Page:Popular Science Monthly Volume 16.djvu/220

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206
THE POPULAR SCIENCE MONTHLY.
Process 1. Process 2.
 
PSM V16 D220 Old Mathematical calculation 1.jpg
 
Process 3.
PSM V16 D220 Old Mathematical calculation 2.jpg The commentator considers this method as difficult, and not to be learned by dull scholars without instruction.
 

Example 2. Multiply 15 by 12.

15 = 4 + 5 + 6. 12 = 2 + 4 + 6.

Process.

PSM V16 D220 Old Mathematical calculation 3.jpg
 

 Example 3. Multiply 30 by 46.

 
Process.
PSM V16 D220 Old Mathematical calculation 4.jpg 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.
PSM V16 D220 Old Mathematical calculation 5.jpg 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.
PSM V16 D220 Old Mathematical calculation 6.jpg 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.