Page:Philosophical Transactions - Volume 003.djvu/17

 parts out of the whole: If it stand upright, and be divided by an upright Plain into two Partitions, imagine it to be a whole Cone, and first, by the method above, find the Segment, as of the whole, and afterwards of the additional Top-Cone, the difference of those two gives the Content of the correspondent Partition.

3. But if the Liquor cut both sides, the Tun leaning, as BCDE, in Figure III. Suppose BAE to be the Triangle through the Axis of the whole Cone, then the Elliptick Cone ACD to the whole ABE is in a Triplicate Ratio of the Side-line AB or AE, to the Geometrical Mean between AC and AD, that is,

As the Cube of the Side-line AB, Is to the Solidity of the whole Cone ABE: So is the Cube of the Geometrical Mean between AC and AD, To the Solidity of the Elliptick Cone ACD.

And this readily follows from the Doctrine of Viviani de Maximis & Minimis, where 'tis demonstrated, that any such Elliptick Cones, cut out of an Upright Cone, that have the Area's of their Triangles through the Axis equal, are equal to each other; and likewise to that Upright Cone which hath the same Area on its Triangle through the Axis on the former Plain thereof; and these Area's he calls their right Canons.

And the mean Proportional by 23. E. 6. finds the sides of an Isosceles Triangle in the Plain of the Axis equal to the Scalene Triangle; and then these Cones are to each other in a Triplicate Ratio of their Axes, Side-lines, or Base-lines, which are proportional to their Axes.

The Area of an Hyperbola being obtain'd, the Solidity of the Hyperbolical Fusa or Spindles (made by the rotation of an Hyperbola about its Base) and their Trunci are computed, according to Cavallieri (in his Geometrical Exercises, printed at Bononia 1647.) and the solid Zones of these Figures may be well taken to represent a Cask.