A spherical vault, whether a hemisphere or a sailvault, is formed of voussoirs. The voussoirs aretypically arranged along circular horizontal rows,

separated from each other by conical surfaces that are

more closed as we advance toward the pole andwhose vertex is always the center of the sphere.Within the same row the voussoirs are all egual orsimilar and they are separated by joints followingvertical planes which converge on the central axis. Inthis way each piece has a spherical inner surface, twotop and bottom bed joints that are cones, and twojoints that are vertical planes. In the unusual case that

the piece appears on the extrados (exterior) of thevault, then the exterior surface could be spherical as

well (figure 1).

Figure 1

Conventional voussoir for a spherical vault

The single coursed ashlar vault

Enrique Rabasa Díaz

We find many exceptions to this geometricaldivision, though most involve slight variations fromarranging the parallels and meridians in a differentspatial position (Fig. 2). In effect, the sail vaults -those

that are sectioned by four perimeter arches in order toadapt to a sguare foundation- on occasion present ageometry that in the 16th century was named «por

hiladas cuadradas» (for its sguare courses), and that,as its name indicates, arranges rows in a sguarehorizontal projection, each time smaller until closingthe surface. Each of the sides of these sguare coursesis a vertical row egual in all ways to those describedabove. So we have here a division also according toparallels and meridians, although with the axis placedhorizontally.1

The usual procedure in the 16th century for thecarving of the voussoirs of a spherical vault is as

follows. Over a block of stone from the guarry, onewould first carve the inner concave surface of thevault (intrados), the one which will remain visible,but not starting from its definitive edges, but ratherworking a portion of spherical surface, to afterwards

mark on it the real contour. This operation, thedefinition of the con tour of that face, would takeplace applying over the face a template, as is typical

in the work of stonecutting. However, as is known, aspherical surface is not developable, so theapplication of a flat template over the sphere is onlyapproximate. In any case, through this one wouldproceed to the carving of perimeter, bed and vertical

joint surfaces.

Proceedings of the First International Congress on Construction History, Madrid, 20th-24th January 2003, ed. S. Huerta, Madrid: I. Juan de Herrera, SEdHC, ETSAM, A. E. Benvenuto, COAM, F. Dragados, 2003.

1680 E. Rabasa

Figure 2Various arrangements of the joint according to parallels and meridians.

As we have said, this in vol ves different surfaces,two cones and two planes, but all can be consideredas surfaces generated by the movement of a radius ofthe sphere, following parallels in the case of theconical beds and meridians in the case of the planes.Therefore these four surfaces can be carved with easeemploying an instrument called a bevel. The bevel is

a type of sq uare that has a curved piece that adapts tothe concavity of the inner surface and another straightbranch that follows the direction of the bed jointtowards the center. Its use is common in the carvingof the voussoirs of the arches, as it allows proof of theorthogonality of the inner surface and the bed joint

and the correct execution of both. Its application tothe spherical voussoirs is a brilliant idea, as itdetermines the shape of more complicated pieces thanthose of an arch.

Finally, if it were necessary to carve the exterior,

one would mark its contour tracing a line parallel tothat of the intrados and confirming the carving with aconcave curved ruler or a concave bevel.

This procedure is of the type that we call direct, asopposed to those that pass through a previous

squared-up stone beforehand. In the 18th centuryvariations on the use of the template for the sphericalvoussoirs were proposed and also the procedure of thesquared-up stone was proposed, but there is no doubtthat in Spain in the 16th century the usual procedurewas that which Ijust explained. That is what is shownin the manuscript of Alonso de Vandelvira, and withmore detail in that of Alonso de Guardia, at the end ofthe century or beginning of the next, and the book of

Francois Derand or the manuscript of the CatalanJoseph Gelabert, in mid-17th century2

We now return to the template of the intrados.Made up of brass or cardboard, it should allow acertain curvature to adapt to, although imperfectly,the spherical surface where it is applied. It is in realitya small piece of an approximated sphere. A spherecan be developed approximately if we suppose that itis divided in segments according to meridians(Fig. 3), and we liken each one of these segments to a

portion of the cylinder. It is a procedure employed incartography. We can al so divide the sphere in

horizontal sections according to the parallels, andsubstitute, in each sector, the spherical surface for thetrunk of cone. These cones, more or less open, havetheir vertex always on the axis of the vault. This is the

