Explanatory Fatigue

So, I had the intention of animating this entire proof. But after I started, I realized that, given the nature of the proof, the complete animation would be thoroughly tedious and confusing. So you at least get Euclid's set up of the proof.

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One nice thing about the animation, though, is that the location of point \(\small\mathtt{C}\) is in fact random as required by the proof. It should be in a different location (within limits) along \(\overline{\small\mathtt{AB}}\) each time you play the animation.


Picking Up

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Euclid continues by showing that CGKB must be a square. Since \(\measuredangle\small\mathtt{G} \cong \measuredangle\small\mathtt{B}\), the sides opposite those angles (\(\overline{\small\mathtt{CB}}\) and \(\overline{\small\mathtt{CG}}\)) must be congruent. And 1.34, which we have covered, tells us that opposite sides of "parallelogramic areas" are congruent, so all of these congruences cascade into CGKB being an equilateral quadrilateral. To show that all of the angles of CGKB are right angles, Euclid uses 1.29 and 1.34. But doing so is unnecessary, as mentioned by Heath in his notes on the translation:

Whereas Euclid still, as in 1.46, seems to regard it as necessary to prove that all the angles of CGKB are right angles before he concludes that it is right-angled, Theon says simply "And it also has the angle CBK right; therefore CK is a square." The shorter form indicates a legitimate abbreviation of the genuine proof; because there can be no need to repeat exactly that part of the proof of 1.46 which shows that all the angles of the figure there constructed are right when one is.

At this point, Euclid simply tells us that "for the same reason \(\small\mathtt{HF}\) is also a square." Wow. Okay, well, 1.34 tells us that \(\overline{\small\mathtt{HD}} \cong \overline{\small\mathtt{GF}}\) and \(\overline{\small\mathtt{HG}} \cong \overline{\small\mathtt{DF}}\). We can use the Vertical Angle Theorem and SAS to show that two adjacent sides of \(\small\mathtt{HDFG}\) are congruent, and since \(\measuredangle\small\mathtt {D}\) is a right angle, \(\small\mathtt{HDFG}\) is a square.


Wrapping Up: Math Is Hard to Read Because It's Hard to Write

The rest is simply too tiring to type up. It is fairly straightforward to show (mentally, not in print) that rectangles \(\small\mathtt{ACGH}\) and \(\small\mathtt{KEFG}\) are congruent. If we use A to represent the length \(\small\mathtt{AC}\) and B to represent the length \(\small\mathtt{CB}\) (or \(\small\mathtt{AH}\)), then we can fairly easily dig out that \(\small\mathtt{(A + B)^2 = A^2 + 2AB + B^2}\).

Ben Orlin, who writes a site called Math with Bad Drawings, has a really nice piece on reading mathematics texts. He explains:

Single words are saturated with meaning; immense focus is required; the diction is exactingly precise… more than anything, reading mathematics is like reading poetry.

This is why good mathematicians always read with a pencil in hand. Passive perusal of mathematics is pretty much worthless. You need to investigate, question, and probe. You need to fill in missing steps. You need to chew for a long time on every sentence, fully digesting it before you move on to the next course of the meal.

And the reason for this is that mathematics explanations, beyond a certain level, are incredibly difficult to write without leaving huge gaps for a typical reader to fill in.


Image mask credit: JérémY