The Apse of the Gothic Cathedral of Tortosa versus Augustine of Hippo’s Civitate Dei

Abstract

This research delves into the influence of St. Augustine on the construction of the Gothic cathedral of Tortosa. The canonical cathedral of Tortosa underwent re-establishment in 1155, which was carried out by Bishop Godfrey who was the abbot of Saint Rufus of Avignon and was governed by Beati Augustini rule. The presence of St. Augustine in the Capitular archives with De Civitate Dei (ACTo-20) from the XII century is examined. This, coupled with a spatial analysis of the liturgical space using laser scanning (TLS), serves to validate the historiographical thesis put forth by Wilhelm Worringer, Erwin Panofsky, and Otto von Simson for understanding the construction of the apse of the Gothic cathedral (1346–1441). This methodology establishes a bijection between patristic and Neoplatonic sources and the interpretation of the liturgical space’s dimensions using statistical systems. This approach addresses the construction of the apse through the incorporation of a heptagon, a geometric figure that is absent in Euclid’s Elementa and Ptolemy’s Almagest. In conclusion, it is determined that both the imagery and metrics employed in the design of a radial heptagonal apse, as well as its cross-section, are influenced by both St. Augustine and the metrics of the Neoplatonics, which remain present in the Chapter Archives.

1. Introduction and Objective

The canon of Tortosa was refounded in 1155 by Bishop Godfrey (+1165), abbot of San Rufo de Avignon, following the rule ‘vivere sub regula Beati Agustini, et iuxta consuetudines Ecclesiae Sancti Ruffi’ (García 1998, pp. 10–13). Regarding la regula, it originally dates back to 1039 when Benedict, Bishop of Avignon, oversaw the functions of the choir and the recitation of the Psalter (Misonne 1963, pp. 471–89). It was governed similarly to other cathedrals in the Catalan territory, such as Vic, Lleida, Girona, and Seo de Urgel (Calvo 2014, pp. 78–79), where the request for books was already ordained within the canonical schedule (Rule 37) (Orozco 1881, p. 36). From this moment on, the construction of the Romanesque cathedral began in 1158, and it was consecrated on 28 November 1178. On 20 April 1346, Bishop Arnau de Llordat (1341–1346) and the Chapter commissioned the magister operis, Bernat Dalguaire (+1347), a new cathedral that was to replace the ecclesia vetulam. (ACTo, NC 1346. 11).

Indirect measurement procedures such as close-range photogrammetry (CRP) and terrestrial laser scanning (TLS) allow for the analysis of the construction reality of the Tortosa Cathedral. This makes it possible to carry out graphic experiments on architectural objects and obtain contrasted explanations relative to the hypotheses based on the geometric and constructive research of seo dertosense1.

The results obtained from these records can be statistically analyzed. These numerical values are subjected to explanatory treatment, attempting to determine the causes and consequences of the construction of the sacred space of the cathedral. The evolution of formal changes is considered a concrete phenomenon established by liturgical and technical changes during the gestation and construction period of the Gothic apse (1346–1441) (Figure 1).

Not only the what but also the why of the causes is sought; therefore, the evolution of this liturgical space and how it has reached its current state are studied. The objective is to create an explanatory model that allows for the observation of the sequences of cause and effect among the promoters, the Chapter and bishop, and the magister operis who have worked on the Tortosa Cathedral. In this way, it is explained, from the perspective of complexity, that this sacred space has been generated by the complementation of both types of knowledge, which were defined as scientia theoria and scientia practica from the perspective of medieval philosophy. This concept spread in Gothic Europe from Domingo de Gundisalvo’s De Scientiis (fl. 1150) and its precedent, al-Fārābī’s “Catalog of the Sciences” (c. 870–950).

Al-Fārābī states that Scientia doctrinali includes arithmetic, geometry, optics, astronomy, mathematics, music, the science of weights, and engineering; it specifies the difference between theory and practice (González 1932, pp. 97–105). Gundisalvo uses these same terms in the De Scientiis (Alonso 1955, pp. 85–112), definitions disseminated to the world of European cathedrals by Vincent of Beauvais (c. 1194–1264) in the Speculum Doctrinale (Mâle 1910, pp. 37–40).

Classical historiography assumes in the scholastic context that the Gothic cathedral is the manifest result of the will of two parties: that of the promoter and that of the builder. The interaction between both figures was highlighted by Wilhelm Worringer (1881–1965) in the Formprobleme der Gotik (Worringer 1911) and by Erwin Panofsky (1892–1968) in Gothic Architecture and Scholasticism (Panofsky 1951). Otto von Simson (1912–1993) initially tackled the question of learned sources (Simson 1952, pp. 6–16), seeking them in Augustine of Hippo’s De Civitate Dei, De Ordine, and Musica, and further developed them in The Gothic Cathedral: The Origins of Gothic Architecture and the Medieval Concept of Order (Simson 1956). Methods utilizing indirect records such as close-range photogrammetry (CRP) and terrestrial laser scanning (TLS) (Figure 2) allow for a precise analysis of the three-dimensional space, enabling the examination of points that were previously inaccessible and allowing them to be contrasted with these historiographical theses.

