2. If the whole denominator vanish in dividing by 2, 5, or 10, the decimal will be finite, and will consist of so many places, as you perform divisions.

3. If it do not so vanish, divide 9999, &c. by the result, till nothing remain, and the number of 9s used will show the number of places in the repetend; which will begin after so many places of figures, as there were 10s, 2s, or 5s, used in dividing.

EXAMPLES.

1. Required to find whether the decimal equal to 2TW be finite or infinite; and if infinite, how many places the repetend will consist of.

3

2 2. a

2110=2118 | 8|4|2|1; therefore the decimal is

finite, and consists of 4 places.

2. Let be the fraction proposed.

3. Let be the fraction proposed.

40

4. Let be the fraction proposed. 5. Let be the fraction proposed.

ADDITION OF CIRCULATING DECIMALS.

RULE.*

1. Make the repetends similar and conterminous, and find their sum as in common addition.

being alike, will be carried to the next, by which means each product will be equally increased, and consequently every four places will continue alike. And the same will hold for any other number whatever.

Now hence it appears, that the dividend may be altered at pleasure, and the number of places in the repetend will still be

the same: thus-90, and, or×3=27, where the number of places in each is alike, and the same will be true in all cases. *These rules are both evident from what has been said in reduction.

2. Divide this sum by as many nines as there are places in the repetend, and the remainder is the repetend of the sum; which must be set under the figures added, with cyphers on the left, when it has not so many placcs as the repetends.

3. Carry the quotient of this division to the next column, and proceed with the rest as in finite decimals.

EXAMPLES.

1. Let 3*6+78*3476+735*3+375+27+1874 be added together.

In this question, the sum of the repetends is 2648191, which, divided by 999999, gives 2 to carry, and the remainder is 648193.

2. Let 5391'357+72 38+187‍21+42965+217*8496 +42176+523+58 30048 be added together,

Ans 5974'10371.

3. Add 9'814+1 5+87*26+083+124 09 together.

Ans. 222 75572390.

4. Add 162+134'09+2*93+97‍26+3*769230+99‍083+1*5

+814 together.

Ans. 501-62651077.

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.

EXAMPLES.

1. Multiply 36 by 25.

*36=3=1

*25=23

1190929 the product.

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.

60

A÷÷÷=4×2=388=183=14229249011857707509881.

the quotient.

2. Divide 319°28007112 by 7645.

3. Divide 234'6 by '7.

4. Divide 13'5169533 by 4'297.

Ans. 4176325. 417632

Ans. 301'714285.

Ans. 3'145.

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 8, and 4 ..

2::8

.. 6.