The Mathematical Work of John Wallis, D. D., F. R. S, (1616-1703) by J. F. ScottReview by: I Bernard CohenIsis, Vol. 30, No. 3 (Aug., 1939), pp. 529-532Published by: The University of Chicago Press on behalf of The History of Science SocietyStable URL: http://www.jstor.org/stable/225533 .

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REVIEWS 529

0.4 of the solar diameter is obscured; as seen from the red Sea, off Massawa, the eclipsed sun was setting, on April I7, I520, over the land of Prester JOHN, and the moon was encroaching on the left limb of the sun, just like the curved tail of a cock or like a black banner. Easter fell, in I520, on April 8; the eclipse occurred on April I7; the fleet remained off the coast of Abyssinia throughout the month of April. The eclipse occurred on a Tuesday, and a Tuesday (of the octave of Ethiopian Easter) is mentioned three pages from the interpolated passage; is a lapsus memoriae or a deliberate attempt to record a quasi-miracle on an Easter Sunday responsible for the confusion of dates ?

The Carta das novas ends with a Portuguese translation " of the letter which the Prester John sent to the King our Lord by his ambassador Matthew in the year fifteen hundred and fourteen." This letter (sent in I508, delivered in I5I4) has been known to scholars through a Latin translation which appeared in the Legatio magni Indorum imperatoris Presbyteri loannis ad Ernanuelem Lusitaniae regem... by DAMIAO DE G6Is (Antwerp, I532). One of the Notes, at the end of the volume under review, proves that the Latin version is based on the printed Portuguese version of I52I. Other early historians have used the Carta das novas, without mentioning it. Now that the text of the Carta das novas is available, historians, geographers, theologians, astronomers, etc. may add their share to the very interesting Notes which appear at the end of this stimulating volume.

A. POGO.

J. F. Scott.-The mathematical work of JOHN WALLIS, D. D., F. R. S, (I6I6-I703). With a foreword by E. N. DA C. ANRADE. XI +240 P., figures, facsimiles, portrait. London, TAYLOR and FRANCIs, I938. I2S. 6d.

Dr. ScoTT's treatise is most welcome as there is a certain strategic necessity for detailed analysis of WALLIS' work. For, to quote Dr. SCOTT, WALLIS' " career was extraordinary in many ways, and not the least of these is the light it throws upon the temper of the world in which he lived. Although it was never granted to him to enrich the world of learning with some outstanding discovery, WALLIS more than once pointed the way to what proved to be new and untraversed realms of thought." One or two examples will suffice to demonstrate this. In the Arithmetica Infinitorum (i656), WALLIS attempted to solve the problem of the quadrature of the circle by the method of Interpolation. (The word " Interpolation " is due to WALLIS.) The chief obstacle that WALLIS failed to surmount in this

problem was the disposition of the fundamental series (-2WR - a )n

which "he knew would have n + i terms so long as n was a positive

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530 ISIS, XXX, 3

integer. If n were i, as the quadrature of the circle demanded, the emergent expression would have 'more terms than one, and less than two,' and even his genius could not see how this impasse could be avoided." The considerations of this difficulty later led IsAAc NEWTON to the discovery of the Binomial Theorem. Or, consider the relation arrived at by WALLIS in the course of the same investigation, which we now call WALLIS' Formula:

7r 2.2.4.4.6.6.8.8.....

2 1.3.3.5.5.7.7.9..... and which he expressed also in the form:

4 9 25 49 8i I21 - -I X - X- X - x - X X 7T 8 24 48 8o I20

WALLIS, unsatisfied by this result, induced his friend Lord BROUNCKER (the first president of the Royal Society) to investigate the problem. The latter had no better success than WALLIS in getting an absolute value, but he did get the beautiful result

7T I

4 I I+

2 + 9

2+ 25

+ 49 2 + etc.

According to CAJORI, although continued fractions were known to the Greeks and the Hindus, BROUNCKER'S expression marks the birth of the modem theory of continued fractions. (For further details see Chapter IV of Dr. Scorr's book.) It is clear that these two important results had their genesis in WALLIS' work.

Of the eleven chapters in the book, Dr. Scorr devotes five to detailed analysis of WALLIS' mathematical works, showing in each case the relation of WALLIS' actual achievement to the work of his predecessors, his con- temporaries, and his successors. It was WALLIS who first effected a synthesis of the method of Indivisibles (as developed by ROBERVAL, FERMAT, ToRRICELLI, and CAVALIERI) and the methods of analytical geometry. His treatise on conics was the first systematic exposition of the application of Cartesian methods to the theory of conic sections as a whole. But what is perhaps of most interest is the Treatise of Algebra (1685, in English). The first section of this work is historical and although the portion of the historical section devoted to the ancients is quite remaikable for its time, the portion devoted to his own century

