The Greeks were instrumental in defining theories of shapes in nature. There were Mathematical theorems, but none corresponded within a formal academic structure. It is in ancient history that the study of mathematics must begin.
Research shows, “The history of mathematics is essentially different from the history of other sciences in its relationship with the history of science because it never was an integral part of the latter in the Whewellian sense. The reason for this is obvious: mathematics being far more esoteric than the other sciences, its history can only be told to a select group of initiates” (Sarton, 1936).
Euclidian Geometry is perhaps considered the foundation of mathematics. The founder of these concepts as one universal structure was Euclid because he took different geometric ideas and transcribed them into one of the most influential books in the history of mathematics. With logic and deductive reasoning, Euclid separated the various theorems into four main areas of study:
- Plane Geometry
- Magnitudes and Ratios
- Whole Numbers
- Solid Geometry (Shuttlesworth, 2010)
The reasoning separating these ideas gave birth to the academic structure of Geometry that promoted advances in Mathematics, Science, Art, Humanities, Astrology, Astronomy, Architecture, and Physics. Analyzing the work of Euclid will show that the Greek Mathematician formulated the theories of Geometry detailed in an academic textbook called Elements that continues to influence society today.
Historical Significance of Geometry
Euclid was a Greek scholar, philosopher, and a mathematician around 300BC, who studied at Plato’s Academy in Athens. Greece was the utopia for a classical evolution of math and science theories because education and philosophy were promoted in that society. He later opened a school of thought in Alexandria where he also taught his beliefs in philosophy and math. Unfortunately, much of the life of Euclid was lost to history because of the rise of Christianity that forbid ancient studies, and it is only through references to his life by other scholars surviving pieces of literature, that we learn of his advancements in Geometric studies. While different mathematicians contributed to the ideas found in Elements, it is Euclid who is credited with forming the ideas into the academic structure still used today.
Euclid did not want to simply understand the different math theories, but he wanted to prove them to show their significance to the world in which he lived. For this reason, his work has been termed Euclidian Geometry (measurement of the earth in one mathematical system). Euclid was not a simple thinker as he was thought to be one of the leading scholars of logical thinking during his time. When society refers to him the Father of Geometry, they are not crediting him with the theories but saying he developed them into one area of study that helped to promote academic advancements. Geometry is a complex area of study, and many people do not fully understand the significance of such a discovery. To understand the importance of Euclid, one must first understand Geometry, and what society has historically used it for to promote the progression of many fields of study.
Geometry is defined as, “a branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids; broadly: the study of properties of given elements that remain invariant under specified transformations” (Merriam-Webster, n.d.). The legacy of Euclid exists within the pages of his written work (translated from Greek into Arabic, Latin, and English) because many scholars, mathematicians, and scientists reflect on his studies as the basis for their own theory studies.
The Mathematics of Euclid: Elements
Around 300BC, Euclid systematically took the random ideas and structured them into structured theorems in a collection of thirteen books he called Elements. The collection of books were so popular that only one book surpassed publication, and that was the printing of the Bible. A person was judged on intelligence if they read and understood the theorems in Elements. So, what was the importance of this book, and why did everyone want to read and study it?
- Thirteen Books of Elements
- Books I – IV, and Book VI: Plane Geometry
- Books V and X: Magnitudes and Ratios
- Books VII – IX: Whole Numbers
- Books XI – XIII: Solid Geometry (Shuttleworth, 2010)
Euclid was meticulous in dividing the books in Elements into specific areas of studies. Books one, two, three, four, and six define plane geometry. Books five and ten show theorems of magnitudes and ratios. Books seven, eight, and nine discuss theorems in whole numbers. Books ten, eleven, twelve, and thirteen all discussed theorems of solid geometry. In total, the thirteen books together created a textbook that has been hailed as the blueprint for geometry.
