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 exterior angles. Taking from each the sum of the interior angles, and there remains the exterior angles, equal to four right angles. PROPOSITION XXVIII. THEOREM. In every parallelogram, the opposite sides and angles are equal. D Let ABCD be a parallelogram: then will AB=DC, AD=BC, A=C, and ADC=ABC. For, draw the diagonal BD. The triangles ABD, DBC, have a common side BD; and since AD, BC, are parallel, they have also the angle ADB DBC, (Prop. XX. Cor. 2.); and since AB, CD, are parallel, the angle ABD BDC: hence the two triangles are equal (Prop. VI.); therefore the side AB, opposite the angle ADB, is equal to the side DC, opposite the equal angle DBC; and the third sides AD, BC, are equal: hence the opposite sides of a parallelogram are equal. Again, since the triangles are equal, it follows that the angle A is equal to the angle C; and also that the angle ADC composed of the two ADB, BDC, is equal to ABC, composed of the two equal angles DBC, ABD: hence the opposite angles of a parallelogram are also equal. Cor. Two parallels AB, CD, included between two other parallels AD, BC, are equal; and the diagonal DB divides the parallelogram into two equal triangles. PROPOSITION XXIX. THEOREM. If the opposite sides of a quadrilateral are equal, each to each, the equal sides will be parallel, and the figure will be a parallelogram. 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 be a parallelogram. A D B For, having drawn the diagonal BD, the 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 (Prop. X.); therefore, the side AD is parallel to BC (Prop. XIX. Cor. 1.). For a like reason AB is parallel to CD therefore the quadrilateral ABCD is a parallelogram. PROPOSITION XXX. THEOREM. If two opposite sides of a quadrilateral are equal and parallel, the remaining sides will also be equal and parallel, and the figure will be a parallelogram. Let ABCD be a quadrilateral, having the sides AB, CD, equal and parallel; then will the figure be a parallelogram. A D B 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 (Prop. XX. Cor. 2.); moreover, the side DB is common, and the side AB=DC; hence the triangle ABD is equal to the triangle DBC (Prop. V.); therefore, the side AD is equal to BC, the angle ADB DBC, and consequently AD is parallel to BC; hence the figure ABCD is a parallelogram. PROPOSITION XXXI. THEOREM. The two diagonals of a parallelogram divide each other into equal parts, or mutually bisect each other. Let ABCD be a parallelogram, AC and DB its diagonals, intersecting at E, then will AE EC, and DE=EB. Comparing the triangles ADE, CEB, we find the side AD=CB (Prop. XXVIII.), the angle ADE=CBE, and the angle E DAE ECB (Prop. XX. Cor. 2.); hence those triangles are equal (Prop. VI.); hence, AE, the side opposite the angle ADE, is equal to EC, opposite EBC; hence also DE is equal to EB. 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, that the two diagonals of a rhombus cut each other at right angles. C BOOK II. OF RATIOS AND PROPORTIONS. Definitions. 1. Ratio is the quotient arising from dividing one quantity by another quantity of the same kind. Thus, if A and B repesent quantities of the same kind, the ratio of A to B is ex The ratios 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 of them may be considered to be divided into some number of equal parts, each of the same kind as the whole, and one of those parts being considered as an unit of measure, the magnitude may be expressed by the number of units it contains. If the other magnitude contain a certain number of those units, it also may be expressed by the number of its units, and the two quantities are then said to be commensurable. If the second magnitude do not contain the measuring unit an exact number of times, there may perhaps be a smaller unit which will be contained an exact number of times in each of the magnitudes. But if there is no unit of an assignable value, which shall be contained an exact number of times in each of the magnitudes, the magnitudes are said to be incommensurable. It is plain, however, that the unit of measure, repeated as many times as it is contained in the second magnitude, would always differ from the second magnitude by a quantity less than the unit of measure, since the remainder is always less than the divisor. Now, since the unit of measure may be made as small as we please, it follows, that magnitudes may be represented by numbers to any degree of exactness, or they will differ from their numerical representatives by less than any assignable quantity. Therefore, of two magnitudes, A and B, we may conceive A to be divided into M number of units, each equal to A': then A=M × A': let B be divided into N number of equal units, each equal to A'; then B-Nx A'; M and N being integral numbers. Now the ratio of A to B, will be the same as the ratio of M× A' to N× A'; that is the same as the ratio of M to N, since A' is a common unit. In the same manner, the ratio of any other two magnitudes C and D may be expressed by Px C' to QxC', P and Q being also integral numbers, and their ratio will be the same as that of P to Q. 2. If there be four magnitudes A, B, C, and D, having such B. D values that is equal to then A is said to have the same ratio A C' to B, that C has to D, or the ratio of A to B is equal to the ratio of C to D. When four quantities have this relation to each other, they are said to be in proportion. To indicate that the ratio of A to B is equal to the ratio of C to D, the quantities are usually written thus, A: B:: C: D, and read, A is to B as C is to D. The quantities which are compared together are called the terms of the proportion. The first and last terms are called the two extremes, and the second and third terms, the two means. 3. Of four proportional quantities, the first and third are called the antecedents, and the second and fourth the consequents; and the last is said to be a fourth proportional to the other three taken in order. 4. Three quantities are in proportion, when the first has the same ratio to the second, that the second has to the third; and then the middle term is said to be a mean proportional between the other two. 5. Magnitudes are said to be in proportion by inversion, or inversely, when the consequents are taken as antecedents, and the antecedents as consequents. 6. Magnitudes are in proportion by alternation, or alternately, when antecedent is compared with antecedent, and consequent with consequent. 7. Magnitudes are in proportion by composition, when the sum of the antecedent and consequent is compared either with antecedent or consequent. 8. Magnitudes are said to be in proportion by division, when the difference of the antecedent and consequent is compared either with antecedent or consequent. 9. Equimultiples of two quantities are the products which arise from multiplying the quantities by the same number: thus, m × A, m× B, are equimultiples of A and B, the common multiplier being m. 10. Two quantities A and B are said to be reciprocally proportional, or inversely proportional, when one increases in the same ratio as the other diminishes. In such case, either of them is equal to a constant quantity divided by the other, and their product is constant. PROPOSITION I. THEOREM. When four quantities are in proportion, the product of the two extremes is equal to the product of the two means Let A, B, C, D, be four quantities in proportion, and M: N ::P: Q be their numerical representatives; then will MxQ= NxP; for since the quantities are in proportion fore N=Mx,or N×P=M×Q. N Q there. Cor. If there are three proportional quantities (Def. 4.), the product of the extremes will be equal to the square of the mean. PROPOSITION II. THEOREM. If the product of two quantities be equal to the product of two other quantities, two of them will be the extremes and the other two the means of a proportion. Let MxQ=NxP; then will M: N:: P: Q. For, if P have not to Q the ratio which M has to N, let P have to Q', a number greater or less than Q, the same ratio that M has to N; that is, let M:N:: P: Q'; then MxQ'= NXP (Prop. I.): hence, Q'= NxP ; but Q= NxP ; con sequently, Q-Q' and the four quantities are proportional; that is, MN: P: Q. PROPOSITION III. THEOREM. If four quantities are in proportion, they will be in proportion when taken alternately. Let M, N, P, Q, be the numerical representatives of four quanties in proportion; so that MN:: P: Q, then will M: P::N: Q. Since M:N:: P: Q, by supposition, MxQ=NxP; therefore, M and Q may be made the extremes, and N and P the means of a proportion (Prop. II.); hence, M: P::N: Q. |