592 



SCIENCE. 



[Vol. I., No. 21. 



sion to meet in reference to rocks of a differ- 

 ent composition a few years ago.^ M. Re- 

 nard's line of argument would prove that a dike 

 in conglomerate had the same origin as the 

 conglomerate itself, — would prove, that, when 

 sandstones and lava-flows are interbedded, both 

 have a common origin. In any volcanic dis- 

 trict we have mingled in inextricable confu- 

 sion lava-flows, ashes, scoriae, dikes, and 

 sedimentary rocks : are these all of common 

 origin because they are associated ? Is a lava- 

 flow, buried b}- the seashore sands, of hke 

 origin with the sand? In our older rocks we 

 have dikes cutting in every direction : are they 

 the same as the rocks they cut? 



The only proof regarding the origin of as- 

 sociated rocks is the relation that thej' bear 

 to one another : the mere fact of association 

 in itself is no proof. 



In another respect M. Renard's argument is 

 faulty, inasmuch as it assumes that all crj-stal- 

 line schists are of sedimentaiy origin. Erup- 

 tive are, as a rule, more subject to alteration 

 than sedimentar}' rocks ; therefore, in propor- 

 tion to their abundance, they are more com- 

 monlj^ found as metamorphic rocks than the 

 others. One of the common metamorphosed 

 characters of eruptive rocks is a schistose 

 structure, and the mere fact that a rock shows 

 such a structure affords no proof of its origin. 

 The writer has seen a well-marked schist 

 cutting in a dike directlj' across the stratifi- 

 cation of a conglomerate, — it was, of course, 

 a metamorphosed basic, eruptive rock, — and 

 be has seen numerous other examples of a 

 similar character. 



The best evidence regarding the origin of 

 the olivine rocks is in behalf of their erup- 

 tive characters, as M. Renard points out : on 

 the other side, positive evidence seems to be 

 wanting, it being rather a matter of personal 

 opinion than facts. In such cases as those 

 examined bj- Professor Bonney, and the one 

 studied b}' the present writer on Lake Supe- 

 rior, the facts and evidence in behalf of their 

 erujDtive origin are clear and explicit. So far, 

 then, as the mineralogical constitution of the 

 St. Paul's rocks go, it points rather towards 

 an eruptive than a sedimentary origin for them. 



Indeed, did it not, it is difficult to see how 

 anj' denudation could take place so far down 

 in the sea, as is here required, when, as M. 

 Renard admits, there is no evidence that any 

 sinking has occurred.^ 



The writer would therefore hold that the St. 

 Paul's rocks offer no evidence in favor of 



their being the remains of a lost Atlantis ; but 

 rather that they are of eruptive origin, like the 

 other Atlantic islands, although probably of 

 earlier date than the prevailing rocks upon the 

 latter. M.-E. Wadsworth. 



THE PASCAL HEXAGRAM. 



The Royal academy of Belgium in 1879, and 

 again in 1881, offered its prize for a solution of 

 the following (Question : "To extend as much 

 as possible the theories of the points and lines 

 of Steiner, Kirkman, Cayley, Salmon, Hesse, 

 Bauer, to the properties which are, for higher 

 plane curves and for surfaces and curves in 

 space, the analogues of the theorems of Pascal 

 and Brianehon (see, for these last, the writings 

 of MM. Cremona, P. Serret, and Folic)." 

 The only contestant for the prize in 1881 was 

 Professor Veronese of the university of Padua, 

 whose work on the subject of the original theo- 

 rems was already well known. To the paper 

 submitted by him, the Belgian academy, 

 advised hy its committee, consisting of MM. 

 Folic, Catalan, and de Tillv, declined to award 

 the prize ; and the paper has, in consequence, 

 been published in full in the Annali di mate- 

 matica (xi., Dec, 1882, 148 p.) with the report 

 of M. Folic, and a commentarj' thereon by 

 Veronese. It is a controversj' of unusual live- 

 liness for a mathematical one. Before entering 

 upon its merits, we shall give a summarj' of 

 the memoir of Professor Veronese. 



The extensions of the properties of the Pascal 

 hexagram hitherto proposed have been special, 

 and not general, and hence are incapable of 

 being carried farther. When, for instance, the 

 six perfectlj- arbitrarj- points on the conic are 

 replaced bj' six generatrices of the hyperboloid, 

 three must be taken from one s3-stem, and three 

 from the other ; and one gets, with this restric- 

 tion, only a single pair of lines, corresponding 

 to one conjugate pair of the twenty- Steiner 

 points. Cremona's extension to a cubic in 

 space, on the other hand, can be obtained b}"- 

 simple projection from the hexagram in a plane 

 conic. To develop these special, uninterest- 

 ing, easy results would not be, according to- 

 Veronese, to answer the proposed question ; 

 so, leaving them one side, he proceeds to the 

 application of a different method, — the theory 

 of substitutions. His method is, in brief, to- 

 represent the six points on a conic bj^ six 

 values of a parameter, whose permutations 

 give, from any figure whatever which they 

 represent, seven hundred and twent}- figures 

 of the same kind, or a divisor of 720. If, foi- 



