F E A B G 1.); therefore the six angles EGD, DGC, CGB, BGA, AGF, FGE are equal to one another. But equal angles at the centre stand upon equal arcs (26. 3.): therefore the six arcs AB, BC, CD, DE, EF, FA are equal to one another and equal arcs are subtended by equal (29. 3.) straight lines; therefore the six straight lines are equal to one another, and the hexagon ABCDEF is equilateral. It is also equiangular; for, since the arc AF is equal to ED, to each of these, add the arc ABCD; therefore the whole arc FABCD shall be equal to the whole EDCBA: and the angle FED stands upon the arc FABCD, and the angle AFE upon EDCBA; therefore the angle AFE is equal to FED: in the same manner it may be demonstrated, that the other angles of the hexagon ABCDEF are each of them equal to the angle AFE or FED; therefore the hexagon is equiangular; it is also equilateral, as was shown; and it is inscribed in the given circle ABCDEF. COR. From this it is manifest, that the side of the hexagon is equal to the straight line from the centre, that is, to the radius of the circle. D Η And if through the points A, B, C, D, F, F, there be drawn straight lines touching the circle, an equilateral and equiangular hexagon shall be described about it, which may be demonstrated from what has been said of the pentagon; and likewise a circle may be inscribed in a given equilateral and equiangular hexagon, and circumscribed about it, by a method like to that used for the pentagon. PROP. XVI. PROB. To inscribe an equilateral and equiangular quindecagon in a given circle. Let ABCD be the given circle; it is required to inscribe an equilateral and equiangular quindecagon in the circle ABCD. Let AC be the side of an equilateral triangle inscribed (2. 4.) in the circle, and AB the side of an equilateral and equiangular pentagon inscribed (11. 4.) in the same; therefore, of such equal parts as the whole circumference ABCDF contains fifteen, the arc ABC, being the third part of the whole, contains five; and the arc AB, which is the fifth part of the whole, contains three; therefore BC their difference contains two of the same parts: bisect (30. 3.) BC in E; therefore BE, EC are, each of them, the fifteenth part of the whole circumference ABCD: therefore, if the straight lines BE, EC be drawn, and B E D straight lines equal to them be placed (1. 4.) around in the whole circle, an equilateral and equiangular quindecagon will be inscribed in it. And in the same manner as was done in the pentagon, if through the points of division made by inscribing the quindecagon, straight lines be drawn touching the circle, an equilateral and equiangular quindecagon may be described, about it: and likewise, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circumscribed about it. SCHOLIUM. Any regular polygon being inscribed, if the arcs subtended by its sides be severally bisected, the chords of those semi-arcs will form a new regular polygon of double the number of sides: thus, from having an inscribed square, we may inscribe in succession polygons of 8, 16, 32, 64, &c. sides; from the hexagon may be formed polygons of 12, 24, 48, 96, &c. sides; from the decagon polygons of 20, 40, 80, &c. sides; and from the pentedecagon we may inscribe polygons of 30, 60, &c. sides; and it is plain that each polygon will exceed the preceding in surface or area. It is obvious that any regular polygon whatever might be inscribed in a circle, provided that its circumference could be divided into any proposed number of equal parts; but such division of the circumference like the trisection of an angle, which indeed depends on it, is a problem which has not yet been effected. There are no means of inscribing in a circle a regular heptagon, or which is the same thing, the circumference of a circle cannot be divided into seven equal parts, by any method hitherto discovered. It was long supposed, that besides the polygons above mentioned, no other could be inscribed by the operations of elementary Geometry, or, what amounts to the same thing, by the resolution of equations of the first and second degree. But M. Gauss, of Göttingen, at length proved, in a work entitled Disquisitiones Arithmetica, Lipsie, 1801, that the circumference of a circle could be divided into any number of equal parts, capable of being expressed by the formula 2"+1, provided it be a prime number, that is, a number that cannot be resolved into factors. The number 3 is the simplest of this kind, it being the value of the above formula when n=1; the next prime number is 5, and this is also contained in the formula; that is, when n=2. But polygons of 3 and 5 sides have already been inscribed. The next prime number expressed by the formula is 17; so that it is possible to inscribe a regular polygon of 17 sides in a circle. For the investigation of Gauss's theorem, which depends upon the theory of algebraical equations, the student may consult Barlow's Theory of Numbers. 14 ELEMENTS OF GEOMETRY. BOOK V. In the demonstrations of this book there are certain " signs or characters" which it has been found convenient to employ. 1. The letters A, B, C, &c. are used to denote magnitudes of any kind. "The letters m, n, p, q, are used to denote numbers only. It is to be observed, that in speaking of the magnitudes A, B, C, &c., we mean, in reality, those which these letters are employed to represent; they may be either lines, surfaces, or solids. "2. When a number, or a letter denoting a number, is written close to "another letter denoting a magnitude of any kind, it signifies that the "magnitude is multiplied by the number. Thus, 3A signifies three "times A; mB, m times B, or a multiple of B by m. When the num"ber is intended to multiply two or more magnitudes that follow, it is "written thus, m(A+B), which signifies the sum of A and B taken m "times; m(A-B) is m times the excess of A above B. 66 Also, when two letters that denote numbers are written close to one an"other, they denote the product of those numbers, when multiplied into 66 one another. Thus, mn is the product of m into n"; and mnA is A mul"tiplied by the product of m into n. DEFINITIONS. 