Euclid's Elements, Book VI, Proposition 1

Proposition 1

Triangles and parallelograms which are under the same height are to one another as their bases.
Let ACB and ACD be triangles, and let CE and CF be parallelograms under the same height. I say that the base CB is to the base CD as the triangle ACB is to the triangle ACD, and as the parallelogram CE is to the parallelogram CF.
Produce BD in both directions to the points H and L. Make any number of straight lines BG and GH equal to the base CB, and any number of straight lines DK and KL equal to the base CD. Join AG, AH, AK, and AL.	I.3

Then, since CB, BG, and GH equal one another, the triangles ACB, ABG, and AGH also equal one another.	I.38
Therefore, whatever multiple the base CH is of the base CB, the triangle ACH is also that multiple of the triangle ACB.
For the same reason, whatever multiple the base CL is of the base CD, the triangle ACL is also that multiple of the triangle ACD. And, if the base CH equals the base CL, then the triangle ACH also equals the triangle ACL; if the base CH is in excess of the base CL, the triangle ACH is also in excess of the triangle ACL; and, if less, less.	I.38
Thus, there being four magnitudes, namely two bases CB and CD, and two triangles ACB and ACD, equimultiples have been taken of the base CB and the triangle ACB, namely the base CH and the triangle ACH, and other, arbitrary, equimultiples of the base CD and the triangle ADC, namely the base CL and the triangle ACL, and it has been proved that, if the base CH is in excess of the base CL, the triangle ACH is also in excess of the triangle ACL; if equal, equal; and, if less, less. Therefore the base CB is to the base CD as the triangle ACB is to the triangle ACD.	V.Def.5
Next, since the parallelogram CE is double the triangle ACB, and the parallelogram FC is double the triangle ACD, and parts have the same ratio as their equimultiples, therefore the triangle ACB is to the triangle ACD as the parallelogram CE is to the parallelogram FC.	I.41 V.15
Since, then, it was proved that the base CB is to CD as the triangle ACB is to the triangle ACD, and the triangle ACB is to the triangle ACD as the parallelogram CE is to the parallelogram CF, therefore also the base CB is to the base CD as the parallelogram CE is to the parallelogram FC.	V.11
Therefore, triangles and parallelograms which are under the same height are to one another as their bases.
Q.E.D.

In a more proper setting out of the proposition, the triangles under the same height would not have a common side, and the parallelograms would not have a common base and side with the triangles. Since triangles on equal bases and in the same parallels are equal (I.36), and parallelograms on equal bases and in the same parallels are equal (I.35), and equals may be substituted in proportions (V.7), Euclid's simplified setting out is sufficient. Nonetheless, a proper setting out does not require a more complicated proof.

The goal of the proof is to show that three ratios, namely the ratio of the lines CB to CD, the ratio of the triangles ACB to ACD, and the ratio of the parallelograms CE to CF, are all the same ratio. That is

CB:CD = ACB:ACD = CE:CF.

The first stage of the proof shows that CB:CD = ACB:ACD. By the definition of proportion, V.Def.5, that means for any number m and any number n that

m BC >=< n CD when m ABC >=< n ACD.

Note that Euclid takes both m and n to be 3 in his proof. Now m BC equals the line CH, n CD equals the line CL, m ABC equals the triangle ACH, and n ACD equals the triangle ACL. So what has to be shown is that

CH >=< CL when ACH >=< ACL.

But that follows from proposition I.38. So the first stage of the proof is complete.

The second stage is easier. Since the parallelograms are twice the triangles, they also have the same ratio.

Other propositions that state fundamental proportions use the same outline for their proofs. Proposition VI.33: arcs of circles are proportional to angles on which they stand; XI.25: parallelepipeds are proportional to their bases; and XII.13: cylinders are proportional to their axes.

On the method of modern analysis

Heath remarked that "some American and German text-books adopt the less rigorous method of appealing to the theory of limits" for the foundation for the theory of proportion used here in geometry. Heath preferred Eudoxus' theory of proportion in Euclid's Book V as a foundation.

It is remarkable how much mathematics has changed over the last century. In the beginning of the 20th century Heath could still gloat over the superiority of synthetic geometry, although he may have been one of the last to do so. Now, near the beginning of the 21st century, synthetic geometry has receded into near oblivion while analysis, based on various concepts of limits, is preeminent.

It took some time to find a foundation for mathematical analysis as solid, or more solid, than geometry. In the 17th century, the time of the creation of differential and integral calculus, geometry was seen as the most dependable justification for calculus. In the first half of the 19th century, the concept of limit was clarified and limits became the foundation of mathematical analysis. Heath's complaint would have been valid then since the theory of real numbers was still without any foundation except a geometric one, which, ultimately was based on Eudoxus' theory of proportion in Euclid's Book V. In the later 19th century Weierstrass, Cantor, and Dedekind succeeded in founding the theory of real numbers on that of natural numbers and a bit of set theory, so that by the beginning of the 20th century, there was a modern foundation for mathematical analysis. All the same, this new foundation could still be called Eudoxus' since the modern definition of real number is the same as his, but in a modern guise.

Use of this proposition

This is one of the most used propositions in the Elements. It is used frequently in Book VI starting with the next proposition, dozens of times in Book X, and and a few times in Books XI and XIII.

Book VI Introduction - Definition VI.Def.4 - Proposition VI.2.