The provision made for exhaustive drill exercises cannot fail to meet the wants of the most thorough and exacting teacher, and at the same time interest and attract the pupil.

No other device can be made so efficient and useful in fixing relations of numbers in the mind of the pupil, and in securing rapidity and accuracy in the performance of work, as these drill tables.

Such of the denominate tables as enter into the business transactions of everyday life, and United States Money are used, to furnish applications in the fundamental rules, and not for the purpose of teaching reduction. Also the fractions ,, etc., too, which occur in daily life, are made familiar by drill tables and applied work, from the commencement.

No more of fractions has been presented than will give the pupil a distinct and correct idea of what fractions are, and their application to simple oral exercises.

To avoid the monotony of too much abstract work, a large amount of applied work, covering a wide scope of easy examples, in which only the natural relations of everyday life are introduced, have been given, and these have been so prepared as to review, and give practice on all previous work.

The author would make special acknowledgment of the valuable services rendered in the plan, arrangement, and compilation of this book, by Prof. Jonathan Piper, of Chicago, whose large experience as an educator, and acquaintance with many of the best teachers and schools of the country, have made him familiar with their needs.

With an earnest desire to add to the facilities for elementary instruction, this little book is confidently submitted to the public.

BROOKLYN, August, 1883.

D. W. F.

SUGGESTIONS.

OOD books aid good teachers. The book cannot contain all that is needed or useful pertaining to the subject, and the best book is the one most teachable and most suggestive. The skillful teacher will enlarge the work suggested by the text-book. Read carefully the following:

1. Do not advance too rapidly.

2. Review daily.

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Repetition is a condition of memory." 3. Seek to cultivate in pupils the habit of self-reliance.

4. Frequently find something to commend. A little judicious praise operates as a great incentive to effort, and stimulates the intellect.

5. Oral and slate work should be carried on together.

6. This book should be used both for seat-work and in recitation.

7. The pupil should first read and solve the questions from the open book. The work should also be prepared on slate or writing-pad, the solution and answer of each question being expressed by the proper signs.

8. Daily oral as well as slate practice should be given in naming and writing sums, differences, products, and quotients.

9. In all written work by the pupil, neatness, rapidity, and accuracy should be insisted on from the start, until it becomes a fixed habit.

10. The names and signs or abbreviations of the measures in common use should also be taught, and be as familiar to the pupil as the signs of operation, and used from the first in all oral and written work.

11. At the same time the pupil should be made familiar with the equal parts of numbers or things, from one half to one tenth inclusive.

12. The practical use of arithmetic is not oral, but mental; therefore, the eye and the hand, rather than the ear and the tongue, are the instruments of work.

13. The eye should be trained to act promptly. The pupil should see results, using the lips and tongue in naming them. As much depends upon this eye-training in all figure work, as in reading, in order to secure rapidity in reaching results.

14. Monotony should be guarded against, and the exercises varied, by plenty of easy slate work.

15. Pupils should be encouraged also to bring their slates from home, filled with neatly arranged work.

16. Every pupil should be provided with a foot-rule divided into inches, and taught to use it until measurements with it are familiar.

17. The same usage should be observed, whenever practicable, in the use of other standard measures, as the yard measure, the liquid pint, quart, and gallon measures, the dry quart and peck measures, the different weights, etc.

18. In estimating the value of the pupil's work, the time taken to do the work, and also the neatness should be noted.

19. Finally, to secure the best results in the least time, and in a manner most in accordance with the laws of mental growth, should be the constant aim of the teacher.

LESSON II.

SIGNS OF NUMBERS.

The signs or symbols of numbers are words, figures, and letters.

Each of the first nine numbers is expressed by a single figure in the Arabic notation, and by letters in the Roman notation, as shown on the preceding page.

The figure 0 is called naught, zero, or cipher.

These ten figures, when combined according to certain principles, can be made to express any number.

The following symbols of numbers should also be learned: