Re: [Phys-L] Conceptual Physics Course

["John Clement" <clement@hal-pc.org>]

As I recall the original source was an article on Newton published in TPT.
But here is an excerpt from
http://www.encyclopedia.com/topic/Sir_Isaac_Newton.aspx

"As to analysis itself, David Gregory recorded that Newton once said
“Algebra is the Analysis of the Bunglers in Mathematicks.”56 No doubt!
Newton did, nevertheless, devote his main professorial lectures of 1673–1683
to algebra,57 and these lectures were printed a number of times both during
his lifetime and after.58 This algebraical work includes, among other
things, what H. W. Turnbull has described as a general method (given without
proof) for discovering “the rational factors, if any, of a polynomial in one
unknown and with integral coefficients”; he adds that the “most remarkable
passage in the book” is Newton’s rule for discovering the imaginary roots of
such a polynomial.59 (There is also developed a set of formulas for “the
sums of the powers of the roots of a polynomial equation.”)60

Newton’s preference for geometric methods over purely analytical ones is
further evident in his statement that “Equations are Expressions of
Arithmetical Computation and properly have no place in Geometry.” But such
assertions must not be read out of context, as if they were pronouncements
about algebra in general, since Newton was actually discussing various
points of view or standards concerning what was proper to geometry. He
included the positions of Pappus and Archimedes on whether to admit into
geometry the conchoid for the problem of trisection and those of the “new
generation of geometers” who “welcome” into geometry many curvcs, conies
among them.61

Newton’s concern was with the limits to be set in geometry, and in
particular he took up the question of the legitimacy of the conic sections
in solid geometry (that is, as solid constructions) as opposed to their
illegitimacy in plane geometry (since they cannot be generated in a plane by
a purely geometric construction). He wished to divorce synthetic geometric
considerations from their “analytic” algebraic counterparts. Synthesis would
make the ellipse the simplest of conic sections other than the circle;
analysis would award this place to the parabola. “Simplicity in figures,” he
wrote, “is dependent on the simplicity of their genesis and conception, and
it is not its equation but its description (whether geometrical or
mechanical) by which a figure is generated and rendered easy to
conceive.”62"

And of course he wrote the Pricipia using geometric proofs.  While he
understood that algebra was useful his distrust was that it concealed the
concepts.  So while he used algebra, his preference was for geometric proof
if possible.  Actually he tended to shy away from mathematics early on, but
conquered his dislike of it.

> From the point of view of students in physics courses the idea that algebra
is too easy is quite valid.  Modeling has found that geometic
interpretations using graphs should be taught first.  Then after this is
mastered algebra becomes a useful tool and students are better problem
solvers.  So having the conceptual courses use graphs heavily is heading
them in the right direction.  Actually using graphs they can easily solve
difficult problems which algebra based students gag on.  This is also the
philosophy in "Minds on Physics" from UMPERG.  The conceptual students
usually never understood algebra.

John M. Clement
Houston, TX

>    Your statement about Newton distrusting algebra for 
> various reasons intrigues me. Do you have a source for that 
> statement? It doesn't accord with most that I've read about or by him.