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The calculus of Gothic architectureMichael R. Hubera

a Department of Mathematics and Computer Science, Muhlenberg College, Allentown Pennsylvania18104, USA

To cite this Article Huber, Michael R.(2009) 'The calculus of Gothic architecture', Journal of Mathematics and the Arts, 3:3, 147 — 153To link to this Article: DOI: 10.1080/17513470903150042URL: http://dx.doi.org/10.1080/17513470903150042

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Journal of Mathematics and the ArtsVol. 3, No. 3, September 2009, 147–153

The calculus of Gothic architecture

Michael R. Huber*

Department of Mathematics and Computer Science, Muhlenberg College, 2400 Chew Street,Allentown Pennsylvania 18104, USA

(Received 20 February 2009; final version received 14 May 2009)

When you look at the facade of the Cathedral of Notre Dame in Paris, what do you see? The rectangular towers,classical Gothic arches, massive domes and stained glass rose windows of this famous cathedral offer excellentexamples of areas for students to calculate via integration. Most modern calculus texts use simple examplesof finding the area of an ellipse or circle when incorporating trigonometric substitution. This article describes anapproach for instructors of single variable integral calculus courses in calculating the areas and volumesof Gothic structures which have incorporated the quinto acuto arch. Further, students do not have to travelfar to find examples of Gothic architecture near their own campuses. Examples and sample calculations areprovided.

Keywords: Gothic architecture; quinto acuto arch; trigonometric substitution; double-dome cupola; integralcalculus

AMS Subject Classification: 00–01

1. Introduction

Are you looking for a good example to apply integra-tion to calculate areas and volumes? Then look at, notout of, the window. In particular, locate an example ofa Gothic window on campus or in your neighbourhoodand ask your students to calculate the area enclosedby the window. When you look at the facade of theCathedral of Notre Dame in Paris, what do you see?The rectangular towers, classical Gothic arches andstained glass rose windows of this famous cathedraloffer excellent examples of areas for students tocalculate via integration (see Figure 1). In addition,these area calculations provide opportunities to applytrigonometric substitution in evaluating the areaintegrals. In an introductory single variable integralcalculus course, students are often required to learnseveral methods of finding antiderivatives and definiteintegrals. These methods include integration by sub-stitution, integration by parts, memorizing tables ofintegrals, learning algebraic identities, trigonometricsubstitution and numerical techniques to approximatedefinite integrals. Most calculus texts use simpleexamples of finding the area of an ellipse or circlewhen incorporating trig substitution [6]. Get yourstudents excited about real applications and show themGothic architecture!

2. Types of Gothic arches

Gothic architecture found its beginnings in France,originating ‘around 1140 in the small kingdom, whichalready bore the name Francia, that occupied the regionbetween Compiegne and Bourges, and that had Paris,the royal city, as its capital’ [9]. The Abbott Suger of St.-Denis (ca 1081–1151), became the initiator of the newspecial buildings in the Gothic art. Suger is often calledthe ‘creator of Gothic,’ as he combined the elements ofBurgundian architecture (with its pointed arch) andNorman architecture (with its ribbed vaults). The resultwas immensely popular, and between 1180 and 1270,about 80 cathedrals were built in France alone.

The defining characteristic of Gothic architecture isthe pointed or ogival arch. This offered flexibilityto the architect, allowing more light into the structurewith the vaulted windows and ceilings than the older,Romanesque arches had (Romanesque arches aresemi-circular in shape). Villard de Honnecourt, athirteenth-century Picard architect in northernFrance, was one of the first to discuss pointed archesand he is believed to be the first to use the term ‘ogive’[2]. Villard authored a manual which contains a set ofdiagrams of masonry techniques. These diagrams weredrawn and annotated by an anonymous followerknown only as ‘Master 2,’ some time around the year

*Email: [email protected]

ISSN 1751–3472 print/ISSN 1751–3480 online

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1250. In his manuscript, Villard describes ogive orlancet arches. As an aside, in the field of ballistics, anogive is a pointed, curved surface used to form theapproximately streamlined nose of a projectile. Thetraditional ogive is formed by creating a surface ofrevolution with a curve that resembles what we todaycall a Gothic arch. Villard’s own description is givenlater.

Before Gothic structures gained popularity, theRomanesque period of architecture dominated thecountrysides, characterized by round arches (insteadof pointed arches), wall mass (instead of flyingbuttresses) and walls with niches (instead of opentracery). The Gothic cathedral emerged as a new form,reaching higher and higher to the heavens. The stainedglass of the ornate rose windows seemed to absorbthe walls, allowing the walls to dissolve into enormouscolourful windows.

