fulness and thoroughness, by clearly defining current mercantile terms, by giving general information on business subjects, and presenting such examples as not only illustrate the theory, but also cover the cases most likely to arise every day in life. Attention is particularly asked to the exhaustive and practical treatment of percentage and its various applications, of interest with its short methods, interest on daily balances, savings-bank business, stock-jobbing, U. S. securities, averaging accounts, foreign exchange, custom-house business, etc. In connection with these subjects and with every other, care has been taken to find out and present the methods actually in use among business men.

Conciseness and clearness in rules and definitions have been aimed at throughout. The rules are approached by preliminary analyses which show the reasons for them, and thus enable them to be understood and remembered. The inductive method, leading progressively from step to step, has been followed in developing the subject. Examples are presented in great number and variety, and enough under every rule are left without answers to make the pupil rely on himself and test his mastery of the subject. They will be found, it is believed, of a grade that properly belongs to a Higher Arithmetic, but not so difficult as to discourage the faithful student.

It is hoped that this work will commend itself to progressive educators by its recognition of all recent changes (and they are neither few nor unimportant) in matters connected with the various subjects of which it treats. It presents the legal rate of interest as now established in the different states, and as changed in several of them by recent legislation; it gives the present par of exchange between the United States and England, as fixed by Act of Congress in 1873; it explains the mode of quoting sterling exchange, based on the new par; it shows the value of foreign monetary units in U. S. money of account, and of foreign coins, as lately proclaimed by the Secretary of the Treasury. In fine, no pains have been spared to make it a fresh, accurate, and reliable text-book, complete and every way adapted to the class

room.

NEW YORK, July 20, 1874.

HIGHER ARITHMETIC.

Definitions.

1. A Unit is one, a single thing.

2. A Number is,

1st. A unit or a collection of units,-as 1, 4; in which case it is called a Whole Number, an Integral Number, or Integer: or,

2d. One or more of the equal parts of a unit,-as, ; in which case it is called a Fractional Number, or Fraction: or,

3d. A combination of an integer and a fraction,as 41; in which case it is called a Mixed Number. 3. A Quantity is anything that can be increased or diminished; as, a number, a line, a surface.

4. Mathematics is the science of quantities.

Arithmetic is that branch of mathematics which treats of numbers. As an art, it teaches how to use numbers in calculating.

5. Numbers are distinguished as Abstract, and Concrete or Denominate.

An Abstract Number is a number not applied to any particular thing or things; as, one, eight.

A Concrete, or Denominate, Number is a number applied to some particular thing or things; as, one pound, eight dollars.

The name of the thing or things to which a concrete number is applied, is called the Denomination. In the last examples, pound and dollars are the denominations.

6. Numbers are also distinguished as Simple and Compound.

A Simple Number is an abstract number, or a concrete number having but one denomination; as, three, three feet.

A Compound Number consists of two or more concrete numbers having different denominations; as, three feet six inches.

7. A Scale is a series of numbers denoting the ratios, or relations, between successive orders or denominations. If these relations are the same, the scale is called Uniform; if not, Varying.

10 cents make 1 dime, 10 dimes 1 dollar, 10 dollars 1 eagle. Here we have a Uniform Scale of 10, or Decimal Scale, as it is called.

4 farthings make 1 penny, 12 pence 1 shilling, 20 shillings 1 pound. Here we have a Varying Scale of 4, 12, 20.

8. An Operation is a process used in calculating.

An operation may be performed according to a Rule,—that is, a prescribed method; or by Analysis, which consists in tracing out relations, and reasoning from what is known to what is unknown, without reference to rules.

A Proof is a method of finding whether the result of an operation is correct.