378 G E O M AB, AC in the first are equal respectively to AC, AB in the second, and the angles included by these sides are equal. Ilence the tri- angles are equal, and the angles in the one are equal to those in the other, viz., those which are opposite equal sides, i.e., angle ABC in the first equals angle ACB in the second, as they are opposite the equal sides AC,and AB in the two triangles. 'l‘lrere follows the converse theorem (Prop. 6). If two angles in a triangle are equal, than the sides opposite them are equ.al,—i.c., the triangle is isosceles. The proof given consists in what is called a. rcductio ad absurdum, a kind of proof often rrsed by Euclid, and principally in proving the converse of a previous theorem. ' It assmnes that the theorem to be proved is wrong, and then shows that this assumption leads to an absurdity, i.e., to a corr- elusion which is in contradiction to a proposition proved before— that therefore the assumption rnade camrot be true, and hence that the theorem is true. It is often stated that Euelid invented this kind of proof, but the method is most likely rmreh older. § 8. It is next proved that two triangles which hare the three sides of the one equal respectively to thoseof the other are identically equal, hence that the angles of the one are equal respectively to those of the other, those being equal which are opposite equal sides. This is Prop. 8, Prop. 7 containing only a first step towards its proof. These theorems allow now of the solution of a number of pro- blerns, viz. :- To bisect a given angle (Prop. 9). To biscct a given finite straight line (Prop. 10). To draw a straight line perpendicularly to a given. straight line through a given point in it (Prop. 11), and also through a given point not in it (Prop. 12). The solutions all depend upon properties of isosceles triangles. § 9. The next three theorems relate to angles only, and might have been proved before Prop. 4, or even at the very beginning. The first (Prop. 13) says, The angles 'which one straight line makes with another straight line on one side of it either are two right angles or are together equal to two right angles. This theorem would have been unnecessary if Euclid had admitted the notion of an angle such that its two linrits are in the same straight line, and had besides defined the sum of two angles. Its converse (Prop. 14) is of great use, inasmrrclr as it enables us in many cases to prove that two straight lines drawn from the same point are one the continuation of the other. So also is Prop. 15. If two straight lines cut one another, the vertical or tqiposite angles shall be equal. §10. Euclid returns now to properties of triangles. Of great importance for the next steps (though afterwards superseded by a more complete theorem) is - Prop. 16. If one side of a triangle be produced, the exterior angle shall be greater than either of the interior opposite angles. Prop. 17, A ny two angles of a triangle are together less than two right angles, is an immediate consequence of it. By the aid of these two, the following fundamental properties of triangles are easily proved :— Prop. 18. The greater side of every triangle has the greater angle opposite to it ; Its converse, Prop. 19. The greater angle of eeerg triangle is subtended by the greater sixle, or has the greater side opposite to it ; Prop. 20. Any two sides of a triangle are together greater than the third side ; And also Prop. 21. If from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle. § 11. Having solved two problcrrrs (Props. 22, 23), he returns to two triangles which have two sides of the one equal respectively to two sides of the other. It is known (Prop. 4) that if the included angles are equal then the third sides are equal; and conversely (Prop. 8), if the third sides are equal, then the angles included by the first sides are equal. From this it follows that if the included angles are not equal, the third sides are not equal, and conversely, that if the third sides are not equal, the included angles are not equal. Euclid now eonipletcs this knowledge by proving, that “if the included angles are not equal, than the third side in that triangle is the greater which contains the greater angle,” and conversely, that “ if the third sides are unequal, that triangle contains the greater angle which contains the greater side.” These are Prop. 24 and Prop. 25. § 12. The next theorem (Prop. 26) says that if two triangles have one side and two angles of the one equal respectively to one side and two angles of the other, eiz., in both triangles either the angles ad- jacent to the equal side, or one angle adjacent and one angle opposite it, then the two triangles are identically equal. This theorem belongs to a group with Prop. 4 and Prop. 8. Its first case might have been given immediately after Prop. 4, but the second case requires Prop. 16 for its proof. § 13. Ve come now to the investigation of parallel straight lines, i.c., of straight lines which lie in the same plane, and cannot be made to meet however far they be produced either way. The in- ETRY vestigation, which starts from Prop. 16, will become clearer if a few names be explained which are not all used by Euclid. If two straight lines be cut by a third, the latter is now generally called a “transversal” of the figure. It forms at the two points where it crrts the given lines four angles with each. 