Saint Augustine acknowledges in the De ordine, libri duo (386) that the Trivium and Quadrivium programs were the basic instruments for understanding the Holy Scriptures. He proposes the formation of man in the arts (De Ord. II.16, 44)2 as they promote the elevation of the spirit on the path toward God. He considers that moderate and rational erudition in the liberal arts makes man more agile and steadfast in encountering the truth of his knowledge3. In the Confessionum, libri tredecim (397), geometry, music, and arithmetic are found, because the ability to understand and the acuity in discerning are gifts from God (Conf. IV, 16, 30)4. He reports that what unfolds reasonably in ordered forms was designated by the name rhythm, which in Latin can only be called numerus (De Ord. II, 14, 41)5. Thus, he sees the need for the study of numbers for the order of music, geometry, and the movement of the stars (De ord. II, 5, 14)6. In the De Genesi ad Litteram libri duodecim (401), he recalls that the number without number is that by which all things are formed. The weight without weight is that by which balance is established, reducing all things to stillness (De Gen. ad litt. IV.8)7. He defines geometry in the beauty of figures, in the figures and dimensions, and in the dimensions and numbers. He also inquires whether lines and spheres or any other form and figure in reality exist as they are contained in the intellect, calling geometry the science that distinguishes and orders these understandings (De Ord. II.15, 42)8.

With these data, the objective is to explore the influence of the thought of Augustine of Hippo (354–430) through the Augustinian Chapter on the established order in the construction of the cathedral, using as reference the codices of the Capitular Archive of Tortosa (ACTo). Saint Augustine himself states in De libero arbitrio libri tres (388–391) that we all desire to be happy and wise, for no one can be happy without being wise (lib. arb. II, 9, 26)9.

The Augustinian model inherited by monks distinguishes between uti and frui10, relating to the use of temporal realities, wisdom, and the enjoyment of the eternal. These terms also applied to the right of usufruct, which can be indirectly applied to the construction of cathedrals. At the threshold of the Renaissance, the Christianized Neoplatonic model is subtly linked to the sensible and rational dimensions of conjectural science. Thus, the canon of Noyon and mathematician, Charles Bovelles (1483–1553), completes this journey by revisiting Saint Augustine to show the Trinitarian dimension of the overcoming of practical or erudite knowledge (Trottmann 2015, pp. 805–16). In this way, the order of Gothic construction allows us to address geometric and proportional issues that stem from the knowledge concentrated in some cases in Augustinian works.

An example is how the heptagonal apse of the Tortosa Cathedral was drawn. Charles Bovelles himself acknowledged in the Livre singulier et utile, touchant l’art praticque de geometrie (1542) that a figure as important for Christian symbolism as the heptagon did not appear in Euclid’s Elements (c. 325–c. 265 BC) (Bovelles 1542, 25v–28r). The French humanist had already discussed and constructed this figure in the Geometricum Introductorium (1503), published in Paris in 1510 (Bovelles 1510, p. 196), a work present in Tortosa (ACTo 300) (Figure 3)11.

2. The Works of Saint Augustine in the Tortosa Cathedral

In the Inventory of the Tortosa Cathedral of 1420, the rule of the Augustinian canon is listed as Regia de sent Agosti [114], Item, Regula canonicorum [205], Regula beati Augustini [213], and Regula de sent Augusti [215], and the works of Saint Augustine are as follows: De civitate Dei [5], Supra psalterium [6], De penitencia [23], Liber vocatus and De penitencia [68], Diversa originalia beati Augustini [98], Liber Questionum beati Augustini [100], and Diversa originalia Augus [110] (Baiges 1999, pp. 3–20). Of particular interest for our analysis are De civitate Dei [5] ACTo 20 from the 12th century and Supra psalterium [6], known as Enarrationes in Psalmos, for which its reference has been lost. In said inventory, Item, un libre appellat Macrobi [183] is preserved in the Commentarii In Somnium Scipionis of ACTo 236 XIII by Macrobius (fl. 400) due to the special teaching relationship between both figures.

In the cataloging by Enrique Bayerri Bertomeu (1882–1958), there is an indication of the following works: ACTo 20, De civitate Dei; ACTo 55, De Trinitate and De Vera Religione; ACTo 86, Contra Plagianos, Liber de spiritu et littera, Liber super Genesi ad litteram, Expositio Symboli, De immortalitate animae; ACTo 110, Meditaciones; ACTo 130, De Corpore et Sanguine Christi; ACTo 131, De Corpore et Sanguine Christi; ACTo 173, De sermone domini de mundis, De munditia cordis; ACTo 195, Exortación del Apocalipsis de San Agustín; ACTo 217, Extractos de la Epistolas de San Jeronimo; ACTo 222, Epístola beati Augustini ad Virginies; and ACTo 230, De Confessione peccatorum, Admonitio, Ex tractatu S. Agustini in Euangelio ubi dicit helemosinas faciendas, De peccato disiderii, Die quidem omni et omni hora et cura omnio continua, Sermo de plasmo XLVIII, Sermi de capitulo euangelii ubi dicit, remitite et remitetur nobis, Epistula pulchra satis, De muliere curva, Sermón De divite feneratore, Trcatatus in Evangelium Iohannis, De Poenitencia, De concordia fratrum, Poentientes sermonus, De moribus Ecclesiae Catholice.

Additionally, there are four manuscripts dedicated to the Rule ACTo 85 Rule of Saint Augustine, ACTo 90 Rule of Saint Augustine, and ACTo 189 Rule of Saint Augustine and two other works related to his figure: ACTo 68, Melliloquium by Fray Bartolomé de Urbino (1472–1517), and ACTo 73, the Life of Agustinus (Bayerri 1962, pp. 615–18). We can also add the incunabulum of the Sermo de Sancta Monica mater sancti Augustini, Calixtus, Sermo de conversione sancti Agustini, Sigibertus, In epistola ad Macedonium de beato Augustine (1486) (Guitarte 1987, pp. 378–90).

