GRADUATION. 



8at nieet of metal, which the prefc-nt method of dividing 

 requires, is feklom, if ever, adopted in the conftruftion of an 

 inftrument of confiderable radius. The fecond book of the 

 " Geometry of the Compafs,'' which has not yet been 

 tranflated into EngHfh, and which therefore may not be 

 known to many of our readers, contains ten problems on the 

 divifion of the circumference of a circle, and one on the man- 

 ner of biferting an arc by a new procefs, that makes no 

 fuperfluous marks, and requires no fecond trial : thefe pro- 

 blems are accompanied by dcmonftrations and coroilariei, 

 that would enlarge our article too much, were we to tranf- 

 late them entirely, but we will give fo much of tlic fubftance 

 ef tliefe problems, as will enable tlie reader lo comprthcnd 

 the nature and extent of the divifion in queftion, which we 

 apprehend will be deemed more curious than ufeful. The 

 ten problems' divide the circle into the following parte, ac- 

 cording to the order in whicli they here iuccr-ed one 

 another, -viz. 4, 8, t2, 24, 48, 9, 10, 120, 20, and 240. 

 Let Ji^. 3, of PIMi; XXI. be a diagram for illullrating the 

 four firft divifions or problems. I^et one beam, or other 

 kind of compafs, defcribe the circle with the extent BA 

 from the centre A, and let this compafs liave its extent pre- 

 ferved unaltered, to reprefent the radius, and be called llie 

 firil comoafs ; with this extent and beginning at 15, as zero, 

 turn over to C, T), E, cf, c, and back to B agam ; now if 

 this is accurately done, B A E will bileft the circL.', B D c/ 

 .^yill trifecl it, and the fix original points will divide it into 

 fix arcs of 60 each ; take now B D, the chord of 120 in 

 the fecond compafs, and preferve this extent unaltered ; from 

 zero B and its oppofite point E, bifeft the point a out of 

 the circle, with this extent ; then will the dillancc a A, taken 

 ia a third compafs, and kept unaltered, be the chord B F 

 of 90 ; alfo of the arc E F, of E. f, and ofyB ; the circle 

 is now divided into four quadrants at the points B, F, E, 

 and f. To obtain an eighth part of the circumference, fet 

 A B, by the compafs N° I, from the point a, out of the 

 circle, to G and H in the arc on each fide ; then the femi- 

 circle is divided into four, and the points g and h, in the 

 other femi-circlc, may be transferred from G to g, and from 

 H to h, with the compafs N 3, or chord of 90", and the 

 whole circle will now be divided into eight equal arcs by the. 

 points B, G. F, H, E, h,j, g. 



