( XIX ) 
I then fiiew, that the Cycloid is a Figure compounded 
of thefetwo; the Semicircle ADol, and theTrilincar 
ADctrhA, lying between the two Curves ADcl and 
Adr, (and thertfjre, to Square any part ot thefe, is 
the fame as to Square the refpedtive part of the C/- 
cloiJ, 
Ilhew further {lltdem.pag.Zo^) that this Trilinear 
is but a diftorted Figure, ( by reafon of the Semicircle 
thruft in between it and its Axis ) w hich being reftored 
to its due Pofition ( by taking out the Semicircle into 
a different Figure, as Figur^x. and thrufting r he Lines 
hB home to the Axis, (j as that BVh^ the lame point) 
is the fame wither a, Fignr. (the Parallelograms 
h^ct,B being fet upright, which in iht Cycloid ftand 
Hoping,- and the Circular Archs ^/S, Figitr. i, becom- 
ing flreighc-lines in Figur, 3. and the Lmes iB being, 
in both, equal 10 the relpedive Archs i5^,every where;) 
which therefore I call Trilineum Reflitutum ( the Trili- 
near reftored to its due Poficion, which Figure I do not 
find that any before me has confidered:) So that to* 
Square any part of this, is the fame as to Square the re- 
fpeftive part of iht Cycloid^ (or of the Trilinear in the 
Cycloid:) That which in xh^ Cycloid lies between two 
Archs of the Circle Generant in different Pofitions , 
anfwering to that which, in the reftored Figure, lies be- 
tween the refpc6llve ftreight-lines. 
And therefore A dD A^—i: d^ i:^Figur. \,^AiDA 
= rd^r, Figur, 3, =K. And AbkdA, -rtk^r, 
Fzgur, =Ah kd A^ rlk^r, Figur, },=s K And 
ikdy Figur. i^=bkd, Figur^i, =R* — s R, Ihidem, 
Cap, 17. B,pag. 756. Where, if h be taken above dkDC 
( pafiing through the Center C, ) thefe Figures are with- 
in the Cycloid^ and within the reftored Figure ; but with- 
out them, if b be taken below that Line, and adjacent 
to the Curve ^^r, in both cafes. 
By 
