The questions which I propose to examine in the following paper, although not themselves dependent on chance, have arisen entirely from games in which it predominates. To determine some method of betting upon a number of successive events, (and the probability of each of which is either equal to, or less than one-half,) by which a profit shall be realised after a considerable number of them have been decided, is a problem which has occupied the attention, and exhausted the efforts, of one set of speculators, as completely as that of the quadrature of the circle has defeated the labours of another. The first and most simple plan, is that of doubling the stake whenever a loss occurs. This is well known, and has been so frequently practised, as to have acquired a peculiar name; it is technically called the martingal; it requires for its success, that the person who employs it have the power of leaving off whenever he please, and that he have the command of an unlimited capital. If the chance of the events happening is one-third instead of one-half, the stake must be tripled.

By the introduction of letters into algebra, to denote all the quantities, both known and unknown, involved in an equation, this very important advantage was gained, that the final equation exhibited both a general rule for the solution of all similar problems, and also the limitations within which such problems were possible.

Man is formed to derive pleasure from a variety of different sources. Many of his pleasures are communicated by the channel of the external senses. Each of these has particular classes of objects that are suited to its gratification ; and these, being diffused in great abundance through the works of nature, or framed by the art and ingenuity of men, become to him a source of frequent and diversified enjoyment.

In the autumn of last year Sir Archibald Geikie, F.R.S., Director-General of the Geological Survey of the United Kingdom, Kindly placed in my hands for determination a number of specimens of Cephalaspis, collected by his officers in the Lower Old Red Sandstone of the neighbourhood of Oban. On examining them, I found that they all belonged to one species, which was, however, new to science.

Accordingly I drew up a brief diagnosis of this new form, which was included by Sir Archibald in his Summary of Progress of the Geological Survey for 1897, and it is now my privilege, with his sanction, to offer to this Society a more detailed description of the species, accompanied with figures.

The experiments from which the following are extracted, make part of a course, instituted at the Royal Military Academy, for determining the resistance of the air to a surface of any form whatever, either plane or curved, moved through it with any degree of velocity. I was induced to undertake these experiments, both for the improvement of my students in the Academy, and with a view to apply the conclusions derived from them towards perfecting the theory and practice of military projectiles, as well as other branches of natural philosophy, in which the pressure or resistance of fluids is concerned: Circumstances, concerning the laws of which, authors on the theory have widely differed; some making the pressure or resistance equal to the weight of a column, whose altitude is equal to the whole height due to the velocity, while others make the altitude very different, either the half or the double of that.

Micromalthidae is a small relictual family of archostematan beetles that is well known for its unusual asexual reproductive system characterised by parthenogenetic, viviparous, larviform females and sterile adults. Here we describe the first micromalthid beetle from mid-Cretaceous Burmese amber (ca.100Ma). Protomalthus burmaticus gen. et sp. nov. differs from all presently known micromalthid beetles by the presence of distinct frontoclypeal, labroclypeal, and medicranial sutures. The finding represents the first Mesozoic record of an adult micromalthid beetle and confirms a Gondwanan distribution of the family during the Cretaceous. A list of micromalthid beetles is provided, together with a key to extant and fossil species.

Glencartholm in Dumfries and Galloway, Southern Scotland, is one of the most important Palaeozoic fossil fish sites in the world, with a fauna containing more than 30 species. Originally discovered in the bank of the River Esk in 1879, further collecting in the 1930s removed all the accessible fossiliferous strata. These strata were not known to occur elsewhere and despite the site's international importance, details of the section including the fossil bearing beds have not been described. An excavation of a new exposure at Mumbie, 50 metres east of the original site, was undertaken by Stan Wood in the 1990s. Sediments were logged over an 18-metre section. Three separate fish beds were identified and more than 200 specimens of ray-finned fish collected, including one possible new species. Differences in scale colour enables fishes from the different horizons to be distinguished. This should allow material collected previously to be more accurately assigned and assist palaeoecological analysis of the site. In addition to ray-finned fish, cartilaginous fishes were collected together with plants, bivalves, crustaceans, scorpions and horseshoe crabs which have not previously been reported from Glencartholm.

This paper examines the astronomical theory of ice ages of James Croll (1821–1890), its influence on contemporaries John Tyndall, Charles Lyell, and Charles Darwin, and the subsequent development of climate change science, giving special attention to the work of Svante Arrhenius, Nils Ekholm, and G. S. Callendar (for the carbon dioxide theory), and Milutin Milanković (for the astronomical theory). Croll's insight that the orbital elements triggered feedbacks leading to complex changes – in seasonality, ocean currents, ice sheets, radiative forcing, plant and animal life, and climate in general – placed his theory of the Glacial Epoch at the nexus of astronomy, terrestrial physics, and geology. He referred to climate change as the most important problem in terrestrial physics, and the one which will ultimately prove the most far reaching in its consequences. He was an autodidact deeply involved in philosophy and an early proponent of what came to be called ‘cosmic physics’ – later known as ‘Earth-system science.’ Croll opened up new dimensions of the ‘climate controversy’ that continue today in the interplay of geological and human influences on climate.

A considerable tropical order of, with but few exceptions, climbing plants, occurring in both the old and new worlds. A few species are found in the cooler regions of North America, Eastern Asia, South Africa, and Australia.

Clinical and pathological observation both show that the dentine of the tooth is very closely connected with the nervous system, and is in consequence highly sensitive. Upon what structures does the sensibility of the dentine depend? In what manner is the dentine connected with the nerves of the pulp so as to become so sensitive to external stimuli?

