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9. "SUERTE DE VARAS" ("Bullfight") by GOYA

Vermeer -- El Greco -- now Goya -- another involved in the Secret!

(In which I explain how I analyzed this one, too -- work in progress, Nov. 06, 2003)

As Painted by Francisco Goya, Spanish, 1746-1828

STEP 1 of my explanation of the discovery of the Grail Geometry in yet another Old Master painting:

In many of these paintings that are composed with the aid of the Grail Geometry we see prominent straight-line features.   In Poussin's "Et In Arcadia Ego" (both I and II) and in David Teniers the Younger's "Saint Antony and Saint Paul" the straight-line features are staffs; in Vermeer we have seen a bowstring ("Lady Standing"), a bow of a viola da gamba (Lady Seated"), the shaft of a geometer's compass (The Geographer), a prominent gutter in a sidewalk (The Little Street), etc. Here in Goya's "Bullfight" the straight line features are lances. As I have explained before, the usual procedure is to look for features such as these and draw exploratory lines through them and parallel to them to see if there is an equilateral triangle underlying the compositional arrangement.  Continued below . . .

The first line I drew is labeled 1--1, beginning at the tip of the lance encircled at "A" -- drawn down through the bottom point of the remarkable little "vee" painted, half-hidden, in black (a big hint for the analyst -- a registration marker used by the painter) and labeled as such and encircled at "B".  I was encouraged to see that this first line I drew was confirmed by its tangency to the top of the head painted by Goya encircled at "C".  There are other confirmations -- features guided by this line -- in particular a very pronounced one within circle "B" where LINE 1--1 clips the sharp point of the bull's horn!  Starting to look very promising!

I expected the lance at circle "A" to make a 60 degree angle with LINE 1--1 (the start of an equilateral triangle, as geometers will understand) -- but this was not so. Therefore I tried exploring the most prominent line of all in this painting -- the one formed by the horseman's lance in the ellipse labeled "D".  I drew LINE 2--2 along this lance, and was delighted to find that the line went -- Bingo! -- right through The Bull's Eye!  Encircled at "E".  Looks like we've got geometry in this one,  gentle viewers!   Continued below . . .

LINE 2--2 is further (and amply) confirmed by going through the bottom of the red ribbon at "F", and the sharp point where the bull's tail contacts his leg at "G".   Not only that -- but LINE 2--2 runs parallel with LINE 1--1. Very promising!

Now either -- or both -- of these lines (1--1 and/or 2--2) could be one side of an equilateral triangle(s) -- a candidate for the Tilted Equilateral Triangle that forms the basis of the Grail Geometry.  In further exploration, I tried LINE 3--3 at 60 degrees to both those lines (see explanation at the "Grail Geometry" button if this isn't self-explanatory).  I was intrigued by the triangular red area between the horse's tail and leg -- so I drew LINE 3--3 tangent to it at "H" and -- Bingo again! LINE 3--3 just clips the edge of the hat at "h"!  With these three lines amply confirmed, we are on our way to drawing a third line to close a tilted triangle.

STEP 2.  I have removed some of the lines and labels of STEP 1 for clarity.  To search for the appropriate tilted equilateral triangle, I have drawn exploratory LINE 4--4 below.

I drew LINE 4--4 from the palm of the outstretched hand down through the tip of the black "V" (within the box labeled "b"), down and running tangent to the shoulder of the reclining horse (at ellipse "I"), and down across the nondescript feature at "J" (is it a flattened corpse in a shroud?).  This line makes a 60 degree angle with each of the other three lines so that the resulting triangles a-b-c  and d-e-c  are equilateral -- and candidates for the principal Tilted Triangle.

Now that these triangles are superimposed on the painting, we see that the mass of people is roughly triangular. Note how the Bull stands apart from the massed people, the horse, and the rider. Yet the lance aimed at the Bull's Eye ties the strangely distracted Bull to the action.  One benefit of searching a painting for lines that an artist may have used to guide his (or her) composition, is the necessity for extended periods of study of every detail in the work.  This prolonged concentration helps me to enter -- in my imagination -- the mind of the painter as he planned and executed his composition. It provides me with clues to the message the artist may have been hoping to communicate.  In this case, I see Goya proclaiming the nobility of the Bull and the wanton, cowardly brutality of the mob -- the act of hiding behind (and upon) the equally noble horse.  Goya comments here on both noble and contemptible beasts.  Fiesta Brava indeed!

 The Grail Geometry is clearly present.  It remains for the appropriate Tilted Square to be associated with one of the tilted triangles. (Note-- if a line is drawn co-linear with the lance held by the standing man, it will intersect the common vertex point "c" of both triangles. A different equilateral triangle may be built in this way. This solves the mystery of why this lance is painted not quite at an angle of 60 degrees with LINES 1--1 and 2--2 (see the exhibit and explanation for STEP 1).  It is not crucial to my argument to present this other triangle, which, nonetheless, is testimony to Goya's complete mastery of the intricacies of the Grail geometry.

STEP 3. The search for the best third line with which to complete the tilted equilateral triangle is sometimes the most difficult one.  While there are any number of possibilities, the best third side is the one that, when the rules of the Grail Geometry are applied, yields the tilted square whose diagonals intersect on the most likely symbolic feature in the painting.

In this analysis, I was obliged, finally, to "work backwards" from the most likely symbolic feature in the painting.  It became clear to me, after much study, that -- like the source of something flowing in the Vermeer analyses of "The Milkmaid" and "Young Woman With a Water Pitcher" -- the gushing flow of blood depicted as coming from the horse's belly would be what Goya would position at the "X marks the Spot", the "PX", the intersection of the diagonals of the proper tilted square.

With this hunch in mind, (see below) I drew (according to rules) one of the exploratory diagonal lines from the point designated "a", enclosed by a little square.  This diagonal line is labeled a--N, and it makes a 15 degree angle with the original LINE 3--3 (refer back to the exhibit for STEP 1 and the GRAIL GEOMETRY button, if necessary). Note that this diagonal line a--N passes directly through the wound in the horse's belly (just in front of the rider's ankle and foot) from which the blood gushes down.  The other diagonal, of course at right angles with a--N, is the diagonal line b--O.

Besides being forced to intersect where I wanted them to intersect (labeled as point "PX" in the exhibit below), do I have any features that confirm the placement of these diagonals? Yes.  Diagonal LINE a--N has positioned the horse's right front hoof for Goya, as well as a corner of the saddle-blanket.  Diagonal LINE b--O has positioned the horse's mouth and nostril, as well as the position of the horse's right back leg, and fixed the position of the front of the rider's boot.

Equipped with the proposed diagonal lines, how do we establish the Tilted Square that is the proper one for them? This became rather easy -- certainly because Goya used the Grail Geometry! -- and it's there for the discovery!  The top corner of the Tilted Square is labeled "a" and is within the little square marker there.  The all-important side of the square is simply found by drawing a line from "a" to "b" -- "b" being the point already established as the bottom of the "vee"(the strange black V) within the little square marker.

Once we have settled on the side a--b for the Tilted Square, the other three sides are readily drawn, yielding the Grail Geometry's Tilted Square labeled here a-b-N-O (see above and below).  Can I point to other confirming features to substantiate and justify the assertion that Goya used this Tilted Square to guide his composition? Yes.

 Look at LINE b--N -- it established for Goya the position of the Bull's head, and it clips the border of the hat lying on the ground, where Goya painted it.  Look at LINE a--O -- it established the position of the hand holding the upright lance and the position of the hair and back of the figure behind the horse.  Continued below . . .

Now for the Tilted Triangle associated with the Tilted Square. According to the rules (see under the "GRAIL GEOMETRY" button), the top side of the square, LINE a--b, is procured from the LINE a--a' (see image above).  LINE a--a' is drawn, according to the rules, at a 15 degree angle to, and the same length as the top side, LINE a--b. Then LINE a'--b'--c' is drawn from a', at an angle of 45 degrees to LINE a--a', always obeying the Grail Geometry. This defines the Tilted Triangle we have been searching for -- it is triangle a-b'-c' (and not triangle a-b-c, as initially guessed!)

