The uppear arch AB, Fig. 9, represents a degree di vided into 12 equal parts, each containing five mi nutes ; and the under arch, CD, a 96th part of the quadrant divided into 16 equal parts ; and LT, the Nonius or subdividing plate fixed to the telescope, and sliding with it in the space between the arches AB and CD. Both these divisions are numbered from left to right, commencing from the intersection of the vertical radius, in order to measure the distances of objects from the zenith ; hut the parts on the Nonius, are number ed the Contrary way, beginning from the line 00, pro duced through the centre of the quadrant. In the fi gure the NoMus EF is so situated, that the upper end of the indt x 00 is not opposite to any one line upon the adjacent arch, but to some point of a 12th part of a de gree intercepted between 50 and 55 minutes. To find the excess above 50, it will be seen by looking back from the index, that a division line of the Nonius, which lies between the numbers 3 and 4, is exactly opposite to a division line upon the adjoining arch, which shows that 31, minutes is to be added to the 50 minutes. For since a degree is divided into 12 equal parts, each con taining 5 minutes, and since the length of the Nonius is made equal to 11 of these parts, and divided into 10 equal pat ts, it appears by the preceding theorem, in counting back again from the coincident division lines to the index, that the first part of the Nonius exceeds the first upon the limb by 1-10th of this latter part, that is by 1-10th of 5 minutes, which is half a minute, and consequently that seven parts of the Nonius, from the coincident division lines to the index, exceed the seven corresponding parts of the arch by seven half minutes or 3' 30".
When no one division line upon the limb is exactly opposite to a division line upon the Nonius, then we must look for that single part of the limb which is so opposed to a single part upon the Nonius, as to be ex ceeded by it at both ends, as shown in the parts G, H, Fig. 9. Then if that part of the Nonius appears to the eye to exceed the part of the limb equally at each end, 15" more must be allowed than if they had coincided at their ends next the index ; and according as the ex cess next the index is judged to be one-third, one-half, double, or ti cble, of the other excess, we must allow 71", 10", 20", 221", respectively. For as the sum of the two excesses is always the same, and is equal to 30", (as is obvious when one of them is diminished to no thing,) the number of seconds to be added will always be to 30", as the excess next the index is to the sum of the two excesses.
The lower arch of the Nonins is divided into tb equal parts, and is equal in length to 17 equal parts upon the opposite arch, and therefore will determine 16th parts of every one of them, by the theorem and method above mentioned. In Fig. 9, the opposite division lines of the Nonius and the lower arch are supposed to coin cide at the end of the 9th part upon the Nonius, which shows that the index cuts off 9-16ths of the opposite part of the arch. And so the length of the arch from the bt ginning of a 96th part of the quadrant, is thus de noted15.9 the lower pointer being past the 15th stroke.
This way of subdividing by a Nonius, is preferable to the common method of drawing diagonals, both be cause the trouble of drawing so many diagonals is en tire ly avoided, and also because they cannot be drawn so exactly by the edge of a ruler, as the lines upon the Nonius ; and lastly, because the intersection of these diagonals with the index ur fiducial edge, (as they call it,) by reason of the great obliquity to each other, can not be determined so exactly by the eye as the coinci dence of two division lines in the Nonius, and the arch, which stands directly opposite to each other.
The object-glass being firmly and permanently fixed in the telescope, the Nonius plate c d, and the collar plate s t, were both screwed fast to the telescope when taken off from the quadrant, and then the line of sight was brought to be parallel to the line c o, drawn through o the centre of the collar p q to c, the beginning of the divisions on the Nonius, in the following manner : The lines s o t and e c f being drawn upon these plates both perpendicular to o c, any distances o t and c f were taken equal to each other, on one side of o c, and any other distances o s and c e (long enough to go beyond the te lescope,) were also taken equal to each other on the op posi side of o c. Through the four points e,s,t,f, the ends of the two plates were filed exactly parallel to o c. Then placing the points t f upon two points nt 7; of an horizontal line, drawn upon a firm plane, a point of a remote object covered by the cross hairs, was mark ed. The telescope being then turned half round, its axis a b, and the opposite points e s of the plates being placed upon the same points in n, another point of a re mote object now covered by the cross hairs, was also marked ; and the telescope remaining fixed, the cross hairs were moved in its focus, till after several repeti tions of the operation, the same point of the object was covered by them in both positions of the telescope; and then the line of sight was exactly parallel to the line o e, supposing the object to be remote. But because smaller marks upon a nearer object are better discerned, the hairs were so adjusted, till in each position of the tele scope they covered a separate mark, the interval of the marks being taken equal to the difference of the heights of the axis of the telescope above the fixed line 771 , as near as could be measured.
The object glass being well centred, the line of sight was first of all made parallel to the plane of the quad rant, as near as it need be, by the measures of the brass work annexed to the telescope ; and then the plane described by the line of sight, turned about the centre of the quadrant, was brought into the plane of the meridian, by observing whether the fixed stars passed over the cross hairs at the same instant of time, as they passed over a meridian telescope, adjusted as above described, and placed so near the quadrant that the two observers would hear each other calling out at the times of the transits. And by the coincidence of these observations upon stars at various altitudes, it appeared that the plane of the quadrant was wrought very true. For it is certain that the meridian plane described by the meridian telescope as turning upon a transverse axis, must be truer than that described by the quadrantal telescope, as guided by the rollers upon the limb.