Perhaps there is no other structure in the body which is so largely supplied with nerves as the pulp of the tooth; even in the smallest fragment we find many nerve fibres. If we take the pulp from the incisor tooth of an ox and examine it after having allowed it to lie in a solution of osmic acid for a few minutes, we can see clearly through the darkened semi-transparent tissue a large blackened nerve trunk passing up the centre of the pulp, giving off on its way innumerable lateral branches, and dividing in a brush-like manner near the upper part of the pulp. All the fine branches are directed towards the periphery of the pulp. In longitudinal sections of the pulp we can see the same in greater detail; many large bundles of medullated and non-medullated nerve fibres running longitudinally near the centre and giving off lateral branches, which are found in great numbers near the periphery and divide into single nerve fibres just under the odontoblastic layer, being specially numerous at the apex of the pulp.

Stan Wood had a gift for finding exceptional Early Carboniferous fossils. Among them are 32 type specimens. His discoveries significantly changed our understanding of the history of life on Earth. Many of the fossils he collected are on display in museums across the UK and the localities he discovered continue to yield important new material. Here we briefly review some of Stan Wood's key achievements and describe the legacy he left.

Shortly after I read to the Society my paper on “Thermal and Electric Conductivity” (Trans. R. S. E., 1878), in which I stated that the results were “by no means final, even so far as my own work is concerned,” I was requested by Sir Wyville Thomson to undertake the examination of the “Pressure Errors of the ‘Challenger’ Thermometers.” This investigation led to another on the “Compression of Sea-Water,” and allied subjects, which is not yet finished. Meanwhile, though I had prepared everything for my promised repetition of the experiments on Thermal Conductivity, the bars formerly used having been nickelised, &c, I found that it would be impossible for me to carry out the investigation. I therefore asked Mr Mitchell, who, as Neil-Arnott Scholar, had already done good and careful work on Thermal Conductivity in my Laboratory, to repeat the experiments under the altered conditions. I put at his disposal all the apparatus which was employed in the former research. The Government Grant Committee allowed a sum for the payment of a computer to reduce the results, and the observations were at once commenced. The results are now laid before the Society, and are probably as good as the method and the thermometers employed can furnish.

Late Eocene fossil Psocodea from the Isle of Wight are studied and classified within three new genera and seven species (Wightipsocus acourti (Cockerell 1921) gen. nov. et n. comb., Wightipsocus intasi gen. et sp. nov., Wightipsocus rossi gen et sp. nov., Awightipsocus jarzembowskii gen. et sp. nov., Awightipsocus megaloi gen. et sp. nov. Awightipsocus minimus gen. et sp. nov. and Wightimicropsocus inexpectatus gen. et sp. nov.) without however attributing them to any family. A key for fossil Psocodea from the Isle of Wight is given.

Benguella is a province of Angola, the chief Portuguese colony in western Africa. From the chief town, Benguella (lat. 12·35° S.) and from Catumbella (lat. 12·25° S.) old-established trade routes go inland on to the Benguella plateau, and across it along the watershed between the Congo and the Zambesi to Katanga and the Upper Congo. The geographical pioneers in this part of Africa showed that the land rises in steps from the coast to a high plateau, which consists of a foundation, of ancient crystalline and sedimentary rocks, and is flanked by younger marine beds along the coast.

The value of the Chronometer for finding the Longitude at Sea, being, by the experience of many years trial, fully established, I am induced to offer to the Royal Society some remarks on the change of rate observed in this instrument, when on ship-board. This change of rate, that had usually been supposed to arise from the motion of the ship, has recently heen attributed, by Mr Fisher, who accompanied Captain Buchan in his Voyage towards the North Pole in the year 1818, ‘to the magnetic action exerted by the iron in the ship upon ‘the inner rim of the Chronometer's balance, which is composed of steel.’ I apprehend, however, that it will be very easy to show, that although the alteration of rate may be, and most probably is, owing to magnetism, yet the magnetic action of the iron in the ship, excepting in cases where chronometers are placed in immediate contact with large masses of iron, can contribute but in a very small degree to the error in question.

In this series of papers it is proposed to describe new or interesting Palæozoic plants, many of which were collected several years ago. Some of these are in the possession of the Geological Surveys of England and Scotland, others belong to personal friends or are in the collections of public museums, while the remainder are in my own collection. They come from various localities and horizons, and a few from coalfields of which papers dealing with their fossil plants have already been published, and they are, in many cases, additions to these lists.

Although great advances have been made during the last thirty years in our knowledge of dilute solutions, there has been no corresponding advance in respect of concentrated solutions. This is primarily due to the fact that hitherto no simple and general relationship has been discovered between the conductivity and the concentration of concentrated solutions of electrolytes. Ostwald's law of dilution holds only for dilute solutions of weak electrolytes, and the formulæ of Rudolphi and Van T'Hoff are applicable only to dilute solutions of good electrolytes. It seems therefore important to inquire whether the difficulty may not be to some extent overcome by an alteration in the mode of representing the facts.

The earliest tetrapodomorph fishes appear in Chinese deposits of Early Devonian age, and by the Middle Devonian they were widespread globally. Evidence for the earliest digitated tetrapods comes from largely uncontested Middle Devonian trackways and Late Devonian body fossils. The East Gondwana Provence (Australasia, Antarctica) fills vital gaps in the phylogenetic and biogeographic history of the tetrapods, with the Gondwanan clade Canowindididae exhibiting a high degree of endemism within the early part of the stem tetrapod radiation. New anatomical details of Koharalepis, from the Middle Devonian Aztec Siltstone of Antarctica, are elucidated from synchrotron scan data. These include the position of the orbit, the condition of the hyomandibular, the shape of the palate and arrangement of the vomerine fangs. Biogeographical and phylogenetic models of stem tetrapod origins and radiations are discussed.