Apollonius of Perga
Book I

Book I presents 58 propositions. Its most salient content is all the basic definitions concerning cones and conic sections. These definitions are not exactly the same as the modern ones of the same words. Etymologically the modern words derive from the ancient, but the etymon often differs in meaning from its reflex.

A conical surface is generated by a line segment rotated about a bisector point such that the end points trace circles, each in its own plane. A cone, one branch of the double conical surface, is the surface with the point (apex or vertex), the circle (base), and the axis, a line joining vertex and center of base.

A section (Latin sectio, Greek tome) is an imaginary "cutting" of a cone by a plane.

The Greek geometers were interested in laying out select figures from their inventory in various applications of engineering and architecture, as the great inventors, such as Archimedes, were accustomed to doing. A demand for conic sections existed then and exists now. The development of mathematical characterization had moved geometry in the direction of Greek geometric algebra, which visually features such algebraic fundamentals as assigning values to line segments as variables. They used a coordinate system intermediate between a grid of measurements and the Cartesian coordinate system. The theories of proportion and application of areas allowed the development of visual equations. (See below under Methods of Apollonius).

The "application of areas" implicitly asks, given an area and a line segment, does this area apply; that is, is it equal to, the square on the segment? If yes, an applicability (parabole) has been established. Apollonius followed Euclid in asking if a rectangle on the abscissa of any point on the section applies to the square of the ordinate. If it does, his word-equation is the equivalent of y 2 = k x {\textstyle y^{2}=kx} which is one modern form of the equation for a parabola. The rectangle has sides k {\displaystyle k} and x {\displaystyle x}. It was he who accordingly named the figure, parabola, "application".

The "no applicability" case is further divided into two possibilities. Given a function, f (x) {\textstyle f(x)}, such that, in the applicability case, y 2 = g (x) {\textstyle y^{2}=g(x)}, in the no applicability case, either y 2 > g (x) {\textstyle y^{2}>g(x)} or y 2 < g (x) {\textstyle y^{2}<g(x)}. In the former, g (x) {\displaystyle g(x)} falls short of y 2 {\textstyle y^{2}} by a quantity termed the ellipsis, "deficit". In the latter, g (x) {\displaystyle g(x)} overshoots by a quantity termed the hyperbole, "surfeit".

Applicability could be achieved by adding the deficit, y 2 = f (x) = g (x) + d, {\textstyle y^{2}=f(x)=g(x)+d,} or subtracting the surfeit, g (x) − s. {\displaystyle g(x)-s.} The figure compensating for a deficit was named an ellipse; for a surfeit, a hyperbola. The terms of the modern equation depend on the translation and rotation of the figure from the origin, but the general equation for an ellipse,

can be placed in the form

where C / B {\displaystyle C/B} is the deficit, while an equation for the hyperbola,

becomes

where C / B {\displaystyle C/B} is the surfeit.
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