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A brief theoretical exposition of the concept of matter as a nowhere dense perfect set of positive measure is developed starting from the observation that matter is largely vacuous in all scales from subnuclear to extragalactic.

Sumner defined a graph to be point determining if and only if distinct points have distinct neighborhoods and he has characterized connected line-critical point determining graphs. Here a short alternate proof of his characterization is provided and arbitrary line-critical point determining graphs are then characterized. Next line-critical point distinguishing graphs are considered; a graph is point distinguishing if and only if it is the complement of a point determining graph. Finally, line-critical graphs that are both point determining and point distinguishing are characterized.

whether we regard the Scripture from a critical point of view or not. ,a \' "1W \\\",' The power of the presentation of Copyright by J. J. Tissot every incident touched upon there is sufficient to command the study and respectful attention of the most radical Christian, or of him who is no Christian even, but is still wise enough to admit that Jesus was one of the greatest of historic figures, from whom has come the greatest worldmovement ever known.

In a recently published paper the conclusion was reached that Stonehenge was built at different periods, and that the Aubrey circle with the bank and ditch represent an older monument. In conclusion it is said ‘that Stonehenge was built after the Aubrey circle is certain’. There is, however, evidence that leads to a different theory, and this being so, it is desirable that it should be stated; otherwise it might appear that those interested agreed with the conclusion, or at least had no evidence to the contrary. A fact used to support the two-date theory is that the end of the ditch overlaps the avenue. There is general agreement that this ditch was levelled and obliterated on the surface in order that it should not obstruct the avenue. The critical point is when this filling-up took place. According to the two-date theory it was only when the later Stonehenge. was built, and therefore a long time after the ditch was first dug. But if the evidence from the excavation is carefully examined it becomes plain that this part of the ditch must have been refilled almost immediately after it was made.

A Secure grasp of Thomas Altizer's theology is not an easy achievement. But if I have correctly understood it, I would argue that it turns on an axis of three themes. These themes will be examined and evaluated here. The critical point I will develop is that the manner in which Professor Altizer has set the stage for what he defends as ‘radical theology’ results in two major deficiences—an inability to confront and wrestle with the reality of evil, and a failure to account for freedom as the source of evil.

1. Introduction. The theory of Tchebycheff approximation for functions of one real variable has been understood for some time and is quite elegant. For about fifty years attempts have been made to generalize this theory to functions of several variables. These attempts have failed because of the lack of uniqueness of best approximations to functions of more than one variable. Indeed, Mairhuber [6] has shown that best approximations cannot be unique unless our functions are defined on a space homeomorphic to a subset of the unit circle. Further negative results have been obtained by Rivlin and Shapiro [11] and the conclusion is that there is no interesting Tchebycheff approximation problem in several variables for which best approximations are unique. Schoenberg [12] gives an account of a theory in several variables, but his assumptions essentially limit the applicability to functions of one variable. In this paper two theories of Tchebycheff approximation to functions of several variables are developed. It cannot be said that they are as satisfying as the theory for one variable, but they are workable theories. For the first theory the concept of a critical point set is given (the definition is somewhat different from that in one variable) and it is shown that the set of critical point sets of a best approximation is unique. A characterization theorem for best approximations is given in terms of critical point sets and other theorems normally found in the theory of Tchebycheff approximation are valid. Similar viewpoints have recently been investigated by Lawson [16] and Rivlin and Shapiro [17]. The second theory defines the strict approximation for functions defined on a finite point set. In addition to the results obtained from the first theory, the strict approximation is unique. It is also a best approximation in the usual sense. There are two basic features of the one variable theory which do not generalize readily. The first of these is the idea of a Tchebycheff set. No counterpart of these sets exists in this paper. A heuristic argument is given that shows that there is no possibility of defining Tchebycheff sets in several variables with the important properties present in the case of one variable. The second feature which does

1. The frontier of Pisidia, where it adjoins Lycaonia, is placed wrongly in my Histor. Geogr. Chs. Q and V. The district was little known, when I wrote: I had traversed it hurriedly in 1882, 1886, and 1890, in each case only on a single hasty route. The excursion of 1882 resulted in placing Anaboura and Neapolis. No name was discovered in the excursions of 1886 and 1890. Prof. Sterrett explored the district very carefully in 1884 and 1885; but the numerous inscriptions, which he found, unluckily did not contain important topographical indications, and he assigned a position much too far north for the city of Pappa-Tiberiopolis. I shared his view on this critical point, with the result that many other towns were drawn away far north of their true situation, because they stood in some relation to Pappa, and when it seemed to lie away in the north, they had to be placed correspondingly. For the same reason, the Pamphylian frontier was drawn too far north.

The supporters of the convergence theory maintain that the economic systems of the East and the West are gradually becoming similar and moving in the same direction. This theory first originated at the beginning of the 1960s and claimed that there was a development toward a "uniform industrial society." Leading exponents of this view were Walt Whitman Rostow (The Stages of Economic Growth, 1960) and Raymond Aron (Dix-Huit Legons sur la Socitte Industrielle, 1963), while the economic aspects of the convergence theory were mainly elaborated by Jan Tinbergen (1963).1 Thereafter, a debate was unleashed: some essays were in agreement with the main points of the convergence theory, while most of them were critical. The following is an attempt to delineate the factors which gave rise to these theories, and to discuss their contents from a critical point of view. Major emphasis will be placed on the debate in the Federal Republic of Germany.

THE amicable discussion between Dr. Romanes and myself, “endeavouring to help in determining the true position of an important question,” has now (in NATURE, vol. xxvii. p. 527) reached a critical point, one seemingly capable of settlement by scientific inquiry, and upon which a brief note may be pertinent.