GRADUATION. 



copy, renders this method of dividing lefs wortliy of imita- 

 tion, as an original method, than it would have been if tlie 

 imperfefiions of the firll portion of the circle liad not been 

 neceflarily transferred to the other, and from that back 

 again. In fhort, we think the wax-work niight have been 

 better employed. 



The elder Troughton (John) whofe dividing was ac- 

 knowledged to be equal to tliat of any of his predecelTors, or 

 contemporaries, ufed the beam-compalTes, like Bird, but 

 rejefted the computation of chords, and the meafures taken 

 from fcales, as being liable to uncertainty in determining 

 the primary points from which the bifLiiiions were to 

 proceed. After ha\'ing defcribed his circle, or rather quad- 

 rantal arc to be divided, he determined the point 60 with the 

 radius, as Graham and Bird had previoudy done, and, having 

 bifefted it at 30 , fet off 30 in addition to the 60' to com- 

 plete the arc of 90 ; he then bifec^cd till he had arcs of 15", 

 and again till he had 7^ 30' in each divilion ; the two marks 

 nearell to 90 were now 82 30', and 86 ij'; but the 

 point 85 20', or hmit of the largeil bifectional arc, lay 

 between thefe two, and could not be obtained by further 

 bifeftions ; the fpa.:e between the two marks in quellion 

 was therefore trifeCted, and the more forward of the two 

 new points was 85 : again, the fpace between this mark of 

 85 , and that of 86 15 , was trifefted ; from which came 

 85 25', as d-noted by the more backward of the two new 

 marks ; and laltly, a fifth part of one of the fub-divided arcs 

 was fet backwards from 85" 25', to 85^ 20', the point 

 from which the 1024 divifions were inferted from o entirely 

 bv bifeftions. The quadrantal arc was then completed 

 from the fub-divifions thus obtained. It may be neceflar)-, 

 however, to obferve, that the marks at firft made by the 

 radius, bifeCtions, trifeclions, 5cc. were none of thtm per- 

 mitted to be permanent, being of no further iif- than to 

 afcertain the individual point 85' 20', from which the fub- 

 fcquent bifeftions were to commence. This method is con- 

 fidered as being preferable to Bird's method of computing 

 the chords and ufing the fcale, inafmuch as it does not 

 depend on fecondary or auxiliary means of afcertaining 

 the primary point in the bifciTtional arc. It has uniformity 

 of means to recommend it in preference to thofe mixed 

 methods that depend partly on computation, and partly on 

 the extended radius. 



The method of dividing a large circle, commonly known 

 by the appellation of Ramfdi'n's method, or the method of 

 coaxing, confilts of Bird's metiiod, and of tint propofed by 

 the due de Chaulnes united : the circle is iirll divided by the 

 beam-compalTes irtto primary points; and the true fituation of 

 each of thefe points is afcertained by oppofitc microfcopes, 

 as the work proceeds, and is reftitied accordingly, by 

 pufhing the points forward or backward a trifle, till they 

 are in their true places. This method, now generally prac- 

 tifed by all the beft dividers, except the prefcnt Mr. 

 Troughton, has not, that we know of, been very particularly 

 defcribed, with references to drawings, &c. though it is 

 capable of confiderable accuracy in the hands of a good 

 workman, who has perfeverance enough to do j .ftice to it. 

 The great number of points that will require to be re£litied, 

 will, notwithltanding the utmoft precaution, render the work 

 irregular in its appearance, and a circular line muft necel- 

 farily pafs tlirough tlie centre of all the points or conical 

 lioles, to render them concentric ; belides, the bifedional 

 arcs defjrm the conical ftiape of the points, by palling 

 through them, and the fubfequent erafjres m\x\\ leave an 

 unevcjinefs in the metal ti'.at cannot but offend a nice eye. 

 Sir George Shuckburgh, iii his paper on the equatorial, calls 



1 



the points that have been enlarged by re(f\ifIcat:on, and 

 burniflied level again, " doubtful or bad points ;" and ihefe 

 bear a confideraVjIe proportion to the whole. «' It would," 

 fays Mr. Troughton, " be a great improvement of this 

 metliod to divide the whole by hand at once, and after- 

 wards to correct the whole ; for a dot, forced to its place, 

 as above, will fcldom allow the compafs-point to reft i« 

 the centrtt of its apparent area ; therefore, otlier dots mad'- 

 from thofe will fcarcely ever be found in their true places. 

 This improvtment alfo pn vents tli corrected dots from 

 being injured or moved by the future application of the 

 compaffes, no fuch application being neceffar)-." 



The circle that is divided by this method is placed hori- 

 zontally to have its firll points made, after it has had it.s 

 circle defcribed from a revolution on its own axis, and 

 then it is placed vertically in a frame, in which it revolve?, 

 and which carries the microfcopes with micruM.eters, that 

 fub-divide, and read to the accuracy of one fecond, in order 

 that the femi-circles, taken from any given oppoGte points, 

 may have their equality afcertained, or their deviation there. 

 from determined previoudy to final rectification. In thefe 

 operations great attention is paid to the temperature of all 

 the metallic parts employed in the work. 



Mr. Ed. Troughton has deviated from the beaten track 

 of his prcdeceffors, and made a road forhimfelf, (probably 

 bffore Ramfden's plan was adopted,) tiiat he has trodden 

 with great fuccefs, and which he has fully defcribed in a pa- 

 per of the Philofophical Tranfaclicns of the year 1809, 

 which gained him Copley's medal. The reafon ihatcaufed 

 him to think for himfelf on this fubjeft, as he has done fuc- 

 cefsfully on many others, fcems to have been this : " With 

 as iLady a hand, and as good an eye,'' fays he, " as young 

 men generally have, I was much difappointcd at finding, that 

 after having made two points, neat and fmall to my hking, I 

 could not bifeCl the diilance between them without enlarg- 

 ing, difplacing, or deforming them with the points of the 

 compaffes." 



This dlfcovery led to the abandonment of the beam-corn- 

 paffes and fpring-dividers, and, the art of turning appearing 

 to have approachi.d the nearell to perfeftion of anv of the 

 mechanical arts, a roller was thought of, which by its revo- 

 lutions might fub-divide the circumf;.rence of a circle rolled 

 over, after the ratio of their refpective diameters was afcer- 

 tained and properly adjufted. When this fpcculation was 

 firit attempted to be realized, fome circumftanccs occurred 

 which could not be certainly inferred from reafoning, a 

 priori, from known data, but of which a perfect know- 

 ledge was neceffary for the confummation of the project ; in 

 the firll place, it was found on trial, that however fmtioth 

 the furfaces of the circle and roller were made, there was no 

 Jlippery afiion, as might have been cxpeCled, but the points 

 of contact aC\ed with each other in an appai~atus l:ke that 

 hereafter to be defcribed, as the teeth of wheel-work of in- 

 definitely fmall dimenfions ; the certainty of this kind of 

 aftion was an indifpenfablc condition ; fccondly, notwiih- 

 lla;iding this ftableiiefs in the motion of the roller, it wa* 

 found to mcafure different portions of the metallic circle 

 with different degrees of accuracy, fome of the meafures 

 being a trifle plus, and others minus, with relpect to the 

 truth : this want of accuracy, which, as we have faid, did 

 not depend on any Hiding of the roller, was expected ti> 

 take place previoully, in a certain undefined degree, by rta- 

 fon of the unequal deiuily uf haJimea-d materi;ds, and of 

 their confequent unequal /^r^/)' ; but thirdly, though there 

 y>-ii found to be a deviation iroin true ineofuivmeDt in indiri- 