To gain twelve equal arcs ot the circle, with N^ 1, or ex- 

 tent equal to radius, from F as the firil point turn over to 

 N, and n, in fucceffion one way, and to O and in fuc- 

 ceflion the other ; and the whole circle will be divided into 

 twelve equal arcs by the points B, N, C, F, D, O, E, 0, il, 

 f, c, n. To double this number of equal arcs, begin at G 

 with the fame extent of radius, and ilep to L and M one 

 way, and to k and / the other ; alfo from H with the fame 

 extent ftcp to I and K one v>-ay, and to m and / the other 

 way, which points will bifeft the former twelve arcs, and 

 make the whole 24. Hitherto we have had occafion for no 

 Other point but a for determining the 24 arcs, and it is fome- 

 what remarkable, that the three extents afcertained thereby, 

 -viz. the radius, the chord of 90 , and the chord of i2o\ 

 are to each other in the fimple ratio of the fquarc roots of 

 I, 2, and 3 refpeftively. In order to fub-divide, without 

 trial, the 24 arcs into 48, another point e within the circle 

 is necefi'ary ; with a fourth compafs take the extent a N, 

 and find the point e by iiiterfettion from B and its oppofite 

 point E ; then N i, or radius, will cut the circumference 

 from this point c in the points jj. and » ; fo (hall K ^z, f* N, 

 M V, and » O be each equal to the 48th part of the whole 

 circumference. To divide the circumference into five equal 

 parts, a third point b within the circle, and in the fame dia- 

 mticr v/^h e, but at the oppoSte fide of the centre, will l^c 



neceflary : to avoid confufion from further fub-divifioti, we 

 will take another fg. (4) with fimilar letters of rcferenccr 

 as the preceding one. With N ' 3, or extent a A of 90% 

 irterfeft at the point b from the points N and O, then with 

 a fifth extent B b fet off from B to Q, and the arc B Q will 

 be the fifth part. of the circumference. To bifeft thcfe five 

 large arts, take the fixth extent A b and fet it from B to P, 

 which will be equal to the arc P Q, and, confequentlv, will 

 be a tenth part of the circumference. Again, w ithout any 

 further operation, the fmall arc () I, already marked, will 

 be the 120th part of the circumference ; and to obtain ^Vth 

 part of the fame, in the quadrant BV_/', the fifth extent 

 B b, fet off from /' to V, will leave the arc B V equal to 

 this quantity. And laltly, to procure 240 equal arcs, take 

 the extent s ■., and fet it over from i to /?, and from I? to <% 

 fo lliall the arc P -5 be the j j^th part of the circumference ; 

 but to fubdi\ide all the circumference by the fmall arc P '^ 

 of I- 30', the arc N G of ly'' mufl. firil be divided into five 

 parts of 3 ' each, without quinqucfeclion ; thus, with the 

 extent Kb = V> V, itep from B to the points P, Q, R, S, 

 and it will fall next in E already marked ; then with A b, 

 as before, begin at L and put in the points q and p fuc- 

 celfively ; alfo from I jmt in tt ; and from O put in (p, by 

 flepping the points j, y, and >' ; thus fliall the points p Y v 

 and ^ lub-divide the arc NG into five equal arcs, and i'lr 

 will be found equal P J, or , \ .th part of the whole. After, 

 having obtained thete five fub-divifions of 3 and 1 30', the 

 firit may be transferred, with the extent N G for inftance, 

 into other arcs of 15 in fucccffion, begimiing with G C firil, 

 and ufmg the points N, p, P, S:c. as they occur, 'ill the 

 whole is gone through, afier which the fmall arcs thus tranf- 

 ferred may be bifected by an extent J N, going a fecond 

 time over the fame points in fucceffion as before. Thus will 

 the whole circumference be fub-divided into 240 parts of 

 each 1" 30'. 



The eleventh or laft problem on this fubjeft, is to divide 

 any arc B C {Jig. 5.) into two equal parts in G. To do 

 this; with radius A B, which has defcnbed the arc B C to 

 be divided, and from the centres B and C, the two extren i- 

 ties of the arc, let the arcs A D and A E be defcnbed ; 

 let A D and A E be each made equal to B C ; then from 

 the points D and E, as centres, and with radius D C or 

 E B, find by intcrfeftion the point F ; lattly, with the 

 radius A F, and from the centres D and E, let the point G 

 be interfered, which will fall in the circumference, if the 

 operation is truly performed, and will allobillft the arc B C 

 as was re(|eii-ed. 



By the lielp of the three remarkable points a, e, and b, 

 in our Jig. 4, the author of the Geoir.etry of the Compafs, 

 has laid down in his twelfth and laft book a dozen equa- 

 tions, fome one of which may be applied to determine, by 

 approximation, almoft. any fmall arc of a circle that fhall 

 cut off a given quantity. It would be tedious to enter here- 

 minutely into his various calctdations, but, for the fake o£ 

 amufing the reader, we will feleit the folufions of a few 

 probh-ms, without the annexed demonllratiuns, which would 

 enlarge our article loo much. 



PKoni.rM I. 



To Jind the' arc of one Jigrte luilL'Out the error of lalj a 

 fecond. 



Soluthii. — I..ct the arc B c, below B, be 55" 30', (in the 

 circle which has been divided into 240 parts of each l^ 30',) 

 take the dillance b-z', and from the pomt e, as a centre, de- 

 fcribe an arc to cut the circumference in fome point Z', and 

 the arc B Z', above B, will be 52 59' \^'.'',, or 53° within 

 about 25'"; then iu tlie divifions between B and V is the 



diviiloa 



