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Chap. XXXI.

PRINCIPLES OF PROPORTION.

1623

west, and the clear width between the plinths about two-thirds of that dimension, and thisis the case with many examples.

The Section through the Nave of Winchester Cathedral is highly deserving of our attention :the clear width of the side aisles is 13 feet 1 inch, and that of the nave 32 feet 5 inches ; theclear width of the building between the outer walls is 80 feet, the thickness of the walls16 feet 10 inches, the projection of the buttress 6 feet, and the thickness of the piers 10feet 8 inches, making for the entire width from north to south 102 feet 8 inches.

The width between the walls forms the base of an equilateral triangle, the apex of whichdetermines the height of the vaulting of the nave; a semicircle struck upon this base, witha radius of 52 feet, determines the intrados of the arches of the flying buttresses on each side,which are admirably placed to resist the thrust opposed to them.

On this section we have endeavoured to apply the principles of Ccesare Cesariano , beforereferred to, to the measurement of mass and void by a method far more simple than thatusually adopted.

By covering the design with equilateral triangles we see the number occupied by thesolids, and can draw a comparison with those that cover the voids : to prevent confusion inthe diagram a portion only of three of the triangles has been subdivided, to show with whatfacility the quantities of the entire figure might be measured, if the several large cqui-laterals were subdivided throughout in a similar manner. The band which extends

Jfig. 3036.

SECTION OP WINCHESTER CATHEDRAL.

from the face of the outer buttress to the centre of the section contains 36 small equilateraltriangles, six of which cover the pier; consequently it occupies on the section one-sixth ofthat quantity ; no further calculation is requisite to find the proportion it bears to thewhole : in like manner the other parts of the section may be compared. Such was the useof equilateral triangles in the middle ages for ascertaining quantity.

The two equilateral triangles which occupy the nave and a portion of the piers arecomprised within the figure called a Vesica Piscis; if the horizontal line drawn at half theheight, uniting the base of the upper and lower triangles, be taken as a radius, and itsextremities as centres, it will he evident that parts of circles may be struck, comprising thetwo triangles within them. Euclid has shown that a perpendicular may be raised or let faitfrom a given line by a similar method, the space between the segments being called afterwardsa nimbus; and there can he no doubt that from time immemorial all builders have usedit: the bee adopts for its honied cell a figure composed of six equilateral triangles, andthis is proved to be the most economical method of construction; the sides of each hexagonare all common to two cells, and no space is lost by their junction. The nearer theboundary line of a figure approaches the circle, the more it will contain in proportion to it

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