Measure and the medieval architect.  Scaling up.

In the three hundred years from c.1060 to c.1360, the men who designed and commanded the building of the great churches cathedrals and abbeys possessed no tape measures, nor any kind of marked ruler. The only measure available in England was the English Foot, a specimen of which, stamped with the Royal mark, was held in every town[1], to be regularly inspected by the King’s officials.  In the rest of Europe the foot measure varied from place to place. Half a foot was the smallest measure.  At work in his drawing office the cathedral architect drew up his plans and elevations on sheets of parchment.  On such a plan the width of the nave might be, say, about four inches, while the actual width to be set out on the building site might be 90 feet, a scaling up of 1:270 .The question that presents itself concerns the means by which the 4 inches on the drawing board was converted to the 90 feet on the ground.  How was any measure on the parchment translated to the site? 

  Bucher, who collected and studied more than three thousand medieval architects drawings, writes:

‘Linear measurements were thus used only as master values for the main modules.     They are for all purposes non-existent in medieval designs.’ (1972, 49).

 

And later (1972, 51, n.38):

      It is difficult to understand discussions of dimension and design scale in medieval design unless one remembers that a. there was no unified system of measurement;  b. space considerations favoured a flexible system;  c. the scale of Paris on the plan is 1:108 while the scale of Orleans on the same sheet is 1:105; the Nuremberg design of Freiburg is 1:94;  d. the scale is sometimes changed within a design to emphasise a detail,  e. the size of the parchment determined the scale in some cases.….All these elements rule out a direct scaled transposition.  Of over 3000 drawings less than ten contain measurements, and even these, as in the St.Gall and Cologne plans, seem added later. 

 

     Could the architect have given instructions to his workmen regarding dimensions?  It would have been impossible for him to do so, either in writing or by word of mouth.  He himself may well have been able to read and write, but his workmen could not.  They were essentially uneducated in modern terms. Even as late as the Dissolution of the monasteries in 1540 fewer than one in thirty children had any education at all (Coulton 1961, 10 to 11), and the great majority of those would have been from the families of the wealthy.   The education of masons lay in their craft skills learned by example and by verbal instruction.  They had no need to read for there was nothing for them to read.    They were also innumerate, counting, if at all, on their fingers. Their wages were in shillings and pence, and the most that one would need would be to count up to twenty for a pound.  To give a workman instructions requiring him to count or calculate would be to invite disaster.  To find foundations for a pillar or buttress in the wrong place would be immensely costly to correct.   The fact that the masons working on any one building would be drawn from many parts of England, speaking many dialects, and even from other countries, used to other units of measure, can only emphasise the impossibility of such a process.   As for their carving, it was all done from templates provided by the architect through the carpenters’ workshop.

        Shelby asks the question:

      Is it possible that such detailed plans and ‘specifications’ were not needed because there was available to medieval masons a system of transferring architectural ideas that is quite unlike anything used in modern building?  This is the argument of those students who emphasize the importance in medieval architecture of Baugeometrie – the art of designing according to certain geometrical formulas, wherein the parts of a building are proportioned to each other and to the whole, with these proportions based on a given geometrical figure: square, triangle, circle, octagon, etc.,

         (Shelby 1964, 392)

 

As will be shown, however, there is a much simpler and obvious alternative method     

     

      It has been suggested that no plans were made on parchment, and that the whole plan for a cathedral would have been worked out on the ground with cords and ropes.   Such ideas take no note of statuary of the period depicting architects holding parchment plans in their hands:

       In the church of Saint-Ouen at Rouen, statues of Alexander and Colin de Berneval portray these two master masons holding in their right hands pairs of dividers that point to architectural sketches held in their left hands. (Shelby 1965, 239-40).

 

     The reason for the scarcity of preserved parchment plans, I would suggest, is that once the foundations of a building had been set out and perhaps the first layer of the plinth laid, the plans were of no further use and the valuable parchment, (some were made so thin as to be transparent), simply scrubbed clean ready for the next assignment.  Harvey (1950, 31) mentions that parchment was prepared by architects by rubbing in powdered bone-meal, which can only mean that they were drawn on by silver-point.

Cennini c.1400 (1899) describes the process of silver-point in his chapters on drawing, all of which are concerned with drawing in silver-point, describing in detail how the bone-meal (from chicken wings) is prepared for silver-point drawing.  It is a very simple technique, permanent but easily washed away, and certainly used by medieval and Renaissance artists up to Leonardo da Vinci, Raphael and even Rembrandt.

     Further on Harvey cites Kletzl: “ who states that the instruments of a leading master might be made of silver, and were bequeathed as treasures to the lodges.” (Harvey 1950, 42).

