Geometry versus math with respect to incommensurate numbers.
"Medieval masons insisted that their whole craft was based on the 'art and science of geometry'. It has been the purpose of this paper to reconstruct the character and content of the geometrical knowledge of medieval master masons from the few literary remains of the masons themselves. As reconstructed from these writings, this geometry scarcely (p. 169) resembles either the classical geometry of Euclid and Archimedes, or the medieval treatise on practica geometriae. Mathematically speaking, it was simple in the extreme; once it is recognized that there was virtually no Euclidean-type reasoning involved, the way is cleared for understanding the kind of geometrical thinking which the masons did employ. This non-mathematical technique I have labeled constructive geometry, to indicate the masons’ concern with the construction and manipulation of geometrical forms. It becomes evident that the 'art of geometry' for medieval masons meant the ability to perceive design and building problems in terms of a few basic geometrical figures which could be manipulated through a series of carefully prescribed steps to produce the points, lines and curves needed for the solution of the problems."1 FOOTNOTE: Roger Herz-Fischler, “Proportions in the Architecture Curriculum”, Nexus Network Journal 3, no. 2, (2001): 163-185.
ALSO SEE Shelby, "The Geometrical Knowledge of Medieval Master Masons", 1072, p. 420. (This is a PDF.)
The proportion of 1 : √2 was used to create unity in a cathedral begun in 1096. SEE Fernie, "The Ground Plan of Norwich Cathedral and the Square Root of Two", 1997.
Primary Sources: Villard, Roriczer, Schmuttermayer, Lechler.
Measurements are always approximate, especially when the figure being constructed is derived from an irrational number like pi or the square root of 2. Irrational numbers are numbers that cannot be written as a fraction using integers. The decimal approximation of an irrational number never ends, and there are no repeating patterns of digits.
Should I include the construction of a square? It's in the Sacred Art Appendix (formerly Constructions).
Villard de Honnecourt (1200-1250)
The drawing, below, shows that Gothic-era artists used grids of squares, presumably to transfer drawings. Restate the following: Villard de Honnecourt was a 13th-century artist from Picardy in northern France. He is known to history only through a surviving portfolio or "sketchbook" containing about 250 drawings and designs of a wide variety of subjects. The portfolio of Villard de Honnecourt, preserved in the Bibliothèque nationale de France (MS Fr. 19093), consists of 33 sheets of parchment containing about 250 drawings. Villard's portfolio ". . . appears to be a model-book, with a wide range of religious and secular figures suitable for sculpture, and architectural plans, elevations and details, ecclesiastical objects and mechanical devices, with copious annotations. Other subjects such as animals and human figures also appear." ALSO: SACRED GEOMETRY figures (See Hiscock, The Symbol at Your Door, p. 348). One drawing shows pentagon construction method based on rotating squares that is inaccurate (See Hiscock, The Symbol at Your Door, p. 222).
The Stonemasons' Geometry
"It is apparent from the varied professional skills and activities of Roriczer, Schmuttermayer, Lechler, and Dürer that the lines of demarcation between the crafts, as represented in the institutional structure of the craft guilds and fraternities, did not keep individuals from crossing over these boundaries in their actual work and in their interest in the technical rules of crafts related to their own. . . . One should remember that Dürer’s father was a well-established goldsmith in Nüremberg, that Dürer himself worked for awhile in his father’s workshop before he was apprenticed to the Nüremberg painter, Michael Wolgemut; and that Dürer indicated clearly in his books that he was familiar with the principles and practices of the masons’ craft." Shelby, Gothic Design Techniques, p. 56.
". . . it was the mason’s craft which developed the technique of geometric constructions that had come to be used by other craftsmen as well. Again Dürer made this point more explicitly in his Four Books on Human Proportions, where he explained the relationship of that book to his earlier Instruction on Measurement: ‘So that my instructions might be better understood, I have previously sent forth a book on measurement dealing with lines, planes, bodies, etc., without which my present system cannot be fundamentally understood. Therefore if anyone wants to understand this art, it is necessary that he be well-instructed beforehand and understand how all things should be set in and drawn out of the basic figure, just as the artistic masons have [done it] in daily use. Without that he will not be able to comprehend my instructions.’" FOOTNOTE: This quote is from Dürer’s Vier bücher von menschlicher Proportion, vol. Aii. FOOTNOTE: The formula for calculating the side of a square is b = (√2/2)a, where b is the current, smaller square in the series of rotating squares and a is the previous, larger square. Shelby, Gothic Design Techniques, p. 58. Is this Volume II, page v?