The most important for its quality and significance is the codex ACTo 20, cataloged for the first time by Heinrich Seuse Denifle (1844–1905) and Émile Chatelain (1851–1933) in their Inventarum codicum manuscriptorum Capituli Dertusensis (1896) as No. 20 of the 12th century, S. Augustini de Civitate Dei libri II-XXII, which notes the absence of the first two booklets (Denifle and Chatelain 1896, p. 7). Subsequently, Josep Maria March i Batlles (1875–1952) found four illustrations (fol. 1r–2v) (March 1916, pp. 351–54). Later, Jordi Rubió i Balaguer (1887–1982) and Ramon d’Alòs-Moner i de Dou (1885–1939) added the initial folios of the codex and the illustration (fol. 5r) (Rubió and Rubió 1914, p. 143).

In the current catalog, it is listed as ACTo 20, Saint Augustine, Bishop of Hippo: Of the City of God. It is a manuscript from the 12th century written on parchment, with 408 folios measuring 370 × 270 mm, and it is placed in a writing box with two columns measuring 280 × 210 mm with 33 lines (fol. 13r). It lacks part of the first book (Bayerri 1962, pp. 157–59). It contains five illustrations cited as follows: the Zodiac (fol. 1r), the Defense of the City of God (fol. 1v) (Figure 4a), the Celestial Jerusalem (fol. 2r) (Figure 4b), the Maiestas Mariae (fol. 2v) (Figure 4c), and the Creation (fol. 5r) (Ibarburu 1984, pp. 93–124). Additionally, it contains 21 initials that head each of the books except for XVIII (Ibarburu 1985, pp. 103–25). In the review through Les manuscrits à peintures de la Cité de Dieu de Saint-Augustin by Alexandre Laborde (1853–1944) with 61 illuminated codices, we can relate the Zodiac (fol. 1r) with the reference from Mss. Franc. 900S-9006, fol. 287v, which is from the Royal Library of Brussels (c. 1410), illustrating that Book VIII is dedicated to natural theology (Laborde 1909, p. 322).

The Defense of the City of God (fol. 1v) (Figure 4a) bears some resemblance to the illustrations of Book I, depicting the sack of Rome by the Goths and showing how God’s mercy tempered the destruction of the City (BnP, Ms. Franc. 22912-13) (c. 1376) and (Ms. Add. 1 5244-45) from the British Museum (1370–1377) (Laborde 1909, p. 191)). The image of Celestial Jerusalem (fol. 2r) (Figure 4b) resembles that of the Heavenly City from Ms. lat. A. 7, fol. 1v of the Metropolitan Chapter Library of Prague’s illustration of Book I (Laborde 1909, Plate IV). The Maiestas Mariae (fol. 2v) (Figure 4c), for which we have no direct reference in the cataloging and which is long considered an Assumption, can be read as an allegory of the Church by virtue of the hierarchical arrangement of Christ, the Virgin, and below them, a figure with a pastoral staff and a book, perhaps St. Augustine, is surrounded by faithful and disciples (Grimaldi 2015, pp. 77–90). The Creation (fol. 5r) could illustrate Chapter 21 of the Creation of the first man and the human race from Book XII, reflected in a large number of codices (Laborde 1909, pp. 193–95).

On the other hand, the architectural layout of the illustrations of the City of God and Celestial Jerusalem (Figure 4a,b) brings us closer to the collection of graphic representations of architecture in the Livre de portraiture (c. 1220–1240) by Villard de Honnecourt (c. 1200–c. 1250) from (BnP, Ms Fr 19093), which consists of 33 parchment folios (0.230 × 0.240)12. Likewise, the vegetal forms and the bestiary depicted in the Zodiac (ACTo 20 fol. 1r) in the representations of Aries, Taurus, Leo, or Capricorn (Figure 5a) bring us closer to the bear and the swan, a figurative representation of Celestial Jerusalem from (BnP, Ms Fr 19093, fol. 4r) (Figure 5b). The têtes de feuilles and the vegetal ornaments (BnP, Ms Fr 19093, fol. 5v) (Figure 5c) bring us closer to the scrolls alternating with foliar motifs, which are symmetrical and heart-shaped motifs from (fol. 1r). The human figures of the Creation of Adam and Eve (Figure 6a) bring us closer to those in (BnP, Ms Fr 19093, fol. 29v) (Figure 6b), an ancient representation of Mercury; to the birds and marine animals of the creation of the fifth day and the terrestrial ones of the sixth day (Figure 6c); and to the lion (BnP, Ms Fr 19093, fol. 24v) (Figure 6d) and eagle for reading the gospel (BnP, Ms Fr 19093, fol. 7r) (Figure 6e).

3. Methodology and Results of Terrestrial Laser Scanner (TLS)

The basic pattern of measurements appearing in the Llibres d’Obra [ACTo] is the cana of eight spans and the span of twelve fingers. The cana of Tortosa is defined in Book IX, No. 15.5 of the Consuetudines Dertosae (1272) [AHCTE: cod. 53, fol. 256r]15. By comparing the unifying documents of the cana of Tortosa with that of Barcelona (24-VII-1593), we concluded that the cana of Tortosa used for the cathedral measures 1.858 m, and the span is 0.2323 m. The methodology proposes a comparative reading of these measurements with the data obtained through a survey conducted using a Leica Scan Station C10 with an accuracy of 4 mm at 50 m and angular accuracy of 12″/12″ (Figure 7).

