956 



SCIENCE. 



[N. S. Vol. IV. No. 104. 



modern mathematics, whicli Newton never 

 touched, and which has only been made imper- 

 atively necessary by the discovery of non- 

 Euclidean geometry. As one of many, take 

 this sentence from his introduction, p. 14, "I 

 am well aware that there are other avenues of 

 approach to the thesis here maintained, — that 

 ' various new mathematical conceptions have 

 been employed by Weierstrass, G. Cantor and 

 Dedekind in establishing three independent and 

 equally cogent theories which should prove the 

 continuity of number without borrowing it from 

 space,' to say nothing of such theories {e. g., 

 Fine's Number- System) as are 'content to get 

 continuity from the line.' The criticism of 

 Fine, here quoted from Halsted's Number, 

 Discrete and Continuous, was acknowledged as 

 valid in a public meeting of the American 

 Mathematical Society, but it applies with equal 

 cogency to Lefevi'e, who here mentions it. In 

 § 41, page 48, Fine outlines an argument for the 

 continuity of the number system from the as- 

 sumption of measurement or obtaining ratio by 

 geometric congruence and the assumption that 

 ' the geometric magnitudes are continuous. ' 



Lefevre does not devote even a single section, 

 nor indeed even a single word, that I can find, 

 to any attempt at proving even any piece of a 

 number system continuous. 



Neither Fine nor Lefevre give any hint that 

 they have yet heard of the fact that the non- 

 Euclidean geometry has made it a life-or-death 

 matter for mathematics to have a continuous 

 number system not based in any way upon 

 geometric congruence, and so absolutely inde- 

 pendent of measui'ement and ratio. 



But this new problem the book under 

 consideration does not pretend to have attacked. 

 More than justified is its modest claim put forth 

 on page 112: "It seems to me something to 

 put neomonic numbers on the same footing as 

 negative numbers, or even numerical fractions." 



George Bruce Halsted. 



Austin, Texas. 



Pioneers of Science in America. With portraits. 



Edited and revised by William Jay You- 

 MANS, M. D. New York, D. Appleton & 



Co. 1896. 



This octavo volume consists of fifty biograph- 



ical sketches of early American scientists with 

 descriptions of their scientific work. Originally 

 they appeared in Appleton' s Popular Science 

 Monthly, but have since been revised, and addi- 

 tions made to them for their present form. The 

 first of these sketches is on Benjamin Franklin, 

 our first great American scientist, and is from 

 the pen of Mr. W. H. Larrabee, of the Popular 

 Science Monthly editorial staflT. The scientific 

 attainments of Franklin have never yet been 

 fully appreciated and Dr. Youmans, who edited 

 the collection, well says in his preface that the 

 present article ' is the first systematic account of 

 what Franklin did in science that has appeared.' 

 The reference on page 10 to Dr. Goode's paper, 

 read at the time of the commemoration of the 

 centennial of Franklin's death by the American 

 Philosophical Society, is suggestive of the fact 

 that for many years Dr. Goode claimed that 

 the beginnings of science in this country could 

 invariably be traced back to Franklin. 



A sketch of the two Bartrams, father and 

 son, follow, and soon after these one of 

 David Rittenhouse, who succeeded Franklin 

 in the presidency of the American Philosophical 

 Society. These, together with the sketches of 

 Benjamin S. Barton, Gerald Troost, Bobert 

 Hare and others, make conspicuous the fact 

 that early in the century Philadelphia 'pre- 

 sented more advantages in science than any 

 other place in the country.' And so, according 

 to the sketch of the elder Silliman, he went 

 there for instruction in chemistry to Hare, and 

 took lectures in botany of Barton. Silliman's 

 able pupils, including Edward Hitchcock, Den- 

 nison Olmsted and Charles U. Shepard, find a 

 place in the book. New York comes in for its 

 full share with the account of Samuel Latham 

 Mitchell, who was the Franklin of the metrop- 

 olis, so universal was his knowledge. Dr. 

 Francis said of him : "In the morning he might 

 be found composing songs for the nursery ; at 

 noon dietetically experimenting and writing on 

 fishes, or unfolding to admiration a new theory 

 on terrane formations ; and at evening address- 

 ing his fair readers on the healthful influence 

 of the alkalies and the depurative virtues of 

 whitewashing." Nor is David Hosack omitted. 

 It was he who in 1801 founded the Elgin Botani- 

 cal Garden in New York City, which property 