"By following the steps prescribed by Roriczer, any mason could, with only his compass and straightedge, construct a solution to the problem, without knowing either the Archimedean theorem or the proofs pertaining to this theorem. The Geometria deutsch thus clearly reveals how medieval masons approached geometrical problems which would seem to require some mathematical calculations, yet they managed to avoid those calculations through step-by-step manipulations of their working tools." Shelby, Gothic Design Techniques, p. 58.
"Long before scaled architectural drawings and shop drawings became a standard part of building construction, medieval masons acquired the ability to translate designs from the head of the master mason to the hands of those actually carving and setting the stones. It was this technique of translating designs which concerned Roriczer and Schmuttermayer." Shelby, Gothic Design Techniques, p. 78.
" . . . the ‘art of geometry’ as the masons understood and practiced it was not the geometry of mathematicians. Rather it was the masons’ way of visualizing potential architectural forms within certain geometrical figures." Shelby, Gothic Design Techniques, pp. 78-79.
Description of the division of a square into thirds, from Hiscock, The Symbol at Your Door, p. 206: "If a square is halved vertically to produce two double squares, the diagonals of the square and each double square intersect each other at a distance equal to one third of the side of the original square."
Re: the technique of setting out the ground plan and elevation of a pinnacle. "Roriczer began by drawing a square. Inside this square he inscribed a second square at a forty-five degree angle to the first, and inside the second square he inscribed a third square in the same manner. He then rotated the second square to make the sides of all three squares parallel. Having obtained the basic outline of the ground plan of the pinnacle by this manipulation of three squares, he proceeded to determine the details of both the plan and the elevation in a step-by-step elaboration of this manipulative technique. The entire process required 234 separate steps, which he illustrated with eighteen figures. In short, this was an application of the technique of quadrature – the famous ad quadratum of the medieval masons – whereby the elevation was derived from the square, the basic geometrical figure of the ground plan. But it cannot be too strongly emphasized that the entire operation consisted simply of the manipulation of geometrical figures through a long series of carefully prescribed steps, and that it was quite devoid of mathematical formulas and calculations." Shelby, Gothic Design Techniques, p. 66.
The rotated squares have a proportional relationship – each square's area is exactly half that of the square in which it is inscribed. This relationship can be visualized as follows: Notice that square ABCD is comprised of square EFGH plus the four right triangles that surround it. Mentally combining two of the triangles into a square reveals that, together, they take up one-quarter of the area of ABCD. So all four triangles represent half of the area of ABCD. The rotated square EFGH must therefore represent the other half of ABCD. To prove to yourself that this is so, notice that the vertical and horizontal lines that bisect square ABCD also divide square EFGH into four right triangles, and that these triangles are identical to the triangles that surround it.
The rotated squares are mathematically related by the factor √2, which is an irrational number – i.e., a number that continues past the decimal point in a seemingly random manner to infinity. But the squares' geometric relationship is easy to express and understand: Each square is half as large as the next-larger square, and twice as large as the next-smaller square.
Hanns Schmuttermayer (active 1487; d. after 1518)
Introduce Schmuttermayer. The following constructions are from Fialenbüchlein (Booklet on Pinnacles), 1489. The only surviving original copy is in the Germanisches Nationalmuseum in Nüremberg.
Medieval stonemasons might not have been able to do the mathematical calculation, but they could have made templates from rotated squares. So why were they using approximation?
Mathes Roriczer (1440-1493)
Introduce Roriczer. The following constructions are from Geometria Deutsch, c. 1497.
Is the set square a template that stonemasons would have used? Any point that you put a point in a semicircle in addition to the endpoints (the points of the diameter) gives a right angle.
Next is a method for approximating the circumference of a circle. Describe it. If the circle's diameter is 3, the construction yields a circumference of 9.429. The mathematical formula (2πr) yields a circumference of 9.425.
