[ 453 ] 
For, in this cafe, the two lines A R and A Qjco- 
incide, and A R is parallel to the diredtrix D N of 
the bafe ; and therefore, ufing the fame fymbols as 
above, the equation, from Fig. 4. will be reduced 
to y a^=.n me a x-\~n m b x 2 ; and from Fig. 5. it will 
come to y a+=.n m c a x — n m b x 2 ; both which fhew 
the curve to be a parabola. 
It may alfo be demonftrated in this manner. It is 
evident, that PO:fflr;:POxMR:«rxMR, that 
is, the ratio of P O to mr is equal to that compound- 
ed of P O : M R, and of M R : m r. But from the 
fimilar triangles MRN, PON, we have P O : M R 
: : NO:NR = «r, and from the triangles BON, 
Br», we have B O : B r : : N O : n r ; therefore P O : 
MR::BO:Br. In like manner, from the equi- 
angular triangles M R A, m r A, there will be 
MR : m r : :R A : r A, and from the triangles ROr, 
Z A R, it is R A : r A ; : O z : r z j therefore M R : 
m r\ ;o z: r z. If then in the ratio of P O x R M 
to m r x M R, we fubftitute the ratio of BO-.B r, 
and of O z:r z, which are equal to P O : M R, 
and MR:ffl r, we fhall have P O \mr \ ;BO xoz: 
Brxrzj which is a known property of the pa- 
rabola. 
And thus I have endeavoured to extend, a little, 
the Theory of the conic fedtions. I have here fhewn 
how two of them may be had from the fedtions 
of this folidj and in the year 1733, I publifhed, 
in my Exercitatio Geometrica> a method of deferr- 
ing all of them on a plane, by the moving of three 
right lines about three given points, two of the inter- 
fedtions being drawn along two other lines given in 
polition * the only hint of which I had from a geome- 
