Substituting 18 as the value of x, and 9 as the value of y, we
have the numerical values of each respectively, as follows: 27 cts.,
9 cts., 162 cts., 2 cts., 2 oranges, and or of his money. By sub-
stituting other numerical values and denominations a large number
of examples may be formed, and, by combination, the ingenious
teacher may find pleasant, and the pupil, profitable employment.

While undue prominence should not be given to this method, it is
important that in the analysis of problems the pupil be continually
required to point out the more important of these elementary ques-
tions, and the formulas involved in them. By this means, he will
become self-reliant, unaccustomed to flee for refuge in every diffi
culty to some RULE; but will depend upon the exercise of his reason
for a solution.

ARRANGEMENT.-It has been objected that, in a logical order of
subjects, fractions should precede denominate numbers, and U. S.
Currency should be placed with decimals, where it logically belongs.
The author would say, in answer, that since no scale can be added or
subtracted without reduction, logic would require reduction to be
taught before addition, and the complete decimal scale before frac-
tions or denominate numbers. He would not follow out this absurdity
but counsel natural instead of a logical order. All will agree that
a child comprehends the simple before the complex. What, then,
is the simplest subject in arithmetic? Certainly it is the integral
portion of the decimal scale. The reduction of this is so simple that
it is omitted, and the fundamental rules are taken next. None will
dispute that the integral portion of the denominate scale is more
simple than the fractional scale, and hence should precede it. An
objection is sometimes raised that fractions are found in denominate
numbers; but, certainly, those few fractions can be more easily ex-
plained than the whole subject of fractions, and the objection is,
therefore, not valid. This arrangement of subjects has the further
advantage, that, after the scholar has passed through the fundamental
rules, he is made to apply his knowledge to business transactions in
the currency which he is daily using, and is thus made to feel that
arithmetic really means something in life. He then gets a thorough
knowledge of the application of the tables of denominate numbers
with especial reference to their use in purchases and sales. The

pupil is better fitted to leave school with a thorough knowledge of
these subjects, and the discipline which the acquisition will give him,
than with a knowledge of the simple operations of fractions and
decimals combined; for the latter he will soon forget, while the former
will constitute a capital at compound interest.

OMISSIONS AND ADDITIONS.—A number of subjects which are usu-
ally introduced into arithmetics are omitted in this treatise for the
purpose of giving more space to those which are more practical,
and many additions are made to the applications of fractions and
per centage.

REVIEW.-The method of review in this series is such that a pupil
entering at almost any stage of progress will be required to review
all the subjects over which the class have passed, before finishing
the book, thus avoiding the necessity of turning the class back on
account of a few of its members.

Subtraction of Fractions,

Multiplication and Division of Fractions,

Reduction of Denominate Fractions,

Comparison of Numbers.