and the angles contained by the two sides equal; therefore the other angles are equal, each to each, to which the equal sides are opposite (g); and so, FKB, FKC, FLC, FLD are equal; and any two of them are equal to any other two: therefore FKB, FKC are equal to FLC, FLD; that is, HKL is equal to KLM. In like manner, it may be shown that each of the angles at H, G, M, is equal to HKL, or KLM: therefore the pentagon is equiangular. Again, because the perpendicular FC bisects KL, the perpendiculars FB, FD bisect HK and LM; for they intercept equal arcs: and since the halves BK, KC, LD are equal, the wholes HK, KL, LM are equal. In like manner, it may be shown, that GH, or GM is equal to HK, KL, or LM. The pentagon is therefore equilateral; and it proves also to be equiangular, and is inscribed in the given circle ABCDE; which was to be done. Recite (a) p. 11, 4; (b) def. 2, p. 16 and cor., also p. 17, of b. 3; (c) def. 10, 1; (d) p. 47,1; (e) ax. 1; 13 P. To inscribe a circle in a given equilateral and equiangular pentagon (ABCDE). Constr. Bisect the angles BCD, CDE, by the straight lines CF, DF; and from the point F, in which they meet, draw FB, FA, FE. Therefore, since, in the triangles BCF, DCF, BC, CF are equal to DC, CF, and they contain equal angles, by bisection, the bases FB, FD are B equal; and the angle CBF equals the angle CDF (a): but CDF is the half of CDE; there fore CBF is the half of CBA: because CBA equals CDE. Therefore ABC is bisected by the straight line BF. In the same way, it may be shown, that the angles BAE, AED are bisected by the straight lines FA, FE. From the point F draw FG, FH, FK, FL, FM, perpendicular upon the sides of the pentagon (b). Now, in the triangles FCH, FCK, the side FC is common; and the angles at C are equal, by bisection; and those at H, K are equal, as right angles (c); therefore the sides FH, FK are equal (d). In like manner, it may be shown, that each of the perpendiculars FL, FM, FG is equal to FH, or FK: therefore, these five straight lines are equal to one another; and a circle described upon the centre F, at the distance of any one of them, will pass through the extreme points of the other four; and touch the sides of the pentagon, where they meet the perpendiculars, in the points G, H, K, L, M (e). But a circle is inscribed in a rectilineal figure, when the circumference touches all the sides of the figure (f): wherefore a circle is inscribed, &c., which was to be done. Recite (a) p. 4, 1; (d) p. 26, 1; (b) p. 12, 1; 14 P. To describe a circle about a given equilateral and equiangular pentagon (ABCDE). Bisect two of the angles of the pentagon, as BCD, CDE, by straight lines CF, DF, meeting in the point F (a): join FB, FA, FE. B A E F D Argument. Because the equal angles BCD, CDE are bisected; the angles FCB, FCD are equal; and they are contained by the equal sides CB, CF; CD, CF: therefore the remaining sides C FB, FD are equal, and also the angle CDF to CBF (6): but CDF is the half of CDE, which equals CBA; therefore FB bisects the angle CBA. In like manner it may be shown, that the angles BAE, AED are bisected by the straight lines FA, FE. Now the five triangles, whose vertices are in the point F, have equal bases; namely, the sides of the pentagon; also equal angles adjacent to the bases (c), as above; therefore the sides FA, FB, FC, FD, FE, are equal to one another; and being drawn from the point F to the angular points of the pentagon, a circle described upon F, at the distance of any one of them, will pass through the five points, and be described about the pentagon; which was to be done (d). 