Andrew Gough's Arcadia

By Geoff Bath

Part One

“In one particular respect we can define Poussin’s conception of Reason more precisely. It was closely bound up with mathematics, and especially with geometry. For the seventeenth century, mathematics was the supreme achievement of human reason because of the absolute certainty of its demonstrations, and it was also a symbol of clarity and order.” Anthony Blunt

So many people have claimed that they alone have solved the enigma of Poussin's Les Bergers d'Arcadie that one is forced to wonder if they can all be looking at the same painting. [1] Anyone who ventures a new approach to the subject, standing in the shadow of so many giants of research and reason, must do so with almost grovelling humility. Furthermore, mainstream thinking on the subject has become so rigid that to suggest there may be an alternative way to view the topic verges on anathema, even heresy.

Equally daunting to the investigator are the many who preach that no geometrical constructs underpin the art of the 16th and 17th centuries other, perhaps, than occasional glimpses of the Golden Section. [2] Some ‘experts’ seem to know that there is positively no geometrical foundation to, and certainly no 'key' embedded in, the works of Poussin and Teniers [3], as is suggested by the potentially fake or, at least, misleading Saunière cipher.

Such attitudes are, presumably, intended to convey to the observer that there is no point in looking any further, which would either be denying established fact or chasing after illusion. However, progress does not proceed from closed minds, any more than enquiry finds reasons to avoid searching for answers to questions unvoiced due to personal bias or lack of initiative.

This work does not claim to have any answers to the vexed question of the existence or not of geometrical formulae in art, it merely presents some observations that would not have surfaced had either of the above extremes of thinking held sway. It is merely a hypothesis for consideration, a suggestion that, perhaps, the Golden Section was not as supremely predominant in the art of the 15th to 17th centuries as has been supposed, and as is currently being insisted upon in certain quarters.

The work may prove to be an unproductive line of enquiry but some observations from a wider field of study are presented here to encourage further exploration and discussion should the subject of pentagonal signature geometry, as outlined here, be considered of any relevance to the solution of this particular aspect of the Rennes-le-Château mystery.

Que Poussin Teniers Gardent la Clef

One prominent element of the enigma of Poussin's Les Bergers d'Arcadie is the assumption that a key to the painting(s) exists but with no knowledge of its form, size and nature, and with nobody, other than Poussin, having seen the lock the key is supposed to fit.

However, if aficionados of the mystery are correct then this may not be altogether true. It is a widely-held assumption that the Shepherds Monument at Shugborough, Staffordshire [4], might bear upon the mystery but, apparently, without considering that the design of this monument may contain the same key as the painting. In this eventuality, the Shugborough tableau would merit careful analysis.

The Shepherds Monument features the central portion of the better-known version of Les Bergers d'Arcadie (hereafter, Bergers 2) but with elements of the earlier version (Bergers 1), in that the design is flipped horizontally, thus preserving the original order of the figures, and with the sarcophagus much as in the earlier painting. [5]

Preliminary analysis reveals that the ratio of the sides of the Shugborough monument approximates to 2:SQR3+1 (two to the square root of three plus one) and that the main feature of the design could, therefore, be a square of two units on the base. After drawing and developing this square, it becomes clear that there may be some potential for investigating the emergence of a surprising feature of the composition.

Figure 1: The geometrical design that emerges from the Shepherds Monument at Shugborough

It will be seen from Figure 1 that the centre of the square on the base lies on the hand of the kneeling ‘blue’ shepherd and that the diagonal passes through the top of the staff held by the standing ‘white’ shepherd. Furthermore, the line of this shepherd's staff is at roughly 75 degrees to the horizontal which, when mirrored, forms a trapezoid and, subsequently, a pentagon or pentagram.

An obvious step, thereafter, is to reverse the original painting (Bergers 2) to determine whether the same square overlay might fit. Initially, there is no obvious indicator as to what might be the common focal point, or the appropriate scale. However, it becomes immediately apparent that the major line through the 'white' shepherd's staff of the Shugborough monument runs almost parallel to the same shepherd's staff in Bergers 2. No surprise there, then!

As the square is tentatively scaled, it is noticeable that its centre moves closer and closer to the finger of the stooping 'red' shepherd, not the 'blue' one kneeling as in the Shepherds Monument. Roughly the same area of the painting as shown at Shugborough can be kept within the square but the centre draws closer and closer to the tip of the finger of the stooping 'red' shepherd, and towards the line marking the join in the masonry of the tomb. Moreover, the prominent trapezoid formed on the ‘white’ shepherd’s staff of the Shugborough monument seems not to fit in this case.

