The requisites for the performance of this work are as fol low scale of equal parts, by which the radius may be measured to 0.001 of an inch, must be provided. My scale is 90 inches long, each inch divided into 10, contiguous to which are nonius divisions, viz. 10.1 inches divided into 100 equal parts, slit wing 0.001 of an inch ; and, by the assist ance of a magnifying glass, of one inch focal length, a third of 0.001 may be taken off by estimation.
Provide five beam compasses, to which magnifying glasses, of not more than one inch focal length, should be applied. Let the longest beam be sufficient to draw the arcs, anti mea sure the radius ; the 2d, to measure the chord of 40'; the 3d, to measure the chord of ; the 4th, 20'; the 5th, 40'; and if a 6th, to measure 15°, be made use of, so much the better.
I have,for the sake of a round number, mentioned above, that the radius of the mural arc in the Royal Observatory at Greenwich is eight feet ; but, as I shall here put down the lengths of the several chords made use of in the dividing, it will be necessary to note the exact radius in inches and deci mal parts.
The radius of the arc of 90°, at the points, = 95.938 inches, from which the following numbers were computed, viz. 49,6615 inches = chord of — 25.0448 inches = chord of — 17.279047 inches = chord of 20' — 7.81186 chord of 40', and 69.80318 inches = chord of 40'. Having drawn the several arcs, between which the divisions were to be cut, the radius and the lengths of the above chords were taken by the beam compasses, which, together with the scale, were laid upon the quadrant, where they remained till the next morning, during which lime the door of the room was kept locked. Before sun-rise I remeasured the radius, which required some correction ; the beam being of white fir, and the scale of brass, which probably contracted, while the beam remained unaltered. The other beam compasses also required correction.
Now the quadrant and scale being of the same tempera ture, the faint arc b, d, was struck, and, with a very fine prick-punch, the point a was made. With the same beam compass, unaltered, I laid off, from a to e, the chord of making also a fine point. frith the chord of 30°, a e was bisected in L. Now, one point of the beam compass, contain ing 60°, was in c ; and with the other was marked the point r, or Next, with the beam compass, containing was bisected c r in n, or 75° From n was laid of the chord of 10° 20', and front r, 46' ; which two last chords joined exactly in g, being the point of 85° 2G'. Now each
degree being to be divided into 12 parts, or every 5', there fore 85 X 12 + 4 = 1024, a number divisible by continual bisections. The last chord computed was 40', with which a g was bisected in o; a o and o g were bisected by trials. But whoever undertakes to divide a large yvadrant will do well to compute also the chord of 20'; but for this chord any of the beam compasses already provided, which win take in the length, may be used. The point g being found as above, I proceeded, by continual bisections, till I had the number required, viz. 1024. To fill up the space between g and r, containing 56 divisions, the chord of 64 divisions was laid off from g towards d, and divided, like the rest, by con tinual bisections, as was also from a towards b. 77ie points 30°, 60°, 75°, and 90°, fell in without any sensible inequality.
Here it is necessary to mention in what manner the bi secting and pointing were performed. Having the chord of 42° 4t/ in the beam compass, and one point placed in a, with the other a faint arch of about 10 of an inch in length was made in o. Again, one point of the beam compass was placed in g, and with the other the aforesaid faint arch was intersected. Here, as in all other places, great care was used to make the points exactly in the arc to be divided, and also in the intersection.
In the following pages, I shall not only describe the far ther process in dividing the mural arc in the Royal Obser vatory, but endeavour also to make the description general.
lithe chord should be taken a little too long, or too short, so that the intersection be made on one side or the other of the arc to be divided, it will not occasion an inequality, provided the point be made in the middle between the two short lines, except at the point of 35° 20', where great care must be used in taking the chords from the scale. Great care must also be used in pointing intersections in general, being more difficult than a single line. But here I must not be understood to mean a single line made by one point of the compass ; for, in all bisections, the place to be pointed must be laid off from left to right, and from right to left ; and it any error arises from an alteration of the beam compass, it will be shewn double.