The Brick Industry Association defines a Gothicarch as ‘an arch with relatively large rise-to-span ratio,whose sides consist of circles, the centres of which areat the level of the spring line’ [3,4]. Figure 2 showsa typical Gothic arch with the spring line, rise, spanand arch soffit displayed. Gothic arches are usuallyclassified into three groups: as a lancet arch, equilateralarch or drop arch, depending upon whether thespacings of the circles’ centres are, respectively, more

than, equal to, or less than the clear span. The spring

line is the horizontal line which intersects the springing

(where the arch begins). The spring line encompasses

the arch’s span, which is the horizontal dimension

between abutments (the total width of the arch from

A to B). The rise is the maximum height (from O to C)

of the arch soffit above the level of its spring line

(usually at the centre of the arch).Lancet arches take their name from the shape � the

tip of a lance. Lancet arches are often grouped

together, usually as a cluster of three or five, often

enclosed under a single external arch. They are seldom

seen by themselves as a single window. Lancet open-

ings may be very narrow and steeply pointed. A fine

example of a structure with lancet arches is England’s

York Cathedral, which has a series of lancet windows,

each 50 feet high and still containing ancient glass.

Looking back at Figure 2, the centre of the arch which

runs from A to C would be on the spring line to the

right of the point B (outside the span).Many Gothic openings are based upon the equilat-

eral arch form, a second type of Gothic arch. In other

words, the radius of the arch is exactly the width of the

opening and the centre of each arch coincides with

the point from which the opposite arch springs. This

makes the arch higher in relation to its width than

a semi-circular arch, which is exactly half as high as it

is wide. The radius is equal to the distance from A to B

in Figure 2. The equilateral arch lends itself to filling

with tracery of simple equilateral, circular and semi-

circular forms. The type of tracery that evolved to fill

these spaces is known as the ‘geometric decorated

Gothic’ and is typified at many French cathedrals,

Figure 1. The Cathedral of Notre Dame in Paris (photo byauthor).

Figure 2. The Gothic arch (photo by author).

148 M.R. Huber


specifically the Cathedral of Notre Dame in Paris andthe Cathedral at Reims. Windows of complex designand of three or more vertical sections are often designedby overlapping two or more equilateral arches.

A third variation of the Gothic arch is the droparch. The drop arch, also discussed in the literatureunder names such as ‘depressed’ or ‘four-centred’ arch,is much wider than its height and gives the visual effectof having been flattened under pressure. Its structure isachieved by joining two arcs which rise steeply fromeach springing point on a small radius and then turninto two arches with a wide radius and much lowerspringing point. This type of arch, when employed asa window opening, lends itself to very wide spaces,provided it is adequately supported by many narrowvertical shafts. Many Gothic structures use the droparch as part of the solid wall, and the drop arch isfilled with windows containing lancet arches. Lookat Figure 1 again: on either side of the rose window,notice the employment of drop arches which containsteeper-arched windows within them. Looking back atFigure 2, the centre of the arch which runs from A to Cwould be on the spring line to the right of the point Obut to the left of B (within the span).

Although the Brick Industry Association only givesthree, there is actually a fourth type of arch, called theflamboyant arch. It is formed by joining four points,the upper part of each main arc turning upwards intoa smaller arc and meeting at a sharp, flame-like point.These arches are mainly utilized for window traceryand surface decoration. The form is structurally weakand has very rarely been used for large openings exceptwhen contained within a larger and more stable arch.It is not employed at all for vaulting. Some of the mostfamous traceried windows of Europe employ flamboy-ant arches and can be seen at St. Stephen’s Cathedralin Vienna, Sainte Chapelle in Paris and the MilanCathedral in Italy.

3. The Quinto Acuto arch and calculation of area

In the thirteenth century manner, a more traditionalapproach arose, which uses the convention that each arcof a Gothic arch has a radius of one less unit of lengththan the span. Known as a quinto acuto or fifth-pointarch, it has a radius-to-span ratio of four-fifths. Thistype of construction falls into the drop arch category.For example, if the span is 5m, the radius of each arc is4m. Figure 3 shows the thirteenth century fifth-pointarch sketch from Villard [2]. Villard labelled his archwith positive integers. You can easily see that the span is5 units and the radius of the arch is 4 units. The quintoacuto arch has been used extensively in the architectureof the last millennium, although there is not much in theliterature on the mathematics behind it.

