contained by the sides of the squares inscribed in, and circumscribed about the circle. 51. If in any circle the side of an inscribed hexagon be produced till it becomes equal to the side of an inscribed square, a tangent drawn from the extremity, without the circle, shall be equal to the side of an inscribed octagon. VII. 52. To describe a circle which shall touch a given circle in a given point, and also a given straight line. 53. Describe a circle touching a given straight line, and also two given circles. 54. Describe a circle which shall touch a given circle, and each of two given straight lines. 55. Two points are given, one in each of two given circles; describe a circle passing through both points and touching one of the circles. 56. Describe a circle touching a straight line in a given point, and also touching a given circle. When the line cuts the given circle, shew that your construction will enable you to obtain six circles touching the given circle and the given line, but not necessarily in the given point. 57. Describe a circle which shall touch two sides and pass through one angle of a given square. 58. If two circles touch each other externally, describe a circle which shall touch one of them in a given point, and also touch the other. In what case does this become impossible? 59. Describe three circles touching each other and having their centers at three given points. In how many different ways may this be done? VIII. 60. Let two straight lines be drawn from any point within a circle to the circumference: describe a circle, which shall touch them both, and the arc between them. 61. In a given triangle having inscribed a circle, inscribe another circle in the space thus intercepted at one of the angles. 62. Let AB, AC, be the bounding radii of a quadrant; complete the square ABDC and draw the diagonal AD; then the part of the diagonal without the quadrant will be equal to the radius of a circle inscribed in the quadrant. 63. If on one of the bounding radii of a quadrant, a semicircle be described, and on the other, another semicircle be described, so as to touch the former and the quadrantal arc; find the center of the circle inscribed in the figure bounded by the three curves. 64. In a given segment of a circle inscribe an isosceles triangle, such that its vertex may be in the middle of the chord, and the base and perpendicular together equal to a given line. 65. Inscribe three circles in an isosceles triangle touching each other, and each of them touching two of the three sides of the triangle. IX. 66. In the fig. Prop. 10, Book IV, shew that the base BD is the side of a regular decagon inscribed in the larger circle, and the side of a regular pentagon inscribed in the smaller circle. 67. In the fig. Prop. 10, Book IV, produce DC to meet the circle in F, and draw BF; then the angle ABF shall be equal to three times the angle BFD. 68. If the alternate angles of a regular pentagon be joined, the figure formed by the intersection of the joining lines will itself be a regular pentagon. 69. If ABCDE be any pentagon inscribed in a circle, and AC, BD, CE, DA, EB be joined, then are the angles ABE, BCA, CDB, DEC, EAD, together equal to two right angles. 70. A watch-ribbon is folded up into a flat knot of five edges, shew that the sides of the knot form an equilateral pentagon. 71. If from the extremities of the side of a regular pentagon inscribed in a circle, straight lines be drawn to the middle of the arc subtended by the adjacent side, their difference is equal to the radius; the sum of their squares to three times the square of the radius; and the rectangle contained by them is equal to the square of the radius. 72. Inscribe a regular pentagon in a given square so that four angles of the pentagon may touch respectively the four sides of the square. 73. Inscribe a regular decagon in a given circle. 74. The square described upon the side of a regular pentagon in a circle, is equal to the square on the side of a regular hexagon, together with the square upon the side of a regular decagon in the same circle. X. 75. In a given circle inscribe three equal circles touching each other and the given circle. 76. Shew that if two circles be inscribed in a third to touch one another, the tangents of the points of contact will all meet in the same point. 77. If there be three concentric circles, whose radii are 1, 2, 3; determine how many circles may be described round the interior one, having their centers in the circumference of the circle, whose radius is 2, and touching the interior and exterior circles, and each other. 78. Shew that nine equal circles may be placed in contact, so that a square whose side is three times the diameter of one of them will circumscribe them. XI. 79. Produce the sides of a given heptagon both ways, till they meet, forming seven triangles; required the sum of their vertical angles. 80. To convert a given regular polygon into another which shall have the same perimeter, but double the number of sides. 81. In any polygon of an even number of sides, inscribed in a circle, the sum of the 1st, 3rd, 5th, &c. angles is equal to the sum of the 2nd, 4th, 6th, &c. 82. Of all polygons having equal perimeters, and the same number of sides, the equilateral polygon has the greatest area, BOOK V. DEFINITIONS. I. 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.' II. 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.' III. "Ratio is a mutual relation of two magnitudes of the same kind to one another, in respect of quantity." IV. Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other. ས. The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth: or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth: or, if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth. VI. Magnitudes which have the same ratio are called proportionals. N.B. 'When four magnitudes are proportionals, it is usually expressed by saying, the first is to the second, as the third to the fourth.' VII. 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. 66 VIII. Analogy, or proportion, is the similitude of ratios." IX. Proportion consists in three terms at least. X. When three magnitudes are proportionals, the first is said to have to the third, the duplicate ratio of that which it has to the second. XI. When four magnitudes are continual proportionals, the first is said to have to the fourth, the triplicate ratio of that which it has to the second, and so on, quadruplicate, &c., increasing the denomination still by unity, in any number of proportionals. Definition A, to wit, of compound ratio. When there are any number of magnitudes of the same kind, the first is said to have to the last of them 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 has to B the same ratio which has to F; and B to C the same ratio that G has to H; and C to D the same that K has to L; then, by this definition, A is said to have to D the ratio compounded of ratios which are the same with the ratios of E to F, G to H, and K to L. And the same thing is to be understood when it is more briefly expressed by saying, A has to D the ratio compounded of the ratios of E to F, G to H, and K to L. 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 N the ratio compounded of the ratios of E to F, G to H, and K to L. XII. 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 that they continue still to be proportionals.' XIII. 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: as is shewn in Prop. XVI. of this Fifth Book. XIV. 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. B. Book v. XV. 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. Prop. 18, Book v. XVI. 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. Prop. 17, Book V. XVII. Convertendo, by conversion; when there are four proportionals, and it is inferred, that the first is to its excess above the second, as the third to its excess above the fourth. Prop. E. Book v. XVIII. Ex æquali (sc. distantiâ), or ex æquo, from equality of distance: when there is any number of magnitudes more than two, and as many others such that they are proportionals when taken two and two of each rank, and it is inferred, that the first is to the last of the first rank of magnitudes, as the first is to the last of the others: Of this there are the two following kinds, which arise from the different order in which the magnitudes are taken, two and two.' XIX. Ex æquali, from equality. This term is used simply by itself, when the first magnitude is to the second of the first rank, as the first to the second of the other rank; and as the second is to the third of the first rank, so is the second to the third of the other; and so on in order: and the inference is as mentioned in the preceding definition; whence this is called ordinate proportion. It is demonstrated in Prop. 22, Book v. XX. Ex æquali in proportione perturbatâ seu inordinatâ, from equality in perturbate or disorderly proportion*. This term is used when the first magnitude is to the second of the first rank, as the last but one is to the last of the second rank; and as the second is to the third of the first rank, so is the last but two to the last but one of the second rank: and as the third is to the fourth of the first rank, so is the third from the last to the last but two of the second rank; and so on in a cross order: and the inference is as in the 18th definition. It is demonstrated in Prop. 23, Book v. AXIOMS. EQUIMULTIPLES of the same, or of equal magnitudes, are equal to one another. II. Those magnitudes, of which the same or equal magnitudes are equimultiples, are equal to one another. Prop. 4. Lib. II. Archimedis de sphæra et cylindro. |