1.1. Review by: D H Lehmer.
Science, New Series 45 (1153) (1917), 115-116.

Dr Stager's tables are intended to furnish the possible number of Sylow subgroups for all groups whose order does not exceed 12,000. For every number within that limit are listed all the divisors which are of the form $p(kp + 1)$, where $p$ is a prime greater than 2 and $k$ is greater than zero. In determining the possible number of Sylow subgroups such divisors must be known before further methods are applicable. Thus from the table we learn that 1,080 is divisible by $3(1 \times 3 + 1), 3(3 \times 3 + 1), 3(13 \times 3 + 1), 5(1 \times 5 + 1), 5(7 \times 5 + 1$ and $5(43 \times 5 + 1)$. From these results we know that for a group of order 1,080 there may be 1, 4, 10 or 40 subgroups of order $3^{3}$ and 1, 6, 36 or 286 subgroups of order 5. The exact number is to be determined by other principles of group theory. The table also gives the expression of each number up to that limit as products of powers of primes.

The making of tables, a tedious and apparently mechanical task, is of the highest importance in all branches of science. It is likely that more fundamental theorems have been discovered by the examination of listed results than by any other means. This is certainly true in the theory of numbers, and it is possible that workers in the theory of groups have not made enough use of this method of investigation. The construction of tables for the theory of groups is especially difficult on account of the great complexity of the material. Only brief tables have hitherto been undertaken and it is to be hoped that Dr Stager's work in this direction may be the beginning of a systematic campaign in this important field.

The construction of an extensive table almost always brings to light hidden relations, suggesting new theorems for investigation. In Dr Stager's tables certain numbers are noted which have no divisors of the sort indicated above. Such numbers seem to resemble primes in many ways, and in particular their "curve of frequency" seems to run roughly parallel to the corresponding curve for primes. Dr Stager has made a study of these numbers, and has added a list of them up to the limit of his table.

The author is to be congratulated upon the completion of so important and formidable a piece of work. While the reviewer has, of course, not checked over any part of the table he has the utmost confidence in the accuracy of the list. The printing has been done by the photographic methods employed by the Carnegie Institution in the publication of the Factor Tables and the List of Primes. Both the author and the publishers deserve the gratitude of every lover of science in putting in the hands of mathematicians results of such permanent value.

1.2. Review by: Anon.
Nature 99 (26 April 1917), 164.

The main object of this publication is to answer the question: Given $n$, the order of a group, what are the possible orders of such Sylow subgroups as it contains? This amounts to finding all divisors of $n$ which are of the form $p(kp + 1)$, where $p$ is prime. For each $n$ up to 11,999 the table gives the complete resolution of $n$ into its prime factors, and the values of $k$ (other than 0 and 2, which do not require entering) corresponding to each prime factor. Each prime value of $n$ is entered in the body of the table in the form $p_{t}$; for instance, the entry $p_{627}$ under 4639 shows that the latter is the 627th prime in order of magnitude, taking $p_{1} = 1$. It is obvious that, apart from its special purpose, this table will be very useful to arithmeticians; every reasonable precaution seems to have been taken to make it accurate, and fortunately the table is of such a kind that every single entry can be tested with very little trouble, and any misprint almost certainly detected, unless a number $n$ has been entered as prime, when really composite. Cases where $p(kp + 1) = n$, and not merely a divisor of $n$, are noted, such numbers are called Ps by the compiler - for instance, $1074 = 3(3 \times 119 + 1)$, so 1074 is a P. On pages xi and xii is a list of these numbers (1 - 12,229) in their natural order; and there are interesting tables and graphs on the distribution of P numbers and primes. Supposing that $\phi(n)$ means the number of primes not exceeding $n$, and $\psi(n)$ the number of P numbers not exceeding $n$, the tables suggest that when $n \rightarrow ∞$ the ratio $\psi(n)/ \phi(n)$ converges to a definite limit not very different from $e$; of course this is a mere guess that might occur to anyone, but at any rate to find a formula for $\psi(n)$ analogous to Riemann's for $\phi(n)$ would be an interesting problem. It may not be superfluous to add that the table does not profess to enumerate actually existent Sylow subgroups for different values of $n$.

