The engine, so far described, is ready to receive the ori ginal graduation of its limb ; and as this operation was done, while work to be described hereafter was preparing, I wiil here explain by what means this most important part of the work was accomplished: to do this, however, with in moderate bounds, I must suppose that the reader is al ready acquainted with my method, as published in the Phil. Trans. for 1809, or that he will turn over to the last Sec tion of this article, where it is fully explained.
In the first step, a roller was placed horizontally in a frame attached to the tripod, having free and steady mo tion round its own axis; this was adjusted so as to be car ried exactly 16 times round, while the engine plate made one revolution, and was itself near the edge upon its upper surface divided into 16 parts. Now, upon turning the plate round, these 16 divisions 16 times told came in succession to the wire of a fixed microscope, and were, by a proper apparatus, transferred. to the surface of the plate, in five lets, at a sufficient distance within the edge to prevent their being disturbed by o.Akilig the teeth. To accomplish the next step, an index was made to revolve upon the arbor of the plate ; it was composed of two branches, each of which carried at its extremity a micro scope with a micrometer ; these had a range of angular motion respecting each other, from a right line to a very small angle. By this index, and these microscopes, the 256 fine dots were examined by a certain bisectional pro cess, from which their individual errors were investigated by computation, and formed into a table. By the help of the table of errors, the future work of racking the limb was prosecuted with as much certainty as could have been done, had the original divisions been inserted without error.
It has already been said that the value of a tooth of the limb should be 10', and consequently their whole number will be 2160 ; now 2160 256 =8 and just so many revolu Cons and parts of the dividing screw will be commensurate with a mean space from dot to dot = an angle of 1° 24' 22".5. In order, therefore, that a comparison between the plate and the screw might be made at every original dot, it became necessary to provide means to ascertain the position of the former at every sixteenth part of a revolu tion. To this end a micrometer head, as large as could be admitted, divided into 16 equal parts, was fixed upon the left end of the screw arbor ; and contiguous to this, was placed a fixed index bearing a fiducial line. For the pur
pose mentioned above, these were all that could be wanted ; but as our dots were erroneous, in order from their appa rent, to determine their true places, a lower subdivision of the head became necessary. Each of the 16 spaces, there fore, was divided into 10 by actual division, and as an eye, practised in such matters, can by estimation accurately ob tain the value of the next decimal figure, it was into the last denomination of subdivision that the table of errors had been reduced, the value of an unit of which, in angu lar measure, is -pis of a second.
The roller was removed when the 256 dots had been transferred to the plate, as were the double index and microscopes from the central arbor, when the position of those dots had been ascertained. Now, the dividing screw was placed in its frame, a micrometer, with a moveable wire fixed to the tripod for viewing the primitive dots, and a winch for turning the screw attached to its arbor on the right : this change of parts being effected, the screw with its frame having free motion in the line of radius, and ca pable of being, by the force of a spring, pressed into con tact with the edge of the plate, or by a screw drawn back wards at pleasure, and the plate itself having free motion round its axis, the important operation of forming the teeth, or racking the circle, was commenced.
It should be premised, that to prevent mistakes, by be ginning an interval at a wrong 16th of the head, which, by making false marks, would occasion much trouble, those parts were numbered 1, 2, 3, &c. to 16, in the order of turning the screw forwards. Corresponding numbers were marked in ink upon the plate opposite to the dots, the ordet of which, from right to left, was 0, 7, 14, 5, 12, 3, 10, I, 8, 15, 6, 13, 4, It, 2, 9. which, repeated 16 times, completed the circle These enabled me to proceed with confidence ; for, in beginning any interval, it was only re quired that the number upon the head should be that w hich distinguished the dot under the wire of the mh•ro scope. In the table of errors, I marked those dots, which were too forwards, with the sign —, and those that were too backward -, because it is evident that a + position of the screw will effect the correction of a— error of the dot, and the contrary.