This article sets out to explore some of the connections between two seemingly distinct mathematical objects: trigonometric functions and the integer sequences composed of the Fibonacci and Lucas numbers. It establishes that elements of Fibonacci/Lucas sequences obey identities that are closely related to traditional trigonometric identities. It then exploits this relationship by converting existing trigonometric results into corresponding Fibonacci/Lucas results. Along the way it uses mathematical tools that are not usually associated with either of these objects.

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Abstract

SummaryThis paper offers a provisional assessment of the development of settlement in part of Cranborne Chase between the Mesolithic and the Late Bronze Age. It builds upon the results of Pitt Rivers' work in this region between 1880 and 1900, as well as more recent excavation and field survey. Special emphasis is placed on three factors: the relationship between activity in this area and settlement both in central Wessex and on the coastal plain; the place of the more prominent ‘public’ monuments in contemporary patterns of settlement and exchange; and the relationship between cemeteries and contemporary living sites. We present the first results from the extensive excavation of two Deverel-Rimbury enclosures and associated barrows, and a new analysis of Pitt Rivers' work on the urnfield at Handley Barrow 24.

Hundreds of early modern forts dot the South Indian landscape, but more is known about their art-historical aspects than how these fortifications were built, maintained, and used. The latter aspects are examined in a comparative analysis of six Mysore hill forts, using East India Company surveys that were prepared in 1802 shortly after British forces took command of these installations from Tipu Sultan's garrisons. These highly detailed inventories show that the hill forts were poorly maintained, inadequately supplied, contained relatively few garrison buildings, and, from the British perspective at least, were inadequately armed to mount a successful defense. At the turn of the century, these forts were functionally obsolete, ill prepared to serve as supply depots, and strategically valuable only to quiet local populations and deny the use of such places to potential enemies.