GRADUATION. 



larger inftniments, and for the finirtiing of the dividing 

 engines thenilclvcs. 



In the year after Mr. Bird had publiihed an account of 

 his method of dividing aftronomical inftruments, the due dc 

 Chaulncs printed in French a folio work, entitled " Nouvelle 

 methode pour divifer les Inllrumens de Mathematique ct 

 d'Allrononiie," and alfo a work proper to be bound with 

 the former, called " D'--fcription d'un Microfcope et de dif- 

 ferens Micrometres deftines a mcfurer des parties circu- 

 laires ou droites avec la plus grande precilion." The former 

 of thefe pamphlets contains 15 plates, and the latter fix, 

 ■which exhibit plans and drawings in different points of view, 

 of all the apparatus, both in pieces and together ; thefe are 

 not neceflary to be copied by us, but we will defcribe them 

 as far as they fall within our prefent purpofe. This new 

 method of dividing is performed by the afliilance of com- 

 pound microfcopes, with crofs hairs in the focus of the eye- 

 glafs of eacli, to be fixed to the circle to be divided in any 

 given fituation, and of moveable pieces of brafs with fine 

 dividing lines marked thereon, which may be fo fixed with 

 wax, as to be adjuftable to the point of interfeclion in the 

 focus of any of the microfcopes, and when duly adjuiled, a 

 Aiding pointril, moving in a complex frame of brals, cuts 

 the line, on the circle to be divided, that fhall correfpond to 

 the point of the circle v.-here the adjuftable divifion is fixed ; 

 though the line cut is not in the fame part of the circle. 

