MATHEMATICAL AND SURVEYING INSTRUMENTS. 67 
bisected by the vertical line passing through the objective. According to 
optical principles, the angle thus obtained and read off on the limb, is just 
half of the actual angle of separation, so that the limb, instead of a gradua- 
tion of 60°, is actually divided into 120 divisions, each one of which repre- 
sents a degree. 
Mayer of Géttingen, and after him, Borda, have improved the sextant by 
employing a full circle instead of the sextant. This forms the reflecting 
circle of Borda, and is represented on pil. 5, fig.39. Here B is the graduated 
circle, placed upon the stand, A, similar to that of the theodolite, and capable 
of receiving a correcting telescope. K is a movable alidade, frequently a 
full circle, as in the theodolite, and provided with a vernier, N, and a 
correcting and clamping arrangement, I. On the alidade is a telescope 
carrier, I'H, with a vertically movable telescope, G, and the objective, M, 
constructed as in the sextant. The dark glasses already mentioned are at 
O. The central-piece, C,is constructed as in the sextant, and carries the 
mirror, L, with the index, C, which has also a vernier, and a correcting and 
a clamp arrangement, D. KE is a lens for reading off the graduation. Pand 
Q are verniers for repeating. This reflecting circle, besides admitting the 
measurement of larger arcs than the sextant, has the advantage of being a 
repeating instrument. 
To make use of the reflecting circle, the alidade, K, is so adjusted that 
one of the two objects is visible through the upper part of the objective, and 
the alidade, C, moved until the mirror, L, reflects the second object into the 
lower half of M. The angle at C is read off, and doubled for the true 
result. The principle of repetition is here the same as in the case of the 
theodolite. 
It becomes necessary to add a word or two in explanation of the vernier 
(pl. 5, figs. 49, 50). The vernier, so called from its inventor, Peter 
Vernier (1600), is an arrangement for reading off small quantities on a 
scale, with great accuracy. Owing to the small size of mathematical 
instruments, the graduations upon them cannot be very minute, and it is 
rarely that quantities so small as { of a line, or 1 of a degree, can be indi- 
cated. A minuter division, so necessary, is attained by the use of the 
vernier. If a certain length, am or an, be supposed to be divided first into 
10 and then into 9 equal parts, one part of the first division will be 9, of a 
part of the second. If both divisions are placed one over the other, then, 
calling the first a, and the second 8, the first part of b will project 7; of its 
length beyond the first part of a, the second part 42,, &c., until the ninth and 
tenth will again coincide. Other numbers besides 9 and 10 may be employed, 
or an arc may be divided instead of the straight line. Suppose the limb of 
an instrument to be divided into quarter degrees, then each such part = 15 
minutes; take 14 of these parts and divide them into 15, then each new 
division will be 1+ of the old, and each old one ;;, or one minute, greater 
than the new. By taking the axis of the alidade as the centre or zero point, 
and describing the given graduation to the right and left of this centre, the 
vernier will be capable of indicating single minutes. ‘Suppose that in mea- 
suring an angle, the zero of the vernier has been found to be between 363° 
67 