15 P. To inscribe an equilateral and equiangular hexagon in a given circle (ABCDEF). Let G be the centre of the given circle, and draw the diameter AGD (a): Again, on the point D, where the diameter meets the circumference, describe a circle to pass through G, and cut the circumference in the points E, A C (b): draw the diameters EGB, CGF; also chords between the points A, B, C, D, E, F, A, in the circumference. If these chords be equai, and their angles equal, the required hexagon is inscribed. Because G and D are centres of equal circles, the radii GE, GD and DE are equal (c): GC, GD and DC are in the same case: therefore, the triangle GED, or GCD is equilateral (d); and being every way isosceles, it is also equiangular (e): therefore, the angle CGD is one third of two right angles (f), EGD is also one third of the same ; and because the straight line EG makes with CF, the adjacent angles equal to two right angles (g), therefore EGF is also one third of two right angles. The chord EF is therefore equal to ED, or DC (h); and these three are placed in the semicircle (i). The opposite, or vertical angles are also equal to these (k): therefore each of the angles AGF, AGB, BGC, is one third of two right angles (f); and the radii GB, GA, GF, being equal (c), the chords AF, AB, BC are also equal to one another; and they are placed in a semicircle (i). Therefore the six chords AB, BC, CD, DE, EF, FA, divide the circumference, cut off equal arcs (1), and are therefore sides of an equilateral hexagon inscribed in the given circle ABCDEF. But, since equal angles stand upon equal arcs (m), the arc AF equals the arc ED; to both add the arc ABCD (n): therefore the whole arc FABCD equals the whole ABCDE; and the angle FED stands upon the former, and the angle AFE upon the latter: therefore the angles AFE, FED are equal. In the same way, it may be proved, that each of the other angles of the hexagon is equal to AFE, or FED. The hexagon (o) is therefore equiangular, and it was shown to be equilateral; and it is inscribed in the given circle: which was to be done. Cor. The side of the hexagon; that is, the chord of one sixth part of the circumference, is equal to the radius, or semi-diameter of the circle. Scholium. To describe an equilateral and equiangular hexagon about the circle and about the inscribed hexagon is the same thing: for if through the points A, B, C, D, E, F, tangents be drawn, touching the circle at right angles to the diameters, the angles of the inscribed hexagon will be in the sides of the one described. Def. 2, 4, b. 4. Tangent literally means they touch. See def. 2, b. 3. 16 P. To inscribe an equilateral and equiangular quindecagon in a given circle (ABCD). B Let AC be the side of an equilateral triangle (a), and AB the side of an equilateral pentagon (b) inscribed in the given circle. Therefore, as it is required to cut the circumference into fifteen equal parts, the chord AC cuts off five, AB cuts E off three, and the difference BC contains two of C those fifteenths. Bisect the arc BC in E (c): therefore, the arc BE, or EC, is one fifteenth F D part of the circumference. Now, if the chord BE, or EC be drawn, and equal chords be placed all around in the circle (d), an equilateral and equiangular quindecagon shall be inscribed in it; which was to be done. And, if through the angular points of the inscribed quindecagon, tangents be drawn, an equilateral and equiangular quindecagon shall be described about the circle, and also about the inscribed quindecagon: for the angular points of the inscribed rectilineal figure shall be in the sides of the one described (e). Recite (a) p. 2, 4; (b) p. 11, 4; (c) p. 30,3; (d) p. 1, 4; (e) def. 36, 1, and Note; def. 1, 2, 3, 4, 5, of b. 1. BOOK FIFTH. Definitions. 1. A less magnitude is said to be a part of a greater one, when the less is a measure of the greater, or is contained a certain number of times in it. 2. A greater magnitude is said to be a multiple of a less one, when the greater is measured by the less, or contains it a certain number of times. A. There is a series of multiples; as, the first, second, third, &c., of which, waving the etymology of the word, the magnitude itself is the first, its double is the second, its triple the third, &c. B. A magnitude may have one, two, or three dimensions, as the case may be; and the proper unit of measure will be a line, square surface, or cube. 3. Ratio is the numerical relation of antecedent and consequent, or the number of times, or parts of times, which the latter contains the former. Or, Ratio is the numerical relation of measure and magnitude, or the number of times which the measure, or a part of it, may be applied to the magnitude. Note. This value of ratio prevails; and the words of several propositions are here changed to correspond with it. 4. Magnitudes of the same kind only, or having some common property, can have a ratio to one another. 