GEMATRIA
BY
FREDERICK BLIGH BOND, F.R.I.B.A
AND
THOMAS SIMCOX LEA, D.D.
ANNOTATED AND TRANSCRIBED BY
PETER WAKEFIELD SAULT
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III.
THE GEOMETRIC CUBIT AS A BASIS OF PROPORTION IN THE PLANS OF MEDIAEVAL BUILDINGS
By F. Bligh Bond, F.R.I.B.A.
In response to an invitation to contribute a Paper on a subject of interest to the profession at large, the writer offers in the following essay a theory which he has for some time past been testing, and which tends in his opinion to explain a principle of proportion found in many mediaeval works for which an adequate explanation seems to have been lacking. Much research and good scholarship will be needed in order to establish his theory and to win it general acceptance — should it be found to merit that reward — but the writer's apology for putting it forward at the present stage of his research is that it answers, or appears to answer, equally the historical as well as the practical and arithmetical tests, and to reconcile in a remarkable way certain doubts and contradictions.
In order to clear the position it will be necessary to state briefly a few wellknown facts accessible to students in recent editions of Gwilt's Encyclopaedia and in works therein referred to. It appears from contemporary records that there were in the Middle Ages two rival geometrical systems for setting out the ground plans of churches and other Gothic buildings, and these were also applied to their cross sections and sectional elevations. These were:—
(1) A system of commensurate squares.
(2) A system of equilateral triangles, which, when contained in
parallelograms, gave a rectangular field or setting.
Our chief authority for these is Cesariano, the sixteenth
century translator (or editor) of the works of Vitruvius. Both systems were habitually applied, and there are records extant of controversies which took place between adherents of the rival schools. The first rule, that of the commensurate squares, is called by Cesariano the rule “a pariquadrato”, and the second, the rule “a trigono”. The first was adopted by the German architects and became more or less identified with their system. The second seems to have been favoured by the Latins, but it will be well not to be too insistent upon this point in the present stage of research.
We now begin to break new ground. The question arises: Were the two methods of planning designed to produce results of a different nature, or were they meant to yield effects approximately similar? The question is of crucial importance, and its answer implies also the solution of some obscure points of mediaeval planning, and the discovery of the principle at stake in these bitter controversies.
In the writer's view, it was one of geometrical perfection, the object being the reproduction of the form of the Rhombus of two equilateral triangles in the greatest degree of accuracy consonant with practical methods of building and harmonious scales of measurement. As to the motive which led the ancients to their preference for geometric truth — that is another question. For the moment we are on safe ground in accepting it as an axiom of their system that they did work where possible on geometric lines, and that from very early times a peculiar respect — even a sanctity — attached to those proportionals which most clearly accorded with the mathematical principles known to Master Masons.
From this it will be inferred that the contest was one
between principle and compromise, the rule “a trigono” being the use of the purist school, and that “a pariquadrato” of the practical builders. The “German” school, logical and practical, preferred to work on a system wherein the measures of length and breadth were commensurate or uniform, whilst their idealist opponents saw in this something approaching a profane disregard of principles instilled into their guilds by the teaching of a tradition so old, so venerable, that to depart from it was architectural heresy. But even they must have perceived and found by experience a limit of possibility in practical working, and hence in Cesariano we find in the instance he gives of the designing of the Cathedral of Milan a reconciliation of the two ideals.
Readers are referred to Gwilt (Ch. IV. sec. 3) for an exposition of this highly symbolic feature, based upon the construction given by Euclid in his first proposition. Among all the select proportionals used by the old builders, this, the ratio of length to breadth in the double triangle, seems to stand apart in a position of preeminence. Not only do we find it reproduced in many approximations in the plans of our own and continental churches, but it is notoriously used in Gothic detail wherever the architectural expression of the best periods reaches its highest point. And its association with certain parts of a church and with statuary of a certain order leaves no doubt of its symbolic intent.
Modern writers have discovered and chronicled many examples of these proportionals in our mediaeval plans. Notably Kerrich, in his communications to the Society of Antiquaries in the second decade of the last century, has furnished us with material for reference, and following him comes Professor Cockerell with his analysis of the works of William of Wykeham, in which the rule “a pariquadrato” seems to find expression, seeing that the ratio he employs is the
most practical of all, and the one which least truly approximates to the ideal proportions of the double triangle, though in buildings of lesser size it would be near enough to exemplify that principle. This is the ratio of Four in breadth to Seven in length. This we will call the “Mason's Convention.”
