COR. III.-If two triangles, ABC, ADE, have two angles of the one, ABC, ACB, respectively equal to two angles of the other, AED, ADE; then the third angle of the one, BAC, shall be equal to the third angle of the other, DAE. (See the last figure.) The s angles; ABC + ACB =LS AED + ADE; Also the s AED + ADE + DAE = two rt. DEм. 1 by Hyp. 2 P. 32. 3 P: 32. 4 Sub. 5 Ax. 3. .. the rem. ▲ BAC = the rem. DAE. Q.E.D. take away the equals ABC + ACB and AED + ADE; SCH.-1. The Corollaries to be deduced from Prop. 32, are very numerous : Lardner's Euclid gives twenty-four. 2. The first Cor. is of universal extent, applying to all rectilineal figures; the second is applicable only to figures that are called convex, in which each int. angle is less than two rt. angles. Some figures, as ABCDE, being concave, at . B, have re-entrant angles, which are greater than two rt. angles: thus the int. angle at B is greater than two rt. angles by the angle ABK. A E K C 3. Without the production of a side, the three angles may be proved to be equal to two rt. angles, by drawing through any angular point a parallel to the opposite side. 4. In an equilateral triangle, each angle measures two-thirds of art. angle, or 60°; and in an isosceles triangle, if one angle is rt. angled, the other two are each 45°, or half a rt. angle. USE AND APP.-1. This Theorem may be employed in Astronomy to determine the Parallax of a heavenly body. Let . C represent the earth's centre, A a point on the earth's circumference AB, and Z the zenith of the station A; S is a star or any heavenly body not in the zenith. : By observation take the angle ZAS, the zenith distance at the earth's surface A if the star S were viewed from C the centre, the angle ZCS would be its zenith distance; and the angle ZCS is less than angle ZAS. For, by P. 32, the ext. angle ZAS is equal to the angles C+ S: thus, the angle S is equal to the excess of the angle ZAS above the angle ZCS. If now, from an Astronomical Table, I learn what is the zenith distance of S as viewed from C, the earth's centre, the difference between angle ZA S and angle ZCS, or the angle ASC, will be the parallax. B 2. By employing the same Theorem we may construct a figure which will give the representative value of the perpendicular height of a mountain, as CD, in the next figure. B A D E At station A with a theodolite, or other suitable instrument, ascertain the angle CAD-suppose 45°; measure AB, 300 feet, and at station B measure the angle CBD, 30°; and from these data construct the figure required. From B draw a st. line of indefinite length BE, and take on it 300 equal parts to A: at. B draw an angle CBD of 30°; and at. A an angle CAD of 45°: the st. lines BC and AC intersect in the point C; and a perpendicular from C to BE, namely, CD, will represent the perpendicular height of the mountain. Take CD in the compasses, and apply the distance to the same scale; it will be found that CD measures 400, feet or yards, according to the unit of length in AB. 3. On the principle that all the interior angles of any rectilineal figure are equal to twice as many rt. angles as the figure has sides, diminished by four rt. angles for the amount of the angles at the centre,— —we are enabled to construct any regular right-lined figure. The angular magnitude of each figure is equal to two rt. angles multiplied by the number of sides, minus four rt. angles; and the remainder divided by the number of sides or angles, gives each single angle of the regular figure. Let S equal the number of sides; 180° the measure of two rt. angles; and 360° the measure of four rt. angles; and, taking the figure below, let A represent the angle at each pair of sides, and C the angle at the centre by radii to the extremities of each side. To construct a regular Polygon: Ascertain by the foregoing formula the angle at C, and the angle at A. 1st. When the side AB is not given-Draw any radius, as CA, and at . C, with another radius of the same length as the first, make the angle C as ascertained by the formula; then describe with rad. CA or CB the circle, and the distance AB will divide the circumference into as many parts as there are units in S: join the points, and the polygon is formed. 2nd. When the side AB is given-At .s A and B make angles equal to angle A, as ascertained by the F formula; and bisect those angles by the st. lines AC and BC meeting in the point C; triangle CAB having the angles CAB and CBA equal, the sides CA and CB are also equal: if now, with CA or CB as radius, a circle be described, AB will cut off as many arcs from the circumference as there are units in S;-draw the chords to the respective arcs, and the polygon will be completed. PROP. 33.