The following is published in The American Mathematical Monthly 21 (1) (1914), 36.

In the spirit of the editorial statement on the first page of this issue, the action recently taken by the mathematics teachers of California, as indicated in the following communication, is most significant. The editors of the Monthly appreciate the compliment and recognise the responsibility which this action implies. We congratulate the teachers of California upon being, pioneers in taking such a bold, forward step.

The communication, under date of 2 January 1914, is from Professor Henry W Stager, of Fresno Junior College, chairman of the mathematics section of the California High School Teachers' Association, and is addressed to the teachers of mathematics in the secondary schools of California. It reads as follows:

At the summer meeting of the Mathematics Section of the California High School Teachers' Association, a Committee, composed of Professor D N Lehmer, Miss S Gilmore, and Professor G A Miller, was named to consider ways and means of broadening the scholarship of secondary school teachers. The Committee recommended the reading of the book, 'Mathematical Monographs', by J W A Young in collaboration with other mathematicians, and further stated: "In connection with this book, or even instead of it, the reading of some elementary but strictly first class journal is also recommended. Where it is possible for a number of teachers to meet frequently for the discussion of problems and short articles this plan has much to commend it. For this purpose the Committee would recommend the 'American Mathematical Monthly'."

The report has been adopted unanimously. The movement for a higher standard of efficiency is one of the strongest attempts to increase the quality of the teaching of mathematics ever made in California. It began with the teachers themselves and only needs your cooperation to make it a success. I commend the report of the Committee to your most earnest consideration.

At this time I wish especially to call your attention to the reading of a first class journal. I have gone over with care the first volume of 'The American Mathematical Monthly' under the new organisation and find it well fitted for this purpose. I am personally acquainted with many members of the Publication Committee and with all the members of the Editorial Committee. The men are in the forefront of the present movement toward a higher standard for the teaching of mathematics in the United States. The 'Monthly' represents the best in its field. I commend it to you, feeling sure that it will prove of very great benefit to you, will help to increase your efficiency, and will be a continual source of inspiration to you.

Yours, for the better teaching, of mathematics,

Henry W Stager.

The Editors of the American Mathematical Monthly write:

The Monthly is pleased to publish this report in full of a secondary association meeting for the following reasons: (1) Because of the advanced type of papers presented; (2) because of the exhibition of serious and thoughtful cooperation between the representatives of the secondary schools and the higher institutions of a great state, with the sole purpose of creating and maintaining high standards in the teaching of mathematics; (3) because of the explicit recognition by a state university of autonomy in the secondary schools with respect to "correlated" or "tandem" teaching of algebra and geometry; and (4) because of the example set by this association with respect to its "reading course" for teachers.

Henry W Stager's Report.

The annual meeting of the Mathematics Section of the California High School Teachers' Association was held at the University of California in the summer of 1915. The attendance was the largest in the history of the section. At the first session, the principal address was given by Professor C J Keyser of Columbia University on the topic: "The Human Worth of Rigorous Thinking. "The keynote of this very inspiring address may be given in the short quotation: "What I wish you to see here is that science, and especially mathematics, the ideal form of science, are creations of intellect in its quest for harmony. It is as such creations that they are to be judged and their human worth appraised." "Certain Aspects of Engineering Mathematics" was the title of the second address by Professor Baldwin M Woods, of the University of California. This paper presented some of the difficulties necessarily attendant upon the efforts of the teacher, a person of supposedly mental type, in instructing the future engineer, who is to be a man of motive type, with particular application to that period of discouragement in the course, when the student has gained a more or less mechanical grasp of the subject and when, on account of the difficulties in attempting to apply his knowledge, he feels he is making no progress. The discussion brought out by this paper was a valuable contribution to the programme.

