171. The Clamp and Tangent Screws. The former of these are used for binding parts of the instrument firmly together, the latter for giving a slow motion when they are so bound. The clamp C tightens the collar O clasping the vertical axis, and thus holds it and the plate attached to it firmly in their places. The other plate, moving on an axis within the former, may, notwithstanding, move freely. When this clamp is tightened, the collar may be moved slowly round by means of the tangent screw T. In its motion it carries with it the axis and attached plate. The clamp D fastens the two plates together. They may, however, when so clamped, be made to move slightly on each other by means of the tangent screw U. If both clamps are tight, the instrument is firm, and the telescope can only be turned horizontally by one of the tangent screws. the clamp C is tight and the other loose, the telescope and upper plate will move while the lower remains fixed. If D is tight and C is loose, the two plates are firmly attached to each other; but the whole instrument can be moved horizontally.

If

Attached to the horizontal axis there is likewise a clamp H and tangent screw I, the purposes of which are similar to those described, the clamp fixing the axis, and the screw moving it slowly and steadily.

172. The Watch-Telescope. Connected with the lower part of theodolites of the larger class there is a second telescope R, the object of which is to determine whether the instrument has changed position during an observation. It is directed to some well defined object, and after all the observations at the station have been made, or more frequently if thought necessary, it should be examined to see whether or not it has changed its position. If it has, the divided arc has changed also. The instrument, therefore, requires readjustment.

173. Verniers. As it would be very difficult to divide a circle to the degree of minuteness to which it is desirable to read the angles, or, if it were so divided, since it would

be impossible for the eye to detect the divisions, some contrivance is necessary to avoid both difficulties. These difficulties will, perhaps, be made more striking by a simple calculation. The circumference of a circle 6 inches in diameter is 18.849 inches. If the circle is divided into 360 19.1 divisions in the space 18.849

degrees there will be

of an inch. If the divisions are quarter degrees there will be 76.4 to the inch; and if minutes, there would be 1150 divisions to every inch. The first and second could be. read; but the third, though it might by proper mechanical contrivances be made, yet it would be almost, if not entirely, impossible to distinguish the cuts so as to read the proper arc. And yet that division is not so minute as is sometimes desirable on a circle of that diameter. The vernier is a simple contrivance to effect this subdivision of space, in a way to be perfectly distinct and easily read.

174. The principle of the vernier will be best understood by a simple example. In the adjoining figure, (Fig. 56,) AB represents a scale with the inch divided into tenths, the figure being on a scale of 3 to 2 or 1 times the natural size.

CD is another scale having a space equal nine of the divisions on AB divided into ten equal parts. This second scale is the vernier. Now, since ten spaces of the vernier are equal to nine of the scale, each of the former is equal to nine tenths of one of the latter. If then the 0 on the vernier corresponds to one of the divisions of the scale, the first division of the vernier will fall of a space or of an inch below the next mark on the scale, the next division

will fall of an inch below, the next 18, and so on. The 0 in the figure stands at 28.7 inches.

If now the vernier be slid up so that the first division shall correspond to a division on the scale, the 0 will have been raised inch. If the second be made to coincide, the vernier will have been raised of an inch. If it be placed as in Fig. 57, the reading will be 28.74 inches.

Fig. 57.

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The student should make for himself paper scales, divided variously, with verniers on other pieces of paper, so that he may become familiar with the manner of reading. them. If his scale is to represent degrees, the portion representing the arc might be drawn as a straight line, for the sake of facility in the drawing. It will illustrate the subject as well as if an arc of a circle were used. He should become particularly familiar with the one represented by Fig. 60, as it is the division most commonly used in theodolites and transits.

175. The Reading of the Vernier. To determine the reading of the vernier,—that is, the denomination of the parts into which it divides the spaces on the scale,―observe how many of the spaces on the scale are equal to a number on the vernier which is greater or less by one. The number of spaces on the vernier, so determined, divided into the value of one of the spaces on the scale, will give the denomination. required. Thus, in Figs. 56 and 57, ten spaces of the ver nier correspond with nine on the scale: the reading is therefore toof of an inch. = ro

If an arc were divided into half-degrees, and thirty spaces on the vernier were equal to twenty nine or to thirty one

spaces on the arc, the reading would be to of 1° = ¿° = 1 minute; or, as it is usually expressed, to minutes. Fig. 60 is an example of this division.

176. To read any Vernier. First, determine as above. the reading. Then examine the zero point of the vernier. If it coincides with any division of the scale as in Fig. 56, that division gives the true reading,-28.7 inches. But if, as will generally be the case, it does not so coincide, note the division of the scale next preceding the place of the zero, and then look along the vernier until a division thereof is found which is in the same straight line as some division on the scale. This division of the vernier gives the number of parts to be added to the quantity first taken out. Thus, in Fig. 57, the 0 of the vernier is between 8.7 and 8.8, and the fourth division on the vernier is in a line with a division on the scale: the true reading is therefore 28.74 inches.

To assist the eye in determining the coincidence of the lines, a magnifying glass, or sometimes a compound microscope, is employed.

When no line is found exactly to coincide, then there will be some which will appear equally distant on opposite sides. In such cases, take the middle one.

177. Retrograde Verniers. Most verniers to modern instruments are made as above described. In some instances, the vernier is made to correspond to a number of spaces on the arc one greater than that into which it is divided. Such verniers require to be read backwards, and are hence called retrograde verniers. Fig. 58 is an example of one of this kind. It is the form that is generally used in barometers. It is drawn to one and a half times the natural size: the inches are divided into tenths, and eleven spaces on the scale correspond with ten on the vernier.

inch. If

The value of one division of the vernier is therefore 0 on the vernier corresponds to a division on the scale, 1 on the vernier will be of an inch below the next on the scale, 2 will be below; and so on. If the vernier is raised so that the 1 on the vernier is in line, it is raised inch; if 2 is in line, it is raised; and so on. The reading in Fig. 58 is 29.7 inches, and in Fig. 59, 29.53 inches.

Fig. 59.

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178. In Fig. 60, the arc is divided by the longer lines into degrees, and by the shorter into half degrees, or 30′ spaces.