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REVIEWS 53I

is one of the greatest distortions in the history of the history of science. For WALLIS takes the point of view that all the major mathematics of the seventeenth century was developed by Englishmen, and that, for example, DESCARTES plagiarized from HARRiOT. Dr. ScoTT analyzes all of the major statements of WALLIS and compares them with statements of other historians, finally to judge WALLIS as falling far short of the standard he had set himself in the introduction to the Algebra. Rather than condemning WALLIS outright, Dr. ScoTT prefers to quote the condemna- tions of other historians such as CANTOR and MONTUCLA, showing, however, that WALLIS' prejudice was directed chiefly against the French mathematicians who, in WALLIS' opinion, had paid but scant attention to British mathematics. According to Dr. ScoTT it was this, coupled with WALLIS' great fear of plagiarism that caused the distortion. (In support of this opinion he quotes the following excerpt from one of WALLIS' letters " I could wish that those of our Nation were a little more forward... in timely publishing their own discoveries and not let strangers reape the glory of what those amongst ourselves are the authors. ") What is more likely to be the case is that WALLIS was embittered against the French mathematicians as a result of the long controversies in which he had been engaged with FERMAT and others of the French school. Dr. ScoTT is a little naive in his approach to the seventeenth century, essentially a century marked by violent controversies. It may, for exam- ple, seem strange to us to find WALLIS and HOBBES engaged in vitriolic controversy over mathematical questions, as a result of their disagreement on theological questions, but such action would certainly not have seemed strange to anyone living at the time.

In conclusion, it must be said that Dr. SCoTT takes great pains to make clear WALLIS' remarkable contributions to the development of modern mathematical nomenclature and notation. It is to WALLIS that we owe the symbol oo for " infinity ", the symbol < for " less than or equal to," the name " hypergeometric series " for the series whose general term is I.3.5 . .-(2n +I) 2 3 .4.6. (2n +

i, etc. This is made especially plain by the set of 2 .4.6 ..... 2n facsimiles of pages from the works of WALLIs and those of his contempor. aries, which forms the first appendix to the book. Another appendix con- sists of 37 pages of paragraph biographies of most of the persons mentioned in the text, "except BOYLE, DESCARTES, FERMAT, GALILEI, LEIBNIZ, NEWTON and PASCAL " who " are sufficiently familiar to the reader as to render their inclusion in the following pages unnecessary." It is strange indeed that Dr. Scorr should believe that such noteworthies as HUYGENS HALLEY and HOBBES are not so " sufficiently familiar." The book is indexed and contains a list of the papers published by WALLIS in the

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532 ISIS, Xxx, 3

Transactions. It is to be regretted that instead of giving a similar list of the mathematical treatises, Dr. Scorr lists only the table of contents of WALLIS' Opera (I693-99). This omission makes incomplete what is in all other respects a complete, accurate, and interesting study.

Harvard University. I BEINARD COHEN.

Hester Hastings.-Man and beast in French thought of the eighteenth century. Inaugural Dissertation, Johns Hopkins University. Re- printed from Johns Hopkins Studies in Romance Literature and Languages, Vol. 27, 297 P. 1936. Price $ 1.25.

This dissertation completes a trilogy on the growth of humanitarian feeling toward animals in the period of its formulation. The other two are HARWOOD'S (1928) Love for Animals and How it Developed in Great Britain and BoAs's (I933) The Happy Beast in French Thought in the Seventeenth Century (Isis 22, 240). The movement culminated in the legislation of the nineteenth century against cruelty to animals and the revulsion against vivisection. Other factors not envisaged in any of these treatises, which were based largely on belles lettres and philosophical discussions, were the rise in interest in animal industry, improvement in breeds, increase of interest in pet animals, especially dogs and cats and the diversification of their breeds, resulting in part from changes in hunting methods and from importations in connection with exploration and increase in travel. BUFFON's Histoire Naturelle both reflected this growth of popular interest in animals and contributed to it. The extended account of domesticated animals in BUFFON'S work and in agricultural encyclopedias which preceded it such as those of MARKHAM and of LIGER reflect this widespread and not unintelligent interest in domestic animals and their proper care. DAUBENTON'S Advice to shepherds is a specialized case of such interest. BUFFON, however, did not sentimenta- lize over any animals and was rather obtuse to aesthetic qualities. His interest lay rather in their activities.

The place of beasts in man's thinking has been interwoven with phil- osophic and religious thought about man himself. The question of the brute's soul and its immortality impinges directly on those of man. Thus BoNNEr (1779-1783) expressed a belief in a future life for the beasts.

The relative positions of beast and man in the scale of perfection were involved in the concepts of the " noble savage " and perforce " the happy beast " of those who oppressed by the miseries of human civilization and ignorant of nature red in tooth and claw, concluded that humanity was not only miserable but degenerate and degraded. On the other hand a more objective and observant group, supported by the tenets

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