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Coefficient matching feels rather natural when solving equations and checking dimensions, however in footnote 2 to "Two alternative derivations of Bridgman's theorem" (Berberan-Santos M N, Pogliani L, J. Math Chem 1999, 26, 255-261; p256) it's mentioned that Descartes shifted from line/area/volume ideas to the basic number line view.

Is this where the idea of coefficient matching between the terms when solving polynomials began, or was it already well established for pure number polynomials?

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  • $\begingroup$ Could you explain what "coefficient matching" is / was? $\endgroup$ – Rory Daulton Aug 7 '16 at 0:48
  • $\begingroup$ @RoryDaulton It may not be the right term, but I was refering to the case where you get, via two routes, say A=ax^2+bx+c, and A=5*x^2 + 7, so c=7, b=0, a=5 (seems obvious?). The comment in the paper implied that historically you could not add a square (e.g. the area of a room) to the length of a line (length of drive way) to a pure number, and even now, we should not add a probability to an angle, despite them being thought of as pure numbers $\endgroup$ – Philip Oakley Aug 7 '16 at 10:13
  • $\begingroup$ This is so in your example only if the two equalities for A hold for at least 3 values of x. For 2 or fewer, the coefficients may well not match. I believe the original view of polynomials was as expressions taking values, not as mathematical objects of their own as in modern algebra. $\endgroup$ – Rory Daulton Aug 7 '16 at 11:44
  • $\begingroup$ @RoryDaulton, when was that change in view about expressions taking values? In the applied sciences there are still some issue regarding quantity calculus and just when polynomial terms can be matched, and which terms are 'just values'. $\endgroup$ – Philip Oakley Aug 8 '16 at 13:51
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Thomas Harriot, 1560-1621 matched coefficients when generating formulas for sums of powers of positve integers.

"Gathering like terms", is, of course, an obvious technique, and was also used in the geometric derivations of these formulas. The link provided goes to the middle of a long article on the history of these formulas; Harriot appears to be the first to use symbolic notation, which allowed him to focus on the coefficients. Descarte, Fermat, and Bernoulli are all later.

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  • $\begingroup$ Thanks @Peter, it's useful to get these points in context, especially when later we get into the 'confusion' between the variables and parameters symbols! I've accepted the answer. $\endgroup$ – Philip Oakley Aug 10 '16 at 11:07

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