Figure 3Developments of the sphere according to cylinders andcones

The single coursed ashlar vault 1681

aproximation chosen by the stonecutters (theapproximation of the segments also will be employed

in certain occasions, especially for the design of thedecoration.) Each one of those cone sections areeasily developed tracing the corresponding arches

with the center in its vertex (Fig. 4). The template ofeach voussoir is no more than one part, more or lesslarge, of this development.

Figure 4Layout for a dome in the treaty of Vandelvira

The prepared template would be applied over thespherical inner surface of the vault to mark itscontour. If we were very strict with the geometricconcepts, we should take care that the template

touches the surface only in the edges corresponding tothe parallels, remaining separate along of the othertwo sides (fig. 5). But the practice shows that, even in

vaults of small span and great curve, this precaution

Figure 5Adaptation of a fiat template to the spherical surface

is excessi ve, and the stone-cutter commits a Iittleerror if he applies the template flattening it withoutcare so that it remains in contact with the stone.3

Thus it happens that, finally, the use ofthe template

and the bevel very easily solves the work of carvingthe voussoirs for spherical vaults. The stone-cutterwould require a different template for each row, butthe templates, and the pieces, of a single row are allthe same. Even if the vault is not very small, thevoussoirs of each row can be of different longitudes,according to the dimensions of the blocks that thequarry provides, but its templates always have the

same form, more or less extended (or appliedsuccessi vely). 4

There are another type of hemispherical vault, thatpresents just one helicoidal row, named by PhilibertDelorme «en forme d'une coquille de Lima,

1682

different ways-, and to support between each twoturns of the helix a ruled developable surface -forwhich there is only one solution, but is not as simplenor evident as when it involves cones and cylinders.

The drawing that we find in the vault en limar;on orlike a snail, of Delorme (figure 6), reappears almostexactly in the manuscript of Jean Chereau(1567-1574) (figure 7). Chereau copies many

solutions from Delorme; in this case it is done in aliteral way, evidently without understanding theproblem, as it duplicates the same mistakes.

In effect, the two drawings trace a spiral in plan,and that spiral is projected vertically in the sphericalsurface. The spiral of the base is drawn in such a way

that the progression of the line in each revolution isconstant. Consequently, the height of each row iscontinually variable. As happens in other layouts ofhemispherical vaults, in the layout of this vault we

have the benefit of the same circumference for thebase and the section, in such a way that in the upper

\,

"

I:Q~:>-!:-<\)I:Q

§ I '~c~ "~$

~ l~~,,,,,~I r=,1:Q1~"

The single coursed ashlar vault 1683

left quadrant we can see the semisection, that projectsthe partitions of the base verticaUy upwards, andshows that the height of the row increases from thepaje towards the equator. Precisely from here is

where the most important error of the layoutoriginates: the heights of the rows resulting in theupper part, numbered 6, 5, 4, 3, are different but

moderately so, whereas the height that would result in

the base of the vault is notably greater than the rest,so that the author (both authors) have decided todivide this first height (1 and 2) in order that itbecomes similar to the rest. Evidently this is not awell-defined solution, because, although we decide tobegin with that divided row, at some point we will

have to return to just one piece, etc.lt is certain that the text of Delorme alludes to the

possibility of employing the same layout to design of

a conical or spherical vault -although the text speaksof a «pyramidal» form, it is 10 be assumed that itrefers to a cone, as it involves, it says, covering a tour

ronde or a spiral staircase. In the case of the cone, thespiral of the base would have projected ontosomething very approximate to an ordinary helix ofconstant step. But the drawing clearly develops thespherical option.

So then, the drawings of Delorme and Chereaucontain operations that any craftsman who reallyfaced the problem would have rejected: to think aboutthe spiral directly like a tracing in plan thentransferred to the space, to accept a row of constantlyvariable thickness, and to solve the problems thatthese decisions generate with unskilled and undefinedclumsiness.

