1. Things which equal the same thing also equal one another.
2. If equals are added to equals, then the wholes are equal.
3. If equals are subtracted from equals, then the remainders are equal.
4. Things which coincide with one another equal one another.
5. The whole is greater than the part.
C.N.4 requires interpretation. On the face of it, it seems to say that if two things are identical (that is, they are the same one), then they are equal, in other words, anything equals itself. But the way it traditionally is interpreted is as a justification of a principle of superposition, which is used, for instance, in proposition I.4. Using this principle, if one thing can be moved to coincide with another, then they are equal. See the notes on I.4 for more discussion on this point.
C.N.5, the whole is greater than the part, could be interpreted as a definition of “greater than.” To say one magnitude B is a part of another A could be taken as saying that A is the sum of B and C for some third magnitude C, the remainder. Symbolically, A > B means that there is some C such that A = B + C. At any rate, Euclid frequently treats these two conditions as being equivalent.
There are a number of properties of magnitudes used in Book I besides the listed Common Notions. Here are a few of them and locations where they are used.
1. | If not x = y, then x > y or x < y. | I.6 |
2. | Not both x < y and x = y. | I.6 |
3. | If not not x = y, then x = y. | I.6 |
4. | If x < y and y = z, then x < z. | I.7 |
5. | If x < y and y < z, then x < z. | I.7 |
6. | If x = y and y < z, then x < z. | I.16 |
7. | If x < y, then x + z < y + z. | I.17 |
8. | If not x > y, then x = y or x < y. | I.19 |
9. | If not x < y and not x = y, then x > y. | I.19 |
10. | If 2x = 2y, then x = y. | I.37 |
11. | If x = y, then 2x = 2y. | I.42 |
Number 3 is an instance of the logical principle of double negation, rather than a common notion. Number 11 is a special case of C.N.2 since doubling is a special case of addition, that is, 2x is just x + x. Some of the others are logical variants of each other, for instance, numbers 1, 8, and 9 are all equivalent to the statement that at least one of the three cases x < y, x = y, or x > y holds. Statement 2 says that two of those cases cannot simultaneously hold. The statement that
First, assume there is a binary relation on a set of magnitudes of the same kind called equality, denoted as usual with an equal sign as in x = y. (This equality is not identity as we want different magnitudes, such as two different triangles, to be equal. Alternatively, we could identify equal magnitudes so that equality is identity.) Assume that equality is what is called an equivalence relation, that is, it satisfies three axioms:
Symmetry: If x = y, then y = x.
Transitivity: If x = y and y = z, then x = z.
Associativity: For each x, y, and z, (x + y) + z = x + (y + z).
Commutativity: For each x and y, x + y = y + x.
We can now define order in terms of addition. Define a binary relation less than by taking x < y to mean that there is some z such that x + z = y. And let greater than just have the opposite order, that is, x > y means y < x. A number of properties of order can be easily proved.
If x = y and y < z, then x < z.
If x < y and y < z, then x < z.
If x < y, then x + z < y + z, and z + x < z + y.
Next, assume an axiom for cancellation:
If x = y and w = z, then x – w = y – z.
(x + y) – y = x.
(x – y) + y = x.
(x – y) – (w – z) = (x – w) – (y – z).
If x < y, then z – x > z – y.
If x < y and w = z, then x – w < y – z.
If x = y and w < z, then x – w > y – z.
If x < y and w > z, then x – w < y – z.
If x does not equal y, then one of them is greater. Let x be greater. Then x + x > y + y, that is, twice x is greater than twice y. But twice x was assumed to equal twice y, the less equals the greater, which is absurd. Therefore x and y are not unequal. Therefore they are equal. Q.E.D.