of life insurance, and the recent British National Insurance Act, make more generally interesting the monograph before us.

The question is raised as to whether the insurance companies are going to be viewed sometime much as the express companies are to-day. When the ratio of management expenses (exclusive of taxes) to premium income for our American companies varies from 8.7 to 28.4, even to 227.6 for a state company, with an average of 22.4; when the company for which this ratio is below 20 is thought to be quite economically managed, one wonders whether the cost of insurance is not much too high. If the government had the monopoly of life insurance, as in some countries, would it not benefit our people as a parcels post does other people? If the company that has cut its cost of management to half of that of most of the best of other companies could flourish since 1759, and if the cost in Germany is still less, why should not our millions of people profit by what national control could save? The similarities between express and insurance companies cannot be considered here, neither can the causes of the great variation in cost of management.

The author of the study, in an introduction of six pages, states the object and aim of life insurance, calls attention to the possible failure to maintain it, and asks the question which has embarrassed many an agent: "If I fall ill and am no longer able to earn anything, who will pay my premiums?" This leads to the matter in hand, l'assurance complémentaire, which provides against the lapsing of a policy on account of temporary or permanent incapacity for labor due to accident or to disease. The premiums are guaranteed and the amount thus paid beforehand is deducted from the amount paid to the benefactor at the time of death of the insured. The history of this sort of insurance is briefly told. Reference is made to the fact that German and American companies sell such a contract. The regular life companies of France are warned by the Controle Central des Compagnies au Ministère du Travail not to enter into such contracts. The author states that the aim of the study is to show that this risk is as rightfully assurable as the risk of death. The companies have made a serious and rational study only as to equitable charges for the new combination. Though definite statistical data are lacking, the data furnished by other branches of insurance may be temporarily used to outline a rudimentary theory.

At present the companies are simply feeling their way to a reasonable tariff.

I have found about four American companies that have some form of select life policy, or clause in a regular policy, covering this field. It may be of considerable interest to many readers to see a sample clause of this sort:

"Waiver of Premiums.-The company, by endorsement hereon, will waive payment of the premiums thereafter becoming due, if the insured, before attaining the age of sixty years and after paying at least one full annual premium and before default in the payment of any subsequent premium, shall furnish proof satisfactory to the company that he has become wholly and permanently disabled by bodily injury or by disease so that he is and will be permanently, continuously and wholly prevented thereby from performing any work for compensation or profit, or from following any gainful occupation. Any premiums so waived shall not be deducted from the sum payable under the policy. Provided that, notwithstanding proof of disability may have been accepted by the company as satisfactory, the insured shall at any time, on demand, furnish the company satisfactory proof of the continuance of such disability; and if the insured shall fail to furnish such proof, or if it shall appear to the company that the insured is able to perform any work or to follow an occupation whatsoever for compensation, gain or profit, all premiums thereafter falling due must be paid in conformity with this contract.

"Without prejudice to any other cause of disability, the entire and irrecoverable loss of the sight of both eyes, or the severance of both hands above the wrists, or both feet above the ankles, or of one entire hand and one entire foot will be considered as total and permanent disability within the meaning of this provision.'

The clear, concise statement characteristic of the whole treatment does much toward holding the reader's interest through the five chapters of the book, as the author treats minutely of the conditions of the complementary insurance and even as he calculates tables of morbidity, of healthy and invalid persons, of value of an indemnity for one and for two insured, of premiums for numerous particular forms of insurance, etc. Though with insufficient statistics the author seems to have presented a rational study of this interesting and very

little known form of insurance, pointing out, it would seem, every exigency that could possibly arise.

CHARLES C. GROVE.

Konstructionen und Approximationen. Von THEODOR Vahlen. Teubner, Leipzig und Berlin, 1911. xii + 349 pp.

ONE who expects to find in this book-Band XXXIII of the Teubner Sammlung a more or less complete list of constructions and approximations with a strong flavor of applied mathematics will be disappointed, as was the reviewer. According to the author it is intended to help bridge the gap which exists between the mathematics of the German gymnasium and university. The latter does not begin its work where the former ends. Various books, notably those by Klein and Enriques, have been published recently which might be studied by those intending to follow the lectures on higher mathematics. The book under review aims to furnish such preparation by having the student actually come in contact with some concrete facts in mathematics and to know these so well that later when the professor during his lecture has him soaring more or less he may still have a point or two of contact with the earth below.

