GRADUATION. 



divided arc, by turning the points over twice each way, 

 from every point already laid down, and then the middle 

 fpace of each five being found equal to the extreme as well 

 as contiguous fpaces r>irpeftively, the whole arc was fubdi- 

 vidcd into points of one degree of diftance from each 

 other. But to enfure a perfett equality among thefe fpaces 

 required great fteadinefs of hand, as well as obfcrvation of 

 the eye, and caution, to preferve the regulated dillances of 

 the points, unaltered during the operation. Of courfe, the 

 marking point was required to be line, and at the fame time 

 well tempered, as well as flrong enough to bear prclfure, 

 which preffure was alfo neccffary to be made in a perpendicu- 

 lar diretlion on the face of the limb. The fub-divifions of 

 the degree fpaces into 12 parts of 5' each, were done firit by 

 trifeftion, and then by bileftion, or vice verfa, in the man- 

 Her we have already defcribcd ; and tlie delicacy of thefe 

 operations, on fo fmall a fcale, required extraordinary atten- 

 tion and care to enfure perfeft equality among the fmalleil 

 lub-divifions, which were now in a ilate to be transferred by 

 lines tending to the centre of the arc to Isc graduated. 

 This was an operation that could not be done well by the 

 Jlraight edge of a rule, and a marking point, or dividing 

 knife that would be liable to deviate a little, notwithftaniing 

 the greateft care ; here another, but fmaller beam-compafs 

 was fubltituted for the ruler, probably for the firlltirae, for 

 the purpofe of transferring the graduated points from the 

 occult arc into the :»rc to be graduated, in the following 

 manner ; fuppofe the points g and e to be intended to be 

 transferred ; becaufe the lines to be cut, as the dividing lines, 

 are required to be in a direftion tending dire£tly towards 

 the centre of the concentric arcs, defcribed on the limb ; 

 the diilance of the cutting point, from the ftationary point 

 ef the compafs, was taken of fuch a length, that the cutting 

 point crofTed the arc to be cut at right angles, or, in other 

 words, the beam was fo fituated, as to become a tangent to 

 the are to be cut ", therefore the diilance of the two points 

 of the beam was regulated by the diftance of the occult line 

 of dots r, h, d, from the arc to be graduated by the ftraight 

 lines, or rather by the curved lines, which in fact were fub- 

 {tituted, and which palfcd without fenfible error for ftraight 

 lines, when the tangent line in queftion was long. From the 

 point or dot ^, the curve /; k was drawn, and from the dot 

 *• the curvey / ; but in fuch a way that a fmall portion only 

 of each, that lies between the circular lines, was cut on the 

 face of the inftrument. In the fame manner all the otlier 

 dots were fucceffively transferred, while each reprefentative 

 «f the numerals 5, 10, 15, &c. were made longer than 

 their intermediate hnes of divillon, and the fubdividing lines 

 were again ftill ftiorter. The vernier carried by the tele- 

 fcopc, when nicely and accurately divided, would detetl 

 any inaccuracy in the fub-divilions thus transferred, by the 

 aid of a magnifier properly adjutted. 



The arc of g6 divifions, with their fub-divifions, was not, 

 properly iYi?^\T.'g, graduated, but divided and fub-divided iu:o 

 portion.'- of fmaller value than degrees, and 5' fpaces ; but as 

 the nu.nib'.-r chofen is divifible continually by the number tiuo, 

 it was completed by eontiiwal bifenions, which method there- 

 Jore requires no further explanation. We are told by Dr. 

 Hmith (.n his Optics) that thefe two arcs were never found 

 So ditf'j.' from each other more than 5" or 6 ' on any part of 

 the lu.ib, butthat when there is fuch difference, the preference 

 ought to be given to the bifefted arc of 96 divifions. 



To prove that the fpaces obtain..-d by the hues of transfer 

 areecjiiaito.thijfe between the correfponding dotsijr points, 

 let e y ar.d g h be joined, alfo of, oh, e, and g ; and the 

 tri,ar,gles r of, g h, will be every way limilar, and equal to 

 Cucii otlicr J ihtrexore, if the common angle (ob be taken 



aw?y from the equal angles e of, g I.', the angles e a anct 

 fo h, that remain, will aifo be cquyl to each other. 



