The next investigator was the late Prof. Cockerell, R. A., who, in his essay before noticed (pages 1007, 1010), considers that Cesare Cesariano "may be said to have done to a great extent in that style what Vitruvius did in the Greek, namely, in discovering many of its fundamental doctrines and principles. More especially does he reveal the estimation in which Vitruvius was held during the middle ages; and the interpretations of his rules attempted by the architects and commentators of that period." Thus, the church in the Castle of Nuremburg, built by Barbarossa in 1158, and the Frauenkirche, probably of later date, in the centre of that city, are exact illustrations of the temple in Antis" of Vitruvius, as given by Cesariano, lib. iii. fol. 52. The use of the work of Vitruvius, about 1284, is also recorded in Galiani's edition of the author, by an amusing story connected with the building of the Castel Nuovo at Naples. "It is needless to produce any further proofs of resemblance," writes J. S. Hawkins, in his History of the Origin of Gothic Architecture, 1813, p. 223, "than to say that, in every Gothic cathedral as yet known, the extent from north to south of the two transepts, including the width of the choir, if divided into ten, as Vitruvius directs (for Tuscan buildings, lib. iv. cap. 7), would exactly give the distribution of the whole. Three arches form the north and three the south transept; the other four give the breadth from one transept to the other. One division of the four being taken for each of the side aisles of the nave. and two left for its centre walls, the complete distribution of the nave is also given. Of the proportion of one-third of the whole width as the height of the columns, the cathedral of Milan is a decided instance. The two transepts together are 110 cubits, the breadth of the choir 28, making together 138; and the height of the columns is 46 cubits." The rules named by Cesariano occur in his Commentary, fols xiv. and xv., and he illustrates them by the plan and section, of Milan Cathedral, which was commenced in 1386. The figures are entitled" ichnographia,"-" orthographia,"-" scenographia," sacræ Ædis Baricephala, Germanico more, a Trigono ac Pariquadrato perstructa," and, "secundum Germanicam symmetriam," and again," per symmetriæ quantitatem ordinariam ac per operis, decorationem ostendere, Germanico more," &c. The first rule, "a Trigono," establishes the respective proportions of the length and breadth of the cross, which are included within the two arcs of 102°, constructed according to the first proposition of Euclid. The fig. 1230. has been commented upon (page 1010) as involving the vesica piscis. Mr. Cockerell continues his remarks by noticing Mr. Kerrich's paper in the Archæologia, xix. p. 353-61, wherein that author uses the figure but does not confess his debt. In all the examples given by him the vesica is applied to the internal length and breadth. The second rule, "a Pariquadrato," is effected by dividing the area comprehended in the vesica, into commensurate squares or bays, on the intersections of which the columns and buttresses are placed. The number of them will be determined by the extent of the plan. Fig. 1230. represents the illustration of the rule in which 14 by 8 are used; in the chapels of Wykeham we have 7 by 4. The plan, Fig. 1231., explains the determination (by the symbol of the resica piscis) of the length and breadth of a church; the subdivision into squares, the position of its piers, &c. Fig. 1232. explains the rule by which the heights of the vaulting, the roof, the spire, &c., are determined, namely, by equilateral triangles erected upon the plan. The woodcut exhibits both a single and a double aisled church. This important fundamental rule will be found to be applicable to cathedrals in England, as at York, Winchester, Worcester, Lichfield, Hereford, Salisbury, Norwich, Exeter, Westminster, Romsey, and others: in Italy, in the church of San Petronio at Bologna. and in most of the works of the architects Lombardi, as San Zaccaria and San Salvatore, at Venice: in France, Fig. 1231, Fig. 1232. Fig. 1230. in the cathedral at Rouen, and in others: and in Germany, in those at Prague, and others. But it is to be noted, continues Professor Cockerell, that another rule of distribution (not yet discovered) is more frequent in the latter countries. The third rule, also a Trigono," is orthographic, and establishes the normal heights in the elevations and sections by equilateral triangles, according to Cesariano. fol. 15. "The application of the first and second ruses in New College Chapel is exact; the whole and the parts are commensurate, as well in the bays or squares as in the subdivi sion of the bays of the windows; of the flanks, as also of the west end. While in All Souls' and Magdalene Chapels, the two copies of the former, the divergences are extreme. Fig. 1233. is the plan and its subdivisions of New College Chapel, Oxford. To our author's own work we must refer the student for the apt remarks and comparison of the three plans, merely adding that the first is three diameters long, while the other two are less than three diameters, by