Duncan F. Gregory,William Walton and the development of British algebra: ‘algebraical geometry’, ‘geometrical algebra’, abstraction

This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on ‘algebraical geometry’ and ‘geometrical algebra’ in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the ‘abstract algebra’ and ‘abstract geometry’ of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to ‘algebraical geometry’ and ‘geometrical algebra’ of the second generation of ‘scientific’ symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s–1840s.     

Butterflies (Lepidoptera: Papilionoidea) of Mount Kilimanjaro: Nymphalidae subfamilies Libytheinae,Danainae, Satyrinae and Charaxinae

This paper, which presents an annotated checklist of the ‘lower Nymphalidae’ (Libytheinae, Danainae, Satyrinae, Charaxinae), is the fourth in a series on the butterfly fauna of Mount Kilimanjaro. Four genera of lower Nymphalidae (Danaus, Amauris, Bicyclus, Charaxes), with a total of 11 included species, are known or believed to occur within the main forest zone, from c. 1800 to 2800 m. Of these, only three species of Charaxes (Charaxes berkeleyi, Charaxes ansorgei, Charaxes xiphares) may be restricted locally to this primary forest. The lower slopes fauna, below 1800 m, is considerably richer, with a total of 11 genera and 41 species listed (8 species of which extend into the forest zone). Possible additional species, dubious earlier records, problems with African subspecies of Danaus chrysippus, a need for more work on certain Satyrinae, and classification of the genus Charaxes are discussed. An identification key to the subfamilies of Nymphalidae, and the 19 genera of Libytheinae, Danainae, Satyrinae, Charaxinae that occur in Tanzania, together with a key to the adults of all the species of these four subfamilies considered to occur or have occurred on Kilimanjaro, with 206 colour images, are included as online Supplementary Information.     

Baronia brevicornis caterpillars build shelters to avoid predation

In at least 18 Lepidoptera families, caterpillars build shelters that mainly serve to regulate microclimate (humidity, temperature) and/or to avoid predation. We aimed to explore the function of the tubular structures built with the leaves of the host plant, Acacia cochliacantha Humboldt and Bonpland ex Willdenow (Fabaceae), by a lepidopteran endemic to Mexico, Baronia brevicornis Salvin. We experimentally evaluated whether tubular structure building behaviour is induced by high temperature or predator odour, and if shelters reduce or enhance predation of B. brevicornis caterpillars. We used Calosoma angulatum as predator. Our analyses showed that caterpillars did not make the tubular structures in response to high temperature. We also found no difference in predators’ visual recognition of sheltered versus unsheltered caterpillars. Caterpillars did not build shelters but they moved more often when exposed to predator odour. Unsheltered caterpillars were more frequently consumed when predators were allowed to interact with sheltered and unsheltered caterpillars. Hence, the tubular structures built by B. brevicornis are most likely a strategy for reducing predation, for example by C. angulatum.     

Diversity and dispersal history of the talitrids (Crustacea: Amphipoda: Talitridae) of Bermuda

Five taxa of talitrid amphipods were found in the archipelago of Bermuda, of which three were recorded there for the first time. Four of these are supralittoral wrack generalists: Platorchestia monodi BOLD:AAB3402, (a unique Molecular Operational Taxonomic Unit according to the Barcode Index Number system), a related species recognized by molecular methods, Platorchestia platensis BOLD:AAA2949, Mexorchestia carpenteri carpenteri BOLD:AAC1491 and Tethorchestia antillensis; and one a terrestrial leaf-litter generalist: Talitroides alluaudi. A key is provided to discriminate between the formally described talitrids of Bermuda. Dispersal mechanisms from the American continent to Bermuda were considered for all taxa based on species distributions along the North American Atlantic coast and also investigated by molecular methods, using genetic population differentiation and haplotype network analysis based on the barcode region of cytochrome c oxidase subunit I gene. For P. monodi BOLD:AAB3402 the genetic results suggest that some dispersal events occurred before human colonization of Bermuda but are equivocal about the source population and therefore the direction of dispersal. Some very recent synanthropic dispersal is possible with this species. For the other two species studied genetically, P. platensis BOLD:AAA2949 and M. c. carpenteri BOLD:AAC1491, the small population samples analysed support dispersal to Bermuda from the American mainland, before human occupation of Bermuda, although the available sample size was limited for these species. The available limited direct, non-genetic evidence supports synanthropic transport for Talitroides alluaudi. Platorchestia monodi BOLD:AAB3402 is found in the same wrack habitat as P. platensis BOLD:AAA2949 on Bermuda, apparently without interbreeding. No evidence was found that driftwood specialist talitrids had become established in Bermuda.