The Elements

[Folio 15v]

Extensive marginal scholia in a very small, cramped hand, likely containing commentary on the geometric proofs.?

...[the line] not enclosed... and the line through point A parallel to the straight line BC? [is] DAE; and since DAE is parallel to BC, and the straight line AB falls upon them, the alternate angles DAB and ABG are equal to each other; again, since DAE is parallel to BC, and the straight line AG falls upon them, the alternate angles EAG and AGB are equal to each other. So that [the angles] DAB and EAG...

31

A geometric diagram for Proposition 31. It shows a point A above a line segment BC. A triangle ABC is formed, and a line DAE is drawn through point A parallel to the base BC.

From these things, it is clear that every triangle figure will make the three interior angles equal to two right angles.

[Folio 16r]

16

...[the] three angles of the triangle are equal to two right angles; for example, if [the angles] BAG and ABG were taken in half, that is [the angles] AGD, [which is] greater than EGD; so that the three [angles] of the triangle [are] when one of the sides is extended, the exterior angle is greater than the two interior and opposite [angles]; which was to be shown.

IN EVERY triangle, when one of the sides is extended,

32

the exterior angle is equal to the two interior and opposite [angles], and the three interior angles of the triangle are equal to two right angles. Let there be a triangle ABG, and let one of its sides, BG, be extended to D; I say that the exterior angle AGD is equal to the two interior and opposite [angles] GAB and ABG, and the three interior angles of the triangle, ABG, BGA, and GAB, are equal to two right angles. For let the straight line GE be drawn through point G parallel to the straight line AB. And since AB is parallel to GE, and the straight line AG falls upon them, the alternate angles BAG and AGE are equal to each other. Again, since AB is parallel to GE, and the straight line BD falls upon them, the exterior angle EGD is equal to the interior and opposite angle ABG. It was shown also that AGE is equal to BAG; therefore, the whole angle AGD is equal to the two interior and opposite angles BAG and ABG.

IN EVERY triangle, the greater side subtends the greater angle.

18

For let there be a triangle ABG having side AG greater than AB; I say that angle ABG is greater than...