A lot of what is going on with his method is using the "FOIL" concept, as it
applies to numbers instead of variables. For example, as in his illustration :

35^2 = 1225 . (I will explain "how come" ?)
Look at 35 as (30 + 5) , then we have
(30+5)*(30+5) = 1225 same as
30*30 + 30*5 + 30*5 + 5*5 = 1225. [@]

So, that first part of finding the first (tens) digit of the sq rt of 1225,
is figuring out the (largest number^2) < or = 12 . That is 3 in this case.
And is the first term in [@].

Then he subtracts that out. Left with
30*5 + 30*5 + 5*5 . [@2]

Then that part about dividing by two is explained, in essence, by the fact
that there are two of the same terms in [@2]

We are seeking a number that :
2 * (30 + "it") + "it"^2 is = or < expression [@2] .

In that case, "it" is 5 , and works out perfectly :)

So the sq rt of 1225 = (30 + 5) = 35 .

Gary Flom
Atlanta Ga. U.S.A.