of substitution give the series of equations,

A - rest = a, B - rest = b, C - rest = c, &c. (1)

Subtracting the second from the first, the third from the second, and
so on, we obtain at once the value of each weight in terms of the
preceding, so that all may be expressed in terms of the largest, which
is most conveniently taken as the standard

B = A/2 + (b - a)/2, C = B/2 + (c - b)2, &c. (2)

The advantages of this method of subdivision and comparison, in
addition to its extreme simplicity, are (1) that there is only one
possible combination to represent any given weight within the range of
the series; (2) that the least possible number of weights is required
to cover any given range; (3) that the smallest number of
substitutions is required for the complete calibration. These
advantages are important in cases where the accuracy of calibration is
limited by the constancy of the conditions of observation, as in the
case of an electrical resistance-box, but the reverse may be the case
when it is a question of accuracy of estimation by an observer.

In the majority of cases the ease of numeration afforded by
familiarity with the decimal system is the most important
consideration. The most convenient arrangement on the decimal system
for purposes of calibration is to have the units, tens, hundreds, &c.,
arranged in groups of four adjusted in the proportion of the numbers
1, 2, 3, 4. The relative values of the weights in each group of four
can then be determined by substitution independently of the others,
and the total of each group of four, making ten times the unit of the
group, can be compared with the smallest weight in the group above.
This gives a sufficient number of equations to determine the errors of
all the weights by the method of substitution in a very simple manner.
A number of other equations can be obtained by combining the different
groups in other ways, and the whole system of equations may then be
solved by the method of least squares; but the equations so obtained
are not all of equal value, and it may be doubted whether any real
advantage is gained in many cases by the multiplication of
comparisons, since it is not possible in this manner to eliminate
constant errors or personal equation, which are generally aggravated
by prolonging the observations. A common arrangement of the weights in
each gro