GRADUATION. 



tlic two ha'.vc-i, to tVie right 3nd left of each piece, may be 

 examined and iidjuflt-d till the bifetlions are complete, and in 

 a rtate to be transferred into the oppofite femi-circle ; for 

 inllance, fnppofe the arc of 60 ' from Z to C to be bifefted, 

 put a moveable ftroke at 30 \ more or kfs, and place the mi- 

 crofcope A over it ; make the table and circle revolve till 

 this llroke 30 ' falls under the fixed microfcope ; in that fitu- 

 Btion, the llroke 60^ ought to bifeft the field of view of mi- 

 crofcope A, and whatever the deviation may be, plus or mi- 

 nus, it muft be adjuiledby trial, partly by moving the mi- 

 crofcope A, and partly by moving the piece with the ftroke 

 on at 30', when brought back again under the microfcope A. 

 In t!ie fame way, any other bifedion may be made within the 

 poiTible opening of the microfcopes. Again, the arcs of 

 30' arc trifefted, as before defcribed, into arcs of lo\ 

 which arc all transferred into tiie oppofite femi-circle in fuc- 

 ceifiun . 



With refpeft to the divifion of the arcs into fmaller quan- 

 tities than thofe of 10% where the microfcopes approach 

 too near together to admit of the neceflary adjuftments, an 

 ingenious mode of fub-dividing is adopted by a property of 

 the number 9, thus; having all the arcs of lo' round the 

 femi-circle laid down, the points 90' and 180^ are of courfe 

 among them, and as thefe are divifible by 9, an opening of 

 nine degrees is taken by trial, which repeated ten times, by 

 means of the moveable ftrokes, will reach from o to 90, 

 when properly adjurted ; befides the 10 ', 20'', 30', &c. there 

 will now be 9 , 18", 27 ', Sec. up to 90', and alfo 99^, 108 ', 

 117'', &c. up to 180°, and the differences or fpaces left be- 

 tween the divifions obtained from io'\ and thofe from 9^ re- 

 peated, will be refpeftively i', 2", 3% &c. up to 9°; now, 

 if the points 9', iS'", 27', &c. which numbers decreafe by 

 unity in the units place, be made in fucceffion points of com- 

 mencement, for the arc of 10^ to be repeated from, they 

 will from 9* produce 19°, 29°, 39 , &c. ; from 18" they will 

 give 28'^, 38 ', 48^ &c. ; and from 27 ' tliey will give 37 ', 47°, 

 57°, Sec. and fo of the reft, till all the fingle degrees are put 

 on by the arc of 10^ thus apphed from the points derived 

 from 9^ and its multiples. The work is performed thus: 

 make tlie opening of the microfcopes again 10, by fixing 

 the on" called A at the tenth degree when zero is at the fixed 

 one, then turn the table and fuperincumbent circle to be di- 

 vided till the ftroke 9 -• cemcs to zero, then a moveable piece 

 ftuck properly under A will be 19 ; bring this to the fixed 

 microfcope, and the place for 29' will be had for the next 

 ftroke, under A again ; and in like manner every individual 

 degree may be marked in fuccedion, by taking thofe ending 

 witli 8, after thofe ending with 9 have been gone through ; 

 and after that, thofe ending with 7, with 6, with 5, &c. till 

 all the numerals are included. If the circle is large enough 

 to admit of divifion to y' by the opening of the micro- 

 fcopes, the numbers 4 and 5 may be fubftituted with advan- 

 tage for 9 and lo, and then tlie opening may be 20' for in- 

 ferting them, inftead of 10 , as in the mode we have de- 

 fcribed. When half degrees are required to be inferted 15' 

 may be bifefted, and the line obtained will be in the middle 

 between 7 ' and 8", from which ftroke, with an opening of 

 15°, all the other half degrees may be marked as before, by 

 the aid of the adjuftable marks to be transferred afterwards 

 into the oppofite femi-circle, and all the ftrokes in that femi- 

 circle may then be transferred back again permanently, fo as 

 to compleat the circle. 



The ule of the numerals nine and ten, or of the four and 

 five, is evidently borrowed from Clavius the Jefuit, whofe 

 problem for thus dividing a right line, or arc of a circle, 

 was publilhed in 161 1. 



