FoMRHI Comm. 2002 John Downing

The 'Trammel of Archimedes' – a Draughting Tool for 16th C.Venetian Lute Makers?

Albrecht Dürer knew the difference between an ellipse and an oval although in his time (and to this day) the two terms were generally used interchangeably (see Comm 1961).

In a letter to his brother, the Bishop of Auxerre, dated 23 May 1533, Jean de Dintville (one of the two subjects in the 'Ambassadors' painting by Hans Holbein) wrote "Je vous prie m'envoyer le portraict du compas auvale duquel m'avez escript; car je suis empesché à comprendre la façon de laquelle il est fait" (Please send me a picture of the oval compass that you wrote to me about because I am quite at a loss understanding how it is made).

Dintville was probably referring to a draughting instrument, otherwise known today as an ellipsograph, designed to draw curves of conic sections such as the ellipse (Note 1). This device – essential for design of instruments such as the astrolabe - was probably the invention of Arabic scholar Abu Sahl al-Quhi who wrote his treatise 'On the perfect compass' in the 10th C. (Note 2).

Renaissance artists Leonardo da Vinci, Dürer and Michelangelo used similar devices in their work also claiming to be their inventors (See Fig.1).

Fig 1

The oval compass is comprised of an adjustable tripod, one leg carrying a pen holder that is free to rotate around the leg and a pen that can freely move longitudinally within the pen holder. As the penholder is rotated around the supporting leg the pen will trace an ellipse onto the horizontal plane where the compass is positioned. (See also Note 3).

So, would an oval compass have been of use to a 16th C Venetian luthier for designing lutes in the form of elliptical curves – unlikely it would seem due to the impractical size of compass that would be required to draw a lute sized elliptical curve and the complexity of set up.

A much simpler device for drawing elliptical curves is the so called 'Trammel of Archimedes' that dates back to the ancient Greeks. This device might very well have been used for lute design but was it?

The basic geometry of the trammel is illustrated in Fig 2. A pen mounted on a rod with its ends A and B sliding along fixed vertical and horizontal guides X and Y will trace an elliptical curve. The size and form of the ellipse depends upon the length of the rod and the position of the pen on the rod.

Fig 2 Trammel Type 1

Another variant of the trammel is shown in Fig 3. It is a bit more complicated but is more compact than the first example. It is, however, likely to be less accurate in operation due to clearances between the guide pins and slots further increased in size by the magnification ratio.

Fig 3 Trammel Type 2

Either varient of the trammel is easy to fabricate and can be made more versatile by providing sliding adjustments to rod lengths, guide pin positions and pen locations.

It is not even necessary to construct trammel hardware. A card strip with the pin and pencil positions marked on it – serving as a trammel rod - may be positioned and its ends moved over drawn XY axes to produce a series of points on an elliptical curve that may then be joined into a smooth curve. This is the method applied (for convenience)in the following examples - the first being the Arnault de Zwolle lute geometry.

The Arnault de Zwolle lute upper soundboard profile is an arc of radius R equal to the soundboard width and the lower soundboard profile is a semi circle. It is ovoid in shape. Nevertheless, an elliptical curve created on the same XY axes of reference using a type 1 'trammel of archimedes' is a reasonably close match to the circular arc of the soundboard - so confirming Dürer's statement "I will call the ellipse an egg line (i.e. an oval) because it is very nearly the same as an egg"(see Fig4). Interestingly, if the XY axes are rotated slightly to positions X1, Y1 an elliptical curve generated from these new coodinates becomes an identical match to the lute soundboard profile (or is very close).

Fig 4

A second example compares the upper sound board profile of a lute by Paduan luthier Michael Harton with an elliptical profile generated on the same XY axes using a type 1'trammel of archimedes'. The lute is the large 8 course lute by M Harton, dated 1599, Cat. # M156 in the GermanischesNationalMuseum, Nurnberg collection – string length 781 mm.

Here (Fig 5) the upper sound board profile is an oval – but a less than perfect match to an elliptical curve (although by creating an elliptical curve on the XY axes rotated to position X1 Y1 as in the first example - the curves do match very closely).

From a practical viewpoint the dimensions of a trammel needed to create an elliptical soundboard profile for this size of lute would require a trammel rod measuring 595 mm in length with the XY guides each measuring 600 mm minimum length – not an impractical size however.

Fig 5

For comparison, Fig 6 shows a proposed upper soundboard geometry of this lute – an oval curve that is an accurate fit comprised of two conjunct arcs of a circle of radii 7 units and 4 units for a soundboard width of 4 units – derived from a full size drawing of the lute.(Note 4)

Fig 6

Conclusion

Although it is possible that 16th C Venetian lute makers may have used a 'Trammel of Archimedes' to determine upper sound board lute profiles (with the XY axes appropriately rotated and offset), it seems far more likely that they would have opted instead for the simpler expedient of creating an oval curve derived directlyfrom conjunct arcs of a circle using compasses or dividers and straightedge – as described by Dürer.

Notes

1) John North author of 'The Ambassadors' Secret' suggests that Dintville may have been referring to a small oval shaped portable sundial/magnetic compass combination. However it is hard to believe that Dintville would have had any problem in understanding how such a device was made compared to the drawing instrument.

2) See 'The Perfect Compass: Conics Movement and Mathematics Around the 10th C' by Thomas deVittori, Université d'Artois, France.

Abu Sahl al-Quhi – Perfect Compass – 10th C.

3) Ellipsograph Patent, Early 20th C.

4) Identical upper soundboard geometries have so far been found on the following lutes by M.Harton - #1808 Bologna Museo Civico, Padua 1599 and Washington, Folger Shakespeare Library, Padua 1598 as well as the descant lute by Wendelin Tieffenbrucker #C39, WienKunsthistorischesMuseum.