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 3'15, 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.

1. 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. 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 § 3A similar remark applies to § 2.

* See frontispiece, fig. 3.

jaws will be given by the outsides vernier. The other pair of faces of these two jaws, opposite to the two plane parallel ones, are not plane, but cylindrical, the axes of the cylinders being also perpendicular to the length of the brass bar, so that the cross section through any point of the two jaws, when pushed up close together, will be of the shape of two U's placed opposite to each other, the total width of the two being exactly one inch. When they are in contact, it will be found that the arrowhead of the vernier attached to the scale marked insides reads exactly one inch, and if the jaws of the calipers be fitted inside an object to be measured-e.g., the internal dimensions of a box-the reading of the vernier marked insides gives the distance required.

Suppose it is required to measure the length of a cylinder with flat ends. The cylinder is placed with its axis parallel to the length of the calipers. The screw A (fig. 1) is then

is then made to bite, so that the attached piece is 'clamped ' to the scale. Another screw, B, on the under side of the scale, will, if now turned, cause a slow motion of the jaw C G, and by means of this the fit is made as accurate as possible. This is considered to be attained when the cylinder is just held firm. This screw B is called the 'tangent screw,' and the adjustment is known as the 'fine adjustment.'

It now remains to read upon the scale the length of the cylinder. On the piece c will be seen two short scalesthe 'outsides' and 'insides 'already spoken of. These short scales are called 'verniers.' Their use is to increase the

accuracy of the reading, and may be explained as follows: suppose that they did not exist, but that the only mark on the piece c was the arrowhead, this arrowhead would in all probability lie between two divisions on the large scale. The length of the cylinder would then be less than that corresponding to one division, but greater than that corresponding to the other. For example, let the scale be actually divided into inches, these again into tenths of an inch, and the tenths into five parts each; the small divisions will then be inch or 02 inch in length. Suppose that the arrowhead lies between 3 and 4 inches, between the third and fourth tenth beyond the 3, and between the first and second of the five small divisions, then the length of the cylinder is greater than 3++', i.e. >3.32 inches, but less than 3++, i.e. <3'34 inches. The vernier enables us to judge very accurately what fraction of one small division the distance between the arrowhead and the next lower division on the scale is. Observe that there are twenty divisions on the vernier,' and that on careful examination one of these divisions coincides more nearly than any other with a division on the large scale. Count which division of the vernier this is-say the thirteenth. Then, as we shall show, the distance between the arrowhead and the next lower division is of a small division, that is 13013 inch, and the length of the cylinder is therefore 3+1% +30 + ro36=3·32+013=3'333 inch.

=

We have now only to see why the number representing the division of the vernier coincident with the division of the scale gives in thousandths of an inch the distance between the arrowhead and the next lower division.

Turn the screw-head в till the arrowhead is as nearly coincident with a division on the large scale as you can make it. Now observe that the twentieth division on the vernier is coincident with another division on the large scale, and that the distance between this division and the first is nineteen small divisions. Observe also that no other Various forms of vernier are figured in the frontispiece.

divisions on the two scales are coincident. Both are evenly divided; hence it follows that twenty divisions of the vernier are equal to nineteen of the scale-that is, one division on the vernier is 8ths of a scale division, or that one division on the vernier is less than one on the scale by 2th of a scale division, and this is rooth of an inch.'

Now in measuring the cylinder we found that the thirteenth division of the vernier coincided with a scale division. Suppose the unknown distance between the arrowhead and next lower division is x. The arrowhead is marked o on the vernier. The division marked I will be nearer the next lower scale-division by Tooth of an inch, for a vernier division is less than a scale division by this amount. Hence the distance in inches between these two divisions, the one on the vernier and the other on the scale, will be

x-1000.

The distance between the thirteenth division of the vernier and the next lower scale division will similarly be

But these divisions are coincident, and the distance between them is therefore zero; that is x=88. 13 Hence the rule which we have already used.

The measurement of the cylinder should be repeated four times, and the arithmetic mean taken as the final value. The closeness of agreement of the results is of course a test of the accuracy of the measurements.

The calipers may also be used to find the diameter of the cylinder. Although we cannot here measure surfaces which are strictly speaking flat and parallel, still the portions of the surface which are touched by the jaws of the calipers are very nearly so, being small and at opposite ends of a diameter.

Put the calipers on two low supports, such as a pair of glass rods of the same diameter, and place the cylinder on end upon the table. Then slide it between the jaws of the

Generally, if n divisions of the vernier are equal to -I of the scale, then the vernier reads to 1/nth of a division of the scale.