Pp. 262-298 (37)
Seymour B. Elk
In the now traditional subdivision of mathematics into algebra, geometry and
analysis popularized by Felix Klein over a century ago, calculus was the first subset of
analysis taught in the educational system. Meanwhile, precisely because many of the uses of
calculus were associated with a geometrical perspective, a special subset of mathematics,
called “analytic geometry”, was formulated and taught (sometimes interdigited and
sometimes as a separate course) concurrent with the underlying foundations of calculus. In
particular, as well as the basic algebra discussed up to this point, an entire chapter is herewith
interjected before continuing on this author’s path to understanding “what is calculus”.
Because one specific geometric figure, the hyperbola, will be shown to play an important role
in that third fundamental level of complexity, exponentiation and root extraction, this treatise
diverts its focus from the l’Hôpitalian aspects associated with extending algebra beyond the
real finite domain in order to probe what may be considered to be a “tangential” path. (Here
the word “tangential” has the layman’s connotation of moving away from a central idea that
is in the process of being developed. This is in contradistinction to any role as a term in
In an attempt to remove the blinders that have been firmly fastened onto the student by the
traditional presentation, this treatise eschews always remaining focused on a single dimension
(be it either a line, a plane or a three dimensional embedding space) and assumes the
perspective of multi-dimensionality. Additionally, even though most of the presentation is in
the domain of real numbers, when appropriate, the place of complex numbers in the overall
scheme of “analysis” is not overlooked. This will be especially useful in understanding the
correlation between traditional (circular) trigonometry of Chapter 2 and a differently focused
“hyperbolic” trigonometry, which will be the focus of Chapter 6.
Analytic Geometry, Conic Sections, Cylinder, Ellipse, Ellipsoid,
Hyperbola, Hyperboloid of One Sheet, Hyperboloid of Two Sheets, Imaginary
Ellipsoid, Linear Equations, Parabola, Paraboloid, Quadric Cone, Quadratic Equations, Rotation of Axes, Translation of Axes.
Elk Technical Associate, New Milford, New Jersey 07646 U.S.A.