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 hisect 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 quinquescction, Air. 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 quinquescction.
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 intersection of the circle with the wire of the microscope is not accurately in the middle of the wire, still, when the beam is reverscd, the point of intersection will lie at an equal distance on the opposite side of the centre, and will consequently he at a given distance from the fixed point of the compass.
