The History of Geometry: How It All Began

Geometry is a mathematical field appertained to the shape of specific objects, features of surrounding space, and the spatial relations amid different objects (Coolidge, 2013). Geometry is among the oldest mathematical branches whose origin is traced back to its invention as a domain of knowledge concerned with spatial relationships. Geometry is among the two primary branches of pre-modern mathematics. Classical geometry incorporated straightedge and compass constructions. Euclid revolutionized the conceptualization of geometry by introducing concepts such as the axiomatic method and mathematical rigor; these concepts are still employed in various mathematical analyses to date. The Elements of Geometry , a book written by Euclid, is widely recognized as the most influential literature of all time (Coolidge, 2013). The contemporary geometric conceptualizations have been generalized to a significant degree of complexity and abstraction; these conceptions have been predisposed to abstract algebra and calculus procedures. The earliest geometrical inceptions may be traced back to around 3000BC. During this era, the people in Ancient Babylonia and Ancient Indus Valley invented the obtuse triangle. Early geometry was an assemblage of empirically developed precepts associated with aspects such as volumes, areas, angles, and lengths that were invented to meet various practical needs in fields such as astronomy, construction, and surveying. 

Egyptian Geometry 

Ancient Egyptians comprehended geometrical conceptualizations, for instance, computing the volume and surface area of three-dimensional structures used in algebra and architectural engineering; this includes quadratic equations and the false position technique. Ancient Egypt’s geometrics concepts were discovered in the RMP (Rhind Mathematical Papyrus) and MMP (Moscow Mathematical Papyrus) (Gray, 2011). The solved examples indicate that the Ancient Egyptians were cognizance with the calculation of area; they computed the areas of various geometric shapes and pyramidal and cylindrical volumes. The Ancient Egyptians inscribed their geometrical problems in different parts, and they often assigned a specific title and data to a particular mathematical problem. In certain texts, the ancient Egyptians would delineate how to solve a specific mathematical problem and ultimately verify its accuracy. The scribes did not employ any variables in explaining various mathematical concepts, and the mathematical problems were usually inscribed in prose. Additionally, the mathematical solutions were inscribed in stages which delineated the entire process. The ancient Egyptians understood that a triangle’s area may be determined by using this formula where h represents height and b represents base. The computations of a triangle’s area appear in the MMP and RMP. Problem forty-nine in the RMP computes the rectangular piece of land’s area. Additionally, MMP’s problem six computes the lengths of a rectangular area’s sides giving the lengths as a ratio. RMP’s problem forty-eight compares a circle’s area (estimated by an octagon) and its confined or circumscribed square. Problem forty-eight solution is then incorporated into problem fifty. The octagonal figure’s area is given by 

Babylonian Geometry 

The Babylonians were familiar with the general principles for determining volumes and areas. The Babylonians estimated a circle’s circumference as three times its diameter. A circle’s area, on the other hand, was estimated as a twelfth of its circumference’s square (C 2 ); this is equivalent to π = 3. A cylinder’s volume was estimated as the product of the cylinder’s height and base. The volume of a square pyramid’s or cone’s frustum was estimated as the product of the pyramid’s height and half the summation of the pyramid’s bases (Leech, 2007). The Babylonians were also conversant with the Pythagorean Theorem. Additionally, a recent discovery revealed that certain Babylonian scribers used π as 1/8 and 3. The Babylonians are also typified by the Babylonian mile conception which is the estimation of distance and is equivalent to approximately 7 miles today. The mensuration of distances was ultimately transformed into a time-mile. The time-mile was used for estimating the sun’s travel, thereby, representing time. Recent studies reveal that ancient Babylonians could have invented astronomical geometry around 1400 years prior to the European’s discovery (Leech, 2007). 

Greek Geometry 

Ancient Greek Geometry 

Geometry was the foundation of the Greek mathematicians’ sciences which attained a significant level of absoluteness and methodological perfection not achieved by any other field of knowledge during that period. The Greek mathematicians extended their geometric range to various types of solids, surfaces, curves, and figures; these mathematicians transformed its techniques from trial-and-error to rational inferences (Leech, 2017). The Greeks acknowledged that the physical objects’ abstractions in geometric studies are approximations and thy coined the axiomatic method conception which is still used to date. 

Pythagoras and Thales 

Various mathematical deductions were ascribed to Thales. Thales inscribed deductive pieces of evidence for five geometric theorems. However, Thales’ deductive evidence was not conclusive. Pythagoras, on the other hand, was among the first individuals to provide a deductive proof to the Pythagorean Theorem, although he did not invent the theorem (Coolidge, 2013). Additionally, Pythagoras together with his students established a profound invention of irrational numbers and incommensurable lengths. 

Plato 

Plato is a highly esteemed philosopher among the Greeks. Despite the fact that he was not a mathematician, Plato’s perception of mathematical concepts was of significant influence. Greek mathematicians, therefore, consented to his idea that geometry should incorporate the use of a straightedge and compass as opposed to measuring tools such as the protractor and marked ruler (Scriba & Schreiber, 2015). He associated the aforementioned measuring tools to a workman’s instruments. Plato’s principle triggered the in-depth research on the practical construction of the straightedge and compass and three ancient construction problems which include how to utilize these tools when trisecting an angle, how to construct or design a cube whose volume is twice the volume of a specific cube, and how to construct a square whose area is equivalent to that of a specified circle. The evidence of these constructions’ impossibility ultimately attained during the nineteenth century, led to the development of crucial precepts concerning a real number system’s depth structure. Aristotle, Plato’s illustrious student, composed a disquisition on the modus operandi of reasoning employed in various deductive proofs; this treatise was later improved during the nineteenth century. 

