8 Preliminaries Ch. 1
1.3.6 Quantities or magnitudes
The Elementsspoke of one segment (then called a line) being equal to or greater than
another, one region being equal to or greater than another, and one angle being equal
to or greater than another, and this indicates that they associated a magnitude with
each segment (which we call its length), a magnitude with each region (which we
call its area), and a magnitude with each angle (which we call its measure). They did
not define these magnitudes or give a way of calculating them, but they gave sufficient
properties for them to be handled as the theory was developed. In the case of each of
them the five common notions were supposed to apply.
Thus inThe Elements,thequantitiesfor which the Common Notions are intended
aredistanceor equivalentlylengthof a segment,measureof an angle andareaof
a region. These are not taken to be known, either by assumption or definition, but
congruent segments are taken to have equal lengths, congruent angles are taken to
have equal measures, and congruent triangles are taken to have equal areas. Thus
equality of lengths of segments, equality of measures of angles, and equality of areas
of triangles are what is started with. Treatment of area is more complicated than the
other two, and triangles equal in area are not confined to congruent triangles. Addition
and subtraction of lengths are to be handled using Common Notions 1, 2, 3 and 4;
so are addition and subtraction of angle measures; so are addition and subtraction of
area.
Taking a unit length, there is a long build-up to the length of any segment. They
reached the stage where if some segment were to be chosen to have length 1 the length
of any segment which they encountered could be found, but this was not actually done
inThe Elements.
Taking a right-angle as a basic unit, there was a long build-up to handling any
angle. They reached the stage where if a right-angle was taken to have measure 90◦,
the measure of any angle which they encountered could be found, but this was not
actually done.
There is a long build up to the area of figures generally. The regions which they
considered were those which could be built up from triangles, and they reached the
stage where if some included region were chosen to have area 1 the area of any
included region could be found. This is not actually completed inThe Elementsbut
the materials are there to do it with.
All this shows thatThe Elementsalthough very painstaking, thorough and exact
were also rather abstract. It should be remembered that the Greeks did not have alge-
bra as we have, and used geometry to do a lot of what we do by algebra. In particular,
considering the area of a rectangle was their way of handling multiplication of quan-
tities. Traditionally in arithmetic the area of a rectangle was dealt with as the product
of the length and the breadth, that is by multiplication of two numbers. However,
reconciling the geometrical treatment of area with the arithmetical does not seem to
have been handled very explicitly in books, not even when from 1600 A.D. onwards
real numbers were being detached from the ‘quantities’ of Euclid.