fore, when measuring from the index of the first scale, the logarithms of numbers from 10 to 100.

A certain length measured on the lower scale gives the logarithm of a number a, say. The same length measured along the upper scale is 2 log a, for the unit of measurement of the upper scale is half that of the lower, also

Thus, to find the square of a number, look out the number on the lower scale, and take the reading on the upper scale which coincides with that found on the lower.

In order to determine the coincidence, a metal slide, called the Cursor, is employed. This is equivalent to a straight-edge at right angles to the length of the scale which can slide along the scale, and thus facilitates the reading of the coincidences.

The rule can be used to find the area of a circle of given radius in the following way :-The area of a circle of radius is 2. The value of log (log 3*142) is marked on the slide. Set this to the index of the upper scale. Set the cursor to the value of r on the lower scale, and note the reading on the upper scale. This corresponds to log 2. Take the reading on the upper scale of the slide which coincides with this, and we obtain the value of = r2.

The cursor may be also used to obtain a continuous product without noting the intermediate steps in the following way-To multiply a, b, c together, read a on the rule; set the zero of the slide to this; set the cursor to on the slide. Move the slide until its index coincides with the cursor, and read on the slide. The corresponding division on the rule gives the value of the product.

The reverse side of the slider in the rule described contains three scales. One of these is a scale of sines, the

second a scale of tangents. These are so divided that when either of them is brought into coincidence with the corresponding scale on the rule, the divisions of the rule give respectively the sine or tangent of the angle read on the slider scale. The upper scale of the rule is used for sines, the lower for tangents. The third scale is one of equal parts, and from it the logarithm of a number can be determined. For set this scale so that its zero coincides with the index of the lower scale of the rule, and read any number, a, say, on this scale. Then, since the distances of the divisions from the index of the scale are proportional to the logarithms of the corresponding numbers, and the whole length of the scale contains 10 divisions, we have the ratio

log a log 10 = distance of a from end: whole

length of scale.

Set the cursor to division a, and take the corresponding reading on the scale of equal parts; let it be x divisions. Suppose that the whole length contains d divisions; then, since log 10

=

I,

log a = x/d.

In the rule already referred to d

=

500, so that

log a = 2x/1000.

This rule also contains a device whereby the logarithms, sines, and tangents may be read without reversing the slider. On the under side of the right-hand end of the scale there is a small opening, on each side of which an index mark is

seen.

When the index of the scale of equal parts, or of sines or tangents coincides with these index marks, it will be found that the scales on the upper side of the rule and slider are coincident.

Now draw out the slider, and note the reading a on the

lower scale with which its index coincides. Note also the reading x on the scale of equal parts.

This last reading gives us the distance the slider has moved that is, the distance between the index of the lower scale and the mark a ; but this distance is proportional to log a, and we have, as before,

log a log 10=x/d,

d being the number of divisions on the scale of equal parts which correspond with the full length of the logarithmic scale.

An exactly similar method applies to finding sines or tangents.

The accuracy obtainable with a slide rule depends partly on the exactness with which it is divided, partly on the possible accuracy of setting. Under favourable circumstances an accuracy of 1 part in 500 is claimed for the rule we have been describing, but this varies in different parts of the scale. Thus suppose we wish to use the rule to multiply 922 by, say, 8:53. There are no divisions between 9.2 and 925, and the actual distance between these divisions is about 75 mm. To set the slider to this so that the error in the result may be 1 part in 500, we have to estimate to about one-fifth of the distance between the marks, or say 15 mm. To do this requires considerable care and practice. Then, again, we have no mark on the slider between 8.5 and 8:55. We have to judge by eye the position of 8:53, and also the division on the scale which coincides with this.

The cursor is of help in this, and it is easy to see that the division required lies between 78.5 and 79. Dividing the distance between these divisions by eye with the aid of a magnifying glass, we get as the result 786. ., and the last figure will be certainly right to 1, which is about 1 in 800 in the result. As another example, suppose we wish to find the circumference of a circle 1752 inches in diameter. To

read the last figure correctly on the scale we have to subdivide to tenths a distance of about 5 mm. ; but an error of 2 in this figure, with a corresponding error in the value of (31416), will only affect the result to 1 part in 500. There are no divisions between 3'14 and 315, but the distance between these two can be subdivided into fifths, and we can set the cursor to 3'142, correct to '002.

The product lies between 5.50 and 5'55, and this distance, which is well over 1 mm. in length, can be subdivided to fifths with certainty. We obtain as the result 5.51, the true value being 5'504.

Or, again, find the angle whose sine is 8.

The divisions in the neighbourhood of 8 on the upper scale, which is used here, are about 75 mm., and we can set the scale with fair accuracy. The angle is seen to be between 53° and 54°. To get it more nearly we have to divide a distance of about a millimetre into parts. We can do this to fifths or sixths, giving an accuracy of, say, 10 minutes, or 1 in 300. For angles above 60° the degree divisions on the scale of sines are very small, while between 70° and 80° each division is 2°, and the divisions corresponding to 80° and 90° are only about 1 mm. apart. The value of sin a changes by about 1 per cent for 1° when a is about 60°, and the setting can be done to about one-fifth or one-sixth of a degree in this position. Thus it will be seen that with care the accuracy of nearly 1 in 500 is attainable over a wide range.

CHAPTER IV.

MEASUREMENT OF THE MORE SIMPLE QUANTITIES.

LENGTH MEASUREMENTS.

THE general principle which is made use of in measuring lengths is that of direct comparison (see p. 2); in other words, of laying a standard, divided into fractional parts, against the length to be measured, and reading off from the standard the number of such fractional parts as lie between the extremities of the length in question. Some of the more important methods of referring lengths to a standard, and of increasing the accuracy of readings, may be exemplified by an explanation of the mode of using the following instruments.

I. The Calipers.

This instrument consists of a straight rectangular bar of brass, D E (fig. 1), on which is engraved a finely-divided scale. From this bar two steel jaws project. These jaws are at right angles to the bar; the one, DF, is fixed, the other, CG, can slide along the bar, moving accurately parallel to itself. The faces of these jaws, which are opposite to each other, are planed flat and parallel, and can be brought into contact. On the sliding piece c will be observed two short scales called verniers, and when the two jaws are in contact, one end of each vernier, marked by an arrowhead in the figure, coincides with the end of the scale on the bar.' If then, in any other case, we determine the position of this end of the vernier with reference to the scale, we find the distance between these two flat faces, and hence the length of any object which fits exactly between the jaws.

It will be observed that the two verniers are marked 'outsides and 'insides' respectively.2 The distance between the

' If with the instrument employed this is found not to be the case, a correction must be made to the observed length, as described in § 3. A similar remark applies to § 2.

'See frontispiece, fig. 3.