2. If all possible triangles having the same base and equal vertical angles be supposed to exist, show that the centres of the inscribed and escribed circles are situated upon the circumferences of determinable circles. 3. Describe three circles, each touching the other two, and the three centres being three given points, in all possible ways. 4. Describe a circle which shall pass through a given point and touch a given line, and a given circle; and assign the limits of possibility. 5. From the three angles of a triangle draw perpendiculars to the opposite sides; and join the intersections two and two: then (a) The perpendiculars intersect in the same point; (c) The circle about the second triangle bisects the sides of the first; (d) The diameter of this circle is half that described about the first triangle. 6. In the figure to IV. 10., produce DC to meet the circle in F, and show that: (a) AC is the side of a regular pentagon inscribed in the smaller circle ACD; (b) Of a regular decagon, in the larger one BED; (c) The angle ABF will be triple of BFD. 7. If there be drawn lines to the alternate angles of a regular pentagon, their intersections will give the angular points of another regular pentagon; and if the alternate sides be produced, their intersections will be the angular points of a third regular pentagon; and the circles circumscribing and inscribed in these pentagons will all be concentric. 8. Divide a right angle into three, four, five, six, eight, ten, twelve and fifteen equal parts. 9. Every regular polygon may have a circle inscribed within it, and another circumscribed about it; and these circles will be concentric. 10. An equilateral figure inscribed in a circle, or circumscribed about it, is equiangular; and an equiangular figure so inscribed or circumscribed is equilateral. 11. A triangle being constituted of the sides of the regular pentagon, hexagon and decagon inscribed in the same circle, will be right angled; and the same is true of the regular inscribed triangle, square, and hexagon. 12. Given the radius of the inscribed circle and the perimeter of the triangle, to construct it in the cases where the third datum is : (a) The sum of the sides; (b) The difference of the sides; (c) The perpendicular on the base; (d) The radius of the circumscribing circle; ƒ) The difference of the angles at the base. 13. If there be equilateral triangles described externally on the three sides of any triangle, the three distances between the vertices of these and the opposite angles of the original triangle will be all equal: And if the original triangle be right-angled, two of the equilateral triangles will together be equal to the third. 14. A regular polygon and a circle concentric with it being given, and from any point in the circle draw lines to all the angles, and likewise lines perpendicular to the sides: the sum of the perpendiculars is constant; and the sum of the squares on all the other lines is constant, wherever in the circumference the point be taken. BOOK V. 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 have a ratio to one another, when the less can be multiplied so as to exceed the other. 5. 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. 6. 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.' 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. Analogy, or proportion, is the similitude of ratios.' 9. Proportion consists of three terms at least. 10. 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. 11. 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, etc., 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 the ratio of B to C, and 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 E 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. 12. 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.' 13. 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 shown in the 16th Prop. of this 5th Book. 14. 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. 15. 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 V. 16. 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 V. 17. 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. 18. Ex æquali (sc. distantia), 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.' 19. 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 the 22nd Prop., Book V. 20. 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 the 23rd Prop. of Book V. AXIOMS. 1. Equimultiples of the same, or of equal magnitudes, are equal to one another. 2. Those magnitudes, of which the same or equal magnitudes are equimultiples, are equal to one another. 3. A multiple of a greater magnitude is greater than the same multiple of a less. 4. That magnitude, of which a multiple is greater than the same multiple of another, is greater than that other magnitude. PROPOSITIONS. PROPOSITION I. THEOR. If any number of magnitudes be equimultiples of as many, each of each; what multiple soever any one of them is of its part, the same multiple shall all the first magnitudes be of all the other. Let any number of magnitudes AB, CD be equimultiples of JA as many others E, F, each of each: whatsoever multiple AB is of E, the same multiple shall AB and CD together, be of E and F together. as Because AB is the same multiple of E that CD is of F, many magnitudes as are in AB equal to E, so many are there in CD equal to F. Divide AB into magnitudes equal to E, viz., AG, GB; and CD into CH, HD, equal each of them to F: the number therefore of the magnitudes CH, HD, shall be equal to the number of the others AG, GB: * Prop. 4, Lib. II., Archimedis de Sphærâ et Cylindro. G D |