GRADUATION. 



tlic gFeatefl: exadlriers, when, by 1-aIfing or depreffing the 

 roller, its comn-.cnfiirntc diameter is found. 



Fi^. J. is a reprefeiitatioii of the apparatus for transferring 

 thediviiions of the roller to the cirele. It coniids of two 

 (lender bars, whicli, being ieen edgewife in the lignre, have 

 only the appearance of narrow hnes ; but looked at from 

 above, t!)ey refemble the letter A. They are fallencd to 

 the main frame, as at VV and Z, by fliort pillars, having 

 alfo the off leg of the angle fecured in tiie fame manner ; Y 

 is a fine conical ftcel point for making the dots, and X is a 

 feeler, whereby the point Y may be preiTed down with an 

 uniform force, which force may be adju lied, by bending the 

 end of the bar jull above the point, fo as to make 

 the dots of the proper ii/.e. The point Y yields moll readi- 

 ly to a perpendicular aiilion, but is amply fecured againd 

 any eccentric or lateral deviation. 



The apparatus fo far defcribed, is complete for laying the 

 foundation, /. e. for making 256 primary dots ; no m.atter 

 whether with perfefc truth or not, as was faid refpefting the 

 divifions of the roller ; precilion in either is not to be ex- 

 pefted, nor wiflied ; but it is of fome importance that they 

 fhoiildbe all of tlie fame fi/,e, concentric, fmaH, and round. 

 Thev fiiould occupy a pofition very near tlie extreme bor- 

 der of the circle, as well to give the grcateil radius pofTible, 

 as that there fiiould be room for the llationary micrcfcope, 

 and other mcchanifm to be defcribed hereafter. 



It mull be noticed that tiicre is a clamp "and adjufting 

 fcrew attached to the main plate oi Jig. i, but as it differs 

 in no refpect from the ufual contrivances for quick and flow 

 motion (feeCuiCLE), it has been judged unneceffary to in- 

 cumber the drawing with it. 



Now the roller having been adjuftcd, with one micro- 

 fcope H upon its proper dot on the circle, and the other 

 microfcope at the lirll divifion on the roller, place the appa- 

 ratus of fig. 5, fo that the dotting point Y may Hand direct- 

 ly over the place which is defigncd for the beginning of the 

 diviiions. In this polition of things, let the feeler X be 

 prelTed down, until its lower end comes in contact with the 

 circle ; this will carry down the point, and make the lirlt 

 impreffion, or primary dot, upon the circle ; unelamp the 

 apparatus, and carry it forward.'? by hand, until another di- 

 vifion of the roller comes near the wire of the microfcope ; 

 then clamp it, and with the fcrew-motion make the coinci- 

 dence complete; where again prefs upon the feeler tort he fccond 

 ^ot ; proceed in this manner until the whole round is completed. 



From tliefe 256 erroneous divifions, by a certain courfc of 

 examination, and by coniputati<ni, to alcertain their abio- 

 lute and individual errors, and to form tliefe errors into con- 

 venient tables, is the next part of the proceis, and makes a 

 very importantbranch of my method of dividing. 



The apparatns mull now be taken off, and the circle 

 mounted in the fame manner, that it will be in the obferva- 

 tory. The two microfeopes, which have divided heads, mud 

 alfo be firmly fixed to the fupport of the inllrumeiit, on op- 

 pofite fides, and then wires brought to bifeft the iirll dot, 

 and the one -vhieh fliouldbe 180 diftant. Now the micro- 

 fcopes remaining fixed, turn the circle half round, or until 

 the firfl microlcope coincides with the oppofite dot ; and if 

 the other microfcope be exaftly at the othei- dot, it is ob- 

 vious that thefe dots are 180 apart, or iu the true diameter 

 of the circle ; and, if they difagree, it is obuous that half 

 tlie quantity by which they difagree, as mcahired by the di- 

 vifions of the micrometer head, is the error of the oppofite 

 divifion ; for the (piantity meafured is that by which the 

 greater portion of the circle exceeds the lefs. It is conve- 

 nient to note tliefe errors -(-or — , as the dots arc found too 



Vol, XVI. 



