GRADUATION. 



Problem I. 



" To cut off from any given arc of a circle a third, a fiftli, 

 a fevcnth, &€. odd parts, and thence to divide that arc into 

 any number of equal parts." 



jExamph I . — To dividethe arc A K B into three equal parts, 

 C A being the radius, and C the centre of the arc. Bifect 

 AB, _/j. 1. of PA?//? XXI. of yljlroiiomical Injlntments, in 

 K, draw the two i-adii C K, C B, and the cliord A B ; pro- 

 duce A B at pleafure, and make B L =: A B ; bifeiA 

 A C at G ; then a rule on G and L will cut C B in E, and 

 B E will be J, and C E j of the radius C B ; on C B 

 witii C E, defcribe the arc Y.ed ; laiUy, fet off the extent 

 E « or D e from B to a, and from a to b, and the arc A K B 

 will be divided into three equal parts. 



Corollary. — Hence, having a feKtant, quadrant, Sec. ac- 

 curately divided, s the chord of any arc iet off upon any 

 other arc of i that radius will cut off an arc fmiilar to the 

 iirft, and containing the fame number of degrees. Alfo, 

 ^d, ith, -'th, &:c. of a larger chord will corillantly cut 

 iimilar arcs on a circle whole radius is jd, jth, 'th, Sec. of 

 the radius of the iiril arc. 



Example 2. — Let it be required to divide the arc A K B, 

 of the fame iigure, into five equal parts, or to find 'th ot 

 .the arc A B. 



Having bifefted the given arc A B in K, and drawn the 

 three radii C A, C K, C B, with radius C I defcribe the 

 arc I n M, which will be bifeftcd in n by the line C K ; 

 then take the extent In, or its equal Ma, and fet it off 

 twice from A to B ; that is, firil from A to a, and then 

 from a to o, and o B will be ' th of the arc A B. Again, 

 fet off the fame extent from B to m, and from m to c, and 

 the arc A B will be accurately divided into five equal 

 parts. 



Example 3 To di\ide the given arc A B into feven equal 



parts. A B being bifeCted as before, and the radii 

 C A, C K, C B, drawn, find (by a problem referred to) 

 the feventh part P B of the radius C B, and with the radius 

 C P defcribe the arc P r N ; then fet off the extent P r 

 twice from A 10-3, and from 3 to 6, and 6 B will be the 

 feventh part of the given arc A B ; the jompaffcs being 

 kept to the fame opening P r, fet it from B to 4, and from 

 4 to I ; then the extent A i will bifett i 3 into 2, and 4 6 

 into 5 ; and thus divide the given arc into feven equal 

 parts. 



It is obvious, that this method of dividing any portion of 

 a circle, into an odd number of equal parts, is lubject to 

 three iources of error in prattice ; iff, the variation of tiic 

 compafs from expanfion ;'2d, the uncertainty of getting the 

 exa& points of interfectidn where the angles are acute ; and 

 3dly, the probable deviation of tlie points in ilepping ; to 

 fay nothing of tlie errors arifing from drawing the right 

 fines, which would, in all probability, exceed all the otiier 

 errors put together, even before they are doubled and 

 trebled, 5cc. by Ilepping. 



PaoBLE.xr 11. 



" To divide a given arc of a circle into any number of 

 equal parts by the lielp of a pair of beam, or otli>r com- 

 paifes, the dilbance of whofe points ihall not be nearer to 

 each other than the given chord," (by Clavius). 



Let A B, yf^. 2, of Plate XXI. be the given circular 

 arc to be divided into a number of equal parts. Produce 

 tbe arc at pleafure ; then take the extent A B, and fet it 

 oil', on tlie prolonged arc, as many times as the given imall 

 arc is to be divided into fmullcr parts, nauielv, to the points 



Vol. XVL 



C, D, E, F, and G. Divide no^v the whole line A G into 

 as many equal parts as are required in A B, ns G H, H It 

 I K, IvL, L A, each of whicii contain i the given line, and 

 one of thofe parts into which the given line is to be divided. 

 For A G is to A L, as A I" to A B ; in other words, A L 

 is contained five times in A G, as A B is in A F ; therefore, 

 fince A G contains A F, and 4th of A B, B L is !th alfo 

 of A B. The"., as G H contains A B, plus F H, which 

 is 4tii of A B, i: I will be the ?ths of A B, D K iths, 

 and C I nhs. Therefore, if we fet off the interval G H 

 from F and H, we obtain two parts between F and I ; the 

 fame interval, or extent, let off from tliefe two points near 

 1, gives three parts between D and K; when fet off from 

 the points from D to K, it gives four parts in C L ; and 

 the next transfer will, from thofe points, give five parts 

 from B to A ; fo that, laflly, the fame extent will give 

 back again the remaining divifions in fuccellion from thofe 

 between B and C. 



This method is liable to fome of the fame fources of 

 error as the preceding method, when compaffes alone are 

 depended on ; but it is ufeful, acconiing to the due de 

 Chaulne's mode of proceeding, and may be fernceable in 

 Ramfden's method of dividing, where tiie points are refti- 

 fied by oppofite microfcopes. In all probability the ver- 

 nier fcale owes its origin to this problem of Clavius, which 

 problem may be varioufly diverfificd, to prevent the necef- 

 iity of fmall extents ; but whenever fmall fpaccs are marked 

 out on an arc by a differential plan of this fort, it is requi- 

 fitc that the extent begun with (liould not alter during the 

 whole proccfs, and alfo that the points, once mark-d, (liould 

 be capable of being refumed with certainty at tl'.e fubfe- 

 quciit transfer. \Vhen, however, an error is made in any 

 divided fpace by a hard particle, or otherwife, thij vm r 

 will recur at every multl^'le of the extent meafurcd there- 

 from. 



This problem of Clavius, it will be remarked, implies 

 the given arc A B to be meafured, or otherwife knows 

 previoufly to the propofed fub-divifion. 



When an entire circle is propofed to be divided int» 

 degrees, the radius, whicli is equal to the chord of 60', 

 affords the means of making fix equal arcs ; and thcfc 

 arcs may be fub-divided to arcs of 15° each bv bifetlioR 

 only ; but to leduce the equal arcs to a iHll lower denomi- 

 nation, recourfe mull be had to either trifeCtion, quinqucfec- 

 tion, computation of the chords, or the differential method 

 originally propofed by Clavius, all wkidi have been already 

 dclcribcd. 



The method of dividing a circle, propofed hv L. Maf- 

 cheroni, is tranilated into French by A. M. Carettf, l7yS, 

 and is contained in the lecond book of his " Geometric du 

 Compas." This method rejects not only the drawing of 

 lines, but all meaiurement from fc.iles, and bileclioiis of an 

 arc by trial, as well as tritections, quinquefeC'tions, &c. 

 but adnfits of ilepping, and fuppofes the extent of a pair of 

 compaffes, once taken, to be atterw.u'ds invariable. The 

 radius of the circle is the balis of .:11 tlie other extents, which 

 are very few in number, conlldering the various divitiont 

 that may be made therewith, and three points determined, 

 one without and two within the circle, afford the means of 

 taking all the meafurements, inlleadof a fcale; coni'equently, 

 any circle divided by .this method mull neceffarily have its 

 plane extended conliderably beyond and witliiH the circular 

 fpace to be divided, which is felJom the cale in a large in- 

 ilrument, wliere a ring or rim is attaclied to radial bars,. 10 

 form, a wheel .for the body of tiie intli:ument,_w!iich con- 

 llrudion contributes equally to ilrengili and ligiiUiefs ; a 

 4 A ' li^t 



