L ? 26 ] 
there follows a Summary of what is moft material ill 
the Treat ifes of Archimedes, concerning the Sphere 
and Cylinder, Conoids and Spheroids, the Quadra- 
ture of the Parabola and the fpiral Lines. The De- 
monftrations are not precifely in the fame Form as 
thole of Archimedes , but are often illudrated from 
the elementary Propofitions concerning the Cone, 
or Corollaries from them, after the Example of 
Pappus, (Coll. Math. Prop. 21ft, Lib. 4.) from whom 
a Proportion is demonftrated, and rendered more 
general, concerning the Area of the Spiral that is 
generated on a fpherical Surface by the Compofition 
of Two uniform Motions analogous to thofe by 
which the Spiral of Archimedes is deferibed on a 
Plane. This Area, though a Portion of a curve Sur- 
face, is found to admit of a perfect Quadrature, and 
this Propolition concludes the Abftrad. He takes 
occaron from thefe Theorems to demonftrate fome - 
Properties of the Conic Sections, that are not men- 
tioned by the Writers on that Subject; and there are 
more of this kind deferibed in the Xlth and XIVth 
Chapters of the Firfl; Book. 
It is known, that if a Parallelogram, circiimfcribed 
about a given Eilipfe, have its Sides parallel to the 
conjugate Diameters, then fhall its Area be of an 
invariable or given Magnitude, and equal to the 
Redangle contained by the Axes of the Figure; but 
this is only a Cafe of a more general Proportion* 
For if, upon any Diameter produced without the 
Eilipfe, you take Two Points, one on each Side of 
the Centre at equal Dillances from it, and the Four 
Tangents be drawn from thefe Points to the Eilipfe, 
thofe Tangents fhall form a Parallelogram, which is 
always 
