MATHEMATICS. PLANE GEOMETRY. [BOOK L PROP, xxn. 



which must now be formally stated, and unhesitatingly 

 admitted as true : it - the principle affirmed in Euclid's 

 twelfth axiom. In lu.-t editions of Eucliil, this is placed, 

 with the other axioms, at the commencement of the 

 Book. It has been here kept out of view till it can no 

 longer be dispensed with ; and this has been done be- 

 cause, in the first place, we should not be called upon 

 to give assent to what concerns anything, of the possible 

 existence of which there may be reasonable doubt. Till 

 he has reached Prop. XXVII. (which proposition is all- 

 sufficient for the construction of Prop. XXXI.), the 

 learner may fairly question whether it be possible for 

 what are called parallel lines to exist : he now knows 

 that parallel or nertr-meetimj lines may be actually drawn . 



In the next place, from the property demonstrated in 

 Proposition XVII ., namely, that any two angles of a 

 tn ingle are together less than two right angles, he knows 

 what he could not know at an earlier stage of hia pro- 

 gress an important particular respecting a pair of meet- 

 ing-liars, crossed by a third line ; namely, that in a pair 

 of meeting linet, the two interior angles on one side of 

 the crossing line, are together left than tiro right angles ; 

 and, aa oonscquence(Prop.XXVlII.), that when these 

 interior angles are together equal to two right angles, the 

 lines crossed must be non-meeting, or parallel lines. It is 

 the convene of the first-mentioned property that he is 

 now to be called upon to receive as true ; viz., that if a 

 straight line, crossing a pair of lines, make the two in- 

 terior angles, on the same side of it, together less than 

 tiro right angles, the pair crossed shall be meeting-lines. 

 This is the twelfth axiom, and is thus expressed in 

 Euclid : 



AXIOM XII. If a straight line meet 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 prolonged, shall at length meet 

 upon that side on which are the angles that are less than 

 two right angles. 



The propositions already referred to enable us to see 

 distinctly what it is that this axiom assumes, and more- 

 over inform us that the assumption is, at least, perfectly 

 consistent with demonstrated truth. That it is necessarily 

 true, must be admitted, upon reflecting for a moment 

 upon that peculiarity of a straight line, really implied in 

 its designation, though not expressly adverted to in its 

 definition its uiuleviating sameness of direction. 



It is obvious, from this uniformity of direction, that if 

 two straight lines, however far prolonged, never meet, 

 then, at no part of their course, can either make any 

 approach towards the other ; for if two straight lines 

 approach one another, their continuance, in the same un- 

 deviating directions, necessitates their meeting, if inde- 

 finiHy prolonged. We cannot doubt this, and yet have 

 an accurate conception of an unlimited straignt line ; 

 since uniformity of direction must enter that conception. 

 It follows, therefore, that parallel lines must be, through- 

 out, equidistant lines. But two distinct straight lines, 

 through the same point, cannot be throughout equally 

 distant from a third ; so that two straight lines, through 

 the same point, cannot both be parallel to the same 

 straight line. It has been seen (Prop. XXVIII.) th.it 

 one (C D) is parallel to another (A U), if the interior 

 angles (BOH, D H G) be equal to two right angles ; a 

 second (H K), which would cause the interior angles 

 (B G H, K H G) to be lets than two right angles, being 

 a distinct line from C D, must therefore meet A B if pro- 

 longed. And this is the assertion of the twelfth axiom. 



PROPOSITION XXIX. THB..REM. 



r f a straight line (E F) fall upon two parallel tirnS'ilt 

 lines ( A B, C D) it make* the alternate angls (A G H, G 

 H D) equal ; and the exterior angle (E G B) - the in- 

 terior and opposite (G II D) upon the same side ; and 

 likewise the two interior angles (B G H, G H D) w/n 

 the same side together "two right angles. [See the pre- 

 ceding diagram.] 



For if A G H be not - G H D, one of them, as A G H, 

 must be the greater. Add the angle BG H to each of 

 them, .-. AUIi + BUU are greater than B G H -f 



G H D. But AGH + BGH-two right angles,* .-. 



IT. 13. BGH + GIID are less than two right 

 + A.. 11. angles, .'. A U, CD, if prolonged, will meet.f 



which is impossible, since (by hypothesis) they are 

 parallel ; .'. A G II is not vnequal to G H D, that is, it is 



IT. ii. emial to it. Again: AGH- KGH,* .-. 

 EG B - G H 1). Add to each of these B G 11, .-. K G B 

 + B G H = G H D + B G H ; but K G B + B G H - two 



+ Pr. is. right angles, t .-.BGH-fGHD tioo riijht 

 angles, . . if a straight line, A-o. Q. E. D. 



PROPOSITION XXX. THEOREM. 



Straight linet (AH, C D) which are parallel to the same 

 straight line (EF) are parallel to each other. 



Let the straight line / 



GHKcut AB, EF, CD: A ' r .' B 



then, because A B is paral- '/ ~ 



lei to E F, the angle A G K JC n/ ? 



Pr. 29. = G H F,* and ~~T~ 

 because E F, C D are also c K / 



parallel, the angle G H F 7 



t Pr. 29. = G K D, t /. f 



Pr. J7. A G K = G K D, . . A B is parallel to C D.* 

 .-. straight lines, <fcc. Q. E D. 



PROPOSITION XXXI. PROBLEM. 



To draw a straight line through a given point (A) parallel 

 to a given straight line (B C). 



In B C take any point D : join A D ; and at the point 



Pr. 23. A in A l) make the angle DA 1C = A D C ;* 

 and prolong E A to F ; then E F shall be parallel to B C. 



Because A D, falling . 



upon EF, BC, makes the \/ 



alternate angles E A D, 



ADC equal, E F is paral- / 



IT. 27. lei to BC,* .-. B -73 c 



through the given point A / 



a line E F parallel to BC is drawn. Wltich was to be 



done. 



PROPOSITION XXXH.-THBORKM. 



Tf a side (B C) of a triangle (A B C) be prolonged, the 

 . .-tfrior angle (A CD) is = the two interior and o 

 site angles (A, B) ; and the three interior angles of every 

 triangle are together = two right angles. 



pr. si. Through C draw C E parallel to A B ;* 

 then the alternate angles 



BAG, ACE are equal, f 

 + Pr. 29. and because 

 B D falls upon the said 

 parallels, the exterior 

 angle E C D is = the in- 

 terior and opposite angle 



Pr. 29. A BC.* Itwaa 

 proved that A C R - B A C, 

 . . the whole exterior angle 



A C 1) = A -f- B, both the interior and opposite angles. To 

 each of these equals add ACB, .-. A(JD + ACB - A 

 + B + AC B: but ACD + AC B -two right angles,* 



Pr. 13. .'. the three angles of the triangle are = two 

 right angles, .'. if a side of a triangle, &c. Q. E. D. 



COR. 1 All the interior angles of any rectilineal figure, 

 together with four right angles, are = twice as many 

 right angles as the figure has sides. 



For, any rectilineal figure A B C D E can be divided 

 into as many triangles as the figure 

 has sides, by drawing straight lines 

 from a point F within the figure to 

 each of its vertices. And, by the 

 preceding prop., all the angles of 

 these triangles are = twice as many 

 right angles as there are triangles ; 

 that is, as there are sides of the 

 figure. But these same angles are 

 equal to the angles of the figure together with the angle* 

 at F, the common vertex of the trungles ; that is, to- 



Pr. 13. gother with four right angles ;* . . all the 

 Cor. J. tingles of the Jigure, togeUktr with four right 