1. A less magnitude is said to be a part of a greater magnitude, when the less measures the greater, that is, when the less is contained a certain number of times, exactly, in the greater. 2. A greater magnitude is said to be a multiple of a less, when the greater is measured by the less, that is, when the greater contains the less a certain number of times exactly. 3. Ratio is a mutual relation of two magnitudes, of the same kind, to one another, in respect of quantity. 4. Magnitudes are said to be of the same kind, when the less can be multiplied so as to exceed the greater; and it is only such magnitudes that are said to have a ratio to one another. 5. If there be four magnitudes, and if any equimultiples whatsoever be taken of the first and third, and any equimultiples whatsoever of the second and fourth, and if, according as the multiple of the first is greater than the multiple of the second, equal to it, or less, the multiple of the third is also greater than the multiple of the fourth, equal to it, or less; then the first of the magnitudes is said to have to the second the same ratio that the third has to the fourth. 6. Magnitudes are said to be proportionals, when the first has the same ratio to the second that the third has to the fourth; and the third to the fourth the same ratio which the fifth has to the sixth, and so on whatever be their number. "When four magnitudes, A, B, C, D are proportionals, it is usual to say "that A is to B as C to D, and to write them thus, A: B::C: D, or thus, A: B-C: D." 7. When of the equimultiples of four magnitudes, taken as in the fifth definition, the multiple of the first is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth: then the first is said to have to the second a greater ratio than the third magnitude has to the fourth and, on the contrary, the third is said to have to the fourth a less ratio than the first has to the second. 8. When there is any number of magnitudes greater than two, of which the first has to the second the same ratio that the second has to the third, and the second to the third the same ratio which the third has to the fourth, and so on, the magnitudes are said to be continual proportionals. 9. When three magnitudes are continual proportionals, the second is said to be a mean proportional between the other two. 10. When there is any number of magnitudes of the same kind, the first is said to have to the last the ratio compounded of the ratio which the first has to the second, and of the ratio which the second has to the third, and of the ratio which the third has to the fourth, and so on unto the last magnitude. For example, if A, B, C, D, be four magnitudes of the same kind, the first A is said to have to the last D, the ratio compounded of the ratio of A to B, and of the ratio of B to C, and of the ratio of C to D; or, the ratio of A to D is said to be compounded of the ratios of A to B, B to C, and C to D. And if A: B::E:F; and B:C::G: H, and C: D::K: L, then, since by this definition A has to D the ratio compounded of the ratios of A to B, B to C, C to D; A may also be said to have to D the ratio compounded of the ratios which are the same with the ratios of E to F, G to H, and K to L. 1 In like manner, the same things being supposed, if M has to N the same ratio which A has to D, then, for shortness' sake, M is said to have to Na ratio compounded of the same ratios which compound the ratio of A to D; that is, a ratio compounded of the ratios of E to F, G to H, and K to L. 11. If three magnitudes are continual proportionals, the ratio of the first to the third is said to be duplicate of the ratio of the first to the second. "Thus, if A be to B as B to C, the ratio of A to C is said to be duplicate "of the ratio of A to B. Hence, since by the last definition, the ratio "of A to C is compounded of the ratios of A to B, and B to C, a ratio, "which is compounded of two equal ratios, is duplicate of either of "these ratios." 12. If four magnitudes are continual proportionals, the ratio of the first 13. In proportionals, the antecedent terms are called homologous to one another, as also the consequents to one another. Geometers make use of the following technical words to signify certain ways of changing either the order or magnitude of proportionals, so as that they continue still to be proportionals. 14. Permutando, or alternando, by permutation, or alternately; this word is used when there are four proportionals, and it is inferred, that the first has the same ratio to the third which the second has to the fourth; or that the first is to the third as the second to the fourth: See Prop. 16. of this Book. 15. Invertendo, by inversion: When there are four proportionals, and it is inferred, that the second is to the first, as the fourth to the third. Prop A. Book 5. 16. Componendo, by composition: When there are four proportionals, and it is inferred, that the first, together with the second, is to the second as the third, together with the fourth, is to the fourth. 18th Prop. Book 5. 17. Dividendo, by division; when there are four proportionals, and it is inferred that the excess of the first above the second, is to the second, as the excess of the third above the fourth, is to the fourth. 17th Prop. Book 5. |