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

SCHOLIUM.

HEnce we have the foundation of Epipedometry or Masuring of Figures that stand on the same Base, and Ichnography; in the Practise whereof this deserves to be taken special Notice of, that to work so much the more Compendiously, you ought to divide your Figure into Triangles, so that (Fig. 31.) 2 of their Perpendiculars may (as conveniently can be) fall on one and the same Base. For thus you'l have but one Base to measure, and 2 Perpendiculars to find the Area of both: But for Ichnography, the distance of the Perpendiculars from the nearest end of the Base must be taken; which we shall supersede in this Place and Discourse more largely on hereafter.

2. This resolution of a Polygon into Triangles may be perform'd by assuming a point any where about the middle, and making the sides of the Polygon the Bases of so many

Triangles; (see the 2d Figure mark'd 31) wherein it is evi∣dent; 1 That all the Angles of any Polygon are equal to twice so many right ones, excepting 4, as the Polygon has sides; for it will be resolv'd into as many Triangles as it has sides, and each of these has its Angles equal to 2 right ones. Subtract∣ing therefore all the Angles about the Point M (which always make 4 right ones by Cons. 2. Def. 8.) there remain the rest which make the Angles of the Polygon. 2 All the external Angles of any right lined Figure (e, e, e, &c.) are always equal to 4 right ones; for any one of them with its Contiguous internal Angle is equal to 2 right ones pr. Consect. 1 of the said Def. and so altogether equal to twice so many right ones as there are Sides or internal Angles of the Figure. But all the inter∣nal Ones make also twice so many right Ones, excepting 4 therefore the external Ones make those 4.