Although some years have elapsed since the first appearance of Professor Halsted’s translations of Lobatchewsky’s Theory of Parallels and John Bolyai’s Science Absolute of Space, they are not so widely known and studied as they deserve to be. There is nothing in the mathematics of these small volumes of a kind to deter any one from reading them; they are short and elementary treatises, disclosing a system of geometry quite distinct from Euclid, by rejecting the axiom concerning parallels. The Science Absolute of Space is a most interesting exposition of one branch of non-Euclidean geometry; but it is unfortunately extremely condensed, and occasionally obscure. It would have been a great advantage if explanatory notes had been added to the English translation.

Consideration of the special case in which the centre of the eccentric circle lies on the straight line whose points of intersection with the conic are required.

In what follows it will be convenient to abandon the system of lettering adopted by Boscovich which we have hitherto followed, and as far as possible to use that common to modern text books.

The claims of Herbart on the attention of the teacher are at last being recognised in this country. Not merely is he deserving of study as one of the founders of modern psychology, not merely does he attract our interest as the central object of veneration of a rapidly increasing school in his native country and in the United States, but he also takes up a position apart from most philosophers, in that he actually devoted a large part of his life to the work of teaching, and deliberately applied his psychology to education. To the readers of this Gazette, the opinions of Herbart, who offers us this rare combination of the theorist of the first rank with the successful teacher of boys and men, will be of additional interest when it is realised that he plays somewhat the same rôle as the staunch advocate of Mathematics in the curriculum that Herbert Spencer does for Natural Science. He urges that “for six hours weekly” Mathematics should be “the beginning, middle, and end” of every system of instruction. And the fact that Mathematik is of wider connotation than we should now award to it will not deprive his remarks of much of their value. It will be as well, however, to bear in mind that he includes in this term all that is not comprised in Geschichte.

As I can pretend to only limited practical experience in the matter of teaching the foundations of Mathematics, and of Geometry in particular, I have thought that the time allowed for my remarks may be advantageously occupied in a rapid survey of what the geometrical method has accomplished, and the estimation in which it has been held at different periods of the history of the development of Mathematical Science. Thus, avoiding expert questions relating to Education, we may begin by considering from a general standpoint what is implied in the science of Geometry, for the right ordering of which, in its educational relations, this Association has been working for a long series of years. It would, I presume, be against the instincts of our members to interpret the subject in any narrow sense. Geometry is usually based on the partly mathematical, partly logical, framework which has descended to us from Greek times, and which is chiefly associated with its earliest great systematization in the work of Euclid. But modern development would perhaps render any definition of its scope incomplete that did not include all methods in formal and physical Mathematics which proceed by direct contemplation of the relations of the subjects considered, instead of through representations of them that are merely quantitative or numerical.

Among the defects to be found in the treatment of the elements of Dynamics in many current text-books, the most obvious is the way in which the “laws of motion” are customarily introduced without any clear explanation as to the base, or axes, relative to which the motion is to be reckoned. Attention is very likely called to the fact that any statement about the motion of a body must, if it is to have any meaning, include the specification of a base; but in the statement of the law of inertia we are apt to find all this thrown to the winds, and the reader left to discover for himself, as best he can, what base is meant to be used. If a straightforward explanation can be given on this point, why is such a mystery made of it? A reader, approaching the subject for the first time, is apt to draw the inference that the choice of a base will somehow or other turn out to be immaterial. Statements may be met with which give encouragement to this view. Thus, in some of the text-books, the law of inertia is said to be merely a definition of force (or of both force and time); and this may be put in such a way that it is difficult to escape the conclusion that we are to recognise as forces simply mass-accelerations relative to an arbitrarily chosen base; although the abandonment of the law of action and reaction, which this would imply, is not really intended. It is more common to find Newton’s laws of motion quoted, and illustrated by examples of motion relative to the earth, the question of a base being left vague. It would be a more intelligible course, and possibly better for some purposes, to begin an elementary treatment of the subject with an approximate theory of motion relative to the earth only. In some books we find a complication produced by preliminary remarks as to none but relative motions being knowable, followed by expositions of the knowableness of absolute rotation and fixedness of direction, without either apology or explanation. In most of the text-books the point of view adopted is rather obscure. An exception must be made in favour of Love’s Theoretical Dynamics, which has the merit of being intelligible; but the procedure employed seems to be unduly ponderous and complicated.

The object of the following paper is to draw attention to a general and powerful but quite elementary method of transformation, by means of which the properties of a conic may be inferred from those of a circle. It has many striking analogies with the method of perspective transformation, and a simple geometrical connection with it. It is, however, of much simpler character for constructive purposes, and though it is intended to show later on how well the method is adapted to serve as an introduction to modern methods, it is hoped that readers will begin by dismissing from their minds all notions of cross ratios, homographic ranges, etc., and regard the theorems presented to them from the point of view of a student who has mastered his Elements of Plane Geometry.

