Pantheon

olden nen U Jema & ui TONS BRUNES S OF THE SECRET Ancient Geometry 257 VOLUME II RHODOS COPENHAGEN INTERNATIONAL SCIENCE PUBLISHERS,

ing. The same division was applied to the half circle that surrounds the altar. But the main aim of our search has (I trust, convincingly) been fulfilled: to illustrate that the design of Cologne Cathedral was built up according to geometric principles that were already hoary with age and use hundreds of years before the cathedral was built. 39 The analysis has picked out in the main the dimensions and proportions that give the cathedral its distinctive appearance and character. It pointed out the framework round which the details were built, and it brought clearly to notice the definite link between the design of the facade and that of the ground-plan, familiar from previous analyses. The Pantheon WALK THROUGH the older quarter of Rome, capital of the ancient world, and you are bound sooner or later to come face to face with one of the most impressive buildings from antiquity: the Pantheon. The building rests like a giant animal amid a conglomeration of tightly packed houses and apartments, which seem loath to yield room for the great structure. From floor to its beautiful, domed ceiling the Pantheon measures almost 45 m. (approx. 140 ft), and from side to side almost 56 m. The main entrance to the building faces a small plaza, and from this opening in the dense housing mass one can stand back and see the grand approach and entrance to the Pantheon in its entirety. The complete frontage, as it stands today, is quite obviously inspired by early Greek temple structures, having eight frontal pillars and richly decorated capitals. Above the columns the sloping roof rises at a somewhat steeper angle than one is accustomed to seeing in a Greek temple, but the style is the same. But comparison with a Greek temple goes no further than the door of the Pantheon. The frontage appears almost to be stuck on as an afterthought to a rounded temple that has an all-enclosing wall instead of a series of columns. We would look in vain therefore for the typical cloister effect found in Greek buildings. As in the case of most monumental structures from early Christian times, the Pantheon has had a dappled history of reconstruction and no longer bears its original appearance. In Fig. 237 we have a picture of the Pantheon, showing its present face. The name Pantheon denotes that it was a temple dedicated to all gods in the community, and history says it was built originally by Agrippa in 27 B.C. as a tencolumn temple, but nothing remains now of the original structure apart from one or two pieces of foundation. These show that the original building was 3.7 m. lower at its base than the present version. The old, original temple burned to the ground about 81 A.D., and Domilian built a completely new style of temple in the ruins of the old. The apparent reason for the new building being placed so much higher than the older seems to be that the burned-out shell 44} & I Fig. 237. of the former temple was simply flattened, rolled and used as the foundation. The new building, too, has since disappeared almost without trace. Historical records indicate that the building had a large circular courtyard at its centre. It would appear to have covered a larger area than the present Pantheon. Its life was short. In 110 A.D. the building, like its predecessor, was razed by fire. The job of planning and rebuilding a new temple was given to Hadrian. Hadrian was not content to produce a carbon copy of the previous temple, but planned a new structure from foundation to roof, taking as his starting point the dimensions of the old inner courtyard. This he made into a circular temple which he topped with a fine domed roof, the ceiling of which he decorated with a peculiar stucco design. The ornamental ceiling does not however bear the same design as the temple capitals or later decorative panels. Its lines and circle look more like a geometric diagram than an architectural decoration. A large hole was left in the centre of the domed ceiling, and through this the light of day streams into the church producing a most unusual shadow effect on the plaster ceiling. We see this in Fig. 238. Hadrian’s reason for erecting a temple which in height exceeded its predecessors but which occupied less area of ground was perhaps that Rome even then was densely built up, and crowded so close to the site of the Pantheon that he was forced to restrict the ground area in order to create a sense of space. There is little doubt that the new structure is smaller than the old in ground area. In the previous building the round courtyard was a part of the building, but only part; the same area is not a part of

