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 hvo 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 supple 
mentary 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 ABC 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. (UK is drawn through A parallel to 
DE, and HDL, KEM through D, E parallel 
to A F meeting the base in L, M respec 
tively. Then the triangles ABF, AFG 
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. I. 43. If a parallelogram is