a Fraction, whose numerator is 1, and whose denominator, the dividing number. Now, if both terms of this Fraction were to be multiplied by this quotient, or circulate, the Fraction, without being altered in value, (§ XLI.) would be changed to one whose numerator would be the circulate itself, and whose denominator, just as many nines as the circulate has places. To illustrate by an example. 999999÷7=142857, and reduced to a decimal=.142857. 47142857 = 14237. Hence, if the circulate .142857 had been given us, and it had been required to find its value in a vulgar Fraction, (that is, to find what vulgar Fraction would produce it,) we might have made the circulate itself the numerator of a Fraction, and just as many nines as it had places, the denominator, thus, 33, and this, reduced to its lowest terms, would have given us the original Fraction, . would give us twice as great a circulate,, three times as great, and so on.

142857

9999999

Hence, to change any pure circulate, or repetend, to its equivalent Vulgar Fraction,

MAKE THE GIVEN REPETEND THE NUMERATOR, AND THE DENOMINATOR AS MANY 98, AS THERE ARE PLACES IN THE REPETEND.

In mixed repetends, the principle is similar. If a repetend begin in the place of hundredths, its value will be the same as before, except that it will be ten times smaller than if it began, like a simple repetend, in the place of tenths. For when it begins in the first, or tenths' place, it is a Fraction of a unit, or whole number. But when it begins in the second, or hundredths' place, it is a Fraction of a tenth. Its value is therefore decreased ten fold. Thus .3 is of a whole number; but .03 is of a tenth 3 of 15%. Af ter finding the Vulgar Fraction, as before, then, we are obliged to divide it by 10, or, (which, (§ LIII., is the same thing,) multiply its denominator by 10; which is done by annexing a cypher. In like manner, if it had begun a place lower still, we should have been obliged to annex two cyphers, and so on; annexing always as many cyphers as there are places between the separatrix, and the first figure of the repetend, or as there are finite places in the decimal. Hence, to find the value of the circulating part of a mixed repetend,

MAKE THE REPETEND THE NUMERATOR; AND FOR THE DENOMINATOR, ANNEX AS MANY CYPHERS AS THERE ARE FINITE PLACES, TO AS MANY 9S AS THERE ARE PLACES IN THE CIRCULATE.

The mode of finding the value of the finite part has already been given. (§ LVIII.) Hence to find the whole value of a mixed repetend, in a Vulgar Fraction,

FIND THE VALUES OF THE FINITE AND CIRCULATING PARTS SEPARATELY, AND ADD THEM TOGETHER.

By the above rule, find the values of .104 .839 .61407 .93 815 7 .6311 .9831 .4762 .83141 .98764 .30942 .111333 .7778889 .63115438 .71324285.

§ LXXI. A single repetend may evidently be regarded as a compound repetend, consisting of as many places as we choose to make it. Thus, 3 is a single repetend. Giving it two places, thus, 33, or three, thus, .333, it becomes a compound repetend. So .97 may be made .97777 or 977777, &c. We may also make a single repetend begin later, or at a lower place, reserving. its higher figures as finite decimals. Thus, .6 may be made .68, or 666, &c.

We may also make the same changes upon compound repetends; making them begin later, thus, .46 changed to .464 or .46464, &c. or extending the number of places, thus, .379 changed to .37979, or to .3797979, &c.; or making both changes at once, thus .432 chan. ged to .4324324, &c. In changing compound repetends, we must be careful to place the points so that the repeating figures, when extended beyond the last point, shall recur in the same order as before. Thus, .862 is inaccurately changed to .86286, and .813 to .81313, the latter numbers, in each case, representing different repetends from those given. By moving the points in this manner, we may render any two repetends similar. Thus .135 and .861 are dissimilar; but .1351 and .861 are similar. Having made repetends similar, they may also be rendered conterminous, by extending their figures to a number of places, that is a common multiple of the numbers of places in the given repetends. For, on making the number of places in any repetend, twice, three times, or any number of times as great as before, will still repeat in the same order as at first. The least common multiple is most convenient.

