March 3, 192 1] 



NATURE 



positions discuss simple cases of the problem of 

 points when there are two or three players ; the 

 method is similar to that of Pascal. The remain- 

 ing five propositions deal with questions relating 

 to dice, after which Huygens gives five exercises 

 without demonstrations, which are left to the 

 reader. Three of these had been proposed to 

 Huygens by Pascal and Fermat. Their solution 

 afterwards occupied Hudde, De Moivre, James 

 Bernoulli, and others, and the generalisations to 

 which they- led had an important influence on the 

 development of the theory of probability. 



Several of the most valuable works of Huygens 

 were published long after they were written, 

 whereby he lost the priority of various important 

 discoveries. Thanks to van Schooten, the 

 treatise on probability was promptly issued, and 

 it remained for more than fifty years the only 

 introduction to the theory. Two English trans- 

 lations appeared, and James Bernoulli reprinted it 

 in his "Ars conjectandi." Huygens continued up 

 to 1688 to occupy himself occasionally with ques- 

 tions arising out of his treatise, and the five exer- 

 cises at the end of it. He never published any of 

 his notes, but they are now printed in the form of 

 nine appendices. The same methods are followed 

 in them as in the treatise. 



The remaining two-thirds of the volume con- 

 tain various mathematical studies from the years 

 1655 to 1666. Among these are some dealing 

 with the theory of numbers, and particularly with 

 the equation known as Pell's, a%2+i=y2^ 

 where a. is an integer which is not a square. 

 Other notes discuss problems of rectification or 

 quadrature, or examine the properties of the 

 cycloid and other curves. Many of the results 

 thus found were pubUshed by Huygens in 1673 

 in his " Horologium oscillatorium, " but without 

 proofs and without any clue to the way in which 

 they were found. The studies now printed for 

 the first time thus form a valuable supplement to 

 that work and throw much light on the methods 

 he employed to discover the results announced in 

 it. A similar case is the "rule for finding 

 logarithms " which Huygens communicated to the 

 Paris Academy in 1666 without explanation or 

 proof, and which was first found in the Archives 

 of the Academy and published by Bertrand in 

 1868. It was suggested by Bertrand that 

 Huygens must have known and used the series 

 \o^{\-\-x)—x-\y?--\-\x^- . . . We see now 

 that this was not the case, but that Huygens used 

 a method founded on an approximate quadrature 

 of the hyperbola deduced from a theorem which 

 he had published in 1651. 



Huygens also contributed to the solution of one 

 of the burning questions of the day, the drawing 

 NO. 2679, VOL. 107] 



of tangents to algebraic curves. His notes on the 

 subject are given in the present volume. He 

 found, however, when the third volume of 

 Descartes 's Letters came out in 1667, that he had 

 been anticipated. This was fully acknowledged 

 by Huygens in a paper published by the Academy 

 in 1693, ii^ which the priority of Sluse and Hudde 

 is recognised. The papers communicated by 

 Huygens to the Paris Academy, and everything 

 connected with them, are to be published in a 

 later volume of the "CEuvres completes." 



J. L. E. D. 



Four Aspects of Parenthood. 



The Control of Parenthood. By Prof. J. Arthur 

 Thomson and Others. With an introduction by 

 the Bishop of Birmingham. Edited by Dr. 

 James Marchant. Pp. xi + 203. (London and 

 New York : G. P. Putnam's Sons, 1920.) 

 75. 6d. net. 



DURING the past seven years the National 

 Birth-rate Commission has been sitting, and 

 it has published two reports, one in 1916, entitled 

 "The Declining Birth-rate: its Causes and 

 Efi'ects," and the other, called "Problems of 

 Population and Parenthood," in 1920. Smaller 

 volumes have already sprung up around these 

 large reports, and they have dealt with certain 

 aspects or phases of the great general question or 

 the falling birth-rate and all it may involve. One 

 of these smaller books is the work before us ; it 

 contains short essays on four aspects of the 

 subject — the biological, the economic, the social 

 and religious, and the Imperial and racial; there 

 is an introduction by the Bishop of Birmingham, 

 and the whole is edited by Dr. James Marchant, 

 who is the secretary of the National Birth-rate 

 Commission itself. 



The biological aspects are considered by Prof. 

 J. Arthur Thomson, of Aberdeen University, 

 whose fascinating works on natural history and 

 sex are an assurance that facts will be found 

 here clearly and attractively stated; and Prof. 

 Leonard Hill, whose research work in physiology 

 gives him every right to speak with authority 

 upon such a subject as the present. Dean Inge 

 and Mr. Harold Cox write on the economic 

 aspects; Dr. Mary Scharlieb, the Rev. F. B. 

 Meyer, and Principal A. E. Garvie repre- 

 sent the social and religious aspects ; and 

 Sir Rider Haggard, the novelist, and Marie 

 Carmichael Stopes, the doctor of science and 

 philosophy, deal with the Imperial and racial side 

 of the matter. All the birds in this little nest of 

 authors are not, however, singing in tune, and, in 



