44 ft. o ins. 10 pts. 5. What is the product of 7 ft. 5 ins. 6 pts. and 3 ft. 8 ins. 9 pts? Anf. 27 ft. 9 ins, 9 pts. I thd. 6 fths. 6. Multiply 39 ft. ro ins. 7 pts. by 18 ft. 8 ins. 4 pts. Anf. 745 ft. 6 ins. 10 pts. 2 thds. 4 fths. 7. Multiply 34 feet 4 ins. 6 fecs. by 29 feet 5 ins. Anf. 1012 ft. 1 in. 10 pts. 5 pts. by 137 ft. 8 ins. 10 pts. 9 thds. 8 fths. 9. Multiply 4 pts. 8. Multiply 368 feet 7 ins. Anf. 50756 ft. 4 pts. 7 ins. 9. Multiply 44 ft. 2 ins. 9 pts. 2 thds. 4 fths. by 2 ft. 10 ins. 3 pts. Anf. 126 ft. 2 ins. 10 pts. 8 thds. Icfths. 11 fths. 10. Multiply 81 ft. 11 ins. 7 pts. by 57 ft. 8 ins. 5 pts. Anf. 4729 ft. 6 ins. 1 pt. 5 thds. 11fths. ARITHMETICAL PROGRESSION. ARITHMETICAL progreffion is when any series or rank of numbers increase or decrease by a common difference; as, 3, 4, 5, 6, 7 increase by the common difference ; and 12, 10, 8, 6, 4 decrease by the common difference 2. t The numbers which form the feries are called the terms of the progreffion. The first and last terms of a series are called extremes, and the other terms the means. There are five things in Arithmetical progreffion which are to be particularly attended to; viz. ft. The first term. 2d. the laft term. 3d. the number of terms. 4th. the common difference. 5th. the fum of all the terms. Any three of thefe being given, the other two may be readily found.. REMARK. Arithmetical progreffion may be divided into ten cafes, each containing two propofitions; but as many of these are beft understood and folved by Algebraic proceffes, they will be omitted, and only fome of the most useful of them inferted here. The first term, the laft term, and the number of terms being given, to find the fum of all the terms. RULE. R U L E. Multiply the fum of the extremes by the number of terms, and half the product will be the answer. EXA AMPLE S. 1. The first term of an Arithmetical progression is 4, the last term 40, and the number of terms 12; required the fum of the series? 40 44 2)528 264 Ani. 3. How many ftrokes do the clocks at Venice (which go on to 24 o'clock) ftrike in the compass of à natural day? Anf. 300. 3. What debt can be discharged in a year, by weekly payments, in Arithmetical progreffion, whereof the first. payment is 18. and the laft 51. 3s. ?—Anf. 1351. 43. 4. Suppofe 100 ftones were placed in a right line, a yard diftant from one another, and the firft ftone to be a yard from a basket; how many miles muft that man travel who gathers them fingly into the basket? Anf. 5 ms. 5 fgs. 36 ps. 2 yds. PROBLEM II. The first term, the last term, and the number of terms being given, to find the common difference. Subtract R U L E. from the number of terms, and divide the difference of the extremes, by that remainder, and the quotient will be the common difference. EXAMPLE S. 1. If the extremes be 10 and 70, and the number of terms 21; what is the common difference, and the fum of the feries? 2. A man had 8 fons, the youngst was 4 years old, and the eldest 32, and their ages increase in Arithmetical progreffion; what is the common difference of their ages? Anf. 4 years. 3. A man is to travel from London to a certain place in 12 days, and to go but 3 miles the first day, increafing every day by an equal excefs, fo that the last day's journey may be 58 miles; required the daily increase, and the distance of the place from London? Anf. daily increase 5 ms.—dist. 366 ms. 4. Given the firft term of a series o and the laft 34, and the number of terms 4, to find the common difference? Anf. 11 The firft term, the last term, and the common difference being given, to find the number of terms. R U L E. Divide the difference of the extremes by the common difference, and add 1 to the quotient for the answer, EXAMPLE S. J. The extremes are 3 and 19, and the common difference 2; what is the number of terms? 19 3 2)16 8+1=9 Anf. 2. A debt is to be discharged by monthly payments in Arithmetical progreffion, the first payment to be il., the last 231., and the common difference zl.; in what time will it be discharged, and how much is the debt? Anf. 12 mos., the sum 1441. 3. A perfon travelling northward from London, went 8 miles the first day, and increased every day's journey 3 miles, till at last he went 50 miles in one day; how many days, and how far did he travel? Anf. 15 days, and travelled 435 ms. 4. A man being asked how many children he had, anfwered, my youngest child is 5 years old, and my eldest 43, and I have increased one in my family every 2 years.. How many children had he? PROBLEM Anf. 20 children. IV. One extreme, the common difference, and the number of terms being given, to find the other extreme. R U L E. Subtract one from the number of terms, and multiply the common difference by the remainder. Then, add the product to, or fubtract it from, the given térm, according as it is lefs or greater, and the fum or difference will be the answer. |