Insights from Euclid’s ‘Elements’

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 ver­tices, you’d need to com­mit things to mem­ory (e.g. tri­an­gles, an­gles) in or­der to read the proof with­out con­tinu­ally glanc­ing at the illus­tra­tion. In a nor­mal equa­tion with vari­ables, it’s .

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 , it turns into tri­an­gle .

Two tri­an­gles are similar when there ex­ists such an af­fine trans­for­ma­tion (i.e., you can scale as well).

Syn­thetic/​analytic

I find it strange that Eu­clid got so far by ax­io­m­a­tiz­ing in­for­mal no­tions with­out any ground­ing in for­mal set the­ory (e.g. ZFC). I mean, you’d get ab­solutely blown away if you tried to pull these shenani­gans in topol­ogy. But ap­par­ently, Eu­clidean ge­om­e­try is suffi­ciently well-be­haved that it ba­si­cally matches our in­tu­itions with­out much qual­ifi­ca­tion?

Area invariance

Book 1, propo­si­tion 35:

This says: sup­pose you draw two par­allel lines, and then make a dash of length 2 on each line. Then, make an­other dash of length 2 on the up­per line. The two par­allel­o­grams so defined have equal area. This is clar­ified in the next the­o­rem.

If you take one of the dashes and slide it around on the up­per par­allel line, the re­sul­tant par­allel­o­grams all have the same area. I thought this was cool.

Notes

  • There aren’t any ex­er­cises; in­stead, I tried to first prove the the­o­rems my­self.

  • Book III treats cir­cles, with won­der­ful re­sults on arcs and their re­la­tion to an­gles. I search for a snappy ex­am­ple, a gem of an in­sight to share, but my words fail me. It’s just good.

  • I read books I, III, IV, and skimmed II. Not all books of the Ele­ments are about plane ge­om­e­try; some are ar­chaic in­tro­duc­tions to num­ber the­ory, for ex­am­ple. Those look­ing to learn num­ber the­ory would do much bet­ter with the gor­geous Illus­trated The­ory of Num­bers.

Forward

Ele­ments is a tour de force. The­o­rem, the­o­rem, prob­lem, the­o­rem, all laid out in con­fi­dent suc­ces­sion. It was not always known that from sim­ple rules you could rigor­ously de­duce beau­tiful facts. It was not always known that you could start with so lit­tle, and end with so much.

Be­fore I found this re­source, I’d checked out sev­eral ge­om­e­try books, all of which seemed bad. To salt the wound, many books were ex­plic­itly aimed at mid­dle-school­ers. This… was a bit of a blow.

How­ever, it doesn’t mat­ter when some­thing is nor­mally pre­sented. If you don’t know some­thing, you don’t know it, and there’s noth­ing wrong with learn­ing it. Even if you feel late. Even if you feel sheep­ish.

Against completionism

I’m glad I didn’t read all of the books, even though they’re beau­tiful. I’d picked up a bad “com­ple­tion­ist” habit – if I don’t read the whole book, ob­vi­ously I haven’t com­pleted it, and ob­vi­ously I’m not al­lowed to make a post about it. Of course.

But I’m try­ing to pick up use­ful skills, to ex­pand the types of qual­i­ta­tive rea­son­ing available to me, to get the most benefit per unit of read­ing. I stopped be­cause I have what I need for my com­plex anal­y­sis book.

Read around

Read­ing rele­vant Wikipe­dia pages /​ other text­books helps me cross-ex­am­ine my knowl­edge. It also helps con­nect the new knowl­edge to ex­ist­ing knowl­edge. For ex­am­ple, I now have a won­der­fully en­riched un­der­stand­ing of the ge­o­met­ric mean.

Over time, as you ex­pand and read more books, you’ll find your­self read­ing faster and faster, un­der­stand­ing more and more sub­sec­tions. I don’t recom­mend learn­ing new ar­eas via Wikipe­dia, but it’s good re­in­force­ment.

Re-de­riv­ing de­pen­den­cies as a habit

Ever since I learned real anal­y­sis, I re­flex­ively re­prove all new el­e­men­tary math­e­mat­ics when­ever I use it. For real anal­y­sis, that meant con­tinu­ally re­prov­ing e.g. when­ever I used that prop­erty in a proof. Did it feel silly and te­dious? A bit, yes.

But with (this) te­dium comes power. I can now re­gen­er­ate a for­mal foun­da­tion for the real num­bers from the Peano ax­ioms, prov­ing the nec­es­sary prop­er­ties about the nat­u­ral num­bers, then the in­te­gers, then the ra­tio­nals, and then the re­als, and then com­plex num­bers too. (But please, no quater­nions!)

With this habit, you con­tinu­ally ask your­self, “how do I know this?”. I think this is a use­ful sub­skill of Ac­tu­ally Think­ing.

Commemoration

In col­lege, I taught my­self a bit of Ja­panese. Through a com­bi­na­tion of spaced rep­e­ti­tion soft­ware and mem­ory palaces, and over the course of three months, I learned to read the 2,136 stan­dard use char­ac­ters. After those three months, I proudly dis­played this poster on my wall:

I look for­ward to an­other beau­tiful poster.

As the ſenſes of ſight and hear­ing can be ſo forcibly and in­ſ­tan­ta­neously ad­dreſſed al­ike with one thouſand as with one, the mil­lion might be taught ge­om­e­try and other branches of math­e­mat­ics with great eaſe, this would ad­vance the pur­poſe of ed­u­ca­tion more than any thing that might be named, for it would teach the peo­ple how to think, and not what to think; it is in this par­tic­u­lar the great er­ror of ed­u­ca­tion origi­nates.