The second step is, to trisect the semicircle, which is to be done by two more of the adjustable pieces, and the two microscopes. With zero under A, place B as nearly at an angle of 60° from it as can be done by estimation or other wise, and put a waxed piece under it ; move back the cir cle so as to bring the waxed piece to A, and put another under B ; again move the circle so as to bring the second piece under A, when, if B should happen to coincide with the line opposite to zero, the thing is done ; but if not, as most likely will be the case, then B must be moved towards the line one-third of the quantity of apparent error, and the operation repeated, as was done for bisecting the circle. The third step is, to bisect the three equal arcs of the se micircle, which is done by similar means ; and at the fourth step by which those last are trisected, arcs of 10° are obtained.
The Duke de Chaulnes' microscopes could not be brought near enough to each other to bisect the arcs of 10°; he therefore had recourse to that of 9°, which he found and used in the following manner. He obtained it by tak ing ten steps with nine additional waxed pieces for each of the two quadrants, which he brought to their places by fre quent trial and adjustment. With this opening between the microscopes, from the tens he set off all the nines for wards, and all the ones backwards. He then resumed the former opening of 10°; and as the operation for ascertain ing the arc of 9^ had left a waxed piece in every interval of 10^, which would in their respective places represent 8°, 7', 6°, &c. to 2° in each quadrant, with the opening of 10° from eight, he set off all the eights, from seven he set off all the sevens, &c. and in this manner put in every single degree of the semicircle.
As all the waxed pieces had their'opposites cut in the other, it was not necessary to have 180 of them ; for it is evident from the mode of procedure, that after the ones and nines had been put in, neither these nor the tens could he wanted. The semicircle of trial has yet upon it no divi sions, and how (the waxed pieces being removed) it was di vided from't he other, has already been mentioned.
It is observed, that when a circle is large enough to al low the microscopes to come so near other as to bi sect the arc of 10°, the numbers four and five may with ad vantage be substituted for nine and ten ; and when half de grees are required, it is proposed to bisect the arc of 15°, and with this opening to put in all the half degrees.
To subdivide the degrees into five-minute spaces, the Duke de Chaulnes proposes to mount a telescope with a vertical wire in its locus, upon the centre arbor of the cir cle, so that it may either revolve with the circle, or concen tric to it by itself, as the case may require ; and let a long beam of wood be provided, accurately dit ided into twelve equal parts, and placed at such a distance as to subtend an angle of one degree ; a thin piece of brass is to be fixed upon the circle under the tracer ; and as the telescope and circle are turned round together, and the wire made suc cessively to coincide with the divisions of the beam, the corresponding strokes arc to be drawn upon the piece of brass : and lastly, this piece is to be placed under the fixed microscope, in order that by means of it and the tracer, every degree may be tilled up.
The ingenious inventor of the above method, like those who preceded Bird, has trade no provision against the er rors arising from expansion ; indeed, his tracer being fixed opposite to the point of trial, subjects his work to the greatest possible error in that respect ; and under this disadvantage, his division of thc first semicircle is no more than a copy of the wax-work ; and, again, the second se micircle is, under the same disadvantage, a copy from the first.
'The method of Ilindley having been in part communi cated to Smeaton in 1741, and fully in 1748, might in our article have preceded sonic of the methods already noticed, but as he received it under the seal of secrecy, it was un known to the public until 1785, in which year Mr Smea ton's paper, before referred to, concerning it, was read to the Royal Society.
Smeaton introduces this method under the full persua sion that vision, even when assisted by glasses, cannot com mand a greater degree of accuracy than to the 4000th par t of an inch ; and maintains, that by contact, the 60,000th be comes equally sensible. Were this true, and contact ap plicable to the graduation of instruments tt ithout drawback. a fine field of improvement would have been opened to the , artist through the paper under consideration.
DIr Smeaton says, << It now conies to be time to open a principle, upon which there is a prospect of effecting such an improvement. I have shewn that a 4000th part of an inch is the ultimatum that we arc to expect front bight, though aided by glasses, when observing the divisions of an instrument. But in the 48th volume of the Philosophi cal Transactions, (p. 149 of this volume), I have shewm the mechanism of a new pyrometer, and experiments made therewith ; whereby it appears, that, upon the principle of contact, a 24,000th part of an inch is a very definite quan tity. I remember very well that I did not then go to the extent of what I might have asserted, being willing to keep within the hounds of credibility ; but on occasion of the present subject, I have re-examined this instrument, and find myself very well authorised to say, that a 60,000th part of an inch, with such an instrument, is a more definite and certain quantity than a 4000th part of an inch is to the sight, conditioned as above specified. The certainty of contact is, therefore, fifteen times greater than that of vision, when applied to the divisions of an instrument : and if this prin ciple of certainty in contact did not take place, even much beyond the limit I have now assigned, we never should have seen those exquisite mirrors for reflecting telescopes that have already been produced.