The third line is on the Rule and characterised by the letters MD, signifying malt-depth, for which reason it has the name of the malt line. This is also a line of numbers beginning at 2150·42, and proceeding from left to right in reverted order with the figures 2, 10, 9, 8, 7, 6, 5, 4, 3, ending at the right hand, as it began at the left, with the number 2150.42.

The Lines on the Second Face of the Rule.

The first line on this face is known by the letter D, being a line of numbers beginning with 1 and ending with 10. On this line are marked with brass pins the following gage points.

1. WG at 17 15 the circular gage point for wine gallons.

2. AG at 18.95 the circular gage point for ale gallons.

3. MS at 46.37 the square gage point for malt bushels.

4. MR at 52-32 the circular gage point for malt bushels.

The second line on this face is the line C on the slide, which is a line of numbers differing nothing from the lines A and B.

NOTE.-On all Rules on which the lines A, B, and C are of single radius, the line D is broken at 3.162, the square root of 10; the first part of the line, viz., from 1 to 3.162 being placed on one side of C, and the other part of it, that is, from 3.162 to 10, being situate on the opposite side of C.

Lines on the Third and Fourth Faces of the Rule.

On these faces are two lines of segments used in finding the ullage of a cask, that is, the quantity of liquor which a cask wants of being full, or the actual quantity of liquor in it when not quite full. These lines are both on the body of the Rule, and alike numbered from left to right with the figures 1, 2, 3, &c. to 10; and from 10 with the figures 20, 30, 40, &c. to 100. One of the lines is marked SL, signifying segment lying; and the other SS, for segment standing. On the first of these faces the line of segments is broken at 4, and on the second at 8. The slide on the third face is called B, and on the fourth C. They are similarly divided, and marked from left to right with the figures 1, 2, 3, 4, 5, 6, 7, 8, 9, at distances in geometrical proportion like the lines A and B.

Underneath the slides are,

1. The gage points, and divisors; as well as the necessary factors for the reduction of liquors of one sort to their equivalent value in another.

2. The malt divisor 2150-42, doubled, trebled, &c. to 9 times, for readily estimating the contents of a floor of malt.

3. A line of inches decimally divided, and numbered from 1 to 12.

4. A line marked Spheroid, being a line of differences for reducing casks of the first variety to a mean diameter; and numbered from 1 to 7.

5. A line numbered from 1 to 6, and marked 2nd

Variety, being a line of differences for reducing casks of that description to a mean diameter.

Besides a Rule of four slides, there are other Rules in use with one, two, or three slides; but these commonly contain only the more useful lines on a Rule of four slides.

Of Estimating the Divisions on the Sliding Rule.

The figures 1, 2, 3, 4, &c. on the lines A and B are all arbitrary, and may be conceived to represent either units, tens, hundreds, thousands, &c., or tenths, hundredths, thousandth-parts, &c. But whatever value is assigned to the first 1 towards the left, the following integral numbers 2, 3, 4, 5, &c. must be taken as twice, thrice, four times, five times, &c. as much: that is, the proportion must always be kept up. For example, if the figure 1 at the beginning of the line represent 10, then 2 will represent 20; or if the figure 1 be considered as, then the figure 2 must be considered as; and so on for all other values.

The value of the integral divisions being thus estimated, the intermediate divisions will be easily understood; for it is evident that, whatever value is assigned to the difference of any two integral adjoining numbers, each of the sub-divisions between them must be the quotient arising from that difference divided by the number of parts between the two numbers. If, for example, 1 at the beginning of the line A, stand for 1, every subdivision must stand for, since the distance

between 1 and 2 is on the scale divided into 50 parts; or if I at the beginning of the line be assumed as 100, then each subdivision will count for two units, two being the fiftieth part of 100, that is, of the difference between 1 and 2. Lastly, if 1 be supposed to express

ten, each of the

longer subdivisions will express 1,

and each of the others will count for

Enough, it is presumed, has been said to enable the learner to find, on any of the lines, the point corresponding to a given number. For the sake of practice, then, let it be proposed to find on the line D (where the divisions between 1 and 2 are all decimal) the point answering to 18.95.

If 1 at the beginning of the scale stand for 10, then it is manifest that 18.95 will be found between 1 and 2, that is, between 10 and 20. For the first figure of the given number, therefore, count 1 at the beginning of the line, and for 8, the second figure, count 8 of the larger divisions: again for 9 tenths count nine of the intermediate or smaller divisions; and lastly, for

take the half of the next small division. At this point is fixed a brass pin marked AG, being the circular gage point for ale-gallons.

If it were required to find 189.5, 1.895, 1895, and 1895, they would all appear at the same point; only in the first instance the value of 1 at the beginning of the line, would be 100; in the second it would be 1; the third, 1; and in the last, 1000.

In like manner is the point that represents any other number, found upon D, or any of the other lines.

Multiplication by the Lines A and B.

The product of two given numbers may be found by those two lines on the Sliding Rule, as follows: To either of the given numbers on A set 1 on B, and against the other number on B is the product on A.

be sufficiently illustrated by two examples.

EXAMPLE 1.

Let the product of 7 by 9 be required.

This will

Set 1 on B to 7 on A, and opposite to 9 on B you have 63 on A: or set 1 on B to 9 on A, and against 7 on B you again have 63 on A.

Here it may be proper to observe that it frequently happens that, when 1 on B is set to the given number on A, the other number cannot, according to the true numeration of the Rule, be expressed on the line B; or, if it can be expressed, it falls beyond the line A.

In such event it is most convenient, after setting 1 on B, to either of the given numbers on A, to take some decimal quotient of the other number; such as one tenth, one hundredth, one thousandth-part, &c., till such quotient can be found opposite one of the divisions on A. Then the product that arises must be taken 10, 100, 1000, &c. times, according as the given number was divided.

The next example will render this plainer.