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

cut into
GEOMETRY 313
four others ADGE, I)F, FGGB, GE, so that DF, GE
are equal, the common vertex G will lie on the diagonal AB.
Heron produces AG to meet GF in H, and then proves that
AHB is a straight line.
Since DF, GE are equal, so are
the triangles D GF, EGG. A dding
the triangle GGF, we have the
triangles EGF, JDGF equal, and
DE, GF are parallel.
But (by I. 34, 29, 26) the triangles
AKE, GKD are congruent,
so that EK=KD
; and by lemma ( 1) it follows that CH=HF.
Now, in the triangles FHB, CHG, two sides (BF, FH and
GC, GH) and the included angles are equal ; therefore the
triangles are congruent, and the angles BHF, GHG are equal.
Add to each the angle GHF, and
Z BHF+ Z FHG = Z CHG + Z GHF = two right angles.
To prove his substantive proposition Heron draws AKL
Then
perpendicular to BG, and joins EG meeting AK in M.
we have only to prove that BMG is a straight line.
Complete the parallelogram FAHG, and draw the diagonals
OA, FH meeting in F. Through M draw PQ, SR parallel
respectively to BA, AG.
314 HERON OF ALEXANDRIA
Now the triangles FAH, BAG are equal in all respects
IHFA = I ABC
= Z CAR (since AK is at right angles to BG).
therefore
But, the diagonals of the rectangle FH cutting one another
in Y, we have FY = YA and L.HFA = LOAF;
therefore LOAF — AGAK, and OA is in a straight line
. with AKL.
Therefore, OM being the diagonal of SQ, SA — AQ. and, if
we add AM to each, FM = MH. .
Also, since EG is the diagonal of FN, FM — MN.
Therefore the parallelograms MH, MN are equal ; and
hence, by the preceding lemma, BMG is a straight line.
(ft)
The Definitions.
Q.E.D.
The elaborate collection of Definitions is dedicated to one
Dionysius in a preface to the following effect
'In setting out for you a sketch, in
the shortest possible
form, of the technical terms premised in the elements of
geometry, I shall take as my point of departure, and shall
base my whole arrangement upon, the teaching of Euclid, the
author of the elements of theoretical geometry ; for by this
means I think that I shall give you a good general understanding
not only of Euclid's doctrine but of many other
works in the domain of geometiy. I shall begin then with
the 'point!
He then proceeds to the definitions of the point, the line,
the different sorts of lines, straight, circular, ' curved ' and
'
spiral-shaped ' (the Archimedean spiral and the cylindrical
helix), Defs. 1-7 ; surfaces, plane and not plane, solid body,
Defs. 8-11; angles and their different kinds, plane, solid,
rectilinear and not rectilinear, right, acute and obtuse angles,
Defs. 12-22; figure, boundaries of figure, varieties of figure,
plane, solid, composite (of homogeneous or non-homogeneous
parts) and incomposite, Defs. 23-6. The incomposite plane
figure is the circle, and definitions follow of its parts, segments
(which are composite of non-homogeneous parts), the semicircle,
the a\jfis (less than a semicircle), and the segment
greater than a semicircle, angles in segments, the sector,