§2i 



SCIENCE. 



[Vol. xXil. Mo. 5^ 



from ku V-S ol* chtl V-s, thus ; ka, ke, ki, 

 ko; ak, ek, ik, ok. It is an excellent ex- 

 ample of how the same elements appear in 

 new combinations with different phonetic 

 values, these being influenced by vowel 

 fluctuation. 

 Figs. 146, 147 = Uch v-s. 

 Fig. 148 = Ka v-s. 

 Figs. 149, 150 = Kan, ka v-s, an v-s. 

 Fig. 151 = Y'ox, iax, yosh, iash, sh, h, xa v-s. The 

 value oi: this element will be demonstrated 

 in Part II. of this article. 

 The phonetic values assigned a series of elements 

 having been given, let me proceed to apply some of 

 them to certain gh'phs beginning with Fig. 160. This 

 glyph, and variants of it, is frequentty found in the 

 codices in connection with a figure which has been des- 

 ignated, for the sake of convenience, "The Long-Nosed 

 God," whom there is good reason to think is kixkulkan. 

 A glance at this glyph shows it to be a representation 

 of an elongated reptile-like head, an ideographic sug- 

 gestion of the serpent god. In the nose we have the 

 elements. Fig. 105 = cha, curved around into a loop-like 

 .end, Fig. 161. The mouth line must not be confound- 

 ed with the parallel earth line. At times small tooth-like 

 squares (Figs. 28, 29, 30) are attached to it — similar to 

 those shown in our i'igs. 31 and 34. They seem like 

 phonetic additions placed to indicate the especial pho- 

 netic value of the eleinent to which they are attached. 

 In this case there are two squares attached = ca. As chi 

 = "niottt" we accept the suggestion as cha or kha (c 

 = k in Maya). It will be observed that the end of the 

 mouth line is somewhat curved upward (see Fig. 162). 

 It might at first be thought the result of accident but 

 an examination of other glyphs (Figs. 171 and 181) 

 shows that this is not the case. Figs. 181, 186, 187, show 

 the mouth element, Fig. 185, connected with a curved 

 line, a motive derived from the life line of the serpent 

 kan. Fig. 192. This line. Fig. 187 a and Fig. 133 has 

 the phonetic value of kan, ka v-s, an v-s. Chan is the 

 evident phonetic value represented by our element. Fig. 

 162. We shajl see it repeated with like value in other 

 face glyphs j^et to be analyzed. Fig. 163 we have 

 assigned the phonetic value of uch v-s, and b}^ vowel 

 fluctuation we obtain cha (see values assigned 

 Figs. 146, 147). The element Fig. 167 = cha or 

 kha or ka. The element shown in Fig. 165 is 

 composed of the perpendicttlar line, Fig. i = ka, and 

 the twisted line. Fig. 135 and Fig. 153^ = ban v-s, ba v-s 

 an, b; its value an is here used, which, placed after 

 ka, = kan or kaan. Fig. 165 by reference to the list 

 at Fig. 45 = o or u. Fig. 166 = cha; it is a variant of 

 the ch'i glyph. Figs. 92 and 93, 114, 115, 116, 117. The 

 Fig. i68Jaas a like value, as our element Fig. 99 = chan 

 or kan. The components. Figs. 169, 170, 170*, by ref- 

 erence to Figs. 7, 8, will be seen to have the value of 

 ka or ca. All of the elements composing this glyph are 

 kan elements recalling "chu-cha-chan" or kukakan. 



Fig. 171 from the Codex Cortesianus is composed of 

 a similar series of kan elements, the three perpendicu- 

 lar black dots to the right of it repeating xo {= three) 

 or chu; so is Fig. 181 with its components, Figs. 182, 

 183, 184, 185, 187; all kan elements arranged into a 

 face glyph. 



In Fig. 172 we have another important face glyph 



which is a composite of kan elements. Curving upward 



around the muuth (= ch'i) Fig. 172'^ is the an curve, Fig. 



U, recalling chan. The element in the nose position, 



g. 173, = cha; see Figs. 109, no, in. The curved 



, Fig. 174, = cha. It is a variant of Figs. 87 and 88 



ny list and appears in many different combinations 



with this value. Some of them will be demonstrated in 

 Part II. of this article. Fig. 175 has already been used, 

 as cha v-s in Fig. 160. The same phonetic value is rep- 

 resented here. Fig. 176 is a series of Figs, i and 2 = 

 ka, and Fig. 177 are variants of Figs. 12 and 13, 14 = 

 an, giving kan. All the elements in this face ghph are 

 kan elements. Where the dotted line Fig. 172^ is pre- 

 fixed to the glyph it gives the hissing sound of x, sh, or 

 ch and the glyph becomes xan or chan. 



Fig. 178 has as one of its components a variant of the 

 kan gtyph. Fig. 172, The face or head is represented 

 in the act of sucking the nipple of a breast. Hoobnelil, 

 Fig. 179. Inside of the outline of the breast, Fig. 179, 

 is the ah prefix. Fig. 180. We have thus recalled by 

 the prefix ah, by the representation of a breast, hoobnil, 

 and by the glyph, a variant of Fig. 172, = kan, the 

 name of the bacab or chak, who represents the cardinal 

 point south, = ah-Hoobnil-kan. The glyph is taken 

 from the series. Codex Troano, Plate 25*, and proves 

 the assignment made by De Rosny to be correct. It is 

 an excellent example of the ikonomatic method of 

 writing used by the ancient Mayas, a similar method 

 being used by the ancient Mexicans, and to use the 

 words of Dr. D. G. Brinton in a letter received b}^ me 

 from that distinguished Americanist, "hence it proba- 

 bly obtained in the Maya." 



[To he continued.] 



DOUBLE SUEFACES. 



BY HENEY I. COAE, CAMBRIDGE, MASS. 



The double surface was discovered by Moebius, prob- 

 a.h\j abovit 1858, and he called attention to some of the 

 peculiarities of the surface as he constructed it, and which 

 has been called after his name, "Blattdes Moebius." Since 

 his time this surface has been studied to some extent, 

 especially by German mathematicians, and many forms of 

 the double surface have been found beside that of Moebius. 

 The most recent work on the subject is by P. Dingledey — 

 "Topologische Studien liber die aus ringfurmig geschlos- 

 seneu Bandern durch gewisse Schnitte erzeugbaren 

 Gebilde" (Leipzig, 1890). In this work Dingledey gives a 

 pretty complete bibliography of the subject. The exist- 

 ence of these surfaces is, however, little known, and it may 

 be of interest to describe the simplest form, aside from 

 any mathematical interest which may be attached to the 

 subject. 



The simplest form of a double surface may be con- 

 structed as follows: Take a strip of paper, whose edges 

 we will denote as in the figure by AB and CD, and bend 

 it imto a ring, at the same time revolving one end through 

 180", so that B will fall on C and D fall on A. Now glue 



the two ends together. AVe shall then obtain a band, 

 which has the distinctive properties that it is bounded by 

 only one edge and has only one surface. In other words, 

 we can pass from any point in the surface of the j)aper to 

 the corresponding point on the other side of the paper 

 without crossing the edge. This is the simplest form of 

 a double surface. 



If, now, we cut our band along the line marked EF iu 

 the figure, it will drop ajsart into a new band of twice the 

 length of the former band, but the new band will no 

 longer be a double surface. The reason for this is obvious. 