41 Fig. 238. Fig. 239. the new building—it is the new building. Historians are not too sure who built the present entrance. It may have been Hadrian, but may possibly have been a later builder, Antonius Pius. Or even a third party. But the builder of the entrance is not of great interest to a geometric study such as ours, since the original facade probably exists no longer. As late as 1606, for example, Pope Barberini had the Pantheon’s bronze porch beams removed and melted down to cannon. The building suffered another “face-lift” in 1747 when, among other things, the windows were altered in size. In spite of these attacks on its originality, the church has retained a composed beauty. It strikes one immediately and inevitably. It is an impressive experience passing from the darkened hall through a narrow opening in the beatencopper door into the dome-shaped church hall. Light pours down from the gaping roof, reflecting on the stucco ceiling, and providing considerably more natural lightsent building to be a brand-new erection by Hadrian around 117—138 A.D. ing than usual for a Catholic church. been, was an ingenious planner and designer. In addition to a wonderful gift of unfettered imagination, he had at his fingertips “the royal art” of geometry. We ought thus to expect to find this monu- The floor is laid in variously shaded marble, shown at its best in the strong daylight. The design resembles a giant chess-board, the squares in turn being laid in mosaic form. There are two motifs: a square with a circle inscribed, and a square with a smaller square inside. The patterns alternate as do the black and white squares on the chess-board (see Fig. 238). Already one senses the presence of ancient symbolism and geometry. Hadrian, whatever else he may have ment to his expertise laid out in accordance with ancient geometric principles. freehand with no indication of source or authority, and cannot therefore be used in a critical geometric study. However, I came across the best material on site, i.e. in the front hall of the Pantheon in a booklet describing the building. Called quite simply The Pantheon, the booklet is the work of Roberto Vichi. Its The rather severe alterations to which the building was subjected in 1606 and later in 1747 leave some doubt about the original design and dimensions of the entrance and windows. These will therefore illustrations are apparently based on accurate measurement of the building, and the sources are quoted of both drawings and photographs. One sectional drawing shows the interior of the building and the structure be omitted from the survey, and we shall of the outer walls. It also shows the pas- In our study of the Pantheon through stick as close as possible to the building sages that lie between the outer wall and the eyes of the ancient geometer, we must ignore the two previous buildings that thought attributable to Hadrian alone. the temple hall. The drawing gives only one (the left) side of the building, as far as the vertical axis. But as we require the whole cross- It was a rather difficult task tracing stood on this site. Of them there is virtusuitable material for an analysis, since ally no trace. We may consider the premost available drawings and plans are in

7 | ye IBAN: 12 | | | | | 26 28 | © 4 Sa | NR 1 È Pi i f i le = Ge Ne 27 29 tI È rev i | = pu à Fig. 241. Fig. 242. We notice immediately that neither the upper side nor the base of this square has been raised and surfaced so often in the past that it is now level with the floor of gram. As regards the upper horizontal (3-4), we are accustomed to seeing it tion reveals however that there were steps round the building at one time. The Pantheon is built in a fairly undulating area of the city, there are steeply E; in 7D À any contact with the building in our diacer placed well above the roof of the building or temple under discussion. But the square’s base-line usually has its place at the foot or base of the building. 23 6 45 THE PANTHEON ANCIENT GEOMETRY il 8 13 5 Fig. 240. section, the illustration was photographed and repeated on the right of the axis in order to produce a complete. picture of a slice of the Pantheon. Apart from the double width, the illustration in the present book, Fig. 239 is almost identical in size to the original. As usual, the first step is to find the constructive basis of the building’s plan. As with most of the other buildings we have examined, there are a number of with the buildings outer wall. We saw this applied to samples of Grecian temples. In Fig. 240 the two verticals were entered flush with the walls, and the problem then remained to find the Pantheon’s horizontal axis. I found this to be at line 2-9, where it is marked both outside and inside the building by a frieze. The inside frieze marks the level at which the inner hall wall changes from the vertical to the possibilities in the Pantheon. Its width exceeds its height, and the natural (and correct) assumption is that Having fixed the horizontal axis, we can go ahead and construct our basic the basic square’s vertical