a by the For one of the Angles being a Rt. 2 182. Suppofition, the Sum of the other two is Rt. 4. 59. But the 4s at the Base of an Ifofceles ▲ are equal; ... each of them must be a Rt. 4, both being equal to a Rt. 4, as hath been juft fhewn. 60. = 2 85. COROLL. 7. An equilateral Triangle being equiangular, each Angle must be equal to one Third of two Rt. Angles; or, which is equivalent, two Thirds of one Right Angle. Pl.1.F.23. 2 78. 568. 86. COROLL. 8. If in any Triangle, one of its Angles is either a right or an obtufe Angle, the other two will be each of them acute. 87. COROLL. 9. All the interior Angles of any rectilineal Figure, together with four Right Angles, are equal to trcice as many Right Angles as the Figure has Sides. For any rectilineal Figure, ABCDE, can be divided into as many Triangles as the Figure has Sides, by drawing Lines from any Point F within the Fi gure, to each of its angular Points. And all the Angles of thefe Triangles are twice as many Rt. Ls as there are Triangles, that is, as the Figure has Sides. And the fame s are the Ls of the Figure, together with the s at the Point F, which is the common Vertex of the Triangles; that is, together with four Rt. s. Therefore this Corollary is true. 88. COROLL. 10. All the exterior Angles of any rectilineal Figure are together equal to four Right Angles. Because every interior ABC, with its adjacent exterior ▲ ABD, is a = 2 Rt. Ls; . all the interior together, with all the exterior s of any Figure, are are twice as many Rt. 4s as the Figure has Sides, that is, by the laft Corollary, they are all the interiors of the Figure, together with four Rt. Ls. all the exterior s are 4 Rt. s. 89. THEOREM 12. The greater Side of every Tri-Pl.1.F.25. angle is oppofite to the greater Angle: That is, in AABC, let the Side AC be greater than AB, then, I fay, the Angle B is greater than the Angle C. For AC being F AB by the Suppofition, let D be a Point in the Side AC, fo that AD be = AB; and fuppofe B, D, joined. Then, ADB being the exterior of A BDC, the LADB r the interior2 77. and oppofite C: But AD being AB, the ADBLABD;. as it has been juft fhewn 59 that ADB is г C, its equal ABD must also be FC, and much more nuft the whole B (which is FA ABD) be гzС. Q. E. D. 90. COROLLARY. Hence, in every Triangle, the greater Angle is fubtended by (that is, is oppofite to,) the greater Side. For by the last Article, the greater Side is oppofite to the greater Angle; confequently, the greater is oppofite to the greater Side. For if any Thing, A, is oppofite to another Thing, B, it is manifeft that the Thing B must be oppofite to the Thing A. Here again we have taken the Liberty to deviate from Euclid's formal Demonftration. 91. THEOREM 13. If two Triangles, ABC, DBC, Pl.1.F.26 bave the fame common Bafe, the two Sides BD, DC, of the infcribed Triangle, taken together, will be less than the two Sides BA, AC, of the circumfcribing Triangle: But the vertical Angle BDC of the infcribed Triangle will be greater than the vertical Angle A of the circumfcribing Triangle. 4 As the Sides BD, DC, are contained within the Sides BA, AC, it is manifeft the Sum of the circumfcribing Sides BA + AC must be r thofe which they circumfcribe, viz. r BD + DC: For whatever Thing includes another, must itself be r that other. This is fo plain, that Monfieur Clairault, a celebra. ted French Geometrician, cenfures Euclid for taking the Pains to demonftrate it. But if a Demonftration be required, as the Number of Axioms ought not to be increased without a Kind of Neceffity, it may be done thus. Suppofe BD produced to meet the Side AC in E. Then in A ABE we have BA+ AET BE, and to each of thefe adding EC, we a find that BA+AE + EC is r BE + EC, but AE · a 7o·· + EC = AC, ·.· BA + AC г BE + EC. Again,, in A EDC we know that CE + EDr DC, and to each of these adding DB, we find that CE + ED + DB г DC + DB; but ED + DB = BE.... CE + BE г DC + DB. But it has been proved that BA+ACT CE + BE, much more then is it r DC + DB. Q. E. D. and oppofite Pl.2.F.I. b BDC is г ▲ A may be thus fhewn. BDC of the ACDE is 'r the interior CED; and the exterior CEB of A. CED, and oppofite BDC is г which is г LA; of the A ABE is r the interior 92. THEOREM 14. If two Triangles ABC, DEF, have two Sides of the one equal to two Sides of the other, each to each, viz. AB=DE, and ACDF; but the contained Angle of one greater than the contained Angie of the other, viz. the Angle A greater than the Angle EDF; that which has the greatest Angle between the equal Sides will have the greatest Bafe, viz. BC will be greater than EF. Of |