August Ferdinand Möbius was born at Schulpforta, in Saxony, in the year 1790. He studied in the Universities of Leipsic and Göttingen, and, at the age of 2G, was appointed extraordinary Professor of Astronomy and Superintendent of the Observatory at Leipsic. There he remained till his death, in 1868, being appointed ordinary professor in 1844. Between the years 1817 and 1868 Möbius wrote his Barycentric Calculus, a Treatise on Statics, another on the Mechanics of the Heavens, and a large number of papers on Mathematical, Dynamical, and Astronomical questions. Most of these papers were contributed to Crelle's Journal, which was founded in 1826. The works of Mobius have recently been collected under the direction of the Royal Scientific Society of Leipsic, and under the editorship of Klein, Scheibner, and Baltzer.

In Bonnycastle's Arithmetic a rule is given for finding any root of a number by approximation which in substance reduces to this statement:—If a and b are nearly equal, then the nth root of a/b is nearly equal to

According to Luther's translation 1 Kings vi. 31 should read— “At the entrance of the choir he made two doors with pentagonal door-posts.” This is probably a wrong translation, for nowhere on Asiatic monuments of this time has a pentagon been found. Prof. A. Merx in Cantor's Vorlesungen über Geschichte der Mathemalik, Vol. i., p. 91.

It is probable that the construction of the regular pentagon is due to the Pythagorean school as a consequence of I. 47. (Ibid, p. 151.)

§1. Since the former paper on this subject was read, Prof. Cantor has published the second volume of his history of Mathematics. This has necessitated various additions to the paper, which can perhaps be best given as an appendix.

On page 413 Prof. Cantor says that the construction of Dürer's pentagon is found in a book called Geometria deutsch, which was lately discovered in the town library at Nürnberg, and gives 1487 as the upper limit to its date. The construction is said to be “mitunverrücktem Zirckel,” the same expression that Schwenter applies to Dürer's solution.

Taking seven figure logarithms it is required to find at what interval from n, a number of five figures, the next entry need be made, in order that any intermediate logarithm may be calculated by the method of proportional parts.

We have

It is sometimes the case that geometrical theorems, which are usually enunciated as properties of the triangle or the quadrilateral, may be stated more succinctly, and in a form that better suggests generalisation, as properties of the complete quadrilateral. Thus the theorem that is usually stated in the following form, “The three straight lines that join the middle points of opposite sides and the middle points of diagonals of a quadrilateral are concurrent and bisect one another at the point of concurrence,” may be enunciated more simply as a property of the straight lines that join the middle points of the three pairs of opposite sides of a complete four-point; and the latter form of enunciation suggests how the theorem may be generalised by projection. So too the figure that consists of four Points, any one of which is the orthocentre of the triangle formed by the other three, is better described as a complete four-point in which opposite sides intersect at right angles, (which may be called an orthic four-point).

In his Theory of Heat, Clerk-Maxwell gives a very elegant and simple geometrical proof of the four thennodynamical relations, and points out that his construction shows that the truth of any one is a necessary consequence of the truth of any other. The usual analytical proof of these relations can be made as simple as Maxwell's geometrical proof, and the fact that any one is a necessary consequence of any other becomes evident when it is considered that they are all deduced by a common process from identical transformations of one equation. The following method of proof seems to me to bring out more directly their necessary interdependence.

In my laboratory we make great use of the method of tracing lines of force described in Glazebrook and Shaw's Practical Physics. The field is modified in various ways; for instance, the fixed magnets are sometimes placed so as to somewhat resemble in their disposition the field magnets of a dynamo: sometimes a circular piece of soft iron is placed in the field, and the effect of its induced magnetization examined. The students like these exercises, and the results (of which some are exhibited) are very beautiful. We use steel bars about 8 cm. long for fixed magnets, and small rather heavy lozenge-shaped needles, about one to two cm. long, as what I shall call pointers.

What is meant will best be understood by the following:-

1. Suppose we have an unlimited straight line XX′ (fig.24) with a finite point O on it. Take two points on the line, C and C1 in the same sense with respect to O such that OC·OC1=a2. Then C and C1 are connected by inversion, and O is the centre inversion. Without loss of generality for what follows we may, for convenience, take a2=12=1, so that OC·OC1=1.

The aim of the scientific teacher is to teach the pupil how to think along scientific lines. By a suitable presentation of the facts of experience he should lead the mind of the learner to form almost intuitively the scientific law or generalisation which embraces them all. We may of course start with the law or formula, and develope it mathematically into all its ramifications. But that reduces itself to mere analytical skill. If carried out faithfully in the elementary teaching of science, it would tend to give the learner an erroneous conception of the whole method of scientific investigation and the meaning of scientific law.

In the second part of Professor Chrystal's Algebra (Exer. V., No. 4) the following exercise is given:-

If 2xyz-x2-y2-z2+1=0, and x,y,z are all real, then all, or none, of the quantities x,y,z lie between-1 and + 1.

§ 1. It is proposed to discuss in this paper partial differential equations involving two independent variables x and y, and a dependent variable z. The method of reduction which is explained can be applied to certain equations involving more than two independent variables, but such application is subject to too many restrictions to be of much general utility.

§ I. On an infinite series of Triad Circles derived from the inscribed circle. Determination of a direct relation between r and the three radii of the nth triad.

Each of the first triad touches two sides of ABC and the inscribed circle : generally, each circle of the mth triad touches two sides of ABC and also touches one of the circles of the (m – 1)th triad.

1. If θ be the eccentric angle of a point P on an ellipse whose semi-axes are a and b and if x, y be the rectangular co-ordinates of a point Q, then PQ2 = (x − acosθ)2 + (y − bsinθ)2. When a = b and x2 + y2 = f2, PQ2 = f2 2afcosθ + a2, if the line from which θ is measured passes through Q. The integral

is a well-known one. I propose in the present paper to consider the more general form which the integral takes when P lies on an ellipse.

In 1746, when he was a candidate for the Chair of Mathematics in the University of Edinburgh rendered vacant by the death of Maclaurin, Matthew Stewart published his first work, Some General Theorems of considerable use in the higher parts of Mathematics. In the preface to it he states that “the theorems contained in the following sheets . … are entirely new, save one or two at most,” but he does not specify the two. They are

The property is that if any quadrilateral, ABCD, skew or otherwise, have its sides AB, DC divided in E, F so that

AE : EB = DF : FC = AD : BO

then the direction of the line EF shall bisect the angle between the directions of BC, AD.

This extension of Euclid's VI. 3 follows immediately from the proposition that if AE : EB = DF : FC, then all such lines as EF are parallel to one plane, namely, the plane parallel to BC, AD; and that they each cut similar lines drawn with reference to BC, AD.