You wrote:
> I know that Fermat and Descartes share the credit for developing
> analytic geometry. But did either of them make the leap into
> three dimensions, or was that someone else?
Here is what I have gleaned from Boyer's
History of Analytic Geometry
On page 81. A short treatise, Isagoge ad Locos ad Superficiem, carried the
problem of loci to 3 dimensions, but it did not make use of
the analytic method. . . The rise of solid analytic geometry did not take
place until about a century later, even though Fermat himself was aware of
the fundamental principle. In a half page work entitled Novus Secundarum
et Ulterioris Ordinis Radicum in Analyticis Usus, he repeated and extended
his discovery of 1629:
There are certain problems which involve only one unknown, and which can
be called determinate, to distinguish them from the problems of loci.
There are certain others which involve 2 unknowns and which can never be
reduced to a single one; these are the problems of loci. In the 1st
problems we seek a unique point in the latter a curve. But if the
proposed problem involves three unknowns, one has to find, to satisfy
the question, not only a point or a curve, but an entire surface. In
this way surface loci arise, etc. Oeuvres v.1, p. 186-7
v.3, p. 161-2
What a pity it is that Fermat did not extend his analytic study to problems
in the latter categories! The whole history of analytic geometry might
have been different, for developments in 3-space later played an important
part in
revising the Cartesian geometry of 2 dimensions; and the analytic geometry
of more than 3 dimensions, at which Fermat seems to hint, was not developed
until
2 centuries later.
Page 93: Descartes then adds the cryptic remark that
if 2 conditions for the determination of a point are lacking, the locus
of the point is a surface which may be plane or spherical or more
complicated. Book II, page 335 in AT
This hint of an analytic geometry of 3 dimensions reappears at the close of
Book II where Descartes indicates that his remarks on plane curves
can easily be made to apply to all those curves which can be conceived of
as generated by the regular movement of the points of a body in
3-dimensional space.
* * *
This volume of Carl Boyer constitutes numbers 6 and 7 of
The Scripta Mathematica Studies,
which was published by
Scripta Mathematica
Yeshiva University
Amsterdam Avenue at 186th Street
New York 33, N. Y.
1956
I would be surprised to learn that it is still in print.
Best wishes,
Sam Kutler