On the 5-point scale current in Estonia, and surely in nearby nations, and familiar to observers of the academic arrangements of the late, unlamented, Union of Soviet Socialist Republics (applying the easy and lax standards Kmo deploys in his grubby imaginary "Aleksandr Stepanovitsh Popovi nimeline sangarliku raadio instituut" (the "Alexandr Stepanovitch Popov Institute of Heroic Radio") and his grubby imaginary "Nikolai Ivanovitsh Lobatshevski nimeline sotsalitsliku matemaatika instituut" (the "Nicolai Ivanovich Lobachevsky Institute of Socialist Mathematics") - where, on the lax and easy grading philosophy of the twin Institutes, 1/5 is "epic fail", 2/5 is "failure not so disastrous as to be epic", 3'5 is "mediocre pass", 4.5 is "good", and 5/5 is "excellent"): 4/5. Justification: Kmo had time to do a reasonably complete and (within the framework of the version 1.0.1, 1.0.2, .. process) reasonably polished job.
- UTC=20160825T0311Z/version 1.2.0: Kmo looked again at his lettering of the rapid (Pappus) proof of pons asinorum, and found that he had at 20160823T0203Z got this wrong! How very annoying. Perhaps the work at 20160823T0203Z was the result of fatigue. Now corrected. - Kmo retained the right to make nunsubstantive tweaks, as here-undocumented version 1.2.1, 1.2.2, 1.2.3, ..., over the coming 48 hours.
- UTC=20160823T0203Z/version 1.1.0: Kmo repaired a very bad error, in which he completely mixed up his lettering in the rapid (Pappus) proof of pons asinorum. He also at some point a little earlier than this repaired other significant errors, involving incorrect lettering of figures. - Kmo retained the right to make nonsubsantive tweaks, as here-undocumented versions 1.1.1, 1.1.2, 1.1.3, ... , over the coming 48 hours.
- UTC=20160823T0003Z/version 1.0.0: Kmo uploaded base version. He retained the right to make nonsubstantive tweaks, as here-undoumented versions 1.0.1, 1.0.2, 1.0.3, ... , over the coming 48 hours .
[CAUTION: A bug in the blogger software has in some past weeks shown a propensity to insert inappropriate whitespace at some late points in some of my posted essays. If a screen seems to end in empty space, keep scrolling down. The end of the posting is not reached until the usual blogger "Posted by Toomas (Tom) Karmo at" appears.]
[In anticipation of follow-on writing, I have now added the necessary "Part A" to last week's title, and the requisite "Part B" to this week's. Additionally, I have now added to "Part A" the necessary section heading, "1. Introductory Comments, on the Inadequacy of Traditional Geometry Textbooks". - I now continue in this "Section 1", for the reader's convenience repeating my last two sentences.]
One could not agree more. But now what, by way of "clear and distinct abstract reasoning", do we actually encounter in traditional presentations of Euclidean geometry, such as in that disappointing 1919 Godfrey-and-Siddons?
Early on, in fact as Godfrey-and-Siddons "Theorem 12", there is the familiar "pons asinorum" proposition regarding triangles that are either isosceles-and-not-equilateral or equilateral: If two sides of a triangle are equal, the angles opposite to these sides are equal. It is given that ABC is a triangle which has the length AB the same as the length AC. The proof is now via the familiar "S.A.S.", in Godrey-and-Siddons's exposition "Theorem 10": If two triangles have two sides of the one equal to two sides of the other, each to each, and also the angles contained by those sides equal, the triangles are congruent. To make S.A.S. applicable, the unhappy pair of authors consider a line bisecting the Theorem 12 triangle at A. Here they say, "Let it cut BC at D."
But (Moise rightly presses this question) how do we know the existence of such a point D? The postulates offered in Godfrey and Siddons do not suffice for an existence proof.
A similar problem, involving an unsubstantiated existence claim, occurs with the Godfrey-and-Siddons discussion of an actual method for bisecting an angle (I quote verbatim, except that for maximum clarity I reletter): "From PQ, PR cut off equal lengths PX, PY"; "With centres X and Y and any convenient radius describe equal circles intersecting at Z." If the existence of equal-length segments PX, PY is charitably granted (this itself requires some discussion), there nevertheless remains the leap to the assertion that point Z exists. How, in general, do we know that given any points X and Y in Euclidean 2-space, there is some distance r such that the circles at X and Y, of radius r, have at least one point Z in common?
