A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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
.
312 HERON OF ALEXANDRIA
triangle which is constructed to be constructed on the same
side of the common side as the given triangle is.
A third class (c) is that which avoids reductio ad absurdum,
e.g. a direct proof of I. 19 (for which he requires and gives
a preliminary lemma) and of I. 25.
(4) Heron supplies certain converses of Euclid's propositions
e.g. of II. 12, 13 and VIII. 27.
(5) A few additions to, and extensions of, Euclid's propositions
are also found. Some are unimportant, e. g. the construction
of isosceles and scalene triangles in a note on I. 1 and the
construction of two tangents in III. 17. The most important
extension is that of III. 20 to the case where the angle at the
circumference is greater than a right angle, which gives an
easy way of proving the theorem of III. 22. Interesting also
are the notes on I. 37 (on I. 24 in Proclus), where Heron
proves that two triangles with two sides of the one equal
to two sides of the other and with the included angles supplementary
are equal in area, and compares the areas where the
sum of the included angles (one being supposed greater than
the other) is less or greater than two right angles, and on I. 47,
where there is a proof (depending on preliminary lemmas) of
the fact that, in the figure of Euclid's proposition (see next
page), the straight lines AL, BG, GE meet in a point. This
last proof is worth giving.
First come the lemmas.
(1) If in a triangle ABG a straight line DE be drawn
parallel to the base BG cutting the sides AB, AC or those
sides produced in D, E, and if F be the
middle point of BG, then the straight line
AF (produced if necessary) will also bisect
DE.
(HK is drawn through A parallel to
DE, and HDL, REM through D, E parallel
to AF meeting the base in L, M respectively.
Then the triangles ABF, AFC
between the same parallels are equal. So are the triangles
DBF, EFC. Therefore the differences, the triangles ADF,
AEF, are equal and so therefore are the parallelograms HF,
KF. Therefore LF = FM, or DG = GE.)
(2) is the converse of Eucl. 1. 43. If a parallelogram is