J graphviz’s Euclid’s Elements

In Sarah Hart’s new book Once Upon a Prime she relays John Aubrey’s account of the philosopher John Hobbes’s¹ first exposure to Euclid’s Elements.

Being in a gentleman’s library Euclid’s Elements lay open, and ’twas the forty-seventh proposition in the first book. He read the proposition. “By God,” said he, “this is impossible!” So he reads the demonstration of it, which referred him back to such a proof; which referred him back to another, which he also read. . .. At last he was demonstratively convinced of the truth. This made him in love with geometry.

She goes on to say that in the 17^th century knowledge of Euclid’s Elements was considered part of a well-rounded education and that gentlemen were expected to know Euclid’s 47^th proposition. Well, there aren’t a lot of “gentlemen” around in the 21^st century. We have lots of obese Chads and decorated truck-driving bands of trumptards, but no gentlemen. But, even in our advanced state of social and intellectual decay, most of us have heard of Euclid’s 47^th proposition. We call it the Pythagorean Theorem.

Lately, as part of a larger delusional project, I have been studying the first six books of The Elements. I haven’t turned into a geometry fanboy like Hobbes, but I must confess that reading great classics like Euclid’s Elements is surprisingly delightful. One delight is the canonical numbering of propositions.² It helps when comparing different versions of The Elements; I’ve been reading three.

1. The Green Lion Elements.

2. Oliver Byrne’s 1847 illustrated Elements.

3. David Joyce’s excellent online version.

My main text is the Green Lion Elements. The Green Lion version is based on Thomas Heath’s monumental and definitive three-volume English translation. It dispenses with Heath’s copious footnotes and commentary and coalesces all thirteen original Elements books into a single attractive volume. The Green Lion Elements is indeed a superb book that should be in every gentleman’s library.

After reading the Green Lion version of a proposition, I pop over to Byrne’s illustrated version and read his presentation of the same canonically numbered proposition. Byrne sought to make The Elements more accessible by replacing tedious proofs with striking, beautiful colored illustrations. He wrote:

This work has a greater aim than mere illustration; we do not introduce colours for the purpose of entertainment, or to amuse by certain combinations of tint and form, but to assist the mind in its researches after truth.

You got to love 19^th-century authors; the poor dears believed there was such a thing as “truth” and that the mind is better filled with it than the relative rubbish we call “our truth” nowadays.

When first published, Byrne’s book was hailed as a graphic masterpiece, a verdict that persists to this day, but unfortunately, the high-quality printing required to render hundreds of colored images made his book too expensive for its intended audience. Consequently, Byrne’s book didn’t impact mathematics education much, but it deeply influenced generations of artists and graphic designers and is still widely revered today. Byrne’s book only covers the first six books of The Elements. This is the main reason I am confining my study to the first six books. Modern admirers of Byrne have completed his work. You can now read all thirteen books in Byrne’s style.

I can personally attest to the effectiveness of Byrne’s style. For me, Byrne’s illustrated proofs are easier to understand than the standard verbose versions. Byrne’s 19^th-century illustrations work for the same reasons the graphic proofs shown on the magnificent YouTube channel Mathematical Visual Proofs work. Horace said it best:

A feebler impress through the ear is made,

Than what is by the faithful eye conveyed.

To keep focused on my larger project I’ve been slowly building up a directed graph (digraph) of the proof “justifications” found in Euclid’s first six books. “Justifications” are the references John Aubrey called out in his story about Hobbes in the gentleman’s library. Typically, proofs refer to prior proofs, which again refer to earlier proofs, and so on right back to what Euclid called postulates, common notions, and definitions. It is this organized comprehensive logical structure that makes The Elements so important. Most of Euclid’s results were known long before him, and “solitary proofs” in many forms, are found in many cultures, sometimes thousands of years before Euclid, but as far as we know, the idea of organizing everything known about a subject, geometry in Euclid’s case, into a coherent deductive logical structure was a unique innovation. It’s not exaggerating to say this logical organizing idea changed mathematics and the world forever. We’re still living in Euclid’s vision and we’re not even sure there is a rational alternative.

