[2] The statement by Galileo that he “fashioned” these first lenses can hardly be taken literally if his very speedy construction is to be credited.

[3] Scheiner also devised a crude parallactic mount which he used in his solar observations, probably the first European to grasp the principle of the equatorial. It was only near the end of the century that Roemer followed his example, and both had been anticipated by Chinese instruments with sights.

[4] He attempted to polish them on cloth, which in itself was sufficient to guarantee failure.

[5] In Fig 13, A is the support of the tube and focussing screw, B the main mirror, an inch in diameter, CD the oblique mirror, E the principal focus, F the eye lens, and G the member from which the oblique mirror is carried.

[6] In fact a “four foot telescope of Mr. Newton’s invention” brought before the Royal Society two weeks after his original paper, proved only fair in quality, was returned somewhat improved at the next meeting, and then was referred to Mr. Hooke to be perfected as far as might be, after which nothing more was heard of it.

[7] Commonly, but it appears erroneously, ascribed to Lord Mansfield.

[8] This was probably due not only to unfavorable climate, but to the fact that Herschel, with all his ingenuity, does not appear to have mastered the casting difficulty, and was constrained to make his big speculum of Cu 75 per cent, Sn 25 per cent, a composition working rather easily and taking beautiful, but far from permanent, polish. He never seems to have used practically the SnCu₄ formula, devised empirically by Mudge (Phil. Trans. 67, 298), and in quite general use thereafter up to the present time.

[9] An F/3 mirror of 1m aperture by Zeiss was installed in the observatory at Bergedorf in 1911, and a similar one by Schaer is mounted at Carre, near Geneva.

[10] More recently his condition proves to be quite the exact equivalent of AbbÉ’s sine condition which states that the sine of the angle made with the optical axis by a ray entering the objective from a given axial point shall bear a uniform ratio to the sine of the corresponding angle of emergence, whatever the point of incidence. For parallel rays along the axis this reduces to the requirement that the sines of the angles of emergence shall be proportional to the respective distances of the incident rays from the axis.

[11] It is interesting to note that in computing Fig. 54a for the sine condition, the other root of the quadratic gave roughly the Gaussian form of Fig. 53.

[12] The curvature of the image is the thing which sets a limit to shortening the relative focus, as already noted, for the astigmatic image surfaces as we have seen, fall rapidly apart away from the axis, and both curvatures are considerable. The tangential is the greater, corresponding roughly to a radius notably less than ? the focal length, while the radial fits a radius of less than ? this length with all ordinary glasses, given forms correcting the ordinary aberrations. The curves are concave towards the objective except in “anastigmats” and some objectives having bad aberrations otherwise. Their approximate curvatures assuming a semiangular aperture for an achromatic objective not over say 5°, have been shown to be, to focus unity

?_r = 1 + (1/(?-?')(?/n - ?'/n'), and ?_t = 3 + 1/(?-?')(?/n - ?'/n')

?_r and ?_t being the respective reciprocals of the radii. The surfaces are really somewhat egg shaped rather than spherical as one departs from the axis.

[13] The doublet costs about one and a half times, and the triplet more than twice the price of an ordinary achromatic of the same aperture.

[14] A very useful treatment of the aberrations of parabolic mirrors by Poor is in Ap. J. 7, 114. In this is given a table of the maximum dimension of a star disc off the axis in reflectors of various apertures. This table condenses to the closely approximate formula

a = lld/f²

where a is the aberrational diameter of the star disc, in seconds of arc, d the distance from the axis in minutes of arc, f the denominator of the F ratio (F/8 &c.) and 11, a constant. Obviously the separating power of a telescope (see Chap. X) being substantially 4.56/D where D is the diameter of objective or mirror in inches, the separating power will be impaired when a > 4.56/D. In the photographic case the critical quantity is not 4.56/D, but the maximum image diameter tolerable for the purpose in hand.

[15] Instruments with a polar axis were used by Scheiner as early as 1627; by Roemer about three quarters of a century later, and previously had been employed, using sights rather than telescopes, by the Chinese; but these were far from being equatorials in the modern sense.

[16] Contributions from the Solar Obs. #23, Hale, which should be seen for details.

[17] A more precise method, depending on an actual measurement of the angle subtended by the diameter of the eyepiece diaphragm as seen through the eye end of the ocular and its comparison with the same angular diameter reckoned from the objective, is given by Schaeberle. M. N. 43, 297.

[18] The angular field a is defined by

tan ½a = ?/F

where ? is, numerically, the radius of the field sharp enough for the purpose in hand, and F the effective focal length of the ocular.

[19] There are binoculars on the market which are to outward appearance prism glasses, but which are really ordinary opera glasses mounted with intent to deceive, sometimes bearing a slight variation on the name of some well known maker.

[20] r the radius of the ring, is given by, r = (15/2)(t'-t) cos Dec., t'-t being the seconds taken for transit.

[21] (For full discussion of this instrument see Chandler, Mem. Amer. Acad. Arts & Sci. 1885, p. 158).

[22] For the principle of diffraction spectra see Baly, Spectroscopy; Kayser, Handbuch d. Specktroskoie or any of the larger textbooks of physics.

[23] The effect on the observed height of a prominence is h = h' sin c/sin t, where h is the real height, h' the apparent height, c the angle made by the grating face with the collimator, and t that with the telescope (Fig. 146).

[24] If A be the brightness of one object and B that of the other, a the reading of the index when one image disappears and the reading when the two images are equal then A/B = tan²(a-). There are four positions of the Nicol, 90° apart, for which equality can be established, and usually all are read and the mean taken. (H. A. II, 1.)

[25] For full description and method see H. A. Vol. 14, also Miss Furness’ admirable “Introduction to the Study of Variable Stars,” p. 122, et seq. Some modifications are described in H. A. Vol. 23. These direct comparison photometers give results subject to some annoying small corrections, but a vast amount of valuable work has been done with them in the Harvard Photometry.

[26] The general order of precision attained by astronomical photometers is shown in the discovery, photographically, by Hertzsprung in 1911, that Polaris, used as a standard magnitude for many years, is actually a variable. Its period is very near to four days, its photographic amplitude 0.17 and its visual amplitude about 0.1, i.e., a variation of ± 5 per cent in the light was submerged in the observational uncertainties, although once known it was traced out in the accumulated data without great difficulty.

[27] Such apparatus is essentially appurtenant to large instruments only, say of not less than 12 aperture and preferably much more. The eye is enormously more sensitive as a detector of radiant energy than any device of human contrivance, and thus small telescopes can be well used for visual photometry, the bigger instruments having then merely the advantage of reaching fainter stars.

[28] E. g., the beautiful astrographic and other objectives turned out by the brothers Henry.

[29] This and several of the subsequent figures are taken from quite the best account of testing objectives: “On the Adjustment and Testing of Telescope Objectives.” T. Cooke & Sons, York, 1891, a little brochure unhappily long since out of print. A new edition is just now, 1922, announced.

[30] Sometimes with ever so careful centering the ring system in the middle of the field is still eccentric with respect to the small mirror, showing that the axis of the parabola is not perpendicular to the general face of the mirror. This can usually be remedied by the adjusting screws of the main mirror as described, but now and then it is necessary actually to move over the small mirror into the real optical axis. Draper (loc. cit.) gives some experiences of this sort.

[31] See also two valuable papers by Sir Howard Grubb, The Observatory, Vol. VII, pp. 9, 43. Also in Jour. Roy. Ast. Soc. Canada, Dec., 1921, Jan. 1922.