Do we find any features that show that Goya did indeed employ Tilted Triangle a-b'-c'? Yes, not only did LINE a'--b'--c' position the head of the figure with the outstretched hand (4), but if extended this line locates the lower left corner of the picture. The important justification, of course, is that the Tilted Triangle I have revealed gives rise to the "X marks the Spot" on the horse's wound, symbolically gushing forth blood.

 Can I point to an interdependent painted feature that additionally justifies the choice of side b'--c' for the principal Tilted Equilateral Triangle of the Grail Geometry? Yes, indeed.  Look at the remarkable and unobtrusive Little Black Registration Marker (vertical arrow below) painted by Goya on the right border of the blood on the horse's belly! (Just below where the diagonal intersects side b'--c', to the left of the PX circle.)  Never would have noticed that!).  It permits him to position again and again the overlay drawings that carry the geometric lines as the painting progresses.  Goya was in his late seventies when he painted this, and he may have grown a little careless about covering his geometric tracks (this one and the "vee"). Not so with Vermeer -- no obvious trace of such markers in his work -- but I would suggest that such might be found with modern technology.

So -- did Goya include "vee" and "MARKER" as registration anchors to facilitate the development of this composition to conform to the lines, nodes, and angles of the Grail Geometry? Yes, indeed -- I think it's been proved beyond the shadow of a doubt by now.

STEP 4. The Tilted Hexagram.  Now that the Tilted Triangle has been identified, it is a simple matter to add another equilateral triangle to make a regular hexagram. I have exhibited it below, based on the geometry identified in STEPS 1, 2, and 3:

I have circled with dashed lines two spots where it is obvious that the Tilted Hexagram of the Grail Geometry, together with its diagonals, has provided anchor points for Goya to position the following painted features: the border of the rider's backside and the position of the rider's lance. The position of the other lance -- the one held upright -- is governed by the intersection of the left side of the Tilted Square (a-b-N-O) with the line of the rider's lance (small dashed-line circle). Note how the diagonal that starts at c' provides the position for the back side of the rider's boot and for his elbow. The diagonal that starts at "a" positions the bent knee of the reclining horse. The diagonal starting at b' has already been mentioned in connection with the two circles it crosses. Goya has exhibited great skill in making this composition register with the Grail Geometry.

Note how the bottom side of the added triangle, when extended down to the right provides a position line for the hat on the ground and for the front hooves of the Bull. We see that the Bull's eye is positioned by the line of the rider's lance, which, in turn, has been positioned by the hexagram and its diagonals. Goya is in firm control of the Grail Geometry -- as will be further exhibited by another hexagram which I have identified below:

The exhibit below demonstrates what a magnificent geometer that Goya really was. He has employed another larger hexagram, built on the triangle a-b-c, as well as another tilted triangle -- the dashed-line triangle A-c'-c'' -- that has positioned the upright lance (whose top is at A).

So -- what appears to be a rather crude composition for "Bullfight" is in reality a tour-de-force of geometric composition! The exhibit of the analysis above should be self-explanatory (albeit quite complex) to those who are fluent in the language of geometry. For those who find any further words of description tedious -- I say just gaze at the diagram and enjoy the visual feast. Isn't that Bull's Eye positioned in a masterly way, making Goya's message as fresh and effective as it was when he painted it in 1824 at age seventy-eight (78) !

  But what of the "supernumerary" features, half-hidden in plain view, such as are typical in Vermeer's work and in the one El Greco analyzed thus far? Is Goya playing the jester?

What's this nebulosity -- this vague and formless mass -- that Goya has painted in the upper right quadrant of his masterpiece?  Could it be a baby seal?  Or is it the long-nosed head of the devil wearing the baby sealskin around his neck? You say you thought it was a dead horse -- the first victim of the Brave Bull?  Now really! Goya could paint a much more convincing dead horse! -- if he wanted to . . .  There are many other bizarre faces half-hidden in this work. Some are unable to see even one!

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