A Historical Introduction to Elementary Geometry and Apollonius’ Conics

Etymology and Historical Foundations of Geometry

• Etymology: The word "geometry" is derived from ancient Greek, consisting of two parts: $ge$, meaning earth or land, and $metria$, meaning measure.
• Euclid of Alexandria: Euclid wrote the Elements of geometry between $330$ and $320$ B.C. This work served as a compilation of the major theorems on plane and solid geometry.
• The Elements: The work is organized into thirteen books and is presented in an axiomatic style. In the first book, Euclid enumerated five fundamental assumptions, known as postulates or axioms, which served as the foundation for proving propositions and theorems in both two and three dimensions.

The Five Postulates of Euclidean Geometry

• Postulate 1: Any two points can be joined by a straight line.
• Postulate 2: Any straight line segment can be extended indefinitely in a straight line.
• Postulate 3: Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. In the context of plane geometry, this circle is tacitly unique. In three dimensions, this postulate defines a sphere.
• Postulate 4: All right angles are congruent.
• Postulate 5 (Parallel Postulate): If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
• Playfair's Axiom: This statement holds only in plane geometry and leads to the same geometry as Postulate 5: Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.

Historical Reception and Mathematical Development

• Euclid's Standing: Euclid's fame was based on the Elements, which was so superior to similar works by Hippocrates, Leon, and Theudius that those earlier works "perished in the struggle for existence." The Greeks referred to Euclid as "the author of the Elements."
• Educational Impact: Even after two millennia, the Elements is regarded by many as the best introduction to mathematical sciences. In England, it was used extensively as a school textbook until the 20th century.
• Misconceptions of Discovery: Some editors falsely suggest Euclid created a finished system "as an armed Minerva from the head of Jupiter," ignoring his use of earlier mathematicians' materials. Very few propositions are Euclid's own discoveries; notably, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him.
• Abraham Lincoln and "Demonstrate": The 16th president of the United States studied Euclid to understand the meaning of "demonstrate" after finding dictionary definitions insufficient. He eventually mastered the first six books of Euclid, enabling him to give any proposition at sight. Lincoln structured his presidential speeches and legal cases based on the six elements of a proposition used by Euclid.
• Post-Euclidean Advancement: Following Euclid and Archimedes, the Hindu-Arabic decimal-based number system, arithmetic, algebra, and modern trigonometry were developed throughout Asia during the first millennia A.D.

The Introduction of Analytic Geometry

• René Descartes: In 1637, Descartes published Discourse on Method: The Geometry, which introduced Analytic (or Cartesian) geometry. This field utilizes real-number algebra and Euclidean geometry theory to describe properties of figures such as lines, polygons, curves, surfaces, and polyhedrons.
• Apollonius of Perga: Although Descartes is credited with invention, it is argued that the subject originated around $240$ B.C. when Apollonius wrote the Conics (Books 1-5). Apollonius described a parabolic conic section using horizontal and vertical displacements from a vertex point.

Fundamental Euclidean Theorems and Proofs

• Theorem 1: Angle Sum of a Triangle: The sum of the interior angles in any triangle is equal to two right angles ($180^\circ$ or a straight angle).
  • Pythagorean Proof: Attributed to the Pythagoreans via Eudemus.
  • Outline: Drawing line $DAE$ through vertex $A$ parallel to side $BC$ and extending segment $BA$ to $BAB'$. Using Euclid's 5th Postulate to show that if $\angle B'AE \neq \angle ABC$, the lines would meet, creating a contradiction. This establishes $\angle ABC = \angle BAD$ and alternate interior angles $\angle EAC = \angle ACB$. Their sum at the vertex identifies the total as a straight angle.
• Theorem 2: Isosceles Triangle Base Angles: In any isosceles triangle, the interior angles opposite the equal sides are equal.
  • SSS Criterion (Proposition 1.8): If two triangles have two sides equal respectively and the base equal to the base, the encompassed angles are equal. Euclid proved this using "superposition," placing one triangle on top of another.
  • Outline: In $\triangle ABC$ where $AB = AC$, let $P$ be the midpoint of $BC$. Then $\triangle ABP \cong \triangle ACP$ by SSS ($AB = AC$, $BP = PC$, and $AP = AP$), meaning $\angle ABC = \angle ACB$.
• Theorem 3: Triangle Inscribed in a Semicircle: Every triangle inscribed in a semicircle is a right triangle.
  • Outline: For $\triangle ABC$ with diameter $AC$ and center $O$, $OA = OB = OC = r$. Thus $\triangle AOB$ and $\triangle COB$ are isosceles. Let $\angle OAB = \alpha$ and $\angle BCO = \beta$. By substitution into the triangle angle sum: $\alpha + (\alpha + \beta) + \beta = 180^o$. Simplifying to $2\alpha + 2\beta = 180^o$, it follows that $\alpha + \beta = 90^o$, making $\angle ABC$ a right angle.
• Theorem on Similar Triangles (Proposition 4 of Book II): Two triangles have corresponding sides proportional if and only if their corresponding angles are equal in measure.
  • $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$

Conic Sections and the Parabola

• Conic Surface Definition: Generated by a fixed vertex and a straight line rotating about the circumference of a circle not in the same plane as the vertex. The line produces two vertically opposite surfaces. The line from the vertex to the center of the circle is the axis.
• Cone Definition: The figure contained by the circle (base) and the conic surface between the vertex and the circle.
• The Parabola as a Locus: Apollonius defined every parabola with vertex $E$ as the locus of all points $L$ for which $EM \times p = ML^2$ for some real constant $p$. Descartes expressed this as the equation $p \times x = y^2$ with the vertex at $(0,0)$.
• Symmetry and Derivation:
  • A parabola has a line of symmetry established via Proposition 1.4 (SAS Criterion).
  • In a geometric derivation using similar triangles and segments related to the cone's properties, the segment length $EM$ is shown to be proportional to the square of segment length $ML$.
  • Derivation involves proportions: $\frac{OR}{BC} = \frac{PR}{BA} = \frac{PM}{PE}$ and $\frac{MR}{EA} = \frac{PM}{PE}$.
  • The final geometric identity is specified as $EH \times EM = MR \times PM$. Substituting $ML^2 = MR \times PM$, the result is $EH \times EM = ML^2$.
  • Setting $EH = p$, $EM = x$, and $ML = y$ yields the Cartesian form: $p \times x = y^2$.