A history of Greek mathematics - Wilbourhall.org

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GEOMETRY 311
Of this class are the different cases of I. 35, 36, III. 7, 8
(where the chords to be compared are drawn on different sides
of the diameter instead of on the same side), III.
12 (which is
not Euclid's at all but Heron's own, adding the case of
external to that of internal contact in III. 11 \ VI. 19 (where
the triangle in which an additional line is drawn is taken to
be the smaller of the two), VII. 19 (where the particular case
is given of three numbers in continued proportion instead of
four proportionals).
(3) Alternative proofs.
It appears to be Heron who first introduced the easy but
uninstructive semi-algebraical method of proving the propositions
II. 2-10 which is now so popular. On this method the
propositions are proved ' without figures ' as consequences of
II. 1 corresponding to the algebraical formula
a (b + c + d + .
.
.)
= ab + ac -f ad + .
.
Heron explains that it is not possible to prove II. 1 without
drawing a number of lines (i. e. without actually drawing the
rectangles), but that the following propositions up to II. 10
can be proved by merely drawing one line. He distinguishes
two varieties of the method, one by dissolutio, the other by
compositio, by which he seems to mean splitting-up of rectangles
and squares and combination of them into others.
But in his proofs he sometimes combines the two varieties.
Alternative proofs are given (a) of some propositions of
Book III, namely III. 25 (placed after III. 30 and starting
from the arc instead of the chord), III. 10 (proved by means
of III. 9), III. 13 (a proof preceded by a lemma to the effect
that a straight line cannot meet a circle in more than two
points).
A class of alternative proof is (6) that which is intended to
meet a particular objection (eWracriy) which had been or might
be raised to Euclid's constructions. Thus in certain cases
Heron avoids producing a certain straight line, where Euclid
produces it, the object being to meet the objection of one who
should deny our right to assume that there is any space
available. Of this class are his proofs of I. 11, 20 and his
note on I. 16. Similarly in I. 48 he supposes the right-angled