Purchase on iTunes

Sample PDF

Sample PDF

Few books in history have affected the development of mathematical, scientific, and philosophical thought more than Euclid's Elements.

The propositions in the first 4 books form the geometric core of the work. Each proposition, intended to prove a particular mathematical statement, is accompanied by a figure. The conclusion established by the proposition forms a premise used to prove the following propositions.

We have turned the 127 original black and white static figures into colorful, interactive apps that illustrate the propositions dynamically.

What translator Sir Thomas Heath calls "one of the noblest monuments of antiquity" is newly dressed for the 21st century, made possible by Geometry Expressions software from Saltire Software.

**Proposition 35.**

*Parallelograms which are on the same base and in the same parallels are equal to one another.*

Let *ABCD*, *EBCF* be parallelograms on the same base *BC* and in the same parallels *AF*, *BC*; I say that *ABCD* is equal to the
parallelogram *EBCF*.

For, since *ABCD* is a parallelogram, *AD* is equal to *BC*. *[Prop. 1.34]*

For the same reason also *EF* is equal to *BC*, so that *AD* is also equal to *EF*; *[C.N. 1]* and *DE* is common; therefore the whole *AE* is equal to the whole *DF*. [C.N. 2]

But *AB* is also equal to *DC*; *[Prop. 1.34]* therefore the two sides *EA*, *AB* are equal to the two sides *FD*, *DC* respectively, and the angle *FDC* is equal to the angle *EAB*, the exterior to the interior; [Prop. 1.29] therefore the base *EB* is equal to the base *FC*, and the triangle *EAB* will be equal to the triangle *FDC*. *[Prop. 1.4]*

Let *DGE* be subtracted from each; therefore the trapezium *ABGD* which remains is equal to the trapezium *EGCF* which remains. *[C.N. 3]*