VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 3Q 



are towards the north, and it is not altogether so smooth nor so well painted. 

 But the carvings areas good, and it appears altogether as stately and magnificent 

 as the former. Besides it has the advantage in age of a whole century of years, 

 as appears from the date of the inscription. It is placed above a nich in the 

 front adorned with handsome borders and cornices, the place, doubtless, of 

 some statue, and probably that of the founder. 



On the Cycloidal Spaces which are perfectly Quadrable. By Dr. JVallis. 



N°217, p. 111. 



It is generally supposed that no part of the semicycloid figure, adjacent to 

 the curve, is capable of being geometrically squared but these two, viz. 1. The 

 segment a /; v, fig. 6, pi. 1, taking a v = -i-Aa, (which was first observed by Sir 

 Christopher Wren, and after him by Huygens and others,) and it is = 4« r = 

 4 r'^ V" 3. — 2. The trilinear A <i d, taking d d, in the parallel dnc, passing through 

 the centre c, which is = r^. 



But it is otherwise, as I have showed in my treatise De Cycloide, and that 

 De Motu ; the figures of which latter I retain here, so far as they concern this 

 occasion, there being other portions of it, equally capable of quadrature. 



In order to which, I there show ( De Motu, cap. 5, prop. 20. A. pp. 802, 803, 

 604) that not only the cycloid is triple to the generating circle, which was 

 known before, but that the respective parts of that are triple to those of this ; 

 which is the foundation on which I build my whole process concerning the 

 cycloid in both treatises, and which is not pretended, that I know of, to have 

 been observed, or known by any person before me ; that is, ^ (3 a a, fig. 6, triple 

 to the sector b a a, taking b |3 parallel to b «, wherever, in the curve a t, we take 

 the point b. 



I then show that the cycloid is a figure compounded of these two ; the semi- 

 circle A D a, and the trilinear a d a t /j a, lying between the two curves a d a and 

 A <f T, and therefore, to square any part of these, is the same as to square the 

 respective part of the cycloid. 



I show further (ibid. p. 804) that this trilinear is but a distorted figure, by 

 reason of the semicircle 'hrust in between it and its axis ; which being restored 

 to its due position, by taking out the semicircle, into a different figure, as 

 fig. 7, and thrusting the lines /; b home to the axis, so as that b v be the same 

 point, it is the same with at«, fig. 8 (the parallelograms b(ix3 being set up- 

 right, which in the cycloid stand sloping, and the circular arcs b [i, fig. 6, 

 becoming straight lines in fig. 8, and the lines b b being, in both, equal to 

 the respective arcs b a every where ;) which therefore I call trilineum resti- 

 tutum, the trilinear restored to its due position, which figure I do not find that 



