In Part I we trace the development of Arithmetic and Algebra from the high cultures of the Egyptian Middle Kingdom and Mesopotamia (1600BC) through Islamic mathematics and early algebra and on to the beginnings of mathematical modernity in the work of Galois in the 1830's. Our style will be to explore mathematical practice and conceptual developments in different historical contexts. In Part II we present the Rise of the Calculus. This is taken from the first work of Archimedes and Apollonius around 200BC onwards, to trace deas on differentiation and integration through to the time of Newton and Leibniz in the early 18th century. Part III is on mathematical logic. We explore the ideas of propositions, logical methods in the axiomatisation of mathematics, and the idea of quantifiers. We explore the interplay between the development of logic and the development of mathematics, including theoretical computing. We discuss the Hilbert programme, first order logic, and the completeness and incompleteness theorems of Goedel, undecidability and independence. We discuss logic programming and artificial intelligence, with brief overview of Prolog. Our approach is not a linear representation of history but a history of the development of the most important ideas in logic. Students will need some mathematical knowledge to attempt the module.