The first step towards sizing the roller, is to compute its diameter according to the measure of the circle, and to reduce it agreeably thereto, taking care to leave it a small matter too large. The second step is, after having brought the roller into its place in the plate BB, to make a mark upon the surface of the circle near the edge, and a similar one upon the roller, exactly opposite each other ; then carrying the apparatus forward with a steady band, until the roller has made sixteen revolutions. If now the mark upon the roller, by having over-reached the one upon the circle, shews it to be much too large, take it out of the frame, and reduce it by turning accordingly : when, by repeating this, it is found to be very near, it may be turned about .001 of an inch smaller on the lower edge, and so far -its preparation is completed. The third and last step is, the use and adaptation of the two micro scopes ; one of these must take its position at II in Fig. 1. viewing a small well-defined dot made for the pur pose on the circle ; the other, not represented in the Fi gure, must also be fixed to the main plate of Fig. 1. as near to the former as possible, but viewing one of the divi sions on the roller. With a due attention to each micro scope, it will now be seen to the greatest exactness, when, by raising or depressing the roller, its commensurate di ameter is found.
Fig. 3. is a representation of the apparatus for transfer ing the divisions of the roller to the circle. It consists of two slender bars, which being seen edgewise in the figure, have only the appearance of narrow lines ; but, when looked at from above, they resemble the form of the letter A. They are fastened to the main frame, as at \V and Z, by short pillars, having also the off leg of the angle secur ed in the same manner; Y is a fine conical steel point for making the dots, and X is a feeler, whereby the point Y may he pressed down with a uniform force, which force may be adjusted, by bending the end of the bar just above the point, so as to make the dots of the proper size. The point Y yields most readily to a perpendicular action ; but is amply secured against any eccentric or lateral deviation.
The apparatus, so far described, is complete for laying our foundation, i. e. making 256 primary dots ; no matter whether with perfect truth or not, as was said respecting the divisions of the roller ; precision in either is not to be expected, or wished ; but it is of some importance, that they should be all of the same size, concentric, small, and round. They should occupy a position very near the ex treme border of the circle, as well to give them the great est radius possible, as that there should be room for the stationary microscope and the other mechanism, which will be described hereafter.
It must be noticed, that there is a clamp and adjusting screw attached to the main plate of Fig. 1 ; but, as it dif fers in no respect from the usual contrivances for quick and slow motion, it has been judged unnecessary to incumber the drawing with it.
Now the roller having been adjusted with one micro scope H upon its proper t:ot on the circle, and the other microscope at the first division on the roller, place the ap paratus of Fig. 3 so that the dotting point Y may stand directly over the place which is designed for the beginning of the divisions. In this position of things, let the feeler X be pressed down, until its lower end comes into contact with the circle ; this will carry down the point, and make the first impression or primary dot upon the circle ; un clamp the apparatus, and carry it forwards by hand, until another division of the roller comes near the wire of the microscope ; then clamp it, and with the screw motion make the coincidence complete ; where again press upon the feeler for the second dot : proceed in this manner until the whole round is completed.
From these 256 erroneous diviSions, by a certain course of examination, and by computation, to ascertain their ab solute and individual errors, and to form these errors into convenient tables, is the next part of the process, and makes a very important branch of my method of dividing.
The apparatus must now be taken off, and the circle mounted in the same manner as it will be in the observatory. The two microscopes, which have divided heads, must also be firmly fixed to the support of the instrument, on oppo site sides, and their wires brought to bisect the first dot, and the one which should be 180° distant. Now, the micro scopes remaining fixed, turn the circle half round, or un til the first microscope coincides with the opposite dot ; and, if the other microscope be exactly at the other dot, itis obvious that these dots are 180° apart, or in the true diameter of the circle ; and if they disagree, it is obvious that half the quantity by which they disagree, as measured by the divisions of the micrometer head, is the error of the opposite division ; for the quantity measured is that by which the greater portion of the circle exceeds the less. It is convenient to note these errors, + or —, as the dots are found too forward or too backward, according to the numbering of the degrees ; and for the purpose of distin guishing the + and — errors, the heads, as mentioned be fore, arc numbered backwards and forwards to fifty. One of the microscopes remaining as before, remove the other to a position at right angles: and, considering for the pre sent both the former dots to be true, examine the other by them ; i. e. as before, try by the micrometer how many divi sions of the head the greater half of the semicircle exceeds the less, and note half the quantity 4- or —, as before, and do the same for the other semicircle. One of the micro meters must now be set at an angle of 45° with the other, and the half differences of the two parts of each of the four quadrants registered, with their respective signs. When the circle is a vertical one, as in the present in stance, it is much the best to proceed so far in the exami nation with it in that position, for fear of any general bend ing or spring of the figure ; but for the examination of smaller arcs than 45°, it will be perfectly safe, and more convenient, to have it horizontal ; because the divid ing apparatus will then carry the micrometers, several per forations being made in the plate 13 for the limb to be seen through at proper intervals. The micrometers must now be placed at a distance of 22° 30', and the half differences of the parts of all the arcs of 45° measured and noted as before : thus descending by bisections to 11° 15', 37' 30", and 2° 48' 45". Half this last quantity is too small to allow the micrometers to be brought near enough ; but it will have the desired effect, if they are placed at that quantity and its half, i. e. 4° 13' 7".5 ; in which case the examination, instead of being made at the next, will take place at the next division but one to that which is the subject of trial. During the whole of the time that the examination is made, all the dots, except the one under examination, are, for the present, supposed to be in their true places ; and the only thing in this most important part of the business, from first to last, is to ascertain with the utmost care, in divisions of the micrometer head, how much one of the parts of the interval under examination exceeds the other, and carefully to tabulate half of their difference.