A history of Greek mathematics - Wilbourhall.org

.
,
^ the square) is .
'
26J-| instead of 27],
have, as the first approximation to */A.
for a second approximation we take
and so on. 2
1
Metrica, i. 8, pp. 18. 22-20. 5.
H
2\i 2^ and 2 T 9^1n
2
«. a 01 my-s
2
the first as 2+ n—x, *
^—7;, the second as 2| —
l
rtl
2.2'
J
2 t\
i(a + /3): v / (a3)-yW): a
2^-
"o> \/^4 >?? ,
then, i(a +b )>^/A> ** or -\ °
324 HERON OF ALEXANDRIA
'Since', says Heron, 1 '
720 has not its side rational, we can
obtain its side within a very small difference as follows. Since
the next succeeding square number is 729, which has 27 for
its side, divide 720 by 27. This gives 26|. Add 27 to this,
making 53§, and take half of this or 26 J J.
The side of 720
will therefore be very nearly 26| §. In fact, if we multiply
26J§ by itself , the product is 720^, so that the difference (in
If we desire to make the difference still smaller than 3^-, we
shall take 720^ instead of 729 [or rather we should take
and by proceeding in the same way we
shall find that the resulting difference is much less than £$'
In other words, if we have a non-square number A, and a 2
is the nearest square number to it, so that A = a 2 + b,
then we
«!=!(«+ -); (D
2
The method indicated by Heron was known to Barlaam and Nicolas
Rhabdas in th'e fourteenth century. The equivalent of it was used by
Luca Paciuolo (fifteenth -sixteenth century), and it was known to the other
Italian algebraists of the sixteenth century. Thus Luca Paciuolo gave
as successive approximations to */6. He obtained
^, ,
and the third as
tt-irir- The above rule - ives l(2+i) = 2|, i(|+-^)-2A,
" • •
20
1 fiilj. l_liP\ 9JL«JL
2 \20^ 48/ — ^1!>60-
The formula of Heron was again put forward, in modern times, by
Buzengeiger as a means of accounting for the Archimedean approximation
to \/3, apparently without knowing its previous history. Bertrand
also stated it in a treatise on arithmetic (1853-). The method, too, by
which Oppermann and Alexeieff sought to account for Archimedes's
approximations is in reality the same. The latter method depends on
the formula
Alexeieff separated A into two factors a , b ,
and pointed out that if. say.