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), YII. 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 proposi 
tions 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 + 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 rect 
angles 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 (eWracm) 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 1. 11, 20 and his 
note on 1.16. Similarly in I. 48 he supposes the right-angled