GRADUATION. 



We might Tiare qnotea feveral other parts of Mr. Trough- 

 ton's paper for the advantage of the reader, but inttead 

 thereof, we recommend tlie perufal of the whole to fuch 

 perfons as are interefted in this fubjed, and (hall conclude 

 our account of the prefent method of dividing by remark- 

 in<T, that it requires not more than one-fourth of the time 

 that Bird's method does, and not much more than one- 

 twelfth that Ramfden's demands, and is applicable to a 

 quadrant or fextant, &c. ; that it may be performed by 

 night, by the aid of an appropriate lamp, as well as by day ; 

 that the work is, as it were, magnified by the feftor ; that 

 any erroneous aftion of the roller, when verified at every 

 interval, docs not extend its influence to any difiaut divilions; 

 that the divifions of the feftor very conveniently convert the 

 256 points into degrees and parts of a degree, while the 

 micrometer head allows for their errors of pofition ; that all 

 tools liable to vary their dimenfions by change of tem- 

 perature are here abandoned ; and laftly, that vijion alone is 

 employed in afcertaining the meafures of the arcs divided ; 

 on which account this method is called dividing by the eye ; 

 which appellation, indeed, might have been given to the due 

 de Chaulne's method ; a metliod which, we underftand, Mr. 

 Troughton was not acquainted with till lately, and from 

 which it will be perceived his differs very effentially. We 

 have no hefitation in aflerting, that this far excels every 

 otUer method of graduating large circles that we are ac- 

 quainted with. 



Soon after Mr. Troughton's paper was read at the Royal 

 Society's room, the late Henry Cavendifli, efq. F. R. S., 

 ivhofe death is an irreparable lofs to the fcientific world, 

 contrived a new method of ufing the beam-compafs, with a 

 view of avoiding the difficulties of jjointing the exaft middle 

 ofafmall fpace between two fcratches, and of ufing that 

 point again without altering its conical figure. As thefe 

 difficnlties had led to Mr. Troughton's new contrivances, 

 Mr. Cavendifh prefumed that a removal of them would ren- 

 der the beam-compafs unobjeftionable; and that future 

 dividers might continue the ufe of an inftrument which long 

 habit has rendered famihar. The method propofed, but 

 which probably has not been adopted, if tried at all, is, to 

 imite a microfcope that has got a micrometer, with the 

 beam-compafs, in fuch a way, that no dots may be necefl'ary 

 at all in dividing or fub-dividing any arc of a circle, but that 

 bifedion, trifeftion, and quinqnefeftlon, may aU be per- 

 formed by "vifion before the dividing llrokes are made on the 

 circle to be divided; we fay circle, becaufe this method is 

 not intended to apply to a quadrant, or any otlier aliquot 

 part of a circle. We will endeavour to explain the method 

 now before us as concifely as poffible. Let C Cl^,'m fg. ^. 