The rise-to-span ratios are easily calculated. Given

a quinto acuto arch with a span of 5m, at the centre of

the arch, the rise is 3.7081m, giving a rise-to-span ratio

of 0.7416. By comparison, a rounded (Romanesque)

arch has a rise-to-span ratio of 0.5 (remember, it is

really a semi-circle).Once again take a look at Figure 1, which shows

the west entrance to the Cathedral of Notre Dame.

Suppose you wanted to calculate the exact area under

the arched doorway (taking into account the innermost

arch only). This classic Gothic architecture was created

by employing the quinto acuto arch.Basically, the quinto acuto arch is composed of two

quarter-circles pressed together to provide relief from

thrust, the weight of the arch above it. This is critical inlarge stained-glass windows, arched doors or domes.

The centres of the arcs are inside the span of the arch

and not directly beneath the tip of the arch. Suppose

the horizontal distance from the centre of the arch

(given by the point (x, y)¼ (0, 0), or the origin) to either

side of the bottom edge of the arch is m metres. This

gives an entrance of 2m metres across. It also means

that the centre of each arc is 2m=5 metres from the

opposite end (see Figure 4). So, the arc starting on

the left side and going upwards and right is centred

at the point (3m=5, 0) and has radius r¼ 8m=5 (found

by adding mþ 3m=5). Its opposite arc is centred at

(�3m=5, 0) and also has radius r. Recall the equation

of a circle as

ðx� x0Þ2þ ð y� y0Þ

2¼ r2,

Figure 3. Villard’s quinto acuto arch [2]. This is a reproduc-tion of the fifth-point arch drawn by Robert Branner(Figure 1: Traditional third- and fifth-point arches, num-bered according to Brutails) from Villard de Honnecourt,Archimedes, and Chartres, J. Soc. Arch. Hist., Vol. 19(3) 1960(RB23), reprinted here with the permission of ShirleyP. Branner.

Journal of Mathematics and the Arts 149


where (x0, y0) is the centre of the circle, and r is the

circle’s radius. Take y0¼ 0 to enter the circle on the

x-axis. Solving for y, we find that

y ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 � ðx� x0Þ

2

q:

We seek to find the area under the curve, above the line

y¼ 0. As a note, we could find the entire area under the

arch by adding any rectangular area of the doors.

Ignoring the doors and just treating the area under the

arch and above the doors, the function y is composed

of two pieces of a quarter circle that are pushed

together before each piece reaches its natural apex.

To determine the area under the arch, we only need

to determine the area under one arc and then double

the area. The area under one arc can be found by

evaluating

Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 � ðx� x0Þ

2

qdx, ð1Þ

over the domain of the arch. This integration requires

a few substitutions. Starting with Equation (1),

let u¼ x � x0. Thendudx ¼ 1, or du¼ dx. Equation (1)

becomes

Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 � u2p

du:

Next, using trig substitution, let u¼ r sin(�). This

requires that du ¼ r cosð�Þd�: Substituting, we have

r

Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 � r2 sin2ð�Þ

qcosð�Þd�,

which can be simplified to

r2Z

cos2ð�Þd�:

Using the half-angle formula, cos2ð�Þ ¼ 12 ð1þ cosð2�ÞÞ,

we integrate

r2

2

Z1þ cosð2�Þð Þd�,

which yields

r2

2� þ

sinð2�Þ

2

� �:

Equation (1) can be solved as

Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 � ðx� x0Þ

2

qdx ¼

r2

42� þ sinð2�Þ½ �: ð2Þ

This would then be evaluated over the limits ofintegration, keeping in mind that

� ¼ arcsinx� x0

r

� �:

4. A sample calculation

Look again at Figure 4. Suppose the horizontaldistance from the centre of the arch (given by thepoint (x, y) ¼ (0, 0), or the origin) to either side of thebottom edge of the arch is 2.5m (m¼ 2.5). This givesan entrance of 5m across. It also means that the centreof each arc is 1.5m from the centre of the arch. So,the arc starting on the left side and going up and to theright is centred at the point (1.5, 0) and has radius 4.Its opposite arc is centred at (�1.5, 0) and also hasradius 4. This gives the following function:

y ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi42 � ðx� 1:5Þ2

q, �2:5 � x � 0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi42 � ðxþ 1:5Þ2

q, 0 � x � 2:5

:

8>>><>>>:

ð3Þ

We will consider the left-hand arc (the first piece ofy above). So, x0¼ 1.5 and r¼ 4. The limits of integra-tion are a ¼ �2:5 and b¼ 0. The area under the left-hand arc becomes

Z0

�2:5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi42 � ðx� 1:5Þ2

qdx: ð4Þ

Figure 4. The radius and the quinto acuto arch (photo byauthor).

