( 4 ° ) 
are particularly confidered as Examples of the Method* 
in the Fir ft Sedlion the Lines of the Second Order are 
confidered ; in the Second, thofe of the Third Order, 
that have a Pentium duplex’, in the 3d Se&ion, the Lines 
of the Fourth Order, and thofe of the Third Order 
that have no Puntfum duplex. in the laft Sedtion there 
are many various Methods of defcribing the Lines of 
any Order. 
In the Second Part, the Curves of the inferior Or- 
ders are made ufe of for defcribing thofe of the higl> 
er kinds. Jn the Firft Sedtion, the Theorems publi- 
(hed by Sir Ifaac Nervton at the end of the Enumeration 
of the Lines of the Third Order are demonftrated In 
the Second Sedtion, Curves are fubftituted in the room 
of (height Lines, in all the Propofitions of the Firft 
Part. From one of thefe Propofitions, Lines of the 
1024th Order may be defcribed by making Angles 
move on (even Conick Sedtions 5 and by three Conick 
Sections more, Lines may be defcribed above the 
1 1 ,000th Order. Laftly, thefe Theorems are applied 
to (hew how the more Complex of the Infinite Order, 
may be defcribed from the more Simple. 
In the Third Sedtion, fome other Methods of defcri- 
bing Curves are confidered, that are not (o general as 
the preceding, but give fometimes more fimple Me- 
thods of defcribing fome few Lines of the Superior 
Orders. Particularly the Epicycloids defcribed by the 
Motion of any Curve, whether Geometric or not, upon 
another equal to it are eafily conftrudted, and feveral 
Infinite Series of them rectified or meafured by Arches 
of more fimple Curves. In this Sedtion, feveral other 
Defcriptions of Curves are treated of, that have been 
propofed by others. In the laft Sedtion, to (hew the 
ufe of Curves in Natural Philofophy, two of the mod 
eminent Problems in Mathematical Philofophy are fol- 
ved 
