Question

Ptolemy must have been aware of a method of trisecting angles by the use of conic sections. Such a method would have enabled him to construct the chord of \(\frac{1}{2}^{\circ}\) given that he knew the chord of \(1 \frac{1}{2}^{\circ}\). Why would Ptolemy not have considered this to be a construction by "geometrical methods"? Can one use such a construction to calculate the chord of \(\frac{1}{2}^{\circ}\) numerically?

Short answer

Answer: Ptolemy might not have considered trisecting angles using conic sections as a "geometrical method" because it involves mathematical objects that cannot be constructed using only a compass and a straightedge, which were the primary tools for constructions in his time. Although trisecting angles using conic sections could theoretically allow calculating the chord of \(\frac{1}{2}^{\circ}\) numerically, Ptolemy wouldn't have considered it a proper geometrical method, and we would need to rely on other methods for estimating the chord length of such small angles.

Step-by-step solution

Explanation of Ptolemy's Geometrical Methods

Ptolemy was a Greek mathematician and astronomer who lived around 2000 years ago. He considered "geometrical methods" to be the process of compass and straightedge constructions that were available during his time. In general, these methods required that all steps in a construction be done using only a compass for drawing circles and a straightedge for drawing lines. Such constructions were exact and did not involve any approximations. The main reason why Ptolemy would not consider trisecting angles by the use of conic sections to be a "geometrical method" is because it involves the use of mathematical objects (conic sections) that cannot be constructed using only a compass and a straightedge.

Conic Sections and Trisecting Angles

Conic sections are the curves obtained by intersecting a cone with a plane at different angles. There are four types of conic sections: circle, ellipse, parabola, and hyperbola. To trisect an angle using conic sections, we need to find curve intersections that split the angle into three equal parts. This method generally involves finding a suitable curve and then locating points of intersection, both of which may require steps that are not possible with a compass and straightedge alone.

Using Conic Sections for trisecting 1.5 degree angle

Let's assume we do know how to trisect angles using conic sections, and we have somehow determined the curve and points of intersection that would enable us to trisect a 1.5 degree angle. We should then be able to find the angle of \(\frac{1}{2}^{\circ}\) and use that angle to find the corresponding chord in Ptolemy's chord table.

Calculating the Chord of \(\frac{1}{2}^{\circ}\) Numerically

To calculate the chord of \(\frac{1}{2}^{\circ}\) numerically, we need to have a right triangle with one vertex at the center of the circle, one vertex on the circumference, and the other vertex at the other side making a half-degree angle with the first side. The chord length we need to calculate will be the distance between the two vertices on the circumference. If we assume a circle of radius 1, then the chord length can be found using the law of cosines as follows: \(c^2 = a^2 + b^2 - 2ab\cos{\theta}\) Where \(a\) and \(b\) are the sides of length 1, and the angle \(\theta\) is \(\frac{1}{2}^{\circ}\). Substituting the values and using the cosine of \(\frac{1}{2}^{\circ}\), we can calculate the chord length numerically. While trisecting angles using conic sections might be theoretically possible, Ptolemy wouldn't have considered it as a proper geometrical method, so we would have to rely on other methods for estimating the chord length of \(\frac{1}{2}^{\circ}\).

Key concepts

Ptolemy's theorem

Ptolemy's theorem is a fundamental theorem in the field of geometry that relates to cyclic quadrilaterals. A cyclic quadrilateral is a four-sided figure where all the vertices lie on a single circle. According to Ptolemy's theorem, the sum of the products of a pair of opposite sides of a cyclic quadrilateral is equal to the product of its diagonals. Mathematically, if we have a cyclic quadrilateral ABCD, then the theorem can be expressed as \[ AC \cdot BD = AB \cdot CD + AD \cdot BC \]This theorem was a significant achievement in ancient Greek mathematics and played a crucial role in the field of trigonometry. Although Ptolemy's theorem primarily pertains to cyclic quadrilaterals, it intersects with trisection problems because of Ptolemy's work with chords. His chord table was a sort of predecessor of modern trigonometric tables. When attempting to trisect an angle, Ptolemy could conceive certain methods through extensions of geometric principles, but he remained bound by the limitations of compass and straightedge construction. Thus, while the concepts of angles and circular measures were very much in his wheelhouse, Ptolemy wouldn't have classified methods involving more complex figures like conic sections as valid geometric methods.

Conic sections

Conic sections are an important concept in both geometry and its applications to real-world phenomena. They are formed by intersecting a cone with a plane at various angles and positions. The different types of conic sections include:

• Circle: Obtained when the intersecting plane is perpendicular to the axis of the cone.
• Ellipse: Formed when the plane intersects the cone at an angle, but does not go through the base.
• Parabola: Created when the plane is parallel to an edge of the cone.
• Hyperbola: Occurs when the plane intersects both halves of the double cone.

These sections are not only fascinating mathematically but also have practical applications in physics, astronomy, and engineering. For example, the orbits of planets are elliptical, and parabolic dishes are used in satellite receivers. In terms of geometry, conic sections provide a means to tackle problems that cannot be solved by using compass and straightedge constructions alone. They offer methods to address classic geometric quandaries, such as trisecting an angle. Though this allows for a solution where traditional approaches fail, Ptolemy and his contemporaries adhered strictly to using only compass and straightedge, thereby considering such solutions not truly 'geometrical.'

Compass and straightedge construction

Compass and straightedge construction is a method of creating lengths, angles, and other geometric figures in an exact way, without measurement tools. This approach was the gold standard for geometrical constructions, predominantly used by ancient Greek mathematicians like Ptolemy and Euclid. The main principles of compass and straightedge constructions are:

• Only a compass and a straightedge are allowed: The compass is used for drawing circles, while the straightedge is used for drawing straight lines.
• No measurements: Unlike using a ruler, the straightedge has no markings.
• Exactness: The constructions must be precise and not rely on approximation.

A famous set of problems regarding these restrictions includes the impossibility of certain tasks, such as trisecting an arbitrary angle, doubling the cube, or squaring the circle. Such tasks are impossible to achieve under these constraints because they generally require solutions that extend beyond the available tools, such as the use of conic sections or other curves. These limitations make compass and straightedge constructions both fascinating and challenging. They represent a beauty of rigor and simplicity in mathematics. However, knowing these constraints can also drive curiosity to explore other mathematical tools and ways to solve problems that appear insurmountable with just a compass and straightedge.