Another method of fub-dividing the circle, when marked 



into degrees, as aboVe defcribed, is alfo propofed as being 

 lefs tedious, and capable of carrying the fub-divifions to a 

 lower denomination ; thus, a focket is made to fit the arbor 

 of the revolving table that carries the circle to be divided, 

 fo tight, that it will revolve with the faid arbor, or without 

 it, as the cafe may be : to this focket a telefcope ii fixed 

 with a vertical hair in the focus of its eye-glafs, that may 

 bifeft any diftant mark to which it may be direfted, then 

 a long ruler, of fix or feven feet in length, is divided, for 

 inftance, into twelve equal parts, fo that it may be read 

 diftinflly ; the diftance of this ruler is fo adjufted from 

 the telefcope, that when placed at right angles to the line 

 of coUimation, it may juft fubtend one degree, which may 

 be firft calculated nearly, and then adjulted by trial, till 

 tlic run of the telefcope over the twelve marks of the ruler 

 be exactly eorrefponding to the run of one degree of the 

 circle under one of the microfcopes. In this fituation of 

 things, the divifions of the ruler, as feen through the tele- 

 fcope, are, with great care and fteadinefs, transferred in 

 fucGelTion, not to one of the degree fpaces itfelf on the 

 circle, but to a pattern-piece of brafs, which may be after- 

 wards fixed with fcrews under the objeft lens of the fixed 

 microfcope, fo as to be applied in fucceffion to each feparate 

 degree, as the circle is made to revolve, during the operation of 

 transferring thefe five minute fpaces, to the oppofite parts of 

 the circle, by means of the pointril. In ftiiCtncfs, the ruler, 

 viewed through the telefcope, ought to be a portion of a 

 circle ; but the arc and the chord of one degree are fo 

 nearly alike, that one may be fubflitutcd for the other, 

 without fenfible error. When fmall circles, or parts of a 

 circle, were propofed by the due de Chaulnes to be divided, 

 they were fixed on the revolving arbor of the table, and 

 the microfcopes and cutting point were fo arranged, that 

 the divifions of a large divided circle were transferred to the 

 fmall one in fucceffion, and the whole apparatus, fo ar- 

 ranged, conftituted an engine for dividing ; but the work 

 could not be performed with fucli expedition as with our 

 modern engines, where flops are fubftituted for the micro- 

 fcopes, and where the touch has greatly the advantage over 

 X.\\e fight. With refpcft to the accuracy of the work per- 

 formed by the due de Chaulnes, we arc greatly inclined to 

 believe, that the flidi.ig mechanifm of tlie cutting-point 

 would not fecure the ftrokes from having a devious diredtion 

 fometimes, arifing from the neceffary hberty that the Aiding 

 parts muft liave had to adn.it of fiee motion, unlefs, indeed, 

 the parts of action were fitted «ith extreme nicety. 



Laftly, tlie vernier fcale was retained by the author 

 before us, as the bcft fub-divider of the 5' fpaces, or other 

 fmall divifions ; but he h.is not conceived it neceffary to 

 defcribe, in detail, how he proportioned it to the divifions of 

 his circle, though it is eafy to conceive how this may be done 

 by the help of microfcopes and the cutting-point. Our 

 remark on this original method of dividing is, that it (liews 

 great ingenuity ; but we conceive, that making an equality 

 in the divided and fub-divided arcs will not always eiifure 

 perfeftion in the angular meafura, becaufe the axis of motion 

 may be out of the centre of the divided circle, and the 

 eccentricity will neither be detefted by tliis method, nor 

 allowed for in the divifions and fub-divifions, unlefs, indeed, 

 two or three equidiftant verniers were ufed for taking an ave. 

 rage of the readings from different fides of the circle. It is 

 fomewhat remarkable that the fame thought that introduced 

 the microfcopes for meafuring the arcs to be divided, did not 

 fubftitutc them for the vernier, in reading off fmall qiraii- 

 tities of a fub-divifion when an inftrument was finifhed ; 

 which praftice was left for Mr. Ramfden afterwards to 

 adopt. The fecond femi-circle, being only the copy of a 



copy, 



