GRADUATION. 



from the points : the beft indnimcnt for this purpofe is the 

 beam-compafs, having both its points conical and very fliarp. 

 Draiv a tangent to the arc b <U fuppofc at e, it will intcrfea 

 the arc .v y in y, this will be the diilance between the points 

 of the be'nm-compars to cut the divifions (nearly) at right 

 angles to the arc b d. Lodge that point of the beam-com- 

 pafs next voiir right hand, in the point r ; let the other fall 

 freelv into the arc xy ; prefs gently with your finger upon 

 the fcrew-hend that fallens the focket (and that is convex), 

 and with the point towards the right haad cut the diviuons. 

 In this manner you mult proceed with the relV.'' 



The intention of transferring the central^ or refting point 

 of the conipafs into a blank tangental line, inftead of fulier- 

 ing the faid point to reft in the refpeftive points made in the 

 faint arc, is, that in the former cafe iliould any alteration 

 take place in the length of the beam, during the operation of 

 cutting, no error is charged on the divilions cut, but the 

 alteration is compenfated by the diilance of the nev.ly made 

 point in the tangeutal hne from the cutting point ; that is, 

 each divition is charged only with fuch a minute error, if 

 any, as arifes from temperature during the acl; of cutting a 

 Tingle Itroke, and if one of the points Ihould break, another 

 may be rLplaced without inconvenience-; whereas,^ by the 

 method of Graham, a broken point produces confidcrable 

 difficulty ; and the whole error ariting from temperature, 

 curinir the whole procefs of cutting all the divifions, is 

 charged not only on the total arc, but likewife in a certain 

 undefined degree on each of the whole divifions. This idea 

 of getting quit of the eil'efts of temperature is ingenious, 

 but has been objefted to as liable to the ill effefts of ilcpping 

 in a certain degree, inafeuch as that a hard partic!e,_ lying 

 under the point of reft, may caufe a little deviation in the 

 diftancej before the ftroke is cut. This objection, however, 

 was not allowed to be realized by Bird, who no doubt mull 

 have obviated it by fome other plan, had he fuund any real 

 inconvenience arifiiig from it. Great care, notwithftanding, 

 nmft have been taken in fctting down, and preiTing on the 

 point at reft, in a perpendicular direftion. Another objec- 

 tion applied to Mr. Bird's fcale, from which the chord-fines 

 were meafured, inafmuch as the fcale itfelf might be eiTO- 

 jieous in fomc places, and would impart its errors to the 

 arcs meafured from the computed chords taken there- 

 :from. 



Befides the arc of 90' , the mural arc by Bird had an 

 arc of 96 divifions, like Graham's, divided by continual 

 bifeAions, till each of the 96 divifions had 16 fub-divifions, 

 as a check on the accuracy of the arc of degiees ; but fub 



a point at one of the angles ; which formed the cutting- 

 point. 



At the requeft of the commiflloners of the Board of 

 Longitude, Mr. Bird, in the year 1767, publiflied his method 

 of dividing in a quarto pamphlet, tor which they rewarded 

 him with 500/. befides 60/. for his plates, after having bound 

 him to inftruft an apprentice of feven years in the art of 

 conttrafting and dividing ailronomical inftruments. 



Mr. Sm.eaton is of opinion that Bird's method of dividing 

 may be improved in one refpett, and exprefies himfelf on 

 the' fubjeft thus : 



" I muft here obferve that I apprehend no quadrant, that 

 has ever undergone a fevere examination, has been found to 

 form a perfeft arch of 90', nor is it at all neceffary it fliould ; 

 the perfeft equahty of the divifions throughout the whole 

 is the firft and primary confideration ; as the proportion of 

 error, when afeertained by proper obfervations, can be as 

 eafily and readily applied when the whole error of the redl- 

 angle is fifteen feconds, as when it is but five. In this view, 

 from the radius taken, I would compute the chord of 16* 

 only. If I had an excellent plain fcale, I would ufe it, 

 becaufe I Ihould cxpedl the deviation from the right angle to 

 be lefs than if taken from a fcide of more moderate accuracy ; 

