was so minute that the whole Bible might be written twenty-seven times in a square inch. Finally, it is claimed that Mr. Whitworth was able to detect differences of one millionth of an inch with a micrometer screw he has constructed.

To measure very minute distances a microscope is often used with a scale inserted in its eyepiece, which is used like a common rule. The absolute size of the divisions must be determined beforehand by measuring with it a standard millimetre, or hundredth of an inch. A more accurate method, however, is the spider-line micrometer, in which a fine thread is moved across the field of view by a micrometer screw, and small distances thus measured with the greatest precision. By using two of these instruments, which are then called reading microscopes, larger distances may be measured, or standards of length compared, as in Experiment 20.

Small distances are also sometimes measured by a lever, with one arın much longer than the other, so that a slight motion of the latter is shown on a greatly magnified scale. Instead of a long arm it is better to use a mirror, and view in it the image of a scale by a telescope. An exceedingly small deviation is thus readily perceptible, and this arrangement, sometimes known as Saxton's pyrometer, has been applied to a great variety of uses. Where we wish to bring the lever always into the same position a level may be substituted for the mirror, forming the instrument called the contact level. Small distances are also sometimes measured by a wedge with very slight taper, but this plan is objectionable on many accounts. In geodesy all the measurements are dependent on the accurate determination in the first place of a distance of five or ten miles, called a base line. Most of the above devices have been tried on such lines; thus the reading microscope was used by Colby in the Irish survey, the wedge in Hanover, and by Bessel in Prussia, the lever by Struvé in Russia, and the contact level is now in use on our Coast Survey. The principle in all is to use two long bars alternately, which are either brought in contact, or the distance between their ends measured each time they are laid down.

Many other physical constants are really determined by a measure of length. Thus temperatures are determined by a scale of equal parts in the thermometer, and here sufficient accuracy is ob

tained by reading with the unaided eye. Pressures of air and water are also measured by the height of a column of mercury or water. Where great accuracy is required, as in the barometer, a vernier is commonly used.

The instrument known as the cathetometer is so much used for measuring heights that it needs a notice here. It consists of a small telescope, capable of sliding up and down a vertical rod to which a scale is attached. The difference in height of any two objects is readily obtained by bringing the telescope first on a level with one, and then with the other, and taking the difference in the readings. A level should be attached to the telescope to keep it always horizontal, but the great objection to the instrument is that a very slight deviation in its position, which may be caused by focussing or turning it, is greatly magnified in a distant object. A good substitute for this instrument may be made by attaching a common telescope to a vertical brass tube, the scale being placed near the object to be measured instead of on the tube, as in Experiment 12.

Although the measurement of the following quantities is directly dependent on the above, yet their importance justifies a separate

notice.

Measurement of Areas. It is difficult in general to determine an area with accuracy, especially where it forms the boundary of a curved surface. If plane, any of the methods of mensuration used in surveying may be adopted. Of these the best are division into triangles, Simpson's rule, and drawing the figure on rectangular paper and counting the number of enclosed squares, allowing for the fractions. Another method sometimes useful is to cut the figure out of sheet lead, tin foil, or even card board, and compare its weight with that of a square decimetre of the same material.

Measurement of Volumes. These are generally determined by the weight of an equal bulk of water or mercury, using the latter if the space is small. The interior capacity of a vessel is measured by weighing it first when empty, and then when filled with the liquid, as in Experiment 19. The difference in grammes gives the volume in cubic centimetres when water is used, but with mercury we must divide by 13.6, its specific gravity. In the same way we may determine the exterior volume of any body by

immersing it and measuring its loss of weight, as when determining its specific gravity.

An easier, but less accurate, method is by a graduated vessel. These are made by adding equal weights or volumes of liquid, successively, and marking the height to which it rises after each addition. The volume of any space may then be found by filling it with water, emptying it into the graduated vessel and reading the scale attached to the side of the latter.

Measurement of Angles. Angles are measured by a circle divided into equal parts, the small divisions being determined by verniers or reading microscopes, as in measuring lengths. A great difficulty arises from the centre of the graduation not coinciding with that of the circle, and on this account it is best to have two or more at equal intervals around the circumference. By taking their mean we eliminate the eccentricity.

