122 Mr. T. S. Davies's Properties 



Through either pair of opposite angles, as V and W, draw 

 any lines KC, ED limited by the sides. The lines DK, EC 

 will intersect in the diagonal S'T', and the lines EK, DC 

 will intersect in the diagonal ST of the intersecting trapezium 

 TS'ST'. 



Prop. VII. 



Let EFDG be any trapezium Fig. 6. 



whatever, and through any point R 

 in either of the diagonals, as in FG, 

 draw lines, each to cut an opposite 

 pair of sides in H, M and N, I ; join 

 also HI cutting the pair of sides EF, 

 EG in O and P and NM cutting the 

 pair FD, DG in K and L. Then 

 OL, PK being drawn, their inter- 

 section Q is in the diagonal FG. 



Dem.— Draw PN, HL intersect- 

 ing in S; then since HLI, NPM 

 are two triangles situated as in Pr. 6, 

 the line joining their intersections S, G will pass through R 

 the intersection of the diagonals HM, NI ; and F, S, R, G, 

 are in a straight line. 



Again, because NOL, HKP, are triangles also situated as 

 in Prop. 6. and intersecting in F, Q, the line FQ passes 

 through S the intersection of the diagonals HL, NP; or Q is 

 in the line FG. o. e. d. 



Cor. When HI passes through E, the points O, P coa- 

 lesce with E and K, L with D ; in which case, the theorem 

 becomes the same with Cor. 3. Pr. 6. 



scribe it on his ensigns ; unto which he did pretend to be admonished in a 

 dream by an apparition of Alexander the Great. And there are many su- 

 perstitious women in these times who believe this to be so lucky a charac- 

 ter, that they always work it upon the swaddling clothes of their young 

 children, thinking thereby to make them healthful and prosperous in their 

 lives. Unto this kind also, some refer the characters that are used in magic, 

 which are maintained to have not only a secret signification, but likewise 

 a natural efficacy." — Works, vol. ii. p. 49. 



Mr. Hobhouse thinks the figure found its way into the Northern super- 

 stitions from its similarity to the hammer of Thor. He remarks that " the 

 English shepherd who never heard of Antiochus, nor saw his coin, still cuts 

 it in the grass" : and perhaps some analogy might be traced by those who 

 are curious in matters of this kind, between the signification of the heraldic 

 mnllet, the cinquefoil, and the rose, when compared with the signification 

 of the figure in the older superstitions. 



Prop. 