Turning now to the Godfrey-and-Siddons proof of the underlying "Theorem 10", the "side-angle-side" congruence principle used in proving their Theorem 12, we find a further gap, of an interestingly different kind. Their proof begins, "Apply triangle ABC to triangle DEF so that A falls on D, and AB falls along DE." Their notion of "applying" conceals subtleties. We in fact need to note (as I think, on my cursory revisit to their book this week, Godfrey and Siddons fail to note) two different situations in Euclidean 2-space.
(a) Triangles ABC and DEF may possibly be such that through some combination, in some order, of zero or more within-the-Euclidean-plane translations of ABC and zero or more within-the-Euclidean-plane rotations of ABC, the vertices and angles of ABC and DEF may be brought into coincidence - as at least one of the conceivable correspondences ABC-DEF, or ABC-DFE, or ABC-EDF, or ABC-EFD, or ABC-FDE, or ABC-FED. In this case, let ABC and DEF be called "within-Euclidean-plane superposable".
Since the just-proffered definition says "zero or more", not "one or more", it is trivially true that for every triangle ABC, ABC and ABC are within-Euclidean-plane superposable. This small fact will soon prove helpful, as a lead-in to Pappus's improvement on Euclid's "pons asinorum" page.
(b) Triangles ABC and DEF may possibly be such that through some combination, in some order, of zero or more within-the-Euclidean-plane translations of ABC and zero or more within-the-Euclidean-plane rotations of ABC and zero or more within-the-Euclidean-plane reflections of ABC through some line, the vertices and angles of ABC and DEF may be brought into coincidence - as at least one of the conceivable correspondences ABC-DEF, or ABC-DFE, or ABC-EDF, or ABC-EFD, or ABC-FDE, or ABC-FED. In this case, let ABC and DEF be called "congruent".
The concept of congruence proffered here matches the traditional plane-geometry textbook concept of congruence, but (this is my main point) is looser than the concept of triangles that are within-Euclidean-plane-superposable.
If triangles ABC and DEF are congruent, and yet are not within-Euclidean-plane-superposable, then ABC can admittedly be brought into coincidence with DEF by taking ABC out of the Euclidean plane, and flipping it over in Euclidean 3-space, and returning it to the Euclidean plane, and doing some translations and/or rotations within the plane.
The distinction is slurred over by Godfrey and Siddons, who unfortunately write of figures (for instance, triangles) that "when applied to one another can be made to coincide (i.e. fit exactly)": such figures, they add, "must be equal in all respects", and "figures which are equal in all respects are said to be congruent". This language is unfortunate. "Can be made to" is not spelled out. The meaning must be "can be made to by a process of translation and rotation in Euclidean 3-space, not necessarily just in Euclidean 2-space". But how is the hapless reader supposed to divine that meaning?
When we leave two-dimensional Euclidean geometry for Euclidean solid geometry, the problem ramifies, becoming an illustration of the topic of "handedness", or (after an ancient Greek root) "chirality" - important in electromagnetics, and if we are not careful leading us into the terrifying world of hyperspace.
For consider two modest solids. One is a Roman "R", with its various straight and curved strokes of some nonzero width, and with the whole letter of some nonzero thickness. The other is a Cyrillic "ya" (the backwards R, the last letter of the Russian alphabet), of the same volume, with matching segment lengths and matching arc radius and matching angles, and so on - being in fact a three-dimensional mirror image of the given R.
The R and the ya are in a straightforward sold-geometry sense - analogous to what we have already developed for Euclidean plane geometry - not only congruent, but even superposable. (As in the two-dimensional case superposition is to be achieved by rotations and translations in 2-space, so in this present, Euclidean solid-geometry, case superposition is to be achieved by rotations and translations in 3-space.)
Imagine, now, the R and the ya to be decorated, each with one small hemispherical dimple in its front surface, where the oblique stroke meets the bowl. If the tiny dimples are scooped out in such a way that the R and the ya remain mirror images, then the two solids are in a straightforward sense congruent-and-yet-not-superposable.
This is like the difference in electromagnetics between a right-handed coil and a left-handed coil. A mirror-image pair of such coils, subject to identical conditions of time-varying magnetic flux, will by Faraday's law of induction develop equal-but-opposite voltages across their terminals.
