c. The effect upon modern arithmetic.

4. The Reasons for Teaching in the Middle Ages.

a. The Church schools.

b. The reckoning schools.

c. The effect upon modern arithmetic.

5. The Reasons Developed by the Renaissance.

a. Influence of commerce.

b. Influence of printing. The crystallization of arithmetic.

c. The effect upon the subject matter of modern arithmetic.

6. The Reasons of To-day.