      Plans had no lasting value.  Elevational drawings on the other hand were kept as examples for show, for promotion, even for display to the public to raise building funds, and also, of course, for the time that elevations could take to complete.  They were drawn in ink, and even then the parchments were often scrubbed off to be used for another purpose.  In this regard Harvey states that “medieval masters made their designs and working drawings piecemeal, as they were required, and commonly built up their great works from many vertical sections or bays horizontally juxtaposed.” (Harvey 1950, 39-40)

       There is no need to invent ways in which the drawing of plans could be avoided, for there is one simple means by which any architect could translate dimensions from his drawing board to his building site to almost any scale, requiring no tape-measures, no instructions verbal or written, and no possibility of a mistake.

       Whenever a medieval architect is portrayed in the art of the period, often as a self-portrait, he is depicted either in his drawing office role as designer, or in his building site role as master-builder. When shown in his designer role he always holds a small pair of dividers Figs. 248,  249,  250,  251), but portrayed on the building site he carries a large pair of dividers (Fig. 254) which are always equipped with curved retaining arms passing through the legs.   One has to ask why two pairs of dividers, one large, about three feet high, and one small about six inches high, and why does he never take his small pair on to the building site?    

       The answer to the problem is simple.  Suppose, as an example, the architect is working to a scale of 1: 150, and needs to set out the width of his nave on the building site.  On the site he will have a centre line marked out east to west.  In his office he will have drawn a centre line down the centre of his plan.  There he will take up his small office dividers and setting one leg on the centre line he will open them to, say, the north side of his nave.  This might be say 2½ inches.  He will then swing his dividers across the drawing board 15 times, a distance that we today would measure as 37½ inches but which the architect of the time cannot define in any way.  He marks it off.

      He then picks up his large out-door dividers and opens them to the distance that he has just marked off on his drawing board.  He puts on his hat and robe and walks out to the building site carefully carrying his large dividers to the centre line, where, placing one leg of the dividers on that line, he swings his dividers 10 times, (i.e. 31¼ feet), to the north, where, with the point sticking into the ground, he orders a peg to be hammered in.  He then repeats the process to the south, and thus the width of his nave width of 62 feet is set out to a scale of 1:150. 

       Every major dimension would be set out in the same manner, using the same two factors of 15 and 10, the pegs being joined with cords until the lines of the foundation trenches to be dug were complete and marked with lime, and ‘recovery points’ set in.  The cords would then be removed to be replaced from the recovery points after the foundations had been dug and the footings completed and levelled. 

      Smaller dimensions, such as those of a pier base, would be dealt with by means of templates, but here again the architect could make for himself templates in leather or canvas to a 1:15 scale from his general plan, which could be taken to the carpenters’ workshop to be enlarged with dividers again by 10 times for the full-size templates to be constructed for the masons’ use.  I would suggest that this could well have been the purpose of the great plaster tracing surfaces.  The smallest elements would be provided for by full-scale templates produced by the architect using geometric processes such as those detailed by Lechler for the design of window mullions.[2]

       Elevations could be dealt with in a similar manner, probably by laying out sections of masonry on the ground.  Carpenters certainly used such a method for Harvey records that “the trusses of Westminster Hall were set out on the ground in 1395 at a place called ‘The Frame’ near Farnham in Surrey”  (Harvey 1950, 31).

       The medieval architect would have needed some modest degree of arithmetical ability to be able to decide on the two factors that would provide him with the degree of scaling that was required for any particular project.  Shelby  found that:

      A review of the educational system, with special attention to the education of master masons reveals that formal schooling played an insignificant role in the technical training of the master mason as architect.  Furthermore the place of mathematics in the curricula of the schools suggest that boys who went into the mason’s craft would have had little opportunity to study the subject seriously even at the highest level of formal schooling that a craftsman might have attended.  (Shelby 1971.b, 238).

 

      But even under the limitations presented by the Roman numerals in universal use throughout most of the Middle Ages, the mathematical skill required would have been very undemanding. By 1400, Geoffrey Chaucer, great friend of Henry Yevely, could write in The Knight’s Tale of ‘Every crafty man that geometrie or arsmetric (arithmetic) can” to describe a gathering of architects.

      At Paris the factors 9 and 12 would give the scaling of 1: 108, while at Orleans 7 and 15 would give 1: 105.

      But what about the scale? How would the architect know what scale to use? 

