GRADUATION. 



or, which gives lefs trouble, wlien the firfl quiiiquefeAinT 

 hne is truly laid down, it may be transferred forward by 

 iifing the fixed hair with it firfl, and then with the fecond, 

 while the third is cutting, and fo forth till they are all mea- 

 fured and cut in futcelllon from the firll mealure repeated. 

 This method is, however, acknowledged to be liable to the 

 double error that may arife, firft, from placing the inter- 

 mediate point, and again, of fub-dividing from it. A third 

 method for quinquefcftions or trlfeftions is laflly propofed, 

 but is confidered inferior to the former plans, and therefore 

 may be paffed over in filence. 



We mull confefs there is fomething fpccious in the theory 

 of Mr. Cavendilh's jiropofed method of dividing, but our 

 opinion, that it would not anfwer well in praftice, has been 

 confirmed by the affertion of a praftical artift, whofe judg- 

 ment in tliis art we eilimate much more highly than our 

 own. Suppofing the point d on the frame not to be altered 

 in ihape or polltion by the large fweeps of the beam from 

 right to left, and vice verfii, alternately, nor yet the point 

 injured, there muft necelfarily be a change of pofition in the 

 liands of the operator, as well as in his whole body, at every 

 backward and forward change of the beam, to the right 

 and left, which circumllance will be not only unfavourable 

 to expedition, but alfo to acctiracy; in facl, the operator 

 mull be both right-handed and left-handed to fncceed with 

 an alternation of operations that require the greatell nicety 

 of management, and in which the hands, the eye, and the 

 diverfified attitudes of the body, have all an equal fliare, 

 and where a falfe llroke once made cannot be well reftified. 

 The bifeiled arcs will not be more affefted by change of 

 temperature probably than by Bird's method, but the 

 quinquefefted arcs may be fenfibly affedlcd, as fome time 

 muft neceifarily be taken up in afcertaining the meafures by 

 vifion, while the operator breatlirs on the beam during the 

 infpcftion of the micrcfcope, and is alio handling it in two 

 places during the wh.cle time of obtaining the fine points, 

 or rather lines, in queilion. 



Soon after Mr. Cavendilh's paper on the manner of divid- 

 ing aftronomical inftruments had been read before the mem- 

 bers of the Royal Society, the Rev. William Lax, A.M. 

 F.R S., Lowndes's profelFor of allronomy in the univerfity 

 of Cambridge, addreffed a paper to the aftrononier royal, which 

 was alfo read and printed in part 2. of the year 1809, im- 

 mediately after Mr. Cavendilh's paper. This paper, as well 

 as Mr. Cavendilh's, was probably brought forward in con- 

 fequence of Troughton's paper, printed in part I. of the 

 fame year. 



