3°t 
M 
of the Binomial Theorem. 
For let AM, AN be two in- 
TsT 
definite straight lines at right 
angles to one another, and 
in AMsetofFAB, BC, CD, 
DE, EF, &c. equal to one 
another, and in number equal 
to the number of units in the 
quantity p * ; and in AN set 
ofFAG, GH, HI, IK, &c. 
each equal to AB, and let 
the number of these parts be equal to the number of units in 
the quantity q. Complete the rectangle K F, and draw 
straight lines parallel to A K, through the points B, C, D, E, 
and let them meet the opposite side K L of the parallelogram. 
Through the points G, H, I, draw straight lines parallel to 
A F, and let them meet F L, the opposite side of the parallelo- 
gram. Then will the whole rectangle K F be divided into 
squares, each equal to G B. Now when p is multiplied into q , 
the number of units in the product is equal to the number of 
units in p repeated as often as there are units in q. But the 
number of squares in the rectangle K F is equal to the number 
of parts in A F repeated as often as there are parts in A K ; 
and therefore, by the above construction, the number of squares 
in the rectangle K F is equal to the number of units in p re- 
peated as often as there are units in q. Hence the number of 
squares in the rectangle K F is equal to the number of units in 
p x q. In the same manner it may be proved that the number 
* When I speak of the number of units in a quantity, I mean the number of units 
in the number measuring that quantity. 
mdccxcv. R r 
