Letters to the Editor re. Mr Beck's article in issue no 7 of The Temple

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Dear Sirs

Mr. Beck seems to have been somewhat carried away by his own enthusiasm (Issue 7, Poussin and the Sacred Angle).  In his eagerness to demonstrate the esoteric properties of his geometrical inventions he weakens his own argument by failing to realise that the construction of “squaring the circle” works with any angle, not just with 55º.        

                                  Fig. 1                            Fig. 2

I illustrate this by using angles of 40º in Fig.1 and of 25º in Fig. 2 but the proof is shown in Fig. 3 (next page) and is as follows:

Take any angle ABC and repeat it at 90º intervals (A₁BC₁, A₂BC₂, etc.).

Bisect each angle (BD₁, BD₂, etc.)

Draw the tangents at points D₁, D₂, etc. to give WX, XY, etc.

As the bisectors are all radii of the circle, it follows that WX is at right-angles to BD₁ and XY is at right angles to BD₂, etc.

Further, since WX = D₄D₂ = the diameter of the circle = XY = YZ = ZW

WXYZ is therefore a quadrilateral figure in which all the sides are of equal length and all the angles are of 90º (i.e. a square).

This is not, of course, Mr. Beck’s square but since A₁ and C₁ are equidistant from D¹, the line ST (passing through A₁C₁) will be parallel to the line WX and similarly, UV (passing through A₂C₂ will be parallel to XY, etc.  Consequently, STUV (Beck’s figure) will always be a square irrespective of the angle originally selected.

                                                                 Fig. 3

Furthermore, Mr. Beck cannot have measured his “exact” angle from the painting because it is utterly impossible to visually measure any angle to the fifth decimal place of a degree (it is about the same as measuring an object one centimetre long from a distance of one kilometre away).  He therefore appears to have calculated the angle of 54.99139º, probably to suit his “interesting” formula.

By using a spreadsheet it is, in fact, very easy to calculate many such “interesting” formulae.  For example:

90 deg(1 qtr circle) - 46.03747 deg = 43.96253 deg

46.03747/43.96253 x 3 = 3.141593 or Pi

and

90 deg(1 qtr circle) – 39.59107 deg = 50.40893 deg

39.59107/50.40893 x 4 = 3.141592 or Pi

and so on.

These defects make Mr. Beck’s otherwise interesting article somewhat unconvincing.

Yours sincerely

Mr Trevor P. Dutt

Dear Oddvar

Your skeptic fails to understand that squaring the circle does not mean simply drawing a square inside or around a circle ( and any elementary math student can figure out that by shifting an angle, any angle, 90 degrees on it’s axis  will produce a square box). What he is failing to understand is in order to square the circle, the internal calculated area of both the circle and the square has to be equal (his demonstration shows a square and circle with different areas altogether). There is absolutely one and only one angle that will produce this result and it is as I indicated, approximately 55 degrees. The symbol (~) means approximate. The 5 decimal place angle I calculated would be needed to produce equal “areas” is very, very close to 55 degrees and seeing as Poussin used this exact same process in another work leaves little doubt of this being some freak coincidence. In any case, it’s great to have people take enough interest to make an effort and try to discredit my work, it may mean that I am actually on to something.

Dear Oddvar

Many thanks for forwarding the comments from Mr. Beck and my apologies for my delay in responding. 

Mr. Beck’s additional comments are most illuminating and it is therefore unfortunate that something of the same was not included in the original text. 

To quote from the original (page 16 in Issue 7 of “The Temple”): 

“…. It occurred to me that by drawing lines connecting the points on these two figures (sic) hands, …. a very interesting angle is created.

“The angle produced by this process is approximately 55 degrees (54.99139 to be exact).”

Nowhere in the original paper does Mr. Beck explain that the more accurate figure is calculated from first principles rather than measured from the painting as is implied by the description.

Furthermore, it was not clear in the original text that Mr. Beck was alluding to the classical problem of “squaring the circle”, indeed his reference to areas seems almost incidental: “The interesting thing about this newly formed square is that the area contained within, calculates out to almost exactly the same area encompassed by the circle.”  This makes the similarity of areas sound more like a casual observation than a central hypothesis of his paper as he now explains.

One other point in Mr. Beck's paper also left me confused.  In his final paragraph he refers to a "likeness of a Templar Cross" - did he actually mean a Hospitaller Cross?

Yours sincerely

Mr Trevor P. Dutt