**Subject: **Re: [HM] Six point Circle

**From: **Ken Pledger (*Ken.Pledger@vuw.ac.nz*)

**Date: **Mon Jun 19 2000 - 18:12:52 EDT

**Next message:**Stacy Langton: "Re: [HM] Mathematics and Music"**Previous message:**Samuel S. Kutler: "[HM] Six point Circle"**Maybe in reply to:**Samuel S. Kutler: "[HM] Six point Circle"**Maybe reply:**Ken Pledger: "Re: [HM] Six point Circle"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

*>
*

*> Is it true that Euler (1765) had a six point circle:
*

*>
*

*> The centers of the three sides of a triangle, and
*

*>
*

*> the footpoints of the three heights.
*

*>
*

The question of why Euler's name is ever attached to the 9-point

circle came up recently in the Hyacinthos mailing list. It may be helpful

to repeat here the message which I posted about it.

*> The attribution to Euler seems to be wrong. I've recently received
*

*> a copy of a very thorough discussion by J.S. Mackay, "History of the
*

*> Nine-point Circle," Proc. Edinburgh Math. Soc. XI (1893), 19-57. Mackay
*

*> begins as follows.
*

*>
*

*>> The earliest author to whom the discovery of the nine-point circle
*

*>> has been attributed is Euler, but no one has ever given a reference to
*

*>> any passage in Euler's writings where the characteristic property of this
*

*>> circle is either stated or implied. The attribution to Euler is simply a
*

*>> mistake, and the origin of the mistake may, I think, be explained. It is
*

*>> worth while doing this, in order that subsequent investigators may be
*

*>> spared the labour and chagrin of a fruitless search through Euler's
*

*>> numerous writings.
*

*>> Catalan in his "Theoremes et Problemes de Geometrie Elementaire",
*

*>> 5th ed. p.126 (1872), or 6th ed. p.170 (1879), says that the learned
*

*>> Terquem attributed the theorem of the nine-point circle to Euler, and
*

*>> refers to the "Nouvelles Annales de Mathematiques", I. 196 (1842). If
*

*>> the first volume of the "Nouvelles Annales" be consulted, it will be
*

*>> found that Terquem has two articles on the rectilineal triangle. The
*

*>> first (pp. 79-87) is entitled "Considerations sur le triangle rectiligne,
*

*>> d'apres Euler"; the second (pp. 196-200) has the same title, but "d'apres
*

*>> Euler" is omitted. In the first article Terquem mentions that Euler
*

*>> discovered certain properties of the triangle, and refers to the place
*

*>> where they are to be found ("Novi Commentarii Academiae ...
*

*>> Petropolitanae", xi. 114, 1765). He says he thinks it useful to
*

*>> reproduce them with some developments, and this is exactly what he does,
*

*>> for the first article is a synopsis of Euler's results, and the second
*

*>> article, which begins with the property of the nine-point circle,
*

*>> contains the developments.
*

*>> Who, then, is the discoverer of the nine-point circle?......
*

*>
*

*> The story then becomes complicated, but perhaps at least I've typed
*

*> enough of Mackay's paper to explain away the Euler myth.
*

Paul Yiu followed up with this:

*> .... this paragraph of MacKay's .... has confirmed
*

*> my impression that Euler did not write explicitly on the nine-point
*

*> circle. Euler's plane geometry papers are collected in volumes 26 -- 29
*

*> of his Opera Omnia, series prima, and it appears that the most natural
*

*> place he would have written about the nine-point circle is
*

*>
*

*> 325. Solutio facillis problematum quorundam
*

*> geometricorum difficillimorum, Nova commentarii
*

*> academiae scientiarum Petropolitanae 11, (1765 -1767)
*

*> pp. 103--123. I26, 139--157.
*

*>
*

*> In his paper, Euler investigated the problem of
*

*> constructing a triangle given O, I, H. In so doing
*

*> he developed many of basic theorem formulae in triangle
*

*> geometry, including the Euler line. But the nine-point
*

*> circle did not appear here and in other shorter papers
*

*> in the ``geometry volumes''.
*

*>
*

*> Best regards.
*

*> Sincerely,
*

*> Paul Yiu
*

Perhaps now some HM subscribers may be able to help lay to rest the

widespread myth about Euler and the 9-point circle.

Ken Pledger.

**Next message:**Stacy Langton: "Re: [HM] Mathematics and Music"**Previous message:**Samuel S. Kutler: "[HM] Six point Circle"**Maybe in reply to:**Samuel S. Kutler: "[HM] Six point Circle"**Maybe reply:**Ken Pledger: "Re: [HM] Six point Circle"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

*
This archive was generated by hypermail 2b28
: Mon Jun 19 2000 - 18:20:05 EDT
*