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

5

Now divided by 3 gives 4,
and - multiplied by 2 produces -.
Hence 2 of A is equal to 1i8.1
And in a similar way it may be shown that of 4 o f -, or any
compound fraction whatever, may be reduced to a single fraction.
Hence any compound fraction is reduced to a single one by multiplying the numerators of the several fractions for a new numerator, and
the denominators for a new denominator.
11. PROP. To reduce fractions to lower, and to their lowest terms.2
A fraction whose numerator and denominator are not prime to
The compound fraction 2 of A is of the same value as 4 of 3, whatever be the value.
of the unit.
It has been shown above that,- of 4 is equal to -. And 4 of 3- denotes that % is>
to be divided into 5 equal parts, and 4 of them are to be taken, or that 2 is to be
divided by 5 and the result multiplied by 4.
Now 3 divided by 5 gives -,A and {- multiplied by 4 produces.w. Therefore;
4 of 2 is equal to -8. Whence 2 of 4 is equal to A of 3.
2 For example, the fraction.- is reduced to lower terms, -, by dividing the
numerator and denominator by 2, a divisor of 12 and 18: and -- is reduced to its
lowest terms, -, by dividing the numerator and denominator by 6, the greatest divisor
of 12 and 18.
It may happen that a fraction, after having been reduced to its lowest terms,
may still be expressed by numbers too large and inconvenient for practical use; a series;
of fractions nearly equivalent in value and in lower terms may be found by con —
verting the given fraction into a continued fraction, and reducing one, two, or more
terms of the continued fraction to a simple fraction.
DEF.-A continued fraction is one which has for its denominator an integer andt
a fraction; and which latter fraction has also for its denominator an integer and a.
fraction; and so on till the series terminates.
The following are examples of continued fractions, consisting of two, three,
and four terms:1 1 1,$3+7. 2+~ 5 5+5+l '
~6 ~~11
The inconvenient manner in which the continued fraction runs across the page,
led the late Sir John W. Herschell to adopt a method of writing the continuedi
fraction in one horizontal line. The sign + being placed below the line which
divides the terms of the fraction, indicates that the following fractional parts of
the expression are each added to the denominators of the preceding fractions in the:
series. Accordingly the continued fractions above may be thus written:
111351111
3+7 ++' 2+4+6 +7+9+ll
To convert the fraction 4 7 into a continued fraction:
Here179 _= 85 9- 1I
443 443 2+17- 85 85 9 +
179 T 9
next 85 l-1 9 4== 1-1
179 179 2+s - t 9 2+s
85 8 1 4 4
Hence by successive substitutions,