In the latter of these instances, we have the principle of repetition truly exemplified, because (as has been justly observed by Sir J. Herschel[7\3],) there is then 'a juxtaposition of units without errour,'--'one vibration commences exactly where the last terminates, no part of time being lost or gained in the addition of the units so counted.' In space, this juxtaposition of units without errour cannot be rigorously accomplished, since the units must be added together by material contact (as in the above case of the threads,) or in some equivalent manner. Yet the principle of repetition has been applied to angular measurement with considerable success in Borda's Repeating Circle. In this instrument, the angle between two objects which we have to observe, is repeated along the graduated limb of the circle by turning the telescope from one object to the other, alternately fastened to the circle (by its _clamp_) and loose from it (by unclamping). In this manner the errours of graduation may (theoretically) be entirely got rid of: for if an angle repeated _nine_ times be found to go twice round the circle, it must be _exactly_ eighty degrees: and where the repetition does not give an exact number of circumferences, it may still be made to subdivide the errour to any required extent. [Note 7\3: _Disc. Nat. Phil._ art. 121.] 18. Connected with the principle of repetition, is the _Method of coincidences_ or _interferences_. If we have two Scales, on one of which an inch is divided into 10, and on the other into 11 equal parts; and if, these Scales being placed side by side, it appear that the beginning of the latter Scale is between the 2nd and 3rd division of the former, it may not be apparent {157} what fraction added to 2 determines the place of beginning of the second Scale as measured on the first. But if it appear also that the 3rd division of the second Scale _coincides_ with a certain division of the first, (the 5th,) it is certain that 2 and _three-tenths_ is the _exact_ place of the beginning of the second Scale, measured on the first Scale. The 3rd division of the 11 Scale will coincide (or interfere with) a division of the 10 Scale, when the beginning or _zero_ of the 11 divisions is three-tenths of a division beyond the preceding line of the 10 Scale; as will be plain on a little consideration. And if we have two Scales of equal units, in which each unit is divided into nearly, but not quite, the same number of equal parts (as 10 and 11, 19 and 20, 29 and 30,) and one sliding on the other, it will always happen that some one or other of the division lines will coincide, or very nearly coincide; and thus the exact position of the beginning of one unit, measured on the other scale, is determined. A sliding scale, thus divided for the purpose of subdividing the units of that on which it slides, is called a _Vernier_, from the name of its inventor.