Let ABCDEF be the given circle; it is required to infcribe Book IV. an equilateral and equiangular hexagon in it. Find the centre G of the circle ABCDEF, and draw the diameter AGD; and from D as a centre, at the distance 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 equiangu lar. F Because G is the centre of the circle ABCDEF, GE is equal to GD: and 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, DEG are equal to one another 3; and Cor. 5. 1. the three angles of a triangle are equal b to two right angles; b 32. x. 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 alfo the third part of two right angles and because the straight line GC makes with EB the adjacent angles EGC, CGB equal c to two right angles; the remaining angle CGB is the third part of two right angles; therefore the angles EGD, DGC, CGB, are equal to one another: and to these are equal d the vertical oppofite angles BGA, AGF, FGE; therefore the fix angles EGD, DGC, CGB, BGA, AGF, FGE are equal to one another. But equal H B C 13. I. D d 15 3. angles ftand upon equal e circumferences; therefore the fix e 26. 3. circumferences AB, BC, CD, DE, EF, FA are equal to one another and equal circumferences are fubtended by equal f f 29. 3. 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 ftands upon the circumference : Book IV. cumference FABCD, and the angle AFE upon EDCBA; 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 shown; and it is inscribed 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 centre, that is, to the semidiameter of the circle. And if through the points A, B, C, D, E, F there be drawn ftraight lines touching the circle, an equilateral and equiangular hexagon fhall be described about it, which may be demonstrated from what has been faid of the pentagon; and likewife a circle may be infcribed in a given equilateral and equiangular hexagon, and circumfcribed about it, by a method like to that used for the pentagon. a 2. 4. PROP. XVI. PRO B. TO infcribe an equilateral and equiangular quindecagon in a given circle. Let ABCD be the given circle; it is required to infcribe an equilateral and equiangular quindecagon in the circle. ABCD. Let AC be the fide of an equilateral triangle inscribed a in b 11. 4. tagon infcribed b in the fame; B E C five; and the circumference AB, which is the fifth part of the whole, contains three; therefore A F D 30. 3. BC their difference contains two of the fame parts: bifect BC in E; therefore BE, EC are, each of them, the fifteenth Book IV. part of the whole circumference ABCD: therefore if the ftraight lines BE, EC be drawn, and straight lines equal to them be placed d around in the whole circle, an equilateral d. 4 and equiangular quindecagon fhall be infcribed in it. Which was to be done. And in the fame manner as was done in the pentagon, if through the points of divifion made by infcribing 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 circumfcribed about it. EL E ELEMENTS OF GEOMETRY. BOOK V. A DEFINITIONS. I. Less magnitude is faid to be a part of a greater magni- Book V. tude, 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 faid to be a multiple of a lefs, when the greater is measured by the less, that is, when the greater contains the lefs a certain number of times exactly. III. Ratio is a mutual relation of two magnitudes of the fame kind to one another, in refpect of quantity. IV. Magnitudes are faid to be of the fame kind, when the lefs can be multiplied fo as to exceed the greater; and it is only fuch magnitudes that are faid to have a ratio to one another. |