340 
requires only 15 entries. But in this problem 
it is the custom to make several solutions in 
succession, in parallel columns ; and in all col- 
umns after the first the criticised method re- 
quires fewer entries than does the suggested 
substitute. The reviewer’s failure to see the 
point is all the more surprising, since, on the 
same page, alongside the first column, is a 
second column, in which only 10 entries are re- 
quired. In fact, if no unnecessary recording is 
done, five entries are sufficient. 
And most teachers of ‘ Practical Astronomy ’ 
will agree in my opinion that the wider publi- 
cation of addition and subtraction logarithms 
has not done away with the desirability of 
‘adapting formule to logarithmic computation. ’ 
The solution of most problems is actually short- 
ened by transforming the equations so that such 
logarithms are not needed. These logarithms 
were well known to Chauvenet, were referred 
to by him, and he made it clear (Vol. I., p. 211) 
when they should be used. In the class of 
problems we are considering, their wider publi- 
cation has not influenced the form of solution 
appreciably with many astronomers, nor does 
it deserve to, for valid reasons. Take the case 
most strongly criticised by G. C. C.—that of 
determining the hour angle ¢ from a measured 
altitude. I have—on five different pages— 
equally recommended using the well-known 
forms tan}? and sin}¢. I understand, and 
every reader of the criticism will understand, 
that G. C. C. would entirely replace these 
‘by the well-known form cost, not only in 
the example solved by me, but in all such 
solutions. A solution through tan }¢ re- 
quires 17 entries, but this method is the 
most accurate and most generally applicable 
of the three. Slightly less accurate and gen- 
eral is the solution through sin3}¢, which 
requires 14 entries; and this is the form most 
frequently used by astronomers. The solution 
through cost requires 13 entries, besides the 
use of two kinds of logarithms, and has the fur- 
ther disadvantage that it is less general than the 
other two forms. In fact, cost should not be 
used at all if ¢ is less than 80°; and the ob- 
server’s position, combined with clouded skies, 
will often make observations under such con- 
ditions desirable. There are many astrono- 
& CIENCE. 
[N.S. Vou. X. No. 245. 
mers, of the greatest experience, who would 
not use the cos¢ formula when ¢ is less than 
45° ; they would employ the forms sin }¢ or tan 
tin preference. To save one or two entries 
at the expense of accuracy and generality of the 
formule, strikes me as being poor astronomy 
and poor pedagogy. 
It is plain that the reviewer regrets the in- 
sertion of an Appendix containing the principal 
‘Formule Resulting from the Method of Least 
Squares,’ ‘with no pretense at their derivation.’ 
The Method of Least Squares is not a branch of 
Astronomy, any more than are Trigonometry 
and Logarithms. It isa method employed in all 
the sciences where quantitative observations 
are made. The formule used in applying the 
method have been appended for ready refer- 
ence, and have been found convenient. There 
is no longer any practical reason for including 
a chapter on this subject, since several small 
text-books on Least Squares are available. 
There is one of some 60 pages written by a gen- 
tleman whose initials are G. C. C. (presumably 
the reviewer)—it is called a ‘Treatise’—in 
which the one fundamental equation of the subject 
is assumed, ‘ with no pretense at its derivation.’ 
The reviewer objects to devoting 2} % ‘‘ of 
the entire treatise to such an antiquated matter 
as lunar distances.’’ As I explained in the 
book, this method ‘‘ is occasionally of consider- 
able importance to navigators and explorers.”’ 
It is sufficient to say that the French Connais- 
sance des Temps devotes about 5 % of its space, 
the British Nautical Almanac about 11 % and the 
American Nautical Almanac more than 133 % 
to the data for solving this problem. 
Likewise, the objectionable 13 % devoted to 
the ring micrometer is introduced with the 
statement that results obtained with it ‘‘ can be 
regarded as only approximately correct, and 
the ring micrometer should never be used with 
an equatorial telescope unless, in case of great 
haste, there is not time to attach the filar 
micrometer and adjust its wires by the diurnal 
motion ;’’ and further. ‘‘that it can be used 
with an instrument mounted in altitude and azi- 
muth, * * whereas a filar micrometer cannot.’’ 
These remarks cover the entire case, and it is 
impossible that they should mislead a student. 
The reviewer has called attention to a real 
