= angle DFE = angle ACB (hyp.). Therefore angle GCB angle ACB, the part to the whole, which is impossible. Hence it is evident that AB and DE must be equal, since an absurdity follows immediately from the supposition that they are unequal, and this being the case we have the other pair of sides required for proving the two triangles ABC, DEF, equal by (I. 4). AB, BC= DE, EF angle ABC= angle DEF .. triangle ABC = triangle DEF. We must now consider the second case, in which the equal sides are opposite equal angles. The same angles being equal, let us take side AB = DE. Then in = В 44 order to meet the conditions of I. 4 we shall have to show that BC= EF. Adopting a similar construction to that of Case I., let us assume that BC is the greater, cutting off BH= EF and joining HA. Then, as in Case I., by I. 4 the triangles BHA, DEF, are equal, and the angle BHA being equal to EFD = ACB; but the impossibility resulting differs, for BHA is not a part of ACB, but you will observe that it is the exterior angle of the triangle HCA, and therefore cannot be equal to the interior and opposite angle at C. Therefore BC = EF, and the proof concludes as in the first case. SUMMARY OF PROPOSITION XXVI., THEOREM 17. To show that triangles are equal if two angles and one side of the one are equal to two angles and one side of the other. Case I. In which the equal sides lie between the equal angles. Case II. In which the equal sides are opposite the equal angles. Cons. Cut off from side assumed to be greater a part equal to the corresponding side in the other triangle, join the extremity of this line with opposite angle. Proof (Indirect).-The new triangle thus formed in each case = ▲ DEF (I. 4), one of its angles (Case I. ▲ GCB, Case II. BHA) = ▲ BCA, which involves the absurdity (Case I.), that the part whole (Case II.), that ext. ▲ of A interior and opposite. = = Hence AB DE (Case I.), and BC EF (Case II.), and triangles ABC, DEF are equal (I. 4).-Q.E.D. This proposition is also ascribed to Thales. It has been proposed to prove the first case by the method of superposition. Let A ABC be applied to ▲ DEF so that BC falls upon EF. Then since BC and EF are equal they will coincide; and since the angles at the bases of the two as are equal, each to each, the sides AB, AC will fall upon DE, DF (produced if necessary). .. The point A must coincide with point D, the only point common to DE and DF. Therefore the As coincide and are equal in all respects. LECTURE XII. ANALYSIS OF THE FIRST SECTION OF BOOK I. INTRODUCTION TO THE THEORY OF PARALLEL STRAIGHT LINES.-PROPS. 27-29. WE have reached the conclusion of the first division of Book I. of the Elements, treating chiefly of the relations and properties of triangles. We will classify and enumerate the results obtained in these twenty-six propositions. I. Of the relations of triangles to each other. We have learnt that triangles are equal in all respects, when the following parts in two or more triangles are equal : (a) Two sides and the included angle (I. 4). (B) Three sides (I. 8). (7) Two angles and the side between them, or the side opposite one of them (I. 26). II. Of the properties of triangles. (a) The relations of sides to angles. Equal angles are opposite to equal sides, and conversely (I. 5 and 6). Hence an isosceles triangle has two equal angles, and an equilateral triangle has three, and if a triangle has two equal angles we know that it must be isosceles, if three, equilateral. The greater side of any triangle is opposite to the greater angle, and conversely (I. 18 and 19). The other sides remaining the same, the greater the vertical angle the greater the base, and conversely (I. 24 and 25). (B) The relations of angles to angles. The exterior angle of any triangle is greater than either of the interior opposite angles (I. 16). F Any two angles of a triangle are together less than two right angles (I. 17). (7) The relations of sides to sides. Any two sides of a triangle are together greater than the third (I. 20). On the same base, and on the same side of it, there cannot be two triangles having their sides terminated in one end of the base equal, and likewise those terminated in the other (I. 7). If two triangles be drawn on the same base, the vertex of one of which falls within the other, the two sides of the triangle whose vertex falls within the other are together less than those of the other triangle, but contain a greater angle (I. 21). (d) Two problems relating to triangles. To construct an equilateral triangle on a given line (I. 1). To construct a triangle having its sides equal to three given lines, of which, in accordance with I. 20, any two must be greater than the third (I. 22). In all sixteen propositions; of the remaining ten, five propositions relate to angles. We know how To draw a rectilineal angle equal to a given angle (I. 23). We know That the angles made by one straight line falling on another = two right angles (I. 13, and its converse I. 14). That the angles made by two straight lines cutting one another, or by any number of lines meeting in a point four right angles. Also that when two straight lines cut one another the vertical opposite angles are equal (I. 15, and cor.). Five propositions relate to straight lines. We know how To draw a straight line from any point equal to a given straight line (I. 2). To cut off from the greater of two straight lines a part equal to the less (I. 3). To bisect a given finite straight line (I. 10). To draw a perpendicular to a straight line from a given point within (I. 11) or without it (I. 12). The second section of Book I. deals with a very important and rather difficult portion of geometry-the theory of parallel straight lines. In it def. 35 and post. 6 are both used for the first time, and as they form the groundwork of Euclid's method of dealing with this subject, it will be well to consider them attentively. The definition referred to tells us that 'parallel straight lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet'—that is, they always remain at exactly the same distance apart. A G E G D In the various theorems which treat of parallel lines we shall be very much concerned with the angles which are made when one straight line cuts two parallels. Let us take any straight line EF, cutting parallel lines AB CD in G and H, you will see that it makes with them eight angles, which, for convenience in reference, are given different names, according to their situation. The four angles which are contained between the parallels, two on each side of the cutting line, are called the interior angles. Those outside of the parallels, two above and two below, are called exterior angles. Again, the interior angles, taken in pairs thus, AGH GHD, and BGH GHC, are called alternate angles. H F The four interior angles' are equal to four right angles (I. 13). It is with these that the sixth postulate deals. It tells us that 'If a straight line meets two other straight lines so as to make the two interior angles on the same side of it, taken together less than two right angles, these lines, being continually produced, shall at length meet upon that side on |