DEM. 4D. 2, 3, H.2. Again, 5 Concl. 7 D. 3, 6. 8 Con, sup. 9 Ax. 10. 10 Ax. 8. ВАС AB coincides with DE, and L / EDF, .. the line AC shall fall on the line DF: But AC being = DF, the C shall fall on the F, and B falling on E, and C on F, the line BC For if, though B falls on E, and C on F, then two st. lines will enclose a space; therefore the base BC does coincide with 11 D.2,5,6, 10 And A B falling on and being equal to DE; 12 Ax. 8. 13 Hyp.1, D.10 14 Ax.8, Concl. AC to DF; and BC to EF, the ABC shall coincide with and equal 15 Hyp. 1, D.7. And in a similar way ACB = 16 Recap. &c. DFE. Q.E.D. SCHOLIUM.-1. This being the first Theorem in the Elements, it is exclusively proved by means of the Axioms. 2. The converse of the 8th Axiom is assumed; namely, that if magnitudes are equal, not merely if they are equivalent, they will also coincide. 3. The equality spoken of in this Proposition, is equality of the sides and of the angles. Triangles may be equal in area, though the sides and angles of the one are not equal to the sides and angles of the other. When the sides and angles mutually coincide, the triangles are named equal triangles; when their areas only are equal, such triangles are called equivalent triangles. 4. Some have taken the 4th Proposition for an axiom. We perceive its truth indeed, almost without demonstration; but the number of axioms in any science should not be needlessly increased; and as this proposition can be naturally established by means of the received axioms of Geometry, it holds its proper place when classed with truths to be demonstrated. It may be more briefly enunciated, thus-" If two triangles have each two sides and their included angle equal, the triangles are equal in every respect." USE AND APPLICATION.-This Proposition contains the first of the criteria by which to infer the equality of triangles, and is applied to various uses: as 1st. Very frequently in all parts of Geometry to establish the equality of triangles. 2nd. To ascertain an inaccessible distance, as AB, the breadth of a lake. With an instrument for measuring angles, take the angle at C formed by the lines AC, BC, from the extremities A and B of the inaccessible distance; and with a chain or other measure of length, find the distances CA and CB. The Representative Values of these measurements must now be taken from a Scale of Equal Parts, and drawn on paper, or on any plane surface: thus, draw a st. line DF of an indefinite length, and at D form an angle, by aid of the graduated semicircle FDE equal to the angle BCA from a scale of equal parts the distance from A to C is represented in proper proportion by the line DE, and the distance from B to C by DF; consequently, on a principle established in the Sixth Book, and which we now assume as a Lemma,-that the sides about similar triangles are proportional, the line E F will represent in the due proportion the distance from A to B and if to the same scale we apply the line EF, that distance on the scale will be the representative measurement of the actual distance A B. N.B.-If the ground near the lake was level enough to admit of setting out the triangle DEF in the actual measurements of CA and CB, we should have EF of the very length of AB, on the principle that two triangles having two sides and their included angle in each equal, have their third sides equal; and if we measure one, as EF, we ascertain the other, AB: but as it is seldom we find the actual surface of a country smooth enough for our purpose, we use the method of Representative Values, or of Geome trical Construction. PROP. 5.-THEOR. The angles at the base of an isosceles triangle are equal to each other; and if the equal sides be produced, the angles on the other side of the base shall be equal. CONSTRUCTION.-Pst. 2. A terminated st. line may be produced to any length in a st. line. P. 3. From the greater line to cut off a part equal to the less. Pst. 1. A st. line may be drawn from any one point to any other point. DEMONSTRATION.-P. 4. If two triangles have two sides and their included angle of one triangle equal to two sides and their included angle of another triangle, the two triangles are equal in every respect. Ax. 3. If equals are taken from equals, the remainders are equal. EXP. 1| Hyp. 1. 2 H.2 & Pst. 3 Concl. 1. 4 " CONS 1 P. 3. 2 Pst. 1. 2. 2 Let ABC be an isosc. A, having the sides. and let the equal sides be produced inde- then the s ABC, ACB, at the base are equal; and the s DBC, ECB, on the other side of the base are equal. On A D take any. F, and make AG=EF; join thes F and C, G and B, by the Is FC and GB. DEM. 1 C.1 & Hyp.1 AF AG, AC = AB, and ▲ A is common, 2 P. 4. 3 D. 2. = ..the side FC GB, and AAFC=AAGB; = AGB. DEM. 4C.1 & Hyp.1 5 Sub. Ax. 3. 6 D. 5 & 2. Again, . AF AG, and AB = AC, = But in As BCF, BCG, BF = CG, and FC 10 Sub. & Ax.3... on taking aways CBG and BCF, 11 Remk. 12 D. 8. 13 Remk. 14 Recap. the and these are the angles below, or on the other side of the base. Wherefore, the angles at the base, &c. Q.E.D. SCHOLIUM. To assist the learner, the original figure 1 is separated into its parts, 2, 2, and 3, 3; and the equality of the triangles proved by Prop. 4. COR.-Every equilateral triangle is also equiangular. Hence, the angles are all equal, A to B, Wherefore, if a ▲ be equilateral, &c. Q.E.D. PROP. 6.-THEOR. If two angles of a triangle be equal to one another, the sides also which subtend, or are opposite to the equal angles, shall be equal to one another. CONSTRUCTION.-P. 3. From the greater line to cut off a part equal to the less. Pst. 1. A st. line may be drawn from one point to another. DEMONSTRATION.-P. 4. If two triangles have two sides and the included angle equal in each, the triangles are in all respects equal. CONS. 2 side A C. For if ABAC, one of them is the greater; let AB be greater than B4 A C. P.3, & Pst. 1 In BA make BD- AC, and join DC. DEM. 1 C. 2 Hyp. 3 P. 4. C DB is made = AC, and BC is common, and ::: / DBC= [ ACB; .. DC AB, and A DBC= AABC. |