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Book 1 of The Elements begins with numerous definitions followed by the famous five postulates. Then, before Euclid starts to prove theorems, he gives a list of common notions. The first few definitions are:
Def. 1.1. A point is that which has no part.The postulates are ones of construction such as:
Def. 1.2. A line is a breadthless length.
Def. 1.3. The extremities of lines are points.
Def. 1.4. A straight line lies equally with respect to the points on itself.
One can draw a straight line from any point to any point.The common notions are axioms such as:
Things equal to the same thing are also equal to one another.We should note certain things.
- Euclid seems to define a point twice (definitions 1 and 3) and a line twice (definitions 2 and 4). This is rather strange.
- Euclid never makes use of the definitions and never refers to them in the rest of the text.
- Some concepts are never defined. For example there is no notion of ordering the points on a line, so the idea that one point is between two others is never defined, but of course it is used.
- As we noted in The real numbers: Pythagoras to Stevin, Book V of The Elements considers magnitudes and the theory of proportion of magnitudes. However Euclid leaves the concept of magnitude undefined and this appears to modern readers as though Euclid has failed to set up magnitudes with the rigour for which he is famed.
- When Euclid introduces magnitudes and numbers he gives some definitions but no postulates or common notions. For example one might expect Euclid to postulate a + b = b + a, (a + b) + c = a + (b + c), etc., but he does not.
- When Euclid introduces numbers in Book VII he does make a definition rather similar to the basic ones at the beginning of Book I:
A unit is that by virtue of which each of the things that exist are called one.
Some historians have suggested that the difference between the way that basic definitions occur at the beginning of Book I and of Book V is not because Euclid was less rigorous in Book V, rather they suggest that Euclid always left his basic concepts undefined and the definitions at the beginning of Book I are later additions. What is the evidence for this?
The first comment would be that this would explain why Euclid never refers to the basic definitions. If they were not in the text that Euclid wrote then of course he couldn't refer to them. The next point to note is that they are very similar to the work which is ascribed to Heron called Definitions of terms in geometry. This contains 133 definitions of geometrical terms beginning with points, lines etc. which are very close to those given by Euclid. In  Knorr argues convincingly that this work is in fact due to Diophantus. The point here is the following. Is Definitions of terms in geometry based on Euclid's Elements or have the basic definitions from this work been inserted into later versions of The Elements?
We have to consider what Sextus Empiricus says about definitions. First note that Sextus wrote about 200 AD and it was believed until comparatively recently that Heron lived later than this. Were this the case, then of course Sextus could not have referred to anything written by Heron. However more recently Heron has been dated to the first century AD and this tells us that Sextus wrote after Heron. The other part of the puzzle we have to consider here is the earliest version of Euclid's Elements to be found. When Vesuvius erupted in 79 AD, Herculaneum together with Pompeii and Stabiae, was destroyed. Herculaneum was buried by a compact mass of material about 16 m deep which preserved the city until excavations began in the 18th century. Special conditions of humidity of the ground conserved wood, cloth, food, and in particular papyri which give us important information. One papyrus found there contains fragments of The Elements and was clearly written before 79 AD. Since Philodemus, a student of Zeno of Sidon, took his library of papyri there some time soon after 75 BC the version of The Elements is likely to be of around that date.
Let us go back to Sextus who writes about "mathematicians describing geometrical entities" and it is interesting that the word "describing" is not used in The Elements but is used by Heron in Definitions of terms in geometry. Again the descriptions he gives are closer to the exact words appearing in Heron than those of Euclid. When Sextus give "the definition of a circle" he uses the word "definition" which is that of Euclid. Sextus quotes the precise definition of a circle which appears in the Herculaneum fragment. This does not include a definition of "circumference" although Euclid does use the notion of circumference of a circle. The later versions of The Elements which have come down to us include a definition of "circumference" within the definition of a circle.
None of the above proves whether the basic definitions of geometric objects have been added to The Elements later. They do show fairly convincingly that the definition of a circle has been extended to include the definition of circumference in later editions of the book. The hypothesis is that Sextus has The Elements and Definitions of terms in geometry in front of him when he is writing and he uses the word "describe" when he refers to Heron and "define" when he refers to Euclid. Even if this is correct it still doesn't prove that the version of The Elements sitting in front of Sextus does not contain basic definitions of geometric objects but it does make such a possibility at least worth debating. What do you think?
One last point to think about. We quoted above:
Def. 1.4. A straight line lies equally with respect to the points on itself.What does this mean? It does seem a strange description for Euclid to give, for it appears to be meaningless. Compare it with the definition of a straight line in Definitions of terms in geometry:
A straight line is a line that equally with respect to all points on itself lies straight and maximally taut between its extremities.Again we ask the reader: do you think that the definition appearing in The Elements is a corruption of Heron's definition and so was added later, or do you think that Euclid gave a rather poor definition which was improved by Heron? Why do neither use the definition of a straight line as the shortest distance between two points?
Article by: J J O'Connor and E F Robertson