150 M.R. Huber


Using Equation (2), the solution to Equation (4)becomes

42

42� þ sinð2�Þ½ �

��0

�2:5

,

with

� ¼ arcsinx� 2:5

4

� :

The exact area under one arc in the arch is then4�� 8 arcsinð38Þ � ð

38Þ

ffiffiffiffiffi55p

, which is approximately6.7101m2. After doubling, the area under the entirearch (and above the spring line) is 13.4202m2. If thisarch were above a set of doors, we would add this areato the area of the doorway, providing the total areaunder the arch. Integrating the right-hand arc over0 � x � 2:5 gives the same answer.

An exercise used in my integral calculus courseis to have students use numerical methods to app-roximate the integral. Using the area integral andDx ¼ 0:1, we get a left-hand sum of 6.5065, a right-hand sum of 6.8773 and a trapezoidal rule sum of6.6919. Using Dx ¼ 0:01, we get a left-hand sum of6.6910, a right-hand sum of 6.7281 and a trapezoidalrule sum of 6.7095, which is a good approximationto the area under one arc obtained above.

Muhlenberg College has a beautiful chapel (EgnerMemorial Chapel � see Figure 5) which is in thetraditional Gothic design. In my integral calculuscourse, I send students to look at its front doorwayand side windows, which are over 20 feet high and4 feet wide. The doorways and windows of EgnerChapel offer a variety of arches for the students tostudy. How much area is determined by the arches andsides of each window? This area must be determinedbefore the stained glass windows and tracery can bedesigned and crafted.

5. What about domes and volumes?

Many domes adhere to what is known as a ‘double-dome’ concept. The hemispherical shape seen from theinside is capped by a revolved quinto acuto dome on theoutside. Michelangelo’s famous dome on Saint Peter’sCathedral in the Vatican was designed to be such anexample [1]. However, construction did not follow theoriginal design, and there is no reference that showsthe outside follows the quinto acuto shape while theinside is hemispherical. The architect of the UnitedStates Capitol in Washington, DC, also used thedouble-dome design. These domes have a shape whichexerts less thrust; Filippo Brunelleschi’s dome inFlorence offers another example [7]. A pointed domeexerts less thrust than a spherical dome. If the force

(weight) on top of the dome becomes too great, its

sides would tend to push outwards. The architects

therefore wrapped tension rings around the sides

of the dome. Similar to hoops on a barrel, the rings

keep the dome’s shape intact, preventing any outward

thrust.Let us compute the volume under a double-dome

cupola. In doing so, we will use the method of slices

and definite integration. First, consider the quinto

acuto arch rotated about the vertical y-axis at the

apex of the arch to form a solid. This creates the three-

dimensional dome. In computing the dome, we:(1) divide the solid region into small sections (vertical

slices) whose volume can be easily approximated;

(2) sum up all the slices to obtain a Riemann sum

which approximates the total volume; and (3) take the

limit as the number of slices tends to infinity. This

yields a definite integral of a single variable with theexact total volume of the solid (dome).

Each slice can be considered a disc of width Dx and

radius given by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 � ðx� x0Þ

2

q,

Figure 5. Egner Memorial Chapel (courtesy of MuhlenbergCollege). Reprinted with permission.



with r and x0 defined in Section 3. In the double-dome

scenario, there is also a hemispherical inner ceiling

visible to the public.For example, suppose the dome has an outside

diameter of 100m and an outside shape governed

by the quinto acuto arch. In addition, suppose that

inside the dome is a hemispherical ceiling open to

the public below, with a radius of 48m. This allows

for stairs to be built between the two sets of walls

so that visitors may climb to the top of the dome.

What is the volume of the space between the two

domes?From Figure 6, we see the outline of the dome with

the inner and outer domes denoted. The distance

from A to B is 100m. The dome is formed by rotating

the curve r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� x0Þ

2þ ð y� y0Þ

2q

around the y-axis

(the dashed line above the point O). We start with

x0¼ 30 and y0¼ 0. We will break the total volume into

a series of thin vertical slices. The volume of a typical

slice is

V ¼ �R2Dx,

where R is the vertical distance (radius) from the y-axis

to the outer quinto acuto curve. In this example, the

radius of the quinto acuto curve is four-fifths of thediameter, or 80m. So, y ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 � ðx� 30Þ2

q, or

y ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi802 � ðx� 30Þ2

q¼ R:

Therefore,

V ¼ �


q� 2

Dx: ð5Þ

We sum the slices from the edge of the dome to its centre(in this case, from x ¼ �50 to x¼ 0m). Taking the limitas Dx! 0 will give us the definite integral where

V ¼ �

Z0

�50


q� 2

dx: ð6Þ

Evaluating Equation (6) using the technique ofSection 3 yields a total volume of 158, 333 1

3�� 497,418.4m3. In comparison, the hemisphere created witha radius of 48m gives a volume of V¼ 2=3�ð48Þ3 �231, 623.1m3. The difference between the two volumescan be easily calculated.