The precision of modern astronomy is almost entirely due to the methods of determining angles with accuracy. This is dependent on two things; first, a good graduated circle, and secondly, a means of pointing a telescope in a given direction, as towards a star, with great exactness. The latter is accomplished by placing cross-hairs at the common focus of the object glass and eye-piece, so that they may be distinctly seen in the centre of the field at the same time as the object. Most commonly two cross-hairs are used at right angles, one being horizontal, the other vertical. When, however, we are to bring them to coincide with a straight line, as in the spectroscope, or in a reading microscope, they are sometimes inclined at an angle of about 60°, that is, each making an angle of 30° with the line to be observed. The latter is then brought to the point of the V formed by their intersection. Still another method is to use two parallel lines very near together, the line to be observed being brought midway between them. The lines may be made of the thread of a spider, of filaments of silk, of platinum wire, or better for most purposes, by ruling fine lines on a plate of thin glass with a diamond, and inserting it at the focus.

There are two methods of graduating circles with accuracy. The first, which is used in Germany, consists in a direct comparison with an accurately divided circle, as when copying scales as

described above. That is, both circles are mounted on the same axis, and the divisions of the first being successively brought under the cross-hairs of a microscope, the graver cuts lines on the second at precisely the same angular intervals. In the second method, which is much quicker but less accurate, the circle is laid on a toothed wheel which is turned through equal intervals by a tangent screw. Both methods are really only means of copying an originally divided circle, as it is called, and the construction of this with accuracy is a matter of extreme difficulty. It is dependent on the following principles. Any arc or distance may be accurately bisected by beam compasses; the chord of 60° equals the radius, and the angle 85° 20′, whose chord is 1.3554, by ten bisections is reduced to 5'. By constructing an accurate scale, laying off 1.3554 times the radius on the circumference, and repeatedly bisecting the arc, we finally divide the circle into 5' divisions. Where great accuracy is not required we may divide circles approximately by hand, as described under the Graphical Method, or more accurately by a table of chords and a pair of beam compasses. When the divisions of the circle are very large we may subdivide them by

2

3

60 30 0
C D

Fig. 4.

4

a scale instead of a vernier. Thus if B AB, Fig. 4, is part of a circle divided into degrees, we may attach a scale CD, divided to ten minutes, and subdivide these into single minutes by the eye. Thus in Fig. 4 the reading is 2° 35'. Much labor is thus saved where the circles have to be divided by hand.

Saxton's pyrometer, described above, is of the utmost value in measuring small angular changes. As the reflected beam moves twice as fast as the mirror, the accuracy is doubled on this account. If the scale is flat, allowance must be made for the greater distance of its ends than the centre. To reduce the reading to degrees and minutes, the formula, tan 2a sd is used, or a .5 tan-1 sd, in which a is the angle through which the mirror turns, s the reading, and d the distance of the scale taken in the same units. Instead of a telescope a light shining through a narrow slit is sometimes used, and an image projected on the scale by a lens, or the mirror itself may be made concave. This plan is adopted

in the Thomson's Galvanometer, and other instruments for measuring the deviations of the magnetic needle.

Very small angles may also be measured by a spider line micrometer attached to the eye-piece of a telescope. This is used to determine the distance apart of the double stars, and other minute astronomical magnitudes. There are other methods, such as divided lenses, double image prisms, &c., but they will be considered in connection with the particular experiments which serve to illustrate them.

Measurement of Curvature. To measure the radius of a sphere, as the surface of a lens, an instrument called the spherometer is used. It consists of a micrometer screw at the centre of a tripod, whose three legs and central point are brought in contact with the surface. By noting the position of the screw, the radius is readily computed, as in Experiment 14.

When the surface is of glass, and the curvature very slight, a much more delicate method is as follows: Focus a telescope on a distant object, and then view the image reflected in the surface to be tested. If the latter is concave, it will render the ray less divergent, and hence the eye-piece will have to be pushed in. The opposite effect is produced by a convex mirror. The amount of change affords a rough measure of the curvature. This method is so delicate as to show a curvature whose radius is several miles.