In France it is probable that the art of instrument-making was at this time less advanced, and therefore the merit of Mayer's principle was more readily appreciated. The celebrated Borda, who was a seaman and navigator, first had an improved instrument on this construction made about 1775,* and published a full description of its form and use in 1787 (` Description et Usage du Cercle de Hamden, par le Chevalier de Borda,' Paris, 1787), to which we refer the reader as a standard and classical work. The accompanying figure will explain Borda's instrument, though it is not precisely similar to that which he recommended.
The index-glass, with its bar, vernier, and tangent screw, and the excentric bar which carries the telescope, horizon-glass, with its vernier and tangent screw, require no explanation. The telescope has a parallel-ruler motion to bring the images to equal brightness, which is, we believe, Bird's construction. Borda holds his telescope by two ears, each of which can be raised or depressed separately, and he has a graduation for each, so as to move them equal quantities. Dollond, in his very beautiful repeating circles, has an up-and-down piece abort the bar. The important and essential conditions of the repeating circle are, that moving one index shall in no way affect the other ; that the clamps when fixed shall not be liable to move from looseness, bad balancing, or bad centering when the position of the circle is changed ; and finally, that the axis of the index-glass and the collar on which the horizon-bar turns shall both be so true, and of such a length, that the motions of each bar are parallel to the plane of the divided circle. The same precautions must be taken as in the sextant in trying the index and horizon glasses, in placing the prismatic edges of the dark glasses up and down alternately, in setting the glasses perpendicular to the plane of the circle, and the telescope parallel to it: and it is scarcely necessary to add, that the directions given above for using dark glasses, equalising brightness, &c., apply to one reflecting instru ment as well as another. The cella into which the dark glasses are inserted, when wanted, are seen between the two glasses and also in front of the horizon-glass.
On looking at Borda's circle as it is here represented, the opening of the angle between the two glasses is towards the spectator ; hence an observer looking through the telescope would see an object directly in the line of the telescope, and some other object, call it A, which lies towards the spectator, by reflexion. Now suppose the index-bar to
be moved through the position of parallelism and until the glasses make the same angle as before, but with the opening from the spectator, it is clear, first, that the angle read off between the first and second positions will be twice the original angle ; and secondly, that the observer, still looking at the same object as before seen directly, will see by reflexion an object on his right hand (call it a). which makes the same angle with the axis of the telescope as A did, but on the other side. Now if we suppose the whole instrument to turn half round upon the telescope as an axis, it is evident that A will be seen exactly as at first, while the Index-bar has been moved forwards twice the angle between A and the axis of the telescope produced. This is exactly the complete observation with Troughton's circle, and thus while we have got double the angle by two observations, we have got rid of index error, and have only two readings to which error of division and reading off can apply.* Now suppose the instrument to be returned to its original position, and, leaving the index-bar securely clamped, move the horizon-bar, which carries the telescope and horizon glass, through the same angle and in the same direction as the index bar has travelled. If the original object be again viewed through the telescope, and the contact between that and A perfected by the tangent screw of the horizon-bar, it is clear that everything is exactly as at starting, except that the index and horizon-bar have each moved over the divided circle exactly twice the angle to be measured. Let the operation which has been described be repeated, and everything will be as at starting, except that the indices will have moved over four times the angle, and it is evident that there is no limit to the number of repetitions except the will of the observer. So that, theoretically at least, the influence of bad division, bad centering, and bad reading off upon the final angle may be reduced below any sensible quantity. There is another very considerable advantage, namely, that there are only two readings off of each vernier 1' for any number of repetitions.