I had one very poor non-sentence, so here it is, CORRECTED.
In doing Euclid's geometry, many theorems have converses
that are very easy to prove by reductio. Of course the Steiner-Lehmus
theorem is the converse of an easy-to prove theorem, but it, the S-L
theorem, is quite difficult for a novice:
Any triangle with equal angle bisectors is isosceles.
Did you try this one out on your students. H.S.M. Coxeter (with Greeitzer)
writes that this theorem
always excites interest.
He calls Jacb Steiner
the great Swiss geometer.
About C. L. Lehmus, they write that the theorem was sent in 1840
by C. S. Lehmus (whose name would otherwise have been forgotten long
ago).
to Steiner.
Is there any information at all about Lehmus?
Best wishes for a fine '99,
Sam Kutler
>At 1.43 am +1200 26-12-98, Roger Cooke wrote:
>> ....
>> First regarding anthyphaeresis, I wrote a computer program to calculate
>> the number of passes in a cycle when computing the square roots of
>> non-square integers. I found that the length of a cycle stays fairly
>> manageable up to 17, then takes a large jump at 19. (Obviously one
>> doesn't need to apply it for 18.) That fact made me wonder about the
>> passage in which Theodorus is said to have stopped at 17. On reflection
>> I decided it was unlikely that Theodorus stopped AFTER doing 17 because
>> of the increase in complexity, especially since Wilbur Knorr made such
>> and elegant argument that he stopped BECAUSE his argument no longer worked
>> for 17. As Sam and David have repeatedly pointed out, there isn't any
>> textual trail, so there is little point in speculating. Still, I found
>> it intriguing. It's no sillier than the "chambered-nautilus" diagram
>> once suggested as an explanation, which as far as I can see has no
>> connection at all with the topic. Has anyone suggested this explanation
>> before?
>
> Yes: Zeuthen and van der Waerden. See B.L. van der Waerden,
>"Science Awakening" I, pp.141-6, especially p.145.
>
>>
>> Second, concerning synthetic geometry (a term I like and still use),
>> I really love the synthetic approach and often teach hyperbolic
>> geometry this way before giving models for it within Euclidean
>> geometry. I like deriving the formulas for the angle of parallelism
>> and solving triangles the "old-fashioned" way. There is something
>> very fascinating about hatching a quantitative chick from a
>> qualitative egg. However, my considered view is that this approach is
>> a pedagogical dinosaur. Students generally hate it, and I've begun
>> to see in my old age that the Greek approach to geometry was ponderous
>> and clumsy....
>
> Well, here's a different experience. For many years I've taught a
>historically-based geometry course to second or third year undergraduates
>who've seen hardly any geometry before. (Around 1970 it disappeared from
>schools in N.Z. just as it did in the U.S.) Those students come to
>Euclidean geometry as a fresh experience, and very obviously enjoy handling
>congruent triangles etc. It may be the first time that many of them have
>been at all confident about constructing proofs for themselves. I remember
>one year when the class was quite resentful at my moving on to Renaissance
>geometry: they would have been perfectly happy to spend the whole course on
>Euclid and Archimedes.
>
> I introduce coordinates in their 17th-century context, and use a
>mixture of synthetic and analytic methods from then on. My choice of
>method tends to be based on pedagogy more than history!
>
> However, simplified Euclid is as good a training ground as ever.
>There are plenty of examples where students can prove things that are less
>obvious from others that are more obvious; which I'm sure helps to boost
>their intuitive belief in the value of proofs. After that, they're more
>likely to appreciate full-bodied axiomatic theories in the modern style. I
>still feel very grateful for my own school training in Euclidean synthetic
>proofs. The central importance of proof in modern mathematics then came as
>no surprise to me, whereas it often alarms and distresses present-day
>students.
>
> Ken Pledger.