We examine now the layout of Alonso deVandelvira (fig. 8). This author draws the spiral ofthebase from the point of the spatial helix that he wantsto obtain. The process consists of first determiningthe height of the row by dividing the section as if itin volved a conventional hemispherical vault in orderto afterwards make those points of division rotatearound the axis at the same time that they uniformlyadvance towards the pole by their meridianoVandelvira divides the circle of the base in equal parts(16 of them) and divides the height of the row in the

same number of parts, projecting them horizontally,in order to trace the spiral line in plan. Each unit ofrevolution corresponds to one part of nearing thecenter. The layout is, naturally, a flat drawing, but it

corresponds with a spatial conception of the line.

:r~'"~.

cA-¡Z~~ ~~~..;';/

A/~

""

,,,

. ,.I

*~'/,/

[]OO~

Figure 8The vault «en vuelta de capazo» in the treaty of Alonso deVandelvira

As a consequence, in Vandelvira's layout theheight or distance between the lines of the upper and

lower bed joints is constant. This helicoidal solutionof the spherica] vault is characterized by itsunnecessary complication, in that it is unavoidablethat all of the pieces will be different; but, within thisgeneral inconvenience, Vande]vira's layout departsfrom reasonable assumptions and attempts a certainuniformity, while that of Delorme arrives at an absurd

solution.On the other hand, neither Delorme nor Chereau

say anything concerning the templates for the

execution of the pieces. Delorme draws them, and itseems that he obtains them starting from a theoreticaltemplate of conventional model, but, as is common

for Delorme, he doesn't explain anything. Although

1684

the layout that Vandelvira proposes to obtain thetemplates could be criticized from a strictly geometricpoint of view, at least the author undertakes andclearly explains this theme.

In effect, we have confirmed that the template forthe carving of a conventional spherical vault resultsfrom the development of a section of coneapproximating the spherical surface. In this case there

is no cone as a substitute for a portion of the sphere.Is unlikely that Vandelvira conceived of the the ruleddeveloped surface mentioned before as a possibleapproximation to the sphere. So that, in this case, as

in others of his treaty, he would obtain the more or

less quadrangular template in a very approximateway, looking for the distances in real magnitude

between the vertices and joining these vertices withlines reasonably near to the straight or curved linesthat we intuitively should obtain.

The intrados have four corners, four vertices in itstemplate, that will not be over the same planeo Afterestimating the distances of the four sides, we willhave to choose one of the two diagonal s, as each onewill result in a different quadrangle. Consequently,what Vandelvira does is not a development, but norcan it be said that it is the real form of the relativepositions of the vertices. The obtained template could

be of value if it is folded by one of its diagonal s,keeping the two parts level, about which Vandelvirasays nothing. Therefore, the triangulation that

Vandelvira uses in order to draw the relative positionsof those four points implies a geometricalappraximation. On the other hand, this template will

be similar to the conventional in that the lines of thelateral joints will appear straight, but the superior andinferior sides, corresponding to the bed joints, arenow developed fram the helicoidalline and not framcircular arcs. Vandelvira draws them, however, ascircular arcs, and it is curious to note that, in doingthis, he does not consider something similar to thecone that passes through the superior and inferiorlines, rather he invents a hypothetical tangent cone for

each one of two parts of the helicoidal line. As aconsequence, the superior edge of a template

coincides with the inferior edge of the template of

the piece that goes above it, which theoretically couldnot occur, as Van del vira occupied himself withexplaining in reference to the hemispherical vault.

It should be recognized that Vandelvira takes manylicenses and departs much from what we now would

E. Rabasa

understand as a strictly correct solution, but also thatthe po sed problem was enormously difficult given theknowledge of that time, and, in this situation,Vandelvira makes a correct use of some resources,finding real magnitudes, imagining hypotheticalcones, etc.

In fact it is very notable that, when the conceptualtools capable of solving this problem did becomeavailable in the 19th century, in contrary to what wewould expect, no one attempted the problem at all.The helicoidal joints of the dome are not evenmentioned. We might as sume that the stereotomy ofthe 19thcentury rejected undertaking an unnecessarilycomplicated problem; however it is clear that they didengage in studying other rather absurd and

sophisticated proposals. The reality is that the correctexecution, which demands the determination of theruled surface and its development, would be complexand not very elegant fram the geometric point ofview.