which Professor Cockerell apprehends that the rule had been lost, or was disregarded, although Chichele's chapel was built by "the King's masons." The chapel at Winchester is upon the same principle, the number 7 including the vestibule, which only occupies one of the divisions instead of two, as at New College; the relation of three diameters is obtained without making the diagram, as in New College Chapel, inclusive of the walls. OXFORD. The application of the third, the orthographic rule, is not traced so distinetly in the elevations and interior Fig. 1233. PLAN; NEW COLLEGE CHAPEL; of New College Chapel, though more exactly in that of Winchester, and we also perceive the value of the principle of the extension of these squares laterally, for the purpose of establishing the height of the ceiling, and of the pinnacles in the east and west fronts." The next exponent of this instructive subject was R. D. Chantrell, who in 1847 read a paper on the Geometric system, before the Institute of British Architects. It was printed, with cuts of the two chief keys of the system, in the Builder for the same year. He first refers to Mr. Kerrich's use of the vesica piscis, explaining it, as in fig. 1234. Where the Fig. 1234 Fig. 1235. Fig. 1236. chancel is separated by an arch, the plan is subdivided by taking the breadth as radius (fig. 1235.), as at Routh Church, near Beverley, the vesica coming sometimes within the walls and western arch, and at others extending to the western face of the arch in the nave, in many works of the 13th century. The apse is sometimes included in, and sometimes excluded from, the vesica. Where the nave and chancel vary in breadth, the base of the triangles equal the breadth of the chancel (fig. 1236.), its length being determined by the vesica, and in each of these cases the breadth of the nave is obtained by framing a similar vesica upon the remaining length. The Anglo-Norman church at Adel, in Yorkshire, is defined upon the extreme length Fig. 1237. internally, as shown in fig. 1237., and The equilateral triangle alone has Fig. 1238. of York, produced a system on the circle. By placing a square or cross on the centre of the circle, dividing it into four equal parts, centres are obtained for vesica (fig. 1238.) of different proportions to those formed by the double triangles. By striking these radiai lines upon each of the four points on the circumference, centres are produced in abundance for quatrefoils, crosses, and other figures applying more especially to tracery. The first vesica gives the proportions of the naves and their aisles of the cathedrals at Durham. Ely. Peterborough, Canterbury and Salisbury, but no others, and cannot tl.crefore be considered an universal system. In 1842 Mr. Chantrell developed a system which includes that of Kerrich's. Its formation is detailed in the journal named, to which we must refer the investigator, as the essay has not otherwise been published. Fig. 1239. will at once show the principle, and if it be drawn out to a very much larger scale it will not appear so complex. Besides the triangles, the points are obtained for many polygons. The six divisions, A A, B B, from the semidiameter are first obtained; and straight lines drawn to each alternate one give triangles. On their intersections, as C C, if lines be continued to the circumference, six centres are given, DD, FF, upon which, with the first radius A B (or of the semi-diameter), strike a second series of segments, and a third set of 12 centres is obtained. The second centres will give two intersecting triangles, completing the first part of the design. Upon the 24 points of the intersecting inner ares, a circle inscribed will determine the inner triangles upon the centres of the first, and the diagram is perfected. For more complex forms, an additional number of centre lines may be drawn upon the remaining intersections. The number 10 was, according to Vitruvius, Plato's perfect number; but the antiPlatonists, with their 6 or the radial division of the circle (A to B, fig. 1239.), could, by the working of their centres, without the necessity of dividing with the compasses, produce the 10, showing that they were the more perfect. as their system combined with all others. The examples named by Mr. Chantrell, in which the system is clearly exhibited," are the rose window in the south transept of York Cathedral; that of Winchester Palace in Southwark (fig. 1190.), but slightly varied and almost undisguised; and the east window of Hawkhurst Church, Kent. Walkington Church, near Beverley, using the entire diagram, affords a simple illustration; whereas Kerrich's plans are proportioned upon the second radial figure produced by the division of the circle, they should be placed upon the base of the great triangle, thus facilitating the operation of giving proportion to a plan. In the composition of the cathedrals at Ely, Lincoln, Canterbury, Norwich, Salisbury, Worcester, Durham, Peterborough, and Winchester, the general proportion is determined by the first of the 24 subdivisions on each side of the centre intersecting the great triangle. The abbeys are all produced on the intersections of the