The object of the papers is to consider two modes of representing points and lines on a spherical surface by points and lines on a plane: one method being by orthogonal projection, and the other by stereographic projection. The authors consider that students of spherical trigonometry ought to be able to accurately draw any figure with which they may have to deal, using one or other of the above methods of projection. The following propositions indicate methods of solving the various problems which would arise in connection with such accurate drawing.

Besides the five regular solids, there are other polyhedra which satisfy some, but not all, the conditions of regularity as usually defined, and so may properly be termed semi-regular. Some of these, which have an interest both as geometrical figures and as naturally connected with the geometrical problems of the partition of space and of crystallography, I propose here briefly to discuss.

The object of the present paper is to invite discussion as to the safest lines upon which to proceed in the teaching of Indices and Surds. And firstly, I am inclined to think that the beginner would get a firmer grasp of the theory of fractional indices (with which alone I am concerned) if a short course of surds, or, at least, of surd-forms, were made to precede; the simple theorem

would suffice, as shown later: and it seems more intelligible than its rival

The extent of the complete domain can be estimated by comparing it with its real part, that is, with the visible universe. For the sake of this comparison, von Staudt speaks as though it were possible to specify the number of real elements in any base, but gives an explicit warning as to the meaning he attaches to the symbol n which he uses for an infinitely great quantity. The number of real points on a line he denotes by n + 1; the one is apparently the ideal point, the n belonging to the actual universe. Now every imaginary pair can be assigned by means of a harmonic cast with one point, A, fixed. If this be (ABA′B′), we have a choice of n(n − 1) pairs for BB′, but no further choice; the number of imaginary points is therefore 2 × n(n−1), that is, n(n−1). Hence if the number of real points on a line is denoted by n+1, the total number of points on the line is n2+1. On an imaginary line of the first kind the points are one real, n imaginary; on an imaginary line of the second kind all are imaginary.

In the preceding discussion of the Galileo-Newton theory, force has been introduced as measured by mass-acceleration, in accordance with the generally accepted procedure. The existence of laws, connecting force with the conditions accompanying it, has been referred to as a fundamental fact which makes the theory possible; though, as a matter of logical order, all details as to such laws have been put into a secondary position. The recognition of the fact that acceleration was the thing, which could be connected by laws with the conditions under which a body moved, has been mentioned as Galileo’s great achievement. This was historically the first and most important, and was probably the most difficult, step to be taken. There is no record of any suggestion of it before Galileo’s time, and no theory of motion of any value had been constructed on any other basis. Perhaps the nearest approach to such a theory was the study of the celestial motions on the basis of circular motion; and this was not capable of progressing beyond the empirical stage.

Apollonius was born at Perga in Pamphylia, in the south of Asia Minor. The date usually assigned to him is 240 B.C., so that he was about seventy years later than Euclid and about thirty years later than Archimedes. He tells us that it was at the suggestion of the geometer Naucrates that he wrote his treatise on Conics, consisting of eight books, and that he composed them somewhat hurriedly, and submitted them to him without revision, simply putting down the different properties as they occurred to him, intending to go over them again afterwards, the reason for his haste being the approaching departure of his friend and guest. He afterwards found time to revise and publish each book separately, and it is in this revised condition that they have come down to us.

As I have been, for some few years past, in the habit of showing my pupils a proof of the formulae for uniformly accelerated motion very like that given by Mr. Dobbs in the October number of the Mathematical Gazette, I venture to add the few following remarks to his.

(i.) A quadric cone is a cone having any conic for base, and a point O on the line perpendicular to the plane of the conic through the focus for vertex.

(ii.) A sphero-conic is the curve in which any quadric cone cuts a sphere whose centre is the vertex of the cone.

In an introductory course on space-geometry, where only a passing mention of sphero-conics seems possible, I have found it useful to show directly the fundamental property that the sum or difference of the focal distances is constant, instead of following the classical method, which consists in deducing the property by the use of infinitesimals, from the fact that these focal distances make equal angles with the curve; this latter fact being in effect a direct consequence of Desargue’s theorem.

Having adjoined the ideal elements to the visible universe, von Staudt proves the usual theorems of projective geometry, noting the different cases that arise from different arrangements of the elements. As the only means of comparing figures is by projection, all the proofs depend on this; in many cases on the theorem that a given cast (Wurf) is unaltered when any two elements are interchanged, provided the other two are interchanged also (G. 59), which can be written symbolically

while if the cast is harmonic, viz. AG harmonic with respect to BD, (ABCD) = (CBAD) = (ADCB).