To inscribe an equilateral and equiangular hexagon in a given circle. Given the circle ABCDEF; it is required to inscribe an equilateral and equiangular hexagon in it. (Const.) Find the centre G of the circle ABCDEF, and draw the diameter AGD; and from D as a centre, at the distance DG, describe the circle EGCH; join EG and CG, and join AB, and produce them to the points B and F; BC, CD, DE, EF, and FA; the hexagon ABCDEF is equilateral and equiangular. and (Dem.) Because G is the centre of the circle ABCDEF, GE is equal to GD; because D is the centre of the circle EGCH, DE is equal to DG; wherefore GE is equal to ED, and the triangle EGD is equilateral; and therefore its three angles EGD, GDE, and DEG are equal to one another (I. 5, Cor.); F E and the three angles of a triangle are equal to two right angles (I. 32); therefore the angle EGD is the third part of two right angles. In the same manner, it may be demonstrated that the angle DGC is also the third part of two right angles; and because the straight line GC makes with EB the adjacent angles EGC and CGB, equal to two right angles (I. 13); the remaining angle CGB is the third part of two right angles; therefore the angles EGD, DGC, and CGB are equal to one another; and to these are equal the vertical opposite angles BGA, AGF, and FGE (I. 15); therefore the six angles EGD, DGC, CGB, BGA, AGF, and FGE are equal to one another. But equal angles stand upon equal arcs (III. 26); therefore the six arcs AB, BC, CD, DE, EF, and FA are all equal; and equal arcs are subtended by equal chords (III. 29); therefore the six chords are also equal to one another, and the hexagon ABCDEF is equilateral. It is also equiangular, for each of the angles stands on the sum of four equal arcs (Schol. IV. 11). (Or, 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 therefore be demonstrated that the other angles of the hexagon ABCDEF are each of them equal to the angle AFE or FED; the hexagon is equiangular); equilateral; and it was shewn to be 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 radius of the circle. And if through the points A, B, C, D, E, and 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 similar to that used for the pentagon. PROPOSITION XVI. PROBLEM. To inscribe an equilateral and equiangular quindecagon in a given circle. Given the circle ABCD; it is required to inscribe an equilateral and equiangular quindecagon in it. (Const.) Let AC be the side of an equilateral triangle inscribed and AB the side of an equilateral and in the circle (IV. 2), equiangular pentagon inscribed in the same (IV. 11); fore, of such equal parts as the whole circumference ABCDF con- being the third part of the whole, bisect B E there straight lines equal to them be placed around in the whole circle, an equilateral and equiangular quindecagon shall be inscribed in it. COR.-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 shall 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. Schol. Whatever regular polygon can be inscribed in a circle, another of double the number of sides may be inscribed in it, by bisecting the arc subtended by a side of the former. It has been proved by M. Gauss, that it is practicable by means of plane geometry-namely, by the intersections of the straight line and circle, or by the resolution of simple and quadratic equationsto inscribe in a circle any regular polygon, provided the number of its sides be a prime number; that is, a number which has no common measure except 1-and be also some power of 2 increased by 1. He has proved more generally-in his Disquisitiones Arithmeticæ —if n, the number of sides of the regular polygon, exceeds by 1 the number which is the product of the prime factors 2a, 3b, 5c,... that the division of the circle into n equal parts, and consequently the possibility of inscribing in it a regular polygon of n sides, can be reduced to the solution of a equations of the second degree, b of the third, c of the fifth, and so on. EXERCISES. 1. Through a given point within or without a circle, to draw a chord that shall be equal to a given line. 2. To draw a chord in a circle that shall be equal to a given line, and parallel to another given line, or inclined to it at a given angle. 3. To draw a tangent to a given circle, so that it shall be parallel to a given line. 4. To draw a tangent to a circle, so that it shall make a given angle with a given line. 5. To find a point in a given line that shall be equidistant from another given point in it and from a given line. 6. To describe a circle that shall touch a given line in a given point, and pass through another given point. 7. To draw a line that shall be a common tangent to two circles. 8. To find a point in a given line that shall be equidistant from another given point and a given line. 9. To describe a circle that shall pass through two given points, and touch a given line. 10. To describe a circle that shall touch a given line in a given point, and also touch a given circle. 11. A regular octagon inscribed in a circle is equal to the rectangle under the sides of the inscribed and circumscribing squares. 12. To inscribe a circle in a given quadrant. 13. To describe a circle that shall pass through a given point, and touch a given circle in a given point. 14. If a quadrilateral circumscribe a circle, its opposite sides are together equal to half its perimeter. 15. If every two alternate sides of a polygon be produced to meet, the sum of the salient angles thus formed with eight right angles, will be equal to twice as many right angles as the figure has sides. 16. If on two opposite sides of a rectangle semicircles be described lying on corresponding sides of their diameters, the muxtilineal space contained by their arcs and the other two sides of the rectangle is equal to the rectangle. 17. To describe a circle that shall touch two given lines, and pass through a given point. 18. To describe a circle that shall touch two given lines, and also a given circle. 19. Inscribe a regular hexagon in a given equilateral triangle, and compare its area with that of the triangle. 20. Describe a circle which shall pass through one angle, and touch two sides of a given square. 21. Given a regular pentagon; it is required to describe a triangle having the same area and altitude. ་ ་་། ་ 22. In the figure (IV. 10) shew that AC and CD are sides of a pentagon in the less circle; and if DC be produced to meet the circumference in F, and FB be joined, FB will be a side of a pentagon in the greater circle. 23. The square on the side of a pentagon inscribed in a circle, is equal to the sum of the squares on the sides of a hexagon and decagon, inscribed in the same circle. FIFTH BOOK. 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. It is also called a measure, or submultiple of the greater. 2. A line, which is a measure of two or more lines, is called a common measure of these lines. 3. Lines that have a common measure, are called commensurable lines; those that have no common measure, incommensurable lines. 4. 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. 5. Equimultiples of magnitudes are multiples that contain these magnitudes, respectively, the same number of times. 6. Ratio is a mutual relation of two magnitudes of the same kind to one another, in respect of quantity; or it is the quotient arising from dividing the first by the second. 7. Magnitudes are said to be homogeneous, or 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. 8. The two magnitudes of a ratio are called its terms. The first term is called the antecedent; the latter, the consequent. 9. A ratio is said to be a ratio of equality, majority, or minority, according as the antecedent is equal to, greater, or less than the consequent. 10. If there be four magnitudes, such that 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, so is the multiple of the third 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. |