The ordinary working eye-pieces, Huyghenian and others, commencing with a magnification of four diameters, are so constructed that they can be conveniently used, as we are accustomed to use them in England, with high powers, Zeiss’s Nos. 12 and 18 compensating eye-pieces being adapted for use with his lower power apochromatic lenses of 16 and 8 mm. The numbering of the eye-pieces is carried out on the plan originally proposed by Professor Abbe--that is, the number denotes how many times an eye-piece, when employed with a given tube-length, increases the initial magnifying power of the objective, and at the same time furnishes figures for their rational enumeration. It is on this basis that the German compensating eye-pieces have been arranged in series, and in agreement with their magnifying power and distinctive numberings of 2, 4, 6, 8, 12, 18. Of these several eye-pieces, 12 is found to be the most useful. The magnification obtained by combining a compensating eye-piece with any apochromatic objective is found by multiplying its number by the initial magnification of the objective, as given in the following proof:--An objective of 3·0 mm. focus, for example, gives in itself a magnification of 83·3 (calculated, for the conventional distance of vision, 250 mm.); eye-piece 12 therefore gives with this objective a magnification of 12 × 83·3 = 1000 diameters. The classification, however, of these eye-pieces, as furnished by Abbe, is dependent upon increase in the total magnifying power of the microscope obtained by means of the eye-piece as compared with that given by the objective alone. The numbering, then, denotes how many times an eye-piece increases the magnifying power of the objective when used with a given body-tube; the proper measure of the eye-piece magnification; and, at the same time, the figures for rational enumeration. [Illustration: Fig. 106.--B and C Achromatic Eye-pieces.] Compensating eye-pieces have been introduced for the correction of certain errors in high-power objectives--those made with hemispherical fronts. All such lenses, whether apochromatic or not, are greatly improved by the compensating eye-piece, but the dry objective and the lower powers are certainly deteriorated. The lower power compensating eye-pieces are Huyghenian, the higher are combinations, with no field-lens, and therefore in working act as a single or positive eye-piece. This is of importance to those who work with low powers--the older forms of objectives. Messrs. Watson and Swift have adopted a new formula for their series of _achromatic eye-pieces_, whereby their magnification and flatness of field are improved. These also bear a constant ratio to the initial power of their objectives.