'l‘hosc of the angles which lie between the. given lines are called interior angles, and of these, again, airy two which lie on opposite sides of the trarrsversal but one at each of the two points are called “ alternate angles." Ve may now state Prop. 16 thus :—If two straight lines 2rlu'<'h meet are cut by a transizersal, tlicir alternate angles are nncqzml. For the lines will form a triangle, and one of the alternate angles will be an exterior angle to the triangle, the other interior and opposite to it. Frorn this follows at once the theorem contained in Prop. 27. If two straight lines which are cut by a trans-2-ersrzl mnlcc altrrrnatr angles equal, the lines cannot meet, liowcrer _]24r they be prurlucul, hence they areparallcl. This proves the existence of parallel lines. Prop. 28 states the same fact in dill'erent forms. If a strniglit line,fulling on two other straight lines, make the ea-trriur angle equal to the interior and opposite angle on the same side of the line. or make the interior angles on the same side togetlwr equal to two right angles, the two straight lines shall be parallel to one another. Hence we know that, “if two straight lines which are errt by a transversal meet, their alternate angles are not equal”; and henee that, “if alternate angles are equal, then the lines are parallel." The question now arises, Are the propositions converse to these true or not? That is to say, “ If alternate angles are unequal, do the lines nreet?" And “if the lines are parallel, arc alternate angles necessarily equal '9” The answer to either‘ of these two questions implies the answer to the other. But it has been found impossible to prove that the negation or the affirmation of either is true. 'l‘hc dillicnlty which thus arises is overcome by Euclid assurning that the first question has to be answered in the aflirmative. This gives his last axiom (12), which we quote in his own words. AXIOM 12.—1_/‘a straight line meet two straight lines, so as to malrc the two interior angles on the same side (f it taken together less than two right angles, these straight lines, being continually protlucal, shall at length meet on that side on ivhich are the angles irhiclt are less than two right angles. The answer to the second of the above questions follows from this, and gives the theorem Prop. 29. If a straight line full on two parallel straight lines, it 7n.alt'es the alternate angles equal to one another, and the exterior angle equal to the interior and (q:posr'tc angle on the same side, and also the two interior angles on the sa me side together equal to two right angles. § 14. 'ith this a new part of elementary geometry begins. The earlier propositions are independent of this axiom, and would be true even if a wrong assrrrnption had been made in it. They all relate to figures in a plane. lut a plane is only one among an irrlinite rmrnber of conceivable surfaces. We may draw figures on any one of them and study their properties. We may, for instance, take a sphere instead of the plane, and obtain “ spherical " in the place of “plane " geometry. If on one of these surfaces lines and figures could be drawn, answer-in_;r to all the definitions of our plane figures, and if the axioms with the ex- ception of the last all hold, then all propositions rrp to the 28th will be true for these figures. This is the case in spherical geometry if we substitute “shortest line ” or “great circle” for “straiglrt line,” “small circle” for “eirele," and if, besides, we limit all figures to a part of the sphere which is less than a lremispliere. so that two points on it cannot be opposite ends of a diameter, and tlrereforc determine always one and only one great circle. For spherical triangles, therefore, all the important propositions 4, 8, 26 ; 5 and 6 ; aml 18, 19, and 20 will hold good. This remark will be sufficient to show the in1p0Ssil)ilil3_' of proving Euclid's last axiom, which would mean proving that this axiom is a consequence of the others, and hence that the theory of parallels would hold on a spherical surface, where the other axioms do hold, whilst parallels do not even exist. It follows that the axiom in question states an inherent difl°cr(-ne: between the plane and other surfaces, and that the plane is only fully clraractcrized when this axiom is added to the other assum p- tions. § 15. The introduction of the new axiom and of parallel lines leads to a new class of propositions. After proving (Pro . 30) that “ two lines which are each parallel to a third areparalle to each other,” we obtain the new properties of triangles contained in Prop. 32. Of these the second part is the most important, viz., the theorem, The three interior (I.7lglc.s' if every triangle are together equal to two right angles. _ As easy deductions not given by Euclid but added by Srmson follow the propositions about the angles in polygons, they are given in Eng ish editions as corollaries to Prop. 32. These theorems do not hold for spherical figures. The sum of the interior angles of a spherical triangle is always greater than two right angles, and increases with the area.
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