EXAMPLE 2.

Required the geometrical mean between 2.5 and

57.6.

57.6

2.5

2880

1152

144.00 (12 mean sought.

1

22)44

44

EXTRACTION OF THE CUBE ROOT.

Separate the number, whereof the cube root is required, into periods of three figures each, beginning at the units place and proceeding towards the left hand in integers, but at the decimal point, and proceeding towards the right hand in decimals, making up, if necessary, the last period, by the adjunction of ciphers. This process will determine the number of figures in the root, and their denomination.

For example, if the cube root of 1404-928 were demanded, the periods would be the three following:

1 | 404 | ·928

Hence the cube root of the number proposed will consist of three places, of which the first two will be integral, and the last decimal.

Having separated the given number into periods, we begin to extract the root by setting the greatest whole cube number contained in the first period, under that period, for subtraction; writing its root as the first figure of the root required.

After subtraction we annex to the remainder the second period, and the remainder thus augmented we term the Resolvend.

Then for a divisor we add to triple the part of the root found, thirty times the square of that part, taking for a dividend the tenth part of the resolvend. The quotient will in general be the next figure of the root, though sometimes too great by one or more units.

Lastly, for a subtrahend, we cube the last figure of the root yet found, and to that cube we add thirty times the square of the last figure, multiplied by the part before it of the root; together with three hundred times the square of that part, multiplied by the last figure. The sum of these three are to be subtracted from the resolvend, and the next period annexed to the remainder for a New Resolvend.

In this manner we go on till every period has been exhausted; after which, if there be a remainder, and. we wish to extend the root, new decimal periods of three ciphers each may be adjoined at pleasure.

As example surpasses precept, we shall illustrate

the Rule by two operations at length: and first let the cube root of 3581.577 be demanded.

The work will be as follows;

3581-577 (15-3 root required.

• The figure 7, is on trial, found to be too great; as is also 6, wherefore recourse is next had to 5, and this figure answers,

EXAMPLE 2.

The cube root of 29631-5 is required.

29 631.500 (30-9 &c. root sought.

It is evident that, by annexing decimal periods of three ciphers each, or supposing them annexed to the given number, the root might be extended at pleasure to any proposed degree of approximation.

As the above Rules and Examples contain every thing in decimals, and the extraction of roots, necessary to our purpose, we shall take our leave of this branch, and next introduce the learner to a valuable instrument called THE SLIDING RULE.

27

DESCRIPTION AND USE

OF THE

SLIDING RULE.

THIS INSTRUMENT is commonly made of box-wood, of various lengths, from six inches to two feet; but these are all in principle the same, and therefore we shall confine ourselves to a twelve-inch Rule with four slides, as being upon the whole one of the most convenient and useful.

The Lines on the First Face of the Rule.

The first line is marked with the letter A, and is called Gunter's line, being numbered from left to right with the figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, at distances in geometrical proportion. In this line are two brass pins, the first at 2150-42, and marked MB, signifying that there are 2150 42 cubic inches in a bushel of malt: the second pin is placed at 282, and marked AG, signifying that 282 cubic inches make an ale-gallon.

The second line on this face is the line on the slide, noted by the letter B, and differing nothing from the line A; for when unity at the end of the one is applied to unity at the end of the other, all the divisions and figures of the one correspond with those of the other; so that the one seems a duplicate of the other. On this line there is a brass pin at 231, the number of cubic inches in a wine-gallon, and marked WG.