triangles and their centres, and the subdivisions for the piers are found in the centre portion of the diagram, with this occasional difference, that transversely the radial lines may either pass through the centres of the piers or come on the outer or inner faces, to conceal the principle on which they were based. Thus M (fig. 1239.) is part of the plan of the nave of Boston Church, Lincolnshire, arranged on the former principle, while N is part of that of Middleton-on-the-Wolds Church, Yorkshire, where the lines come on the inner face of the piers. "For the elevation, proceeding with the double spherical triangle upon the centres longitudinally, and the variations before noticed, transversely, the various heights were obtained for the pillars; and the subdivisions by the spherical triangles upon them gave arches, capitals, and bases, triforia, tracery, mouldings of every description, and due proportion to each feature. I have every reason to believe," concludes Mr. Chantrell, "that this system will apply to the works of all ages that can be tested by sound geometric principles." The results of the investigations published by E. Cresy in 1847 are added in Sect. VI. F. C. Penrose, in his investigations at Lincoln Cathedral in 1848, for the Archæological Institute of Great Bri'ain and Ireland, urges that "the tendency towards the system of designing on the square, with greater or less degree of approximation, is found to occur in so many churches that it is a law which had great authority with, at least, the more orthodox of the middle age architects, although they did not scruple to modify it when they saw occasion." He decides that the nave of Lincoln Cathedral was formed on this system on the intersection or intended intersection of the piers, and coinciding with the outside of the main walls. The choir seems to be built upon the true system of squares, which are of the same size as those of the nave, but the greater width of the former allows of the squares coinciding with the inside of the walls. The height of the choir appears to be obtained, as is so frequently the case, from that of an equilateral triangle, whose base lies within the walls. The height of the nave is obtained by a square placed within the same limits, which, though less symbolical, is more commensurate." He thinks that if the length of St. Hugh's choir could be recovered, the whole length from east to west was then such that it included the transepts within a resica piscis. He also conceives (for reasons he states) that the architect to the presbytery had access to the original drawings prepared for the earlier parts of the building. The ratio of the voids to solids appears to be more remarkable than is to be found in any vaulted building in Europe, at least among the larger structures. Very careful measurements taken immediately above the plinths give voids 1056, supports 107, or the former nearly ten times the supports, including in the latter, the external buttresses and walls; and including in the voids, the clear internal area of the church. 66 Mr. Penrose gives the measurements of the heights of various parts taken by him with great exactness, and this height he divides into 26 parts, which will be found to agree in on exceedingly accurate manner with the principal divisions of the bays. In Bourges Cathedral, he states the height of the vaults agrees with that of an equilateral triangle whose base occupies the breadth from centre to centre of the external walls. Some of the heights may be obtained from parts of this triangle, and others from integral numbers of French feet. In the cathedral at Metz, the height is 130 French feet=1386 English feet. If this be divided into 300 parts, various proportions of them determine heights. The cathedral of Ratisbon appears to be founded on the triangle taken as that at Bourges. The ratio of its height to length is as 3 to 8; and is 1042 English feet high. or 110 Bavarian feet. These results are well worth further consideration, from the well-known conscientious manner of taking measurements adopted by Mr. Penrose. list An early investigator, Mr. W. P. Griffith, published in 1847-52, numerous essays on this subject, as named in the of books in the GLOSSARY addendum. He exhibits the adaptation of the square, set square and diagonally, one u on another (as in fig. 1073.) to the church of the Holy Sepulchre at Cambridge, fi. 1240. and also of the triangle, for early churches, as at Little Maplestead, fig. 1241. Westminster Abbey Church, and the cathedrals at Salisbury, Winchester, and Rochester, are based upon a triangle whose base being the width of the nave, including the walls, is placed upon the centre of the central CHURCH OF THE HOLY tower, and three of these triangles will include the SEPULCHRE, CAMBRIDGE. length of the nave. Ely Cathedral, Redcliffe Church, Bristol, and Bath Abbey Church, are proportioned in a similar manner. "We must insist," he writes, "after a primary figure of form or unit has been given, that each part Fig. 12. CHURCH AT LITTLE MAPLESTEAD. Fig. 1240. produced shall bear a proportion to each other, and to the original unit. Although the equilateral triangle