68 MATHEMATICS. 
and 362°, or that the angle is greater than the former and less than the 
latter. To determine the differences, find what division of the vernier 
accurately coincides with a division of the limb; let this be the tenth on the 
right side. We know that each part of the limb is one minute greater than 
any part of the vernier, consequently, for 10 divisions of the limb, we must 
add 10 minutes; then, 36° 30’+10 = 36° 40’. If the divisions thus met, had 
been to the left of the centre, the number of minutes thus ascertained would 
have to be subtracted instead of added. The same reasoning applies in the 
case of the rectilineal vernier. 
It still remains to mention levelling instruments, used to determine the 
direction of a horizontal plane. Of these, the simplest is the common 
masons level ( pl. 5, fig. 56). It is known that every line hanging plumb 
must be perpendicular to a horizontal plane, and that a line from the vertex 
of an isosceles triangle to the middle of the base, will be perpendicular to 
the base. Upon these principles rests the determination of a horizontal 
position by the foot level. This is an isosceles triangle, abc, from whose 
vertex depends a plumb line, cf. If this be placed upon a board, and one 
end elevated or depressed until the plummet hangs opposite to the middle 
of the arc, de, the base of the triangle will be horizontal. This instrument 
can only be used for short distances—12 feet at the most—and for greater 
lengths, other means must be employed. The first to be mentioned is the 
water level ( fig. 40). This consists of a simple stand, A, upon which, by 
means of a socket, B, is placed a tin tube, CD, bent at right angles at both 
ends. In each end, two glass tubes, KE and F, are cemented water tight. 
If this water level be filled with water until it enters and partially fills the 
glass tubes, the upper surface of the water in both of these will stand in the 
same horizontal plane, independently of the position of CD. The horizontal 
plane thus indicated may be prolonged at pleasure, by sighting forwards at 
a given signal. An adjustable objective, IK, placed exactly over the middle 
of the instrument, is sometimes employed to furnish to the eye a more 
convenient point of reference, and to enable it to do with but one surface. 
The movable diopter ( fig. 41) is, however, better calculated for this purpose. 
B is the bent part of the level, E: the glass cylinder, cemented in the collar, 
a; the support, 0, is attached to this collar, and in it a rack, cd, may be 
moved up and down by means of the pinion, k. The diopter is seen at f; 
its aperture, g, is divided into two parts by the horizontal thread, Ai. At 
the commencement of the operation, the two diopters are brought to an 
equal height with the surface of the water, in which case sights may be 
taken over the threads, instead of over the surface of the water. In some 
cases the tubes inclose floats, supporting diopters, with a horizontal cross 
hair in each; these hairs must both be at precisely the same height above 
the water. 
The mercurial level of Keith (figs. 42-44) is much more complete than the 
water level. This consists of a wooden box, AB, in which is a canal, EF, 
filled with mercury to a certain height, and then covered tight. At the two 
ends of the canal are rectangular boxes, C and D, into which mercury 
flows from the canal. GH is a bottom, separating the upper part, IK, of 
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