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).

We have already explained pp. (25-31) that there are two types of uniform three-dimensional space, viz., finite or elliptic space, in which the geodesies or straight lines are closed and of constant length, and infinite or hyperbolic space, in which the geodesies are unclosed and of infinite length. Euclidean or parabolic space may be regarded as a special third type, or as the dividing case between the two general types; but it is best to regard it as a variety of hyperbolic space, owing to the reality of the infinite. In elliptic space all geodesies through a point either return without intersecting again, or have at most one other fixed point of intersection. In hyperbolic space it does not appear necessary that a geodesic should consist of only a single branch, though this is of course the simplest case; if the geodesies have several branches, all geodesies through a point will have a fixed point on each branch, and two geodesies which intersect will have as many points of intersection as they have branches. When we consider two-dimensional space we find that, even in the elliptic type, the geodesies may have any number of intersections, and may be unclosed and infinite. The proof to the contrary on p. 31 does not apply when the geometry is confined to a surface.

It is far too much the custom now to rely on the analogy of algebra to justify the introduction of imaginaries into geometry. Analogy, however, is no justification unless we first prove the exact correspondence of the fields of investigation. In analytical geometry the identification of the two fields is permissible, and is easily explained; but in pure geometry any reference to algebra, expressed or implied, is irrelevant and misleading. The elements of pure geometry have no dependence on calculation.

The Sixth General Meeting of the Association for the Improvement of Geometrical Teaching appointed a Committee for Geometrical Conies, and one for “Higher Plane Geometry, including such subjects as Transversals, Projection, Anharmonic Eatio, etc.”

Dr. Hirst in his Presidential Address has said of the latter subjects, “Until these notions become more familiar ones, I, for my part, believe that Geometrical Conies will always remain in its present unsatisfactory condition. . . . It will be, of course, a question for this Association to decide whether, pending the introduction of the more thorough treatment based on the notions to which I have alluded, some improvement may not be introduced into the subject of Geometrical Conies, as at present understood, with a view of enabling examiners at all events to examine with greater facility and purpose. At present the definitions are so multiform and the sequence of propositions so varied in different text-books, that it is found to be an exceedingly difficult task to examine satisfactorily in the subject at all.”

I am venturing to call the attention of the Association to a difficulty that I have to contend with, as a private tutor, when assisting boys fresh from school either to continue their mathematical studies or to prepare for some of the public examinations. It is of course impossible for me to assert positively that this difficulty is experienced by tutors generally, but the strong conviction that it is felt by others as well as by myself has led to my bringing the matter forward.

The method usually given in text books on Algebra for finding the present value of an annuity to last for a given number of years, or the accumulated amount of an annuity left unpaid for a given time, consists in writing down the present values or amounts of the several payments, and summing them by the usual formula for a geometrical progression. The following method, by which the result is found from first principles without the use of formulae, is so simple and instructive that I am rather surprised that the idea does not appear to have suggested itself to teachers and writers of text books before now, or, if it has done so, that it has received so little notice.

Edward Wright was probably born about 1560, and he died in 1615. He was a fellow of Caius College, Cambridge. The chief source of information about him, besides his own published works, is an obituary notice in a Latin paper preserved in the college library. He furnishes a notable instance of the conditions under which mathematical science was pursued in his time. The main object kept in view was to supply material needs of life, principally at this period in all matters connected with navigation, and the promoters and supporters of research in this direction were not the universities, but the trading and mercantile communities. Wright was appointed by the East India Company their Lecturer on Mathematics, at a salary of £50 a year. He was mechanical tutor to the eldest son of James I., Prince Henry, who died in 1612 at the age of nineteen. For his pupil’s instruction he designed and made a wonderful piece of mechanism, displaying all the movements of the heavenly bodies then known, and constructed to indicate these correctly for 17,100 years. It lasted, in fact, but a very few years. During the troubles of the civil war it was cast away as old rubbish, and in 1646 was found by Sir Jonas Moore, and deposited in its dilapidated state at his house in the Tower.

Pythagoras was born in Samos about 580 B.C. The statements regarding his masters are destitute of solid historical foundation, and so are the accounts of his travels. It is known, however, that he visited Egypt. He may have taught in Samos, but whether he did not obtain sufficient consideration among his fellow-citizens, or whether he wished to withdraw from the tyranny of Polycrates, he removed to Croton about 540 B.C. There he founded an association, less scientific than politico-religious, which eventually became the nucleus of the aristocratic party.

It may be useful to compare with the results proved above, the interesting and comparatively little known properties of the Stereographic Projection of the sphere so far as they affect its practical application to spherical triangles. This projection, in which we take a point S on the sphere as the vertex of projection, and the central plane parallel to the tangent plane at S as the plane of projection, has certain great practical advantages, as shown in (1) and (2) below; and obviates the difficulty of drawing an ellipse.

The idea of Relative Motion is one that presents a good deal of difficulty to the beginner in Mechanics. The following Notes are offered to readers of the Gazette as indicating a mode of presenting the subject which I have found to be helpful to learners. A clear grip of the working ideas, rather than a discussion of their logical basis, is what is here aimed at.

This short paper is supplementary to that read to the Association by Mr. Langley (Jan. 1892); that also being supplementary to one read by Mr. Hayward (Jan. 1889).

The aim of this note is to point out a certain looseness of method and haziness of conception which seems to me to exist in the treatment of the above subjects in our current text-books, and to suggest improvements.

The functions in question are those which involve only the differences of their arguments ; and the object of this note is to suggest that an elementary discussion of these functions would be a valuable addition to the usual school course in algebra.

In his memoir on the Schwarzian derivative, Cayley gives the name ‘diaphoric’ to these functions (Art. 13). Invariants, seminvariants, covariants of quantics are diaphoric functions of the roots. Norms of complex numbers and ‘circulants’ are diaphoric functions, a particular case of which already appears in books on algebra under the properties of cube roots of unity.

In a recent review of Prof. W. B. Smith’s “Infinitesimal Calculus,” fault is found with his statement that “In all strictness there can be no motion at an instant, and hence no speed (or velocity) at an instant. The concept of speed (or velocity) or motion will not combine with the concept of instant (or point of time) to form a compound concept.” Part of the reviewer’s criticism is as follows: “If we allow that motion at an instant is impossible, how are we to escape from Zeno’s paradoxical conclusion that all motion is impossible? How can I move from one place to another during a minute, say, if at every instant of that interval motion is impossible?”

The learned professor of Higher Geometry in the university of Genoa has undertaken to write a treatise on The Exact Sciences in Ancient Greece. The treatise will consist of five books, the first being devoted to the period of preparation, the second to the period of greatest splendour, the third to the period of decadence or the epoch of the commentators, the fourth will trace the mathematical discoveries incidentally made by the Greek astronomers and surveyors, and the fifth will treat of the arithmetic of the Greeks.

The following note is intended to draw attention to a curious relation which I arrived at, some years ago, between the lengths of the common tangents of two intersecting ellipses, and the lengths of the arcs between their points of contact.

The property proved may be regarded as an extension of Graves’ Theorem [SALMON; Art. 399] and is deduced from that theorem in conjunction with a lemma proved in this note.