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sir CHRISTOPHER WREN, knt.

appears in the Ailes of Dioclesian' s Baths, and in some respect also in the 'TcmplumPads.

The different Forms of Vaultings are necessary to be considered, either as theywere used by the Ancients, or the Moderns, whether Free-maJ'ons , or Saracens.

The Romans , though they sometimes used a Hemisphere, where the Room wasround; or Half-hemispheres, as in the Exedra of the Baths, or the Tribunesof Temples and Basilica , yet generally they used a plain cylindrical Vaulting,where the Walls were parallel; or Cross-vaulting, where two Cylinders inter-sect in Diagonals, as in the Pemplum Pads ; and in all the Theatres in thePassages under the Steps. The Moderns, whose Arches were not circular, butmade of Sections of Circles, used commonly another sort, where the Spandrilsresting upon the Pillars, sprang every way round as their Arch rose. It is noteasy to give a geometrical Definition, but by calling it a circular inverted Cone(A), resting upon its Apex (B); (C) the Middle, they filled up with Tracery- Fig. v >work, for which this Way gave them great Opportunity of divers Variations,which I need not insist on. Another Way, (which I cannot find used by theAncients, but in the later eastern Empire, as appears at St. Sophia, and by thatExample, in all the Mosques and Cloysters of the Dervises, and every where atpresent in the East) and of all others the most geometrical, is composed of Hemi-spheres, and their Sections only: and whereas a Sphere may be cut all mannerof Ways, and that still into Circles, it may be accommodated to lie upon allPositions of the Pillars. Let E be a Cupola or Hemisphere, resting upon four Fig. III.Pillars A B C D, from whence arise the four Arches, to which the Sections,being Semicircles, must join on all Sides, whether A B be equal to B C or not.

Cut the Hemisphere again horizontally, the Section will be an entire Circle,touching in the Keys of the Arches, and G H K L will be Spandrils resting Fig. IV.upon the Pillars, yet still are Parts of the Hemisphere; and if the horizontalCircle be taken away, you may build upon that Circle an upright Wall, whichmay bear a Cupola again above, as is done at St. Sophia and St. Peter's, and atall the Churches at Rome. I question not but those at Constantinople had it fromthe Greeks before them, it is so natural, and is yet found in the present Seraglio,which was the episcopal Palace of old; the imperial Palace, whose Ruins stillappear, being farther eastward. Now, because 1 have for just Reasons followedthis way in the vaulting of the Church of St. Paul's, 1 think it proper to shew,that it is the lightest Manner, and requires less Butment than the Cross-vaulting,as well as that it is of an agreeable View; and, at the same time, 1 shall shewhow the Centers of Gravity are to be computed. To shew that it requires lessButment than the diagonal Cross-vaulting, 1 will compare them both together,without any perplexed Demonstration, as follows.-

It is evident that the Spandrils, or loading of the diagonal Cross-arches, wheretwo cylindrical Vaults meet, must be an inverted Pyramid, whose Basis is a Pa-rallelogram, with two Sides strait, and two circular; and wherever it be cuthorizontally, it will be cut into like Parallelograms: now, in the other easternWay of Vaulting by Hemispheres, the Spandrils are the Solids, which are leftwhen a Hemisphere is taken out of a Half-cube; each of these also must be asort of inverted Pyramid, whose Bases and Sides are circular, and wherever itis cut horizontally, it is cut into Pieces of Circles.

What these are that give the Butment of Arcades in the several Forms ofArches may be geometrically determined, for Example in the Roman Way ofCross-arches.

Let A BCD represent the whole Vaulting between sour Pillars, then efg Fig.vi.will represent the Quarter of this Vaulting resting upon D. Now, because thesolid Half-cylinder C D is cut off by the Half-cylinder B D, it is evident thewhole Crofs-vault will be equal to one Half-cylinder, whose Dameter is BD,