Special attention is invited to the method of presentation given in the teacher's edition. This is arranged at the beginning of each subject, just where it is required, and contains definite and full instructions regarding the order in which the subject should be presented, the points that require special attention and illustration, the kind of illustrations that should be used, a method for drill exercise, additional oral exercises where required for the teacher's use, and such other instructions as are necessary to form a complete guide to the teacher in the discussion and presentation of each subject.

The plan adopted of having a separate teacher's edition avoids entirely the injurious course usually pursued of cumbering the pupil's book with hints and suggestions which are intended strictly for the teacher.

Attention is also invited to the Properties of Numbers, Greatest Common Divisor, Fractions, Decimals, Compound Numbers, Business Arithmetic, Ratio and Proportion, Alligation, and Square and Cube Root, with the belief that the treatment will be found new and an improvement upon former methods.

The author acknowledges with pleasure his indebtedness to Prof. D. H. MACVICAR, LL.D., Montreal, for valuable aid rendered in the preparation of the work, and to CHARLES D. MCLEAN, A. M., Principal of the State Normal and Training School, at Brockport, N. Y., for valuable suggestions on several subjects.

POTSDAM, September, 1877.

M. MACVICAR.

Art. 1. Numbers are expressed by means of ten figures,

567

Figures, 0 1 2 3 4 5 6 7 8 9

Names, Naught, One, Two, Three, Four, Five, Six, Seven, Eight, Nine.

Observe regarding the ten figures:

1. The naught is also called cipher or zero, and when written alone stands for no number.

2. The other nine figures are called digits or significant figures, and each stands for the number written under it.

3. Any number of objects not greater than nine is expressed by one figure.

Thus, 2 boys, 5 girls, 7 pens, 9 desks, 4 windows.

2. PROP. I.-Numbers from nine upwards are represented by means of the nine digits and the cipher, by regarding objects as arranged in groups of different orders thus,

1. We regard one more than nine objects as a group which we call Ten, and represent by 1 and 0 thus,

9 and 1= 1 ten, written 10.

Observe, that any digit written, as the 1 is in 10, in the second place from the right, represents the number of tens. Hence 20 means 2 tens, 30 means 3 tens, and so on.

2. We regard 10 tens as a group of a higher order, which we call Hundred and represent thus, 100.

Observe, that any digit written, as the 1 is in 100, in the third place from the right, represents the number of hundreds. Hence 200 means 2 hundred, 300 means 3 hundred, and

so on.

3, Higher Orders of groups are formed in the same manner as Tens and Hundreds, by regarding ten of any order as one of the next higher. The number is also represented by writing each new order one place farther from the right.

Thus Ten hundred make 1 Thousand, and the number is written 1000. Hence 2000 means 2 thousand, 3000 means 3 thousand, and so on.

4. Observe, the position which a figure occupies determines the name of the order whose number it expresses.

Thus, in 379, the 9 stands for the number of Units, or single things; the 7 for the number of Tens, or groups of ten single things, and the 3 stands for the number of Hundreds, or groups of ten tens.

5. Observe, also, that when one or more orders are wanting in a number, their places are filled by ciphers. Thus, 5 hundred 7 is expressed 507; 8 hundred 3 tens is expressed 830.