It may be worth mentioning that with the circle of Borda the measurements may be made by moving the telescope contrary to the order of divisions, that is, if the first observation is made with the face to the right hand instead of the left. There is no advantage in this modification, except perhaps that, as the screws aro handled a little differently, any discrepancy between an angle measured in the two ways would show an error in one or both the methods. It would be advisable, where several series are taken, to vary the direction in which the telescope is brought to its new position, and that in which the circle is turned upon the star. The tangent screw might be worked either constantly one way or the other, or alternately, but we should not expect any sensible difference in the results if the instrument is well made and the observers careful. If two steady observers can bo found, the double altitude of any star may be repeated, with the genuine Borda and its two telescopes, exactly as a terrestrial anglelthat is, by measuring the angle between the star seen directly and by refiexion from mercury. This would in theory appear to bo the most perfect application of the instrument, as flexure ham nn effect upon the angle measuresl, being equal and in the same direction in both posi tions of the telescopes. The level is not wanted for this observation nt all, but the circle must be set truly vertical A single observer might indeed make the same observation with one telescope, as the level would show and measure any shifting of the circle. And here again the effect of flexure is eliminated from the observed double altitudes. Finally, double nadir distances may be observed of a star reflect"! from a mercurial horizon, exactly as double zenith distance* aro observed in the ordinary method. This last species of observation may have the advantage of being sometimes more convenient, but the chief reason why it is pointed out is, that the effect of flexure upon double nadir distances is equal, but in a contrary direction, to its effect on double zenith distances ; the observer has thus an easy mode of ascertaining whether flexure exists, and of measuring its amount and law. We suppose that the apparent nadir distance of a reflected star is the same as the zenith distance of the star seen directly. The horizontal points of all the large circles which we know, vary to some extent with the altitude of the star observed, which seems contrary to what is hero supposed. This laet-mentioned dis crepancy is one of the most perplexing points in modern practical astronomy. The amount is, however, so small as scarcely to vitiate sensibly any conclusion drawn from reflected observations with so small an instrument as a repeating circle.
What precedes refers almost entirely to the repeating principle when applied to measurements In altitude. Instead of the circle of Bonin and its two telescopes, later artists have given a motion in azimuth to their theodolites, in addition to the motion of the tele scopes and verniers, by which the angle can be measured precisely as in 3layer'm original instriunent. But in several interments which we have seen there is a considerable probability of disturbing the circle clamp while moving the telescope or using its tangent screws. It is to secure the detection of any similar motion that a watch telescope, as it is called, is frequently added to English theodolites, and is indeed required for many, spoiled as they are by an unnecessary adjustment to zero, to save indolent or ignorant surveyors a simple subtraction. A watch telescope can scarcely be applied to a repeating theodolite, and we think that the repeating tripod may be so made as to be free from any objection. This was first constructed by Mr. Dollond on
the suggestion of the late Astronomer Royal, and has been found very useful in the trigonometrical survey of Ireland.
The three foot-screws of an ordinary theodolite are placed in the three notches which are seen on the table of the tripod. This upper part turns heavily on a stout short centre, it is fixed by a clamp at a, and there is a tangent screw at n for giving slow motion. The steady action of the tangent screw is secured by a spring at c. It is evident that if, in using an ordinary theodolite, n (the right hand object) is first bisected, the circle read off, and then t (the left hand object) is bisected, the circle being again read of that the difference between the two readings is the angle to be measured, if the instrument has not been shifted, and if the bisection, reading off, &c., are perfect. Now suppose the whole instrument to be taken up and set down exactly concentric with its first position, without any other alteration, but with the telescope on a ; if L be a second time bisected, the difference between this latter and the preceding reading will be also the angle to be measured, and therefore the vernier will have passed over twice the angle, reckoning from the beginning. If, instead of this impracticable taking up and setting down again, the stand on which the theodolite rests can be turned round concentrically so that is is bisected by the telescope, the theodolite itself being untouclxl, it is clear that the operation :s equivalent to that just described, and consequently that the telescope being brought on t by its own motion, the measure is obtained of twice the angle required, and the process may be continued ad libitum, until the errors of read, lag off and of division are eliminated. The only precaution to be observed is, that the stand must receive no angular motion from the motion of the telescope, and this is easily effected by giving a certain massiveness to the stand and a considerable heaviness to its motion, while the telescope and its verniers revolve as lightly as possible. The observer should satisfy himself as to this perfect independence of the two motions and the stability of his repeating tripod by taking a set of twenty repetitions of an angle, always moving the telescope for ward in the order of the divisions, and a second set of twenty of the same angle, carrying the telescope round the contrary way. The two results should agree if the stand has no motion ; and if they do not, the upper motion must be lightened or the lower be loaded till they do agree. We should not feel satisfied to use the repeating stand if a motion of the telescope a little ruder than necessary affected its posi tion although undamped ; and the stand which is here figured fulfilled this condition very well with a 12-inch theodolite. As the absolute coincidence of the axes of the tripod and theodolite cannot be obtained, the angle should be repeated at least once round the circle, and, if the case requires it, until the multiple angle' is very nearly equal to one, two, or more circumferences. The original repeating tripod as designed by Mr. Pond was considerably higher than that figured here. (See Pearson's ' Practical Astronomy,' plate xxix., fig. 7, and voL ii., p. 513.) A little greater nicety is thus given to the adjustment of the vertical axis of the tripod, which is not necessary, and the snugness of the present stand is, we think, more than an equivalent. The axis of the tripod must first be set vertical, either by a level of its own or the theodolite level, exactly as is described in the adjustment of the vertical column of Borda's circle. For the subsequent adjustments, &c., of the theodolite, see Tneonotrre.