not equal to it; therefore the angle BAC is greater than the angle EDF. Wherefore, if two triangles, &c. Q. E. D. 常 PROP. XXVI. THEOR. If two triangles have two angles of the one equal to two ungles of the other, each to each; and one side equal to one side, viz. either the sides adjacent to the equal angles, or the sides opposite to equal angles in each; then shall the other sides be equal, each to each, and also the third angle of the one to the third angle of the other. B D CE F Let ABC, DEF be two triangles which have the angles ABC, BCA equal to the angles DEF, EFD, each to each, viz. ABC to DEF, and BCA to EFD; also one side equal to one side; and first let those sides be equal which are adjacent to the angles that are A equal in the two triangles, viz. BC to EF: the other sides shall G be equal, each to each, viz. AB to DE, and AC to DF, and the third angle BAC to the third angle EDF. For, if AB, DE be unequal, one of them must be greater than the other. Let AB be the greater of the two, and make BG equal† to DE, and join GC: therefore, because BG is equal to DE, and BC+ to EF, the two sides GB, BC are equal to the two DE, EF, each +Hyp. to each; and the angle GBC is equal to the angle DEF; therefore the base GC is equal to the base DF, and the triangle GBC to the triangle DEF, and the other angles to the other angles, each to each, to which the equal sides are opposite: therefore the angle GCB is equal to the angle DFE: but DFE is, by the hypothesis, equal to the angle BCA; wherefore also the angle BCG is 11 Ax. equal to the angle BCA, the less to the greater, which is impossible: therefore AB is not unequal to DE, that is, it is equal to it; and BC 43. 1. tHyp. #1. 1. +Hyp. equalt 1 +Hyp. to EF; therefore the two AB, BC are equal to the two DE, EF, each to each; and the angle ABC is equalt to the angle DEF; therefore the base AC is equal* #4. 1. to the base DF, and the third angle BAC to the third angle EDF. +3 1. +Hyp. tHyp. B D HC E Next, let the sides which are A opposite to equal angles in each triangle be equal to one another, viz. AB to DE: likewise in this case, the other sides shall be equal, AC to DF, and BC to EF; and also the third angle BAC to the third angle EDF. For, if BC, EF be unequal, let BC be the greater of them, and make BH equal to EF, and join AH: and because BH is equal to EF, and AB to† DE; the two AB, BH are equal to the two DE, EF, each to each; and they contain equalt angles; therefore +4. 1. the base AH is equal to the base DF, and the triangle ABH to the triangle DEF, and the other angles to the other angles, each to each, to which the equal sides are opposite: therefore the angle BHA is equal to the angle +Hyp. EFD: but EFD is equal to the angle BCA; +1. Ax. therefore also the angle BHA is equal to the angle BCA, that is, the exterior angle BHA of the triangle AHC is equal to its interior and opposite angle BCA; which is impossible:* wherefore BC is not unequal to EF, that is, it is equal to it: and AB is equalt to DE; therefore the two AB, BC are equal to the two DE, EF, each to each; and they containt equal angles; wherefore the base AC is equal to the base DF, and the third angle BAC to the third angle EDF. Therefore, if two triangles, &c. Q. E. D. 16. 1. tHyp. +Hyp +4. 1. PROP. XXVII. THEOR. If a straight line falling upon two other straight lines makes the alternate angles equal to one another, these tiro straight lines shall be parallel. Let the straight line EF, which falls upon the two straight lines AB, CD, make the alternate angles AEF, EFD equal to one another: AB shall be parallel to CD. For, if it be not parallel, AB and CD being produced will meet either towards B, D, or towards A, C: let them be produced and meet towards B, D in the point G; therefore GEF is a triangle, and its exterior angle AEF is *16. 1. greater than the interior and opposite angle EFG ; +Hyp. but it is also equalt to it, which is impossible; therefore AB and CD being produced do not meet towards B, D. In like manner it may be demonstrated, that they do not meet towards A, C: but those straight lines which meet neither way, though produced ever so far, are A F B G parallel to one another: therefore AB is parallel *31 Def. to CD. Wherefore, if a straight line, &c. Q. E. D. PROP. XXVIII. THEOR. If a straight line falling upon two other straight lines makes the exterior angle equal to the interior and opposite upon the same side of the line; or makes the interior angles upon the same side together equal to two right angles; the two straight lines shall be parallel to one another. Let the straight line EF, which falls upon the two straight lines AB, CD, make the exterior angle EGB equal to the interior and opposite angle GHD upon the same side; or make the interior angles on the same side BGH, GHD together equal to two right angles: AB shall be parallel to CD. Because the angle EGB is +Hyp. equal to the angle GHD, *15. 