It is worth noting that this type of vault appears inthe copy of the treaty of Vandelvira conserved in theEscuela de Arquitectura de Madrid, but not inthe copy kept at the Bibloteca Nacional (by FelipeLazaro de Goiti).5 Vandelvira presents this vau]t as

a variant of the vuelta redonda or conventionalhemispherical vault, and in the copy at the Escuela deArquitectura it appears after the col!ection of ribbedhemispheres. The copy of Felipe Lazaro de Goiti

omitted al! of the ribbed vaults, as much the gothicones as the rib and panel vaults (enrejada), probablythinking that they had to do with an old tradition. Ifthis type of vault (vuelta de capazo) was situated inthe original as it is in the copy at the Escuela deArquitectura, that is to say, a continuation of the rib

and panel vault hemispheres, it is not strange that itwould be left out together with this group.

Perouse de Montclos (1981) explains that thisparticular method and other similar games are nomore than unrealized fantasies of Delorme andChereau. It is certain that we do not know of anyFrench examples, and, in any case, it is evident that

the drawing of De]orme/Chereau can not have anyrelation with a real example that was known verydirectly by them, because as 1 have argued, it presentsimportant practical prablems.

Similar layouts of spirals in a circle are found in themanuscript of Hernan Ruiz el Joven; it is not clear,however, if this is a layout for a vault of this type,

The single caursed ashlar vau!t 1685

since it involves Archimedes' spiral (in some casesthe integration of a few), that open or separateprogressively as they depart from the center, which

would notably exaggerate the defect found inDelorme. It would be possible, however, to have arelation with the ionie volute; in fact one of thesespirals is applied to this function. In any case, the

usual layout of the ionic volute is a spiral ready-madewith circumferential arcs, of di verse centers, startingfram an initial small circumference.

1 have commented that the way to trace the spiralof the base in Delorme/Chereau allows the step to beconstant. This is done with a layout very similar tothat of the ionic volute, employing circumferentialarcs. In this case it started fram the circle of thecentral keystone, employing the two points of thecircle that coincided with the horizontal axis. Thespiral begins then with a first semicircular arc from

the center of the left, continues with anothersemieircular arc from the center right, etc. It is easy toverify that in this way the distance between turns is

constant and equal to double the diameter of thekeystone. We can also find a spiral of two centers inVillard de Honnecourt's notebook, which has beenput in connection with the layouts of pointed arches

(Bechmann 1991, 225-230).

The type of the vault de capazo (or en limaqon)appears later in the book of Milliet-Descales (1674),figure 9, and in that of T. V. Tosca (1707), figure 10.It is known that Tosca also copies Milliet-Deschalesin almost all of his exposition on the geometry ofstonecutting. In this case the original French layout isas problematic as that of Delorme, and Tosca's copyis even worse.

In effect, Milliet-Deschales lays out a spiral evenmore similar to that of the ionic volute, as it employsfour centers, the four axial points of the initial circle,but in doing this obtains, as in the case of the volute,a progressively wider step as it departs from thecenter. In this way the prablem of the row's height ismade worse, and the author no longer occupieshimself with drawing the section, where again thefirst row would appear disproportionately taller thanthe rest.

Tosca copied the drawing, which is identical in amultitude of details, but takes two centers, and thesepoints are, unlike those of Delorme, one on the cercle(that is marked 1) and the other in his center. In this

way there are not four eireurnferential ares in eaeh

S,i.'.z¡,.. ,tjI_Ji- tffim.r,t-.

