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. PROPORTION. With the slide in the ordinary position and with the indices of the C and D scales in exact agreement, the _ratio_ of the corresponding divisions of these scales is 1. If the slide is moved so that 1 on C agrees with 2 on D, we know that under any number _n_ on C is _n_ × 2 on D, so that if we read numerators on C and denominators on D we have C 1 1·5 2 3 4 ───────────────────────────────────────── D1 2 3 4 6 8. In other words, the numbers on D bear to the coinciding numbers on C a ratio of 2 to 1. Obviously the same condition will obtain no matter in what position the slide may be placed. The rule for proportion, which is apparent from the foregoing, may be expressed as follows:— RULE FOR PROPORTION.—_Set the first term of a proportion on the C scale to the second term on the D scale, and opposite the third term on the C scale read the fourth term on the D scale._ EX.—Find the 4th term in the proportion of 20 ∶ 27 ∷ 70 ∶ _x_. Set 20 on C to 27 on D, and opposite 70 on C read 94·5 on D. Thus C 20 70 ───────────────── D 27 94·5.