The New International Encyclopædia/Vernier

VERNIER, vẽr′nĭ-ẽr′. A scale invented by the French geometrician Pierre Vernier (q.v.), by which linear or angular magnitude can be read with a much greater degree of accuracy than is possible by mere mechanical division and sub-Fig. 1. retrograde or reverse vernier

division. The principle is essentially shown by the following examples: Fig. 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 ten of its subdivisions are equal to eleven of the smallest divisions of the scale; then each division of the vernier is equivalent to 1.1 of a scale division; and consequently if the zero-point of the vernier (Fig. 1 ) be opposite 11 on the scale, the 1 on the vernier is at 0.9 (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 the scale, then on the vernier is one-tenth to the left of the next division on the scale; if 4 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 applied to instruments by being carried at the extremity of the index limb, the zero on the vernier being taken as the index-point; 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 mechanical graduation permits, and the number of the division of the vernier which coincides with a division of the scale supplements this result by the addition of a fractional part of the smallest subdivision of the scale. Thus in Fig. 2 suppose the Fig. 2. retrograde vernier showing method of reading.
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 zoro-point of the vernier is 0.4 of a division to the left of 15°, and that the correct reading is 15.4°. 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 be read to twentieths ; and a still greater increase in the length of the vernier would secure further accuracy. Verniers like the above in which the number of its divisions is less than the corresponding number on the scale are called retrograde or reverse verniers. But some instruments are provided with direct verniers, that is, those in which the number of divisions exceeds the corresponding number on the scale. The principle of operation is the same as in the retro-

Fig. 3. direct vernier.
grade veniier, 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

Fig. 4. direct vernier showing method of reading.

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 comes 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 n the number of divisions on the vernier, then nv = (n − 1)s for the direct vernier and nv = (n + l)s for the reverse vernier. Therefore s − v = $\scriptstyle {\frac {1}{''n''}}$⋅s, v − s = $\scriptstyle {\frac {1}{''n''}}$⋅s respectively, which shows the comparative size of the divisions 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′, s − v = 1/n⋅s = 1′ 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 nonius. The latter is so called from its inventor, Petrus Nonius (Pedro Nuñez), a Portuguese mathematician (1492-1577), who described it in a treatise Dee Crepusculis Liber Unus (Lisbon, 1542). It consists of 45 concentric circles described on the limb, and divided into quadrants by two diameters intersecting at right angles. The outermost of these quadrants was divided into 90, the next into 89, the third into 88, etc., and the last into 40 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-limb; and whichever of the 45 circles it crossed at a graduation, on that circle was the angle read; for instance, if it cut the seventh circle from the outside as its forty-third graduation, the angle was read as 43/84 of 90°, or 46° 4′ 171/7″. Consult Ludlow, “Subscales, Including Verniers,” in Van Nostrand's Engineering Magazine (1882).