Fig. 60.
The accurate division of a circle into two equal comma-shaped areas of the special shape presented by the "Tomoye" of the Japanese (Fig. 54) and the rotating "Great Monad" of Chinese cosmogony (Fig. 55), is effected by describing within a given circle two circles each having its diameter equal to a radius of the enclosing circle. The two inscribed circles touch one another at the centre of the latter, but do not overlap. The area of the enclosing circle is thus divided into four areas, a, b, c and d (see Fig. 60, A). The areas a, b are the two inscribed circles. Each of the residual areas c, d is called (as Sir Thos. Heath, F.R.S., kindly informs me) an "arbelus" by ancient Greek geometricians—a name used for a rounded knife used by shoemakers. The comma-shaped bent cone or pine is formed by the fusion of one of the two small circles with one of the adjacent arbeli (Fig. 60, B). The figure so formed which to-day is loosely spoken of as a "bent cone," a "pine," or a "comma," has never, so far as I can ascertain, received a name in geometry, nor in the language of decorative design or pattern-making. Nor has the S-like line made by the two semicircles separating the contiguous "pines" or "commas" received any designation though vaguely indicated by the word "ogee." The comma-like areas might conveniently be called "streptocones," and their S-like boundary "a hemicyclic sigmoid." As shown in Fig. 56, by drawing a second hemicyclic sigmoid of the same dimensions at right angles to the first, the circle is divided into four smaller streptocones. By using sigmoids or half-sigmoids of a curvature of a different order from that of the hemicyclic one, but of a precisely defined nature, the circle may be divided into three, six, eight or more equal "streptocones" of graceful proportions, some of which have been used either in series as borders in metal work (for circular dishes and goblets) or as detached or grouped elements in pattern-designs (stone-work tracery, embroidery, woven and printed fabrics).
Apart from this development of the "streptocone" as an important feature in decorative work, it is not without interest in connection with the probable importance and significance of the Japanese double streptocone, as we may call the Tomoye, to note some of its geometrical features. Referring to the Fig. 60, it is obvious that each of the paired streptocones is equal in area to half the enclosing circle, also that each of the two inscribed circles (a, b) has an area of one-fourth of that of the enclosing circle—and that each arbelus (c, d) has also an area one-fourth that of the enclosing circle and is equal in area to each of the inscribed circles (a, b). Each of the two constituent "streptocones" is made up of a complete circle capped by an "arbelus" equal in area to it (namely, one-quarter of that of the big circle). It is obvious that the area of the arbelus formed in a semicircle by two enclosed semicircles which are contiguous and of equal base as in Fig. 60, is equal to that of a circle the diameter of which is the vertical line drawn from the apex of the arbelus to the arc of the semicircle (Fig 60). This is true whether the enclosed contiguous semicircles have chords of equal or unequal length (Fig. 60). This fact was known to the Greek geometricians, as I am informed by Sir Thos. Heath.
FOOTNOTE:
[8] I am indebted for the figures (not the diagrams) illustrating Chapters XVII., XVIII., XIX. to the report by Mr. Thomas Wilson on the Swastika—in the Smithsonian Reports, 1894. Those interested in this subject will find a vast store of information in that report.
CHAPTER XX
COAL
COAL is so much "a matter of course" in our daily life that most people are only now, when its supply is becoming precarious, anxious to know something of its nature and history. By the word "coal," or "coles," our ancestors understood what we now distinguish as "charcoal," prepared from wood by the "charcoal-burner," or "charbonnier," as the French call him. What we now call "coal" was known to them as "sea-coal," and, later, as "black" or "stone cole," to distinguish it from "brown coal," known nowadays as "lignite," though the name "stone coal" is locally applied in England to that very hard kind of black coal also called "anthracite," of which jet is only an extremely hard and dense variety found in small quantities in the oolitic strata of Whitby, Spain, and other localities.