Bisect (9. 1.) the angles BCD, EA : And because the angle HCF is cCDE by the straight lines CF, DF, qual to KCF,and the right angle FHC and from the point F, in which they equal to the right angle FKC; in meet, draw the straight lines FB, FA, the triangles FHC,FKC, there are two FE: Therefore, since BC is equal to angles of one equal to two angles of CD, and CF common to the triangles the other, and the side FC, which is BCF, DCF, the two sides BC, CF opposite to one of the equal angles in are equal to the two DC, CF; and each, is common to both; therefore the angle BCF is equal to the angle the other sides shall be equal, (26. 1.) DCF: therefore the base BF is equal each to each; wherefore the perpen(4. 1.) to the base FD, and the other dicular FH is equal to the perpendiangles to the other angles, to which cular FK; In the same manner it the equal sides are opposite ; there may be demonstrated, that FL, FM, fore the angle CBF is equal to the FG are each of them equal to FH or angle CDF: And because the angle FK: Therefore the five straight lines CDE is double of CDF, and that CDE FG, FH, FK, FL, FM are equal to is equal to CBA, and CDF to CBF; one another: Wherefore the circle CBA is also double of the angle CBF; described from the centre F, at the therefore the angle A BF is equal to distance of one of these five, shall the angle CBF; wherefore the angle pass through the extremities of the ABC is bisected by the straight line other four, and touch the straight BF: In the same manner it may be lines AB, BC, CD, DE, EA, because demonstrated, that the angles BAE, the angles at the points G, H, K, L, AED, are bic M are right angles ; and that a sected by the straight line drawn from the extremistraight lines ty of the diameter of a circle at right AF,FE: From angles to it, touches (16. 31.) the the point F circle : Therefore each of the straight draw (12. l. lines AB, BC, CI), DE, EA touches FG, FH, FK, the circle ; wherefore it is inscribed FL, FM per in the pentagon ABCDE. Which pendiculars to was to be done. the straight lines AB, BC, CD, DE, M B B H қ PROP. XIV. PROB. To describe a circle about a given equilateral and equiangular pentagon. Let ABCDE be the given equilate. position, that the angles CBA, BAE, ral and equiangular pentagon; it is AED are bisected by the straight required to describe a circle about it. lines FB, FA, FE: And because the Bisect (9. 1.) the angles BCD, angle BCD is equal to the angle CDE by the straight lines CF, FD, CDE, and that FCD is the half of and from the point F, in which they the angle BCD, and CDF the half meet, draw the of CDE; the angle FCD is equal to straight linesFB, FDC; wherefore the side CF is equal FA, FE, to the (6. 1.) to the side FD: In like manpoints B, A, E. ner it may be demonstrated that FB, It may be de FA, FE, are each of them equal to monstrated, in FC or FD: T'herefore the five the same man. straight lines FA, FB, FC, FD, FE, ner as in the are equal to one another; and the preceding pro circle described from the centre F, A B at the distance of one of them, shall the equilateral and equiangular penpass through the extremities of the tagon ABCDE. Which was to be other four, and be described about done. PROP. XV. PROB. To inscribe an equilateral and cquiangular hexagon in a given circle. Let ABCDEF be the given circle: to one another: But equal angles it is required to inscribe an equilate- stand upon equal (26. 4.) circumfer. ral and equiangular hexagon in it. ences ; therefore the six circumferen Find the centre G of the circleces AB, BC, CD, DE, EF, FA are ABCDEF, and draw the diameter equal to one another : And equal cirAGD; and from Das a centre, at cumferences are subtended by equa the distance DG, describe the circle (29. 3.) straight lines ; therefore the EGCH, join EG, CG, and produce six straight lines are equal to one anthem to the points B, F; and join other, and the hexagon ABCDEF is AB, BC, CD, DE, EF, FA: The equilateral. It is also equiangular; hexagon ABCDEF is equilateral and for since the circumference AF is equiangular. equal to ED, to each of these add Because G is the centre of the circle the circumference ABCD; therefore ABCDEF, GE is equal to GD: And the whole circumference FABCD because D is the centre of the circle shall be equal to the whole EDCBA: EGCH, DE is equal to DG ; where. And the angle FED stands upon th. fore GE is equal to ED, and the tri- circumference FABCD, and the angk angle EGD is equilateral ; and there- AFE upon EDCBA; therefore the fore its three angles EGD, GDE, angle AFE is equal to FED: In the DEG are equal to one another, be- same manner it may be demonstrated cause the angles at the base of an that the other angles of the hexagor isosceles triangle are equal; (5. 