      I would suggest a simple answer.  His employer, abbot, bishop, whoever, decides on one important dimension, say the width of the transept from north to south, as say 200 feet.  The architect has a one foot rule.  He marks off one foot across his parchment as the width of the transept and knows that his plan will be to a scale of 1:200.  When he is ready to set out his plan on site he will march his small compass twenty times across his board for each particular dimension, open up his large outdoor compass to the size arrived at and swing it ten times across the site.    Or ten times for his small compass and twenty times for his large compass, whichever is most convenient, for both will produce a scaling up of 1:200.    This would have been, I would suggest, at least the basis of the medieval architect’s system of measurement and scale

       The 1: 94 scaling reported for Freiburg is a problem, for the only factors of 94 are 2 and 47, which are impracticable.  If it were 1: 96 there would be no problem, but if the scale of 1: 94 is correct we are led to consider a most interesting possibility concerning the possible use of proportional dividers.  Proportional dividers with fixed point, and thus a fixed ratio, were available to Roman masons, and an example was found in a mason’s workshop in Pompeii.  A similar instrument is reported to be illustrated in a Leonardo da Vinci  document c.1495.  

      The proportional divider is an extremely simple instrument, consisting of a normal pair of divider legs extended beyond the joint to form another longer pair of pointed legs.  Nothing could be simpler and one cannot believe that such a simple device was totally unknown to medieval architects, if they ever actually found a need for such an instrument.  So could it have been used by the Freiburg architect to deal with his scale of 1: 94?

       The important quality of any fixed joint proportional dividers is that they can provide only one ratio.  We must therefore ask the question: “What one ratio would be of special value to a medieval architect?”  Any simple arithmetical proportion would be of no special value because such a proportion could easily be obtained by the use of normal dividers.  Therefore one should expect a pair of proportional dividers to provide an important ratio that was not easily obtainable from normal dividers.  The ratio 1: Ö2 is the one that comes immediately to mind.

       Looking again at the 1:94 ratio judged to have been used at Freiburg one may take the next logical step of supposing the architect to have proportional dividers with the ratio of 1:1.414 or Ö2, bearing in mind that such a ratio was easily obtained by measuring the short legs against the side of a square, and the long legs against the diagonal.   Such an instrument would save the architect from having to use numerical approximations such as the 5:7 that were used by Lechler and his comtemporaries as their short cuts to Ö2.

    The method of use for such an instrument is simple: using the short legs of one’s special fixed-proportion dividers one opens them to, say, 2 inches on one’s plan.  Turning the dividers over and using the long legs we now have an opening of 2.828 inches.    Swinging this across the drawing board 6 times we create a distance of 16.97 inches.   Putting down our proportional dividers and taking up one’s “great” outdoor dividers, one opens them to the 16.97 inches that one has measured out on the board, and proceeding to the building site we swing them across 11 times to a distance of 186.6 inches or 15 feet 6½ inches, a scale of 93¼ times the drawing office 2 inches.  Not quite 1: 94 but near enough to make such a system feasible.

     A pair of Ö2 proportional dividers would not have been acquired simply for such an operation. Its main purpose would have been to provide short cuts to geometrical developments involving squares within squares and other Ö2 developments that are generically essential elements of ad quadratum geometry.  The construction of Schmuttermayer’s row of Ö2 squares would have been very quickly created using such an instrument (Fig. 35).

       It was supposed at one time that the dividers portrayed on the Libergier tomb were proportional dividers, but this was shown to be erroneous by Colombier in 1953, cited by Shelby (1965, 240).  In fact it should have been obvious from the first that they could never work as proportional dividers.  They would have required points on the upper arms that would have needed to be very much longer.  As Morgan B.G. (1961) has shown, the square depicted on the tomb is almost certainly traced around the architect’s original own square, and the same must be said of the dividers portrayed.  Judging the figure of Libergier to be about six feet tall and enlarging a published image of his dividers to the same scale I was able to construct a full-size model of his dividers.  In Fig. 260 it will be seen that however large or small the opening of the points of the dividers, the opening between the upper arms, being almost level with the joint, changes very little indeed.  There is no way that such an instrument could be used as a proportional divider.  What one does discover is that these are a very superior “new improved” pair of normal dividers, for whatever the width to which they are opened, its points change very little from the vertical.  This allows for much greater accuracy than that possible with ordinary dividers when opened wide at a great angle to the drawing board.  It prevents the opening becoming altered inadvertantly by pressure on a wide opening angle.  It also avoids the possibility of the points slipping when opened widely, and the problem of holes widening through being used at an angle.  The rounded arms at the top are there simply to fit the instrument comfortably and reliably into the hand and to allow one handed control, an important asset.  The instrument is simply a fine design by an expert geometer, almost certainly Libergier himself, a man of recognised genius.  My model is made of particle board but a fine silver pair would be a very acceptable acquisition.

       To return to the main subject concerning the system of ‘walking’ the small dividers across the drawing board followed by ‘walking’ the large dividers across the site in accord with calculated numerical factors in order to achieve a particular given scale of transfer of dimensions from board to building, it is worth noting that such a method would easily be kept secret. The architect would be seen marching his dividers across the building site, but no-one would see what he had previously done in his drawing office, and would not understand it if they did.  It would seem very mysterious that everything came out absolutely correctly on the ground.