The English Perpendicular, in which William of Wykeham worked, was in some respects a reversion to practical principles, though it must not be assumed that the scheme of symbolic representation by number and proportion, into which Durandus gives us a guarded insight, was superceded by the later builders altogether. It probably tended with time to become overelaborate, and for that reason a reaction would sooner or later be inevitable, by which processes of simplification would assert themselves.
To find the geometrical principle in its more perfect expression an examination must be made of the works of the
best period, that is to say, the time when architectural achievement had reached its highest point and decadence had not set in. This would be the twelfth century and the early thirteenth century of English work. And reference should be made to examples of the most careful character and least altered by subsequent builders.
It has been the writer's good fortune to discover an almost perfect example of such a principle in the Lady Chapel of Glastonbury Abbey, a structure whose history would be all in favour of a perfect symbolic expression, since the extreme and scrupulous care used by its builders in designing it on ancient lines is on record. Kerrich alludes to this building and gives diagrams and dimensions, but he was not accurately informed as to its true proportions, which have only been recoverable by careful measurement owing to the violence to which the building has been subjected and its consequent partial collapse in width at one end. A plan of this chapel is given, from which it will be seen that the figure of the Vesica is present in a form so nearly accurate as to leave only the most insignificant margin of error, and the figure is repeated in duplicate on the main axis.
The true proportion of the double triangle is as 1 in breadth to the square root of 3 in length, or as 1 : 1·73205. A building 40 feet in breadth would thus have a proportionate length of 69·282 feet or, say, 69 feet 4 inches.
St Mary's Chapel has a width, measured on the west face, of 40 feet 1 inch, but as some of the joints have opened it may be taken to have been intended originally for 40 feet. This is the measurement outside the plinth of the angle turrets. In the length there is another open joint to be allowed for, and for this we must deduct nearly an inch. The present measurement is 69 feet 7 inches, and with the corresponding deduction comes to 69 feet 6 inches. If the conventional 4 to 7 had been
employed, it must have been about 6 inches longer.
But the inner measurement, which is the breadth between the buttress faces on centre of north and south walls, is computed to have been as nearly as possible 37 feet, and hardly even now deviates from that figure. The proportionate length brings us in this case to the face of the west wall at the level of the cill beneath the recessed wallarcade, and this is the most natural point from which to calculate a dimension of length.
Although the corresponding line on the east wall is now missing, owing to the removal of the central section for the inclusion of the Galilee — which was done in the fourteenth century — the evidence of the original length of the chapel is attested clearly enough by the remaining data.
The superior accuracy of the inner Rhombus, which is, after all, the measurement of the actual framework of the walling, gives us a suggestion of another and closer approximation to truth than the “Mason's Convention” first described. This nearer approach is represented by the ratio 37 : 64.
Allusion may be made here to a remarkable fact noted constantly by the writer in his measurements of the various parts of the fabric of Glastonbury Abbey and that of the other foundations and walls discovered. It is that the whole scheme of the Abbatial church and buildings appears to be planned upon a series of commensurate squares of 37 feet, or more accurately speaking, of twice 37, that is, 74 feet. The west wall of St Mary's Chapel marks the western limit of this great series of squares, which are figured in a plan contributed to the Somerset Archaeological Society's Proceedings for 1913. The reason for the choice of this number of feet (or inches) as the unit of general measurement is still under investigation and will take some time to determine, as it is by no means clear that some second standard or measure, different from the English foot or yard, was not employed, such as the ell of 37 inches, which was used at Gloucester as a land measure, and may have been used
also by the Glastonbury builders.
But this is a digression from the main issue. We have now two approximations to a geometric rule: one rough, but extremely simple and convenient (4 : 7), capable of lending itself to the making of handy builders' measuring rods, and the second a finer adjustment but inferior in convenience, since 37 inches is a long scale, and a relatively clumsy one to deal with,* the compensating feature in this case being that the longer proportional 64 is capable of subdivision by repeated bisection to an extent which would surely have commended itself to the practical mason.
