We have no difficulty in conceiving and adequately describing determinate possible motions which we have never witnessed and which we never shall witness. We have merely to assign a determinate direction and a determinate velocity. But we could not possibly have formed the concept of such a colour as blue or such a shade as sky-blue unless we had perceived instances of it, no matter how much we had reflected on the concept of Colour in general or on the instances of other colours and shades which we had seen. It follows that, even when we know that a certain kind of secondary quality (e.g., colour) pervades or seems to pervade a region when and only when such and such a kind of microscopic event (e.g., vibrations) is going on within the region, we still could not possibly predict that such and such a determinate event of the kind (e.g., a circular movement of a certain period) would be connected with such and such a determinate shade of colour (e.g., sky-blue). The trans-physical laws are then necessarily of the emergent type.This should remind anyone of Jackson's famous Knowledge argument involving the physicist Mary who knows the correct physical theory of color, but who lacks color experiences. I would not say that the arguments are identical, of course. Jackson is arguing against physicalism, while Broad is arguing against what he calls mechanism. But both physicalism and mechanism have reductive implications, and the appeal to color experience in both cases to block this sort of reduction is quite similar.
This made me wonder how widely known this sort of overlap is, and if there are obvious antecedents that Broad is drawing on. Brie Gertler notes in her Encyclopedia of Philosophy article on the Knowledge argument that "Arguments in the same spirit had appeared earlier (Broad 1925, Robinson 1982)". And one finds a section on the Knowledge argument in Kent Gustavsson's entry on Broad in the Stanford Encyclopedia of Philosophy. Broad is also noted in the entry on the Knowledge argument by Martine Nida-Rümelin. So, maybe it is well-known.
It appears to me that Broad is drawing on the obvious chapter in Mill's Logic, especially Mill's remark that "from no knowledge of the properties of those substances could we ever predict that it [the tongue] could taste, unless gelatine or fibrin could themselves taste; for no elementary fact can be in the conclusion, which was not in the premises" (Bk. III, ch. vi, section 1). Indeed this style of argument could perhaps be traced back to Locke's Essay and God's power to superadd the power of thinking to matter (Bk. IV, ch. III, section 6).
Update (Feb. 13): Chalmers has drawn my attention to two prominent discussions of Broad's knowledge argument: Stoljar's introduction to There's Something About Mary and Chalmers' own "Consciousness and its Place in Nature".