3. In the fact that generally the pupil is prepared for the demonstration of each rule by the time he arrives at it.

Thus the pupil is taught as early as Simple Division to distinguish between the abstract quotient arising when divisor and dividend are both concrete, and the concrete quotient resulting from an abstract divisor. This distinction is applied in several subsequent rules. The rules in which the "rate per cent." is used as the standard of comparison, are introduced by general applications of the term. The pupil is by this means greatly assisted in understanding its special application in the calculation of Interest, Discount, Value of Stocks, etc., etc. The chief exception to this course will be found in the case of the rule for finding a Vulgar Fraction equivalent to a repeating decimal, given in Part I. of the work, the proof of which is necessarily deferred until we treat of Progression in Part II.

4. In the plan of grouping together rules which depend on the same principles.

The adoption of this plan has sometimes induced a departure from the usual order of succession of the various rules; but the arrangement adopted seems the most rational and systematic. For instance, the calculation of averages, involving as it does the rules of addition and division only, is given immediately after Simple Division: this has enabled us to impart increased variety to the exercises on the simple rules. Again, most of the sums usually given in Proportion may be worked without any knowledge of the properties of ratios upon which the demonstration of that rule is based. The method of solving such questions by Multiplication and Division given at page 50 is intelligible to the student at an early stage; and by introducing the method at that point we are enabled to postpone to a later and more fitting stage the full treatment of the principles of Proportion, and to avoid the prevalent error of giving a mechanical rule, without demonstration or any direction as to when it may or may not be applied. After Proportion naturally follows the rule for finding proportional parts, including that class of questions usually dignified with titles of Single and Double Fellowship. We have accordingly inserted it in that place.

5. Questions arranged in series and questions on the principles are frequently introduced.

Examples of the former will be found on pages 54, 56, 116, etc. In these series each question suggests one step in the process employed in the solution of the general problem with which the series concludes.

We may also further notice that the Properties of Numbers are treated of at some length, and that the rules of Logarithms are fully explained. The student is thus not only enabled to use this abridged method of computation, but also to understand the principles on which the method is based. A table of Logarithms is added to the chapter on the subject.

The PRACTICAL portion of the work will be found distinguished by the following characteristics :—

1. The practical nature of the questions.

Questions that arise in business are given in all the exercises; and rules that are chiefly serviceable in commerce, such as the Weights and Measures, both English and Foreign, Practice, Exchange, Discount, Annuities, Stocks, and Life Assurance, are very fully explained, and followed by numerous and varied exercises.

2. The graduation of the exercises.

Great care has been taken to effect this, so that the pupil shall have but one difficulty at a time to encounter. For example in the rule of Simple Division will be found the following stages :-FIRST, Sums are given in which the divisor is contained in each figure of the dividend. SECOND, Sums in which a remainder is carried from one figure to the next. THIRD, Sums with larger divisors, but in which the pupil will not be troubled with a remainder. FOURTH, A new notation for division is introduced, which prepares the way for the consideration of Vulgar Fractions. FIFTH, Sums with remainders. SIXTH, The method of cancelling. SEVENTH, A number of general questions complete the rule.

3. The number and variety of the exercises.

The work will be found to contain a greater number of Exercises by far than any work of the same dimensions; and provision is further made whereby the number may be

indefinitely extended by the teacher or student. The answers to the numbered exercises are given in the Book of Answers, and simple methods of verification are added to the additional exercises, which number half-a-million. The utmost facility is thus afforded for practice in calculation.

4. The care taken to avoid stating any rule in a form likely to mislead.

The pupil often finds on arriving at the higher branches of Mathematics that he has to unlearn certain things taught in his book on Arithmetic. For example, it is no uncommon thing for pupils to be led into error by such phrases as "feet multiplied by feet give feet;" "feet multiplied by inches give inches," etc.; which can only be understood as abbreviated modes of describing the process of finding an area by taking a certain number of square feet or inches a certain number of times. These and all similar misleading expressions are avoided.

5. The numerous methods for economising labour in long calculations.

In order not to interfere with the general arrangement of the rules, the large number of approximate and contracted methods are presented together in the second part of the work, but they may be given by the teacher if necessary in connection with the more general rules to which they severally refer.

It only remains to add that care has been taken to make the collection of answers as far as possible correct, but it is almost impossible in dealing with so large a number of exercises entirely to exclude errors. If any teacher using the book who may discover any such error will send the correction to the publisher, he will confer an obligation on the author.

ARITHMETIC.

Numeration and Notation.

1. ARITHMETIC teaches us the use of numbers.

2. Notation is the art of representing numbers by figures or letters.

3. There are two methods of Notation: the Arabic and the Roman.

4. The

ARABIC NOTATION.

Expresses numbers by the figures 1, 2, 3, 4, 5, 6, 7, 8, 9. 5. A single object or thing is called a unit.

6. A greater number of units than nine cannot be expressed by one figure alone.

7. The same figures are made to represent numbers greater than nine, by placing them more or less to the left of the position occupied by the unit, every remove to the left increasing the value of a figure ten times.

8. The symbol 0, called nought, cypher, or zero, is used to keep a certain place vacant, or show that it is not occupied by any significant figure.

B

Tens.
Hundreds.
Thousands.

Hds. of Thous.

Units.

Tens of Thous.

9. The following table gives the names of the different places.

7 6 4, 3 9 8, 7 4 6, 2 5 1, 6 4 3, 5 8 6

Write in words the following numbers

(a) 8, (b) 9, (c) 2, (d) 5, (e) 10, (f) 6, (g) 3, (h) 4, (i) 7, (k) 15, (1) 19, (m) 20, (n) 27, (o) 39, (p) 170, (g) 101, (r) 300, (s) 504, (†) 353, (u) 24, (v) 371, (w) 208, (x) 816, (y) 204, (*) 329.

Ex. 2.

10 In reading off a number consisting of more than three figures, it is necessary to divide it into periods of three figures each, commencing with the unit figure.

11. The second period of three figures represents so many thousands.

Read the following:

(a) 86000, (b) 25000, (c) 103000, (d) 605000, (e) 78000, (f) 9000, (g) 86720, (h) 25020, () 103103, ()605201, (1) 78101, (m) 9009, (u) 20017, (v) 34007, (w) 300002, (x) 2002, (y) 34002, (*) 10072.

Ex. 3.

12. The figures beyond the first two periods of three figures, as far as the fifth period, are millions, those in the fifth and sixth periods, billions, &c.

Read (a) 86000000, (b) 25000000, (c) 103000000, 9000000, (e) 86071302,`` (ƒ) 781201012, (g) 8070740, (h) 9098009, (i) 260801, (k) 27108013, (7) 1047100006125, (m) 420670212, (n) 71800002, (o) 7369073, (p) 46803,004 (q) 307405, (r) 472003765, (s) 514000514, (t) 4126359, (u)317962518, (v) 310070654, (w) 4726342179, (x) 2007003, (3) 500500500, (2) 270027002700.