dictated the general proportions, the square and pentagon were found very useful in the details. The chapter houses of Wells, York, Salisbury, and Westminster, are proportioned by jo`nt squares forming an octagon; and those of Lincoln, Westminster, Worcester, and others, by two conjoint pentagons, forming a decagon." He illustrates the formation of the plan of Salisbury Cathedral, both on the square and on the triangle; but, as noticed respecting Milan Cathedral, although the square appears to suit best for the plan, the elevations appear to have been set out upon the triangle. Fig. 1242. shows the system 19fect applied by him to the plan of Sefton Church, Lancashire. A comparison of the number of equilateral triangles, as named by Mr. Griffith, in fixing the height of buildings, will be as follows. viz., Westminster Abbey, 6; King's College Chapel, 4; Lincoln, S; Hereford Cathedral, 4; Peterborough; 3; Lichfield, 3; Exeter, 4; Worcester, 3; Bristol, 3. The loftiness of Westminster Abbey is attributed to the cause that the cloisters adjoining (similar to double aisles, as originally intended) being included in the Fig. 1242. CEFTON. 100 base of the triangle of the transverse section, therefore the height of the abbey is more than the cathedrals. The chancel of Bristol Cathedral has no triforium, and is accordingly less in height. These buildings having been based upon the equilateral triangle, that figure will alone be a key to them, and it will be futile to try the square. In Westminster Abbey that figure mostly abounds (in trefoils, hexafoils, dodecafoils, &c.); while Salisbury Cathedral, being based on the square, that figure and its products will be found chiefly employed (in tetra'oils, octafoils. &c.). This building is 4 squares high. Dr. Henszimann, in his Remarks on his alleged discovery of the constructional laws of mediæval church architecture, read at the Institute of British Architects, 1852, states that the architects of old did not employ much reckoning in their constructions, but used geometric forms.-In studying the churches, I became persuaded that out of a ground-line or sum, considered as a basis, there can be developed, either by a geometrical or algebraical method, between 30 and 60 sums or lines, corresponding to the size, age, and importance of the building, and there is, with very few exceptions, not a structural member, be it large or small, the proportions of which are not defined by one of these lines or sums, or excep. tionally by their multiples or divisions." He published the first portion of his elaborate system in 1860; this, together with the extensive system put forward by D. R. Hay, of Edinburgh, must be left to the reader to investigate from the books themselves. The last of the investigators with whose system we shall trouble the student is W. White, who published it in the Ecclesiologist for 1853. He has perceived that each architectural period has its own appropriate order of rules, and this in minute accordance with an intelligible system of development. Thus, in the Norman period, the general proportions of the plan are reducible to the square, and the relative proportions and positions of the minor parts chiefly by the equilateral triangle. As architecture progressed the square disappeared, and to the outline and detail was applied the triangle. In the middle of the 14th century, as art declined, the triangle was forgotten, and a system of a diagonal square was taken up. Since then mathematical proportions have been chiefly employed, especially that of the diagonal of the square, fig. 1243. "The figures applicable to the setting out of mediæval buildings are these: 1. the square; 2. the equilateral triangle; and 3. certain ares described upon diagonals and bases of the same." Thus, in Norman work, the proportion of a square placed lozengeways from the ends of which a vesica piscis is struck (fig. 1244.) is in common use. In first pointed work, the proportion is that of a square touching the head and sill (fig. 1245.). The system shown in fig. 1246. seems chiefly used in lancet windows and works of tat period, the height being first determined. The proportion in fig. 1247. is used in traceried first pointed, the vesica, giving the width, being obtained from the apex of an equilateral triangle. The proportions figs. 1248. to 1251. predominate in middle pointed; those of II. and V. the same, only in the latter period the rule is applied to the determining of the lights or bays instead of the whole opening, and is applied to the centres of the mullions and not to the sides only. All these proportions appear to have been equally well known in all early times, but in the middle pointed period they gradually became more complicated, and are consequently more difficult to trace out. In third pointed they can hardly be found, and in obtuse third pointed they quite VIL Fig. 1252. VIII. AA Fig. 1254. CE Fig. 1253. disappear, the proportions shown in figs. 1243. and 1252. taking their place. lateral triangle of 60°, D E F (fig. 1253.) used to obtain one point, is often accompanied by the angle formed of 30°, E F G, to obtain another relative point; each equal subdivision |