# Page:Popular Science Monthly Volume 16.djvu/220

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THE POPULAR SCIENCE MONTHLY.
 Process 1. Process 2.

 Process 3. The commentator considers this method as difficult, and not to be learned by dull scholars without instruction.

Example 2. Multiply 15 by 12.

15 ${\displaystyle =}$ 4 ${\displaystyle +}$ 5 ${\displaystyle +}$ 6.﻿12 ${\displaystyle =}$ 2 ${\displaystyle +}$ 4 ${\displaystyle +}$ 6.

Process.

﻿Example 3. Multiply 30 by 46.

 Process. This method was called "crosswise," from the manner in which the partial products to be added were obtained. It is not improbable that our present sign of multiplication was derived from the crossing of the lines in this process, as being somewhat indicative of the operation.

Here is a larger example worked by the cross-method:

Example 4.﻿Multiply 456 by 456.

 Process. To indicate the successive steps the linking-lines are numbered, so as to show the groups in which the products are to be taken for addition.

 Process. Let the products be found and properly grouped and 456 added mentally, and one will better appreciate how much 456 depend upon mere mechanism in our own mental operations.