We may now enquire what other fractions will yield us good conventional working elements for the setting out of buildings to the rhombic proportion? In what other manner could the architects of the former time have divided their measuring rods so as to secure the desired result without undue difficulty or inconvenience? There are several fractions giving a near approximation to truth, but only one or two which could be called convenient. The best of these is the ratio 26 to 15, which is alluded to in a work which will be found in the R.I.B.A. Library, entitled The Canon — An exposition of the Pagan Mystery perpetuated in the Cabala as the Rule of all the Arts (London: Elkin Matthews, 1897). This work, being very mystical and illarranged, is not freely consulted, but it is evident that the author was well read in his subject and has been most painstaking in his collection of evidence on such points as connect the ancient rules of building with geometrical symbolism. We gather from his pages that he considers the ratio 26 : 15 was well known and of high repute among the ancients in some such connection as the tradition inherited by the mediaeval builders from remoter
* In the Gloucester records it is viewed as a yardandaninch, or thumb, and spoken of as “virga cum police interposito”.
antiquity would point out. The Louvre Cubit of 28 digits, unequally divided as 13 : 15, might suggest a practical application of this principle.
We have yet, however, to find proof of the use of these proportionals either in the known history of our building measures or as evidenced in the masonry of our schools of builders. But this may well be due, as regards the lastmentioned, to a want of definiteness in research. It is a point which would clearly have easily eluded any investigation not based upon the a priori conception of the existence and use of such a proportional. In this respect it is hoped that the present essay may stimulate research.
But another and very fair adjustment of scales to integral number in inches offering a near approach to the rhombic ratio is that of 11 to 19 — with its complementary ratio 19 to 33. Take 33 inches as your “yard”, and the cross measure 19 inches, which we may describe as a “cubit”, and you have a very workable pair of measures, since 19 inches is 18 + 1, and 11 inches is 12  1. We do not say that there is any warrant for the assumption that such a pair of measures was actually in use, but in all these cases it is well to remember that the masons were undoubtedly in the constant habit of halving, doubling, or otherwise devising simple and compound fractions of their normal standards, and indeed, our own foot of 12 inches is a case in point, seeing that it is onethird of the real standard, that is to say, of the yard. And in this connection it is sufficiently clear that a cubit was often employed which was 1½ feet in length, or half a yard, whilst there is also evidence of an 11inch foot.†
The title chosen for this Paper must now be justified. We
† The ratio 2×19 : 3×11 (38 : 33) also subsists in our land measures as it is the ratio of the nautical mile to the statute mile (2026·67 yards : 1,760 yards).
are dealing with a proportional which, as already shown, bears to the 12inch foot the approximate relation of Ö3 : 1. Theoretically it is the Mean Proportional between the foot and the yard, but in practice it could not be so if any arithmetical convention were used for harmonising it with the other standards, since any artificial adjustment, if it increased the difference between the foot and the geometric mean, must correspondingly decrease that which lies between this mean measure and the yard, and vice versa. In this case the proportionals would appear as:
Foot  Cubit  Yard  
4  :  7  :  12 
Hence greater accuracy would be desirable. There is a triplet of measures to be kept in reasonable harmony. Our yard and foot are ancient measures — so ancient, indeed, that the mere statement of their probable antiquity is apt to excite scepticism. But the fact remains that they are in close geometrical relation with the principal measure employed by the builders of antiquity — i.e., the cubit of 7 palms; and if this geometrical relation be admitted, then they may be claimed to have been in all probability the original standard from which that cubit was derived, seeing that they do, as a matter of fact, happen to be in the strictest sense of the word geometrical measures — that is to say, measures of the earth's axis. For it has been stated on good authority that our inch, the unit upon which the foot and yard are based, represents within a close approximation a fraction of ^{1}/_{500,000,000} part of the earth's axial length. Sir J. Herschel, in a letter to the Times dated 30th April 1869, says that the inch appears to have deviated from geometric accuracy by the loss of just ^{1}/_{999} of its length. Sir C. Warren, however, in his work on the Ancient Cubit seems to regard even this as an overestimate. In any case the loss is so minute as to be negligible in practice, and the amazing fact is that the standard, if originally derived from the earth's
measure should have been so well maintained. The cubit, which is the mean between our foot and yard, is strictly 20·7846 inches, and is the side of a square whose area is 432 square inches, equivalent to an area of 24 inches × 18 inches.
