Trigonometry, at its core, is the branch of mathematics that deals with the relationships between the sides and angles of triangles. While today we use it in everything from navigation systems to computer graphics, the story of trigonometry stretches back millennia, rooted in the practical needs of ancient civilizations and blossoming into a sophisticated field of study thanks to groundbreaking contributions from various cultures. Let’s delve into the fascinating history of trigonometry, exploring its origins and evolution.
Ancient Origins: Proto-Trigonometry in Egypt
The earliest traces of trigonometric concepts can be found in ancient Egypt. Far from abstract theory, their understanding emerged from practical necessities, particularly in construction. Consider the massive pyramids – their precise angles and slopes speak volumes about the Egyptians’ geometric knowledge. The term “seked,” found in the Rhind Papyrus dating back to around 1800 BCE, is a key example of this “proto-trigonometry.”
The seked was essentially a measure of the slope of an incline. Problem 56 of the Rhind Papyrus illustrates this clearly, asking for the seked of a pyramid 250 cubits high with a base side of 360 cubits. The solution, expressed in palms per cubit, is equivalent to the ratio 18/25. This ratio, the “run-to-rise,” is, in effect, the cotangent of the angle between the pyramid’s base and its face. This demonstrates that the Egyptians possessed a working knowledge of numerical relationships within triangles, a foundational step towards what we recognize as trigonometry. Their focus was practical application, calculating slopes for construction, rather than developing a formal mathematical system.
Greek Foundations: Hipparchus and the Table of Chords
Trigonometry as a systematic branch of mathematics truly began to take shape with the ancient Greeks. Hipparchus of Nicaea (c. 190–120 BCE) is considered a pivotal figure, often hailed as the “father of trigonometry.” His major innovation was the construction of a table of chords, a precursor to our modern trigonometric tables.
Hipparchus’s approach was to inscribe triangles, whether planar or spherical, within circles. Each side of the triangle then became a chord of the circle. To solve for the parts of the triangle, he focused on the relationship between the length of a chord and the central angle that subtended it. Essentially, he aimed to define the length of a chord as a function of the angle or arc width. This became the central task of trigonometry for centuries to come. While primarily an astronomer interested in celestial triangles, Hipparchus also understood plane trigonometry. In his era, these relationships were expressed geometrically, relating chords and angles directly, as the symbolic notation for trigonometric functions wouldn’t emerge until the 17th century.
Ptolemy’s Almagest: A Comprehensive Trigonometric System
Building upon Hipparchus’s work, Claudius Ptolemy (c. 100–170 CE) further advanced trigonometry in his monumental work, the Almagest. This 13-book compendium on astronomy became the cornerstone of astronomical understanding for over 1400 years. Trigonometry was essential for Ptolemy to develop his geocentric model of the cosmos.
Chapters 10 and 11 of Book I of the Almagest are dedicated to constructing a table of chords. This table provided the length of a chord in a circle as a function of the central angle subtending it, for angles from 0° to 180° at half-degree intervals. This table was effectively a sine table. The relationship can be understood by considering the radius r, arc A, and chord length c: c = 2r sin (A/2). Ptolemy, using the Babylonian sexagesimal system (base 60), worked with a standard circle radius of 60 units, simplifying the relationship to c = 120 sin (A/2). Thus, his table provided values proportional to sin (A/2), and by extension, sin A. Using his refined trigonometric tools, Ptolemy improved geographical measurements and enhanced models of celestial movements, solidifying trigonometry’s place in both astronomy and mathematics.
Transition to Modern Trigonometry
While the ancients laid the groundwork with chords and geometric relationships, modern trigonometry evolved with the introduction of trigonometric functions like sine, cosine, and tangent, and their analytical properties. These functions, initially derived from the chord concept, became fundamental tools in various fields beyond astronomy, including physics, engineering, and computer science.
In conclusion, the answer to “What Is Trigonometry?” is multifaceted. It is a mathematical discipline focused on triangle relationships, but it’s also a story of human ingenuity, evolving from practical problem-solving in ancient Egypt to a sophisticated system developed by Greek mathematicians like Hipparchus and Ptolemy. Their work not only advanced astronomy but also paved the way for the wide-ranging applications of trigonometry we rely on today.
