Andreae and the Squaring of the Circle

NOTE: This page is under construction and will be amended during June 2005.

Plate 9 of Andreae's Collectaneorum Mathematicorum Decades XI (1614) has a geometrical construction which illustrates an instance of squaring the circle:

... which was found in the Bible, has the approval of mathematicians; and it is a square of side 75 which is equal in area to a circle whose diameter is the square root of the sum of the squares of 75, 39 and 4. (Hæc Circuli quadratura inter sancta inventa, Mathematicorum testimonium habet, estque quadratum de 75, æquale areæ circuli, cujus diameter est radix quadrati compositi ex quadratis de 75, de 39, & de 4.)

 

The source of this construction was the Templum Ezechielis which was published in 1613 by Andreae's tutor and friend, Matthias Hafenreffer. In some respects this can be seen as a Lutheran response to the Jesuit Villalpando's Ezechielis Explicatio (1596-1604). Interestingly, while Villalpando's work was supported by Philip II of Spain, who granted 3,000 scudi for engraving the prints of his reconstruction of the Temple of Ezekiel, some of the engravings for Hafenreffer's book were made by Johann Valentin Andreae while he was lodging with Hafenreffer in 1611. We can illustrate the connection between them and the circle-squaring diagram as follows

 

This is a slightly simplified version of the view of the temple described by Ezekiel (chapters 40 - 44 which was engraved for Hafenreffer by Andreae. It has stylistic similarities to the plate which Andreae subsequently engraved to illustrate his own Christianopolis.

 

This is a plan view of the temple complex above, simplified further to show no more than the walls which divide the square up, and the gateways.  The design is built up from modular units of 100 cubits square.

The plan is now further simplified into a linear geometrical construction. In the diagram at the top of the page construction lines have been added to find the centre of the square, which acts as the centre of the circle whose radius is found using the triangular construction shown above.

 

 

As to the 'squaring of the circle', it is of course impossible using this kind of plane geometry. Nonetheless the values given by Andreae are very close: a square with base 75 has an area very close to that of a circle whose diameter is √ (75² + 39² + 4²). Put another way, equating the areas of this square and circle yields a good approximation to the value of  π (= 3.14158056...), so it is not surprising if it had at least some support from mathematicians of the calibre of Kepler and Mästlin.

 

Andreae certainly came to be sceptical about attempts to square the circle. As early as Turbo (1616) he was bracketing 'squaring the circle' with 'perpetual motion',¹ and  making fun of Naometrian calculations, for which an (autobiographical?) artist is required "... to illustrate serpents, dragons ... and the fifteen different Jerusalems" and each sits "... doing calculations with little crosses, spirals, angles, suns, moons, stars, serpents, circles, flying angels, crowns, pillars, candelabra, ... (using) animal numbers, sacrificial numbers, sabbatical numbers, Jerusalem numbers and the Key of David."²  It may be noted that Ezekiel 40, with which this exercise begins, makes much of the special length of cubit used to measure the proportions of the temple.

 

By 1618 in Menippus Andreae places  "... the squaring of the circle" alongside "... the perpetual motion machine, or the Grand Catholicon..." in opposition to the practical science of the farmer.³ A year later in Mythologiae Christianae Andreae was lumping together "mathematicians who had perfected the squaring of the circle" with promoters offering the secret of "perpetual motion...  multiplying machines ... vessels for diving under water, ships for crossing over the land ... the Philosophers' Stone, Universal Medicine, the Panacea, an everlasting light, pellucid gold, malleable glass, rejuvenation..."⁴  None of these was perhaps known to be impossible, but they were associated with crooks, swindlers and the deluded.

*

Regardless of the sense in which Andreae took the circle-squaring supposedly embodied in Ezekiel, he was of course familiar with the non-mystical side of mathematics. The value of π he used for this kind of purpose was the familiar approximation given by 22/7. This can be seen in Collectaneorum Mathematicorum where his plate on regular plane and solid figures includes the dimensions of the earth:  

 

It can be seen that if the circumference of the earth (ambitus terræ) is 5400 German miles⁵ and the diameter is 1718 2/11 miles, then Andreae is using the ratio of 22/7 for π.

This appears to be confirmed by the values he gives for the other measurements, such as the volume (solidetas) of the earth 2,662,560,000 cubic miles, though it is difficult to judge where rounding off has taken place. As Hafenreffer's book shows, mathematicians of his day were perfectly well aware that 22/7 was inaccurate; but the calculations also show that using a more accurate value of  pi was arithmetically difficult. 

This allows us, finally, to say something about Christianopolis, the design of which owes a certain amount to Andreae's experience when working on Templum Ezechielis. Andreae says that the circular temple at the centre of the community is 316 feet in circumference. His plan makes it clear however that the diameter of this temple is 100 feet, so the circumference should be 314 feet.⁶ In all probability the printer misread Andreae's manuscript.

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¹  Turbo 4th Interlude

²  Turbo act IV, scene 4

³   Menippus  72 'Idiota'.

⁴   Myth.V, 32 'Nundinae' pp 259-261

⁵   The German mile (Meile) varied considerably from one principality to the next. In Württemberg the mile was about 7,421 metres long, while in the neighbouring state of Baden it was more like 8,889 metres.

⁶  Christianopolis  ch.82