[1] This series, usually written in the equivalent form $\frac {\pi}{4} = 1-\frac{1} {3}+\frac{1}{5}-\frac{1}{7}+\cdots$, is due to Leibniz. We'll see how to use this as the basis for some fancy numerical tricks in section 3.5.3.
[2] Notice that we have used block structure (section 1.1.8) to embed the declarations of pi_next and pi_term within pi_sum, since these functions are unlikely to be useful for any other purpose. We will see how to get rid of them altogether in section 1.3.2.
[3] The intent of exercises 1.311.33 is to demonstrate the expressive power that is attained by using an appropriate abstraction to consolidate many seemingly disparate operations. However, though accumulation and filtering are elegant ideas, our hands are somewhat tied in using them at this point since we do not yet have data structures to provide suitable means of combination for these abstractions. We will return to these ideas in section 2.2.3 when we show how to use sequences as interfaces for combining filters and accumulators to build even more powerful abstractions. We will see there how these methods really come into their own as a powerful and elegant approach to designing programs.
[4] This formula was discovered by the seventeenth-century English mathematician John Wallis.
1.3.1 Functions as Arguments