Chapter 17 Proposition 36

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

Next, we try to prove Proposition 35 and 36, which its methods are not difficult. Proposition 35 is proved by subtracting equal quantities and adding equal quantities to equal quantities, while Proposition 36 builds on the previous proposition and only needs to find a suitable medium as an intermediate bridge.

It is interesting to see the expansion of the concept of equality. The act of comparison - equivalence - was never defined; it seems to be taken directly from the real-life concept and used. Most of the things we proved before were equal because they overlapped, that is, it was hoped that the parts proved to be equal could be used as a continuous whole, with each uncuttable part corresponding to the other. Because overlapping objects are separable, then after motion, the two objects that first overlap and then do not overlap are equal. Here the separation motion has an assumed premise that its separation must be without any loss. For example, translation, flip, and symmetry of these movements, we assume that in the abstract geometric world, will not produce any loss of energy, which will have the equivalence after the separation of the movement. (For a displaced object, there is also such a premise of motion, we assumed that the object has no loss in displacement, otherwise it would not be the same before the motion and it would be the same after the motion)

The first step of equivalence is from coincidence to partial separation, which is the same base in Proposition 35, and then comes the complete separation, which has become equal in Proposition 36 At the same time, in both propositions, it is implied that the case of detachment from the shackles of shape, the same base but the angles are not different, and the parallelogram with the same base between the same set of parallel lines, which does not have the same shape, and their equivalence is also proved in Proposition 35, is done by splicing, which means that we can combine two parallelograms by cutting them and the parts they are divided into, are identical in shape, thus introducing the equivalence of the whole. This is an example of changing perspective and going back to the local and microscopic parts for proof. If we endorse such a leap of logic, then geometry will break through the limits of our vision and take us on to situations approaching infinity. Imagine a parallelogram close to a straight line with a fixed base and a side close to infinity in length. Although it is not known exactly how long it is, its area is known because it can be known from other parallelograms between the same base and the same set of parallelograms, a jump from infinity to finite that is well worth thinking about.