The foregoing examination of the nature of circulating decimals, will afford us some useful rules for conducting arithmetical operations upon them. Single repetends, in their true value, have been seen to be ninths. Hence, for ADDITION when there are FINITE DECIMALS, AND SINGLE REPETENDS, or SINGLE REPETENDS ONLY,

I. MAKE THE REPETENDS CONTERMINOUS, EXTENDING THEM ONE PLACE BEYOND THE LONGEST, FINITE DECIMAL; CARRY FOR 9 INSTEAD OF 10 FROM THE RIGHT HAND COLUMN; AND IN OTHER RESPECTS ADD AS THE RIGHT HAND FIGURE OF THE SUM WILL BE A REPETEND.

USUAL.

In case there are compound repetends, having made them conterminous, it is evident that, if they were extended farther still, (as they might be,) there would often be something to carry from those figures so extended; and this number carried would be the same as that carried forward from the first place of the repetends. Hence, when there are FINITE DECIMALS, AND CIRCULATES, or CIRCULATES ONLY,

II. MAKE THE REPETENDS SIMILAR AND CONTERMINOUS, COMMENCING 'THEM ONE PLACE BELOW THE LONGEST FINITE DECIMAL; CARRY TO THE RIGHT Hand figure OF THE SUM, THE SAME NUMBER THAT IS CARRIED FORWARD FROM THE FIRST PLACE OF THE REPETENDS; AND IN OTHER RESPECTS ADD AS USUAL. THE SUM OF THE CIRCULATING Figures,

WILL BE THE CIRCULATE OF THE AMOUNT.

NOTE. If the circulate found by adding be a series of 9s, it becomes equal to the denominator of its equivalent vulgar Fraction, and of course its value is 1; which may be added to the next higher place, and the repetend neglected.

Multiplication is a repeated addition. Hence, for MULTIPLICATION, when the MULTIPLIER IS FINITE, and the MULTIPLICAND A SINGLE REPETEND,

I. MULTIPLY AS USUAL, CARRYING FOR 9 FROM THE PRODUCT
THE RIGHT HAND FIGURE OF EACH PAR-
OF THE REPETEND.
THE TOTAL PRODUCT

TIAL PRODUCT WILL BE A REPETEND.
MUST THEREFORE BE FOUND BY RULE.I. FOR ADDITION.

That is, the partial products must be made conterminous, before adding, and, in adding we must carry for 9 from the right hand column. The last figure of the total product will then be the repetend. When the MULTIPLIER IS FINITE, and the MULTIPLICAND A CİR

CULATE,

II. MULTIPLY AS USUAL, CARRYING TO THE RIGHT HAND FIGURE OF EACH PARTIAL PRODUCT, THE NUMBER WHICH IS CARRIED FORWARD FROM THE FIRST PLACE OF THE CIRCULATE. EACH PARTIAL PRODUCT WILL HAVE AS MANY CIRCULATING THE TOTAL PRODUCT MUST FIGURES AS THE MULTIPLICAND.

THEREFORE BE FOUND BY RULE II. FOR ADDITION.

The case in which the multiplier is a repetend, remains to be considered. Here we have no means of proceeding decimally with Hence, when the MULTIPLIER IS A REPETEND,

accuracy.

III. CHANGE THE MULTIPLIER TO ITS EQUIVALENT VULGAR FRACTION; MULTIPLY BY ITS NUMERATOR, AND DIVIDE BY ITS DENOMINATOR.

NOTE. If the multiplicand be also infinite, this rule anticipates a case of division to be mentioned. (Rule 1.) It may however be mentioned here, that, when, in ordinary cases, we should annex cyphers in dividing, we must annex the repeating figures of the number divided instead of them; and so continue to do until the decimal repeats. This will determine the number of figures in the final circulate.

In Subtraction it is evident, that, when the repetends are similar and conterminous, if that of the subtrahend be greater than that of the minuend, we shall be obliged to borrow 1 from the next higher place. And this would also be the case, if the repetends were extended below the last point. This would make the remainder 1 less. Hence, for SUBTRACTION, when EITHER OR BOTH NUMBERS ARE REPETENDS,

PREPARE THE NUMBERS AS IN ADDITION, AND SUBTRACT AS USUAL, OBSERVING TO DIMINISH THE REMAINDER BY 1, WHEN THE REPETEND OF THE SUBTRAHEND IS GREATER THAN THAT OF THE MINUEND.

In division, when the dividend is infinite, it is plain, that since the repeating figures may be extended to any distance, those figures will occupy the places of the cyphers which would otherwise be

annexed to continue, the division. Hence, for DIVISION, when the DIVISOR IS FINITE, and the DIVIDEND A REPETEND,

I. DIVIDE AS USUAL, ANNEXING THE REPEATING FIGURES OF THE DIVIDEND, IF NECESSARY, INSTEAD OF CYPHERS; AND SO PROCEED UNTIL THE QUOTIENT REPEATS.