1. and the angle EGB equal to the angle AGH, therefore the angle +1. Ax. #27. 1. *Hyp. #13. 1. tl Ax. 13 Ax. +27. 1: AGH is equal to the angle GHD: and they are the alternate angles; therefore AB is parallel to CD. Again, because the angles BGH, GHD together are equal* to two right angles, and that AGH, BGH together are also equal to two right angles; therefore the angles AGH, BGH are equalt to the angles BGH, GHD: Take away the common angle BGH; therefore the remaining angle AGH is equal to the remaining angle GHD: and they are alternate angles: therefore AB is parallelt to CD. Wherefore, if a straight line, &c. Q. E. D. Scholium. This proposition and that which immediately precedes it plainly shew the possibility of the existence of parallel lines, or lines which, however far they may be prolonged, never meet each other; and the next step in the theory is the deduction of their properties. This next step cannot, however, be taken without the aid of a principle specially introduced for the purpose, and confessedly unauthorised by anything hitherto established. Up to the present point Euclid has developed the several truths which he has announced, from the definitions of the things to which they refer; and this development has uniformly been effected by the purest reasoning, every argument resting ultimately upon the self-evident propositious called Axioms, for the admission of which he has stipulated at the opening of the subject, and against which nobody understanding the terms in which they are expressed ever feels the slightest objection. These simple principles are found, however, insufficient for carrying on the reasoning beyond the present point; and although numerous contrivances have been thought of by the moderns for obviating the difficulty which at this stage interrupted the progress of the ancient geometer, yet it has never been satisfactorily removed. The additional principle assumed by Euclid, as necessary to be admitted before he can proceed with his subject, and which he calls an Axiom, is as follows. AXIOM XII. If a straight line meets two straight lines so as to make the two interior angles, on the same side of it taken together, less than two right angles, these straight lines, being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles. It is readily seen that this proposition is no other than the converse of Proposition xvii., which proves that, If a straight line BD meet two straight lines BA, CA which themselves meet, these meeting lines make with the former the two interior angles on the same side, viz. ABC, ACB, which, taken together, are less than two right angles: a proposition which, however clearly it may help us to see what is really implied in the "twelfth axiom," furnishes us with no ground for admitting that "axiom" as a truth. This difficulty in developing the theory of parallels some have attributed to the definition of parallels-from the negative property of their never meeting: yet as this property fully and sufficiently characterises this class of lines, we are inclined to trace the perplexity to a different source-to the vagueness and insufficiency with which a straight line is defined. That we have a correct notion of a straight line, independently of definition, is unquestionable. This is indeed proved by the fact that when we come to the so called definition of Euclid-which definition ought to give additional clearness and precision to our pre-formed notion if it were obscure, we feel that what that notion necessarily comprises is not fully expressed. The conception of a straight line necessarily comprises that of length-mere linear extension,-and uniformity of direction,* this latter quality being the only distinguishing peculiarity of straight lines.t Now, it is an inference readily admissible from this uniformity of This conception is implied in our assent to the second postulate, the meaning of which obviously is, "That a terminated straight Jine may be prolonged to any length in the same direction.” The proper character of a definition is very correctly described by Dr. Whately, in his excellent treatise on Logic. "Such defini |