The method of examination made ufe of by profelFor Lax 

 may be fainiliarly explained thus, without entering into any 

 mathematical formulx to mark the valae of particular ex- 

 prefiions, that might puzzle the ordinary reader : two mi- 

 tiofcopes were adapted by frame-v.-ork, fo as to be capable 

 of adjuftment to any given points of the circle to be exa- 

 mined, and one of them, having a micrometer attached to it, 

 was capable of meafuring minute differences between a given 

 arc of 90 , 60 , or 45 , &c. and any of the following mul- 

 tiples of that arc taken in fuccelTion ; confequently, if all the 

 faid differences marked + or — , as the cale might be, ba- 

 lanced one another exaftly, when the whole circle was gone 

 through, the meafure of the firll arc was a Ihuidard mealure, 

 or the exaft fradlion of the whole that it ought to be ; if 

 nut, a proportional part of the balance of errors would be- 

 long to it, and muft be allowed for accordingly. It is not 

 ncceftary to be more minute in explaining the principle of exa- 

 mination before us, as it is of no i.fe in graduating a circle, 

 but i« only intended as a correftion on bad graduation, like 



the French repeating principle, to which it bears fome ana- 

 logy. The author's idea of afcertaining the fractional value 

 of each divifion in \m inftrument fuccelTively, which he calcu- 

 lates will coft him an expenditure of 98 hours, is founded on 

 an opniion that aftronomical inftruments in general d»j not 

 poffefs that degree of accuracy in their divifions that thcmakeri 

 of them affirm ; but he docs not appear to be aware that 

 as rigid an examination as his own is praclifed by all the bed 

 inftrnmcnt-makera in the final adjuftment of the divifions, 

 and that too by means of micrometrical microfcopes ; fo that, 

 if a pollerior examination is found neceflary, it muft, gene- 

 rally fpcaking, be neceft'ary in confcquencc of fome change of 

 figure tliat the inftrument has undergone from carriage, ac- 

 cident, or temperature ; in any of which cafes the profcftbr'* 

 mode of examination will be very defirable. But any one 

 who propofes to take the trouble of making a table of all 

 the pofitive and negative errors in the divifions of an indiffer- 

 ent inftrument, will do well to have the whole of the origi- 

 nal paper before him at the time, by which every ftep in the 

 procels will be pointed out as it is wanted Without meaning 

 to difcourage any attempt of this nature, for the propofai 

 of which altronomers are much indebted to the learned pro- 

 feftbr, we profefs a belief that three or four readings of any 

 individual obfervation by as many microfcopes properly ar- 

 ranged round the divided circle, in ftationary fituations, will 

 anfwer every purpofe of accuracy, ami corred all the ufual 

 fources of error that ex'ijl at the tinu of making the obferva- 

 tion ; for when any table of errors is completed with the great- 

 eft care, it ought only to apply to obfervatioiis made while 

 the inftrument remains in the fame predicament as it was when 

 the table was conliructed. Mr. Trougliton, who is engaged 

 with the conftrudion of his fix-feet circle for the Royal Ob- 

 fervatory, intends, we underlland, to adopt the ufe of four 

 microfcopic readings, two oppofite each other, which will 

 corredl for eccentricity finiply confidered, and two at i 20'' 

 dillanct' from one of the former ones, which w.ll check the 

 bifettional dividing, and corretl the inaccuracies of divifion, 

 if any, as well as the effefls of any change of figure in the 

 circle by unequal temperature, in the room where it may be 

 ufcd. This propofai, coming from fo flcilful and experienced 

 an artill as Mr. Troughton, we confider as a ftrong argu- 

 ment in favour of our opinion, which we cxprefTed in favour 

 of three readings in our article Cihcle, before the paper in 

 queftion was publidied, or known to us. 



" I find," fays Mr. Lax, " that I can read off, to a cer- 

 tainty, within lefs than thrte-foiirths of a feeond, and hence I 

 conclude that I could examine the divifions of my lircle (of 

 one foot radius) without being liable to a greater error than 

 9 63 feconds.'' This is Hated on calculation to be the extreme 

 limit ; which, according to the dodrinc of chances, can very 

 (Lldom happen, but if one-half of this error is llkelv to 

 happen in any one divifion of the « hole circle, as read oS 

 by the microfcopes, the refnlt will be, that the ermrsof the 

 table may be as great as thofe of a well-divided inftrument, 

 fuppofing it to preferve its ligure unaltered after examina- 

 tion In another part of tlie paper before us, the author 

 fays, " we may likewife obferve tliat by this examination we 

 ftiall not on'y be fecured agiiuft the errors of divifion, but 

 againft thuie which urife from bad centering, and fri m the 

 imperfcdt figure of the circle, and which in general are of 

 to« great a magnitude to be neglcctij," From this obferva- 

 tion we mull neceflarily conclude, tluit the diftancc between 

 the microfcope and micrometer, ufed in nuafuring, was not 

 the meafure of the included a>x, but of the a;if»<;<jr quaitiily 

 at the centre oppofite that arc : for it mull te evident, 

 that, in cafe the a:iis of motion is not prccifely the centre 



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