Having now given a faithful detail of every part of the process of dividing this circle, I wish to remind the read er, that, by verification and correction at every interval, any erroneous action of the roller is prevented from extending its influence to any distant interval. It will be farther ob served, that the subdividing sector magnifies the work : that by means of its adjustable arc, it makes the run of the roller measure its corresponding intervals upon the circle ; and, without foreign aid, furnishes the means of reducing the bisectional intervals to the usual division of the circle. Farthermore, the motion of the wire of the micrometer H, according to the division of its head and corresponding table of errors, furnishes the means of prosecuting the work with nearly the same certainty of success, as could have happened, had the 256 points been (which in practice is quite impossible) in their true places.
Now, the whole of my method of dividing being per formed by taking short measures with instruments which cannot themselves err in any sensible degree, and, inas much as those measures are taken, not by the hand, but by vision, and the whole performed by only looking at the work, the eye must be charged with all the errors that are committed until we come to cut the divisions ; and, as in this last operation the hand has no more to do than to guide an apparatus so perfect in itself, that it cannot be easily made to deviate from its proper course, I would wish to distinguish it from the other methods, by denominating it, dividing by the eye.* The number of persons at all capable of dividing origin ally have hitherto been very few : the practice of it being so limited, that in less than twice seven years, a man could hardly hope to become a workman in this most difficult art. How far I shall be considered as having surmounted these difficulties, I know not ; but if, by the method here reveal ed, I have not rendered original dividing almost equally easy with what copying was before, I have spent much la bour, time, and thought, in vain. I have no doubt, indeed, that any careful workman who can divide in common, and has the ability to construct an astronomical instrument, will, by following the steps here marked out, be able to di vide it, the first time he tries, better than the most expe rienced workman, by any former method.
If, instead of subdividing with the roller, the same thing be performed with the screw, it will not give to dividing by the eye any very distinctive character : I have practised this on arcs of circles with success, the edge being slightly racked, the screw carrying forward an index with the re quisite apparatus, and having a divided micrometer head ; the latter answers to the subdividing sector, and being used with a corresponding table of errors, forms the means of correcting the primitive points ; but the roller furnishes a more delicate action, and is by far more satisfactory and ex peditious.
It is known to many, that the six feet circle, which I am now at work upon for our Royal Observatory, is to be di vided upon a broad edge, or upon a surface at right angles to the usual plane of division: the only alterations which will on this account be required, arc, that the roller must act upon that plane which is usually divided upon ; which roller, being elevated or depressed, may be adjusted to the commensurate radius without being made conical, as was necessary in the other case. The apparatus, similar to the other, must here be fixed immoveably to the frame which supports the circle : its position must be at the vertex, where also I must have my station: and the instrument it self must be turned around its axis, in its proper vertical position, as the work proceeds. The above may suffice, for the present, to gratify those who feel themselves inter ested upon a subject, which will be better understood, if I should hereafter have the honour of laying before the Roy al Society a particular description of the instrument here alluded to ; a task which I mean to undertake, when, after being fixed in the place designed for it, which I hope will be effected at no very distant period, it shall be found com pletely to answer the purposes intended. See CIRCLE.
Should it be required to divide a circle according to the centesimal division of the quadrant, as now recommended and used in France, we shall have no difficulty. The 100° of the quadrant may be conveniently subdivided into 10 each, making 4000 divisions in the whole round. The 256 bisectional intervals, the two tables of errors, and the man ner of proceeding and acting upon them, will be exactly the same as before, until we come to cut the divisions ; and for this purpose we must have another line divided upon the sector. For part of the circle being equal to ° 2 4 " 5 22 5'.4 of the usual angular measure = 151 5 sions ; and just so many will be equivalent to one of the in tervals or the circle. The value of one of the great divi sions of the sector will be 1° 26' 24", and that of the nth parts, which are to be annexed to the right and left as be fore, will be lb' 48", therefore divisible by the engine. Should any astronomer choose to have both graduations upon his instrument, the additional cost will be a mere trifle, provided both were done at the same time.