While in recent years the market has been flooded with small books of mathematical tables, each of which reproduces most of the defects of the others, there are few tables, if any, which meet all requirements when it is necessary to use logarithms for operations involving both multiplication and division. In such cases logarithms of reciprocals are constantly needed. Few tables contain these, a notable exception being the tables at the end of Carson and Smith's Algebra. On the other hand, tables of reciprocals are often given, which are rarely of use except in geometrical optics, and only confusion is occasioned by the separation of the tables for logarithms and antilogarithms, and of those for sines and cosines. Either the table of antilogarithms or that of logarithms may be banished with advantage.

If we retain the table of logarithms, it is impossible to obtain logarithms of reciprocals without performing an additional operation, which increases the work and the risk of error. Further, in the lowest parts of the scale the differences are large, and it is difficult to pick out the number which represents the logarithm correctly to four or five places of decimals as the case may be.

The past year has been one of comparative inactivity. The new Committee did not come into office until May, so that no work of importance was possible before the mid-summer vacation. Since that time, owing to the outbreak of the war, the conditions have not seemed suitable for entering upon any work of an important or controversial character. The Committee is now considering what work it can most fittingly undertake under existing circumstances.

If we seek approximations to in the form

of such a nature that the fraction gives, on performing the division, an infinite series whose first 2r+1 terms are identical with the first 2r+1 terms of the Binomial Expansion for we shall obtain 2r equations of the form

1. 1. In previous reports issued by the Association, general reforms have been suggested which apply to the teaching of Arithmetic as well as of other subjects, and some detailed suggestions have been madewhich applyexclusivelyto Arithmetic. The present report deals at greater length with Arithmetic only—its relation toother subjects, its scope and its treatment.

2. 2. There are certain parts of the subject-matter of the Arithmetic Course that are regarded by many teachers as of little educational value to the ordinary student.The Committee suggest that these should not be considered a necessary part of the Mathematical Course.This suggestion in no way interferes with a teacher who, either from personal preference or local considerations,wishes to take a more detailed course ; at the same time, it allows others to omit certain parts of a more specialised character. The Committee hope that in the course of time the parts omitted will cease to be required by examining bodies.The omissions will be discussed in detail below.

3. 3. The Committee recommend that the time set free by the elimination of a part of the usual subject-matter should be spent, partly in securing a higher standard of accuracy in straightforward computation and partly in extending the range of the ordinary school course in Mathematics.

The directions in which this course may be extended are discussed in detail in a recent report issued by the Association entitled “A General Mathematical Syllabus for Non-Specialists in Public Schools.”

There are two ways in which the solution of a particular linear differential equation may “fail” although the solulion of a more general equation obtained by replacing certain constants by parameters is complete.

where D as usual stands for d/dx.

For the general equation

the perfectly general solution is

A, B being independent arbitrary constants, but if we attempt to apply this solution to the particular equation (l), we find in the first place that the coincidence of n with l and m renders the first term infinite, and in the second place that the coincidence of m with l leaves us with only one effective constant, A + B. The method by which in the commoner textbooks the passage from the general solution to that of a particular equation is made in such cases as this is unconvincing.

When I had the honour of being made President of the Mathematical Association, I did not foresee the unusual responsibility which it entailed. It was my intention to take as the theme of a presidential address the consideration of some aspect of those special subjects to which my own researches have principally been directed. Events have forced me to abandon that intention. It is useless to discuss abstract questions in the midst of dominant practical preoccupation. We callnot disregard the present crisis in European civilization. It affects every function of life. In the harder struggle for existence which lies before the nation, all departments of national effort will be reviewed for judgment. The mere nccessity for economy in resources will provoke this reformation.

Four meetings of the Committee have been held during the year. A circular has been sent to the Senior Mathematical Master of each of the Public Schools, asking whether he desires that all boys entering his school should be acquainted with the same method for the multiplication and division of decimals, and if so, whether he is in favour of the methods given in the circular. The investigation has been completed, and the result is being sent to the General Committee.

When my previous paper on the “Dissection of Rectilineal Figures” was published (Mathematical Gazette, October, 1914, vol. vii. p. 381), I had not seen Mr. H. M. Taylor’s paper in the Messenger of Mathematics, 1905, vol. xxxv p. 81. He gives the four-part dissection of a pair of triangles, and two varieties of this; the three dissections being all of one type. He also gives the three-part and other dissections of a pair of parallelograms. He also gives a four-part dissection of a triangle and a parallelogram (equivalent, by projection, to that of a triangle and a square) which, for the case of an equilateral triangle and a square, was published in 1905 by Mr. H. E. Dudeney, This dissection, which I did not know before, is also applied by Mr. Taylor to the case of a quadrilateral and a triangle with a side equal to a diagonal of the quadrilateral. As I have already given two types of dissection of a triangle and a square, with four parts in their fundamental case, which I have called the first and second types, I will call this, which is shown in Fig. 17, the third type.

The following is a simple method of establishing the formula for the determination of the duration of day, at any latitude, for a known declination of the sun.

terminator or limit of the sun's illumination of the earth (at right angles to the line showing the direction of the sun's rays), ά is the point at which the terminator, άO'c', cuts the elevation of the line of latitude λ.

Referring to the plan, α and α are the points in plan corresponding to ά in elevation ; 0 is the plan of the polar axis.

§ 11. We are now in a position to understand how Napier completed his Tables.

We take, to begin with, his First Table. The logarithm of the first term is zero. From § 10, Theorem I., the logarithm of the second term lies between r-s and (r-s)r/s; that is, between 1.000 000 0 and 1.000 000 I. If this is taken as the A.M. of these two numbers, we have for the logarithm of the second term in that Table the value 1.000 000 05. The remaining terms in that Table form with these two a G.P., so that the logarithms have a common difference 1.000 000 05, and the logarithm of the last term can be taken as 100.000 005.

During the past two years the Public Schools Special Committee has been considering a report on the teaching of Arithmetic. One of the chief principles underlying the report may be put shortly as follows. The Arithmetic course in the past has contained a good deal that a very large nurnber of teachers regard as unprofitable, or at any rate as less profitable than other subjects that might be substituted for it. It is proposed that such parts of the Arithmetic course should not be set in the compulsory Arithmetic paper in Pass Examinations, so that those people who do not wish to teach them are not compelled to do so.

My special object is to consider how extensive the tables should be for school use, and how far pupils who have used fairly complete sets of tables at school are likely to be handicapped in examinations for which scantier tables are provided, or tables less conveniently arranged, and to what extent steps could be taken to minimise this handicap. It is to be hoped that most of the important examining bodies will before long adopt one or other of the completer sets, as it seems a pity to give the candidates needless difficulty over mere computation. I confine my remarks to 4-figure tables. And, first, as to the tables which are the most essential.