After digraph’ing the first two books of The Elements, I decided I needed to compare my handcrafted squiggles to someone else’s. This is what led me to David Joyce’s online Elements. If you browse any of Joyce’s pages, say Proposition 47, you will notice the explicit marginal justifications that set Hobbes on his way. Unlike Hobbes, we can peek under the hood. If you look at the HTML page code, you’ll see the justifications are marked in <div> elements. With such clearly marked references, it’s a simple matter to extract Joyce’s <div>’s and generate a graphviz digraph of Euclid’s first six books. Mining web pages is a dark art practiced by geeky nerds wielding magic software tools like the wonderfully named Beautiful Soup. I’ve used Beautiful Soup but sometimes, just sometimes, it’s easier to write a few lines of code than wrestle with frameworks like Beautiful Soup. The following J verb sucks out Joyce’s web page justifications.

eucjoycejust=:3 : 0

NB.*eucjoycejust v-- extract justifications from Joyce proposition html.
NB.
NB. monad:  blcl =. eucjoycejust clHtml
NB.
NB.   hdr=. 'https://mathcs.clarku.edu/~djoyce/elements/' 
NB.   htm0=. gethttp hdr,'bookI/propI47.html'
NB.   eucjoycejust htm0

NB. justifications in html
cc=. CR,LF,TAB
(,'a'&geteleattrtext"1 (] '</div>'&beforestr;.1~ '<div class="just">' E. ]) y -. cc) -. a:
)

By blowing over all the web pages with verbs like:

eucjoycebkdeps=:3 : 0

NB.*eucjoycebkdeps v-- justifications from Joyce book html files.
NB.
NB. NOTE: use (wget) or (curl) to download the files at:
NB.
NB. https://mathcs.clarku.edu/~djoyce/elements/
NB.
NB. monad:  btcl =. eucjoycebkdeps blclHtmlFiles
NB.
NB.   NB. justifications in first three books
NB.   bks=. 'bookI/propI*html';'bookII/propII*.html';'bookIII/propIII*.html'
NB.   eucjoycebkdeps ; 1&dir&.> (<'~temp/') ,&.> bks

;(<@justfile@winpathsep ,. eucjoycejust@read)&.> y
)

You can build graphviz code like this file on GitHub that collects the justifications in the first six books of The Elements. By loading this graphviz code into J’s graphviz addon you can generate SVG digraphs like euclid_digraph_books_1_6.svg. SVGs can be easily converted to PDFs. Here is a PDF of all the postulates, common notions, definitions, and propositions found in the first six books of The Elements. The nodes are color-coded to distinguish postulates from propositions, definitions, and common notions easily. Large digraphs quickly lose their informative nature. euclid_digraph_books_1_6.pdf looks more like the work of spiders on acid than anything else. Still, the web links embedded in the document are handy. Clicking on any graph node will take you directly to the relevant page on Joyce’s Elements site.

The following J verb follows digraph connections back to their source.

eucpropback=:4 : 0

NB.*eucpropback v-- generate reverse proposition digraph.
NB.
NB. dyad:  cl =. clNode eucpropback clGv
NB.
NB.   path=. jpath '~JACKS/eucgvuts/'
NB.   gv=. read path,'euclid_digraph_books_1_6_dependencies.gv'
NB.
NB.   NB. typical use
NB.   gt=. 'I.47' eucpropback gv
NB.   gf=. jpath '~temp/euclid_i_47_dependencies.gv'
NB.   (toHOST gt) write gf
NB.   graphview gf

gs=. s: gc [ 'gpr gpo gc'=. eucnctsparse y
'no such node' assert (rn=. s: <x) e. 1 {"1 gs

NB. work backwards in dependencies
dn=. 0 2 $ s:<''
whilst. #rn do.
  sn=. gs #~ (1 {"1 gs) e. rn
  dn=. dn , |."1 sn
  rn=. (0 {"1 sn) -. 0 {"1 dn
end.

title=. 'Proposition ',x,' Dependencies'
title fmteucgv gpr;gpo;<5 s: |."1 dn
)

Running this verb on Proposition 47 generates this reverse sub-digraph that retraces the steps Hobbes took back in his 17^th-century gentleman’s library.

The code referenced in this post is available on GitHub at: https://github.com/bakerjd99/jacks/tree/master/eucgvuts

For more information read the jupyter notebook eucgvuts_notebook.ipynb.