It seems that similar examples can be found in stereochemistry. The picture of amino-acid "enantiomers" at the beginning of https://en.wikipedia.org/wiki/Chiral is suggestive.
The language of Godfrey and Siddons notwithstanding, there is no sense in which the dimpled R and the dimpled ya "can be made to coincide" - unless, indeed, we leave Euclidean 3-space for the terrifying, unvisualizable, world of Euclidean 4-space.
A more satisfying approach to the "Theorem 12" of Godfrey and Siddons is one pioneered a couple of decades after Godfrey and Siddons, by Harvard authority George David Birkhoff (1884-1944; he is considered one of the major American mathematicians of his era). Birkhoff was writing for a school-mathematics community, as a collaborator with educator Ralph Beatley, in a book bearing the title Basic Geometry. Their third and final edition was printed, if not earlier, then at any rate in 1959, at the publishing house Chelsea. A sign of excellence is that this same edition was reissued in 2000 by the American Mathematical Society, under ISBN 978-0-8218-2101-5.
Introduce (says Birkhoff-with-Beatley; I paraphrase from memory) S.A.S. as a postulate, in the form "Let triangles ABC and DEF be congruent in the sense that AB and DE are of equal lengths, and AC and DF are of equal lengths, and the internal angles at A and D are of equal measure; then the sides BC and EF are of equal lengths, and the sides CA and FD are of equal lengths, and the internal angles at B and E are of equal measure, and the internal angles at C and F are of equal measure."
We can now rapidly prove "Theorem 12", without recourse to the problematic claim that a bisector of the apex angle intersects the triangle base. If sides AB and AC are of equal length, triangle ABC satisfies the S.A.S. postulate, in that ABC is congruent with itself (in the protasis, we innocently take D to be A, cunningly take E to be C, cunningly take F to be B); with S.A.S. assumed, and with the S.A.S. protasis found to be satisfied, each part of the apodosis can now be asserted. In particular, the internal angle at B is now guaranteed to have the same measure as the internal angle at C.
Moise has not only the virtue of rigour, but the virtue of classical erudition. For in pointing this terse proof out, and remarking that Euclid's proof runs by contrast to the length of a whole printed page, he addes that the terse proof was known in antiquity to Pappus.
Moise has also a third virtue, namely the ability to be witty at the expense of computer scientists. He points out that the simple proof weas inadvertently rediscovered in a comp-sci lab:
Not many years ago - or so the story goes - an electronic computing machine was programmed to look for proofs of elementary geometric theorems. When the pons asinorum theorem was fed into the machine, it promptly printed Pappus' proof on the tape. This is said to have been a surprise to the people who had coded the problem; Pappus' proof was new to them. /.../ if you want a machine to get the idea that the triangles /... / in the SAS postulate are supposed to be different, you have to say so explicitly. It didn't occur to anybody to do this, and so the machine proceeded, in its simple-minded way, to produce the most elegant proof.
What I have just just quoted is from pp. 105 and 106 of Moise's final, third, edition, published in 1990. The first edition, from an earlier era in computing, has the same anecdote, but with some additional disparaging remark on "vacuum tubes".
I have the time and patience to cite one further example of Godfrey-and-Siddons's logical failings, concerning their already-cited notion of "equality". I have quoted them as asserting that figures that can be made to coincide are "equal in all respects". Well, as they say around the samovar in my imaginary Nikolai Ivanovitch Lobachevsky Institute of Socialist Mathematics (across that imaginary potholed Siberian street from the "Alexandr Stepanovitch Popov Institute of Heroic Radio"): "Vot MEENZ 'equal'?"
[To be continued next week, I hope, as "Part C". - To anticipate a little, "equal" has to be unpacked in the manner of mathematical logician Frege, with "x equals y" having just one correctly clear meaning: "x is the same object as y". I guess I will be citing here Frege's dictum that the only acceptable mathematical sense for "equals" is the one in which the heavenly body Hesperus "=", i.e., is the same planet as, the heavenly body Phosphorus: the thing you occasionally see in the evening sky, and are wont to call Hesperus, is the same entity, being the planet Venus, as the thing you occasionally see in the morning sky, and are wont to call Phosphorus.]