Figure 9

«Spiraliter testudine efformareo>, by Claude Francois

Milliet-Deschales

Figure 10

Hemisphere that clases «a manera de roscao>, by Tomas

Vicente Tosca

1686

turn, rather only two, and one of them is traced from

the center of the circumference, producing concentricsemicircles with it. So then, the spiral form isachieved in Tosca in the strangest of ways, since thehalf of the base is exactly like a vault of conventionalround rows, while the other half is charged withconstructing the helices. The text just refer to thelayouts of conventional spirals, as if the subject were

irrelevant.These two authors draw the template of the vault

faces by a somewhat different process to that ofVandelvira, although also approximate, departingfrom an ideal straight template. The Spanishmanuscript of Portor y Castro,6 copying Tosca, offers

the same drawing (Fig. 11). I do not know of otherlayouts for this pattern.

( ,~ f,~.t-J.' Q.-~.k ~/ 1M

,

"'''''' .

,

~ ~;"':\f "'~,. :

~

. ;.../'JW"

;7 ¿ -:: ~fr.-,,(.c.r_:4dv¿'''.i ~J.,(.k ~.J..A "",.j" ,"

~.-¡,¡c,,,;;t;,, j!.;:9~,jJu..»..!lJ./r/:'Ji.l"7A7':;.fíl..1-~.~{if;~.ut::t :J~

~'~~{,,"y ~.!..' ..-,.g.~~~ 4~~;r~ ~,,¡~~,f+:,{ ~~¡--.~. - J''';~!: ~fr:::~~'~;QJ?~'/...~ 1

c#';"'"~:.t~.

":'ft:'~~f'z;~ ~'; "'~:,4';'. ,:,¿tf., (~~.~

"~-~-Jf"" -~ I :>-..

,.

X/

- ,°",

,

The single coursed ashlar vault 1687

practical solution along such models. But that ofVan del vira, despite its multiple approximations andlicenses, offers a buildable pattern. This is espacially

true if we keep in mind that accepting approximatesolutions was a habitual custom in the stonecutting ofthe 16th and 17th centuries.

Greater demand for precision is require for anotherof the examples of this type of vault found in Spain,the one situated in the spiral stairway, named caracolde Mallorca, that leads to the right tower of thePalace of the Guzmanes de Leon, figures 13 and 14.The two towers of this principal fa

1688

On the other side, in this same fac,:ade we find a mostcurious example of vis de Saint Gilles. So the relationbetween the spiral staircase and the vault de capazocontinues.

Figure 15Helicoidal vault in the portico of San luan de los Caballerosin lerez de la Frontera

We don't know the origin of these patterns.Although the proposal of Vandelvira may be muchbetter and more real than that of Delorme, nothingimpedes that it be this drawing first that, widelypublished, would provoke interest in Spain. It doesn't

seem that the Florentine tradition of drawing helicesin the brick domes would be relevant; it involvesloxodromical curves generated by a herringbonepattern, and they are different from the vault de

capazo as much constructively as formally. Weshould consider, however, two other possibleinfluences. On one hand the Byzantine tradition ofexecution of spherical vaults with ceramic piecesprepared along one or various helicoidal rows. Thisis, for example the dome of San Vital de Ravena. And

furthermore we know of the existence of an examplein stone much earlier, in Anatolia.

In effect, the Sultan Han near Aksaray (1229), figure16, was built as a rest area for the caravans, with

commercial and religious function. As usual for thistype of construction, it has a bedroom space of great

dimensions, similar to the Christian temples, and

formed in this case by five naves separated by columnsand covered by barre] vaults. In the center of theprincipal nave was constructed a vault of this type,

over squinch arches covered with a pyramid also of

E. Rabasa

stone. Its architect was original!y from Damascus; it isunavoidable to record here the opinion of Viol!et-le-Duc, who was convinced that al! Westem stereotomy

had its origin in the admiration of traveling Christianstowards the pattems of the Syrian construction. Thereare at least two other smal! domes or calotas, ofreduced dimensions, with this type of pattem, coveringthe central part of domes in the hospital annex to themosque of Divrigi, that date also to the 13thcentury.7

Figure 16Dome in the Sultan Han near Aksaray (Turkey)

Final!y we have lO remember that this type ofpattern is a solution enormously more complex thanthe conventional solution for a hemispherical vault. Ineffect, the solution by round rows demands only one

template per row, and in each row the pieces areequal. Common sense leads Vandelvira to adopt a

uniform height for the he]icoidal row, but even so thepieces and all of its templates are different and require

an individualized layout. It is to be supposed that inpractice the constructed vaults were made with an

added license to those other geometrical licensesa]ready explained: since the design of al! of the

templates would be laborious, complicated andsusceptible to confusion, one wou]d employ the sametemplate to a whole series, maybe to al! of the ashlarsin one turn of the helix.

