THE ELEMENTS OF GEOMETRY. PART I. Of Right Lines and Rectilineal Figures. DEFINITION I. A rectilineal (or right-lined) Angle is that which is formed by two right lines meeting at the same point, but not lying in the same right line. Thus in the annexed plate, the right lines A B and A C which meet at the point A, but are not in the same right line, form a rectilineal angle. D B It is to be observed that by an angle is not meant the surface between the lines which form it; for though we increased that surface by producing the lines (to D and E, for instance), the angle will still remain the A same in magnitude. By an angle in fact is meant the degree of increasing width, or separation, E between the lines which form it; this depends, not upon the length, but upon the direction of the lines. The right lines (as AD, AE) which form an angle are usually called its sides or legs; and the point (A) where they meet is called its vertex. In order to specify the vertex of an angle we place a letter at it, and specify the letter in order to specify the angle itself, we place a letter at each side of it and another at the vertex; and specify the two former letters with the other between them.. Thus, in the above plate, we specify the vertex by mentioning the letter A ; and we specify the angle itself by mentioning the letters BAC or CAB. DEF. II. Figures are bounded portions of space. DEF. III. A plane figure is a plane surface bounded by one or more lines. DEF. IV. A plane rectilineal figure is a plane surface bounded by right lines. LESSON I. DEF. V. A rectilineal triangle is a plane figure bounded, by three right lines. It is so called because the three lines form with each other three angles. A B If we conceive a triangle to stand upon any one of itssides, that one is called the base of the triangle, and the other two are called its legs or sides. Also the angle opposite to the base is called the vertical angle; and its vertex the vertex of the triangle. Thus, in the above figure, if we conceive the triangle to stand upon the side AC, this side is called the base; the angle at B is called the vertical angle; and the point в the vertex of the triangle. [See NOTE B.] ARTICLE 1. If there be two triangles which have two sides of the one equal, respectively, to two sides of the other; and likewise the angles contained by those sides equal to one another:—then, the bases or third sides of these triangles are also equal. E AA Let ABC, DEF be two triangles, having the side BA equal to the side ED, and the side BC equal to the side EF, also the angle ABC equal to the angle DEF. Then, also, the base AC must be equal to A the base DF. B 2 DEMONSTRATION. Conceive the triangle ABC so applied to the triangle DEF, that the point в may be on the point E, and the side BA upon the side ED; moreover, that the side BC may lie towards the same hand as the side EF. Then, the point в falling on the point E, and the side BA on the side ED, the point A would necessarily fall on the point D, because the sides BA, ED are equal. Likewise, the side BC would fall upon the side EF, because the angles ABC, DEF are equal in width. The point c would likewise fall on the point F, because the point в falls on the point E, and the sides BC, EF are equal. Therefore, as it has been shewn that the points a and c would coincide with the points D and F respectively, the right lines AC and DF would coincide exactly throughout their whole extent, else they would enclose a space between them, which two right lines cannot do, as is manifest *. Hence, inasmuch as AC and DF would coincide exactly, they must be exactly equal. This was the assertion of the present Article. PROBLEMS. DEF. 1. An equilateral triangle is a triangle whose three sides are equal to each other. Observation. One extremity of a definite right line remaining fixed, if the line be made to revolve about this point, it is evident that the other extremity will trace out a line which is every where equally distant from the fixed point: and that if the line revolve progressively to its first direction, the line traced out will return into itself, so as totally to include a surface. Thus, if AB be the right line, A its fixed extremity, the line BCDB will be traced out by the progressive revolution of AB round the point A, through the points C, D, to its first direction AB. Also every point of this line BCDB will be equally distant from a. DEF. 2. A Circle is a plane figure bounded by one line, such that all right lines drawn B A from it to one and the same point within the circle, are equal to each other. A xiom 1. Two right lines cannot completely enclose a space. ART. 2. In such triangles as above described, those angles at the bases which are opposite to equal sides, are respectively equal. In the above figures the angles at A and D, opposite the equal sides BC and EF, are equal; also the angles at c and F, opposite the equal sides BA, ED, are equal. DEM. In the preceding demonstration it was shown that the sides of the angle BAC would coincide exactly with the sides of the angle EDF; also that the sides of the angle BCA would coincide exactly with the sides of the angle EFD. Hence, the angles BAC, EDF are necessarily equal; and also the angles BCA, efd*. This was the assertion, &c. ART. 3. Such triangles as above described are equal, in every respect, to each other. For in the same demonstration it was shown that the three sides of the triangle ABC would coincide exactly with the three sides of the triangle DEF. Hence these triangles must be in every respect equal*. This was, &c. [See NOTE C.] DEF. 3. In a circle, the bounding line is called the Circumference, and the point from which it is every where equally distant is called the Centre of the circle. PROB. I. To describe an equilateral triangle on a given finite right line. Let AB be the given finite right line. It is required to describe an equilateral triangle upon it. CONSTRUCTION. With the point a as a centre, and the line AB as distance, describe the circle DBE. Also, with the point B as a centre and the line Draw the right lines BA as distance, describe the circle DAF. Ax. 2. Magnitudes which coincide exactly with each other are equal· This theorem, which is made the fourth proposition in Euclid, should rightly be the first, inasmuch as it is the foundation of Geometry, the corner-stone upon which the whole superstructure of geometrical science rests. Yet viewing it without fear or prejudice, we see how nearly where these circles intersect. lateral. Then, the triangle ACB is equi DEMONSTRATION. AS DBE is a circle, of which a is the centre, Ac is equal to AB, by DEF. 3; and as DAF is a circle, of which B is the centre, BC is equal to BA, by the same definition. Hence AC and BC, being both equal to AB, are equal to each other, and the three sides of the triangle are equal *. This was what was required to be done by the present Article. PROB. II. From a given point to draw a right line equal to a given finite right line. Let A be the given point, and вc the given right line. It is required to draw from the point a a right line equal to BC. H G F D CONS. Draw the straight line AB from the given point to either extremity of the given line, and upon AB describe the equilateral triangle ADB, by PROB. I. With в as a centre and BC as distance describe the circle CEF, and produce DB through the point B till it meets the circumference at E. With the point D as a centre, and the distance DE, describe the circle EGH, and produce DA through the point A till it meets the circumference of this circle at 1. Then the right line AI drawn from the point A is equal BC. B DEM. DI and DE are equal, because D is the centre of the circle EIG, by DEF. 3; therefore, taking away from each the equal sides DA and DB, AI remains equal to BE t. But as B is the centre of the circle ECF, BE is equal to BC. Hence AI is equal to BC. This, &c. Ax. 3. Things which are equal to the same thing are equal to each other. † Ax. 4. If from equal things we take away equal things, respectively, the remainders will be equal. By Ax. 3. |