# How is magnitude defined in Euclid's elements?

I thought to avoid working with irrational numbers ancient Greeks introduced the concept of magnitudes. This allows mathematicians to do a rigorous mathematics based on geometry for 2000 years. However, what I do not understand is that how could they rigorously define things like addition and multiplication for magnitudes without relying on numbers. By rigorously I mean to only use the concept of magnitudes and nothing else.

For example for magnitude x, why was it acceptable to define x+x=2x? shouldn't 2 in this case be defined in terms of magnitudes?

• What do you mean by "define $x+x=2x$"? Certainly you will find no statement written in this notation in Euclid. And what is this equation meant to be a definition of, exactly? Apr 3, 2017 at 4:24
• @EricWofsey, Thanks! Yes you are absolutely right. This is meant to be the definition of adding a magnitude by itself. I am trying to make sense of the definitions given here (aleph0.clarku.edu/~djoyce/java/elements/bookV/defV5.html) but I do not understand how the concept of numbers is freely used in these definitions.
– abk
Apr 3, 2017 at 4:40
• What on earth makes you think that the ancient Greeks avoided working with irrational numbers? Have you considered actually reading sources like Euclid's Elements rather than asking ill-informed questions on MSE about Greek mathematics? If you did, you would find that Euclid devotes all of Book X to incommensurate magnitudes: i.e., magnitudes whose ratio is irrational. Apr 3, 2017 at 20:07
• @RobArthan, there's no need to be rude. The whole point of this site is for people to ask questions. Apr 3, 2017 at 21:27
• @mweiss: to whom have I been rude? The question begins with an incorrect statement and my comment to abk is correcting that statement. My comment to abk is a little sarcastic, but it is not rude. Apr 3, 2017 at 21:36

Magnitude is not really a defined term in the first few books of Euclid, despite the fact that the concept is used informally beginning with the very first proposition of Book I; it is used to capture the informal notion of (variously) length, area, or volume. If $AB$ is a line segment, for example, then an expression like "twice $AB$" simply means "a line segment formed by adjoining two segments, each separately congruent to $AB$". If $ABCD$ is a rectangle then "twice $ABCD$" means "a rectangle formed by adjoining two rectangles, each separately congruent to $ABCD$". When Euclid says something like "rectangle $ABCD$ is equal to twice triangle $PQR$" he means that two identical copies of triangle $PQR$ have area equal to rectangle $ABCD$. Proving such claims does not require that magnitudes be interpreted as numerical quantities; it does, however, require some axioms about magnitudes (things like a whole being equal to the sum of its parts, for instance).

• thanks that makes sense but shouldn't expressions like "twice AB" be well defined. These statements do not seem to be coming from Euclid's axioms of geometry.
– abk
Apr 3, 2017 at 5:13
• What do you mean by "well defined"? Euclid's Elements are not rigorous by modern standards. Apr 3, 2017 at 5:19
• Right got it! Thanks. I thought he needed to have everything built up on his axioms, but probably they didn't have or need that level of rigor back then.
– abk
Apr 3, 2017 at 5:25
• He thought he was being rigorous. But whippersnappers since have been poking holes and pointing the rigor failures ever since. Apr 3, 2017 at 7:00
• @mweiss: your answer is on the border-line between plain wrong and just badly mis-informed. Have you read the first few paragraphs of Book VI of Euclid's Elements? See my answer below for more information about why you really ought to read some more Euclid. Apr 3, 2017 at 21:00

I am treating this as a reference request. The first reference you want to consult is Euclid's Elements. If you can get Heath's translation, then it contains some excellent commentary, but there are good versions online too, e.g., a nice translation by R. Fitzpatrick.

Particularly important to dispel your misunderstandings are:

Book V, which expounds Eudoxus's theory of proportion. This begins with what is effectively a relative definition of the concept of magnitude: a magnitude is something like a length or an area that you can add, subtract and compare geometrically. Particularly important is definition V.5, which tells you how to compare two ratios between arbitrary magnitudes. Dedekind's approach to defining the real numbers is a direct descendant of this definition.

Book VII, which is about elementary number theory. It begins with a definition of number, i.e., positive integer. I don't believe that numbers where intended to be thought of as magnitudes, but the theory of proportions also applies to numbers and so numbers and magnitudes can be compared. Euclid has no notion of algebraic expression, but he can say of magnitudes $x$ and $y$ that "$x$ is to $y$ as $1$ is to $2$", meaning $y = 2x$.

Book X, which is about incommensurable magnitudes, i.e., magnitudes $x$ and $y$ such that $x/y$ is irrational. It begins with a definition of Euclid's algorithm for arbitrary magnitudes. Theorem X.2 analyses the termination condition for this algorithm and concludes that it terminates on inputs $x$ and $y$ iff $x/y$ is rational.

You should then bear in mind that Euclid's Elements is not an enyclopaedic compendium of all ancient Greek mathematics. It was probably intended as something more like a graduate textbook. There are lots of good sources on the history of mathematics that you should look into if you are interested in the subject. The MacTutor page on Euclid could be a good place to start in this case.

• While this is all correct, it is also correct that Euclid begins using the informal notions of "whole number multiple of..." (and the related notion of "unit fraction of..." in Book I. For example in I.9 we have "to cut a rectilinear angle in half", and in I.10 we have "to cut a straight line [segment] in half." At this point in the text the notion of "half" as a numerical quantity is not formally defined; from context, it clearly means "two parts that are equal", and yet "two" is not defined either. (contd) Apr 3, 2017 at 21:19
• Likewise Prop I.13 refers to "angles that are equal to two right angles". Prop I.47 (the Pythagorean Theorem) speaks of a square being equal to the sum of two other squares. All of this precedes the formal definitions of proportion and number found in the later books. (contd) Apr 3, 2017 at 21:21
• This kind of arithmetic of magnitudes seemed to me to be precisely what the OP was asking about, and what I was referring to in my response. Apr 3, 2017 at 21:22
• @mweiss: your answer begins "Magnitude is not really a defined term in Euclid". That statement is hard to justify given the depth of the analysis of the concept of magnitude in the later books. It is grossly unfair to the Greek mathematicians to draw snap conclusions based on a reading of the first few books of Euclid with 2,000 and more years of hindsight. Apr 3, 2017 at 21:32
• @RobArthan Thanks for the references.
– abk
Apr 3, 2017 at 23:09