6. Classroom discussions

This technique may seem trivial at first glance.However, compared with the examples in most currentcalculus textbooks, this problem provides an alternativeto the common approach of teaching trigonometricsubstitution for calculating the area of an ellipse orcircle. Dealing with inverse sine functions is not trivialfor most students. These texts tell us that trigonometricsubstitution can be used to transform complex integrals,but there is usually no practical example for studentsto transform. In contrast, the Gothic architectureproblem provides a real-world context of the inte-gration technique. I have used this approach forthe past two years in the classroom with success.My students compare and contrast Gothic windowareas with Romanesque window areas in other build-ings on campus. I find that they appreciate putting thetechniques of integration in context of the ‘real world.’They also gain an appreciation for the trigonometricintegration technique when comparing their solutionsto numerical methods (as explained in Section 4).

In addition to calculating the areas and volumesof windows and domes, we can connect mathematicsto other features of the Gothic tradition. Most intro-ductory architecture courses deal with thrust and forcescarried by flying buttresses and Gothic spans.Regarding the scope of this article, the beautiful traceryin Gothic windows can offer numerous examples ofgeometry and area, using the ideas of accumulation and

Figure 6. The double-dome concept and St. Peter’s Basilica(photo by author). See insert for colour version of this figure.

152 M.R. Huber


calculus or without using calculus. Geometry and

trigonometry problems are abundant in almost all

examples of Gothic architecture. How can we calculate

the area of a rose window, for example? See [5,8] for

treatises on this subject.Finally, for further exploration, we can discuss

the reason that Gothic architecture was incorporated

in the first place, in the context of architecture. The

massive weight of the building itself was a concern for

architects, and we can model the forces in the mathe-

matics classroom. The principal forces acting upon

arches usually result from vertical dead and live loads

and wind loads. By incorporating a Gothic arch, the

load is spread to surrounding masonry, decreasing the

thrust above the window or doorway. For a sample

calculation of loads carried by an arch, see [3]. Thrust is

a force governed by Newton’s second and third laws

of motion. When a system exerts a mass (load) in one

direction the accelerated mass will cause a proportional

but opposite force (thrust) on that system. The calcu-

lations of load are beyond the scope of this article,

but I have had the discussions in my mathematical

modelling course, and my students have discovered that

the thrust is inversely proportional to the rise-to-span

ratio.

References

[1] G.C. Argan and B. Contardi, Michelangelo: Architect

M.L. Grayson and N. Harry, trans., Abrams Publishers,New York, 1993.

[2] R. Branner, Villard de Honnecourt, Archimedes, and

Chartres, J. Soc. Arch. Hist. 19(3) (1960), pp. 91–96.[3] The Brick Industry Association, Technical notes 31A:

Structural design of brick masonry arches, 1986. Available

at http://www.bia.org/BIA/technotes/t31a.htm. Accessedon 30 November 2007.

[4] The Brick Industry Association, Technical notes 31: Brickmasonry arches, 1995. Available at http://www.bia.org/

BIA/technotes/t31.htm. Accessed on 30 November 2007.[5] P. Calter, Squaring the Circle: Geometry in Art and

Architecture, Key Curriculum Press, Berkeley, CA, 2008.

[6] D. Hughes-Hallett, A. Gleason, W. McCallum,D.E. Flath, P.F. Lock, D.O. Lomen, D. Lovelock,B.G. Osgood, T.W. Tucker, D. Quinney, K. Rhea, and

J. Tecosky-Feldman, Calculus: Single Variable, JohnWiley & Sons, Hoboken, NJ, 2005.

[7] H. Saalman, Filippo Brunelleschi: The Cupola of Santa

Maria del Fiore, A. Zwemmer Ltd, London, 1980.[8] M. Sykes, A Sourcebook of Problems for Geometry Based

upon Industrial Design and Architectural Ornament, DaleSeymour Publications, Parsippany, NJ, 1900.

[9] R. Toman (ed.), Gothic: Architecture, Sculpture, Painting,Tandem Verlag, GmbH, China, 2007.



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