Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides.

Let *AC* and *CF* be equiangular parallelograms having the angle *BCD* equal to the angle *ECG.*

I say that the parallelogram *AC* has to the parallelogram *CF* the ratio compounded of the ratios of the sides.

Let them be placed so that *BC* is in a straight line with *CG.* Then *DC* is also in a straight line with *CE.*

Complete the parallelogram *DG.* Set out a straight line *K,* and make it so that *BC* is to *CG* as *K* is to *L,* and *DC* is to *CE* as *L* is to *M.*

Then the ratios of *K* to *L* and of *L* to *M* are the same as the ratios of the sides, namely of *BC* to *CG* and of *DC* to *CE.*

But the ratio of *K* to *M* is compounded of the ratio of *K* to *L* and of that of *L* to *M,* so that *K* has also to *M* the ratio compounded of the ratios of the sides.

Now since *BC* is to *CG* as the parallelogram *AC* is to the parallelogram *CH,* and *BC* is to *CG* as *K* is to *L,* therefore *K* is to *L* as *AC* is to *CH.*

Again, since *DC* is to *CE* as the parallelogram *CH* is to *CF,* and *DC* is to *CE* as *L* is to *M,* therefore *L* is to *M* as the parallelogram *CH* is to the parallelogram *CF.*

Since then it was proved that *K* is to *L* as the parallelogram *AC* is to the parallelogram *CH,* and *L* is to *M* as the parallelogram *CH* is to the parallelogram *CF,* therefore, *ex aequali K* is to *M* as *AC* is to the parallelogram *CF.*

But *K* has to *M* the ratio compounded of the ratios of the sides, therefore *AC* also has to *CF* the ratio compounded of the ratios of the sides.

Therefore, *equiangular parallelograms have to one another the ratio compounded of the ratios of their sides.*

Q.E.D.

This proposition is a generalization of the basic formula for the area of a rectangle, that is, the area of a rectangle is the product of its length and width. Such a formula depends on predetermined units of length and area so that the unit area is the area of a square whose sides have length equal to the unit length. Euclid and other Greek mathematicians did not use predetermined units of length or area, so they expressed this formula as a proportion. We would state that proportion as saying the ratio of the area of a given rectangle to the area of a given square is the product of the ratios of the lengths of the sides of the rectangle to the length of a side of the square. Of course, Euclid would say that without using the words area and length as follows: the ratio of the a given rectangle to a given square is the product of the ratios of the sides of the rectangle to a side of the square.

Note that his terminology for a product of ratios involves “compounding the ratios.” A natural generalization of the ratio of a rectangle to a square is the ratio of a rectangle to a rectangle. A broader generalization is the ratio of one parallelogram to another parallelogram having the same angles. That gives the generalization as stated in this proposition.

Early in this book was the proposition VI.1 generalizing I.35 which said that parallelograms with the same height are proportional to their bases. Finally, in this proposition we have the full statement about areas of rectangles and parallelograms.

In Book XI there are analogous statements for volumes of parallelepipeds. For instance, Proposition XI.33 states that similar parallelepipeds are to one another in the triplicate ratio of their corresponding sides. That statement for parallelepipeds is analogous to this one for parallelograms.