This provides a close approximation. Pi was also an approximation, until calculus.
Accurate, still in use.
A vesica piscis is constructed, then . . . This is accurate. The value of this construction is that you don't have to change the width of the compass legs. More importantly, you know what the length of a side is going to be in advance. (Regular polygons have equal sides.)
This is an approximation. The heptagon cannot be constructed with dividers (compass) and straightedge, the stonemasons' tools. It is the smallest regular polygon with this property.
De inquisicione capacitatis figurarum, late 14th or early 15th century
De inquisicione capacitatis figurarum is another fifteenth century treatise on constructive geometry. The author is unknown. Like Roriczer's Geometria deutsch, it contains instructions for finding the square that has the same area as a given triangle.only pertains to equilateral triangles. The construction in De inquisicione capacitatis figurarum can begin with any triangle. It yields an exact result, not an approximation.
Draw a rectangle around the triangle the sides of which are equal to its base and height. Bisect the rectangle horizontally, then extend the triangle's base line on one side by half of its height – the dotted arc shows how this is accomplished using a compass. Next, draw a semicircle centered on the extended base line. Finally, draw a vertical line from the end of triangle's base line on the extended side up to the semicircle. This line measures the side length of the square that has the same area as the triangle.
Rodrigo Gil de Hontañón (1500-1577
The significance of Gil's plan is its use of overlapping squares for setting out a building. According to Hiscock, this is not quadrature, and it is unique among his published architectural plans. The point of the overlapping squares, it seems, was to get around the builders' inability to draw large circles. Utilizing only squares to set it out, the building constitutes an amazingly good approximation of a golden rectangle. The sanctuary is a root 2 rectangle, for which Gil has an unique construction method that involves swinging a couple of relatively small arcs on the outside in order to position a square. Hiscock describes the sanctuary an octagon, which it is not. The ellipse in the plan appears to be a symbol for using intersecting arcs (circles) to locate points. The following information is from Hiscock's The Symbol at Your Door, p. 380. "Some of [Rodrigo Gil's design methods for plans and sections of different types of church] became incorporated, somewhat haphazardly, into the Compendio de arquitectura y simetria de los templos by Simon Garcia in 1681, a commentary on which appeared in a modern edition. One plan in particular is of interest because it has dimensions in unidentified measures inscribed on it, and it consists of an aisled basilica, with a square crossing, three nave bays, and a sanctuary enclosed by five sides of an octagon. . . . All the bays in the main body of the church represent the musical consonances: 1 : 1 for the crossing, 1 : 2 for its side bays, 2 : 3 for the aisle bays, 3 : 4 for the nave bays. Moreover, the whole is a golden rectangle, exactly based on the Fibonacci numbers 8 : 13, tenfold, at a time when the golden section was being celebrated in Renaissance literature as the divine proportion. Intriguingly, no dimension is given for the depth of the sanctuary. Apart from this missing dimension, the whole plan could be laid out either in the tracing-house or on site using the dimensions provided without the aid of geometry, but there is no way of similarly setting out the sanctuary. Since geometry is needed for this, it follows that the whole layout should be capable of being set out geometrically and this can be achieved by using only the square in two stages, the first comprising just four steps for the body of the church, the second for setting out the sanctuary in six steps. The first stage starts with the square crossing, followed by a series of overlapping squares, in a system not to be confused with quadrature."
The plan is constructed is as follows: The width of the church is 2 squares. The width of the nave is one of these squares centered between the double squares. The aisles, therefore, are half a square wide. The length of the church is determined as follows. Two squares, each three-quarters the width of the church, are drawn below the double squares. The area of overlap is the nave. Two identical squares are drawn below the overlapping squares, bisecting them vertically. In other words, the first set of overlapping squares overlaps the second set. The result is 3 identical nave sections. A church in the shape of a golden rectangle can be set out in this manner without using a compass. Gil's plan instructs the builders to use the center square at the top to set out the sanctuary as follows: Place a compass point on one of its upper corners and draw an arc from the square's center to its upper edge. Between that point and the corner opposite, draw a small square in the corner of the square. Then, beginning at the base of the small square, lay out a square the same size as the center square. The sanctuary thus created is a root 2 rectangle.
1 Note 1.
2 Note 2.