AS PRACTISED BY THE ARABS. 115 



Lemma 7. If two rows of figures be written mutually parallel, but 

 so that the units of the first row be opposite the n tb place of the second, 

 the tens of the first opposite to place n -j- 1 of the second, the hundreds of 

 the first opposite to place n -f- 2 of the second, &c.and the rows be in this 

 situation added by the common Rule of Addition, this is equivalent to 

 the first row having added (the second row) X (10 n_1 ). 



Thus if there be written for the first row, 56,789. 



for the second row, 123. 



58,019. 



Then since 3, the units place of 123 is written opposite the tens or 

 2d place of 56,789, and since 2 — 1 — 1, so this is equivalent to 56,789 -f- 

 123 X 10 1 — 56, 789 -f 1,230. 



Similarly if there be written 56,789. 



123. 



69,089. 



Then since the 3 is here written opposite the hundreds, or 3d place 

 of 56,789, and since 3—1 — 2, so this is equivalent to 56,789 -j- 123 x 

 10 s — 56,789 -j- 12,300. 



Similarly if there be written 56,789. 



123. 



179,789 



This is equivalent to 56,789 + 123,000- 



G 1 



