In the bisection of an arc, the distance between the point and the axis of the microscope, is to be taken as nearly as may be to the chord of half the arc, and the joint adjusted, so that the cross wires and point may both at the same time coincide with the line of division. With the beam to the right, by moving the frame upon the circle, the wires of the microscope must be brought to coincide with the point that marks the left boundary of the arc, and then a faint mark across the line of division must be cut with the point. The beam is next turned to the left, and the point which bounds the arc on the right is, by turning the frame, to be set to the microscope, and another mark across the line made as before. It is evident, that if the open ing between the microscope and point was exactly equal to the chord of half the arc, the two marks would coincide upon the line of division ; hut if that opening was too great or too little, the marks would cut each other without or within the circle. But in neither of these cases would Mr Cavendish put in a dot. Instead of which, when in sub sequent division these marks are to be used, as well as when the final strokes are cut, he would place the wire of the microscope by estimation in the middle between them, where they cut the line of division.
In quinquisecting an arc, the opening being taken as near as possible, equal to the chord of a fifth part of it, bring the microscope to one extremity of the arc, and with the point make a mark across the line ; bring the micros cope to the mark just made, and with the point make another mark, &c. until four are put in ; change the posi tion of the beam, and from the other extremity of the arc set off four marks, as before. If the chord in the above operation was not correctly taken, it is evident that there will now he four double marks, and the spaces between them equal to each other, and five times greater than the en-or of the opening. The real point of quinquisection, reckoning from one end of the arc or the other as the opening, was or—, will be S, 4, 4, and 4 of the space be tween them ; and these Mr Cavendish would also use by estimation in the subsequent progress of the work. Mis takes in making a wrong estimation of the spaces are to be prevented, by making proper marks upon the circle opposite to them. Mr Cavendish has given three different ways of proceeding ; but that which we have described, is the one which he himself thinks the best.
To give a general idea of Mr Cavendish's method was all that we intended. To follow hint through his whole paper would be useless; for, notwithstanding that much ingenuity is displayed in pointing out such errors as he foresaw it would be liable to, and in contriving means to obviate them, we consider it as altogether inconsistent with practice, and inelegant in design.
Immediately after Mr Cavendish's paper, we find one by Professor Lax of Cambridge, in the form of a letter to Dr Maskelyne. It is entitled, On a method of examining
the Divisions of Astronomical Instruments.
The learned professor sees no reason why astronomers should trust to the ability and integrity of artists, when, by means of a proper apparatus, they have it in their own power to examine and note the error of every division of an instrument. Mr Lax is in possession of au altitude and azimuth circle, or one foot radius, made by Mr Cary, and it is to this instrument that his examining apparatus is adapted ; but the computation of error is expressed in ge neral terms.
His apparatus consists of an arc fixed to the frame of the instrument, exterior to, and concentric with, the elide, and which stands still while the circle is turned round. The arc contains about 90", and a microscope, which slides upon it from cud to end, may be clamped to any part of it. The microscope regards the divisions of the circle, and is used in combination with one of the reading micrometers : the former has an inclination of about 30° to the latter, in order that both of them may be made to coincide with one and the same division of the circle ; by means of which contrivance, any opening between them from 0 to 90° may be taken. Professor Lax finds the error of the division 180° by help of the two reading mi crometers, exactly as Troughton did, but in every other step the examination is carried on in a way quite different from that pursued by the artist.
The second step is performed when zero of the circle is brought to one of the micrometers, and the microscope fixed to the exterior arc at the division of 90°, by bring ing in succession to the microscope the divisions 180° and 360°, and comparing the first arc of 90° with the other three arcs of the difference of which having been measured with the micrometer, and distinguished by -1- or —, affords data for computing their respective errors. In like manner, the first arc of 60° is to be measul eel against al; the other five arcs of 60°, precisely as the first arc of 90° was measured against all the other arcs of 90°. And again, the first arc of 45° is to be measured against all the other seven arcs of 45°. So far the Professor pro ceeds before sunrise, in order to avoid the effect of expan sion ; the rest, on account of the small arcs that are used, may be done at any time. The arc of 3o° may now be measured against every succeeding arc of 30' in the first, third, fourth, and sixth arcs of 60° ; and let the length be determined from a separate comparison with the arc of in which it is comprehended, and not from a general com parison with all the four. The arc of 15° must then be measured against every succeeding arc of 15° in all the arcs of 30°, except the second, fifth, eighth, and eleventh, and the value of each deduced from a comparison with the arc of in which it is contained.