54 PHILOSOPHICAL TRANSACTIONS. [aNNO 1786. 



mounted ; in like manner as I suppose the index and cutting frame to have been 

 removed, to make room for the mounting of the screws. I now consider the 

 teeth of the dividing plate, so formed, as having all the equality that the present 

 known state of human art has pointed out; and the whole convertible on the axis 

 mandrel on which it has been originally formed, and the central hole of the plate 

 concentric with it : I therefore consider the ground faces of the teeth of the plate 

 as the actual divisions. It now remains to show how they are to be transferred, 

 to form the divisions of an instrument. 



Preparation of the Dividing Plate Jor Graduating Instruments. — If a small 

 cylinder of hard steel be duly polished, and made of a size so as just to chock in 

 between the extremities of the teeth, then the centre of that cylinder will be a 

 fixed point, in respect to the circumference of the wheel : -if another cylinder be 

 applied in like manner, at the distance of a number of divisions, suppose it a prime 

 number, so as to cross all former divisions, viz. 17 or 19, then the middle of the 

 line joining the centres of the two cylinders will remain in the direction of the 

 same radius, though one of them should force in a minute quantity farther than 

 another ; and if a point be assumed in the direction of a tangent to a circle at this 

 middle-point, then though both the cylinders should drop in a minute quantity 

 farther at one time than another, yet the middle-point would remain at the same 

 distance from the point in the tangent ; provided that point was removed to a 

 competent distance, as to 5 or 6 inches. On this principle I would construct 

 an index, the two cylinders being fixed in a frame, convertible about the middle- 

 point, and to be centred in the end of the lever, representing the tangent ; then 

 this lever being again convertible about the point in the tangent line, the middle- 

 point would always have a fixed distance from the point in the tangent, and there 

 hold it steadily fast ; the tangent point being placed on the fixed block before- 

 mentioned. 



Use of the Dividing Plate in the Graduation of Instruments. — Our dividing 

 plate is now ready for the reception of an instrument ; suppose it a quadrant, 

 whose radius however must not exceed the radius of the dividing plate : it is to be 

 laid on the face of the dividing plate, and a weight, or weights, equivalent to that 

 of the quadrant, is placed on the opposite side, to balance it. It must also be 

 supposed, that the quadrant is made with a view to be divided by this engine ; and 

 consequently that the central cylinder is so well adapted, and nicely fitted to the 

 centre hole of the quadrant, that the centre cylinder can be removed, in order 

 for the limb to be divided, and again replaced without sensibly altering its centre. 

 This being the case, let a piece of metal be turned, to apply to the quadrant, 

 perfectly like its centre cylinder at the upper end, and turned nicely to fit the 

 central hole in the dividing plate, at the lower end; then, the quadrant being fixed 

 with proper fastening screws, I would cut the divisions with a beam compass; and. 