sides run flush square: 3-4-5-6. curved dome. This is in fact also true of the Pantheon. But whereas we normally analyse a facade drawing or photograph of a particular building, with the Pantheon we have a sectional drawing, a slice down through the middle. And therein lies the explanation. Facade plans normally show the outside steps leading up to the actual temple entrance, but these steps do not show in a sectional drawing. We may therefore assume that base-line 5-6 marks the bottom of these steps that lead up to the temple. But there are no steps leading up to the front of the Pantheon! We saw this clearly in the photograph in Fig. 237 taken from the square in front of the building. This apparent absence of approach steps is caused by the fact that the square has II 4* the Pantheon’s porch. A further inspecsloped streets all around. The street, for example, running behind the Pantheon is about 10 m. higher than the church’s base. To protect the building against encroachment from surrounding houses, the authorities built a wall. Outside it lies bustling Rome. In Fig. 241 we see a view af the righthand side of the Pantheon towards the rear of the building, illustrating clearly the “moat” between Pantheon and the adjacent neighbourhood. Fig. 242 is a photograph of the building’s left side, showing a similar variation in ground level. In Fig. 243 the camera has caught a view along the right side of the building towards the front, and we see here that recent authorities have placed steps from ground level up to the temple’s porch— proving that the base of the building lies

ANCIENT GEOMETRY obviously a planner of the ancient geometric school. Our survey continues with the entry of the square on the circle’s rectangle, which we place centrally in the diagram. Even more lucidly we see the building’s plan open before our eyes. The square is marked by lines 14-15 and 16-17. The former, we see, indicates the internal height of the main hall, passing along the uppermost curve of the dome. When we inscribe a circle in square 14-15-16-17 we see that it follows exactly Fig. 243. in effect much lower than the level of the front plaza. This is evidence that our base-line 5-6 in Fig. 240 is properly positioned, and proof, too, that a geometric analysis can uncover or rediscover factors that only an intimate search on site can establish. Factors which may be totally absent from the drawing we analyse. We now add to Fig. 240 the acuteangled triangle 6-7-5 in order to mark off the circle’s rectangle. The latter is seen as 10-11-12-13. The basic square’s proportions were selected so as to run flush with the extreme outside of the building, i.e. the projecting sills that encircle the Pantheon in the form of a frieze. Now that we have entered the circle’s rectangle we can establish another part of the plan: the rectangle’s vertical sides form the basis of the positioning of the ring of pillars that run around the outside edge of the main hall. If concrete indication were required that the Pantheon was planned according to the principles of ancient geometric diagrams, the positioning of these pillars certainly whets the appetite. Hadrian was the sweep of the dome and takes in part of a colonnade halfway up the inside of the hall. Other drawings of the Pantheon exist, showing a circle drawn in a similar position, but that circle lies entirely within the hall and does not touch the surrounding columns. But if the arc of a circle is to follow precisely the curve of the dome, it must be drawn as indicated here. A circle with either a shorter radius or a different centre will not match the curve of the ceiling exactly. The other drawings are not in detail, and give the distinct impression that the hall has been sketched round a circle rather than a circle placed within the building. Later analysis will prove that the circle I have arrived at is more likely than any other to be correct, since it marks a number of factors both in the sectional view of the building and in its ground-plan. We enter the sacred cut in the square on the circle’s rectangle. The respective lines are 18-19, 20-21, 22-23 and 24-25, Line 18-19 plays an important part in the building’s structure, representing the height of the outside vertical walling. This is the level from which the domed roof starts. The lower horizontal sacred cut, 20-21, is also marked both inside and outside the THE PANTHEON building. Inside, we see it as the height of the niches placed between the columns bordering the main hall. The actual marking is represented by a frieze and can perhaps best be seen in Fig. 238. The external marking is the edge of the sill that encircles the building. Fig. 238 also illustrates clearly the stucco ceiling in the main hall. The design is composed of five concentric rings of geometric figures, the rings reducing in size towards the centre. The smallest of the five stops short some distance from