454 The Achromatic Telescope. 



In the construction of achromatic object-glasses opti- 

 cians, however, do not all adopt the same mode of pro- 

 cedure. There is much room for mathematical ingenuity in 

 devising curves, the combination of which is likely to produce 

 good definition in the image, and different proportions are pre- 

 ferred by different artists. The chromatic error is removed 

 very simply in theory by a due proportion between the focal 

 lengths of the two lenses ; though in practice it may involve 

 some troublesome adjustment ; but the correction of the sphe- 

 rical aberration is a more complicated affair. The problem is 

 what is styled an indeterminate one ; that is, it may be equally 

 solved in many ways ; and some condition must be fixed upon 

 to limit it : much in the same way as, if it were merely pro- 

 posed to find two numbers whose sum should amount to ten, 

 there would be five answers equally correct; but if it were 

 required that one of these numbers should be half as large 

 again as the other, it would admit of only one answer, six and 

 four. Different conditions have been assumed by different 

 mathematicians, but not all equally advantageous in practice. 

 Clairaut made the two internal surfaces of equal curvature; 

 but this has no advantage beyond convenience to the workman, 

 and his formulee do not suit ordinary glass. Frauenhofer cor- 

 rected the aberration not only for the axis, but as far as possible 

 for lateral rays, so as to produce a very fine field ; but in doing 

 this he seems to have sacrificed the accurate union of his ex- 

 ternal pencils. It is indeed impossible on theoretical grounds 

 to combine the central, the marginal, and the intermediate 

 rays precisely in the same point ; but in Frauenhofer's con- 

 struction the marginal rays were left so much to take care of 

 themselves, that his glasses would not bear a ring-aperture, 

 i. e., having the centre stopped out ; and we read of two 11 -inch 

 object-glasses, presumably by Merz, which gave no well-defined 

 image when only one-fifth of the surface was left open round 

 the margin. Herschel II. proposed a formula which would 

 render the correction equally perfect for celestial and terrestrial 

 objects; but this would be of but little advantage to astro- 

 nomers ; and the construction is rendered less valuable by the 

 exact correspondence required between theory and workman- 

 ship, a correspondence which practical opticians know to be 

 matter of considerable uncertainty. Barlow, therefore, pro- 

 posed a fresh formula, with the condition that such practical 

 deviations should have the least possible influence upon the 

 result ; and this limitation seems the most sensible of all ; 

 though whether glasses so ground admit of the high perfection 

 occasionally, and it may be said fortuitously, obtained in the 

 working of other formula;, is perhaps not certainly known. 

 As old Dr. Kitchener tells us, " in every department of art it 



