DISCOVERY 



185 



shown to be the guiding principle in much Italian and 

 late Norman work down to the end of the twelfth 

 century, while the more subtle figure of the pcntacle, or 

 five-pointed star, is illustrated in ecclesiastical buildings 

 at Rouen. An ancient cathedral or other important 

 architectural work was in effect, then, a crystalline 

 structure whose ultimate molecules had a definite shape, 

 determined in each case by the root figure adopted in 

 its design. 



Nothing need be added to the full explanation given 

 in the above-named works, though I maj' be permitted 

 to refer to a more recent example as an indication of the 

 method to be adopted in trying to find the controlling 

 figure of the design. The Manchester Free Trade Hall 

 was built so recently as 1856. In Baines's History of 

 Lancashire the admirable great hall, noted for its 

 acoustic properties, is described as being 134 feet long 

 by 78 feet wide. Dividing one of these numbers bj- 

 the other, we find that they are in the ratio of 172 to i. 

 Now, in the diamond-shaped figure formed by two 

 equilateral triangles placed base to base, the foundation 

 of the Vesica Piscis, the ratio of length to width is 1-732 

 to I. In a chamber whose length is 134 feet, the width, 

 if proportioned in this ratio, should be 77 feet 5 inches 

 nearly, and the agreement with the published dimen- 

 sion is evidently so close as to justify the belief that the 

 design was intentionallj' based on traditional methods. 

 In the same chamber the height and width are in the ratio 

 of 2 to 3, a proportion which will be referred to later. 



It may be suggested here that a convenient way of 

 detecting these and other root figures is to rule a sheet of 

 tracing-paper into small triangles or squares as the 

 case may be, and apply it to the published plans or 

 elevations of the building under examination. 



As to the reasons which led to the adoption of such 

 figures, there is little doubt that in their first origin they 

 were used from purely utilitarian motives. In Ars 

 Quatiior Coronatorum, in 1906, I showed that, having 

 regard to the want of instruments of precision, the equi- 

 lateral triangle offered many practical advantages to 

 the early builders in the setting-out of the work on the 

 site as well as in the preparation of the preliminary 

 plans. Great accuracy could be attained by its use 

 on the ground, even when working with long \\ooden 

 trammels or with stretched cords, and the setting-out 

 of right angles was easily effected by constructing 

 equilateral triangles on either side of a given line, base 

 to base, and joining their apices. The intersecting arcs 

 forming these two triangles constitute the correct form 

 of the Vesica Piscis, around which manj' mystical 

 associations have accumulated. 



Instead of the architectural use of the above figures 

 being founded on reverential and religious motives, 

 consideration seems to indicate that their first adoption 

 was a matter of practical utility, and that the associa- 



tion therewith of the various religious and philosophic 

 trinities and recondite mystical sjTiibolism was the 

 result of a profound veneration for the wonderful pro- 

 perties of the geometriced figures, and. perhaps, a wish 

 to preserve as a trade secret the knowledge of their use. 



There is one root figure, a triangle possessing valuable 

 properties of the nature referred to, that has so far 

 escaped the notice of investigators, although when 

 once pointed out it is impossible to avoid meeting 

 its presence in multitudinous examples. I refer to the 

 Pythagorean triangle, or triangle which has its sides in 

 the proportion of 3, 4, 5. From the examples here- 

 after given it will be seen that this remarkable figure 

 has formed a basis of design in architectural design 

 work from the earliest days of Egyptian civilisation to- 

 a period of less than a hundred years ago. Its propor- 

 tions can be traced in the measurements of ground 

 plans and elevations of buildings, in the pitch of roofs, 

 in the proportions of doors and windows, diamond- 

 leaded panes, ventilating grilles, and other details of 

 construction. 



It may be remarked at this point that the proofs of 

 the statement made below are matters of simple arith- 

 metic, and to avoid any charge of dealing too s\-mpa- 

 thetically with figures in order to support a novel 

 theory, I have in every case given the authorities from; 

 whom the measurements have been derived, so that 

 the reader may satisfy himself of the value of the 

 evidence offered. 



The utility of this particular triangle for the purpose 

 is easily demonstrated. It is the most suitable one for 

 the setting-out of a right angle on a large scale— not the 

 only one with sides in a definite ratio of whole numbers, 

 but the one whose measures are easiest to carry in 

 the memory and whose form best lends itself to accuracy 

 of construction. The following triangles, amongst 

 others, can also be used for the setting-out of a right 

 angle ; 



5 12 13 



7 24 25 



8 15 17 

 16 63 65 

 20 21 29 



but with the exception of the last the 3, 4, 5 ratio gives 

 a better intersection of lines, and angles more fitted for 

 practical setting-out. It is in regular use to-day in 

 land surveying and in setting-out buildings. It also, 

 possesses other rather extraordinary properties which 

 it is not necessary to detail here. 



Two of the root triangles, having sides of 3, 4. and 5. 

 unit§ in length, when laid together in the manner shown 

 in Fig. I, form a rectangle measuring 4 by 3, and this 

 proportion is traceable in many old buildings. Let us 

 begin with St. Peter's at Rome. In the Architectural 

 Director, a book on classic architecture by Mr. Joha 



