Sec. 1.3 Euclid’sThe Elements 7
- If equals be added to equals the wholes are equals.
- If equals be taken from equals the remainders are equal.
- Magnitudes which coincide with one another, [that is, which exactly fill the
same space] are equal to one another. - The whole is greater than its part.
1.3.4
The Elementsattempted to be a logically complete deductive system. There were ear-
lier Elements but these have not survived, presumably because they were outclassed
by Euclid’s.
The Elementsare charming to read, proceed very carefully by moderate steps and
within their own terms impart a great sense of conviction. The first proposition is to
describe an equilateral triangle on[A,B]. With centreAa circle is described passing
throughB, and with centreBa second circle is described passing throughA.IfC
is a point at which these two circles cut one another, then we take the triangle with
verticesA,B,C.
It is ironical that, withThe Elementsbeing so admired for their logical proceeding,
there should be a gap in the very first proposition. The postulates and common notions
did not make any provisions which would ensure that the two circles in the proof
would have a point in common. This may seem a curious choice as a first proposition,
dealing with a very special figure. But in fact it is used immediately in Proposition 2,
from a given point to lay off a given distance.
The main logical lack inThe Elementswas that not enough assumed properties
were listed, and this fact was concealed through the use of diagrams.
1.3.5 Congruence
Two types of procedure inThe Elementscall for special comment. The first is the
method ofsuperpositionby which one figure was envisaged as being moved and
placed exactly on a second figure. The second is the process ofconstructionby which
figures were not dealt with until it was shown by construction that there actually was
such a type of figure. In the physical constructions, what were allowed to be used
were straight edges and compasses.
The notion ofsuperpositionis basic to Euclid’s treatment of figures. It is visu-
alised that one figure is moved physically and placed on another, fitting perfectly. We
use the termcongruentfigures when this happens. Common Notion 4 is to be used in
this connection. This physical idea is clearly extraneous to the logical set-up of prim-
itive and defined terms, assumed and proved properties, and is not a formal part of
modern treatments of geometry. However it can be used in motivation, and properties
motivated by it can be assumed in axioms.