Let \(r(N)\) denote the representations of an even number \(N\) as the sum of two primes, so that Goldbach's conjecture is that \(r(N)\) is positive for \(N>2\). In the use of the Hardy-Littlewood method to tackle the problem, the contribution to \(r(N)\) from the major arcs gives what may be expected to be the dominating term, to which an associated singular series is attached. In this posthumous paper the author applies an elementary method to express \(r(N)\) in terms of simple arithmetic functions, from which he isolates easily what can be expected to be the principal term. He then uses quite a direct method to show that this principal term also has the singular series featured in the Hardy-Littlewood method. Contemporary mathematicians had remarked that the author possessed a gift for seeing straight to the kernel of a problem, even in a subject that was new to him. The present paper is of remarkable interest in that it does provide some new insight into a problem that must have fascinated the author throughout his life.