In this paper I propose to describe a scheme of progressive taxation which has been introduced in Australia in the “Income-Tax Act, 1915” (Commonwealth of Australia). As a novel illustration of the use of the Integral Calculus it may be of interest to readers of the Mathematical Gazette. Incidentally I desire to point out the need for care in using the term “rate of tax” when such a progressive tax is described. In the schedules of this Act the words are used in at least two different senses. And if the curves of the second degree, or any degree, are to be referred to in Acts of Parliament—a step the wisdom of which certainly may be questioned so long as the mathematical knowledge of the average man remains what it is—the nature of the curves might with advantage be more clearly stated than is done in this Act.

Before reviews devoted to particular branches of science had come into existence, results obtained by individual research were diffused by means of letters to those who were likely to be interested. At a later date they were published in encyclopaedic journals, in literary periodicals, and in the Proceedings of the various academies. This makes it extremely difficult to obtain accounts both accurate and complete of much that has been written in this or that particular subject. To assist in removing this difficulty, and to bring within the reach of the student the results of historical research, I have suggested, for instance, an examination of the periodicals, etc., which, while not exclusively devoted to science, may nevertheless contain material of the greatest value to us in the history of science. The suggestion had the honour of receiving the approval of one eminent bibliographer, and, in his turn, he did for three rare German periodicals what I had done for two of the most celebrated Italian literary journals.

After Geometrical Similitude has been retained by making two structures, animal, machine, or steamship, from the same drawing but to a different scale, there is still the remaining variable of the material of construction to be taken into account ; and a further question arises of the Mechanical Similitude in a calculation of the relative velocity and stress in the parts.

The first statement of the problem is to be found in the Principia, Book II, Section VII, Prop. XXXII, Theorem XXVI.

In its application to the steamship the principle goes by the name of Froude's Law; it is applied to infer the performance of the full size steamer by calculation on the record of the model in the experimental tank.

In continuation of the Mathematical Gazette, p. 389, September 1914, on Linear Dynamics, the investigation is resumed here of the rolling and pitching of the body of a carriage on the springs, such as experienced in a railway carriage, tram, or motor car, and the effect can be studied as realised ; but most instructively in the motibus, where the action is felt on a more violent scale, vibrating with dynamical interest and poetry.

1. Begin by supposing the carriage body lowered down and deposited on four springs at the corners of a rectangle, sinking down vertically on the springs an equal distance c suppose, the permanent average set of the springs.

A vertical oscillation of the body will then synchronize, as on p. 393, with a simple equivalent pendulum of length c, assuming the Law of the Spring (Hooke's, the Linear Law).

Colin Maclaurin was one of the most distinguished of the brilliant coterie of Scottish mathematicians that marked the course of the eighteenth century. He was born at Kilmodan in Glendaruel, Argyleshire, in February, 1698, and was the youngest of a family of three sons. His father, John Maclaurin, who was minister of the parish and a man of mark in his day, died when Colin was but six weeks old. His mother, after residing for some time in Argyleshire, removed to Dumbarton, and died in 1707 when Colin was nine years of age. The orphans were taken in charge by their uncle, Daniel Maclaurin, minister of Kilfinnan, who faithfully discharged his duty to them.

Farming is still an important industry in England, although neglected and depressed by comparison with Manufacture and Navigation ; so it is useful still to teach the old agricultural units, mile, furlong, chain, perch, acre, rood, . . inherited from earliest civilisation through the Roman and Greek, and based on physical unchangeable qualities, nob likely to be displaced by an artificial Metric System.

Consult the lecture to the Society of Arts, December 1915, by Sir Charles M. Watson, on The Origin of the English Measures of Length, and Herodotus Ii 149 on Greek Measures.

But it is surprising no mention is made in the Report of the geographical units employed universally by the sailor of all countries, based on the measurement of the Earth, and the 24 hours of the day.

In these units there is never mention of the diameter or radius of the Earth, or of the military land-mile. The day being divided into 24 hours, and the equator also, longitude is reckoned in hours, each of 15 degrees or 60 minutes of time on the chronometer, and the degree is divided into 60 minutes of angle or 15 seconds of time. The length along the equator or a meridian of one minute of angle is then made the geographical sea-mile, le mille marin.

We turn now to the question of the generality of the solution of equation (2) obtainable by means of equation (3), that is, to the problem of the construction of integrals of the equation

In passing, we remark that it is shewn by the last example that integrals of equation (2) deduced from different functions satisfying the conditions of equation (3) may differ, so that a number of independent functions satisfying equations of the form of (3) may lead to an equivalent number of independent solutions of (2), or may lead to any smaller number other than zero.

The Annual Meeting of the Mathematical Association was held at the London Day Training College, Southampton Row, W.C., on Saturday, 9th January, 1915, at 2.30 p.m.

The President, Sir George Greenhill, F.R.S., was in the chair.

Another function of the pentagon dissection, Fig. 19, is to provide, by means of an additional cut in each pentagon, an eight-part type of dissection of a pair of quadrilaterals. This, which I will call the first type of eight-part dissection of quadrilaterals, is shown in Fig. 24. The dissection of the pair of quadrilaterals is shown in full lines, and dotted lines are added to show a pair of auxiliary pentagons, the lettering being the same as in the previous figures. A dissection of the pair of pentagons is shown, and there is an additional cut in the first pentagon joining T to the middle point X of the base AB, and an additional cut in the second pentagon joining W to the middle point Y of the base CD. The other new lines shown in the figure are merely consequential on the formation of a pair of quadrilaterals by means of these two cuts. There is no relation between the dimensions of the quadrilaterals except the equality of their areas. Each quadrilateral is divided by six lines, two or more of which are broken ; four of these lines are half sides of the other quadrilateral, and the other two are half diagonals, one of one quadrilateral and the other of the other quadrilateral.

About a year ago, to solve a puzzle to which my attention had been drawn, I found it necessary to obtain an approximate numerical solution of extreme accuracy to a set of four trigonometrical equations in four variables. The method by which I succeeded in avoiding some of the labour involved in a direct application of the classical method of approximation I have described elsewhere, but it has been suggested that an account of it will interest readers of the Mathematical Gazette. In this paper the method is introduced by means of a particular pair of equations in two variables, but it is applicable to any set of n independent equations in n variables, whatever integer n may be.