By placing the centre of the square upon the tip of the finger of the stooping 'red' shepherd it becomes clear that a trapezoid of a different shape is formed by the staff of the 'white' shepherd. Suffice it to say that there is a potential geometry underlying Bergers 2 sufficiently similar to that of the Shepherds Monument to suggest that the designer of the bas-relief was aware of the importance of the staff and of this particular square of the painting.

Figure 2: The two pentagonal schemas, Shugborough (left) and Bergers 2 (right)

Figure 2 presents the two schemas as they appear from the foregoing analysis. The small in-filled circles indicate the different points of origin of the staffs. At Shugborough, the tip of the staff is at 45 degrees from the centre, and in Bergers 2 it is at 142.5 degrees (that is, 37.5 degrees, or half of 75 degrees, ‘north of west’.) The distinctive pentagons seem, thereafter, to emerge quite naturally.

The portion of the painting represented at Shugborough seems to be the inner square of a construction pertaining to Bergers 2. However, the side of this square cuts through the kneeling 'blue' shepherd's left foot in the painting, but at Shugborough the foot has, apparently, been redrawn so this fails to happen. At Shugborough, the intent seems more to enclose the figures within an arch formed by the lower semi-square and the upper semi-circle. In both, the standing 'white' shepherd's staff seems to be at 75 degrees but sourced at different points. The objective in both cases would seem, therefore, to be to outline a pentagon, but not of regular form.

Further analysis suggests that the perceived angle of the pentagon’s left face would be correct at 105 degrees, as close inspection reveals that the top of the staff appears to have been painted to one side of the line and the bottom to the other. The fact that a similar device would seem to have been used in Bergers 1 hints at deception or, at least, an attempt to muddy the waters.

Figure 3: The major portion of the design showing the main and subsidiary squares and pentagons

The Shugborough pentagon has internal angles of (105, 105, 112.5, 112.5, 105 degrees, from base to apex) and Bergers 2 has two possible pentagons, one of (105, 105, 105, 105, 120 degrees) and the other of (105, 105, 108.75, 108.75, 112.5 degrees). The strange angles of the latter are due to this pentagon being in a circle perfectly inscribed within the former.

In order to recreate the design, it was found easier to work backwards from the assumed centre of the main circle (at the tip of the stooping 'red' shepherd's finger) than to attempt to move forward as if from a blank canvas. One of the major problems is the confusion over the size of the canvas and the availability of a good image of the painting out of its frame. Added to this, the canvas has been increased in size, and it is not known whether what we have at present is as Poussin originally planned it.

The pentagon is easily constructed, bearing in mind that the tips of the staff are at 142.5 and 232.5 degrees, the base angles being 105 degrees and the three 'containing' sides of the trapezoid the same length. The shepherd's staff provides a trapezoid into which a circle is introduced and another trapezoid within. The large dashed circle on the shepherd's staff in Figure 3 is extremely close to, if not matching, the inked line reported by Andrews and Schellenberger [6]. Note that this lies between two significant horizontal lines of this construction.

The inner pentagon could have the function of linking the four figures of the painting at the main points of the trapezoid, marking shepherd 1 (white) on the left side, shepherd 2 (blue) on the right calf, shepherd 3 (red) on the left foot and the shepherdess also on her left side.

More on Composition

This leaves the problem of the 'red' shepherd's staff, which Poussin painted before the tomb. The staff must, therefore, be highly significant, but in what way? It is fairly easy to determine that it lies at some 85 degrees to the vertical, which suggests that a similar pentagonal construction might be possible based on this staff. [7]

A clue may be provided by another Poussin painting, Summer, or Ruth and Boaz. This depicts a guard with a lance at much the same angle - and a similar pentagon may be formed by matching the angle of the lance and the trunk of the tree about the vertical centre line. Note also the ‘indicator’ of the pointing finger and the ‘confirmers’ of the line marking the chasm, upper right, and the bridge across it, the stone at bottom right and the ‘correspondence’ of the line of the uncut corn. [8]

Figure 4: Poussin’s Summer, or Ruth and Boaz

In this painting the pentagon emerges, once more, from basic ‘constructional’ techniques, and the geometry is relative to the size of the canvas. The pentagon has internal angles of (95, 95, 112.5, 112.5, 125 degrees).

