308 Afr. Robertson’s ;z<?w Demonstration 
It is well known to mathematicians that the theorem has 
been repeatedly proved, either by induction, by the summa- 
tion of figurate numbers, by the doctrine of combinations, 
by assumed series, or by fluxions ; but that multiplication is a 
more direct way to the establishment of the theorem than 
any of these, cannot, I think, be doubted. Proceeding by 
multiplication, we have always an evident first principle in 
view, to which without the aid of any doctrine, foreign to the 
subject, we can appeal for the truth of our assertions, and 
the certainty and extent of our conclusions. 
3. If p , q, be any two quantities, the product arising from 
the multiplication of p by q is equal to the product arising 
from the multiplication of q by p* For magnitudes being to 
one another as their equimultiples, p xq : 1 xq : :p : 1, and 
q xp : 1 xp : : q : 1. But 1 xq — q, and 1 x />=/>, and there- 
fore, placing for ex sequali in a cross order, 
p x q : q : 1 
qxp :p : 1. 
Consequently,/* x q : 1 : : qxp : 1, and therefore pq—qp. 
Hence it follows that the product arising from the multi- 
plication of any number of quantities into one another, conti- 
nues the same in value, in every variation which may be 
made in the arrangement of the quantities which compose it. 
Thus p, q> r, s, being any quantities, pqrs=pqr x s = spqr = 
spq x r~rspq=rsp x q—qrsp—qr x s xp—qr xp x s=qrps, See. 
And if x-\-a~p, jr-f- b—q, x-\-c~r, x-\-d=s, — &c. 
then x-j -a x x~\-b xx-\-c xx>| -d x x-\-e—pqrst^=x-\-axx-\-bx 
• When I speak of the multiplication of quantities into one another, I mean the 
multiplication of the numbers into one another which measure those quantities. 
