SUBTRACTION OF CIRCULATING DECIMALS.

RULE.

Make the repetends similar and conterminous, and subtract as usual; observing, that, if the repetend of the subtrahend be greater than the repetend of the minuend, then the figure of the remainder on the right must be less by unity, than it would be, if the expressions were finite.

1. Turn both the terms into their equivalent vulgar fractions, and find the product of those fractions as usual.

2. Turn the vulgar fraction, expressing the product, into an equivalent decimal, and it will be the product required.

1. Change both the divisor and dividend into their equivalent vulgar fractions, and find their quotient as usual.

2. Turn the vulgar fraction, expressing the quotient, into its equivalent decimal, and it will be the quotient required.

EXAMPLES.

1. Divide 36 by '25.

*36=3=

*25=33

+2=123=368=1383=14229249011857707509881.

PROPORTION IN GENERAL.

NUMBERS are compared together to discover

the relations they have to each other.

There must be two numbers to form a comparison; the number, which is compared, being written first, is called the antecedent; and that, to which it is compared, the consequent. Thus of these numbers 2: 4 : 3:6, 2 and 3 are called the antecedents; and 4 and 6, the consequents.

Numbers are compared to each other two different ways; one comparison considers the difference of the two numbers, and is called arithmetical relation, the difference being sometimes named the arithmetical ratio; and the other considers their quotient, and is termed geometrical relation, and the quotient the geometrical ratio. So of these numbers 6 and 3, the difference or arithmetical ratio is 6-3 or 3; and the geometrical ratio is or 2.

If two or more couplets of numbers have equal ratios, or differences, the equality is named proportion; and their terms similarly posited, that is, either all the greater, or all the less, taken as antecedents, and the rest as consequents, 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 couplets 2, 4, and 8, 16, taken thus, 2, 4, 8, 16, or thus, 4, 2, 16, 8, are geometrical proportionals.* Proportion is distinguished into continued and discontinued. If, of several couplets of proportionals written in a

* In geometrical proportionals a colon is placed between the terms of each couplet, and a double colon between the couplets; in arithemetical proportionals a colon may be turned horizontally between the terms of each couplet, and two colons written between the couplets. Thus the above geometrical proportionals are written thus, 2 : 4 :: 8: 16, and 4 : 2 :: 16: 8; the arithmetical, 2 4:: 6 2::86.

8, and 4

series, the difference or ratio of each consequent and the antecedent of the next following couplet be the same as the common difference or ratio of the couplets, the proportion is said to be continued, and the numbers themselves a series of continued arithmetical or geometrical proportionals. So 2, 4, 6, 8, form an arithmetical progression; for 4—2—6———4—8— 6=2; and 2, 4, 8, 16, a geometrical progression; for =1= 18--2.

But if the difference or ratio of the consequent of one couplet and the antecedent of the next couplet be not the same as the common difference or ratio of the couplets, the proportion is said to be discontinued. So 4, 2, 8, 6, are in discontinued arithmetical proportion; for 4—2—8—6—2, but 8-2-6; also 4, 2, 16, 8, are in discontinued geometrical proportion; for 162, but 1-8.

Four numbers are directly proportional, when the ratio of the first to the second is the same, as that of the third to the fourth. As 2: 4:3: 6. Four numbers are said to be reciprocally or inversely proportional, when the first is to the second, as the fourth is to the third, and vice versa. Thus, 2, 6, 9, and 3, are reciprocal proportionals; 2: 6:: 3 : 9.

Three or four numbers are said to be in harmonical proportion, when, in the former case, the difference of the first and second is to the difference of the second and third, as the first is to the third; and, in the latter, when the difference of the first and second is to the difference of the third and fourth, as the first is to the fourth. Thus, 2, 3, and 6; and 3, 4, 6, and 9, are harmonical proportionals; for 3-2 =1 : 6—3—3 :: 2 : 6; and 4—3—1 : 9—6—3:: 3 : 9.

of four arithmetical proportionals the sum of the extremes is equal to the sum of the means.* Thus of 2.4::68

* DEMONSTRATION. Let the four arithmetical proportionals be A, B, C, D, viz. A“ B:: CD; then, A—B—C—D, and B+D being added to both sides of the equation, A—B+B+D —C—D+B+D; that is, A+D the sum of the extremes C+B the sum of the means. And three A, B, C, may be thus expressed, A B :: B. C; therefore 4+C=B+B=2B.

Q. E. D.

the sum of the extremes (2+8)= the sum of the means (4+6) 10. Therefore, of three arithmetical proportionals, the sum of the extremes is double the mean.

Of four geometrical proportionals, the product of the extremes is equal to the product of the means.* Thus, of 2:48:16, the product of the extremes (2×16) is equal to the product of the means (4x8)=32. Therefore of three geometrical proportionals, the product of the extremes is equal to the square of the mean.

Hence it is easily seen, that either extreme of four geometrical proportiouals is equal to the product of the means divided by the other extreme; and that either mean is equal to the product of the extremes divided by the other mean.

SIMPLE PROPORTION, OR RULE OF THREE.

The Rule of Three is that, by which a number is found, having to a given number the same ratio, which is between two other given numbers. For this reason it is sometimes named the Rule of Proportion.

It is called the Rule of Three, because in each of its questions there are given three numbers at least. And because of its excellent and extensive use, it is often named the Golden Rule.

RULE.t

1. Write the number, which is of the same kind with the answer or number required.

* DEMONSTRATION. Let the proportion be A: B:: C: D, and let ===r; then A=Br, and C=Dr; multiply the for

A C

B D

mer of these equations by D, and the latter by B; then AD= BrD, and CB=DrB, and consequently AD the product of the extremes is equal to BC the product of the means.s.-And three may be thus expressed, A: B:: B: C, therefore AC=Bx B=B2. Q. E. D.

† DEMONSTRATION. The following observations, taken col