The total error of the methodological process (Et) is the sum of the evaluation of the uncertainties of the work (Ei), the observation and data collection (Ed), the computer processing of these points (Et), and those derived from the appreciation of the layout of the work (Er), for which its value (Et = ±0.083 m) and calibration with respect to the three canes of Tortosa represent an error of ±1.502% with respect to this pattern. The most characteristic values of the data obtained from the apse are statistically analyzed so that the relative error (e_r) will be equal to the most probable value (c_a) plus or minus the root mean square error of the mean (c_cm).

After the statistical analysis of the data carried out by the magister operis, possible connections of the sources that inspired this fabrica by the Chapter are sought in the existing works of the ACTo. The outline of Guarc (Factory—No. 49) (c. 1345–1380) (0.917 × 0.682 m) (Figure 8a) is preserved, representing the plan of a project of the apse and part of the body of a cathedral nave, with an annexed chapel with exterior access. On the parchment, a base of auxiliary lines is observed prior to the final outlines (Figure 8b), for which its analysis allows for the establishment of an interpretative methodology of the geometric layout of the apse (Figure 8c) (Lluis i Ginovart et al. 2013, pp. 325–48).

3.1. The Metrology of the Floor Plan of the Apse

The apse with a double ambulatory was built between 1374 and 1441, embracing and replacing the previous Romanesque cathedral from the outside. The first phase (1383–1424) involved the construction of the belt of radial chapels, executed in a sequential manner (Figure 9a–c). The second phase determined the construction of the ambulatory (1424–1434) symmetrically covering the axis of the presbytery (Figure 9d–f), which is balanced on the pilar major (Figure 9d). Finally, the closure of the presbytery (1435–1441) occurred with the first placement of the keystone on the main pillar (1439) (Figure 9g) and the subsequent construction of the vaults (Figure 9h).

The total width of the ambulatory is the statistical value of the inner face of the enclosing wall, obtained as the most probable value c_a(rg1–10 = 17,500 m) with a root mean square error of the mean (c_cm = 0.065 m), resulting in an interval [17.565–17.435 m] within the reference value (17.423 m), which is equivalent to 75 spans (Figure 10) (Appendix A,

Table A1. The measurements of the magnitudes of the pillars of the apse.

APSE PILLARS DIMENSIONS
Dimensions of Radial Chapel Pillars
Pillars	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10
F ex/m,	1.049	1.923	1.992	1.980	2.004	2.002	2.010	2.002	1.896	1.046
F ey1/m,	2.387	0797	0.861	0.767	0.765	0.748	0.788	0.738	0.806	2.362
F ey2/m,		0.903	0.780	0.769	0.783	0.732	0.760	0.763	0.795
F s, /m2	1.611	1.677	1.755	1.742	1.769	1.684	1.787	1.709	1.681	1.593
B ex/m,	1.375	2.456	2.551	2.540	2.564	2.562	2.570	2.562	2.456	1.375
B ey1/m,	1.464	1.079	1.029	1.044	1.048	1.012	1.040	1.043	1.075	1.458
B ey2/m,	1.483	1.086	1.063	1.052	1.060	1.027	1.069	1.018	1.087	1.461
B s, /m2	3.013	3.695	3.726	3.741	3.782	3.668	3.807	3.689	3.695	2.984
Dimensions of presbytery chapel pillars
Pillars	P11	P12	P13	P14	P15	P16	P17	P18	P19	P20
F ex/m	1.928	1.641	1.663	1.653	1.624	1.682	1.634	1.666	1.652	1.928
F ey1/m	0.964	0.966	0.756	0.75	0.774	0.77	0.776	0.744	0.758	0.947
F ey2/m	0.954	0.699	0.774	0.77	0.777	0.735	0.787	0.774	0.987	0.963
F s,/m2	2.364	1.563	1.41	1.391	1.391	1.405	1.41	1.402	1.642	2.354
B ex/m	2.238	2.201	2.23	2.213	2.034	2.092	2.194	1.911	1.902	2.238
B ey1/m,	1.119								0.833
B ey2/m,	0.954							0.849		1.118
B s, /m2

and

Table A2. The measurement of the layout in the plan of the apse.

MEASUREMENTS APSE OF THE CATHEDRAL OF TORTOSA
Chapels of the Ambulatory
Width of apse chapels	c1	c2	c3	c4	c5	c6	c7	c8	c9		ca	ec	em	ccm	er (+)	er (−)	V. ref
Apse chapels	5.451	5.441	5.607	5.598	5.597	5.601	5.659	5.444	5.445		5.538	0.09	0.225	0.075	5.6132	5.463	5.575
Radial chapels		5.441	5.607	5.598	5.597	5.601	5.659	5.444			5.564	0.086	0.214	0.081	5.6448	5.483	5.575
Apse ambulatory radius
Apse ambulatory radius	rg1	rg2	rg3	rg4	rg5	rg6	rg7	rg8	rg9	rg10	ca	ec	em	ccm	er (+)	er (-)	V. ref
Gyrola radius	17.445	17.419	17.591	17.627	17.570	17.566	17.399	17.499	17.439	17.445	17.500	0.082	0.205	0.065	17.565	17.35	17.423
Presbytery Radius
Presbytery radius	rp1	rp2	rp3	rp4	rp5	rp6	rp7	rp8	rp9	rp10	ca	ec	em	ccm	er (+)	er (--)	V. ref
Presbytery radius	6.137	6.265	6.281	6.273	6.285	6.268	6.296	6.276	6.271	6,.37	6.249	0.000	3.162	0.029	6.278	6.219	6.272

).

