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By Stephen Gaukroger (ed.)

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All these arguments serve to link rule 4B to a likely context for the development of universal mathematics between March and November 1619. In effect we have now argued for the dating of rules 4A and 4B from two different directions. In the section on reading rule 4, rule 4A 54 Descartes: philosophy^ mathematics and physics was plausibly— but not definitively— linked to the winter of 1619/20. Rule 4B was shown on rather stronger textual evidence, derived from Weber, to be a precursor of rule 4A and a model for it.

Algebra allows complex problems to be reduced to the ease of ‘direct’ deduction; and the theory of equations, symbolically expressed and manipulated, permits the derivation, articulation and codification of general schemas of solution for increasingly broad classes of problems.

If from the fact that we know the first [term] and the last to be connected with each other in a certain way, we should want to deduce the nature of the middle terms which connea them, we should then be following an order that was wholly indirect and upside down. But because here we are considering only involved inquiries, in which the problem is, given certain extremes, to find certain intermediaries by the inverse process of reasoning, the whole of the device here disclosed will consist in treating the unknowns as though they were known, and thus being able to adopt the easy and direct method of investigation even in problems involving any amount of intricacy’.

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