A history of Greek mathematics - Wilbourhall.org

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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.
Q.E.D.
(ft)
The Definitions.
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,