Scholium. When this proposition is applied to polygons which have re-entrant angles, each re-entrant angle must be regarded as greater than two right angles. But to avoid all ambiguity, we shall henceforth limit our reasoning to polygons with salient angles, which are named conver polygons. Every convex polygon is such, that a straight line, drawn at pleasure, cannot meet the sides of the polygon in more than two points. PROPOSITION XXVII. THEOREM. If the sides of any polygon be prolonged, in the same direction, the sum of the exterior angles will be equal to four right angles. Let the sides of the polygon ABCDFG, be prolonged, in the same direction; then will the sum of the exterior angles a+b+c+d+f+9, be equal to four right angles. For, each interior angle, plus its exterior angle, as A+a, is equal to two right angles (P. 1). But there are as many exterior as interior angles, and as many of each as there are sides of the polygon: hence the sum of all the interior and exterior angles, is equal to twice as many right angles as the polygon has sides. Again, the sum of all the interior angles is equal to twice as many right angles as the figure has sides, less four right angles (P. 26). Hence, the interior angles plus four right angles, is equal to twice as many right angles as the polygon has sides, and consequently, equal to the sum of the interior angles plus the sum of the exterior angles. Taking from each the sum of the interior angles, and there remains the sum of the exterior angles, equal to four right angles. A a B/b PROPOSITION XXVIII. THEOREM. In every parallelogram, the opposite sides and angles are equal each to each. Let ABCD be a parallelogram: then will AB=DC, AD=BC, the angle A=C, and the angle ADC=ABC. For, draw the diagonal BD, dividing the parallelogram into the two triangles, ABD, DBC. Now, since AD, BC, are parallel, the angle ADB DBC (p. 20, c. 2); and since AB, CD, are parallel, the angle ABD=BDC: and since the side DB is common, the two triangles are equal (P. 6); therefore, the side AB, opposite the angle ADB, is equal to the side DC, opposite the equal angle DBC (p. 10, s.), and the third sides AD, BC, are equal: hence, the opposite sides of a parallelogram are equal. Again, since the triangles are equal, the angle A is equal to the angle C (P. 10, s.) Also, the angle ADC composed of the two angles, ADB, BDC, is equal to ABC, composed of the corresponding equal angles DBC, ABD (A. 2): hence, the opposite angles of a parallelogram are equal. B A C Cor. 1. Two parallels AB, CD, included between two other parallels AD, BC, are equal; and the diagonal DB divides the parallelogram into two equal triangles. Cor. 2. Two parallelograms which have two sides and the included angle in the one equal to two sides and the included angle in the other, each to each, are equal. D Let the parallelogram ABCD, have the sides AB, AD, and the included angle BAD equal to the sides AB, AD, and the included angle BAD, in the next figure; then will they be equal. A For, in each figure, draw the diagonal DB. By the last corollary, the diagonal divides each parallelogram into two equal triangles: but the triangle BAD in one parallelogram, is equal to the triangle BAD in the other (P.5): hence, the parallelograms are equal (A. 6). PROPOSITION XXIX. THEOREM. If the opposite sides of a quadrilateral are equal, each to each, the equal sides are parallel, and the figure is a parallelogram. D Let ABCD be a quadrilateral, having its opposite sides respectively equal, viz.: AB=DC, and AD=BC; then will these sides be parallel, and the figure a parallelogram. For, having drawn the diagonal BD the two triangles ABD, BDC, have all the sides of the one equal to the corresponding sides of the other; therefore they are equal, and the angle ADB, opposite the side AB, is equal to DBC, opposite CD (P. 10, s.); therefore the side AD is parallel to BC (P. 19, c. 1) For a like reason AB is parallel to CD: therefore, the quadrilateral ABCD is a parallelogram. A PROPOSITION XXX. THEOREM. If two opposite sides of a quadrilateral are equal and parallel, the other sides are equal and parallel, and the figure is a parallelogram. Let ABCD be a quadrilateral, having the sides AB, CD, equal and parallel; then will the figure be a parallel ogram. A For, draw the diagonal DB, dividing the quadrilateral into two triangles. Then, since AB is parallel to DC, the alternate angles ABD, BDC are equal (P. 20, c. 2); moreover, the side DB is common, and the side AB=DC; hence, the triangle ABD is equal to the triangle DBC (P. 5); therefore, the side AD is equal to BC, the angle ADB=DBC, and consequently AD is parallel to BC (P. 19, c. 1); hence, the figure ABCD is a parallelogram. D PROPOSITION XXXI. THEOREM. The two diagonals of a parallelogram divide each other into equal parts, or mutually bisect each other. Let ADCB be a parallelogram, AC and DB its diagonals, intersecting at E; then will AE-EC, and DE= EB. = Comparing the triangles AED, BEC, we find the side AD=CB (P. 28), the angle ADB CBE, and the angle DAE=ECB (p. 20, c. 2); hence, these triangles are equal (P. 6); consequently, AE, the side opposite the angle ADE, is equal to EC, opposite CBE, and DE opposite DAE is equal to EB opposite ECB. D B E Scholium. In the case of the rhombus, the sides AB, BC, being equal, the triangles AEB, EBC, have all the sides of the one equal to the corresponding sides of the other, and are therefore equal: whence, it follows, that the angles AEB, BEC, are equal, and therefore, the two diagonals of a rhombus bisect each other at right angles. BOOK II. OF RATIOS AND PROPORTIONS. DEFINITIONS. 1. PROPORTION is the relation which one magnitude bears to another magnitude of the same kind, with respect to its being greater or less.* 2. RATIO is the measure of the proportion which one magnitude bears to another; and is the quotient which arises from dividing the second by the first. Thus, if A and B represent magnitudes of the same kind, the ratio of A to B is expressed by B A A and B are called the terms of the ratio; the first is called the antecedent, and the second, the consequent. 3. The ratio of magnitudes may be expressed by numbers, either exactly or approximatively; and in the latter case, the approximation may be brought nearer to the true ratio than any assignable difference. Thus, of two magnitudes, one may be considered to be divided into some number of equal parts, each of the same kind as the whole, and regarding one of these parts as a unit of measure, the magnitude may be expressed by the number of units it contains. If the other magnitude contain an exact number of these units, it also may * See Davies' Logic of Mathematics: Proportion, § 267. |