In Book 1 of Euclid’s Elements, the second of the Five Common Notions reads (Fitzpatrick 7):
| Greek | English |
|---|---|
| βʹ. Καὶ ἐὰν ἴσοις ἴσα προστεθῇ, τὰ ὅλα ἐστὶν ἴσα. | 2. And if equal things are added to equal things then the wholes are equal. |
This Common Notion is included in all extant manuscripts and modern editions of the Elements. It was considered a genuine Euclidean axiom by Proclus and Heron (Heath 223).
In the last article in this series we discussed at length what sorts of things Euclid is referring to in his Common Notions and in what ways they can be equal to one another. Some mathematicians restrict the term to magnitudes, such as the length of a line segment, the measure of an angle, the area of a plane figure, or the volume of a solid figure:
These common notions, sometimes called axioms, refer to magnitudes of one kind. The various kinds of magnitudes that occur in the Elements include lines, angles, plane figures, and solid figures. (Joyce)
Proclus takes a broader view, but in the following passage he probably means that geometers apply the Common Notions to magnitudes while other scientists apply them to other quantities:
These axioms are common, but each individual uses them with reference to his specific subject-matter and to the extent which his subject-matter demands. One man applies them to magnitudes, another to numbers, another to intervals of time. In this way, although the axioms are general, they lead to specific conclusions in each science. (Morrow 154)
But what does Euclid mean when he speaks of adding one thing to another?
The verb he uses is προστίθημι [prostithēmi], to add. This is the usual word employed by ancient Greek mathematicians when referring to the arithmetic operation of addition (Liddell & Scott 1322). But Euclid never formally defines how things like line segments or angles are to be added together. For example, in Proposition 1:13, the first in which he invokes Common Notion 2, he is talking about various angles that arise when one straight line stands upon another. The literal translation of what he states is laconic to the point of obscurity. Note how Thomas Heath and Richard Fitzpatrick’s translation are far from literal:
αἱ ἄρα ὑπὸ ΓΒΕ, ΕΒΔ τρισὶ ταῖς ὑπὸ ΓΒΑ, ΑΒΕ, ΕΒΔ ἴσαι εἰσίν.
Thus, the ones under ΓΒΕ and ΕΒΔ are equal to the three under ΓΒΑ, ΑΒΕ, ΕΒΔ. (Euclid)
therefore the angles DBA, ABC are equal to the three angles DBE, EBA, ABC. (Heath 275)
Thus, the (sum of the angles) CBE and EBD is equal to the (sum of the) three (angles) CBA, ABE, and EBD. (Fitzpatrick 18)
In Proposition 1:35, Euclid adds a triangle to a trapezium (trapezoid). As this is the first time he adds two figures, Thomas Heath appends a long comment, which begins:
Equality in a new sense.
It is important to observe that we are in this proposition introduced for the first time to a new conception of equality between figures. Hitherto we have had equality in the sense of congruence only, as applied to straight lines, angles, and even triangles (cf. 1:4). Now, without any explicit reference to any change in the meaning of the term, figures are inferred to be equal which are equal in area or in content but need not be of the same form. No definition of equality is anywhere given by Euclid; we are left to infer its meaning from the few axioms [ie Common Notions] about “equal things.” It will be observed that in the above proof the “equality” of two parallelograms on the same base and between the same parallels is inferred by the successive steps (1) of subtracting one and the same area (the triangle DGE) from two areas equal in the sense of congruence (the triangles AEB, DFC), and inferring that the remainders (the trapezia ABGD, EGCF) are “equal”; (2) of adding one and the same area (the triangle GBC) to each of the latter “equal” trapezia, and inferring the equality of the respective sums (the two given parallelograms). (Heath 327-328)
Heath notes that the French mathematician Adrien-Marie Legendre introduced the term equivalent [équivalent] to express Euclid’s broader sense of equality, restricting the term equal [égal] to congruence (Heath 328).
If the different magnitudes that arise in geometry can be measured by numbers, then Euclid’s adding of things becomes simple arithmetic. But Euclid and his contemporaries were well aware that many geometric magnitudes are incommensurable—that is to say, the “numbers” that measure them are irrational. In fact, Book 10 of the Elements is devoted to such magnitudes. How does one add irrational numbers?
This question is moot, however, as Euclid never attempts to measure magnitudes by rational numbers, even in cases where the magnitudes are commensurable. For example, the unit of measurement for angles is the right angle. Euclid never says that the sum of the three angles of a triangle is equal to 180° or 2π radians. What he actually says is that the three internal angles of a triangle are equal to two right-angles (Proposition 1:32).
In the Elements Euclid is very sparing in his use of this Common Notion. Three commentators on the Elements, Richard Fitzpatrick, Thomas Heath and David Joyce, only cite it explicitly in the following propositions:
Common Notion 2 is not cited explicitly by any of these commentators in the remaining twelve books of the Elements. Of course many later propositions build upon those of Book 1, so this Common Notion is often implied in the proofs. For example, Thomas Heath invokes it in his discussion of Proposition 6:14, though not in Euclid’s actual proof of this proposition (Heath 2:218).
And that’s a good place to stop.
