ilently fhall be thefe ; the 3 zd, and 4/^^, of the book^ 
moft of the 2d, and y^/^ books, the jji, and 16th of the 
d^^, with their Corollaries. In order to demonftrate 
the szd, I fuppofe it known what is meant by an angle, 
triangle, circle, external angle, parallels, and that the 
meafure of an angle is the arch of a circle intercepted 
between its fides, that a right angle is meafured by a 
quadrant, and 2 right angles by a femicircle. I fay then 
(ini^^. ij that in the triangle ABC, the external an- 
gle B C E is equallto the 2 oppofite internal ones ABC, 
B A C 5 for let a circle be drawn, C being the Center, and 
BC the radius, and let CI> be drawn parallel to A B, 
thofe 2 lines being alwayes sequidiftant will both have 
the fame inclination to any saline falling upon them, 
that is (by the definition of angle) they will make equal 
angles with iti for if any part of CD (for inftancej did in- 
cline more toBC then didAB,upon that very account they 
would not be parallel, it follows therefore that the angles 
ABC, BCD are equal alfo BA C= D CE, becaufe AE 
falls upon 2 parallel, but the external angle BCE= 
BCD 4 DCE which were before proved to be equall to 
ABC, BAC (Q:,E. D.) hence may be infer'd as a corollary, 
that the S angles of every triangle are equal to 2 right 
ones, for the angles ACB 4. BCE are meafured by a femi- 
circle and therefore equal to 2 right angles j Corollaries 
alfo from hence are the 20th 22^ and 3 i/of the 3^^book 
which contain the propertiesof circles, who fe dedudion 
from hence being moft natural and obvious, 1 omitt. 
In order to demonftrate the^jth, Ifuppofe the mean- 
ing of the terms made ufe of to be known; and that 
2 angles or fuperficies are equal when one being put on 
the other, it neither exceeds, nor is exceeded: this being 
allowed, I fay the fides about the right angle are either 
equal or unequal, if equal (as in Fzg. 2.) let all the fquares 
be defcribed, the whole figure exceeds the fquare of the 
/:^/?(?^y5^«/:^/^BCby the 2 triangles M,U, and exceeds alfo 
the fquares of the other 2 fides AB. AC. by the 2 trian- 
L 2 g^es 
