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them, the author proposes a means of obviating that inconvenience, 

 by substituting a microscope instead of one of the points ; and he 

 describes a method of proceeding, in which there is no need ever to 

 set the other point into any division already made. 



The beam to be employed for this purpose must have a fixed point 

 at one extremity, and at the other a centre of motion, round which 

 the length of the beam may revolve as radius. A microscope is to 

 slide in a groove along the middle to any required distance from the 

 point ; and in order that these may both be over the circle at the 

 same time, the centre of motion must be capable of adjustment, that 

 it may be fixed at a greater or less distance from the centre of the 

 circle, according to the magnitude of the arc intercepted between 

 the point and microscope. 



In dividing by continual bisection, the microscope is first to be 

 removed from the point to a distance nearly equal to the chord of 

 the half-arc ; and when the centre of motion has been duly adjusted, 

 and the wire of the microscope is made to bisect the dot at one ex- 

 tremity, a faint scratch must be made with the point. 



The beam having next been turned half round, and the dot at the 

 other extremity brought under the wire of the microscope, a second 

 scratch is made with the point, which, if the distance has been taken, 

 will be very near the former ; and the wire of the microscope will 

 easily be placed midway between them in the further process of bi- 

 section, which is again performed in the same manner, after the po- 

 sition of the microscope and of the centre of motion have been duly 

 altered. 



In laying down the real divisions from the marks thus made, the 

 centre of motion must be so placed that the whole length of the beam 

 may become a tangent to the circle ; and when the microscope has 

 been fixed close to the point, and the first dot brought under it, the 

 first division is to be marked, and the rest in succession till all are 

 made. 



Since the entire arc of a circle cannot be divided to degrees with- 

 out trisection and quinquesection, Mr. Cavendish describes three 

 methods of quinquesection. which it would be difficult to render in- 

 telligible without reference to the figures which accompany his paper; 

 and he also makes an estimate of the comparative accuracy attain- 

 able in bisection, trisection, and quinquesection. 



As it would be difficult to place the centre of motion accurately, 

 so that the point and axis of the microscope shall both be in the 

 circle in which the divisions are made, it becomes necessary' that 

 the wire of the microscope should be placed truly at right angles 

 to the length of the beam ; for then, although the point of intersec- 

 tion of the circle with the wire of the microscope is not accurately 

 in the middle of the wire, still, when the beam is reversed, the point 

 of intersection will lie at an equal distance on the opposite side of 

 the centre, and will consequently be at a given distance from the 

 fixed point of the compass. 