the hole in the centre oi the roof. As near as I can ascertain, the point at which the fifth and inner ring is met by a smooth area of plaster is marked by the vertical sacred cut, lines 22-23 and 24-25. The junction is indicated at the intersection of the curved dome and the two sacred cut lines. The combination of the sacred cut in square 14-15-16-17 creates another smaller square. If we again enter the sacred cut in this latter square, we find that the two vertical lines, produced upwards to the ceiling, mark the diameter of the large “sky-light” in the centre of the roof. Line 27-26 is produced to point 30, and 29-28 is produced to point 31 showing the diameter. The upper sacred cut (horizontally) is 32-33 and indicates the top of the upper row of windows in the building. The analysis of Fig. 240 has provided us with details of a number of important 49 latter is again 14-15-16-17. Thus far the diagram is identical with Fig. 240. The next step is to construct the basic square’s half-size version in the centre of the diagram. This is done by joining the intersections of the basic square’s inscribed circle and diagonals. The required square is 34-35-36-37. We execute the sacred cut in this square and observe first the placing of the upper horizontal cut. It is produced across the building as line 38-39. In the previous analytical diagram we found that line 18-19 marked the upper edge of the frieze or sill that tops the outside walling. Here we find that 38-39 indicates the underside of the same sill. The thickness of this projection is thus determined by the distance between the two upper horizontal sacred cuts in squares 14-17 and 34-36 respectively. The lower horizontal sacred cut is also produced across the building as 40-41. It indicates the top of a lower projecting frieze. The underside of the same frieze was marked in the previous analytical diagram by line 20-21, ie. the sacred cut in square 14-17. Thus the thickness of this sill, too, is determined by the distance between the two (lower) horizontal sacred cuts in squares 14-17 and 34-36. The base of the latter square, line 36-37, forms the floor of the main hall. The vertical sides of the half-size square, lines 34-37 and 35-36, appear to have been the determining factors in fixing the We may thus assume that our choice of net width of the main hall just as the sides of the basic square marked the gross width of the whole building. We notice that the first-mentioned set of lines touch basic square was correct. the projecting sills of the inside columns Our next diagram, Fig. 244, is constructed in the same manner as Fig. 240. in the same way as lines 3-6 and 4-5 run flush with the friezes of the outer col- The basic square is 3-4-5-6, in which we umns. inscribe the circle and acute-angled tri- In the horizontal lines of the sacred cut in the inner square, lines 38-39 and 40-41, we can find the relationship with the bafeatures in the Pantheon. Most of the diagram’s lines were used by the architect to place some factor or other in the building. angle 6-7-5, following this with the circle’s rectangle 10-11-12-13. The square on the

ANCIENT GEOMETRY of the wall; 44-45 is the inside of the wall; 14-16 the outside dimensions of the colonmarks nade in the main hall; and 34-37 the effective width of the main hall. This is a typical example of geometric diagrams being applied as a form of static y curve, in the same way as a present-da calrapid make engineer uses graphs to culations of height/weight/strength ratios, 424 etc. The ratio within the diagram remains constant, of course, irrespective of the size of the finished building. The deciding factor is the length of the basic square’s baseline. The smaller the basic square, the slimmer the wall; the larger the square, the thicker the wall will be. The ability to appreciate these ratios depended to a great extent on the ex- 38 perience and wisdom of the master builder, and it was also a matter of routine in applying these experiences. But I have no doubt at all that builders in ancient times followed certain rules and ratios laid down by ancient geometry when they had to determine bearing thickness, strength, proportions, etc., in a building. It was a matter of knowing the properties of the basic Ja 1345 1l 13 5 Fig. 244. sic square’s inside circle that decided the dimensions of the temple’s outer walls. And the factor that determined the width of the upper “shelf” or sill from which the domed roof rises. The sides of the basic square coincide with the extreme tip of this ledge and, as we have seen, indicate the temple’s total width. To decide on a thickness for the outer wall Hadrian then drew a vertical line through the intersection of sacred cut 38-39 in the half-size square and the basic square’s inside circle. The vertical is seen as line 42-43 and marks the outside of the main wall. That determines the outside of the wall, but how did he