If the slide be inverted in the rule but with the same face uppermost,
so that the Ɔ scale lies adjacent to the A scale, and the right and left
indices of the slide and rule are placed in coincidence, we find the
product of any number on D by the coincident number on Ɔ (readily
referred to each other by the cursor) is always 10. Hence, by reading
the numbers on Ɔ as decimals, we have over any unit number on D, its
_reciprocal_ on Ɔ. Thus 2 on D is found opposite 0·5 on Ɔ; 3 on D
opposite to 0·333; while opposite 8 on Ɔ is 0·125 on D, etc. The reason
of this is that the sum of the lengths of the slide and rule
corresponding to the factors, is always equal to the length
corresponding to the product—in this case, 10.
It will be seen that if we attempt to apply the ordinary rule for
multiplication, with the slide inverted, we shall actually be
multiplying the one factor taken on D by the _reciprocal_ of the other
taken on Ɔ. But multiplying by the _reciprocal of a number_ is
equivalent to _dividing_ by that number, and _dividing_ a factor by the
_reciprocal_ of a number is equivalent to _multiplying_ by that number.
It follows that with the slide inverted the operations of multiplication
and division are reversed, as are also the rules for the number of
digits in the product and the position of the decimal point. Hence, in
multiplying with the slide inverted, we place (by the aid of the cursor)
one factor on Ɔ opposite the other factor on D, and read the result on D
under either index of Ɔ. It follows that with the slide thus set, any
pair of coinciding factors on Ɔ and D will give the same constant
product found on D under the index of Ɔ. One useful application of this
fact is found in selecting the scantlings of rectangular sections of
given areas or in deciding upon the dimensions of rectangular sheets,
plates, cisterns, etc. Thus by placing the index of Ɔ to 72 on D, it is
readily seen that a plate having an area of 72 sq. ft. may have sides 8
by 9 ft., 6 by 12, 5 by 14·4, 4 by 18, 3 by 24, 2 by 36, with
innumerable intermediate values. Many other useful applications of a
similar character will suggest themselves.