...the proposition is a fundamental one, we will prove it. Suppose two triangles, of whatever form, to have two sides of the one equal to two sides .of the other, each to each; and the angle contained between those two sides in the one triangle to be equal to that which is contained...

...triangles would therefore be equal in all their parts. And we say universally, — When two triangles have two sides of the one equal to two sides of the other, each to each, and the angle contained by these two sides of the one, equal to the angle contained by the two sides...

...considered by Euclid. Of these seven combinations, six of them belong to the case of two triangles, having two sides of the one equal to two. sides of the other, each to each, and one angle to one angle, viz. those to which equal sides are opposite. This case will be fully discussed...

...AE has been cut off equal to C the less. Which was to be done. PROP. IV. THEOREM. IF two triangles have two sides of the one equal to two sides of the other, each to eacji ; and have likewise the angles contained by those sides equal to one another, they shall likewise...

...triangles would therefore be equal in all their parts. And we say universally,— When two triangles have two sides of the one equal to two sides of the otlicr, each to each, and the angle contained by these two sides of the one, equal to the angle contained...

...any other figure, curvilineal or otherwise, which has the same perimeter. PROP. 39. If two Mangles have two sides of the one equal to two sides of the other, eac/i to each, and the angle contained by the two sides of the first a right angle, but the...

...polygon is less than the circumference of a great circle, (art. 32.) 49. PROP. If two spherical triangles have two sides of the one equal to two sides of the other, each to each ; and have likewise the angles contained by those sides equal to one another, the two triangles shall...

...the given rectilineal angle DCE- Which was to be done. PROP. XXIV. THEOR. If two triangles have fwo sides of the one equal to two sides of the other, each to each, but the angle contained by the Iwvsidesof the one prettier limn the angle contained by the two sides...

...give Euclid credit for the particular use of the word ' a£iav*a,' is by him called ' xoivn Iwoia,' two common words, implying ' that which is in the...one equal to two sides of the other, each to each, Sic.' The phrase in italics is not an English idiom, but the literal translation of the Greek Ixserega...

...reasoning, the like may be proved of all other triangles under the same conditions ; that is to say, which have two sides of the one equal to two sides of the other respectively, and also the angles between those sides equal to one another. Wherefore, universally,...