4X3 (the index of the given power)=12, 1k divif. 27X 27X27=19683=2d fubtrahend. 27X27=229(next inferiour power) and, 729X3(=index of the given power)=21872d div. 273X273+273=20346417=3d fubtrahend. 2. What is the biquadrate root of 34827998976? Anf. 431,9+ OF PROPORTION IN GENERAL. Numbers are compared together to difcover the relations they have to each other. There must be two numbers to form a comparison : The number, which is compared, being written firft, is called the antecedent; and that, to which it is compared," the confequent. Numbers are compared with each other two different ways: The one comparison confiders the difference of the two numbers, and is called Arithmetical Relation, the difference being sometimes named the Arithmetical Ratio; and the other confiders their quotient, which is termed Geometrical Relation, and the quotient, the Geometrical Ratio. Thus, of the numbers 12 and 4; the difference,or Arithmetical Ratio, is 12-48; and the Geometrical Ratio is 12 4 If two, or more, couplets of numbers have equal ratios, or differences, the equality is termed proportion; and their terms fimilarly profited, that is, either all the greater, or all the lefs taken as antecedents, and the rest as confequents, are called proportionals. So, the two couplets, 2, 4, and 6, 8, taken thus, 2, 4, 6, 8, or thus, 4, 2, 8, 6, are arithmetical proportionals; and the two couplets, *RATIOS are, here, always confidered as the refult of the greater term of comparison diminished, or divided by the less; not regarding which of them be the antecedent. couplets 2, 4, and 8, 16, taken thus, 2, 4, 8, 16, or thus, 4, 2, 16, 8, are geometrical proportionals.* Proportion is diftinguished into continual and difcontinual. If, of feveral couplets of proportionals, written down in a feries, the difference or ratio of each confe quent, and the antecedent of the next following couplet, be the fame as the common difference or ratio of the couplets, the proportion is faid to be continual, and the numbers themfelves, a feries of continual proportionals, or an arithmetical or geometrical proportion. So 2, 4, 6, 8, form an arithmetical proportion; for 4-2-6-4 8-62; and, 2, 4, 8, 16, a geometrical propor16=2. tion; for "NOTE. It is common to read the geometricals 2:4:8; 16: Thus, 2 is to 4 as 8 is to 16, or as 2 to 4 fo is 8 to 16. But if the difference or ratio of the confequent of one couplet, and the antecedent of the next couplet be not the fame as the common difference or ratio of the couplets, the proportion is faid to be difcontinued. So 4, 2, 8, 6, are in difcontinued arithmetical proportion; for 4-2=8 -6=2=common difference of the couplets, 8—2=6 difference of the confequent of one couplet and the antecedent of the next; alfo, 4, 2, 16, 8 are in difcon. 4 16 tinued geometrical proportion: For- 2 8 16 2= common ratio of the couplets, and--8-ratio of 2 the confequent of one couplet and the antecedent of the next. ARITHMETICAL * Four numbers are faid to be reciprocally or inversely proportional, when the fourth is less than the fecond, by as many times as the third is greater than the first, or when the first to the third, is as the fourth to the fecond, and vice verfa. Thus, 2, 9, 6 and 3 are reciprocal proportionals. ARITHMETICAL PROPORTION. THEOREM If any four quantities, 2, 4, 6, 8, be in Arithmetical Proportion, the sum of the two means is equal to the fum of the two extremes. And if any three quantities, 2, 4, 6, be in Arithmetical Proportion, the double of the mean is equal to the fum of the extremes. THEOREM 2.-In any continued arithmetical proportion, 1, 3, 5, 7, 9, 11, the fum of the two extremes, and that of every other two terms, equally diftant from them, are equal: Thus, 1+1=2+9=5+7. When the number of terms is odd, as in the proportion, 3, 8, 13, 18, 23, then, the fum of the two extremes, being double to the mean, or middle term, the fum of any other two terms, equally remote from the extremes, must likewife be double to the mean. THEOREM 3-In any continued arithmetical propor tion, 4, 4+2, 4+4, 4+6 4+8, &c the laft or greatest term is equal to the first or leaft more the common difference of the terms drawn into the number of all the terms after the firft, or into the whole number of the terms, lefs one. - THEOREM 4. The fum of any rank, or feries, of quantities in continued arithmetical proportion, 1, 3, 5, 7, 9, 11, is equal to the fun of the two extremes multia plied into half the number of terms.* ARITHMETICAL The fame thing also holds, when the number of terms is odd, as in the feries, 4, 8, 12, 16. 20; for then, the mean, or middle term, being equal to half the fum of any two terms es qually distant from it on contrary fides, it is obvious that the value of the whole feries is the fame as if every term the Of were equal to the mean, and therefore is equal to the mear half the fum of the two extremes) multiplied by the whole on nber of terms; or to the fom of the extremes multiplied by half the number of terins. or The fum of any number of terms of the arithmetical feries of odd numbers, 1, 3, 5, 7, 9, &c. is equal to the fquare of that number. For X ARITHMETICAL PROCRESSION. Any rank of numbers, more than two, increasing by a common excefs, or decreafing by a common difference, is faid to be in Arithmetical Progreffion. If the fucceeding terms of a progreffion exceed each other, it is called an afcending feries or progreffion; if the contrary, a defcending feries. So & So. I 2.4 6. 2 I 2. 4 8 16. S 8. 6. 4. 2. 1. 8, Ac is an afcending arithmetical series. 32, &c. is an afcending geometrical series. 0, &c. is a defcending arithmetical feries. 32. 16 8. 4. 2. I, &c. is a defcending geometrical feries. The numbers which form the feries, are called the terms of the progreffion. Note. The first and last terms of a progreffion are called the extremes, and the other terms the means Any three of the five following things being given, the other two may be easily be found. 1. The first term. 2. The laft term. 3. The number of terms. PROBLEM I. The first term, the laft term, and the number of terms being given, to find the common difference. For, o or the fum of I term =12 or 1 1+3 or the fum of 2 terms=22 or 4 4+5 or the fum of 3 terms 32 or 9 9+7 or the fum of 4 terms=42 or 16 169 or the fum of 5 terms=52 or 25, &c. RULE. Whence, it is plain, that let there be any number whatever, the fem of its terms will be equal to its fquare. EXAMPLE. The first term, the ratio, and number of terms given, to find the fum of the feries. A Gentleman travelled 29 days; the first day he went but 1 mile, and increased every day's travel 2 miles: How far did he travel? 29X29-841 miles the Anfæer. RULE. Divide the difference of the extremes by the number of terms lefs i, and the quotient will be the common difference fought. I. EXAMPLES. The extremes are 3 and 39, and the number of terms is 19; What is the common difference? 39) Ex- 36 Divide by the number of terms less I=19—1=18)36(2 Anf. 2. A man had 10 fons whose several ages differed alike; the youngest was 3 years old, and the eldest 48; What was the common difference of their ages? 3. A man is to travel from Boston to a certain place in 9 days, and to go but 5 miles the first day, increafing each day by an equal excefs, fo that the laft day's jour ney may be 37 miles; required the daily increafe? PROBLEM 2. The firft term, the last term, and the number of terms given, to find the fum of all the terms. RULE.-Multiply the fum of the extremes by the number of terms, and half the product will be the anfwer. EXAMPLES. |