The prime object of the use of the Mean Proportional in measures would appear to be to provide a Standard of Area, of square form, from which other spaces of equivalent area might readily be derived. There is evidence of the highest antiquity for the practice of obtaining equal areas with diverse proportionals. It is found in the ancient Indian “Shilpashastras”, or rules of religious art; and Professor Petrie notes such a custom as controlling methods of the Egyptian builders.* In the King's Chamber of the Great Pyramid are recorded, in linear measure, the roots of the simpler arithmetical values, such as are employed for this purpose. These roots would appear to have been among the more guarded traditions of the ancient builders. In the case of the mediaeval workers, however, there does not at present appear sufficient evidence that their object was the equalising of areas of floor or wall space, but it is more likely that the practice of employing these proportionals had become so interwoven with their traditions, and so hallowed by time and religious association, that it had taken on a purely symbolic implication. This would be preeminently so in regard to the use of the triangular proportions, as the history of the Vesica shows plainly enough.
But, more than this, it must always be borne in mind that a practice of this sort may be grafted on original necessities of the craft, and that the mason, in order to set out his square and perpendicular lines, must necessarily have made use of the equilateral triangle on each side of his base line of standard length, and would thus obtain a third measure which would be the geometric mean between his two principal units of linear measure.
* Pyramids and Temples of Gizeh, pp. 194, 199, 200, 220, and p. 181 ref.
The triangular ratio does not appear definitely in Egyptian monuments, which rather follow the laws of the numbers 2 and 5 and their roots — the proportions of the side of the square to its diagonal and of the rightangled triangle whose sides are as 2 : 1, and the hypothenuse consequently Ö5, since 2^{2} + 1 = 5, the square on the same. Nevertheless, in Egypt appears in later times a wellmarked triplicity of measure, similar to our own, the extreme proportions being in their case the Mahi, or Nile cubit of 20·76 inches (or thereabouts) and its triple, the Xylon, or staffmeasure; whilst their unrecorded mean would be as nearly as possible our yard of 36 inches. The tripling of measures had, without doubt, a religious — probably a trinitarian — significance, and in the choice of the two leading units we may recall the proportions of the first Hebrew Temple, with its single and double square areas, the Holy of Holies and the Holy Place — together, three squares in length. It would be well if, in the case of some of our own unspoilt Early churches, a careful measurement of the floor areas of the nave and quire (often square) could be made, with the object of testing the principle involved, by finding the side of a true square of the same total area.
To revert to our own units of length. A witty Frenchman once said, “If God did not exist, we should have to invent Him,” and the writer would be disposed to make a similar observation with regard to our Inch, that, if it did not exist, we should have to discover it — in order to explain the measures of the ancient world, to reconcile their apparent incompatibilities, and introduce any coherence amongst them. For truly our measures would appear the only possible nucleus of a stable system to which these others could be linked, unless we are to be content with a merely physical — i.e., corporal — origin for all. Such human measures are well known and admitted, but their use argues a perpetual variability, and does not logically exclude the geometrical theory, any more than the counting of fives and tens on the fingers excludes a quite distinct geometrical basis for the denary system of
notation. (This geometrical foundation can be shown and is of the greatest interest). Some measures are strictly geometric (in the sense of terrestrial measure). Others are in mathematical relation with these (as the cubit of seven palms). Many are ascertained to have a counterpart in the measures of the human frame. The two systems coexist, blend, and harmonise. But we must make our choice as to those which we deem original, and those which we think derived. In the writer's view the most reasonable working hypothesis is that the Inch, Foot, and Yard are the original series, and the Cubit, a measure of acknowledged variability, the secondary or derived one.
The argument from antiquity is fortified by the facts in the following table, which shows the units required for setting out areas equivalent to that of a given square, with sides proportioned as 2 : 1, 3 : 1, and 5 : 4.
(1)  Assumed Original Unit, on the primitive sexagesimal system: 
Sides. 