To determine the measurement of the nine radial chapels (c₁…c₉) and reviewing the auxiliary lines of Guarc’s parchment (T6.2) and (T6.3), the most probable value is (c_a(c1–9) = 5538 m). The root mean square error is e_c = 0.090 m, resulting in a relative error (e_r(c1–9)) with a range of 5.613–5.463 m. Similar results are obtained with the seven radial chapels (c₂–c₈), where the reference values yield a range of 5.645–5.483 m, close to 24 spans (5.575 m), which are three canas (Figure 11). The centers of the presbytery pillars (P₁₁–P₂₀) and the 10 radii (rp₁–rp₁₀) are situated at a most probable distance of c_a = 6249 m, with a relative value between the range of 6.278–6.219 m and within the measurement of 6.272 m; they have 27 spans, which is half of the 54 spans of the radius where the radial chapels are traced (Figure 12). The centers of the pillars P₁₂ y P₁₉, homologous to the new designs of (P₁₁–P₁₂) and (P₁₉–P₂₀), with displacements P₁₁ and P₁₂, are 0.107 m and between P₁₉ and P₂₀ (0.168 m).

3.2. The Metrology of the Section of the Apse

To determine the measurement of the height of the nine radial chapels (c₁…c₉), the zenith of the ribbed vault of the radial chapels (c₁–c₉) is taken as a reference, and it is located at the neck of the keystone, with the most probable value of the chapels (h_ca(c1–9) = 10,469 m) obtaining a relative error (e_r(c1–9)) in the range of 10.581–10.357 m. The range of the statistical result falls within the metrological measurement of 45 spans (Figure 13) (Appendix A, Table A1 and Table A2).

The height of the nine vaults of the ambulatory (h_d1…h_d9) has a most probable value (h_da(c1–9) = 16.007 m). The relative error (e_r(c1–9)) ranges from 16.070 to 15.940 m within the range of 69 spans, and it is close to 70 spans, which is within the methodological error (Figure 14). The section of the presbytery vault does not have a direct measurement since the presbytery keystone casts a shadow over the neck of the constructive element. Taking references from the points (TLS), knowing point A (22.908 m), and given that the arch has a quarter section, we can deduce, by extending said arch, the keystone center, which would be situated at a height of 23.237 m above the presbytery floor. Taking a metrological value of 100 spans (23.230 m), a value similar to the indirect result determined by (TLS) is obtained, and it is where the keystone with the Coronation of the Virgin is located (Figure 15).

Analyzing the height of the radial chapels (45 spans) in relation to the proportionality of the Gothic section theory with respect to the metrology of the floor plan, it is observed that at the midpoint of the ambulatory (27 + ½ 27 = 40.5 spans), a proportion is formed between the height of the chapel and the ambulatory (45/81), expressing the development of a ratio of (9 ÷ 5) (Figure 16a).

The reference value of the ambulatory keystones [16.070–15.940 m] falls within the range of 69 spans. Considering that the layout of the plan for (9 ÷ 5) would need to be 58 + 1/3 spans, we observe that it is far from the value of the finish of the apsidal chapels. If we check for the ratio (9 ÷ 6), the height of the ambulatory vaults would need to be at 70 spans (Figure 16b,c). The main keystone is situated at 100 spans over a total width of 150 spans from the ambulatory (Figure 16d,e). With these values, we can propose a plan layout of the measurements [81–105–150 spans], elevated in height corresponding to 45–70–100 spans. The presbytery keystone representing the Coronation of the Virgin has a larger crown diameter of 2.030 ± 1.502% m, neck diameter of 1.625 m ± 1.502% m, and a height of 0.902 ± 1.502% m, with a total elevation of 1.280 ± 1.502% m. The section is 2.110 m² and has a volume of 3.640 m³, with an apparent density of 26.90 kp/m³ and a weight of 97.916 KN (9.8 Tn).

4. A Discussion of the Results

In the design of a Gothic apse, the essential element is the determination of the number of chapels given that, on the one hand, they are the element of uniqueness that has its origin in the Gothic liturgy of the Prochiron, vulgo Rationale divinorum officiorum (1291), written by the Bishop of Mende, Gulielmus Durandus (1230–1296). This liturgy is known in Tortosa through ACTo 58 from the end of the 13th century and the incunabula of Rome (1477) (ACTo No. 258) and Venice (1482) (ACTo No. 290). On the other hand, the chapels are a modular element in construction as they allow for economic sponsorship and act as a buttress element (Puig i Cadafalch 1923, pp. 65–87). From a metric point of view, the radial and lateral chapels must be equal since they are part of the sequence of the pilgrimage that circumnavigates the apse (Font i Carreras 1891, pp. 9–14). In the case of Tortosa, as with the Augustinian canonical church of Girona, in 1312, nine chapels were built in its apse (Street 1865, pp. 318–39)16, which was obligated by a contract, and seven of them are located within the perimeter of the circular crown of the presbytery; the other two are located on the axis of the existing Romanesque cathedral (Figure 17).