Book I defines basic plane geometry that teaches triangles, area, and parallels. Euclid included definitions of lines, points, straight lines, surface, the edges of any surface are lines, plane surface, plane angles, rectilinear, right and perpendicular, obtuse angle, acute angle, boundary, figure, circle, center, diameter, semicircle, rectilinear isosceles triangle, and acute-angled triangle. Euclid goes on to define a square, an oblong, a rhombus, and a rhomboid. He defined parallel lines as straight lines with the same plane and lines do not meet the other in both directions. Book I’s postulates include drawing straight lines from point to point, producing finite straight lines that continued, define circle, center of circle, and radius of circle, right angles equal to other right angles, straight lines meeting other straight lines redefines them as interior angles. Euclid also made five notions and 48 propositions. The definitions defined the geometric concepts. The postulates are axioms Euclid believed did not require proof. The common notions are also axioms that are less common than the postulates, but they defined angles, plane figures, and solid figures (Clark University, 1998). Book I may very well the most important part of the thirteen books as Euclid defined the beginnings of geometric teaching and structure.
Book II defines geometric algebra. Euclid included two definitions on rectangular parallelogram (containing two straight lines with a right angle), and gnomon (the diameter of two compliments in a parallelogrammic area). The fourteen propositions in book two define algebraic expressions (Clark University, 1998). It is complemented by Book I in that plane geometry is the basis for Book II’s quadratic expressions. However, it is very much like the algebra used in modern society today.
Book III teaches circle theorems. The eleven definitions define equal circles, circles touching lines, circles touching other circles, segments of circles, angles of segments, straight lines with angles, sectors of circles, and similar segments of circles. In this book, there are 37 prepositions that proves circle theorems. The different theories include finding the center of a circle, circumference of circle with two points, straight lines through center, different centers when two circles combine or touch, points outside of a circle when straight line is drawn through a circle, points inside a circle, less than two points when a circle cuts in another circle (Clark University, 1998).
Book IV defines figures containing inscribed and circumscribed constructions of circle, squares, triangles, and pentagons. The book includes seven definitions defining rectilinear figure that are inscribed in figures, circumscribed and pass through angles, inscribed in a circle, touches the circumference of a circle, or fitted in a circle when it ends at the circumference of a circle (Clark University, 1998). Euclid used sixteen propositions to prove these theories. Book four uses the work found in books one and three to support these theorems.
Book V defines abstract proportions in theorems on ratio and proportions. The definitions introduce ideas like magnitude, multiple, ratio, magnitudes with the same ratio, proportional, equimultiples, proportions with three terms, three magnitudes with proportional values with duplicate rations, corresponding antecedents, alternate ratios, inverse ratios, rations taken jointly, conversion of a ratio, ex aequali, and perturbed proporation (Clark University, 1998). Euclid provided proof in the twenty-five propositions in book five. Half of the propositions define multitudes of magnitudes or ratios, and the other half of them define ratios.
Book VI defines the theorems of geometric figures and proportions. The definitions in this book define similar rectilinear figures, reciprocally related figures, extreme and mean ratios, and height of figure (Clark University, 1998). There are thirty-three propositions included that define the propositions between lines, figures, angles, and circumferences.
Book VII defines number theorems. The twenty-two definitions include vocabulary like unit, number, part, multiple, even number, odd number, even-times-even number, even-times-odd number, odd-times-odd-number, prime number, prime numbers, composite numbers, relatively composite, multiply, plane, solid, sides, square number, cube, proportional, similar plane, solid numbers, and perfect numbers (Clark University, 1998). Euclid’s propositions included ratios used as properties of prime numbers.
Book VIII continues propositions and number theory in introducing twenty-seven additional propositions. These theories include theories involving advanced propositions on relatively prime numbers, proportions of ratios, proportions of square numbers, cubes with triplicate ratios, squares with duplicate ratios, measures of squares, cubic measures of numbers, numbers with similar planes, and similar solid numbers (Clark University, 1998).
