It will here be necessary to complete the description of the remaining part of the apparatus. And, first, a little instrument which I denominate a subdividing sector pre sents itself to notice. From all that has hitherto been said, it must have been supposed, that the roller itself will point out, upon the limb of the instrument to be divided, spaces corresponding to others previously divided upon itself, as was done in setting off the 256 points : but, to obviate the difficulty of dividing the roller with sufficient exactness, recourse was had to this sector : which also serves the equally important purpose of reducing the bisectional points to the usual division of the circle. This sector is repre sented of half its dimensions by Fig. 5, Plate CCLXXXIII. It is formed of thin brass, and centered upon the axis at A, in contact with the upper surface of the roller t it is capable of being moved round by hand ; but, by its fric tion upon the axis, and its pressure upon the roller, it is sufficiently prevented from being disturbed by accident. An internal frame BB, to which the arc CC is attached, moves freely in the outer one, and by a spring D is push ed outwards, while the screw E, the point of which touches the frame B, confines the arc to its proper radius. The arc of this sector is of about four times greater radius than It is convenient in the formation of the table of real er rors, that they should be inserted in the order of the num bering of the degrees on their respective quadrants ; al though their computation necessarily took place in the order in which the examination was carried on, or ac cording to the arrangement in the table of apparent er rors. The first dot of the first quadrant having been as sumed to be in its true place, the first of the third quad rant will err by just half the difference found by the exa mination ; therefore these errors are alike in both tables. The real error of the first dot of the second quadrant comes out in the first exami.le ; that of the fourth was the roller, and upon it are divided the spaces which must be transferred to the instrument, as represented on a mag nified scale by Fig. 4. Now, the angle of one of the spaces of the circle will be measured by sixteen times its angu lar value upon the sectorial arc, or 22° 30'; but this does not represent any number of equal parts upon the instru ment, the subdivisions of which arc to he 5' each ; for 1° 24' -- is exactly 164, therefore so many divisions are 5 exactly equal to a mean space between the dots, the er rors of which have been tabulated. Let, therefore, the arc of the sector be divided into 16 spaces of 1° 20' each, and let a similar space at each end be subdivided into eight parts of 10' each, as in Fig. 4 ; we shall then have a scale which furnishes the means for making the true divisions, and an immediate examination at every bisectional point.
I have always divided the sector flom the engine, be cause that is the readiest method, and inferior to none in point of accuracy, where the radius is very short ; but, as it is more liable than any other to centrical error, the ad iustment of the arc by the screw E becomes necessary : by that adjustment, also, any undue run in the action of the roller may be reduced to an insensible quantity.
When the utmost degree of accuracy is required, I give the preference to dividing by lines, because they arc made with a less forcible effort than dots are ; and also because, if any small defect in the contexture of the metal causes the cutter to deviate, it will, after passing the defective part, proceed again in its proper course, and a partial crookedness in the line will be the only consequence; whereas a clot, under similar circumstances, would be al together displaced. But, on the other hand, where accu racy has been out of the question, and only neatness re quired, I have used dots ; and I have done so, because I know that when a dot and the wire which is to bisect it are in due proportion to each other, (the wire covering about two-thirds of the dot,) the nicest comparison possible may be obtained. It may be farther observed, that division by lines is complete in itself; whereas that by dots requires lines to distinguish their .value.
On the upper side of Fig. 1. is represented the appara tus for cutting the divisions. It consists of three pieces JKL, jointed together so as to give to the cutter an easy motion, for drawing lines directly radiating from the cen tre, but inflexible with respect to lateral pressure ; d d are its handles. The cutting point is hidden below the mi croscope H; it is of a conical form, and were it used as a dotting point, it would make a puncture of an elliptical shape, the longer diameter of which would point towards the centre. This beautiful contrivance, now well known, we owe to the ingenuity of the late Mr Hindley of York ; it was borrowed by Mr Ramsden,* and applied with the best effect to his dividing engine.
