VERNIER, a scale, by which linear or angular magnitude can be read off with a much greater degree of accuracy than is possible by mere mechanical division and subdivision, derives its name frdm its inventor, Pierre Vernier, " capitaine et chastellaine pour sa majeste au chasteau Dornans," who gave a description of it in a tract published at Brussels in 1631. The principle of this invention is essentially as follows: AB (fig. 1) is a portion of the graduated scale of an instrument showing divisions and subdivisions; ab, a small scale (called the vernier), made to slide along the edge of the other, and so divided that ten of its subdivisions are equal to eleven of the smallest divisions of the scale AB,' then each division of the vernier is equivalent to lylii of a subdivision of AB; and , consequently, if the zero-point of the vernier be (fig. 1, A) opposite 11 on AB, the 1 on the vernier is at 915a below 11), 2 on vernier is at below11), etc. Also, if the vernier be slid along so that 1 on it coincides with a division on the scale, then 0 on the vernier is one tenth above the next division on the scale; if 4 on the vernier coincide with a division on the scale, the 0 of the vernier is four tenths above a division. 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 off is to he performed, the position of the zero point, with reference to the divisions of the scale, gives the result as correctly as the mechanical graduation by itself 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 (fig. 1, B), suppose the scale-divisions to be degrees, then the reading by the graduation alone gives only a result between 15° and 16°; but as the 2d division of the vernier coincides with a grad uation on the scale, it follows that the zero-point is of a division above 15°, and that, therefore, the correct reading is 15.2°. IL will be at once seen that by merely increasing the size of the vernier, as, for example, making 20 divisions of it coincide with 21 on the scale, the latter may he read off to twentieths; and a still greater increase in the size of the vernier would secure further accuracy.

The above is the vernier as proposed by its inventor, and as it was employed for long after his time; but in the more recently constructed astronomical and geodesical instruments a vernier is employed .which has one graduation snore (8g. 2) than the corresponding portion of the scale. A little con sideration will show that the only effect of this modification is to enable the vernier to be graduated toward the same direction as the scale, and thus save a little confu sion in the reading off. In small instruments, or where the utmost accuracy is required, a small magnifying lens is fixed over the wider, to enable the observer, in cases where no two graduations coincide (which is generally the case), to estimate the 'amount of error introduced by assuming that the two graduations which approach nearest to coincidence actually coincide. • • 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 thenenius. The latter, so called from its inventor, Petrus Nonius (Pedro Nunez), a Portuguese mathematician, who described it in a treatise De Crepusculm Olyssipone, published in 1542, consists of 45 concentric circles described ou 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 46 equal parts, giving on the whole a quadrantal division into 2,532 separate and unequal parts (amounting on an average to about 2' intervals). The edge of the bar which carried the sights passed, when pro duced, through the center, and served, consequently, as an index•limb; and whichever of the 45 circles it crossed at a graduation, on that circle was the angle read off; for instance, if it cut the 7th circle from the outside at its 43d graduation, the angle was read of as 11 of 90°, or 46° 4' 17f".