    Are there any problems likely to arise from the projected method of site measurement?  Its great property is that it is impossible to make a serious mistake, for every vital measurement is made by the architect, and he is the man responsible for the whole work.  Could he miscount his swinging of his great dividers ten times across?   If he did he would have only himself to blame and display his incredible incompetence to the world.  Impossible.   Could his large dividers alter in angle as he walks them around?

The answer is that they could not, for it is to prevent such an occurrence that his large dividers are, without exception, furnished with their curved retaining bars.  They are constructed to pass through the leg of the dividers in order to create the necessary friction (Fig. 256).  A light hammer tap there on the leg would tighten things up as and when required.  The illustrations to follow show that all portrayals of architects on site depict them carrying large compasses, about three feet high, all equipped with at least one retaining bar, while portrayed in their drawing office they hold only the small compass with no retaining bar.

     Why should the architect require a retaining bar on the large compass that he uses on site?

      Clearly because he needs to retain the particular dimension that the compass is set to, and that can only be that that dimension will be measured out a number of times, which is what happens when he swings it along a line to measure out a scaled up distance.  In his office he does not need such a retaining bar on his small drawing compass because he is constantly changing its setting.  When measuring the earth a retaining bar is needed.  When measuring parchment it is not.

       There is another even more cogent reason for the retaining bar on the outdoor compasses.  Take two metal compasses, one six inches high, the other three feet high.  The weight of the large compass will be 216 times that of the small compass, and each leg will weigh 216 times the leg of the small compass.  But the frictional area of the  joint of the large compass will  be only 36 times that of the small compass, and the pressure for the legs to fall together will be 6 times as great as that of the small compass. Therefore the necessity of the retaining bar.

     The only problem with the method lies in the start of the process, namely opening the small office dividers to the dimension on the plan, for no matter how sharp his eyesight, however sharp his divider points and however steady his hand, a difference of

one hundredth of an inch at this point would multiply to perhaps an inch and a half on the ground.  This is supposing that the lines on his drawing are almost invisibly thin and straight.  But this liability to slight inaccuracies fully explains such minor discrepancies as the 26foot width of Peterborough’s north portal and the 26foot 4 inches of the south portal.

     Thus the method can be insignificantly inaccurate, although it cannot make significant mistakes.  The universal existence of small discrepancies between dimensions taken from the fabric that should theoretically be equal is thus clear verification of the use of the system of scaling up that is here described.  They would not occur if tape measures existed and were used.  If the factorial two compass method here described was used, they would be unavoidable.

      Another product of the system is the number of dimensions produced on the ground that are not whole numbers of feet.  In fact it would be unusual to find a dimension in whole numbers of feet, whereas if some kind of foot-rule were used such dimensions would be common.. 

     What is the probability that such a scaling system was in universal use during the middle ages?  In the late middle ages when inches came into general use as twelfths of a foot, and finely divided rulers and Arabic numerals became available, methods would change, but the fact is that if a modern architect were to be faced with the problem of translating a drawn plan at a particular scale on to a building site with no possibility of error, without the use of tape-measures or marked rulers, but with only two pairs of dividers, one small, six inches (152mm.) high, the other large, three feet (0.91m.) high, and a set square, (or a 3,4,5 triangle of rope), he would have to use this method.  There is no other.

 

THE TEST

It was suggested to me that it would be worthwhile to try out the method in practice.  A good and clever friend quickly made a superb pair of compasses three feet in height.   They were taken outdoors and opened to about two feet. ( Being a medieval architect I was, of course, unable to define the width of the opening).  The first test was carried out on concrete. The weather was cool and grey.  A metal tape-measure was laid on the ground and opened to its full length of twenty-six feet.  One point of the compasses was carefully placed at the beginning of the tape, and the compass was then swung eleven times across the ground alongside the tape, and the arrival point was recorded – 6446 millimetres.   The process was repeated eight times, and the results were all the same, never varying by more than one millimetre.   But then the distance covered seemed to be getting a little shorter by 4 or 5 mm, which caused me some concern until it was realised that the sun was now shining warmly on the tape which was lengthening in the heat.  The distance measured out by the compasses had not changed.  This fact was confirmed by repetition, and the medieval compasses were found to be extremely accurate and far more reliable than the modern metal tape-measure.  

     The test was repeated on a grass lawn, on soft soil.  The result was a difference  of two centimetres, less than an inch, in comparison with the measure made on concrete over more than twenty feet.  (Fig. 262)

 



[1] Zupko (1997)

[2] It is of possible relevance that Renaissance architects used such a system of two factors to divide a dimension,  such as the height of a column, into a large number of parts, as seen in several pages of Batty Langley’s The Builder’s Directory (1767). (Fig. 290).