4.1. The Metrology of the Apse and Augustine of Hippo

The magister operis had to solve three geometric problems: first, the utilization of a method for drawing the heptagon; second, constructing the geometric figure without knowing the center because it was occupied by the ecclesia vetulam; and finally, proportionally solving the relationship between the radial chapels and the two chapels in the straight section. In the bijective analysis, with respect to theorica y practica, which is the comparative reading between the construction and the ACTo, we have direct sources in the inventory of De civitate Dei [5], ACTo 20, Supra Psalterium [6]. We also have Enarrationes in Psalmos (Baiges 1999, pp. 3–20) and the De Trinitate (ACTo 55, fol. 1r–172r); the De libero arbitrio (ACTo 86, fol. 33v–59v); and the De genesi ad litteram (ACTo 86, fol. 59v–124v) (Denifle and Chatelain 1896, pp. 7–26).

The number seven does not appear in the early works of the Soliloquia (386) and De ordine (386). It is in the De quantitate animae (388) that he introduces it more and more into his discourse due to the influence that came to him through the catechesis for baptism and the sermons of St. Ambrose (340–397) (Cilleruelo 1953, pp. 510–11). In De Civitate Dei (ACTo 20), he refers to the number seven in relation to the seventh day as fullness and rest after Creation (Civ Dei. XI.31) (ACTo fol 168r–169v)17. On the seventh day, the same is seven times, and it manifests as God’s rest and the sanctification of this day. The author warns that the first total odd number is three, which, together with four, forms the septenary.

In De genesi ad litteram (ACTo 86, fol. 59v–124v), the reference to the number seven is made through the number three, which is part of this number, along with eight and nine. Seven can be divided into three and four (De Gen. ad Litt. IV.2,2)18. In Enarrationes in Psalmos (392), it is related to the Ten Commandments, which are divided between the three that show love to God and the seven that show love to their neighbor (Enr. Psal. 32. 2, 6)19.

Regarding the seven radial chapels, the figure of the heptagon, as recognized by Bovelles, does not appear in Euclid’s “Elements”. The Elementa as translated by Adelardo of Bath (1075–1166) in 1142 considers the layout of regular polygons in Book IV20, but it fails to mention either the heptagon or the mathematical syntax in Ptolemy’s (ca. 85–165) Almagesto, which was translated by Gerardo of Cremona (1114–1187) around 117521. The heptagonal layout by means of geometrical instrumentation was first refuted by Kepler (1571–1630) in his Harmonices Mundi, Libri V (1619), in the Propositio. XLV (Kepler 1619, 32–40)22, and later by Gauss (1777–1855) at the end of his Disquisitiones Arithmeticae in Section VII, Propositions 361–366 (Gauss 1801, pp. 454–63).

We know the methodology of constructing the heptagon through the Geometria Deutsch (1472), attributed to Hans Hösch von Gmünd (fl. 1472) (Heideloff 1844, pp. 96–97), and the Geometria Deutsch (1488) by Matthäus Roriczer (+c. 1495) (Roritzer 1999, pp. 56–60). In these works, the side of the heptagon is determined as the height of an equilateral triangle with a side equal to the radius of the circumference. This method has come down to us through Underweysung der Messung (1525) by Albrecht Dürer (1471–1528), a consequence of the corollary of the construction of the pentagon (LII.15) with a value of √3/2 (Dürer 1525, 26r–27v). The origin of this method can be traced back to the Kitāb fī mā yaḥtāju al-ṣāni‘ min al-a‘māl al-handasiyya (c. 993–1008) (Book on those geometric constructions which are necessary for craftsmen) by Mohammad Abu’l-Wafa Al-Buzjani (940–998), and its reception in Latin Europe occurred through Ibn Yunus, Kamal al-Din (1156–1242) in the court of Emperor Frederick II (1194–1250) (Raynaud 2012, pp. 34–83). Guarc’s method determines the diameter of the circumference through the side of the regular polygon and its ratio (18:8), unlike Euclid’s “Elements,” Book IV, where the side is a consequence of the inscribed circumference (Heath 1908, pp. 88–111). Knowing the measurement of the chapel (c_i) results in the determination of the ambulatory (4,5 c_i) (Figure 18a), or vice versa is also possible given the width (2 r_i) and the radial chapel (1/4,5 2r_i) (Figure 18b).

We can speculate about the geometric method that might have been used to replant the radial chapels based on Abu’l-Wafa’s (c. 993–1008) approach, starting from the height of the triangle (√3/2 = 0.866), or other medieval approximations such as the 7:6 ratio provided by Gerbert of Aurillac (c. 950–1003) with a measurement of 0.8571 (Bubnov 1899, pp. 43–45). Gabriele Stornaloco in 1391 utilized [8:7] 0.875 (Ackerman 1949, pp. 84–111) or the 9:8 ratio of 0.889 provided by Antoni Guarc (c. 1345–1380). The relationship between the side of the tetradecagon (ci) and the radial axes of the ambulatory (rdi) is analyzed. The most probable values for the radial chapels (c_a(c2–8) = 5564 m) and the radius (c_a(rd1–9) = 12,553) have a relationship (c_a(c2–8))/0.5 c_a(rd1–9) = 0.886, which is closer to Guarc’s resolution. The chapel has a statistical value of three canes and is situated over a radius of 54 spans. Guarc’s ratio is mentioned in Macrobio’s Comentarii In Somnium Scipionis in ACTo 236 (fol. 1r. 61v) with the term “epogdo” (9/8) (ACTo 236 fol. 36v), as well as in the translation and commentary of Plato’s “Timaeus” by Calcidio (fl.350) where the epogdo is expressed as (1 + 1/8) (ACTo 80, fol. 150r–150v). This completes the issue with a fragment from Book VII of Marcianus Capella’s “Geometria” (fl. 430) (ACTo 80, fol.160v–161r) where the commensurable measure is rhētós.