fix the inside? He took the area between the basic square and the circles rectangle (area 3-10-11-6) and split it in two pieces down the centre. The dividing line is seen as 44-45. The actual process of division was carried out in square 16-11-6, by means of the simple diagonal cross. Thus we have the five outer vertical lines of the Pantheon: line 3-6 is the temple’s total width; line 42-43 the outside square. It was naturally a stage that he and his predecessors had approached very slowly over the centuries. Whereas in the dawn of geometric thinking and draughtsmanship the actual lines and diagrams were a sacred subject, revealing wisdom of occult geometry and numbers, it gradually over thousands of years became a subject applied to more 47 etc. They went ahead with geometric symbols, diagrams, plans and finished structures that could support ten, twenty times the required weight—and more. The Great Pyramid of Egypt, for example, is more a mammoth memorial to ancient geometry than a building planned for the sake of economics. For its effective interior is nothing compared with the tremendous mass of material used in the building’s construction. As the religious builders gained more experience, studied their finished efforts, returned for a second look at their geometric plans, and speculated on whether such massive walls, roofs, etc., were strictly necessary, so the style of building altered. More space was allowed within the structure, dimensions were slimmed down, columns became tapered, new materials were made available—and geometric diagram becamé even more than before a tool of the builder. Once he had experimented with a particular height/thickness ratio and had found a suitable set of lines in his favourite geometric symbol, the builder stuck to it. He had found a successful shape from which to advance. He could afford to be bold. But always he came back for a second look at the geometric diagram. Hadrian was certainly a brilliant and experienced builder. A less experienced, more timorous planner would have been tempted to select line 3-6 as the outside of the main wall and not, as we have seen in the analysis, line 42-43. But the difference between the two lines meant, for practical spheres—such as the building example, a considerable saving in buildsite. For thousands of years Temple brethren had gone about the business of fixing the dimensions of a building by the principles of ancient geometric symbols. But in the beginning they knew nothing of (and gave little regard to) the most economic form of structure, wall thickness, arch curve, In Fig. 245 the basic square is subdivided 10 x 10 in the same manner as we saw in Chapter Ten. Plato, we reing materials. And cost was presumably also a factor worth remembering in those days, too. member, split the square on the circle’s rectangle 8 x 8, and by producing his

3a 10 Da x& - | % Nu SS & Ni o 7/ I 7 K-— AA NO IN / # FEN x of # \ Pe À x AR È /‘ :46 di ee à di gif FT À À1 ES Set à Cr 1 N 4 € x d ÈM VR | 7 \ S 62 IN IN | _ *% a Vs SW \ 9 sl de 1 ER, IN | wl NE \£ Se = Ss ae Si U fe à Nab PIX x PR 60 x 07 A Al WA x sé è| 1| 12 IS NA A Py a LA € we À geAni ae 1| Dee x N IN SAN SR | Na a Pd % | EN z A 4 N 3 E SS NZ N vd à >dà > ix ie. N4T a. __ ye + VI 2) / dA soli Ble sé | H ZEN a 2 he N / 53 À pe / 2 14 = se Kn se NE ti = Hoe i bici ¥ A / À Le iN ght +> di 4% SF >57 tal lower edge of the corinthian-style capi in the inner colonnade. The representative marking of line so 58-59 is most interesting. The line lies s no low in the building’s plan that it play but e, ctur stru the in constructive part 2°9 i Se AE SI 1 > 2 6 N Tue CHOICE of starting point for an ana- Pai X N 11 what lower than the building itself, it has no real marking there, but perhaps acted as the level of the foundation or the steps which originally led up to the temple. These three surveys of the building’s to sectional view ought to be sufficient andemonstrate the guidance given by cient geometry in planning the structure of the Pantheon. Pantheon’s ground-plan N VA square’s base-line, which is part of the original construction. Since it lies somei ar 7 of nevertheless, to point out the existence the in made been has mark a the line, main waist of the pillars that surround the church hall. Line 16-17 is the same as seen in Figs. e 940 and 244. It is the base of the squar is this on the circle’s rectangle when placed centrally in the diagram. Finally we come to line 6-5, the basic i PIZZE po Pa n 55 N i “4 ie Sd Sa ; between of the windows. the Line 56-57 marks, as we can see, ; è 4 : far as points 52 and 54 is equal to the hei ee 2 more clearly seen in Fig. 238. As can be ascertained, the distance té < Va into the These are gallery above the main hall. —-349 da Pa the windows and niches backing di ze ie As + 3 sill of Line 54-55 indicates the lower 7 Po. basic square. and its In Fig. 240 it was numbered 2-9 . ysis anal that in value was illustrated Sat Se I mc D rg m, running place or value in the diagra outside. the merely through a frieze on line 52-53 h wit We are already familiar as the horizontal axis of the =e KENN mS i ays ZN Pi è 48k— = è i u ? x= 3 =< ! 