It does not appear that Graham took any meafures ta 

 guard againft, or even to detect the errors that his method 

 of dividing is liable to, from variations of temperature in 

 his quadrant and beam, during llie time that the operation is 

 going on, and from the correfpondiiig variations of length 

 in the metals, according to their relpetiive expanfibilities ; 

 nor is it quite certain that he was aware t.'f the probable ex- 

 tent of fuch errors, feeing he conftradtcd the frame-work of 

 his inftruments of iron, and had his circle to be divided of 

 brafs. In Dr. Bradley's zenith feCtor made by Graham, 

 Dr. Maflvelyne has caufed an iron limb to be fubilitutcd for 

 the original brafs one, and has had the points of divifion in- 

 ferted on iluds of gold, to avoid the errors that arole from 

 the unequal expanfibility of the dillereiit metals. 



We might here mention Mr. H. Hindley's plan of dividing 

 a circle by a tootlied circular plate and endkfs fcrew in form 

 of an engine, about the year i 740, but that we think his 

 method of dividing and drilling the holes of his plate, by 

 bending a ftraight flender bar of brafs into a circle, and 

 transferring the holes therefrom, cannot be depended on 

 where much accuracy is required in the divifions of a circle. 

 It was originally intended, and is much better calculated, 

 for dividing and cutting the notches between the teeth of a 

 wheel ; but the reader may fee the plan defcribed, and fome 

 improvements on it propofed by Smeaton, in his paper already 

 mentioned, as contained in the Philofophical Tranfaclions of 

 London, in the year 1 785. 



Jeremiah, the fon of Jonathan SifiTon, was of Graham's 

 fchool of dividing, and did nearly as much juftice to the 

 method he adopted, as Graham iiimfelf, probably ; and his 

 nice care, and perfevenng affiduity, have claffed him among 

 tiie firft dividers of his time ; but we are not aware that he 

 was the inventor of any original contrivance, except, perhaps, 

 that lie applied a triple index to fonie of his inftniments, 

 one of which had the vernier, and each of the others had a 

 finglc line or ftroke drawn at one-third of a circle from each 

 other, and from zero of the vernier, which might aft as 3 

 check on the eccentricity of the circle, as well as on the 

 inequality of its divifions ; though it does not appear cer- 

 tain that thi.s was the original intention, as the three props of 

 his vernier-bar in his theodolites required it to be triple ; 

 but in his circular indexes one ftroke only was made, and 

 that oppofite the vernier : the importance of a triple vernier 

 has not been noticed particularly by any one, previoufly to 

 the time of Mr. Ed. Troughton's introduction of tlie triple 

 vernier into his circular inftruments. Mr. Ludlam, however, 

 fays, that SifTon very early rejected the method of trifefting, 

 and that of lleppiRg too. " Having (by means of the radius 

 and bifeftionsorly) divided his quadrant into three arches 

 of 30 each, he fet off in each of thefe arches the clierd of 

 2 I 20', or 256 times five minutes. This cliord was taken off 

 a fcale of equal parts, and was checked by the chord of 

 8 40', both chords together filling up the arch of 30°. The 

 arch of 21 20 was divided by continual bif'eftions into 

 arches of five minutes each." This defcription, fays Mr, 

 Ludlam, in a note to page 4, of his Introdudion and Notes 

 on Mr. Bird's method of dividing, was given by Jeremiali 

 Sifton, in a private letter dated Mav 20, 1766 ; and according 

 to the fame letter it appears alfo, th?t SifTon placed the 

 fixed or central point of his compafs in a blank tarigental 

 line, as hereafter defcribed by Bird, during the operation 

 of transferring the divided points into linear divifions ; but 

 as Bird has m the year 1767 publifhed thefe proceffes as 

 originally his own, and as he worked for the Silfons pre- 

 vioully to 1766, we are difpofed to conlider Dirdj and not 



either 