It might have been mentioned sooner, that in the in stance which I have selected as an example of my dividing, the operation took place when the season of the year, and the smoke of London, had reduced the day to scarcely six hours of effective light; and rather than confine nty la bours within such narrow limits, I determined to shut out the day-light altogether. Fig. 7. shows the construction
of the lanterns which I used. A very small wick gave sufficient light, when kept from diverging by a convex lens ; while the inclining nossel was directed down exact ly upon the part looked at, and the light, having also pass ed through a thin slice of ivory, was divested of all glare.
I enter into this description, because, I think, I never saw my work better, nor entirely to so much advantage, as in this instance ; owing, perhaps, to the surrounding dark ness allowing the pupil of the eye to keep itself more ex panded, than when indirect rays are suffered to enter it. The heat from a pair of these lanterns was very inconside rable, and chiefly conducted along with the smoke up the reclining chimney.
Previous to cutting the divisions, the parts now describ ed must be adjusted. The cutting apparatus must be plac ed with the dividing point exactly at the place where the first line is intended to be drawn, and clamped, so that the adjusting screw may be able to run it through a whole interval. The microscope II must be firmly fixed by its two pillars b. b to the main frame, with its micrometer head at zero ; and with its only wire in the line of the ra dius, bisecting the first of the 256 dots. And it should be observed, that the cutting frame and this must not vary respecting each other, during the time that the divisions are cut ; for any motion that took place in either would go undiminished to the account of error. The microscope I is also fastened to the main frame ; but it is only required to keep its position unvaried, while the divisions of the sector pass once under its notice ; for it must have its wires adjusted afresh to these divisions at every distinct course. The microscope I has two wires, crossing each other at an angle of about 40° ; and these are to be placed so as to make equal angles with the divisions of the sector, which are not dots but lines. The sectorial arc must also be ad justed to its proper radius by the screw E, Fig. 5.; i. e. while the main frame has been carried along the circle through a mean interval shewn by II, the sector must have moved through exactly 16-1ths of its divisions, as indicated by 14 Things being in this position ; after having given the parts time to settle, and having also sufficiently proved the permanence of the micrometer H and the cutting frame, with respect to each other, the first division may be trade; then, by means of the screw for slow motion, carry the ap paratus forward, until the next line upon the sector comes to the cross wires of I; you then cut another division, and thus proceed until the 16th division is cut, 20'; Now, the apparatus wants to be carried further, to the amount of pits of a division, before an interval is complete ; but at this last point no division is to be made ; we arc here only to compare the division on the sector with the cor responding dot upon the instrument. This interval, how ever, upon the circle, will not be exactly measured by th( corresponding line of the sector, which has been adjusted to the mean interval, for the situation of the dot I° 4 is too far back, as appears by the table of real errors, by divisions of the micrometer head. The range of the screw for slow motion must now be restored, the cross wires of II set back to divisions, and the sector moved back by hand, but not to the division 0, where it began before ; for, as it left off in the first interval at iths of a division, it has to go forwards 4th more, before it will arrive at the spot where the 17th division of the instrument 1° 25' is to be made, so that in this second course it must begin at ath short of O. Go through this interval as before, mak ing a division upon the circle at every one of the 16 great divisions of the sector ; and 11 should now reach the third dot, allowing for a tabular error of — 10.2 when the divi sion .t-ths of the sector reaches the cross wires ol I. It would be tedious to lead the reader through all the variety of the sector, which consists of eight courses ; and it may he sufficient to observe, that at the commencement of course, it must he put back to the same fraction of a division which terminated its former one ; and that the wire ol the micrometer II must always be set to the tabular error be longing to every dot, we end one interval and begin anodic'. The eight courses of the sector will have car ried us through Ad part of the circle, 11° 15', and during this time the roller will have proceeded through half a re volution ; for its close contact with the limb of the circle does not allow it to return with the sector when the latter is set back at every course. Having in this manner pro ceeded, from one interval to another, through the whole circle, the micrometer at last will be found with its wire at zero, on the dot from which it set out ; and the sector, with its 16th division, coinciding with the wires of its micro scope.