 of Plate XVIII. of JJlronomtcal Jnjlruments, be the circle to 

 be divided, in which we will retain the original letters of 

 reference, and B B B a frame rcffing on its plane fo ftea- 

 dily as to be without fhake, but notwithftanding to have 

 tlijL- power of Aiding fmoothly round the circumference to 

 any given point, to which it may be adjuited by a flow 

 motion, and there clamped ; let d J be a beam-compafs, 

 having its cutting point adjullable near «, and let m be a 

 microfcope with two parallel hairs, one fixed, and the other 

 moveable by the micrometer, made fo as to flide :Jong the 

 beam of the compafs into any required fituation. Let a 

 point d on the frame be fo cholen, that the line d m i may 

 lie in the direftion of the chord of half the arc to be bifect- 

 ed, when bifeftion is ufed, in which cafe the whole length 

 from d to i mull be very nearly equal to the whole chord of 

 ihe faid arc, and then both the centre of the field of view 

 of m, and alfo the point J, will fall in the circle to be 

 divided. It is not faid how the fixed or central point d is 



to be fupported, and kept perfeftly i?^eady, but as its dif- 

 tance from the centre of the circle jnuft necefiarily depend 

 on the length of the chord d S, it is propofed that the piece 

 of metal that conftitutes the fupport fhould be adjullable 

 by a Hiding motion of fome fort, and that the beam near J 

 fliould rell on a prop to guard the cutting point, when de- 

 mitted low enough to make a ftroke. Suppofe now F and 

 f to be the extreme points of an arc to be bifefted fomewhere 

 near <p ; after having placed the microfcope at or near the 

 middle of the beam, with refpetl to the two points (/and i', 

 and fixed the point on the relling place of d, flide the frame 

 till the fixed hair in the center, which mud be exaftly per- 

 pendicular to the line d in ,5, interfetls the circle at F; then, 

 lowering the point S from its prop, make a faint fcratch; 

 in the next place turn the beam-compafs a httle raifed at 

 the end S, round its centre d, till it is found over the circle, 

 at the other fide at D, where it mull rell on a prop for a 

 time; flide now the frame forwards, and adjufl till the fixed 

 hair cuts the point f, and there fix it ; the fecond fcratch 

 mull now be made, which will be over, or fliort of the firlt, 

 accordingly as the microfcope has been beyond or fiwrt of 

 the middle of the dillance d ^. It is not propofed to make 

 a dot between thefe two llrokes, but to bifeCl the fpacc 

 between with the hair, by the eye, or, if neceiTary, by the 

 moveable hair of the micrometer, after it has meafnred the 

 whole of this fpace ; the bifecling line being now the true 

 place of ^, which we fuppofe brought back to the microfcope 

 at m, this interfccElion is the extreme point of the bifeftional 

 arc F (f or f (p, which mull be bifcfted fucceffively in hke 

 manner, when the point k has been adjufted. When the arcs 

 become fmall, a crooked point is propofed to be ufed at 0, 

 that it may not be in the way of the microfcope ; or other- 

 wife, if that plan will not fucceed, to adopt what he calls 

 Mr. Troughton's method of bilefting an odd number of 

 contiguous divifions, which was alfo done, as we have feen, 

 by the due de Chaulnes and others. In cutting the lines 

 of divifion, the microfcope and point J' are propofed to be 

 very near together, and, if pollible, fo near at one fide of 

 one another, that the flroke cut may coincide with the 

 fixed hair, when the beam has an angular motion giveuj as 

 a check on the liability of the microfcope ; and lallly, the 

 point d mull be fo taken that the line d a may now be a 

 tangent to the circle at the cutting point. 



If the whole circle could be graduated by continual 

 bifeftion, this accwunt is all that would-be necefiary for ex- 

 plaining the method of dividing here propofed, but, as tri- 

 feiition and quinqnefeftion are neceffarily introduced, either 

 before or after bifeftion, where the computation of chords 

 is rejcdled, we mull give alfo an account of thefe procefFes, 

 which we believe are- original. Suppofe the arc a a, in^^. j, 

 to be propofed for quiuquefeftion, the equal meafures qj', 

 f e, e d, d b, and b a, are laid down in fucceffion, by bring- 

 ing a to the microfcope, and making y~nearly one-fifth of the 

 whole, then by bringing up^'to the microfcope, and mark- 

 ing e, &c. in the fame way, beginning at k, the points /S, 

 cf, i, and f, are made with the fame extent, wliich it ap- 

 pears was with too fmall an opening. Now the true paint 

 of the firll quinquefec\ion from k will be between a and;/?, 

 and diltaut from /3 jull one-fifth of the differential fpace & b, 

 and the fecond point will be two-fifths of i ,■/ froin •;, and 

 fo on for the rell; therefore, in cutting the dividing llrokes^ 

 the micrometer mull mealurc the wiiole difl'erential fpace, 

 and then allow for the due proportion, after which it mull 

 be brought into the place of the fixed hair, to bifeCl the 

 fcratch before covered by it, and then the dividing line may 

 be cut: in the fame way allowance may be made for ?-tlis of 

 the fmall fpacc a </, before a fecond dividing ilroke is matje; 



or. 



