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.*

Then, since *CB, BG,* and *GH* equal one another, the triangles *ACB, ABG,* and *AGH* also equal one another.

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.

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.*

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.*

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.*

Therefore, *triangles and parallelograms which are under the same height are to one another as their bases.*

Q.E.D.

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

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

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

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.

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, in 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.