many from the left of the divisor, for a new divisor; and, in dividing by this new divisor, instead of annexing a figure to the remainder, omit one on the right hand of the divisor, but observe to take in the carriage arising from it, by the quotient figure multiplied by, as in Contracted Multiplication. Proceed in this manner, dropping a figure of the divisor, at each division, till it be exhausted.

EXAMPLE.

Divide 291.439765 by 27.65138725, retaining only 3 decimal places in the quotient.

27.651,38725)291.439765(10,539

27651

1492

1382

110

83

27

25

EXERCISES.

2. Divide 27457.55 by 32.1755, retaining only 4 decimal places in the quotient.

3. Divide 15.1275 by 9.813275, retaining only 2 decimal places in the quotient.

4. Divide 89.12543 by 12.34567, retaining only 4 decimal places in the quotient.

5. Divide 857.6543218 by 27.1234567, retaining only 5 decimal places in the quotient.

6. Divide 351.7 by 4125.6539725, retaining only 6 decimal places in the quotient.

It was observed, at page 65, that numbers were diminished when multiplied by proper fractions, but increased when divided by them. Hence multiplication, by fractions, has the same effect as division, by integers; and division, by fractions, as multiplication of integers; therefore, if any number be multiplied by, or .5, the result will be the same, as when that number is divided by 2. Every integer has a decimal corresponding to it, which may be used in a similar manner: this decimal has received the name of the reciprocal of the number, and may often be employed with advantage, instead of the number itself, both in performing multiplication and division. (See page 3, Art. 23.)

To find the reciprocal of any number, divide 1, with ciphers annexed to it, by that number, and the quotient (after there is no remainder) is the reciprocal.

EXAMPLE.

Required the reciprocal of 25.

25)100(.04 Reciprocal.

100

The product of any number, multiplied by .04, will be the same as the quotient of that number divided by 25.; and the quotient of any number, divided by .04, will be the same as the product of that number multiplied by 25.

EXAMPLE.

3875

.04

25)3875(155 = 155.00

25

137

125

125

125

When large numbers are to be multiplied, or divided by large numbers, and frequent use to be made of the same multiplier or divisor, it is preferable to use the reciprocal, instead of the number itself.

It is evident, from what has been exhibited of Decimals, that some are never complete, though extended to any number of places; others, that are finite, consist of so many places, that it would be extremely tedious and troublesome to apply them in calculations. In cases where the decimal extends to a great number of places, the three or four first places may be used and the others neglected, which will not materially affect the result, except the integer be very valuable; and, when this is the case, one or two more places may be retained But, for the purposes of business, three or four places are sufficiently exact.

Those Vulgar Fractions which have 2, 5, or any power* of these numbers, for their denominator, produce finite decimals, and the number of places is denoted by the exponent of the power. If the numerator be 1, the decimal, or reciprocal, of any power of 2, is the same power of 5; and the decimal, or reciprocal, of any power of 5, is the same power of 2: thus, the decimal for .25=52, and 15.04=2; this arises from the radix of the scale being 10, and the one of these figures being the reciprocal of the other, when the radix is divided by either of them: for=2, and '=5; these are the only figures that will divide 10, (the radix of the decimal scale) without a remainder, and no numbers except these, or their powers, will measure 1, with any number of ciphers annexed; therefore no other denominators can produce finite decimals.

2. The denominators 3 and 9, produce pure repeaters; and, when the denominator is 9, the decimal consists of the numerator repeating. (See pages 29 and 30).

3. When the denominator is a power of 2 or 5, multiplied by 3 or 9, the decimal is mixed repeater; and the number of finite places is denoted by the expouent of that power of 2 or 5, which is multiplied by 3 or 9. This will plainly appear, if the nume

* Numbers produced by the successive multiplication of any figure, into itself, are called powers of that number; and the small figure, usually placed over auy number, to show how many times it is multiplied into itself, is called the exponent of the power. (See page 3, Art. 25.)

rator be divided by the component parts of the denominator; for the first division, by 2 or 5, quotes a finite decimal, and the second, by 3 or 9, quotes a pure repeater, after the figures of the dividend, or first, quotient is exhausted. When the numerator is 1, the first quotient is either a power of 5 or 2; and, if the second divisor be 3, the remainder of the second division is the same as the result of that power, when the 3's are cast out.Thus, if the 3's be cast out 22=4, the result is 1; out of 23= S, the result is 2; out of 2-16, the result is 1; and, in general, the result of the even powers is 1, and of the odd powers 2: it is similar with the powers of 5 If the exponent of the power be even, and the division continued, the repeating figure will be 3; if odd, it will be 6. If the second divisor be 9, the repeating figure will be the same as the result of the first quotient, when the 9's are cast out.

4. All denominators, that are multiples of 2 or 5, and their powers, by any other numbers, except 3 and 9, produce mixed circulates; for, if the division be performed by the component parts of the denominator, the first divisor, being a power of 2 or 3, quotes a finite decimal, and the second quotes a circulate, after the figures of the first quotient are exhausted.

5. All denominators, of which 2, 5, and their powers, are not component parts, (except 3 and 9) quote pure circulates. The number of places in the circulate cannot exceed the denominator diminished by 1, but it often consists of much fewer places. If the denominator be any power of 3, the number of places is found, by dividing it by 9.

The form of all decimals depends upon the denominator of the vulgar fraction from which they are derived; and, therefore, the decimal remains the same, if the fraction be in its lowest terms, whatever the numerator be.

It would be a useful exercise for the student of this branch of Arithmetic, to find the reciprocal of very number, betwixt 1 and 30, dividing by the component parts of the number, when it can be done; for, by this means, he cannot fail to acquire a knowlege of the nature of decimals, especially if he keep in mind the foregoing observations.

The Arithmetic of Decimals is one of the most important branches of the science of numbers, being equally useful in philosophical and commercial calculations. To enable the student

to have a still clearer view of this subject, the following observations on Arithmetical Scales have been added*:

ARITHMETICAL SCALES.

To assist the mind in forming clear and accurate ideas of numbers, they have been arranged or formed into classes.

A certain number of units are conceived to form a class of the lowest kind, and an equal number of these classes, to form another of a higher kind; and superior ranks of classes are formed, in the same manner, as far as occasion requires.

This regular method renders our ideas of numbers exact, and our progress in their right application, in the various uses to which they are applied, comparatively easy.

The number of units, which constitutes a class of the lowest kind, is termed the radix, or root of the scale.

There does not seem to be any number naturally adapted to this purpose, to the exclusion of others. The number 10 has, however, been universally employed by all nations who have cultivated this science, and it is probably the most convenient for general use.

Additional figures might easily have been invented, so as to have carried the scale to 12 or 15, or any other number; but, by increasing the characters beyond certain limits, and applying to each of them an appropriate sign, we should fall into that very complexity which it is the object of a distribution of numbers, into limited periods, to avoid. On the other hand, though by employing fewer characters, we might render the operations

*It was thought superfluous to give examples of the application of decimals to the solution of questions in Proportion, as a sufficient variety will be found in the commercial part of this work.