50 PHILOSOPHICAL TRANSACTIONS. [aNNO 1786. 



chaps to 59 holes inclusive, which will fill up the whole space from o to the 6oth 

 division before obtained. You now again have recourse to the straps, and 

 placing them one on each side the hoop-plate, you put the pin through the 120th 

 hole of the straps, and through the hole marked o on the hoop- plate; and regu- 

 lating the steel plates to the hoop-plate as before, you drill and form a hole 

 with the steel plates, which will correspond with the 120th hole on the hoop- 

 plate; and afterwards filling up the 59 holes wanting, by means of the chaps, 

 you then have all completed to the 120th division, which is -ji^- of the whole circle. 



You then proceed, in like manner, with another set of 120 holes; that is, 

 placing the 6oth hole of the straps to the 120th hole of the hoop-plate, and 

 from it producing the 180th hole; you, in like manner as before, fill up this 60 

 with the chaps; and afterwards placing the 120th hole on the straps in the 120th 

 hole on the hoop-plate, you will obtain the 240th hole; so that filling up this 

 last set of 60 divisions, you have obtained 241 holes, including 240 spaces or 

 divisions of the hoop; and repeating this process 10 times more, you will in like 

 manner obtain 1441 holes, comprehending 1440 spaces.* And this process, 

 being carried on in temperate weather, the manner of working produces 12 simi- 

 lar operations, wherein the materials and tools concerned will not only be subject 

 to very little change of temperature, but that change, whatever it is, will be 

 nearly similar in each set of 120 holes: we may therefore infer, that the greatest 

 inequality, or indeed any that can be sensible, must be at every 60 divisions, that 

 is, between the 59th and 6oth, and between the 1 19th and 120th, both which 

 will be equally repeated 12 times in the whole length which is to compose the 

 circumference of a circle, and which will thus be checked 12 times in the circum- 

 ference, and 12 times more at the intermediate distances; that is, with 12 master 

 checks, and 12 subordinate ones, in the whole round. 



It is proper here to observe, that in M. Roemer's method even 60 divisions 

 could scarcely be trusted in an affair of great accuracy, on account of the ob- 

 jections already made, arising from the points having such slight hold in the 

 surface of the brass ; but here the parts are held so exceedingly firm, and the 

 operation carried on with so much power, that any small inequality in the hard- 

 ness of the brass, or irregularity of surface, cannot be supposed to affect the 

 place of the centre of the hole ; nor will any small inequality that may be sus- 

 pected from the wear of the steel holes sensibly affect the centre of the hole, to 

 which every thing is ultimately referred. 



Method of Joining the Hoop. — A more happy thought than that of Hindley's, 



* It will be proper, for reasons hereafter to be mentioned, to continue the divisions to 20 holes 

 more, making in the whole 146 1 holes, — Orig. 



