Re: [HM] It fascinated Leibniz

Julio Gonzalez Cabillon (jgc@adinet.com.uy)
Wed, 09 Sep 1998 00:27:31 -0300

At 06:47 AM 06/09/1998 -0400, Samuel S. Kutler wrote:

| According to Boyer
|
| A HISTORY OF MATHEMATICS, 1968 edition, page 439,
|
| the harmonic triangle
|
| fascinated Leibniz:
|
| 1/1 1/2 1/3 1/4 1/5 1/6 . . .
|
| 1/2 1/6 1/12 1/20 1/30 . . .
|
| 1/3 1/12 1/30 1/60 . . .
|
| 1/4 1/20 1/60 . . .
|
| 1/5 1/30 . . .
|
| 1/6 . . .
|
| Is there an account of the harmonic triangle anything like the short book
| on the Arithmetic Triangle of Pascal?
|
| Did this triangle originate with Leibniz?

Dear Sam,

Yes, I think the "harmonic triangle" did originate with Leibniz. He even
coined that term (and if you leaf through the Leibnizian _Mathematischen
Schriften_ (vol V) you will find exactly the same (arithmetic and harmonic)
triangles that appear on page 439 of Boyer's "A HISTORY OF MATHEMATICS"
(1968 edition).

There is a brief article titled "Leibniz' triangle" [_Delta (Waukesha)_,
vol. 4, pp. 75-85, 1974] written by John Niman which may be of your
interest. As far as I've read, Niman suggests convincingly that Leibniz
devised his triangle in attempting to find the sum of the infinite series
whose terms are the reciprocal triangular numbers, a problem posed to
Leibniz by Huygens in 1672.

Another paper, "Leibniz et le triangle harmonique", written by Maria Sol
de Mora Charles, has been reviewed for ZfM by Ivo Schneider, who fortunately
inhabits this list, so if he finds the time he may give us an idea, I
presume (and I would be most grateful!), whether this article is worth the
try or not.

Best wishes,
Julio