Means of Measurement.—A micrometer becomes an indispensable instrument to those who make sidereal observations of an exact character. Without such means it will be impossible to determine either positions or distances except by mere estimation, and this is not sufficiently precise for double-star work. With a reliable micrometer[53] excellent results may be obtained, especially with regard to the varying angles of binary systems. Frequent remeasurement of these is desirable for comparison with the predicted places in cases where the orbits have been computed. In this department of astronomy the names of Herschel, South, Struve, Dawes, Dembowski, Burnham, and others are honourably associated, and it is notable that refracting-telescopes have accomplished nearly the whole of the work. But reflectors are little less capable, though their powers seem to have been but rarely employed in this field. Mr. Tarrant has lately secured a large number of accurate measures with a 10-inch reflector by Calver, and if care is taken to secure correct adjustment of the mirrors, there is no reason why this form of instrument should not be nearly as effective as its rival. Mr. Tarrant advises those who use reflectors in observing double stars “to test the centering of the flat at intervals during the observations, as the slightest shift of the large mirror in its cell will frequently occasion a spurious image which, if it by chance happens to fall where the companion is expected to be seen, will often lead to the conclusion that it has been observed. In addition to this, any wings or the slightest flare around a bright star will generally completely obliterate every trace of the companion, especially if close and of small magnitude, and such defects will in nine cases out of ten be found to be due to defective adjustment. Undoubtedly for very close unequal pairs the refractor possesses great advantages over a reflector of equal aperture; in the case of close double stars the components of which are nearly equal there appears to be little, if any, difference between the two classes of instruments; while for any detail connected with the colour of stars the reflector certainly comes to the fore from its being perfectly achromatic.” These remarks from a practical man will go far to negative the disparaging statements sometimes made with regard to reflectors and stellar work, and ought to encourage other amateurs possessing these instruments to take up this branch in a systematic way.

Dividing Power.—This mainly depends upon the aperture, and it was made the subject of careful investigation and experiment by Dawes, who found that the diameters of the star-disks varied inversely as the aperture of the telescope. Adopting an inch as the standard, he ascertained that this aperture divided stars of the sixth magnitude 4″·56 apart, and on this base he constructed the following table:—

Aperture
in inches.		Least
separable
distance.
″		Aperture
in inches.		Least
separable
distance.
″
1·04·56 6·50·70
1·62·857·00·65
2·02·287·50·61
2·252·038·00·57
2·51·828·50·536
2·751·669·00.507
3·01·529·50.480
3·51·3010·00·456
3·81·2012.00·380
4·01·1415·00·304
4·51·0120·00·228
5·00·9125·00·182
5·50·8330·00·152
6·00·76

Dallmeyer, the optician, confirmed these values by remarking:—“In all the calculations I have made, I find that 4·33 divided by the aperture gives the separating power. Thus, 4·33 inches separates 1 ″.” But a good deal depends upon the character of the seeing and upon other conditions. A large aperture will sometimes fail to reveal a difficult and close comes to a bright star when a smaller aperture will succeed. This is due to the position of the bright diffraction-ring, which in a large instrument may overlap the faint companion and obscure it, while in a small one the ring falls outside and the small star is visible[54]. Dawes concluded that “tests of separation of double stars are not tests of excellence of figure,” and he gave much valuable information with regard to micrometers and double-star observations generally in the ‘Monthly Notices,’ vol. xxvii. pp. 217-238. This paper will well repay attentive reading.

Number of Stars.—In the northern hemisphere there are about 5000[55] stars perceptible to the naked eye. This is less than an observer would suppose from a casual glance at the firmament, but hasty ideas are often inaccurate. The scintillation of the stars and the fact that many small stars are momentarily glimpsed but cannot be held steadily have a tendency to occasion an exaggerated estimate of their numbers. Authorities differ as to the total of naked-eye stars. Sir R. S. Ball says “the number of stars which can be seen with the unaided eye in England may be estimated at about 3000.” Gore gives 4000. Backhouse mentions 5600 as visible in the northern hemisphere. Argelander, who has charted 324,188 stars between 2° S. of the equator and the N. pole, gives the following numbers of stars up to the 9th magnitude:—

1st.2nd.3rd.4th.5th.
20651904251100
6th.7th.8th.9th.
320013,00040,000142,000

With every decrease in magnitude there is a great increase in numbers, and if this is extended to still smaller magnitudes down to the 15th or 16th we can readily understand that there exist vast multitudes of fainter stars. Paul Henry, of the Paris Observatory, says there are about 1,500,000 stars of the 11th mag., and Dr. Schönfield, of Bonn, gives 3,250,000 as of the 11½ mag. It is probable that by means of photography and the largest telescopes considerably more than 50 millions of stars may be charted, but this is a mere approximation. Roberts has photographed 16,206 stars within an area of four square degrees in a very rich region of the Galaxy near η Cygni, and Gore computes that were the distribution equal to this over the whole firmament the number of stars would reach 167 millions. He also remarks that in the Paris photographs of the Pleiades, 2326 stars are shown in a space covering about three square degrees, and this gives for the entire sky a total of 33 millions. It is, however, manifest that unusually plentiful spots in the heavens cannot be accepted as affording a criterion of the whole.

Magnitudes.—According to Argelander’s figures, above quoted, each magnitude exhibits a rise of about 300 per cent. But authorities rarely agree as to scale, as the following comparison between Sir J. Herschel and Struve will show:—

H.S. H.S.
4·03·6 11·09·3
4·54·111.59.6
5·04·612·09·8
5·55·0512·510·0
6·05·513·010·18
6·55·9513·510·36
7·06·414·010·54
7·56·8514·510·71
8·07·315·010·87
8·57·716·011·13
9·08·117·011·38
9·58·518·011·61
10·08·819·011·82
10·59·120·012·00