7 4 LN a: NER Ds | ir Nr, lat a i % 4 49 61 63 13 ; ‘5 Fig. 245. lines of division to the basic square, divided the latter 10 X 10. In the present diagram the circle’s rectangle is 10-11-12-13, and its square lies at the bottom of the figure: 46-47-1311. We divide the latter square with verticals, horizontals and diagonals as shown previously, and produce these to meet the sides of the basic square. The extensions are shown as broken lines. | We shall now examine the horizontal lines of 10-part division—and the result is surprising. Almost all of them are responsible for marking some dimension or other in the building. The top line, 3-4, is the constructive starting point. It lies outside the building and therefore has no place in the actual structure. The next line, 14-15, runs flush with the inside ceiling of the dome, and is in fact the same line (in Figs. 240 and 244) as the top of the centrally placed square on the circle’s rectangle. We examine next line 46-47, which runs through the upper half of the dome. From this level down to the next line, 48-49, we note that the outside of the dome is broken in a series of seven steps. Line 50-51 has apparently no special lysis of the Pantheon’s ground-plan must, as we know from past experience, be geometrically linked with the building’s facade (or in this case its cross-section). Since the Pantheon is a circular building and since the building’s complete elevation was contained by the basic square al in the earlier analysis, it would be natur e squar this of ns nsio dime the fer to trans to the ground-plan. baIn Fig. 246 we see this taken as the e entir sic square: 1-2-3-4. It embraces the ional Addit circular part of the building. the structures at the front and rear of bathe from main building are excluded in a sic square. In order to include these basic the ruct const geometric study, we square’s double-size version: 5-6-7-8. We notice first that lines 1-2 and 3-4 r in were used as the determinating facto s room gular trian the of h dept the fixing on at the front and rear of the building either side of the vertical axis. We see, too, that line 5-6 completely of encloses the ground-plan at the rear the building, while the lower line of the outside square, 7-8, appears to run aimsis, lessly through the porch. Later analy the of h dept the however, will prove that porch was determined by other factors verthan the basic square’s double-size sion. The sacred cut is executed in the large iouter square as 9-10 and 11-12 (vert lzonta (hori 15-16 and 13-14 cally) and Immediately we notice that the verti

cals indicate the width of the porch. They run precisely through the centre of the two outer rows of columns. ed cuts The combination of the four sacr re in the creates as usual a smaller squa -20. centre of the diagram: 17-18-19 If we can imagine this square flipped heon’s over in the direction of the Pant ge” entrance, with line 19-20 as the “hin 19-20or axis, we achieve a new square: re squa new the that 17A-18A. And we see ance entr te ple com the takes in exactly lay arrangement, including the part that 7-8. 5-6re outside squa the Thus we have covered the area of the ains cont 5-6 e Lin lan. nd-p grou whole re rear, the vertical sides of the basic squa (1-4 and 2-3) contain the long sides of the building, and line 17A-18A contains the entrance. In square 17-18-19-20 we execute the ed sacred cut. We are particularly interest -21 21A are se The . in the two vertical lines of and 23A-23. Produced to the entrance th wid l tota the k mar the building, they of the approach—passing through the that middle of the two rows of pillars re squa h the oug thr and ance line the entr door the of side er eith on ars pill butt-end way. The extensions are to points 22 and 24 respectively. ’s We have concentrated in the Pantheon and al zont ground-plan so far on the hori rvertical lines of the diagram and the inte play of the various squares. is But a circular building of this type diin ably ider cons ed naturally influenc mensions by the circles within and outwe with the respective squares. A square, one les: circ tive truc cons two recall, has r described around the outside, the othe re. squa the in d with ribe insc Before examining the applications of the circle, however, we require to add one more square to our diagram. It is the n as basic square’s half-size version (see al, 35-36-37-38), and its associate circles have ing’s an important role to play in the build lay-out. We now have four squares placed conbecentrically within each other, and we of that with s circle their gin our study of e. the outside squar Square 5-6-7-8’s inner circle was a dethe ciding factor in planning