Page:Philosophical Transactions - Volume 004.djvu/19

 Radio Circuli comprehensorum, mensura. Concerning which he tells you, that Archimedes squared that Spiral, which was made b an equal motion both in the Radius and Circumference of the Circle: that Stephano Angeli hath done the like, when the Motion in the Radius is equal, but in the Circumference according to any degree of Acceleration; which gave him occasion to render this Doctrine easie and Universal by reducing it to one Analysis, when the motion is accelerate according to any degree either in the Radius or Circumference; and hence resolves this Probleme; In Circulo describere Spiralem ex talibus motibus compositum, ut Circulus ad spatium Spirale habeat rationem datam numeri ad numerum. And applies the same Doctrine in

Chap. 3. to another sort of Infinite Spirals.

Chap. 2. He treats De mesnura spatiorum, curva & recta Contentorum, & eorum Centri Æquilibrii; applying the former Analysis or Algebraick Calculation thereto.

Chap. 5. Treats De Puncto flexus contrarii in Conchoide Nicomedis prima: which Point he determines by the Intersection of a Parabola, whose Axis is situated in the same Line with that of the Conchoid; or by a Cubick Parabola, whose Axis is parallel to the Base of the Conchaid, and Vertex the same with the Pole of the Conchoid; and hence invents innumerable other Conchoids of like properties, and finds the Curve, passing through those points of flexure, that are made by Infinite Conchoids, described about the same common Pole and Base, which in the Common Conchoids he finds to be the Perimeter of the Cubick Parabola here mentioned: But in his own new Conchoids, it is the antient Cissoid; extended beyond a Quadrant and running Asymptotick: And he finds also the round Solids made by the Rotation of these infinite Curves; and of the Cissoid Line, about their Base Lines or Asymptotes equal to finite Solids.

Chap. 6. The Author considering; that Vincenzo Viviani in Book De Maximis & Minimis found, that if there were innumerable Parabola described, having the same Axis and Vertex common, if from any point in that Axis, the shortest Lines were drawn to those Parabola's, all those points of Incidence would fall in an Ellipsis; and the Authors Analysis taught him, that the Prop. was Universal, wheresoever the point be assigned, from Rh