### Full text

Introduction . 7 thereby - , which although not eaual to one other , yet their Sum , that is , taken altogether , will be equal to four right Angles . For as much as the Angles ON are leflencd , the other oppofite Angles M P will be enlarged , and therefore their Sum will be as before . 4 . If again , as in Fig . 4 , there be another Line ST drawn Parallel to X Z , then there will be four more Angles , 1 , 2 , 3 , 4 , made in all Refpedts equal to the former , MNOP ; for M is equal to 1 , N is equal to 2 , O is equal to 3 , and P is equal to 4 , as may be demonstrated at large . Thus far then being known of Lines , and the gles they form by thefe Pofitions one to another ; let a Triangle be formed , as in Fig . 5 , and the Side A H be extended as far as E , it is plain by Step i , 2 , 3 , that the internal Angle A , and the external Angle D taken together , are equal to two right ones : If therefore it can be proved , that the Angles B and C amount to juft the lame as the Angle D , then the Proportion will be demonilrated . To this Pur - pofe therefore , the beft Way is to try to divide D into two Angles in fome fuch Manner , if poiTible , as to be commenfurable feparately to B and C : And how this may be done , appears from Step 4 , by drawing a Line through the Vertex , as XZ , which lhall be parallel to the Bafe BC , as in Fig . 6 , and will cut the external Angle into K and L . Now by the fame Reafon that N is equal to 2 , Fig . 4 , or that there is an exaft Equality between the correfpondent Angles made by each Parallel ; for the very fame Reafon , I fay , K is equal to B in Fig . 6 , and L equal to C . If then the Sum of ALK is equal to two right Angles , as before proved , and B added to K is equal to K added to L , then the Sum of B 4 ABC