Cone is the third Part of a Cylinder up- X 'or dron. Corol. 1. As a Tetraedron is no other To measure than one Sort of Pyramid, and therefore an Octaeits Solidity is to be found by this Theorem the 2d; so an Octaedron, being no other than two Tetraedrons or Pyramids join'd together at their Bafes; hence the Content of the common Bafis being found, and multiplied by a third Part of the Height of the Octaedron, (which is made up of the two Heights of the two Tetraedrons or Pyramids, into which the Octaedron is divifible,) will give the Solidity of the O&taedron. caedron Corol. 2. Forafmuch as a Dodecaedron To measure may be conceiv'd to be made up of a Dodetwelve Pyramids, whofe feveral quinquangular Bafes are all equal one to the other, and make the twelve several Sides or Surfaces of the Dodecaedron, and whofe Vertex's confequently meet together in the Center of the faid Solid hence, having found the Area of one To measure an dron. quinquangular Bafis, and multiplied it into one Third of Half the Height of the Solid, you will have the Solidity of on Pyramid. And this being multiplied b 12, will give the Solidity of the whol Dodecaedron. Corol. 3. The Icofaedron being made Icofae. up of twenty equal Pyramids, whole Bafes are fo many equal Triangles in the Surface of the Icofaedron; and whofe Vertex's confequently meet in the Center of the faid Body; it follows, that the Area of one of the twenty Triangles be ing found, and multiplied by a third Part of (the Height of one Pyramid, e) Half the Height of the Icofaedron, the Product will give the Solidity of one Pyramid, or 20th Part of the said Icofaedron. And, confequently, this Product being multiplied again by 20, will give the Solidity of the whole Icofaedron. And thus there has been shewn the Method of Measuring, as of other Solids, fo of the five Regular (or Plato nick) Bodies. For the Hexaedron or Cube is one Sort of Parallelepiped, and fo its Menfuration is taught in Theotem 1. And the Tetraedron is one Sort of Pyramid, and fo its Menfuration is taught in Theorem 2. And the Menfutation of the other three regular Bodies, is taught in the other three Corollaries Theorem 2. THEOREM III. Parallelepipeds and (other) Prisms, and alfo Cylinders, if they have the fame or equal Bafes, are one to the other as their Altitudes; or if they have the fame or equal Altitude, are one to the other as their Bases. This might be accounted for, as to Parallelepipeds and other Prifms, from the Relation they bear to Parallelograms and Triangles, between which there is the like Proportion, (as has been fhewn, Chap. 1, Theorem 7.) But I fhall rather account for it from the Manner of producing Parallelepipeds and Prisms, according to Defin. 19, of this Chap. 3; because, hereby the like Proportion between Cylinders will alfo be accounted for. According then to Defin. 19, a Parallelepiped is produ ced by multiplying a Parallelogram into a right Line; which Parallelogram may be conceiv'd to be the Bafis of the Parallelepiped, and the right Line its Altitude. Wherefore Parallelepipeds, which have the fame or equal Bafes, are (or may be conceiv'd to be) produced produced by the fame or equal Paralle lograms. And therefore, if the fame or equal Parallelograms be multiplied into the fame or equal Lines, it is evident, that the Parallelepiped thence produced must be equal. But if the fame or equal Parallelograms be mul tiplied into Lines of different Lengths, fuppofe one twice as long as the o ther, then it is evident, that the Par allelepiped produced by the Multiplication of a Parallelogram upon the longer Line, muft be twice as big as the Parallelepiped produced by the Multiplication of the fame, or an equal Parallelogram into the fhorter Line. And fo of any other Proportion. To illu ftrate the Matter in Numbers. Suppose the Parallelograms to be each fix Foot square, and the Lines to be each two Foot long. It is evident, that the Parallelepipeds, produced by the Multiplication of the faid Parallelograms into the faid Lines, will each be twelve Foot cubical, and therefore equal. But, fuppofe one of the faid equal Parallelograms to be multiplied into a Line two Foot long, and the other equal Parallelogram be multiplied into a Line twice as lòng, viz. four Foot long; it is evident, that the Paral lelepiped, lelepiped, produced by the former Multiplication, will contain twelve cubical Feet and the other Parallelepiped, produced by the latter Multiplication, will contain twenty-four Feet cubical; and confequently, the Parallelepipeds will be one to the other, as are (the Lines by which the Parallelograms were multiplied, i. e.) their Altitudes. In like Manner, fuppofe one Parallelogram to contain fix Feet square, and the other three, and both to be multiplied into a Line four Foot long; it is evident, that the Parallelepiped, I produc'd by the former Multiplication, will contain twenty-four cubical Feet and the Parallelepiped, produced by @ the latter Multiplication, will contain twelve cubical Feet; and confequently, the Parallelepipeds, having the fame or equal Altitude, will be one to the other, as are (the Parallelograms multiplied, i. e) their Bafes. And the like may be illuftrated after the fame Man ner as to all Prifms, and alfo Cylinders. From whence the Truth of this Theorem ad evidently appears. 91 Corol. Pyramids, and alfo Cones, if they have the fame or equal Bafes, are one to the other as their Altitudes or if they have the fame or U 4 equal Al titudes, |