The centers of the pillars in the presbyteries (P₁₁–P₁₂) and (P₁₉–P₂₀) of the new design correspond to a radius of 27 spans, half of the 54 spans of the presbytery (2 × 27). In turn, the layout of the ambulatory was carried out at 108 spans (2 × 2 × 27), a measurement related to the number nine in Book XVIII (ACTo 20, fol. 294v–295r). This reference takes into account the number of the twenty-seven verses, defining this number as three raised to the cube since three times three is nine and in turn transforming it from width to height: three times nine is twenty-seven (Civ. Dei XVIII.23,2) (ACTo 20, fol. 295r)23. Nine is also referenced in “De Genesi ad Litteram” (ACTo 86, fol. 59v–124v) in relation to its divisors of three and six, recognizing the three parts of three, which is one-third of nine (De Gen. ad Litt. IV.2, 2)24.

The width of the cathedral of 150 spans is determined using the statistical value of its radii (17.423 ± 0.065 m (17.423 m)), which is equivalent to 75 spans. At the end of the codex Supra psalterium [6] from a reference of the old inventory, which we have interpreted as Enarrationes in Psalmos in the commentary on Psalm 150, Augustine of Hippo provides a substantial apology for this number25. Firstly, he considers the number 15, from which 150 is formed, since the number 15 in relation to the simple numbers is a relation of the number 150 with respect to the tens because it comprises fifteen times ten. The number 15 symbolizes the conformity of the two Testaments. In the Old Testament, the Sabbath is observed, which signifies rest; in the New Testament, the Lord’s Day signifies resurrection. The Sabbath is the seventh day of the week and is Sunday, which immediately follows the seventh. The number 50 contains a great sacrament within itself, as it consists of weeks with the addition of one day, as if it were the 8th, which completes the number 50. Seven times seven results in the number forty-nine, to which one is added to form fifty. The number 50 contains great symbolism: Counting from the day of the Lord’s resurrection, on the fiftieth day, the Holy Spirit came upon those who were gathered in Christ (Enr. Psal. 150, 1). He adds that 50 refers to penance, 100 to mercy and judgment, and 150 to the praise of God in His saints as we progress towards eternal life (Enr. Psal. 150, 3). A similar conception appears in the Liber numerorum qui in sanctis Scripturis occurrunt (612–615) (Book of Numbers) attributed to Isidore of Seville (c. 560–636), where this number takes on significance, through the Holy Scriptures, as a perfect number and of those who are predestined by God toward eternal life. (Lib. Num. 27,109)26 (Pardillos 2000, pp. 285–304).

The metrological condition is based on the chapel with 24 spans (3 × 8 spans) and 3 canas, with a total width of 150 spans [21–27–27–27–27–21] (Figure 19a). Given that the radial chapels have a depth of 21 spans, the metrological construction is 3 × 7, 3 × 9, 3 × 9, 3 × 9, 3 × 9, 3 × 7 spans, allowing for the tracing of the apse without the circumference that inscribes it (Figure 19b). In the background, there is an intrinsic relationship between seven and nine, which is recognized in De Musica libri sex (387), in the numerical articulation of the half-feet: (2 + 2) + (2 + 1) = 7 and (2 + 2) + (2 + 3) = 9 (De Mus. V. 4, 7)27.

4.2. The Metrology of the Apse Section and Augustine of Hippo

Analyzing the height of the radial chapels (45 spans) in relation to the proportionality of the theory of the Gothic section with respect to the metrology of the floor, an initial relationship of (9 ÷ 5) is expressed. The reference value of the keys of the ambulatory [16.070–15.94 m] is within the range of close to 70 spans, with a floor layout of (9 ÷ 6). Finally, the main key is situated at 100 spans above a total width of 150 spans from the ambulatory relative to the base of the inner wall of the facade. With these values, we can propose a floor layout of the measurements [81–105–150 spans], elevated in height in a corresponding metrological manner to [45–70–100 spans] (Figure 20a). The evidence of a change in proportionality of the model by Pascacio Xulbi (b. 1383–1441) and Juan Xulbi (b. 1416–1428) with respect to the initial model entailed a structural and visual change, where the image of the Coronation of the Virgin will not only have a central vision but will also visually preside over the chapels of the ambulatory in a centripetal manner (Figure 20b). At the change in section and the proportion 3:2, sesquialtera appears in the commentary of Timaeus by Chalcidius (fl.350) (ACTo 80 146r–155 v), as in Comentarii In Somnium Scipionis of ACTo 236 (fol. 1r. 61v) by Macrobius.

