botany and zoology, to geography and geology, even to logic, metaphysics, 

 and theology, valuable services, some of which must be held in everlasting 

 remembrance. No man would select from among the successors of Descartes, 

 Leibnitz, and Newton, twenty names of those who had shown the greatest 

 genius in pure mathematics, down to the year 1875, without including Pcirce. 

 Even the reader who knows nothing of pure mathematics must admire the 

 wonderful genius and the sublime self-knowledge of the man, who, when all 

 the scientific world was rapt in admiration of Leverrier, the creator of in- 

 visible astronomy, who had said to Galle, " Point your telescope to such a 

 spot, and you shall see a planet never yet beheld by mortal eye, but revealed 

 to me by the eye of faith guided by mathesis," calmly said, "Leverrier de- 

 serves all praise as a mathematician, but Galle's discovery is only a happy 

 accident : Leverrier's planet does not exist, and the planet seen by Galle is 

 an entirely different body." Edward Everett, then president of the Academy, 

 asked Peirce to withhold his remark from publication, saying that no words 

 could express the improbability of his statement. "But," replied Peirce, 

 "it is still more improbable that there can be an error in my calculations." 

 Time has long since demonstrated that our American geometer was right. 



A few weeks after this great mathematical triumph I met, in State Street, 

 Boston, the historian Jared Sparks, and he remarked to me that he con- 

 sidered Leverrier's calculation, and Galle's discovery, among the most im- 

 portant events in all recorded history. The effect, said he, upon the general 

 human mind, will be enormous, in the confidence which it will produce, the 

 impulse which it will give to every department of science. Wonderfully has 

 this prediction of President Sparks been fulfilled ! 



A yet more remarkable prediction by Peirce still remains unfulfilled, and 

 ages may pass before even its partial accomplishment. About ten years ago 

 some papers of his were published by the generosity of a few of his friends 

 and pupils. They contained an investigation of sixty or seventy kinds of 

 mathematical language, that is. of sixty or seventy kinds of algebra, a dozen 

 or more of which were very simple. All these kinds were discovered by 

 him in his er.f eavor to answer the question, What conditions must be ful- 

 filled by any algebra? In solving this question he confined himself under 

 some restrictions, so as to narrow the field, and even then found the multi- 

 tude of algebras, that is, of mathematical languages, which I have mentioned. 

 Of these, only three had ever been used by mathematicians ; those three had 

 given employment to men of genius for centuries ; those three had led to all 

 the marvellous triumphs of the science of this nineteenth century ; the others 



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