the rear of link ve ructi const the g bein n, d-pla groun ebetween the basic square and its doubl size version. Being the inside circle of the large, outthe side square, it is the outside circle of same This next (basic) square: 1-2-3-4. square’s inside circle, we see, follows the curve of the outer wall of the Pantheon and was obviously the factor that determined the outer dimensions of that wall. de This circle is simultaneously the outsi versize halfe’s squar basic circle of the sion: 35-36-37-38. This square’s inner circle provides the absolute net floor area es in the large main hall. The circle touch on mns colu large two the of de outsi the either side of the altar (on the left of Fig. 938). The actual perimeter of the inner hall is broken by a series of recesses and projecting walls. Our attention turns now to the inner square created by the combination of sacred cuts in the large outer square. The ve small square is 17-18-19-20. We obser of back the s form e how its outside circl cting proje the into cut s the small niche wall all the way round the main hall. The rs arc of the same circle positions the pilla remain six the of each of h at the mout cesses in the hall. As we saw with earlier Greek temples, the intersections of existing lines and squares often provide opportunities for entering new lines. One such intersection in the Pantheon’s ground-plan is the meeting of the square created by the sacred-cut combination (17-18-19-20) and the basic square’s half-

\ this is 9-11-12-10, and horizontally s in 16-15. When we enter the diagonal | \\ IN 13-14- 0, etc.) these two rectangles (i.e. 9-12, 11-1 ugh thro tly we observe that they pass exac N in the the middle of the eight windows outer wall of the church. This is a construction we have not seen ? previously. What are its characteristics ain cert a ess poss to lines for Is it not usual geometric property? A close examination will, however, reveal that these lines do indeed have a purpose, a geometric birthright. Hadrian did not include them in his plan accidentally. The diagonal lines under discussion have the following property: their intersections coincide with the sacred cut in any square sharing the same centre and axis as the basic square. Whether this discovery was made by fai hy b 2 -1) ia | Hadrian or whether by some earlier geometer is difficult to state categorically. Only the existence of another, earlier building showing the same construction would prove the latter. a - 372 A al? -1) B a - $12 That the discovery is correct is illustrated in Fig. 247 which shows the geometric construction and corresponding arithmetical calculation. AADB ACOD = b a(Z-1) 2 2 bW2-1) 2 Fig. 247. size version (35-36-37-38). Their junction produces four tiny squares, one of which is 33-17-34-35. Taking the diagonal (33-34) of this square as the basis of a new circle with the same centre as the previous, we see how the new circle follows the inside sweep of the main wall of the circular hall. And the distance between this circle examined are so much a part of the structural plan of the Pantheon that they represent a reality that cannot be ignored. and that drawn within the basic square of his work. At the centre of the large square the gives the total thickness of the wall. The various concentric circles we have There can be no doubt that they were part of the original ground-plan sketched out by Hadrian and his fellow-builders. Not only was Hadrian well aware of the structural value of geometric symbols, he also made full use of them in this sample sacred cut forms a rectangle. Vertically We move on in the analysis of the ground-plan to a new diagram, having obtained the majority of the principal dimensions in the church from the preceding diagram. The object of the next two analyses is to demonstrate how a similar analysis of the same squares provides further information on the recesses and column spacing within the main hall. In Fig. 248 we start with the basic square 1-2-3-4, and construct its half-size version 35-36-37-38. This is a repeat of the previous analytical diagram, but in this case we go down one stage further, constructing yet another half-size square: 39-40-41-42. 