-THEOR. The st. lines which join the extremities of two equal and parallel st. lines towards the same parts, are also equal and parallel. CONS.-Pst. 1. A st. line may be drawn from one point to another. DEM.-P. 29. A st. line falling on two parallel st. lines makes the alternate angles equal. P. 4. Two triangles having in each two sides and their included angle respectively equal, are equal in every other respect. P. 27. If a st. line falling on two st. lines make the alternate angles equal, the two st. lines are parallel. L D B Hence, in the two As ABC, DCB,... AB = CD, And . st. line BC with st. lines AC, BD, .. the st. line AC is || to the st. line BD; Wherefore, the st. lines which join the extremi- Q.E.D. SCH.-A Parallelogram is a four-sided figure of which the opposite sides are equal and parallel, and the diagonals join opposite angles. USE AND APP.-The principle contained in this Theorem enables us to ascertain the perpendicular height of a mountain AG, as well as the distance from the base to the foot of the perpendicular, CG. Take a large right-angled triangle, ABD, and putting one end at A, let the side DB, by means of a plummet, be brought to the perpendicular; DB shows the altitude AH, and D A the horizontal distance BH: repeat E D B F A the process as often as the case requires ;—then DA+EB, &c., equals the horizontal distance CG; and DB+EC, &c., the altitude AG. PROP. 34.-THEOR. The opposite sides and angles of parallelograms are equal to one another, and the diameter bisects them; that is, divides them into two equal parts. DEM.-P. 29. A st. line falling on two parallel st. lines makes the alternate angles equal. P. 26. If two triangles have two angles and a side of one equal to two angles and a side of the other, the other sides and angle are respectively equal. Ax. 2. If equals be added to equals, the wholes are equal. P. 4. If two triangles have two sides and the included angle equal in each, they are altogether equal. DEM. 1 byH.&P.29. BC meets the || s AB and CD, L .. the ABC = the alt. ▲ BCD; 2 H. & P. 29. and 3 D. 1 & 2. 4 P. 26. 5 D. 1 & 2. 6 Ax. 2. 7 D. 4. 8 Concl. .. the Hence, s ABC, ACB = 4s BCD and CBD, and their adjacent side BC is common to the As ABC, BCD; .. ▲ BAC=▲ BDC, side AB = side CD, L And. ABC = ▲ BCD, and ▲ CBD = ▲ ACB, .. the whole ABD = the whole ACD; and ▲ BAC=▲ BDC; ..the opp. sides and angles of s are equal. 9 H. & D. 1. Also, . AB = CD, BC is common, and ABC 10 P. 4. 11 Recap. .. the ABC= the ABCD. and diam. BC divides the into two equal parts. Wherefore, the opposite sides and angles of parallelograms, &c. Q.E.D. SCH.-1. In the last figure, the two diagonals, AD and BC, bisect each other. 2. The converse to Prop. 34 is-If the opposite sides or opposite angles of a quadrilateral figure be equal, the opposite sides shall be parallel; i.e., the figure shall be a parallelogram. 3. Both the diagonals, AD and BC, being drawn, it may, with a few exceptions which require subsequent propositions, be proved, that a quadrilateral figure which has any two of the following properties, will also have the others : 10. The parallelism of AB and CD; 50. The equality of the angles A and D ; 80. The bisection of BC by AD; 100. The bisection of the area by BC. These ten data, being combined in pairs, will give 45 distinct pairs; with each of these pairs it may be required to establish any of the eight other properties, and thus 360 questions, respecting such quadrilaterals, may be raised. These questions will furnish to the student useful geometrical exercises. "The 9th and 10th data require the aid of subsequent propositions."-LARDNER'S Euclid, p. 49. USE AND APP.-1. The construction and accuracy of the parallel ruler depends on Prop. 34. 2. A finite st. line may be divided into any given number of equal parts. EXP. 1 Data. Given a st. line AL, and the number of parts, as four; to divide AL into 2 Quæs. EX D CONS. 1 by Pst. 1. C From one extremity B A of AL, as A, draw an indefinite st. line AX, 2 P. 3 & Pst. 1. take AB, and make BC, CD, DE each AB, 3 P. 31. 4 Sol. 5 P. 31. DEM. 1 by C.3&5,P.34 2 P. 29. 3 P. 26. and join EL; 3. On the same principle the Sliding Scale, called from the inventors the Vernier or Nonius, is constructed. This scale is very useful, as in the |