GRADUATION. 



error, taken from the table of half differences, of the dot 



„. . , ■ o -^ b . , 



under inveftigation ; then is + f = its real error. 



But as this fimple cxpreffion may not be fo generally iinder- 

 ftood by workmen as I would wifli, it may be iiecefTary to 

 fay the fame thing lefs concifely. If the real errors of the 

 preceding and following dots are both + or both — , take 

 half their fum, and prefix thereto their common fign ; but if 

 one of them is + and the other — , take half their differ- 

 ence, prefixing the fign of the greater quantity : again, if 

 the apparent error of the dot under inveftigation has the 

 fame fign of the quantity found above, give to their fum the 

 common fign for the real error ; but if their figns are con- 

 trary, give to their difference the fign of the greater for the 

 real error. I add a few examples. 



Example I . 



For the firft point of the fecond quadrant. 

 Real error of the firft point of the firft quadrant o.o 



Real error of the firft point of the third quadrant — 6.9 

 Half fum or diifereuce - - - — 3.4 



Apparent error of the dot under trial - +12.2 



Real error .... -j- 8.8 



Example 2. 



For the point 45' of the fecond quadrant. 



Real error of the firft point of the quadrant -\- 8.8 



Real error of the laft point of the quadrant — 6.9 



Half difference - - - +0.9 



Apparent error of the dot under trial - — 8.9 



Real error .... — g.g 



Example 3. 



Point 88''. 6, or laft point of (he third quadrant. 

 Real error of the point 84°.4 of the third quadrant — 2 l.o 

 Real error of the point 2 .8 of the fourth qua- 

 drant ... — 2.9 

 Half fum - - ' . " " — 11.9 

 Apparent error of the dot under trial - — 4.0 

 Real error ... — ij.9 



Example 4. 



Point 88^.6, or laft of the fourth quadrant. 

 Real error of the point 84". 4 of tlie fourth qua- 

 drant - ■ - - - — 21.6 

 Real error of the point 2 ,8 of the firft quadrant — 10.2 

 Half fum - - - — 15.9 

 Apparent error of the dot under trial - + 9.^; 

 Real error . . . _ 5.^ 



It is convenient, in the formation of the table of real 

 errors, that they fhould be inferted in the order of the num- 

 bering of the degrees on their refpedlive quadrants ; although 

 their computation neceffarily took place in the order in which 

 the examination was carried on, or according to the arrange- 

 ment in the table of apparent errors. The firft dot of the 

 firft quadrant having been affumed to be in its true place, 

 the firft of the third quadrant will err by juft half the dif- 

 ference found by the examination ; therefore thefe are alike 

 in both tables. The real error of the firft dot of the fecond 

 quadrant comes out in the firft example ; that of the fourth 

 was foupd in like manner, and completes the firft line. It is 

 convenient to put the error of the divifion 90 of each 

 quadrapt at the bottom of each column, although it is the 

 fame as the point o*"" on the followiog quadrant. The line 



of 45^ is next filled up ; the fecond example ftiewsthis ; but 

 there is no occafion to dwell longer upon this explanation, 

 for every one, who is at all fit for fuch purfuits, will think 

 what has already been faid fully fufiicient for his purpofe. 

 However, 1 will juft mention that there can be no danger in 

 the formation of this table, of taking from a wrong line the 

 real errors which are to be the criterion for finding that of the 

 one under trial ; becaufe they are in the line next to it ; the 

 others which intervene in the full table not being yet in- 

 ferted. The laft courfe of all is, however, an exception ; 

 for, as the examining microfcopes could not be brought near 

 enough to bifeft the angle 2 48' 45", recourfe was had to 

 that quantity audits half; on wliich account the examination 

 is profecuted by ufing errors at two lines diftance, as is ftiewn 

 in the two laft examples. 



When the table of real errors is conftrufted, the othef 

 table, although it is of no further ufe, fhould not be thrown 

 away ; for, if any material miftake has been committed, it 

 will be difcovered as the operation of dividing is carried on; 

 and in that cafe the table of apparent errors muft be had re- 

 courfe to, indeed not a figure ftiould be deftroyed until the 

 work is done. Refpefting the angular value of the num- 

 bers in thefe tables, it may be worth mentioning, that it is 

 not of the leaft importance; icoof them being comprifed 

 in one revolution of the micrometer fcrew ; and in the in- 

 ftance before me, 56 of them made no more than a fecond, it 

 is not pretended that one of tliefc parts was feen beyond a 

 doubt being fcarcely .jjji^. dtli of an inch, much lefs the 

 tenths, as exhibited in the tables ; but as they were vifible 

 upon the micrometer heads, it was judged beft to take them 

 into the account. 



Having now completed the two firft feftions of mv me- 

 thod of dividing ; namely, the firft wiiich confifts of making 

 256 fmall round dots ; and the fecond in finding the errors of 

 thofe dots, and forming them into a table ; I come now to 

 tlie t!:ird and laft part, which confiils in ufing the erroneous 

 dots iii comparifon with the calculated errors, lo as ultimately 

 to make from them the true divifions. 



It will here be neceffary to complete the defcription of the 

 remaining part of the apparatus. And firil a little inftru- 

 ment which I denominate a fub-dividing feftor prefcnts it- 

 felf to notice. From all that has hitherto been (aid, it mufl 

 have been fuppofed that the roller itfclf will point out, upon 

 the limb of the inllrument to be divided, fpaces corre- 

 fponding to others previoufly divided upon itfelf, as was done 

 in fetting off the 256 points ; but, to obviate the difficulty 

 of dividing the roller with fufiicient exaftnefs, recourfe 

 was had to this feftor ; which alfo fervcs the equally im- 

 portant purpofe of reducing the bifedtional points to the 

 ufual divifion of the circle. This feftor is repreftnted ia ' 

 full dimenfions hy fg. 2; it is formed of thin brafs, and 

 centered upon the axis at A, in contaft with the upper fur- 

 face of the roller; it is capable of being moved round by 

 hand ; but by its friftion upon the a.iis, and its prefiure upon 

 the roller, it is fufficiently prevented from being difturbed by 

 accident. An internal frame B B, to which the arc C C 

 is attached, moves freely in the outer one, and by a fpring F) 

 is puflied outwards, while the fcrew E, whole point touches 

 the frame B, confines the arc to its proper radius. The 

 arc of this feftor is of about four times greater radius than 

 the roller, and upon it arc divided the fpaces, which muft be 

 transferred to the inllrument, as reprefented on a magnified 

 fcale hy Jig. 6. Now the angle of one of the fpaces of the 

 circle will be meafured by iixteen times its angular value 

 upon the fedlorial arc, or 2i 30'; but this does not repre- 

 sent 



