Elementary arithmetic, with brief notices of its history... by Robert Potts.

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DEF. 3. A mixed number consists of an integer and a fraction.
DEF. 4. A simple fraction consists of one numerator, and one.denominator.
DEF. 5. A compound fraction is defined to be the fraction of a
fraction, or of more than two fractions.
)EF. 6. A complex fraction is defined to be one that has an integer,
a fraction, or a mixed number for its numerator or denominator.
DEF. 7. An integer may be expressed in the form of a fraction by
placing the integer for the numerator and unity for the denominator.
2. PROP. A simple fraction may be considered Cas representing the quotient
which arises fromt dividing the ne6ber above the line by the number
below it.1
For the expression 3 has been assumed to indicate the quotient
arising from the number 3 divided by 4.
It has also been assumed to denote three-fourths of any unit.
Now 3 divided by 4, or - of 3, is three times as great as 4 of 1.
It follows that 3 divided by 4 must be equal to - of 1,
and the expression properly represents either of them.
3. PROP. A fraction is mzltiplied by any nemzber by multiplying the
-2nmercator by the znumber and retaining the denominator.2
Thus any fraction 2!- multiplied by 7 gives —;
for in each of the fractions -j and - 4,
the unit is divided into fifteen equal parts;
and 14 of these parts are taken in one case and 2 in the other;
and since 14 parts are 7 times 2 parts,
it follows that the fraction 14 is 7 times 2 -or that - multiplied by 7 gives 4.
4. PROP. A fiaction is mzltiplied by any nzmber by dividing the denomzi-:nator by the number and retaining the numerator.
Thus the fraction 3- multiplied by 4 gives —;
for the unit in I is divided into 20 equal parts,
and the unit in 3 into 5 equal parts.
in this case be applied in a sense different from that which expresses a number of
parts less than the whole; as - being greater than 1 or A, it would be absurd to
speak of the former being a fraction of the latter. The meaning, however, of such an
expression as, is, that the unit is divided into 4 equal parts, and this expression 5
denotes 5 of such equal parts; and thlis notation of fractions leads to no absurdity
whatever. It cannot, however, be denied, that there is an inaccuracy of language, if
a large quantity be stated to be a fraction of a smaller one.
1 This may be illustrated by any concrete number whatever. Let the unit assumed
be a lineal yard or 36 inches, then 3 units will contain 108 inches or equal parts,
and the 3 units or 108 inches divided by 4 gives the quotient 27 inches. Again, 4 of
the unit or 36 inches will be 9 inches, and 4 of the unit will be 27 inches. Hence
it appears that 3 yards when divided by 4, or I of 3 yards, is the same as -
of one yard.
2 If a fraction be multiplied by a number equal to the denominator the product is
equsal to the numerator.