5. The first of four magnitudes has the same ratio to the second which the third has to the fourth, when equimultiples of the first and third, also of the second and fourth, being taken; if the multiple of the first be greater than that of the second, the multiple of the third is greater than that of the fourth; if equal, equal; and if less, less. 6. Magnitudes which have the same ratio are called proportionals; of which it is usually said, "the first is to the second as the second is to the third; or, the first is to the second as the third is to the fourth. Note. The is to, as above, is expressed by a colon, thus, (:), the as by two colons, thus (: :). 7. The ratio of one couplet (or antecedent and consequent) is less than the ratio of another couplet, when the quotient of the former consequent, divided by its antecedent, is less than the quotient of the latter consequent divided by its antecedent. 8. When three terms or magnitudes are proportionals, the ratio of the first to the third is the duplicate, or square of the ratio of the first to the second. 9 When four terms are continued proportionals; that is, when the second is the consequent of the first, the third that of the second, and the fourth that of the third; then the ratio of the first to the fourth is the triplicate or cube of the ratio of the first to the second. Such ratios are called compound. 10. And when any number of magnitudes of the same kind are in a certain order, however different the ratios of the couplets may be, the ratio of the first to the last of them is the continual product of all the ratios; namely, the product of all the antecedents for an antecedent, and the product of all the consequents for a consequent. 11. In proportionals, taken, two and two, from different series, or from remote terms of the same series, the odd terms, namely, the first, third, fifth, &c., are the antecedents; and these are said to be homologous; so, likewise, the even terms, viz. the second, fourth, sixth, &c., which are the consequents. Geometers use the terms permutando, or alternando, invertendo, componendo, dividendo, convertendo, ex æquali distantia, ex æquo, and ex æquali, in proportione perturbata, vel inordinata, to signify various changes in the order, or magnitude of proportionals, and still preserving the equality of the ratios, in which proportion consists. The sense of these terms is expressed in the following examples: The use of the marks +, -, ×, ÷, :, : :, and, is generally known. Example 1. By permutation, or alternately; when, of four proportionals, as A: B:: C: D, comes A: C:: B: D. See p. 16 of b. 5. Ex. 2. By inversion; when, of four proportionals, as A: B::C: D, comes B: A::D:C. See p. B. of b. 5. Ex 3. By composition; when, of four proportionals, as A: B :: C: D, comes A+B:B::C+D: D. See p. 18 of b. 5. Ex. 4. By division; when, of four proportionals, as A: B::C: D, comes A-B:B::C-D: D. See p. 17 of b. 5. Ex. 5. By conversion; when, of four proportionals, as A:B::C: D, comes A: A-B::C:C-D. See p. E of b. 5. Ex. 6. From equal distance in order; when, of two ranks of proportionals, as A, B, C, D, and E, F, G, H, taken, two and two, in order, namely, A: B:: E: F;—B:C::F:G; and C:D::G: H;-it comes to be inferred, that A: D:: E: H. See p. 22 of b. 5. Ex. 7. From equal distance out of order; when, of two ranks of proportionals, as A, B, C, D, and E, F, G, H, taken, two and two, in a cross order, namely, A: B::G:H;-B:C::F:G; and C:D :: E: F; it comes to be inferred, that A: D:: E: H. See p. 23 of b. 5. Axioms. 1. Equimultiples of the same, or of equal magnitudes, are equal to one another. 2. Those magnitudes are equal to one another, of which the same, or equal magnitudes, are equimultiples. 3. A multiple of a greater magnitude is greater than the same multiple of a less. 4. One magnitude is greater than another, of which a multiple is greater than the same multiple of the other. |