Returning to Bergers 2, by applying a process of pentagonal design similar to that employed with the standing 'white' shepherd’s staff, and basing the construction upon a point on the horizontal ‘centre’ line directly below the tip of the staff, it will be found that a line projected at a right angle to the ‘red’ shepherd’s staff passes across the tip of the kneeling 'blue' shepherd's finger. This point can also be seen to lie on the line from the bottom right point of the inner pentagon (formed indirectly from the ‘white’ shepherd’s staff) to the centre. The point of intersection is on the letter ‘R’ of the inscription.

A similar procedure can then be applied to the staff of the kneeling ‘blue’ shepherd, which is found to bear at some 110 degrees to the horizontal. The centre of the enclosing circle is, once more, at a right angle to the staff on the ‘centre’ line (and passing through the point of intersection of the ‘blue’ shepherd’s finger, as above). The radius in this case is determined by the ‘constructed’ base of the third staff. The resulting pentagon has internal angles of (110, 110, 105, 105, 110 degrees).

Figure 5: The positioning of the ‘white’ staff (left) and the ‘blue’ staff (right) in Bergers 2

The centres of all three circles defining the staffs lie on the ‘centre’ line, and it may be appreciated how the three interact. The ‘white’ staff, on the left, effectively sources the other two, both of which are pinned by the two lines that define the location of the ‘R’ under the kneeling 'blue' shepherd's finger. This point determines the line of the middle staff and, by extension, that of the right-hand staff, this pentagon having internal angles of (95, 95, 117.5, 117.5,115 degrees).

Figure 6: The placing of the ‘red’ staff in Bergers 2 (left) and the location of the main points and features of the design (right)

It should be noted that the constructional lines are slightly longer than the staffs, and that the origin of the top of the ‘red’ shepherd’s staff is at the same level as that of the ‘white’ shepherd.

I have attempted to determine what dimension might form the basis of the painting and conclude that this could be an outer square of 30 pouces (the French inch) on the base or, perhaps, an inner square of 28 pouces (possibly with SQR2 as 10:7). One pouce is 27.07 mm, or about 1.066 inches. The width and height of Bergers 2 can be variously constructed and determined but in Figure 7 the canvas size is calculated at 34.86 x 44.68 pouces (944 x 1210 mm) based on a square of 30 pouces. [9] The 'viewing' frame is assumed to be 31.59 x 44.68 pouces (855 x 1210 mm) having the ratio of 1:SQR2 in line with the main construction. Needless to say, the suggested canvas size and viewing frame are inconsistent with the Cornford pentagon.

Figure 7: A potential design framework for Bergers 2 based on SQR2 (ad quadratum)

The height here is merely a suggestion, as there are a number of ways to derive the assumed measure of around 94 cm. Any analysis is hampered by the lack of a clear statement of the actual size of many paintings, both in and out of the frame, with official and accurate images - and no knowledge of what Poussin intended! [10]

Figure 8: Bergers 2 with the overlay of significant compositional features derived geometrically

It may be appreciated that, as in Figure 8, it is unnecessary to complete the three pentagons because the line of each staff is the first face to be produced, or ‘constructed’. The inked line on the ‘white’ shepherd’s staff, as marked upon the original painting, is shown here in white.

“The idea of connecting painting with mathematics was not, of course, new – it had been one of the foundations of Renaissance aesthetics – but Poussin believed in it with such fervour that it produced new and astonishing results in the paintings of the 1640’s, of which the most striking example is the Holy Family on the Steps.” Anthony Blunt

Figure 9: Poussin’s Holy Family on the Steps, National Gallery of Art, Washington

The analysis in Figure 9 is, assuredly, not specific to any geometry Blunt had in mind but is certainly a demonstration of what he intended to convey. It is unnecessary to search for a constructional geometry here because the eye provides both the framework and the focus in any event. Of a certainty, it’s there, but is it as formal as suggested, and is the setting of the staff and the derivation of the ‘platform’, both arising from the geometry, purely accidental? Am I, perhaps, reading too much into this? Further analysis of other paintings connected with the mystery may reveal some common features and could, perhaps, assist in developing a testable hypothesis vis-a-vis the Saunière cipher and Rennes-Le-Château.

Read Part Two