* The florin may often be used with advantage in calculations, e.g., 63 articles at 16/- each = 63 @ 8 florins each = 504 florins = £50 „ 4 florins = £50 „ 8/-.

* The term “decimals” is used throughout to include all numbers expressed decimally whether they be integral, fractional, or “mixed” : thus 34, .34, and 3·4 are all in this sense decimals.

* It is sometimes objected to the teaching of decimal fractions before vulgar fractions that this is not the historical order of development. The same argument would require us to go back a step further and operate only with fractions having unity for numerator. No one, however, nowadays proposes to substitute (say) for before operating with it. Of course it is not suggested that pupils should not use fractions such as ½ , ¼, ⅓, &c., till they have mastered decimal notation completely ; what is proposed is that decimals should be taught without any reference to vulgar fractions, e.g., .1 is not to be defined as .

† In many of the schools of Germany decimal fractions are introduced before vulgar fractions, their system of weights and measures being used in illustration, e.g., 7 metres 2 decimetres = 7·2 metres.

* Professor Everett—Discussion on the Teaching of Mathematics at the British Association, Glasgow, 1901.

† From the point of view of the present paper this phrase is unfortunate, as all the places in a number are equally entitled to be called decimal. The expression, however, is firmly established. It may be accepted and extended to the left. E.g., in 516·37, the decimal places of 7, 3, 6,1, 5, are respectively 2, 1, 0, −1, −2. [The characteristic of a logarithm of a number is minus the decimal place of the most important (or first significant) figure of the number.]

* Parts of a mil must be considered when the sum of money is to be multiplied, and the product is not to be divided by a number as large as, or greater than, the multiplier; also, when the sum of money is a divisor, and more than 3 decimal places are required by the method of page 57.

In these cases the parts of a mil may be obtained as a decimal, by treating the last 2 decimal places (or their excess over 25, 50, or 75) as pence, and reducing these to mils for the next two places, and so on.

E.g., approx., = £·31041 more nearly, = £·3104166 still more nearly, = £·310416666, &c., &c.

The proof is left to the reader.