 As this method may be confidered as the ground-work of 

 Ramfden's method of dividing a large circle, and has not, 

 that we know of, been tranilated into our language, we will 

 give the fubttance of tliat part of the work which more imme- 

 diately relates to our prefent fubject, without following the 

 author through all his minute detail?. The circle, which 

 the due de Chaulnes propofes to divide by his original method, 

 is the table of his dividing engine, but as it is equally ap- 

 plicable to any circle that may be fixed concentrically on it, 

 we will fuppofe it an agronomical circle for altitudes or azi- 

 muths that is to be divided by his method. In the firll 

 place, he propofes to have from 30 to 40 thin pieces of 

 fmooth brafs about one-third of an inch long, and onc-fixth 

 broad, having each a fine ilroke drawn acrols, perpendicular 

 to the long fides and juft deep enough to be feen ; and fe- 

 condly, three compound m.icrofcopes are to be provided, 

 one to hejjxed diametrically oppofite the pointril, or cutting 

 point, that is carried in and out, /. e. towards and from 

 the centre of the circle by an adjuftable frame, and the 

 other two are adjuftable to any given points along the 

 plane of the circle to be divided. When the circular hues 

 are ftruck on, the adjuftable microfcopes, which we will call 

 A and B, are placed as oppofite to each other, in a diame- 

 trical line, as can be guefled, or roughly meafured by any of 

 the ordinary means, and a moveable divifion on a piece of 

 brafs already defcribed, is put by means of wax under each 

 of the two microfcopes, and moved by hand, till the lines, 

 which muft be in a radial direftion, fall refpeClively under 

 the centres or intcrfecting points of the fields of view ; tliefe 

 microfcopes, it muft be underftood, are fixed, not on the 

 table which Ijears the circle, and which is made fo as to re- 

 volve on a long vertical axis, but on a fixed or ftationary 

 furrounding frame, that is unconnefted with the table itfelf, 

 when the table is at liberty to revolve ; fuppofe now the 

 table and circle on it to be carried half round, while the 

 microfcopes remain fixed, in fuch a way that the moveable 

 divifion which was under the microfcope A, falls under the 

 centre of microfcope B ; in this fituation, if the oppofite 

 divifion falls under the centre of A, the circle is truly 

 bifedled ; but as this is nut Hkely to be the cafe at the firft 

 trial, the quantity that is over or fliort muft be adjufted, one 



half by the moveable divifion, and tlie other by the micro- 

 fcope, and the operation of revcrfing muft be repealed, and 

 the reclifications made, till the circle is found to be truly bi- 

 fected, both backwards and forwards, by the coincidence 

 of the divifions with the interfered points in each focus of 

 the microfcopes. The two microfcopes A and B may row 

 be removed, and the third or fixed microfcope muft be made 

 faft over one of thefe two dividing ftrokes, after the pointril 

 has been brought precifcly to occupy the place of tlie other, 

 in which fituation the pointril and the fixed microfcope will 

 ftand fo diametrically oppofite each other, that whenever an 

 adjuftable line is brought to bifeCl the field of view of this 

 microfcope, the pointril will be ready to cut a line on the 

 oppofite fide of the circle, and whenever a fet of adjuftable 

 divifions are properly placed, by the wax, on one (cmi-cir- 

 cle, the lines in the otlier femi-circle, directly oppofite thefe rc- 

 fpe(itively,^can be permanently cut, before the moveable pieces 

 are taken off, and afterwards thefaid linesfocut maybe brought 

 fucceffively under the fixed microfcope, in order that their 

 oppofites, where the moveable pieces were ftuck on, may, 

 in like manner, be cut ; fo that one-half of the circle will b-." 

 fulBcient to be divided and fub-divided by the adjuftable 

 pieces, feeing that each of its divifions and fub-divifions, 

 brought in fuccefiion under the fixed microfcope, may be 

 inftantly transferred by the pointril, into the oppofite femi- 

 circle, and alfo thefe in their turn transferred back again. 

 Let us now fee how the femi-circle is divided and fub-divided 

 by means of the two microfcopes A and the fixed 'one, B 

 being no longer wanted ; the firft operation is t!ie trifeClion 

 of it into arcs of each 60 ; for this purpofe, wliile the poin- 

 tril remains at T, (Ji^. 3. of Plate XVIII.) one of the points 

 cut, 11/2. at 180S and the fixed microfcope refts at Z, or 

 zero, /. e. the firft point of the circle, to be figured o or 360 ; 

 two moveable pieces are ftuck on at C and D refpcctively, 

 fo as to trifetl the femi-circle Z B T very nearly, and the 

 microfcope A is placed directly over C, fo as to view its 

 ftroke or line at the point of interfeclion in the focus of the 

 eye-glafs ; there it is made faft for the prefent ; the line C is 

 now brought under the fixed microfcope at zero, by making 

 the circle revolve, ar.d fuffored to remain ; the piece D has 

 its ftroke put under the microfcope A, as before was the cafe 

 with piece C, now at zero ; the circle is again made to re- 

 volve till the ftroke of D is under the fixed microfcope at 

 zero ; in this fituation the point T, or 180, will have been 

 brought forward 120', or 60' twice over, provided the 

 pieces C and D were truly placed, and in this cafe an eye, 

 looking into the microfcope A, will fee this dividing line bifeA 

 the field of view ; but as tliis is not likely to happen at the 

 firft trial, the microfcope muil be moved one third of the 

 error now ftiewn, and each of the pieces C and D another 

 third, accordingly as the portions of the femi-circles Z C, 

 C D, and D T taken together, are found to be more or lefs 

 than an exac^ femi-circle. The operation we have here de- 

 fcribed is repeated again and again, till the three equidif- 

 tant arcs amount to an exacl femi-circle ; after which they 

 are rendered permanent, by ftrokes made fucceffively on 

 the oppofite femi-circle, at the points c and d, which tri- 

 feCt the femi-circle T A Z, when tlie adjuftable ftrokes C 

 and D are in fuccedion exactly fixed in the centre of the 

 field of view of the fixed microfcope. The diftance between 

 the two microfcopes, which is here 60 , is called the owning 

 of the microfcopes, which are confidered as the two poinn 

 of a pair of dividers, and the arc between them is therefore 

 denominated accordingly. 



The next flep is to bifect the arcs of 60^ each, into two of 

 30 , by an opening of 30 ; to do this one moveable piece, 

 iUick at the middle of each arc of Cxf, will be requifite, and 



the 