The class of books having in view preparation for the university is decidedly different for Germany than for the United States. To illustrate this we might mention that the first 75 pages of the book under review are devoted to having the student obtain definite notions concerning the fundamental principles of projective geometry. Special emphasis is placed in all of its phases, both algebraic and geometric, on the interpretation of the cross-ratio. Good drill work, all of it. Of course, it couldn't be included in the lectures given later-that would seem too much like teaching.

Throughout the book the various aspects of the solutions of the three famous problems of antiquity are presented and many references to the literature on the subject given. Interesting metric cubic constructions in which algebra and geometry are closely correlated are cited. Approximate solutions of cubics and biquadratics are obtained geometrically and the limited range of constructions possible with ruler and compass pointed out. In this connection are included several solutions, ancient and modern, of the duplication of the cube and tri

section of an angle and various pieces of apparatus designed to solve the same problems are described and their theory discussed. Later some interesting constructive approximations are given.

The properties of various transcendental curves are used to obtain approximately an nth root and to divide any angle into n equal parts. The division of the circle and of the arc of the lemniscate into n equal parts for special values of n with the aid of ruler and compass alone is discussed.

A development of attempts to arrive at the value of from the time of Ahmes to that of Lindemann is presented. This leads naturally to mechanical quadrature and rectification.

Under the heading of analytic approximations are included such titles as Taylor's series, Lagrange's interpolation formula, exponential series with application to the quadrature of the hyperbola, De Moivre's theorem, indeterminate forms, and the determination of by the use of series.

π

The discussion of the irrationality of and 2 brings out the methods used by famous mathematicians of old. The book closes with the proof of the transcendental nature of e and . There is much concrete work in algebra and geometry throughout the book, consequently a chance for errors, many of which have been listed in an appendix of two pages.

Student's mathematical clubs in our universities desiring some interesting material for the rounding out of a course in mathematics would find the volume rich in suggestions. ERNEST W. PONZER.

NOTES.

THE sixth regular meeting of the Southwestern Section of the American Mathematical Society will be held at the University of Kansas on Saturday, November 30. Titles and abstracts of papers to be presented at this meeting should be in the hands of the chairman of the programme committee, Professor J. N. Van der Vries, University of Kansas, by November 8.

THE annual meeting of the Society will be held this year at Cleveland, Ohio, in affiliation with the American association

for the advancement of science, on Tuesday, Wednesday, and Thursday, December 31-January 2. The winter meeting of the Chicago Section will be merged with the annual meeting. Titles and abstracts of papers should be sent to the Secretary of the Society, 501 West 116th Street, New York, on or before December 10.

THE twentieth summer meeting of the Society will be held at the University of Wisconsin early in September, 1913. At the seventh colloquium of the Society, held in connection with this meeting, courses of lectures will be delivered as follows: By Professor W. F. OSGOOD: "Selected topics in the theory of analytic functions of several complex variables." By Professor L. E. DICKSON: "Certain aspects of a general theory of invariants, with special consideration of algebraic and modular invariants."

THE concluding (October) number of volume 34 of the American Journal of Mathematics contains: "Simple groups from order. 2001 to order 3640," by L. P. SICELOFF; "On certain orthogonal systems of lines and the problem of determining surfaces referred to them," by A. E. YoUNG; " Wallace's theorem concerning plane polygons of the same area," by W. H. JACKSON; "On harmonic functions," by R. A. HARRIS; "Curves on quintic scrolls," by F. B. WILLIAMS; "On the structure of forms, and the algebraic theory of n-lines," by O. E. GLENN.

THE opening (September) number of volume 14 of the Annals of Mathematics contains the following papers: "Threedimensional chains and the associated collineations in space," by HAZEL H. MACGREGOR; "Determination of the constants in Euler's problem concerning the minimum area between a curve and its evolute," by E. J. MILES; "Theorems on reducible quantics," by O. E. GLENN; "A determinant formula for the number of ways of coloring a map," by G. D. BIRKHOFF.

AT the meeting of the Edinburgh Mathematical Society on June 14 the following papers were read: By Professor H. S. CARSLAW, "A problem in the linear flow of heat discussed from the point of view of the theory of integral equations"; by D. G. TAYLOR, "Linear substitutions and their invariants"; by WM. BRASH, "Two general results in the differential calculus."