It was carried as the banner of Antiochus, King of Syria (surnamed Soter, or the Preserver), in his wars against the Gauls (260 B.C.). Among the Cabalists, the star with the sacred name written on each of its points, and in the centre, was considered talismanic; and in ancient times it was employed all over Asia as a charm against witchcraft. Even now, European troops at war with Arab tribes, sometimes find, under the clothing, on the breasts of their slain enemies, this ancient emblem, in the form of a metal talisman, or charm. The European Göethe puts these words into the mouth of Mephistopheles: "I am hindered egress by a quaint device upon the threshold,--that five-toed damned spell." I shall set forth the geometric significance of this star, as far as my general subject warrants me, and show that it is the _geometric emblem of extreme and mean ratio_, and the _symbol of the Egyptian Pyramid Cheops_. A plane geometric star, or a solid geometric pyramid, may be likened to the corolla of a flower, each separate side representing a petal. With its petals opened and exposed to view, the flower appears in all its glorious beauty; but when closed, many of its beauties are hidden. The botanist seeks to view it flat or open in its geometric symmetry, and also closed, as a bud, or in repose:--yet judges and appreciates the one state from the other. In the same manner must we deal with the five pointed star, and also with the Pyramid Cheops. In dealing with so quaint a subject, I may be excused, in passing, for the quaint conceit of likening the interior galleries and chambers of this pyramid to the interior whorl of a flower, stamens and pistil, mysterious and incomprehensible. Figure 67 (page 101), is the five pointed star, formed by the unlapping of the five slant sides of a pyramid with a pentagonal base. Figure 70 (page 106), is a star formed by the unlapping of the four slant sides of the pyramid Cheops. The pentagon GFRHQ, (_Fig._ 67) is the base of the pyramid "_Pentalpha_" and the triangles EGF, BFR, ROH, HNQ and QAG, represent the five sides, so that supposing the lines GF, FR, RH, HQ and QG, to be hinges connecting these sides with the base, then by lifting the sides, and closing them in, the points A, E, B, O, and N, would meet over the centre C. Thus do we close the geometric flower Pentalpha, and convert it into a pyramid. In the same manner must we lift the four slant sides of the pyramid Cheops from its star development, (_Fig._ 70) and close them in, the four points meeting over the centre of the base, forming the solid pyramid. Such transitions point to the indissoluble connection between plane and solid geometry. As the _geometric emblem of extreme and mean ratio_, the pentangle appears as an assemblage of lines divided the one by the others _in extreme and mean ratio_. To explain to readers not versed in geometry, what extreme and mean ratio signifies, I refer to Figure 65:-- Fig. 65.