If the divisor be infinite, the division cannot be performed deci mally. Hence, when the DIVISOR IS A REPETEND,

II. CHANGE THE DIVISOR TO ITS EQUIVALENT VULGAR FRACTION; MULTIPLY BY ITS DENOMINATOR, AND DIVIDE BY ITS NUMERATOR.

NOTE. For short repetends, the preceding rules are very useful. But when repetends are long, the processes are tedious, and the value of the lower figures so small, that they may be neglected, without occasioning any important error. The same may be remarked of long finite decimals. We have before mentioned that decimals not carried out in full, are marked with + and—, according as they are too small or too great, and called APPROXIMATES. Particular rules may be given for their calculation, which the limits of this work will not permit us to insert. The common rules are sufficient for ordinary purposes; and those who are curious to examine the subject further, must resort to more extensive works. We subjoin a few examples.

1. Add 3.6; 78.3476; 735.3; 375; .27 and 187.4

A. 1,380.0648193

2. Add 5,391.357; 72.38; 187.21; 4.2965; 217.8496; 42.176; .523; and 58.30048

3. Add 9.814; 1.5; 87.26; .083 and 124.09

A. 5,974.1037i

A. 222.75572390

4. Add 162.162; 1.5; 134.09; 2.93; 97.26; 3.769230; 99.083

NOTE. The pupil will observe, that as either factor may be made the multi

plier, questions like the last may be solved, by rule II. or III. at pleasure.

11. From 85.62 take 13.76432

12. From 26.43 take 25.2

A. 71.86193

A. 1.2i

13. From 476.32 take 84.7697
14. From 3.8564 take .0382

15. Divide 319.28007112 by 764.5
16. Divide 274.6 by .7
17. Divide 18.56 by .3

18. Divide 24.081 by .386
19. Divide .36 by .25

A. 391.5524.

A. 3.81

A. .4176325

A. 301.714285

A. 55.69

A. 62.32383419689i

A. 1.4229249011857707509881

DUODECIMALS.

§ LXXII. Decimals, we have seen, consist of a series of numbers decreasing towards the right by tens, as far as we choose to carry them. We are now about to speak of a class of numbers, decreasing in a similar manner by twelves. Of course, the calculations made upon them must be similar, in many respects, to those upon decimals. They are called DUODECIMALS, from the Latin word duodecimus, which signifies twelfth. Duodecimals are commonly used only for measuring length, surface, and solidity.

A linear, square or solid foot is, therefore, considered the unit, or whole number. The lower denominations, or orders, are, of course, Fractions of a foot.

The twelfth part of a foot, of any kind, whether a solid, a square, or a linear foot, is called a PRIME; the twelfth part of a prime, is called a SECOND; the the twelfth part of a second, a THIRD; the twelfth part of a third, a FOURTH, and so on, as far as we choose to go.

Now, as primes are 12ths, seconds, 12ths of 12ths, or 144ths, thirds, 12ths of 144ths, or 1,728ths, fourths, 12ths of 1,728ths, or 20,736ths, &c. it is plain that we might write these numbers, as vulgar Fractions, with their denominators. But, since we know that they increase and decrease regularly by 12, we may, exactly as in decimals, omit the denominators, and write each order in a separate place. This is the mode, in which duodecimals are actually written, and, to distinguish the orders from oneanother, primes are marked with an accent, thus': seconds with two accents, thus": thirds with three accents, thus "" and so on. Then 3 feet, 2 primes, 10 seconds, 11 thirds, 5 fourths and 6 fifths are written

32′ 10′′ 11" 5""" 6""""

Instead of these marks, the the Roman numerals i, ii, iii, iv,,&c. are sometimes employed. These are generally substituted when the number of marks becomes large. Thus the above example may be written,

3 2 10 11 5iv 6 or 3 2′ 10′′′ 11′′ 5 6

ADDITION and SUBTRACTION of duodecimals may evidently be performed like addition and subtraction of other compound numbers, or like the same operations in decimals, except that we are 10 carry and borrow by 12, instead of by 10, from one denomination to another.

MULTIPLICATION of duodecimals may be likewise performed like Multiplication of decimals, observing the same rule in carrying.