the number is to be taken on the hither edge, and the root will in either case stand opposite on D.

On the second face of some Rules the line D slides between two lines, the one marked C, and the other marked E. In this case the line C is of double radius, the line D of single radius, and the line E of triple radius. The numbers on E are respectively the cubes of the numbers on D when unity on the one is set to unity on the other: hence if the beginning of the line on the slide D be set even with the beginning of the lines C and E, the whole forms a table of cubes, roots and squares; that is, on E are the cubes, on C the squares, and on D the roots.

Extraction of the Cube Root by the Lines D and E.

If unity at the beginning of D be set to unity at the beginning of E, the cube root of any number on E

may be found by inspection on D. Yet there is some difficulty in knowing on which of the three radiuses on E the given cube should be taken.

To obviate this difficulty observe the following directions:

DIRECTION 1.

If the given number consists of 1, 4, or 7 places of integers, find it on the first radius of E, and opposite to it will be its cube root on D.

DIRECTION 11.

If the given number consists of 2, 5, or 8 places of integers, look for it on the second radius of E, and you have the cube root against it on D.

DIRECTION III.

If the given number consists of 3, 6, or 9 places of integers, it must be found on the third radius of E, and directly against it on D, will, as in the other cases, be seen the root required.

or cube roots of any number be extracted with sufficient accuracy on the lines C, D, and E; but it is obvious that, to arrive at expedition and certainty, practice is necessary.

Reduction of a Vulgar to a Decimal Fraction, by the Sliding Rule.

Set 1 on B to the denominator on A, and opposite to the numerator on A will be the decimal required on B.

The reason of this will be apparent if we reflect on what has been said under the head of Division, and on the method of reducing a vulgar fraction to a decimal by the pen.

But it is evident that in reducing any proper fraction to a decimal, the quotient will not be found upon B, if 1 at the beginning of the line be set to the denominator on A: hence it is necessary to set 10 on B to the denominator on A, taking th part of the quotient for the decimal required.

In illustration let the following examples be proposed :

4

EXAMPLE 1.

Reduce to a decimal fraction.

Setting 10 on B to 4 on A, we may observe opposite to 3 on A, 7.5 on B; now 7.5 divided by 10 becomes 75 the decimal required.

EXAMPLE 2.

Reduce to a decimal fraction.

Setting 10 on B to 25 on A, and looking against 1 on A, we see '4 on B; but 4 divided by 10 becomes '04 the decimal required.

Here if we set 10 on B to 14 on A, we cannot find 9 on A, unless the Rule be of double radius. To obviate this inconvenience, let us multiply both numerator and denominator by 2, and the fraction will become 1. Then setting 10 on B to 28 on A, and looking opposite 18 on A, we have 6·43 on B.

18

28

Now the tenth part of 6.43 is 643 the decimal equivalent to

9

In like manner may any other vulgar fraction be reduced to its equivalent decimal by the Sliding Rule.

OF FACTORS.

When a divisor and dividend are given, it is sometimes convenient to perform the operation by means of a multiplier that shall by multiplication give the same result, that the given divisor would give by division. For

instance, if 525 were given to be divided by 25, the result will be the same whether 525 be divided by 25 or multiplied by the decimal corresponding to, which has been found in the second example above to be ⚫04.

A multiplier of this description is in Gaging technically called a FACTOR.

To find a FACTOR by the Lines A and B.

From the definition we have given of a FACTOR, it is manifest that the proportion will be, As the given divisor is to unity, so is unity to the factor sought. Wherefore to ascertain the fourth term of this proportion by the Sliding Rule, we must set the given divisor on A to unity on B, and opposite to unity on A will be the factor on B.

In treating of Division and Reduction of Vulgar to Decimal Fractions, enough has been said to preclude the necessity of further explanation of the particular cases that may occur, since if sufficient attention has been paid to the preceding examples, every difficulty will vanish. The exercises following will, however, serve for practice.

In solving this example we count 1 at the beginning of A for 100; then 125 are two of the large, and two