Book IX continues number theory. In this book, only thirty-six propositions are stated. The theories define multiples of numbers with similar planes, multiplication of square numbers, multiplication of cubic numbers, continued proportions of prime and relatively prime numbers, even subtracted from even is even, odd subtracted from even numbers are odd, odd subtracted from odd is even, even numbers subtracted from odd numbers are odd, odd multiplied by even is even, and odd multiplied by odd equals an odd number. The rules go on to define odd numbers measure by even numbers to reduce a number by half, odd numbers using relatively prime numbers to double, and theory on even and odd multiplication of numbers (Clark University, 1998).
Book X defines the theories on the classification of incommensurables. The book includes four definitions that introduce commensurable and incommensurable measurement in straight lines and squares, rational and irrational numbers, first/second/ third/fourth/fifth/sixth binomial, first/second/third/fourth/fifth/ sixth apotome (Clark University, 1998). Euclid included one hundred and fifteen propositions that include theory continued to define how magnitudes either are commensurable (with measurement) or incommensurable (no common measurement). Book ten proves that straight lines have measurable (finite) and unmeasurable (infinite) numbers that can be rational and irrational. Binomials are two numbers used to produce an expression, and apotome defines the properties of lines. The higher degree of binomial and apotome (first/second/third/fourth/fifth/sixth) will produce more complicated algebraic expressions.
Book XI defines solid geometry theory. It contains twenty-eight definitions that introduce vocabulary like solid, face of solid, right angles of straight lines, inclination of straight line, inclination of a plane, similarly inclined, parallel planes, similar solid figures, equal and similar solid figures, solid angle, pyramid, prism, semicircle with fixed diameter, axis a of sphere, center of a sphere, diameter of a sphere, right triangle, cones that are right, obtuse, or acute-angled, axis of cone, base, rectangular parallelogram, axis of cylinder, bases, similar cones and cylinders, cube, octahedron, icosahedron, and dodecahedron (Clark University, 1998). The propositions again show how straight lines parallel straight lines, perpendicular lines, straight lines with angles, parallel lines with planes, solid angles, and parallelepipedal solids define theorems of solid geometry.
Book XII defines measurement and figures to which both were defines in a previous book. The book in twelve contains only eighteen propositions to define more complex geometric structures like polygons, pyramids, cones, cylinders, and spheres.
Book XIII defined theories of regular solids. Like book XII, XIII relied on only eighteen propositions as further proof of solid geometry. Straight lines, rational straight lines, equilateral pentagons, equiangular pentagons, hexagons, decagons seemed to further define Euclid’s straight line theory. In addition, he wrote about the rules for constructing pyramids, octahedrons, cubes in spheres, icosahedron, and dodecahedron. Based on Book X’s theories on apotome, he shows how to construct the five figured shapes (straight line theory) for comparison in this part of Elements.
The thirteen books collectively define the basis for geometry studies. The theories further define the basic principles of how geometry should be used by mathematicians. For many centuries, Euclidean Geometry has been used to influence many fields of study.
The Real-World Applications of Euclid’s Elements
Society can see the influence of mathematics by studying the history of technology, art, nature, engineering, astronomy, medicine, fashion, architecture, and physics, and other fields. In the 9th century, a Persian mathematician named Mohammed ibn-Musa al-Khowarizmi, used Euclid’s work on Algebra found in Book II to develop algorithms that multiplied and divided numbers quickly. Today, algorithms are used in technology in many different ways. One successful use of algorithms can be found in policing. In Memphis, Tennessee, in 2005, the police began using a new system called CRUSH (Criminal Reduction Utilising Statistical History) to target crime areas. Algorithms helped to predict where the crimes would take place. Research shows, “It’s putting the right people in the right places on the right day at the right time,” said Dr. Richard Janikowski, an associate professor in the department of criminology and criminal justice at the University of Memphis, when the scheme launched” (Guardian News, 2016). By using this form of math, crime fell by 24% that was contributed directly to the program. Another program that uses algorithms is the US National Security Agency (NSA). In addition, electronic banking, internet searches, and trading all depend on algorithm programs. Dr. Panos Parpas said, “They’ve been used for decades – back to Alan Turing and the codebreakers, and beyond – but the current interest in them is due to the vast amounts of data now being generated and the need to process and understand it. They are now integrated into our lives” (Guardian News, 2016). It further defines how important Algebraic Geometry is to modern society.