forward or too backward, according fo tfie numbering of 



the degrees ; and for tlie purpofe of ditiinguilliing tlic -^ 

 and — errors, tlie licids, as meiitioiicd before, arc nuirbiTf J 

 backwards and forwards to fifty. Chie of the microfeopes 

 remaining as before, remove the ollicr to a pofition at riglit 

 angles ; and, confidoring for the prefeiit Loth the former 

 dots to be true, examine the others by tliem ; /. r as before, 

 try by the micrometer how many divifions of tlic head the 

 greater half of the femi-circlecKceeds the lefs, and note half 

 the quantity -f or — as before, and do the fame for the 

 other femi-cirde. One of the micrometers mud now be fct 

 at an angle of 45- with the other, and half the differences 

 of the two parts of each of the four quadrants regiftered 

 with their refpedive figns. When the circle is a vertical one, 

 as in the prefent indance, it is much the bed to proceed fo 

 far in the examination with it in that pofition, for fear of 

 any general bending or fpring of the figure ; but for tl«» 

 examination of fmaller arcs than 45 ', it will be perfectlv faff 

 and more convenient to have it horizontal ; bccaufe the divid- 

 ing apparatus will then carry the micrometers, feveral per- 

 forations being made in the plate B for the limb to bo fecn 

 through at proper intervals. The micrometers mud now be 

 placed at a didance of 22 ~ 30', and the half differences of 

 the parts of all the arcs of 45 meafured and noted as be- 

 fore ; tluis defcending by biledions to 1 T 15', 5 37' 30", 

 and 2' 48' 45". Half this lad quantity is too fmall to al- 

 low the micrometers to be brought near enough ; but it will 

 have the defired effect if they arc placed at that quantity and 

 its half, J. f. 4- 13 "".5, in which cafe the examination, in- 

 dead of being made at the next, will take place at the nc:^t 

 divifion but one, to that which i.". the f ubiect of trial. During 

 the whole of the time that the examination is made, all the 

 dots, except the one under examination, are for the prefent 

 fuppofed to be in their true places ; and the only thing in 

 this moll important part (-f the bufinefs, from firfi to lad, is 

 toafcertain with the iitmofl care, in divifions of the micro- 

 meter head, how much one of the parts of the interval under 

 oxan.ination exceeds the other, and carefully to tabulate the 

 lialf of their difference. 



I will fuppofe that every one who attempts to divide a 

 large adrononiical indrument, will have it engraved firft. 

 Dividing is a mod delicate operation, and every coarfcr one 

 dionld precede it. Dende.^ it." being numbered is particularly 

 nfeful to dillinguidi one dot from another; thus in the two 

 annexed tables of errors, the fide columns give fignilicant 

 names to every dot in terms of its value to the nearefl tenth 

 of a tk'giee, and -the millaking of one for anotlicr is rendered 

 nearly impodible. 



The foregomg examination furniflies materials for the can- 

 flnidtion of the table of half didcrences, or apparent errors. 

 The fird line of this table confids of two varioties ; ;". e. the 

 micrometers were at iSo didance forobtakiing the numbers 

 which fill the columns of the tirll and third quadrants ; and 

 at 90 for tliofe of the I'econd and fourtli quadranti:. Tlie 

 third variety makes one line, and was ohtairicd w ivli a didance 

 of 45 ; the fourth confifls of two line.-, with a diilance of 

 22 30'; the fifth of four lines, with a diftaiice of U' 15 ; 

 the iixth of eight lines, with a didance of 5 37' 30'; the 

 leventh of fixteen lines, with a didance of 2 48 45'; and 

 the eighth and lall variety, bcuig the reniaiuvltir of the tabl6» 

 confills of ihirlv-two lines, and was obtained with a diltance 



of 4" '3'-" 5' 



The uible of apparent errors or half diiferences, jufl ex- 

 plained, furnilhes data for computing the table of real errors. 

 'i'he rule is this ; let a be the real error of the pivcetling 

 dot, and b that wf the following oue, a;id c llie apparx'nt 

 3 Z etTiir, 



