GRADUATION. 



of the divided circle, the an, fimply confidered, will not and unlefs the fteady pins fill their holes complttely, thefc 



be the true mealure of the on^/-, feeing one fide of the pins will, in all probabihty, be drawn afide by the fixing 



circle will be farther from the centre than the other, fcrews in fome of the poutions. fr »i„ ...u„.,i .„ ....i '-..^ 



For i 



to the 



centre of motion, placed a fmall quantity out of the centre 



of the graduated circle, in this cafe the angles fubtended by 



the iiril and fourth qnadrantal arcs will be greater than 



thofe fubtended by the fecond and third ; but if tlie hne 



from Cjo ' to the centre of motion (liouldpafs through the centre 



of the divided circle, then the angles fubtended by the tirll and 



fecond quadrantal arcs will be greater than thofe lubtended 



by the third and fourth. Wlieu, however, the microfcopes 



have a crofs iiair, as well as a radial one, the length of the 



raiiiiis as well as of the air may be meafured, and confe- 



quently the true quantity of the fubtended angle. 

 - The great fimiliarity between the due de Chaulne's method 



of dividing a circle, and of profeffor Lax's method of ex- 

 amining one when divided, cannot but Itrike the reader ; 

 they both ufe microfcopes, one with a micrometer attached, 

 and both take multiples of the arc under confideration ; but 

 one correds the errors by adjuft menf during the att of 

 dividinsr, and the other calculates and tabulates tlicm for fub- 

 lequent corrections 



Neither will the l!!:'^nefs pafs unnoticed between the ap- 

 paratus employed by Mr. Lax, and that for the purpofe of 

 examining llandardmeafures, defcribed by fir Geo. Shuck- 

 burgh (Phil. Tranf. I7(;S) ; but Mr. Tronghton, we 

 know, acquits the learned profeffor of borrowing any thing 

 from him. 



If the wheel is iruly racked 

 ^nli"ance,"'if w'e"fiippofe the line that paffcs from zero in the firft inftance by a careful divider, and the ileady 

 le centre of the divided circle, (liould alfo pafs through the pins placed with precificn, the reverfing of the rim will not 



" " difturb the coincidence of the two rows of teeth all round, 



which is the thing ultimately defired ; but if the original 

 divifion was bad, and had both negative and pofitive devia- 

 tions from true divifion, in dilTerent parts of the circle, then 

 we have a doubt whether the rim will produce the defired 

 effetf of equahzing them : a good fcrew will of itfelf equa- 

 lize contiguous teeth, when fifteen teeth or upwards are 

 afted on at the fame time ; but it will not equalize diftant 

 teeth, wliich the rin^. profefles to do. In order, however, to 

 anfvver the purpofe of a fs!j-corrc8cr, the rim muil be com- 

 petent to meet all cafes of bad divifion that are liable to 

 occur ; let us fuppcfe, in the firft place, that whenever 

 large teeth are made by the divider on one fide of the 

 wlicel, teeth correfponding in fmalhiefs are to be met with 

 at the oppofite fide, in this cafe, which is the moft favourable 

 to the operation of the rim, the errors being oppofed to each 

 other,by an oppofite pofitionof therim, will correct each other, 

 provided they arc of fimilar quantity, and the rim will pro- 

 duce its defired cftecl ; but, fecondly, if equal errors happen 

 to be in the fame direction they will not be corrected at all ; 

 and, thirdly, if unequal errors be in the fame direftion, they 

 will be correfted by only half their difference ; nay, arcs 

 of the wheel, that are true at firft, will, we conceive, be 

 vitiated, unlefs they fall in with arcs of the rim that are 

 alfo true. How far a change of temperature will affect 

 this long procefs we will not affirm ; but we hefitate not to 

 ■ In the fpring of 1810, Mr. James Allan, of Blewitt's o-ive it as our opinion, that an upper plate, or wheel, revolv- 

 Buildings, Fetter Lane, London, received the gold medal ing on the fame centre as the wheel itfelf, would be lefs liable 

 from the Society of Arts, at the Adelphi, for a new method to central error than the rim with fteady pins, however care- 

 of equalizing the teeth on the edge of an engine plate for fully fixed ; for fteady pins could not then caufe any fenfible-. 

 dividing fextants. Sec , which would have been defcribed eccentricity in fixing at the different pofitions. 



moft properly under the article ExciNE, but as it was not 

 then made known, we will take fome notice of it in this 

 place, though the Society's account is not yet publifiicd. 

 The profefTed improvement of Mr. Allan's method of rack- 

 ing, confifts in its enabling a bad divider of a circle to correft 

 his divifions, by what he cal's ?i f:lf-corrcYling proceis. The 

 contrivance is this: the ufi>al wheel or circular plate that is 

 racked, has a brafs rim of about two inches breadth, and of 

 the fame diameter and thicknefs as the wheel, rabbeted over 

 it, fo as to appear apart of the fame wheel; four fteady 

 pins, at the exaft diftance of a quadrant from each other-. 



It would have been defirable if Mr. Allan had given* • 

 with his account of the racking, his method of making a per- 

 feft fcrew, which is cfTential in a dividing engine. We lately 

 had an opportunity of examining one of his engine-divided 

 reflecting circles, and found, however, the oppofite verniers 

 very well correfponding with each other all round the cir- 

 cumference. 



Gr.-VDUATION of a Circle geovu-lncally. After the de- 

 fcriptiun v>-e have given in our preceding article of the 

 various practical methods of graduating allronomical inllru- 



keep the rim in its place by the aid of fevrral equi-diftant ments, it may, perhaps, feem fuperfluous to give a further 



fcrews, and the rim will therefore admit of four pofitions account of the methods that geometricians have propofed 



on the wheel. When the wheel and rim together have for graduating a circle theoretically ; but which have been 



been faintly racked in the ufual way (fee ExGlXE by of little or no ufe in the practical divifion of circles, or cir- 



Ramfil.n) one of the other pofitions is given to the rim, cular arcs, on inftruments. It is not fufficient for tlie pur- 



fuppofe the oppofite one, and then the operation of pofc of the inftrument-maker, that a probh'.n be true merely, 



racking by the fcrew is refumed ; when the teeth are a but that the means necefiary for its adoption in pradtice be' 



little formed, another pofition is given to the rim, and not fnbjeiEt to the introduction of error. Tlie ancient 



the racking refumed as before to a certain extent, and, after geometry implies the pofTibillty of making a ftraight line 



feveral changes have been gone through, it is prefumed in pradtice through two given points, whicli cannot be 



that the teeth become nearly equalized, and are exadtly effected with certainty by a ruler and pointed too!, howevvr 



equal to one another at the conclufion. How far this will carefully applied, while the modern geometry of Mafche- 



be precifely the cafe we will not pretend to afllTt. We roni, the Italian, rejedts the ufe of lines altogether, and fiib- 



conceive it to be a very nice operation to drill four holes ftitutcs the compafles only in the divifion of a circle. Vk\- 



in the wheel and rim fo exactly equidiftant and concentric, will gratify the curiofity of the reader with a few fpecimei.i 



that a reverfed pofition fliall not affect their relative fittings, of each method in fucceffion. 



and if there is the leaft deviation in this refpeft, it will be I„ Adam's geometrical and graphical effays are the f -] 



charged as an error on the divifions. The fcrews alfo pro- lowing problems, wliich imply the practicability of drawn., 



fefs 10 be placed fo as to admit of reverfing in pofition, a ftraight line through two given points. 



8 Pkou- 



