This paper is the second of a multi-part examination of the creation of the Babylonian mathematical lunar theories known as Systems A and B. Part I (Britton 2007) addressed the development of the empirical elements needed to separate the effects of lunar and solar anomaly on the intervals between syzygies. This was accomplished in the construction of the System A lunar theory by an unknown author, almost certainly in the city of Babylon and probably early in the 4th century B.C. (...) The present paper focuses mainly on System A and the likely process of its construction. The first three sections are largely descriptive – first of the basic concepts which underlie the theory; then of the component schemes comprising the theory; and finally of two distinctive texts which suggest how the theory was constructed. The crux of the paper is Sect. 4, which describes how the theory seems likely to have been constructed. Here the crucial insight in separating the effects of lunar and solar anomaly appears to have been recognizing that – of all the measurable intervals bounded by eclipses – only 235 months exhibits a variation due solely to lunar anomaly, and that by means of an elegant mathematical model the amplitudes of 223 and 12 months could be deduced from its amplitude. The rest of the section describes the likely details of the derivation of the Φ ~ Λ scheme, and the extension of the methodology to the other components of the theory. It concludes with a demonstration that Φ and its dependent schemes were anchored through the Φ ~ W scheme to the syzygy on –403 Aug 18 (GN 7391) which concluded the shortest 6-month interval in the first 24 saros cycles since –746 (and in fact in the 900 years separating Nabonassar and Ptolemy). The next three sections address a number of largely technical details and amplifications of the theory, beginning with the schemes describing the variation of lunar velocity (column F) in Sect. 5. Section 6 addresses issues concerning the interpretation of Φ and the so-call Saros Text (BM 36705), while Sect. 7 discusses System B’s corresponding treatment of the effects of lunar anomaly, illustrating both its derivative nature and mathematically less rigorous structure. Section 8 examines the accuracy of the two theories, showing that the System A theory was both remarkably accurate and superior to System B. The final section offers some brief remarks on the power and elegance of the mathematical treatment of the problem by the author of System A. (shrink)