The single coursed ashlar vault 1689

In any case, the idea of the helicoidal row, which isreasonable in brick or ceramic pieces, becomesunnecessarily complicated in stone. That isn't theonly case. Often more complicated and difficultpatterns than the conventional ones have beenproposed. But this, which happened especially in the19thcentury, frequently was done with the intention of

showing an elegant idea from the geometric point of

view. Or rather, as occurs with the curious variants ofwaterspouts between the 16th and the 18thcenturies, inorder to develop a showpiece of a mechanical morethan formal type. In this scope the vault de capazo is

a very early and atypical example of this unnecessarycomplication. Maybe the intention of introducing thehelix archetype in the stone work existed in someexamples, but very probably the mentioned cases aredisplays directed to the stone-cutting guild itself.

NOTES

1. 11'onc were to look for a difference, it could be found inthe comers where the pieces. in the shape of L,correspond to two rows. Although frequently, including

at El Escorial, this form is avoided and the pieces 01' the

comers would be resolved in a similar manner to the

rest.

2.

3.

1 have justified these statements in Rabasa (2000).

I have confirmed that this is true in the construction of

a small dome in collaboration with the Centro de los

Oficios de León. It had only a one meter span, but for

that very reason the pieces have great curvature.

11'all 01' the voussoirs 01' a row have the same length, thetemplate is made for this size. But if the dome is 1arge.

the distribution 01' joints is not established in advance,and each piecc has a different length, adapted to the size

01' the blocks from the quarry. In this case the templateis easily slid over the surface of the stone to mark theperimeter of the face 01' intradm¡ when the length varies.

4.

5. It is the manuscript MS.I2.719 01' the BibliotecaNacional de Madrid and R.I O of the Biblioteca de la

Escuela de Arquitectura de Madrid, that is a reproduced

facsimile by Barbé-Coquelin (1977).

Manuscript from 1708, conserved in the Biblioteca

Nacional de Madrid with the call number Ms. 91 14.

1 am grateful to Profesor Aysil Yavuz for calling this tomy attention. Drawings 01' these vaults can be seen in

Yavuz (1993,165-192).

6.

7.

REFERENCE LIST

Barbe-Coquelin De Liste, Genevicve. 1977. Tratado de

Arqllitectllra de Alot1S0 de Vandelvira, Libro de traras de

cortes de piedras. Albacete. Caja de Ahorros, 1977.

Bechmann, Roland. 1991. Villard de Honnecollrt. Paris,

Picard.

De l'Orme, Philibert. 1567. Architectllre. Paris. (facsimil de

la edición de 1648 en Bruselas: Pierre Mardaga, 1981).Milliet-Dechales, Claude Francois. 1674. Tractatus XIV. De

lapidum sectione. In ClIrslls sell mllndus mathematiclls.Lyon, Anissonm.

Pérouse De Montclos, Jean Marie. 1981. L'architecture á la

franraise, París: Picard.

Pinto Puerto. Francisco. 1998. «Las esferas pétreas»,

doctoral thesis. Sevilla.

Rabasa, Enrique. 2000. Forma y construcciÓn en piedra: De

la cantería medieval a la estereotomía del siglo XtX,Madrid: Akal.

Tosca, Thomas Vicente. 1707-15. Compendio mathematico, Va1encia, Antonio Bordazar. (1721-27, 1757;

Tratado de arquitectura civil, montea y cantería y

reloxes, Valencia, Hermanos Orga, 1794. Facsímil en

Valencia: París-Valencia, 1992)

Yavuz. A. T. 1993. The characteristics 01' star vaults inSeljuk Anatolia. In Structural Repair ami Maintenance of

Historical Buildings lI/, edited by C.A. Brebbia and

R. J. B. Frawer. Southampton Boston: Computational

Mechanics Publications.