1.) ABCDEF are each of them equal to and the three angles of a triangle are the angle AFE or FED: Therefore equal (32. 1.) to two right angles; the hexagon is equiangular; and it therefore the angle EGD is the third is equilateral, as was shewn; and it is part of two right angles : In the same inscribed in the given circle ABCDEF manner it may be de Which was to be done. monstrated, ihat the Cor. From this it is manifest, that angle DGC is also the the side of the hexagon is equal to the third part of two right straight line from the centre, that is, angles : And because to the semi-diameter of the circle. the straight line GC And if through the points A, B, C, makes with EB the D, E, F, there be drawn straight adjacent angles EGC, lines touching the circle, an equilateCGB equal (13. ..) to two right ral and equiangular hexagon shall be angles ; the remaining angle CGB is described about it, which may be dethe third part of two right angles; monstrated from what has been said therefore the angles EĞD, DGC, of the pentagon; and likewise a circle CGB are equal to one another : And may be inscribed in a given equilato these are equal (15. 1.) the verti- teral and equiangular hexagon, and cal opposite angles BGA, ÁGF,FGE: circumscribed about it, by a méthod Therefore the six angles EGV, DGC, like to that used for the pentagon. CGB, BGA, AGF, FGE, are equal PROP. XIV. PROB. To inscribe an equilateral and equiangular quindecagon in a given circle. Let ABCD be the given circle; it same parts : Bisect (30. 3.) BC in E; is required to inscribe an equilateral therefore BE EC are, each of them, and equiangular quindecagon in the the tifteenth part of the whole circumcircle ABCD ference ABCD: Therefore, if the Let AC be the side of an equilateral straight lines BE, EC be drawn, and triangle inscribed (2. 4.) in the circle, straight lines equal to them be placed and XB the side of an equilateral and (1.4.) around in the whole circle, an equiangular pentagon inscribed (11.4.) equilateral and equiangular quindecain the same ; therefore, of such equal gon shall be inscribed in it." Which parts as the whole circumference was to be done. ABCDF contains fifteen, the circum- And, in the same manner as was ference ABC, being the third part of done in the pentagon, if, through the the whole, con points of division made by inscribing tains tive; and the quindecagon, straight lines be the circumfer drawn touching the circle, an equilaence AB, which teral and equiangular quindecagon is the fifth part shall be described about it : And likeof the whole, wise, as in tie pentagon, a circle may contains three; be inscribed in a given equilateral and therefore BC, equiangular quindecagon, and cir. their difference contains two of the cumscribed about it. y IV. the third the duplicate ratio of that Magnitudes are said to have a ratio which it has to the second. to one another, when the less can XI. be multiplied so as to exceed the When four magnitudes are continual other. proportionals, the first is said to V. have to the fourth the triplicate raThe first of four magnitudes is said to tio of that which it has to the se have the same ratio to the second, cond, and so on, quadruplicate, which the third has to the fourth, &c. increasing the denomination when any equimultiples whatsoever still by unity, in any number of of the first and third being taken, and proportionals. any equimultiples whatsoever of the Definition A, to wit, of compound second and fourth; if the multiple of ratio. the first be less than that of the se. When there are any number of magcond, the multiple of the third is also nitudes of the same kind, the first less than that of the fourth ; or, if is said to have to the last of them the multiple of the first be equal to the ratio compounded of the ratio that of the second, the multiple of which the first has to the second, the third is also equal to that of the and of the ratio which the second fourth; or, if the multiple of the has to the third, and of the ratio first be greater than that of the se- which the third has to the fourth, cond, the multiple of the third is and so on unto the last magnitude. also greater than that of the fourth. For example, if A, B, C, D, be four VI. magnitudes of the same kind, the Magnitudes which have the same ra- first A is said to have to the last D tio are called proportionals. the ratio ccmpounded of the ratio N.B.