53 This new square is divided 3 X 3 in the same manner as executed in the Greek temple analyses. The simplest method of doing this is to enter the diagonal cross and the acute-angled triangle. The intersections of these two figures indicate the 3-part dividing lines. Vertically these lines are 47-48 and 49-50. We see immediately that they indicate in the rear wall the width of the semi-circular niche in which the altar is situated. The lines run through the centre of the two columns flanking the entrance. At the opposite side of the temple the corresponding lines mark, as I believe was intended, the net width of the hall entrance. The same lines horizontally (43-44 and 45-46) show the same thing: they indicate the width of the two niches at right and left of the main hall. The 3-part division just executed constructs a new small square at the centre of the diagram, and in this square, too, we enter the 3-part dividing lines. They are (horizontally) 51-52 and 53-54. The vertical lines are not required. We recall that 3-part division was the most common mode of spacing pillars in earlier Greek temples. If we produce lines 51-52 and 53-54 across the diagram, we see that they were apparently responsible for positioning the columns in the niches on the right and left of the hall. Fig. 249 shows in effect the same diagram, the difference being that the inner square has been swung through 45°. In other words, square 39-40-41-42 corresponds in the new diagram to square 55-56-57-58. We see how the 3-part division of this square was used to mark the maximum width (at the rear) of the four rectangular recesses in the main hall. When this width is linked to the centre of the diagram in the form of radii we see that the

angle matches perfectly the angle of the recess wall. The small inner square created at the centre of the diagram by the lines of 3-part division is also divided 3 x 3, and we see how the resultant lines appear to have been the factor that positioned the two columns placed before each of the four rectangular recesses. We see this carried out, for example at recess 69-70. The dividing lines are extensions of 63-64 and 65-66. As in previous analysis, a mass of detail is still hidden geometrically in the diagram, but we have nevertheless traced as many of the main lines as permit us—as long as we remember the order and manner in which Hadrian applied the symbols—to reconstruct the chief structural features of the Pantheon. We require neither measurements nor drawings. Only a full appreciation of the circles and squares of ancient geometry. A continued analysis would certainly provide us with a wealth of additional information. The diagrams have not been exhausted, possibilities are many. But as CHAPTER FIFTEEN The Golden Section versus the Sacred Cut emphasised previously, the aim and intention of this book is not to explain every line of any specific building. It is to demonstrate the presence of ancient geometry in planning a particular building. We HAVE made free and frequent use throughout this book of a completely new term in geometry: the (by now) familiar Sacred Cut, a label applied by the author. This geometric newcomer distinguishes itself from other similar terms in that it is not simply a geometric or mathematical phenomenon; it was a concrete factor in the sphere of building right from the earliest days of religious constructions through instructions for Moses’ tabernacle in the Midian desert, and we have seen it employed as undoubtedly one of the principal motives of design in building temples of antiquity and religious structures of the Middle Ages. In fact the sacred cut has proved itself the corner-stone. of the entire building industry of ancient times in precisely the design and construction work were wrested from the hands of the religious orders and passed instead to professional builders ignorant of the training and tradition of the cloister and temple. We have studied the practical application of the sacred cut in the planning same way as it represents the main factor in ancient geometry itself, the subject to which this book is devoted. The ancient system of geometry can almost be regarded as a direct development of meditation on the sacred cut and the sacred number seven, and in the same way as these latter it has remained part of the early Church’s occult teaching. The phenomenon arose, as we saw, at an immeasurably early stage in Man’s mathematical speculation, and assisted the developing geometer to calculate the circumference of the circle to within an error of less than 1 %. The sacred cut has the property of being explicable to any intelligent observer devoid of mathematical knowledge and experience as we understand it today. The concept of the sacred cut is so devastatingly simple, requiring only a primitive system of numbers and a reasonably intelligent operative to apply it successpicked it out from among the building was observed and recorded so early in time as far as the Middle Ages. Notwithstanding that the sacred cut existed as one of the chief factors in apportioning dimensions to almost all monumental structures of the past and can be traced and revealed in those examples of such structures as (from the point of view of preservation) lend themselves to study, the Temple and later the Church were completely successful until now in keeping secret from uninitiated the mathematical knowledge that the term conceals. Indeed the sacred cut and ancient geometry, still hidden from the casual observer, died an unnoticed death the day that building, of the Great Pyramid of Egypt, we have fully, that it is scarcely surprising that it