2. Let 5391·357+72·38+187·21+4·2965+217·8496 +42·176+‍523+58′30048 be added together.

Ans. 5974-10371.

3, Add 9.814+15+87·26+083+124·09 together.

4. Add 162+134·09+2·93+97·26+3·769230+99·083

+1.5+814 together.

Ans. 222.75572390.

Ans. 501-62651077.

SUBTRACTION of CIRCULATING DEgimals.

RULE.

Make the repetends similar and conterminous, and subtract as usual; observing, that, if the repetend of the sub-trahend be greater than the repetend of the minuend, then the right-hand figure of the remainder must be less by uni ty, 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

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.

4+3=4×29=159=11873=14229249011857707509881

the quotient.

2. Divide, 319-28007112 by 764′5.

Ans. 4176325.

3. Divide

3. Divide 234'6 by 7.

4. Divide 13'5169533 by 4*297.

Ans. 301-714285.
Ans 3*145.

OF 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 conse quents, 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 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 down in a serics, 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 = 1/10 = 2.

6

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 8-26; also 4, 2, 16, 8, are in discontinued geometrical proportion; for 2, but =8.

1 6

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; 2:6:39.

Three or four numbers are said to be in harmonical proportion, when, in the former case, the difference of the first and

*In geometrical proportionals a colon is placed between the terms of each couplet, and a double colon between the couplets; in arithmetical 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 arithmet4: 68, and 4 ..2::8

ical, 2

"

6.

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 1:6-33:26; and 4-31:9-6=3 ::3:9.

*

Of four arithmetical proportionals the sum of the extremes is equal to the sum of the means. Thus of 2 46.. 8 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 (2X 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 proportionals is equal to the product of the means divided

* DEMONSTRATION. Let the four arithmetical proportionals be A, B, C, D, viz. A.... B :: C. D; 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:: BC; therefore A+C=B+B=2B. Q. E. D.

DEMONSTRATION. Let the proportion be A: B:: C: D, and let; then A Br, and C=Dr; multiply the former of these equations by D, and the latter by B; then ADBrD, and CB Dr B, and consequently AD the product of the extremes is equal to BC the product of the means. And three may be thus expressed, A: B:: B: C, therefore A C=Bx B=B'. Q. E. D.

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