Book IV. See N. 2. 5. I. b. 32. I. d. 15. 1. O infcribe an equilateral and equiangular hexagon in a given circle. Let ABCDEF be the given circle; it is required to inscribe an equilateral and equiangular hexagon in it. Find the center G of the circle ABCDEF, and draw the diameter AGD; and from D as a center, at the diftance DG defcribe the circle EGCH, join EG, CG and produce them to the points B, F; and join AB, BC, CD, DE, EF, FA. the hexagon ABCDEF is equilateral and equiangular. Because G is the center of the circle ABCDEF, GE is equal to GD. and because D is the center 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, DEG are equal to one another, because the angles at the base of an isofceles triangle are equal a. and the three angles of a triangle are equal to two right angles; therefore the angle EGD is the third part of two right angles. in the fame 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 F with EB the adjacent angles EGC, CGB C d A B D H c. 13. 1. equal to two right angles; the remaining angle CGB is the third part of, two right angles; therefore the angles E EGD, DGC, CGB are equal to one another. and to these are equal the vertical oppofite angles BGA, AGF, FGE. therefore the fix angles EGD, DGC, CGB, BGA, AGF, FGE, are equal to one another. but equal angles ftand upon equal circumferences; therefore the fix circumferences AB, BC, CD, DE, EF, FA are equal to one another. and equal circumferences are fubtended by equal f ftraight lines; therefore the fix ftraight lines are equal to one another, and the hexagon ABCDEF is equilateral. It is alfo equiangular; for fince the circumference AF is equal to ED, to each of these add the circumference ABCD; therefore the whole circumference FABCD fhall be equal to the whole EDCBA. and the angle FED stands upon e. 26. 3. f. 29. 3. the circumference FABCD, and the angle AFE upon EDCBA; Book IV. therefore the angle AFE is equal to FED. in the fame manner it may be demonftrated that the other angles of the hexagon ABCDEF are each of them equal to the angle AFE or FED. therefore the hexagon is equiangular. and it is equilateral, as was fhewn; and it is infcribed in the given circle. ABCDEF. Which was to be done. COR. From this it is manifeft, that the fide of the hexagon is equal to the straight line from the center, that is, to the semidiameter of the circle. And if thro' the points A, B, C, D, E, F there be drawn straight lines touching the circle, an equilateral and equiangular hexagon fhall be described about it, which may be demonftrated 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. O inscribe an equilateral and equiangular quinde- See N. cagon 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 fide of an equilateral triangle inscribed a in the a. 2. 4. circle, and AB the fide of an equilateral and equiangular penta gon infcribed in the fame; therefore of fuch equal parts as the b. 11. 4. whole circumference ABCDF contains fifteen, the circumference ABC, being the third part of the whole, contains five; and the circumference AB, which is the fifth part of the whole, contains three; therefore BC their difference con-B tains two of the fame parts. bifect c E A F c. 30. 3. drawn, and straight lines equal to them be placed around in the d. 1. 4. whole circle, an equilateral and equiangular quindecagon shall be Book IV. And in the fame manner as was done in the pentagon, if thro1 the points of divifion made by inscribing the quindecagon, ftraight lines be drawn touching the circle, an equilateral and equiangu→ lar quindecagon fhall be defcribed about it. and likewise, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circumfcribed about it THE ELEMENTS OF Book V. A EUCL I D. BOOK V. DEFINITIONS. I. LESS magnitude is faid to be a part of a greater magnitude, II. A greater magnitude is faid to be a multiple of a lefs, when the greater is measured by the lefs, that is, when the greater con⚫tains the less a certain number of times exactly.' III. "Ratio is a mutual relation of two magnitudes of the fame kind See N. "to one another, in respect of quantity." IV. Magnitudes are faid to have a ratio to one another, when the lefs can be multiplied fo as to exceed the other. V. The first of four magnitudes is faid to have the fame ratio to the fecond, which the third has to the fourth, when any equimultiples whatsoever of the firft 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 Book V. than that of the fecond, the multiple of the third is also greater See N. than that of the fourth. VI. Magnitudes which have the fame ratio are called proportionals. N. B. When four magnitudes are proportionals, it is usually 'expreffed by faying, the first is to the fecond, as the third to the fourth.' VII. When of the equimultiples of four magnitudes (taken as in the 5th Definition) the multiple of the firft 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 faid 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 lefs ratio than the firft has to the second. VIII. "Analogy, or proportion, is the fimilitude of ratios." IX. Proportion confifts 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 fecond. XI. When four magnitudes are continual proportionals, the first is faid to have to the fourth the Triplicate ratio of that which it has to the second, and so on Quadruplicate, &c. increafing 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 fame kind, the firft is faid to have to the last of them the ratio compounded of the ratio which the firft has to the fecond, and of the ratio which the second has to the third, and of the ratio which the third has to the fourth, and fo on unto the last magnitude. For example, If A, B, C, D be four magnitudes of the fame kind, the first A is faid to have to the laft 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 faid to be compounded of the ratios of A to B, B to C, and C to D. And if A has to B, the fame ratio which E has to F; and B to C, the fame ratio that G has to H; and C to D, the fame that K |