Another real world application of Euclid’s mean ratio theorems from Book V and VI is found in the work of Leonardo Da Vinci (1509AD). Today, society terms this work as the Golden Ratio that many artists used to produce art that was both beautiful and balanced. Da Vinci used the work of Fibonacci’s nature theory (1200AD) directly, but his work was influenced by Euclid’s work on ratios of 1:1.618. The use of Euclid is found in Da Vinci’s Golden Pyramid, Golden Rectangle (Mona Lisa), Golden Ratio (Last Supper), and Golden Pyramid (Madonna on the Rocks) that shows how ratio, mean, and straight-line theory influenced art.
Da Vinci’s work led to other advances in math and science. However, his art influenced his own thoughts on anatomy, technology (scissors), and engineering (parachutes and helicopters). Many of his ideas were just thoughts he put on paper, but many inventors and scholars founded modern day mathematics based on his ideas. However, in 2001, a group of engineers took one of his sketches called the Golden Horn Bridge that was 240 meters long. The bridge was never built. In 2016, students at Eindhoven University of Technology, in Juuka, Finland, brought Da Vinci’s ideas to life in a project called Bridge of Ice. While Da Vinci imagined stone, these students are using frozen water to construct the pedestrian bridge.
For 2,300 years, Euclid’s work has been the foundation for academic studies and scientific advances. As Euclid laid the groundwork for the future, many men and women used his work to promote the society in which they were a part of. In so doing, it created the society we live in today where Euclidean geometry is used in every part of our lives from banking to technology. Many of the greatest mathematicians, scientists, and artists lay their success at the feet of Euclid’s Elements. The books also dared men to prove him wrong which led to the discovery of Non-Euclidean Geometry. To date, some might argue that Euclid remains the most influential mathematician in history. His work dares men to question, to use logical thinking, and to dream of the possibilities the world offers. Because his work was transcribed from papyrus rolls by many different men over time, it also offers people who love math to continue to question and test his theories. In so doing, it leads new minds to rethink old idea that often leads to new discoveries in math. It also dares people today to answer questions people have yet to ponder. In so doing, his work continues to influence the world in which we live.
Clark University. (1998). Euclid’s Elements. Department of Mathematics and Computer Science. Retrieved on February 19, 2016, from website http://aleph0.clarku.edu/~djoyce/java/elements/elements.html
Guardian News. (2016). “How Algorithms Rule the World.” Retrieved on March 10, 2016 from website https://www.theguardian.com/science/2013/jul/01/how-algorithms-rule-world-nsa
Huffington Post. (2016). Students Build a Bridge Based on A Leonardo Di Vinci Sketch. Image. Retrieved on March 11, 2016, from website http://www.huffingtonpost.com/entry/students-build-a-bridge-based-on-a-leonardo-da-vinci-sketch-using-nothing-but-ice_us_568aa0bfe4b06fa68882e1eb
Library of Congress. (n.d.). Euclid, Elements. In Greek. Parchment. Ninth century. Image. Retrieved on February 27, 2016, from website http://www.loc.gov/exhibits/vatican/images/math01.jpg
Geometry. (n.d.). Retrieved February 27, 2016, from http://www.merriam-webster.com/dictionary/geometry
Mathematician Pictures. (n.d.). Leonardo Da Vinci and The Golden Ratio. Image. Retrieved on March 10, 2016, from website http://www.mathematicianspictures.com/LEONARDO_DA_VINCI_GOLDEN_RATIO_GOLDEN_MEAN/Leonardo_Da_Vinci_Golden_Ratio_Golden_Mean.htm
Sarton, G. (1936). The study of the history of mathematics (pp. 43-53). Cambridge, Mass: Harvard University Press.
Shuttleworth, Mary. (May 12, 2010). Euclid, the Father of Geometry. Retrieved on February 19, 2016, from website https://explorable.com/euclid