The second session was devoted entirely to the report of the Committee on entrance requirements, of which Professor Henry W Stager, of Fresno Junior College, was chairman. The report was followed by an exceedingly lively discussion, in which every person present took some part, among those present being several members of the mathematical faculty of the University of California. The essential arrangement was heartily concurred in by all, the main objections being only to minor points. At the close of the discussion, the report was adopted with only one dissenting voice, and the Section recommended the adoption of the requirements as -outlined by all the universities and colleges in the state. The committee was continued for the purpose of promoting the adoption of the report by the proper authorities. The plan in its entirety follows:

I. Elementary Mathematics, 2 credits (two full years' work).

The fundamental operations of algebra, including the laws of exponents for positive and negative integers, synthetic division, the various methods of factoring, with applications, simultaneous equations of the first degree with problems involving their solution, simple quadratic equations (solution by factoring especially), linear functions. An important aim in this requirement should be to acquaint the pupil with the notion of functionality, mainly through the early and continuous use of graphical methods.

The usual theorems and constructions of elementary plane geometry, with the simple trigonometric ratios and their applications in connection with the treatment of similar triangles, the different methods for determining π. The solution of original exercises, including problems in loci and applications to mensuration, should be emphasised.

The topics of this requirement may be given in succession, or in a correlated course, at the option of the teacher.

II. Advanced Mathematics, Part 1, 1 credit (one full year's work).

Supplementary studies in plane geometry, including topics in modern geometry, and the fundamental propositions of solid and spherical geometry. The ability to apply geometry to practical problems is important in this requirement.

The development of the general formulae of plane trigonometry, with applications to the solution of triangles and the measurement of heights and distances; also the fundamental formula of spherical trigonometry, with applications.' Practise in computation with logarithmic tables.

III. Advanced Mathematics, Part 2, 1 credit (one full year's work).

Determinants of the second and third order; synthetic division; remainder and factor theorems; quadratic equations; both single and simultaneous (both, graphical and algebraic treatment); theory of exponents; radicals; ratio and proportion; variation; arithmetic and geometric progressions; elementary theory of logarithms, with practise in computation; complex quantities; theory of quadratic equations; the binomial theorem for positive integral exponents.

The fundamental methods of plane analytic geometry. The straight line and circle, and the simpler properties of the conic sections. Problems in loci. The graphical solution of equations.

The topics of this requirement may be given in succession, or in a correlated course, at the option of the teacher.

This requirement should be given in the fourth year.

The Committee on Progress, consisting of M A Plumb, of the California School of Mechanical Arts, San Francisco, Miss S L Gilmore, of Antioch, R E McCormick, of Bakersfield, G E Mercer, of Palo Alto, and Professor H W Stager, of Fresno, presented the revised Reading Course for 1915-1916, with the preface that the purpose of the course was to arouse a greater interest in mathematics rather than to increase the teacher's knowledge of mere mechanical methods of presentation. The course follows:

A. Books for careful reading and study.

I. Books of easy grade.
1. Cajori: A History of Mathematics.
2. Smith-Karpinski: The Hindu-Arabic Numerals.
3. Whitehead: An Introduction to Mathematics.
4. Lodge: Easy Mathematics.

II. Books of medium grade.
5. Ball: Mathematical Recreations and Essays.
6. Beman and Smith: Klein's Famous Problems in Elementary Geometry.
7. Young: Fundamental Concepts of Algebra and Geometry.
8. Fine: The Number-System of Algebra.

III. Books of more advanced grade.
9. Young: Monographs on Modern Mathematics.
10. Manning: Geometry of Four Dimensions.

B. Books for reference.

1. Moritz: Memorabilia Mathematica.
2. Poincaré: Science and Method. Translated by Francis Maitland.

C. Journals.

1. The Mathematics Teacher, Syracuse, New York.

Especially commended to those teachers who teach only elementary algebra and plane geometry, or who are not specialists in mathematics.

2. The American Mathematical Monthly, Chicago, Illinois.

Especially commended to teachers of trigonometry and advanced algebra and to all teachers of mathematics in colleges.

The session adjourned with the election of Principal A G Grant of the Ferndale High School as Chairman of the Section for the ensuing year.