6lU PHILOSOPHICAL TBANSACTIONS. [aNNO J 693-4. 



lating to the practice of such buildings, as how, by means of the lathe and 

 cylindrical auger to nnake models of this, as well as of five other cupolas ; and 

 for this purpose he constructs some other curious problems ; the demonstrations 

 of all which are omitted by the author, but will easily appear from what is de- 

 livered below. 



It is manifest that the four windows in the hemisphere, constructed as above, 

 are figures that are equal, similar, and alike placed : it only remains therefore 

 to show that the remaining part of the hemispherical surface is capable of a 

 true geometrical quadrature. Now at the point E, conceive a line, equal to 

 AE, to be erected perpendicular to the plane ACBD; and on the periphery 

 A C B D let there be an erect cylindrical surface of the same height. It is well 

 known, that a portion of the spherical surface contained between any two 

 planes parallel to the circle ACBD, is equal to the portion of the cylindrical 

 surface between the same planes ; and that like portions of these rings, cut off 

 by planes mutually intersecting in the perpendicular erected at E, are also equal. 

 Now by drawing innumerable planes parallel to the base ACBD, if in the 

 cylindrical surface parts be conceived to be described in the aforesaid manner, 

 equal to the corresponding spherical parts, that which is represented by the per- 

 foration of the superficies, and taken away from the opposite side, is equal to 

 it. Hence it appears, that the remaining surface after the perforation, is equal to 

 the remaining cylindrical surface, excepting that which is determined by the said 

 innumerable planes, and opposite to that which is taken away. Let there be 

 drawn then any diameter P M, cutting the periphery A H E in any point H ; 

 join AH, and through H draw R T perpendicular to AB, and parallel to C D 

 drawn through E, meeting the periphery A C B D in R and T, and the peri- 

 phery A HE in I. On the diameter RT describe a semicircle, cut in S and Q 

 by H S and I Q perpendicular to R T ; and conceive the plane of this semicircle 

 to be erected perpendicular to the circle ACBD. Whence the periphery 

 R S Q T will be in the hemispherical superficies, and the right line H S, now 

 perpendicular to the plane ACBD, will be the latitude of the perforating 

 cylindrical superficies above the point H of the base. And the same for every 

 point of the perforating cylindrical surface, viz. that its latitude to the surface of 

 the sphere above any point H in the base, is the right line H S drawn as above. 

 But H S is equal to H A, the sine of the arc A M, because each of them is a 

 mean proportional between MH and HP, the one in the circle MAP, the 

 other in a great circle of the sphere passing through the points M S, and P. 



If, in the perpendicular to the plane ACBD erected at E, from E there be 

 taken a right line equal to H S or H A, and from its extremity there be drawn 

 lines parallel to P M and VN ; then the plane drawn through them will be 