The De civitate Dei (413–426) addresses the judgment in Book XX (ACTo 20 fol. 333r–359r), dedicated to the two resurrections and the thousand years of the apocalypse (Civ. Dei XX.7) (fol. 337v–338v). It uses the thousand years to designate the duration of the world, employing it as the fullness of time. The number one thousand is the cube of ten, and ten times ten is one hundred. This is a flat figure, and to make it solid, it is necessary to multiply one hundred by ten, and we already have one thousand (XX.7, 2)28. The carving and roughing of the key attributed to Bartomeu Santalínia (b. 1420–1440) exhibit a neck (1.625 m ± 1.502%) within the range of 7 spans and a height (1.280 m ± 1.502%) within 5.5 spans, where the ten vaults and the eleven transverse arches converge. The crown of the ten angels surrounding it has a measurement (203.00 m ± 1.502%). Given the geometric difficulty of stone carvings, hypothetically, it could be thought to have had a diameter of ten spans such that the measurement of key 10, when multiplied by 10, is situated at 100 spans, and when it is multiplied by 10, it becomes 1000, representing the fullness of time.

The iconography of the Coronation of the presbytery corresponds to the enthroned Virgin, situated to the right of her Son (Puigarnau 2023, p. 914), who is being crowned with glory and honor and holds an orb in her hand (Hebrews, 2:9); she also exhibits a gesture of reverence before the King of kings and Lord of lords (Revelation, 19:16), both sharing the throne surrounded by ten angels held in their flight (Figure 21b).

The Coronation of the Virgin concludes with the episodes of her Dormition and her Assumption into heaven, which are frequent themes in iconography attributed to Melito of Sardis (+c. 180)31. It was popularized in the passage Summa aurea (1261–1266) by Jacobus de Voragine (ca. 1229–1298), Assumptio Beatae Mariae Virginis Cap. CXIX, which states, “Come from Lebanon, you who are to be crowned.” (Voragine 1999, pp. 479–83), and this was referred to in the 1420 cathedral’s inventory: Item, liber vocatus Flos sanctorum, in pergameno, in latino, qui fuit honorabilis domini decani Dominici Meseguer [38]; Item, Summa Aurea, in pergameno, cum littera minuta, spessa, cum cohopertis virmilis [40]; and Item, quidam liber vocatus Flos sanctorum [48] (Baiges 1999, pp. 3–20). The image of the cathedral (Figure 21b) is consistent with the illumination of the Horae ad usum parisiensem (1375–1425) (BnP, NAL 3093-2, fol. 75v) (Figure 21a).

The Coronation also appears to illustrate Book XXII of De Civitate Dei, for which its object is the eternal happiness of the saints in order to attain the City of God, thus giving solidity to the faith in the resurrection of bodies, where it is also linked to the Final Judgment, given that the crown is of life (Revelation, 2:10) and where the purely human woman is the first to fulfill God’s project for the divinization of man.

The ten angels surrounding the coronation appear in De Civitate Dei (BnF, Franc. 24 fol. 262v) (15th century) (Figure 21c), accompanied by bishops, the Pope, and the Saints. The keystone of the presbytery is protected by eleven winged dragons arranged on the crossing arches converging at the main keystone. The meaning of the dragon, symbolizing the combat to be overcome by Virtue, is that it is a protective element of the temple and the faithful who seek refuge in it (Figure 22).

The placement of the keystone took place with a Solemn Mass celebrated on 27 September 1439. The chosen date is the first Sunday after the forty days of the Assumption, which could have been better chosen by taking the Legenda aurea as a reference: “When my dormition occurred, all the apostles were by my side. They were precisely the ones who, with utmost reverence, carried my body to the tomb. Forty days after my death, I resurrected” (Voragine 1999, p. 482). The document says, “Today, Sunday, 27 September 1439, with the Altarpiece of the Cathedral open and well adorned, all the bells ringing, which had rung last night for the occasion... Here the office was performed, and the Mass very solemnly of the Assumption of the Virgin Mary” (O’Callaghan 1887, pp. 17–20).

5. Conclusions

The Gothic cathedral of Tortosa, conceived from 1346 onwards, is the last of the Catalan Gothic cathedrals. Through the results obtained, its peculiar characteristics are visualized both in terms of its architectural order and its construction systems within the southern Gothic context. The principles in ecclesiastical replacement since 1155 by Bishop Godofredo (+1165) of vivere sub regula Beati Agustini, et iuxta consuetudines Ecclesiae Sancti Ruffi, render the figure of St. Augustine present both in intellect and in the iconography of the vault keystones. The abundant work still preserved in the eleven codices and thirty-two works is the greatest idolatry of the promoters of the cathedral. With the principle of numbers used as the ordering of creation by St. Augustine on the one hand and, on the other hand, the results obtained via quantitative methodology focused on the constructive reality analysis of the Tortosa Cathedral through close-range photogrammetry (CRP) and terrestrial laser scanning (TLS), it can be concluded that the construction was executed with great precision and with an error of less than 1.502%.

These direct sources coincide with the large dimensions of the cathedral: 150 span wide, 100 span high, and keystones measuring 10 spans in diameter, surrounded by 10 angels and the 7 radial chapels. The main works of Chalcidius and Capella (ACTo No. 80) and Macrobius (ACTo 236) also appear in this library. What is particularly interesting is the tonal ratio (9/8) of the epogdo used by Macrobius’ Comentarii In Somnium Scipionis, who was the teacher of Augustine of Hippo and who appears in the codex (ACTo 236 fol. 36 v), and the ratio is used for the layout of the seven radial chapels. This thesis is in line with the principles of classical art historiography originating from Wilhelm Worringer (1911), Erwin Panofsky (1951), and especially Otto von Simson in Order (1956).