Presumably, I was taught geometry as a child. I do not remember.

Recently, I'd made my way halfway through a complex analysis textbook, only to find another which seemed more suitable and engaging. Unfortunately, the exposition was geometric. I knew something was wrong – I knew something had to change – when, asked to prove the similarity of two triangles, I got stuck on page 7.

I'd been reluctant to tackle geometry, and when authors reasoned geometrically, I'd find another way to understand. Can you blame me, when most geometric proofs look like this?

Distasteful. In a graph with $n$ vertices, you'd need to commit $O (n^{3})$ things to memory (e.g. triangles, angles) in order to read the proof without continually glancing at the illustration. In a normal equation with $n$ variables, it's $O (n)$ .

Sometimes, we just need a little beauty to fall in love.

Welcome to Oliver Byrne's rendition of Euclid's Elements, digitized and freely available online.

Elements

Propoſitions are placed before a ſtudent, who though having a ſufficient underſtanding, is told juſt as much about them on entering at the very threſhold of the ſcience, as gives him a prepoſſeſſion moſt unfavourable to his future ſtudy of this delightful ſubject; or “the formalities and paraphernalia of rigour are ſo oſtentatiouſly put forward, as almoſt to hide the reality. Endleſs and perplexing repetitions, which do not confer greater exactitude on the reaſoning, render the demonſtrations involved and obſcure, and conceal from the view of the ſtudent the conſecution of evidence.”

Thus an averſion is created in the mind of the pupil, and a ſubject fo calculated to improve the reaſoning powers, and give the habit of cloſe thinking, is degraded by a dry and rigid courſe of inſtruction into an unintereſting exerciſe of the memory.

~ Oliver Byrne

Equality and Similarity

Old mathematical writing lacks modern precision. Euclid says that two triangles are "equal", without specifying what that means. It means that one triangle can be turned into another via an isometric transformation. That is, if you rotate, translate, and/or reflect triangle $A$ , it turns into triangle $B$ .

Similarity is a bit more lenient, because you can rescale as well:

My favorite characterization of similarities is:

As a map $f : R^{n} \to R^{n}$ , a similarity of ratio $r$ takes the form $f (x) := r A x + t$ , where $A \in O_{n} (R)$

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