“When four magnitudes are pro- of A to B, and of the ratio of B to portionals, it is usually expressed C, and of the ratio of C to D; or, by saying, the first is to the se- the ratio of A to D is said to be • coud, as the third to the fourth.' compounded of the ratios of A to VII. B, B to C, and C to D. When of the equimultiples of four And if A has to B the same ratio magnitudes (taken as in the fifth de- which E has to F; and B to C, the finition), the multiple of the first is same ratio that G has to H; and greater than that of the second, but C to D, the same that K has to L; the multiple of the third is not then, by this definition, A is said greater than the multiple of the to have to D the ratio compounded fourth; then the first is said to of ratios which are the same with hare to the second a greater ratio the ratios of E to F, G to H, and than the third magnitude has to K to L: And the same thing is to the fourth ; and, on the contrary, be understood when it is more the third is said to have to the briefly expressed, by saying, A has fourth a less ratio than the first has to D the ratio compounded of the to the second ratios of E to F, G to H, and K VIII. to L. ' Analogy, or proportion, is the simi- In like manner, the same things being • litude of ratios.' supposed, if M has to N the same IX. ratio which A has to D; then, for Proportion consists in three terms at shortness sake, M is said to have le28t. to N, the ratio compounded of the X. ratios of E to F, G to H, and K When three magnitudes are propor- to L. tionals, the first is said to have to XII. XVIII. In proportionals, the antecedent terms Ex æquali (sc. distantia), or ex æquo, are called homologous to one ano- from equality of distance; when ther, as also the consequents to one there is any number of magnitudes another. more than two, and as many • Geometers make use of the follow- others, so that they are proportion • ing technical words to signify als when taken two and two of ' certain ways of changing either each rank, and it is inferred, that • the order or magnitude of pro- the first is to the last of the first 'portionals, so as that they continue rank of magnitudes, as the first is ' still to be proportionals. to the last of the others : * Of this XIII. • there are the two following kinds, Permutando, or alternando, by per- • which arise from the different or mutation, or alternately. This • der in which the magnitudes are word is used when there are four s taken two and two.' proportionals, and it is inferred, XIX. that the first has the same ratio to Ex æquali, from equality. This term the third, which the second has to is used simply by itself, when the the fourth: or that the first is to first magnitude is to the second of the third, as the second to the the first rank, as the first to the sefourth: As is shewn in the 16th cond of the other rank; and as the prop. of this 5th book. second is to the third of the first XIV. rank, so is the second to the third Invertendo, by inversion : When there of the other; and so on in order, are four proportionals, and it is in- and the inference is as mentioned ferred, that the second is to the in the preceding definition; whence first, as the fourth to the third. this is called ordinate proportion. Prop. B. Book 5. It is demonstrated in 2ed Prop. XV. Book 5. Componendo, by composition; when XX. there are four proportionals, and it Ex æquali, in proportione perturbata, is inferred, that the first, together seu inordinata, from equality, in with the second, is to the second, perturbate or disorderly proporas the third, together with the tion. This term is used when fourth, is to the fourth. 18th Prop. the first magnitude is to the second Book 5. of the first rank, as the last but one XVI. is to the last of the second rank; Dividendo, by division; when there and as the second is to the third of are four proportionals, and it is in- the first rank, so is the last but two ferred, that the excess of the first to the last but one of the second above the second, is to the second, rank; and as the third is to the as the excess of the third above the fourth of the first rank, so is the fourth, is to the fourth. 17th Prop third from the last to the last but Book 5. two of the second rank; and so on XVII. in a cross order: And the inference Convertendo, by conversion; when is as in the 18th definition. It is there are four proportionals, and it demonstrated in the 23d Prop. of is inferred, that the first is to its ex- Book 5. cess above the second, as the third to its excess above the fourth. Prop. E. Book 5. |