886

THEORY AND PRACTICE OF ENGINEERING.

Book II.

Fig. 1200

D y I*

NC o M I

Fig. 1201.

Fig. 1202.

To divide Figures of four Sides into several equal partsuniting at a point on one side ; as to divide the four-sided figure A B C D into two equal parts uniting at E:reduce the quadrilateral A BCD to a triangle, A F D,and bisect the line I) F in G : draw GE, and throughA aline parallel thereto IIA; draw HE, and it willdivide the four-sided figure into two equal parts.

To divide Figures of four Sides into several equal partsuniting at a point in their superficies: as to divideA B C D into two equal parts, uniting at the point E :divide the trapezoid A BCD into two equal parts,uniting by a line, F G, passing through the point E ;reduce the trapezoid ABC D, to a triangle A II D, andF B C G to a triangle FIG: draw A K parallel toD H: elevate the triangle FI G to the same height asA H D, and you will have the triangle LMG: bisectthe base D II in N, and draw A N, forming the twoequal triangles A II N and AND: set the base M G ofthe triangle LMG on the line HN of the triangleA II N, and remark if these two lines are equal: butsince the base M G is shorter by the distance O N, setoff this distance O N from G to P, and draw L P,forming the triangle I. G P : then remarking the figureF B CQ only contains I M G, or its equal, All (), youmust add to this figure the triangle A ON or LG P,lowering the height to E on the side DII, as is thetriangle EGQ, which will give you the irregularpentagon FBCQE for one half, and AEFQD forthe other half of the trapezoid A B C D.

To divide Figures of four Sides into several equal partsby lines parallel to one of their sides: as to divide thetrapezoid HIK L into two equal parts by a lineparallel to HL: reduce the trapezoid IIIKL to atriangle HML: bisect its base LM in N: prolongL K and H I till it cuts L K prolonged in O : find amean proportional between O L and ON, as P Q,which set off from O to R, and draw R S parallel toII L ; it will divide the trapezoid into two equal parts.

To divide Pentagonal Figures into several equal partsabutting at one angle: as the regular pentagonA B C D E into two equal parts abutting on the angleA. Reduce the pentagon ABCDE to a triangle,

A F G : bisect the base G F in I, and draw A I, it willdivide the pentagon ABCDE into two equal parts.

To divide Pentagons into several equal parts, abuttingat a given point on their sides; as to divide theirregular pentagon ABCDE into two equal partsuniting at the point F. Reduce the pentagon to atriangle, AGE: bisect the base E G in II, and drawAH forming the triangle AHE: half the triangleAGE, or the pentagon ABCDE: then lower thetriangle A H E to F, as FIE: draw F D, and throughI its parallel I K, and draw F K, which will divide thepentagon into two equal parts.

To divide Pentagonal Figures into several equal partsuniting at a point in their superficies : as to divide theirregular pentagon ABCDE into two equal parts,abutting at the point F. Divide the pentagon into twoequal parts by a line passing through the given point,as G H ; then reduce the pentagon to a triangle, AI E,and the figure GBCDH to a triangle G K H, whichyou must elevate as high as A I E, as is done in thetriangle LMII: divide the base El into two equalparts in N, and draw N A, which will form the triangleAIN: set the base I N on the base M H ; it will begreater by the distance HO: draw OL forming thetriangle LMO : lower the triangle EHO to the point Fig. 1205 .

F, as F H P, which will give you G B C D P F for half the pentagon ABCDE.

Fig. 1203.

Fig. 1204.