The mode of measuring an angle with this instrument is as follows :—Let the two objects be st (that to the right) and L (that to the left). Set the fixed bar to some angle from 10 to 20 degrees to the right of a by the motion of the tube on the axis, and clamp the axis-screw firmly; then, by the motion of the upper bar alone, bisect R with the telescope. Take, with a pair of compasses, the distance between the dots, apply the distance to a scale of chords, and you have the angle between the fixed bar and the object R. Call this angle O. Now, by the motion of the upper bar alone, bisect the object L. It is clear that, if the distance between the dots were'again measured, and the angle deduced, as before, from the scale, we should have a measure of the angle required + 0. But instead of measuring at present, let the telescope be brought back on a, by unclamping the axis-screw and moving the whole instrument on its axis ; when this is satisfactorily per formed, clamp the axis, and bisect L exactly as before, by moving the upper bar and telescope alone. The angle between the bars as deduced from measuring the chord between the dots will now clearly be twice the angle required + 0. Let the operation be performed so many times—eight, for instance—that the bars are nearly in their original position with regard to each other, and let the distance between the dots be measured and the corresponding angle be deduced from the scale of chords, which suppose to be 0, 0 being larger than 0. If this last-mentioned angle had been 0 exactly, it is clear that the bar would have come round exactly to its original position after having moved through 360°; but as it has besides moved over an angle = 0-0, the whole angle moved through is which is also eight times the angle to be measured : hence the angle subtended at the spectator by a and L is A (360° +0-0). By continuing this process of stepping several times round, there seems to be no limit to the accuracy with which an angle can be measured, except that which depends on the imperfection of the telescope, the indistinctness of the objects, or the uncertain lateral effect of terrestrial refraction. Mayer used a scale of chords, probably because he was thus able to construct the instrument himself, and could dispense with any circular are or divisions. We do not see that he has noticed one slight inaccuracy, namely, that as the dots lie in different planes, the distance between them is not the actual chord of the angle required, but is the hypothenuse of a right-angled triangle, the altitude of which is the thickness of the upper bar, while the base is the chord required ; but this error is easily allowed for, and, when the angle to be measured by the compasses is of a tolerable size, is scarcely worth considering. If we conceive the plane of the lower bar extended and changed into a divided circle, while the upper bar becomes a vernier at each end, we should probably have the instru ment Mayer would have proposed, had it been in his power to employ a tolerable mathematical-instrument maker. Mayer says that he invented this instrument eight years before the publication of his memoir.
The reward proposed by the English parliament for any means by which the longitude at sea could be determined, stimulated Mayer to perfect the method of lunar distances. For the successful solution of this problem two things are required—tables correct enough to predict the true place of the moon at any future time, and an instrument for measuring the distance between the moon and star with sufficient accuracy. Mayer fulfilled the first condition by his celebrated Lunar Tables, one copy of which was sent to the Lords of the Admiralty in 1755, and a later, improved up to his death (1762), forwarded by his widow in 1763. For measuring the distance between the moon and star he proposed an instrument similar to Haclley'a sextant, but in which the angle can be repeated or multiplied without intermediate readings off, similar in principle to the instrument just described.
Mr. Troughton says (article Circle,' Brewster's Cyclopzedia ') that Bird was employed to make reflecting circles after Mayer's idea, but his dividing was so excellent, that the entire circle was thought useless, and the sextant preferred, as having a larger radius, and being lighter and handier.
In 1787 the Chevalier de Borda published his ' Description et Usage du Circle de Rtlflexion,' in which he proposed a modification of Mayer's circle, so alight that at first sight it would almost seem trivial, but which gives an unquestionable superiority to this above every other form of reflecting instrument when well made and skilfully and per sereringly used. We shall return to Mayer and Borda's construction of
the repeating reflecting circle in the article SEXTANT, as those instru ments cannot be understood until the principle of reflecting instru ments has been explained.
The date of the invention of the repeating circle which is the proper subject of this article, is somewhat uncertain : it is later than that of the reflecting circle. One was constructed in 17S7, and employed in eonnecting the meridians of Paris and Greenwich. (See ' Mem do l'.Aceshten. ; a Memoir by Le aware, 1797 ; and a memoir by Quoin', 179q.) The ' Connoiasance des Tema; An VI. (l797-$), contains the plate and description of a repeating circle which was made by Lenoir for the astronomer La Lands. When the French government under took the measurement of an arc of the meridian from Dunkirk to Barcelona, the commission to whom this operation was entrusted resolved to employ the repeating circle.
This Is one of the most complicated as well as ingenious of existing instniments, and obtained an Immense reputation, from being the only instniment employed in the geodesic:II and astronomical observations of the great measurement of an are of the meridian, on which the French have founded their modern system of measures, weights, and money. Since that time the construction has been altered by different artists, but not always with advantage.
clamp holding ao near the axis of motion has little power, and there is room for getting at the acrow-head, while the slow-motion clamp Is out of the way of the observer when be requires it for bisec tion. The large weight behind is a counterpoise, and the small level above is for setting the circle vertical. There is a clamp at c, which bites on the semicircle to make this adjustment and prevent sits being deranged. We will describe the process of observing with the in atrument when the object is a star at or near its meridian altitude.
Supposing everything to be adjusted, that is. the axis and circle both vertical, the observer bisects the star with the telescope, ho or an assistant having previously brought the level nearly to the correspond ing points of its scale. The level is now read oft giving it time to settle if wanted. We suppose its graduation to be In seconds, and reckoned outwards from the centre of the scale. The verniers aro read off, the instrument turned half-round on the vertical axis, the telescope clamp released, and the star again bisected by the telescope, using its peculiar clamp and tangent screw ; and finally, the level is again read oft This operation is precisely the same in all circles having an azimuthal motion, and it is clear that If the vendors were again read off, the difference between the first and second readings would be (after it is corrected for the indication of the level) twice the zenith distance of the star. Let the circle be now reversed, tho level clamp and circle-axis clamp be released, and the whole circle moved in its plane till the telescope points to the star, and let the star be bisected again by using the axis clamp and its tangent screw only. The level mist be brought back to be horizontal while this is doing, and be actually damped before the final bisection of the star is made. We conceive that this must be done at twice, even by two observers; and it may be done at twice by one, though' in a longer time. If the reader has fully understood the process, he will see that the instru ment is precisely as at the commcncmcnt, except that the telescope and its verniers have travelled over the circle, an arc equal to twice the zenith distance of the star. A repetition of the operation will carry the telescope verniers over four times the distance, and by con tinning the process the final arc read off may be made any number of times twice the zenith distance of the star. If the series stops after ten such processes, the are travelled over is twenty times the simple zenith distance. Let the verniers be now read oil, then subtracting the first reading from the last, and dividing by twenty, the result will In this figure the general form of the instrument is shown tolerably well, but some of the essential 'notions are at the back of the circle, and these are drawn on a larger scale in a second diagram. The whole circle turns round on the vertical column, which has an inner axis of steel, with goal fittings at the top and bottom. It is usual and proper to make these fittings with great care, but it is not an essential con dition to accuracy in the performance of the instrument.