206 



THE POPULAR SCIENCE MONTHLY. 



Process 1. 



13 5 

 12 12 12 



Process 2. 

 1 3 



1 2 



60 

 G 



1 6 







Process 3. 

 135 

 12 



10 

 11 

 5 

 1 



1G20 



The commentator considers 

 this method as difficult, and 

 not to be learned by dull schol- 

 ars without instruction. 



Example 2. Multiply 15 by 12. 



15 = 4 + 5 + 6. 12 = 2 + 4 + 6. 

 Process. 



Process. 



1610 



35 



X 



46 



Example 3. Multiply 30 by 46. 



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 multiplica- 

 tion was derived from the crossing of the lines in this pro- 

 cess, as being somewhat indicative of the operation. 



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

 Example 4. Multiply 456 by 456. 

 Process. 



ft 



To indicate the successive steps the link- 



ing-lines are numbered, so as to show the 



groups in which the products are to be taken 



for addition. 



2 7 



Process. 



456 added mentally, and one will better appreciate how much 



456 -^e depend upon mere mechanism in our own mental opera- 



207936 tions. 



.Q 3 6 



Let the products be found and properly grouped and 



