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

2. If any equimultiples or equisubmultiples of the first and third
terms of a proportion be taken, and the same or any other equimultiples or equisubmultiples of the second and fourth be taken, the
four numbers will form a proportion.
3. The sum or difference of the first and second terms is to the
first, as the sum or difference of the third and fourth terms is to the
third.
4. The sum or difference of the first and second terms has the same
ratio to the second, as the sum or difference of the third and fourth has
to the fourth.
5. The sum of the first and second terms is to their difference, as
the sum of the third and fourth terms is to their difference.
6. If the corresponding terms of two proportions be multiplied
together, or divided, the products will form a proportion.
4. A proportion may be considered either as direct or inverse.
A proportion is direct when the same relation exists directly between
fhe first and second terms as between the third and fourth; as 3, 6,
5, 10, are in direct proportion; for the ratio of 3 to 6 and of 5 to 10 is.~; it follows that 3 bears the same direct proportion to 6, as 5 bears to
10; or 3: 6:: 5: 10.
A proportion is inverse when the first and second terms are directly
proportional to the inverse of the third and fourth terms; as 2, 6, 9, 3,
are in inverse proportion. For the direct ratio of 2 to 6 is }, the same
as the inverse ratio of 9 to 3; it follows that 2 bears to 6 the same
direct proportion as the inverse of 9 bears to the inverse of 3, or that
*.. ~.~1 1.
2: 6:: I.
The nature of direct proportion may be exemplified, if the names
of cause and effect may be given to those numbers in the proportion
which depend directly on one another, whatever may be the causes or
effects, provided only that they can be expressed by numbers. As, for
instance, a sum of money may be considered the cause, and what is
gained by it the effect: as if ~100 produce a gain of ~5; at the same
rate ~500 will produce a gain of ~25. Here the first cause bears the
same ratio to the second cause as the first effect to the second effect.
The nature of inverse proportion may be illustrated in the same
manner. In this case, however, the causes bear an inverse ratio to
the distances or the times involved in the question, in which the effects
are the same; as, for example: If 12 men can perform a certain work
in 5 days, 6 men can perform the same in 10 days. Here the units
of cause are inversely proportional to the times. Also in the case of
the straight lever, when kept in equilibrium on a fulcrum; the two
weights are inversely proportional to their distances from the
fulcrum.
In questions, however, in which the different causes produce different effects in given times, the time is an important element in the
calculation, as 1 man in 3 days can produce the same effect as 3 men