VERNIER, vi.leni-er. A scale invented by the French geometrician Pierre Vernier (q.v.), by whieh linear or angular magnitude can be read with a mneh greater degree of accuracy than is possible by mere mechanical division and sub division. The principle is essentially shown by the following examples: lig. 1 is a portion of a graduated scale of equal parts with a vernier below, which is made to slide along the edge of the scale, and is so divided that its sub divisions are equal to eleven of the smallest visions of the scale; then each division of the vernier is equivalent to 1.1 of a scale division; and consequently if flie zero-point of the vernier ( 1) he apposite 11 on the scale, the 1 on the vernier is at 9.9 (1.1 to the left of 11), 2 on the vernier is at 8.8 (2.2 to the left of 11), etc. Also, if the vernier be moved along so that 1 on it coincides with a division on tile scale, then 0 on the ver nier is to the left of the next division on the scale; if .1 on the vernier coincides with a division on the scale, the 0 is four-tenths to the left, of a division as in Fig. 2. The vernier is ap plied to instruments by being carried at the ex tremity of the index limb, the zero on the vernier being taken as the and when the reading is to be performed, the position of the zero-point, with reference to the divisions of the scale, gives the result as correctly as the mechan ical graduation permits, and the number of the division of the vernier which coincides with a di vision of the scale supplements this result by the addition of a fractional part of the smallest sub division of the scale. Thus in Fig. 2 suppose the scale divisions to be degrees, then the reading by the graduation alone gives a result between 15° and 16° ; but as the fourth division of the vernier coincides with a graduation on the scale, it follows that the zero-point of the vernier is 0.4 of a division to the left of 15°, and that the correct reading is 13.4'. It will be seen that by merely increasing the length of the vernier, as, for example, making 20 divisions of it coin cide with 21 on the scale, the latter may lie read to twentieths; and a still greater increase in the length of the vernier would secure further ac curacy. Verniers like the above in which the number of its divisions is lesa than the corre sponding number on the scale are called retro grade or rerersc verniers. But some instru ments are provided with direct verniers, that is, those in which the number of divisions exceeds the corresponding number on the scale. The prin
ciple of operation is the same as in the retro grade vernier, except that one must look forward along the vernier to find the coinciding line. Fig. 3 shows a direct. vernier, and the principle of its construction is the same as for reverse verniers, only the vernier division is greater by a tenth of a scale division instead of being smaller. Fig. 4 shows a direct vernier where the coincidence conies at 3 giving a reading of 5.3. In general, if v is the length of a vernier division, s the length of a scale part, and a the number of divisions on the vernier, then ay = (a 1)s for the direct ver nier and 2112 = (n + 1 )s for the reverse vernier.
Therefore s v = 1 . s, v s = respective?), ly, which shows the comparative size of the divi sions of the two scales and to what fraction of a division any vernier will read. E.g. we wish a direct vernier, attached to a scale graduated to read half-degrees. to read minutes; what must be the relation between a vernier and a scale division? Here s = 30', and, since the vernier is to read minutes, s v = 1' and v = 29'. v 1 . s = and n = 30. Therefore, a space equal to 29 scale divisions is to be subdivided on the vernier into 30 equal parts.
Of the various methods for subdivision which were in use before the introduction of the vernier, the most important were the diagonal scale (q.v.) and the imains. The latter is so called from its inventor, Pclrus .Vonins ( Pedro Nufiez), a Portuguese mathematician (1492 1577), who described it in it treatise De Crepusealis Libre ens (Lisbon, 1542). It consists of 43 concen tric circles described on the limb, and divided into quadrants by two diameters intersecting at right angles. The outermost of these was divided into 90, the next into 89, the third into 88, etc., and tbe last into 46 equal parts, giving, on the whole, a quadrantal division into 2532 separate and unequal parts (amounting on an average to about 2' intervals). The edge of the bar which carried the sights passed, when produced, through the centre, and served as an index-hi oh; and whichever of the 45 circles it crossed at a graduation, on that circle was the angle read; for instance, if it cot the seventh circle from the outside as its forty-third gradu ation, the angle was read as of 90 ',. or 46° 4' 17 V. Consult Ludlow, "Subscales, including Verniers," in Van Nostrand's Engineering Maga z.ine (1882).