 but if not, the equality of the divifions would not be affect- 

 ed, though taken from any common diagonal fcale. This 

 chord, fo prepared, I would lay off five times in fuccef- 

 fion, from the primary point of o given, which would com- 

 pleat 80 ; I would then bifecl each of thofe arches of 16^, 

 as prefcribed by Mr. Bird, and laying off one of them be- 

 vond the Beth, would give the 88th degree, proceeding then 

 by bifediion, till I came to an arch of 2~, laving that off from 

 the 88th degree, wouldgivethepointof 90°. Proceeding ftill by 

 bifettion, til! I had reduced the degrees into quarters of ij' 

 each, I would there flop, as from experience I know that 

 when -divifions are over clofe, the accuracy of them, even by 

 bifeftions, cannot be fo well attained, as where they are mo- 

 derately large. If a fpace of y'^ths of an inch, which is 

 a quarter of a degree, upon an eight feet radius, is thought 

 too large an interval to draw the index over by the micro- 

 meter fcrew, this may be fhortened by placing another line, 

 at the diftance of one-third of adivifion on each fide of the 

 index line, in which cafe the fcrew will never have to move 

 the index plate more than one-third of a divifion, or 5', and 

 the perfect equality of thofe fide lines from the index line may 

 be obtained, and adjulled to 5' precifely, by putting each 

 of the fide lines upon a little plate, capable of ad'iuitment to 

 its true diftance from the middle one, bv an adjufting fcrew. 



hich 



fequent dividers of ailronomical inftruments have found this The above hint is not confined to the chord of 16 

 fuperfluous, as bein<r in the opinion of others, as well as of prohibits the fub-divifion going lower than 15' ; for if it be 



Jeremiah Sifibn, a check only on bad dividing, and as ren- " 



dering the reduction of obfervations troublefome, when made 

 thereby. 



The vernier was retained by Bird as the beft method he 

 knew of fub-dividing the 5' fpaces into quantities of 30" each, 

 ■which was the fmalleft angle he profelled tomeaiiu-e; and 

 to effect his purpofe he made ten divifions on his vernier equal 

 to eleven on the limb of the inftrument, firft by computing 

 the chord of thirty-two parts, and then by taking ten out 



of thofe parts, when bilecled, as the proper divifions for Jhpping, arifing from Roemer's method, afterwards propofes 

 the vernier ; thefe, being made in points, were alfo transfer- to lay down the chord of 64", or of 85^ 20', from compu- 

 ed by a tangental line into linear divifions by the bcam-com- 

 pafs, as before defcribed ; but great care was taken, that 

 the ftroke zero on the vernier was drawn from the qua- 

 cranl's centre, precifely parallel to the hne of colhmatiou of 

 the telefcope. Mr. Ludlam fays, that the cutting-point 

 which SifTon ufed was flat in the knife-edge form, but that 

 Bird's was a triangular prifm, with a flope ground down to 



required to have divifions equivalent to 5' upon the limb it- 

 felf, then I would compute the chord of zf 20' only ; and 

 laving it off four times from the primary point, the lall 

 ■vs'ould m.ark out the divifion 85 ' 20', pointed out by Mr. Bird, 

 fupplying the remainder to a quadrant from the bifefted 

 divifions as they arife, and not by the application of other 

 computed chords." Mr. Smeaton, fenliblc, however, that 

 this method of laying i6- five times over, or 21° 20' four 

 times over, may be objected to, as liable to the errors of 



tation all at once, and then to bifeCt, and complete the 

 quadrant from the bifefted divifions, which Mr. Bird him- 

 felf prefcribed as a good method for Hadley's fextants and 

 cCtants. But thele plans are now fuperfeded for inftruments 

 of fmall radius by the dividing engine, which performs the 

 work with great expedition, as well as accuracy, and the 

 original methods of dividing the circle are only ufeful for the 



kirgcr 