36" × 36" = 1296 square inches  36" = 1 yard  
(2)  First derivative (for a double square):  
(36 × Ö2) : (36 × ^{1}/_{Ö}_{2})unit, 36 × ^{1}/_{Ö}_{2}  = 25"·45  
(Cf. Royal Cubit of Persia, Chaldea, and Judah.)  
(3)  Second derivative (for a triple square):  
(36 × Ö3) : (36 × ^{1}/_{Ö}_{3})unit, 36 × ^{1}/_{Ö}_{3}  = 20"·78  
= Egyptian Royal Cubit.  
(4)  Third derivative (for a rectangle proportioned as 5 : 4): 

(36 × ^{Ö}^{5}/_{2}) : (36 × ^{2}/_{Ö}_{5})unit, 36 × ^{Ö}^{5}/_{2}  = 40"·248  
= Egyptian Metric Yard (early form). 
These results err but very slightly on the side of excess, the average in each case being a trifle lower in the case of known examples, but, being geometrical, are liable to modification to suit any arithmetical adjustment of scales desired.
How small, comparatively, are the differences between them and actually discovered units will be seen by the following table:—
(1) Yard. 36" Eng. 36" Gt. Pyramid (Great Step) (Practically alike) 
(2) Chaldean Cubit. 25·45 Geometric 25·38 Maximum found 00·07 defect.  
(3) Egyptian Cubit. 20·78 Geometric 20·76 Nilometer 00·02 defect. 
(4) Old metre. 40"·248 Geometric 40"·23 Egyptian 00·018 defect. or 00·009 on the half metre. 
The Egyptian Royal Cubit was of seven palms, each measuring from about 2"·91 to 2"·95 according to the size of the digit (·727 to ·737). The “Pyramid” palm yields a digit approaching the latter.
40 digits of ·735 — or 12 palms = 29"·4, again a derivative of the yard, since 36 × ^{Ö}^{6}/_{3} = 29"·40.
The division of the Royal Cubit into seven palms seems good evidence of the presence of the seven as a proportional, either as 5 : 7 : 10, or (more probably) as 4 : 7 : 12. The cubits vary from about 20"·5 to 21". Sevenfourths of 12" is 21", and twelvesevenths of 12" is 20"·57, which is nearly the Louvre Cubit.
The Cubit of 20"·63, found by Professor Petrie in the King's Chamber, seems to be the most representative cubit of this order. These 28digit cubits are Royal, or Temple, Cubits.
The Common cubit is of six palms.* The European cubit seems to accord very nearly with the latter, and become readily identified with the halfyard of 18". The Seven stands out prominently as a proportional both in ancient and mediaeval usage. With the Egyptians it appears to have ruled the relation of square and diagonal, since the side of a square is very nearly as 5 : 7, or as 7 : 10 of its diagonal.
We can express this ratio as  5 : 7 : 10 
or as  7 : 10 : 14 
12 : 17 : 24 
and the sum of the two sets of proportionals gives us the better adjustment (12 : 17) found in Roman work.
To summarise our conclusions — in the old and mediaeval systems of measure
we can discern at least three main standards:
(1) The Yard, with a senary division representing the old sexagesimal system.
(2) The Metre, representing the Ö5 and Ö2
derivatives, with a decimal division.
(3) The Cubit, representing the Ö3 derivatives, with the
proportionals 4 : 7 : 12, and a consequent division into 28 (4 × 7),
or into 84 (·7 × 12).
Of these the first is the overt measure in English work, the 21" cubit being a latent proportional. In Egypt the cubit is the overt measure, and the yard is implicit only, or scarcely to be detected in the monuments. The second, the old metre, is common to both systems, and though abolished by statute in this country since 1439, is still well represented by our decimal system of land measure, where the unit is 3"·96 (+·06), giving us the fathom of 79"·^{22}/_{10} of a yard. Were our Metric Yard still in use, it may be that our debt to Egypt in the matter of measures would more readily have been recognised.
* Usually about 18"·24. It unites in square measure with the cubit of 20·63, forming a rectangle of 376 square inches, equivalent in area to one square Euboic cubit, and almost precisely ^{7}/_{12ths} of the area of one square Royal Cubit of 25·38 (= 644 sq. ins.). With the square “Remen”, which is ^{4}/_{12ths} of the same area, we have again the triplicity of 4 : 7 : 12, this time in surface measure.
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