rom one to two
hectograms is easily appreciable. A match lighted in the day-time makes
no increase in the illumination, and so on.

A mathematical analysis shows that from the law of FECHNER it follows
that the impression increases in _arithmetical_ progression (1, 2, 3, 4,
...) simultaneously with an increase of the intensity in _geometrical_
progression (_I_, _I_^2, _I_^3, _I_^4, ...). It is with the sight the
same as with the hearing. It is well known that the numbers of
vibrations of the notes of a harmonic scale follow each other in a
geometrical progression though, for the ear, the intervals between the
notes are apprehended as equal. The magnitudes play the same role in
relation to the quantities of light as do the logarithms to the
corresponding numbers. If a star is considered to have a brightness
intermediate between two other stars it is not the _difference_ but the
_ratio_ of the quantities of light that is equal in each case.

The branch of astronomy (or physics) which deals with intensities of
radiation is called _photometry_. In order to determine a certain scale
for the magnitudes we must choose, in a certain manner, the _zero-point_
of the scale and the _scale-ratio_.

Both may be chosen arbitrarily. The _zero-point_ is now almost
unanimously chosen by astronomers in accordance with that used by the
Harvard Observatory. No rigorous definition of the Harvard zero-point,
as far as I can see, has yet been given (compare however H. A. 50[3]),
but considering that the Pole-star ([alpha] Ursae Minoris) is used at
Harvard as a fundamental star of comparison for the brighter stars, and
that, according to the observations at Harvard and those of HERTZSPRUNG
(A. N. 4518 [1911]), the light of the Pole-star is very nearly
invariable, we may say that _the zero-point of the photometric scale is
chosen in such a manner that for the Pole-star _m_ = 2.12_. If the
magnitudes are given in another scale than the Harvard-scale (H. S.), it
is necessary to apply the zero-point correction. This amounts, for the
Potsdam catalogue, to -0m.16.

It is further necessary to determine the _scale-ratio_. Our magnitudes
for the stars emanate from PTOLEMY. It was found that the
scale-ratio--giving the ratio of the light-intensities of two
consecutive classes of magnitudes--according to the older values of the
magnitudes, was approximately equal to 21/2. When exact photometry began
(with instruments for measuring the magnitudes) in t