Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

DEFINITION XXI.

THere remain those Bodies to be consider'd which are call'd Regular, which correspond to the Regular Plane Figures; and as those consist of equal Lines and Angles, so these likewise are comprehended under Regular and Equal Planes meeting in equal solid Angles; and as those may be Inscribed and Circum∣scribed about a Circle, so may the latter likewise in and about a Sphere. But whereas there are infinite Species of Regular Plane Figures, there are only five of Regular Solids; the first whereof is contained under four Equal and Equilateral Trian∣gles, whence it is nam'd a Tetraedrum; the second is terminated by six equal Squares, and thence is call'd Hexaedrum, and other∣wise

a Cube; the third being comprehended under eight Equal and Equilateral Triangles, is call'd an Octaëdrum; the fourth is contained under twelve Regular and Equal Pentagons, and so is nam'd a Dodecaëdrum; the fifth, lastly, is contained under twenty Regular and Equal Triangles, and is thence nominated an Icosa∣ëdrum. Besides these five sorts of Regular Bodies there can be no other; for from the concourse of three Equilateral Triangles arises the Solid Angle of a Tetraëdrum, from four the Solid An∣gle of an Octaëdrum, from five the Solid Angle of an Icosaëdrum; from the concourse of four Squares you have the Solid Angle of an Hexaëdrum; from that of three Pentagons you have the Solid Angle of a Dodecaëdrum; and in all this Collection of Plane Angles, the Sum does not arise so high as to four Right ones. But four Squares, or three Hexagons meeting in one Point, make precisely four Right Angles, and so by Consect. 2. Definit 8. would constitute a Plane Surface, and not a Solid Angle. Much less therefore could three Heptagons or Octagons, or four Pentagons meet in a Solid Angle, to form a new Regular Body; for those added together would be greater than four Right An∣gles. But now, for the Measures of these five Regular Bodies, take the three following

CONSECTARYS.

I. SInce a Tetraëdrum is nothing else but a Triangular Py∣ramid, and an Octaëdrum a double Quadrangular one, their Dimension is the same as of the Pyramids in Schol. of De∣finit. 17.

II. The Solidity of an Hexaëdrum may be had from Consect. 3. Definit. 13.

III. A Dodecaëdrum consists of twelve Quinquangular Py∣ramids, and an Icosaëdrum of twenty Triangular ones, all the Vertex's or tops whereof meet in the Center of a Sphere that is conceived to circumscribe the respective Solids, and consequent∣ly they have their Altitudes and Bases equal: Wherefore ha∣ving found the Solidity of one of those Pyramids, and multi∣plied it by the number of Bases (in the one Solid 12, in the o∣ther 20) you have the Solidity of the whole respective Solids.