INTRODUCTION.

THE advantages of Mental Arithmetic are so great that it ought to form one of the principal parts in the education of all classes. In preparing for the commercial transactions of life, its claims are the very first: but, independently of this, when properly taught, it is one of the best means of disciplining the mind, the demonstrations of number being so much easier than those of magnitude, and consequently more suitable for children.

In teaching, the first principles and examples must be made plain to the senses, by the use of the slate, or the black board. The first must be seen by intuition, and so on for all the rest: intuition the most thorough. The pupil must not advance one step into the regions of darkness, every succeeding operation should be based upon something previously known. For this purpose, the examples and rules are so arranged as the first shall require but one operation of mind, the second two, and so on. The examples under each rule are divided into two parts, to be learned in two distinct courses. The tables are all placed at the end of the book, and should be learned as they are wanted in the study of the rules, but learned most thoroughly, for upon a knowledge of tables and data depend the power of the Mental Arithmetician, and enable him to answer questions which, to the uninstructed, appear truly astonishing: an attainment not to be accomplished by a few lessons, learned in an irregular manner, but by a thorough training of the mental faculties, and clear views of the principles of

DEFINITIONS.

1.-Arithmetic is the science of numbers, and teaches the art of making calculations. The figures made use of are the following:-1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

2.-MENTAL ARITHMETIC is the art of calculating by the mind, without the aid of symbols.

3.-NUMERATION teaches how to read, and notation how to write any number of figures..

4.-ADDITION teaches how to add two or more sums into one; as 8, 7, 12, 6 added together, the sum is 33.

5.-SUBTRACTION teaches how to take a less number from a greater, and shows the remainder or difference; as 7 taken from 22, the remainder is 15.

6.-MULTIPLICATION shews what one number will amount to when repeated a certain number of times; as 10 repeated 6 times will be 60.

7.-DIVISION teaches how to separate a given number into a certain number of parts; as 18, divided into 3. parts, each part will be 6.

8.-The following signs are used to express these rules :+ Addition; - Subtraction; × Multiplication; Division; = equality; as 8+4=12. 8-35. 6 x 6 = 36, 3666.

9.-An Integer or Unit is one whole number; as 1 yard, 1 shilling, or 1 penny.

10.-A part of 1 is called a fraction, and is thus written; (three eighths), or (three fourths). The bottom figure shews how many parts it is broken into, and is called the de-. nominator; the top figure shews how many are taken, and is called the numerator.

SECTION 1.-ELEMENTARY RULES.

11.—In teaching the elementary rules it is best to begin with very low numbers, such as can be represented to the senses, and demonstrated to the eye by tangible objects, such as nuts, counters, marks on the slate, or black board, or upon an abacus. Children find great difficulty in comprehending abstract numbers; all questions put to them should be after the manner of the first examples under each rule, until they have clear views of number. The examples given here (and all through the book) are only specimens, a great many more should be given, and, advancing by very easy steps, from those which are more simple to those which are more difficult. Those who wish for any assistance in this department will find it in Knight's Arithmetic for Children.

12.-Addition (4).-James had 5 apples; George had 4. How many had they?

Add 8 to 5; 6 to 7; 9 to 11; 15 to 6; 25 to 7; 16 to 8; 27 to 6; 34 to 11; 7 and 8 to 15; 9 and 7 to 18; 8 and 6 to 24; 11 and 10 to 36; 6 and 22 to 34.

13.-Subtraction (5).-James had 8 nuts; the squirrel stole 4. How many had he left?

Take 3 from 5; 7 from 12; 9 from 16; 3 and 4 from 18; 8 and 2 from 45; 7 and 8 from 32; 15 and 8 from 46; 24 from 55.

14.-Multiplication (6).-How many eyes have 5 boys? How many fingers have 4 boys? How many legs have three horses ?

Three times 4; 6 times 8; 5 times 15; 3 times 25;

6 times 32; 8 times 75; 4 times 55; 3 times 122; 4 times 324; 7 times 426; 8 times 284; 4 times 3626.

15.-Division (7).—If 3 classes contain 24 boys, how many are there in each ?

784 by 4;

SECTION 2.-OF FRACTIONS.

16.-Fractions (10) are parts of units. One of the best methods of teaching them is to take a line upon the black board, and divide it into an equal number of parts, and compare it with other lines of the same length, yet divided into different proportions, something like the following:

4ths 8ths 16ths 3ds 6ths 9ths 12ths

The pupils should be questioned upon the various lines, and the relation of the various parts to a whole line, or to each other, something in the following manner:-which is 4, or 4, or 4, or , or, or 3, &c. How many in ? How many 8ths in ? How many 9ths in? How many 12ths in ?

17.-Another method of teaching, is to take any particular number, and divide it into different parts, in the following manner:-as of 12; 4 of 12; of 12; of 12; of 20; of 20; of 20; of 20; 4 of 20; of 20; 1 of 20 ; of 24; of 24; of 24; of 24; of 24; of 24; of 24; of 24; of 24; of 24; 3 of 24; 7 of 24.

18. The following is a very good method of shewing the relation between fractions, and of developing the intellectual powers:

of 12 equal how many 6ths of 12? of 122; and 824 Ans. 12ths of 24 of 20 equal how many equal how many 5ths of 20?

Solution: 3 of 12=8; of 24 equal how many 10ths of 20? of 16

19.-Reduction.-The numerator and denominator of a fraction being multiplied or divided by the same figure, the value of the fraction remains the same.

20.-When the numerator is larger than the denominator, it is called an improper fraction, and is reduced by dividing the numerator by the denominator; as 3543.

21.-To add or subtract fractions, they must have like denominators, then add all the numerators together, and divide the sum by the common denominator; or for subtraction take the less numerator from the greater.

22-Addition.-Add

= 1 or (19) 11⁄2.

=

to . 7+5=12; then 12÷8

Add to; to; to; to; 1⁄2 to ; † to 4;

Ans.from; from ; from;

; from; from 11.

Note. The denominators in these questions have all a ratio to each other, and such only can be answered mentally by children. The general rules for the reduction of fractions to a common denominator are too complex to be inserted here.

24.-Multiplication. To multiply a fraction by a whole number.-RULE. Multiply the numerator by the whole number; as x 6 or (20) 4; × 4 = 2 or 14.

Multiply by 8; by 10; by 4; 7 by 3: by 5.

25.-To multiply a whole number by a fraction.-RULE. Multiply by the numerator, and divide by the denominator.Note: Divide first, when you can without any remainder.16 x ; 16÷÷8=2×7 to 14: or 16 × 7 = 1128

= 14. Ans.

12 by ; 18 by ; 15 by † ;

Multiply 20 by ; 24 by;