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should state and prove the propositions orally, using a pointer to indicate on the figure every line and angle named. He should be encouraged, in reviewing each Book, to do the original exercises ; to state the converse of propositions ; to determine from the statement, if possible, whether the converse be true or false, and if the converse be true to demonstrate it; and also to give well-considered answers to questions which may be asked him on many propositions.

The Teacher is strongly advised to illustrate, geometrically and arithmetically, the principles of limits. Thus a rectangle with a constant base b, and a variable altitude 2, will afford an obvious illustration of the axiomatic truth contained in [4], page 88. If x increase and approach the altitude a as a limit, the area of the rectangle increases and approaches the area of the rectangle a b as a limit; if, however, x decrease and approach zero as a limit, the area of the rectangle decreases and approaches zero for a liinit. An arithmetical illustration of this truth would be given by multiplying a constant into the approximate values of any repetend. If, for example, we take the constant 60 and the repetend .3333, etc., the approxi. mate values of the repetend will be o, 18, 38, 133887, etc., and these values multiplied by 60 give the series 18, 19.8, 19.98, 19.998, etc., which evidently approach 20 as a limit; but the product of 60 into } (the limit of the repetend .333, etc.) is also 20.

Again, if we multiply 60 into the different values of the decreasing series, 30, 310, 3000, 50000, etc., which approaches zero as a limit, we shall get the decreasing series, 2, }, to, zoo, etc. ; and this series evidently approaches zero as a limit.

In this way the pupil may easily be led to a complete comprehension of the whole subject of limits.

The Teacher is likewise advised to give frequent written examinations. These should not be too difficult, and sufficient time should be allowed for accurately constructing the figures, for choosing the best language, and for determining the best arrangement.

The time necessary for the reading of examination-books will be diminished by more than one-half, if the use of the symbols employed in this book be permitted.


January, 1879.


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