Hellenistic Geometry 

Euclid 

Euclid formulated a discourse in thirteen chapters entitled The Elements of Geometry where he delineated geometry in a perfect axiomatic form which was later referred to as the Euclidean geometry (Ferrarello, Mammona, & Pennisi, 2016). The disquisition is not a compilation of all the Hellenistic mathematicians’ insights about geometry at that period. Moreover, Euclid composed eight more advanced geometric publications. Euclid’s treatise commences with the elucidation of terms, basic principles of geometry (postulates or axioms), and the general quantitative precepts which form a basis for the logical deductions of geometric proofs. 

Archimedes 

Archimedes developed techniques that are significantly similar to the analytic geometry’s coordinate systems and integral calculus’ limiting procedure (Ferrelera, Mammona, & Pennisi, 2016). 

Vedic India 

India’s Vedic era has a geometric tradition commonly exhibited in the building of elaborate altars. Ancient Indian inscriptions on this particular subject include the Sulba Sutras and Satapatha Brahmana . According to Scriba and Schreiber (2015) the Sulba Sutras consisted of the ancient extant articulations of the Pythagorean Theorem, albeit the Old Babylonians had initially discovered the concept. The Sulba Sutras were composed of a listing of Pythagorean triples; these are specific sets of Diophantine equations. Additionally, they contained inscriptions regarding the circling of a square and the squaring of a circle. One of the most common Sulba Sutras is known as the Baudhayana Sulba Sutra ; it contained examples of simplified Pythagorean triples, for instance, (7, 24, 25), (8, 15, 17), (5, 12, 13), and (3, 4, 5) (Scriba & Schreiber, 2015). The Baudhayana also contained an account of the Pythagorean Theorem for a square’s sides and a general description of the Pythagorean Theorem for a rectangle’s sides. According to Dani S.G, a famous Indian mathematician, the Babylonian cuneiform plaque (Plimpton 322) inscribed c. 1850BC bears 15 Pythagorean triples typified by significantly large listings; this includes (13500, 12709, 18541), a primeval triple (Gray, 2011). Dani’s report indicates that the Indians were conversant with the subject (Pythagorean Theorem). 

Classical Indian Geometry 

Multiple geometric problems are contained in the Bakhshali manuscript; this manuscript incorporates a decimal place value approach using a dot to represent the number zero (Scriba & Schreiber, 2015). Aryabhata incorporates the calculation of volumes and areas in his literature, Aryabhatiya . Brahmagupta developed his astronomical studies, Brahma Sphuta Siddhanta , with sixty-six Sanskrit compositions that were sub-divided further into two primary sections: Practical mathematics and basic operations. Practical mathematics incorporates aspects such as mathematical series, brick stacking, and plane figures. Basic operations incorporate concepts such as barter, proportions and ratios, fractions, and cube roots. Brahmagupta’s theorem proposes that if a cyclic quadrilateral is made up of perpendicular diagonals, the perpendicular line is drawn from the diagonals’ intersection point to any quadrilateral sides often bisects the obverse side. Brahmagupta also incorporated the formula for determining a cyclic quadrilateral’s area in chapter twelve and a comprehensive delineation of rational triangles. 

Chinese Geometry 

The earliest definitive geometric study in China was done by Mo Jing, an ancient Mozi philosopher. Mo Jing delineated various physical science concepts and provided a significant level of knowledge in the mathematics field which presented an atomic description of the geometric point. He argued that a line may be divided into various parts and the section of a line which cannot be divided further into smaller segments forms the terminal end of a line commonly referred to as a point (Coolidge, 2013). Mo Jing proposed that a point may be at a line’s terminal end or the beginning of a line. China witnessed significant advancements in mathematics during the Han Dynasty. The Suàn shù shū was among the oldest mathematical literature in China to delineate geometric progressions in the Western Han period (Coolidge, 2013). Zhang Heng, the astronomer, inventor, and mathematician, utilized geometrical formulas to resolve various mathematical problems. Although the rough approximations for π were specified in Zhou Li, Zhang Heng, the first Chinese mathematician to establish a concerted attempt in developing a significantly accurate π formula, estimated π as 730/232; this is approximately 3.1466. 

However, Zhang used a different formula for π when determining the volume of a sphere. He used the square root of ten; this is approximately 3.162. Additionally, Zu Chongzhi improved the precision of the estimation of π to amid 3.1415926 and 3.1415927 (Leech, 2007). The Nine Chapters of Mathematical Art , a book edited by Liu Hui, a third-century mathematician contained multiple problems typified by the application of geometry, for instance, the determination of the surface areas of circles and squares, utilization of the Pythagorean Theorem, and the computation of the volumes of objects with three-dimensional shapes. The book presented an exemplified proof or evidence of the Pythagorean Theorem, contained in an inscribed conversation amid Shang Gao and the Ancient Duke of Zhou on the right-angled triangle’s properties and the Pythagorean Theorem. 

In the conversation, the two mathematicians incorporated mathematical aspects such as the astronomical gnomon, square and circle, and the mensuration of distances and heights. Liu Hui delineated π as 3.141014. Liu Hui also incorporated the concept of mathematical surveying in the calculation of distance mensuration of surface area, width, height, and depth (Leech, 2007). With regards to solid geometry, Liu proposed that a wedge-shaped figure typified by a rectangular base with sloping sides may be sub-divided into a tetrahedral wedge and pyramid. Moreover, Liu propounded that a wedge-shaped figure typified by a trapezoid base with sloping sides could be adjusted to create two tetrahedral wedges divided by a pyramid. He further delineated Cavalieri's volume principle and Gaussian elimination concept. Other ancient advances in geometry may be seen in the Islamic Golden Age, and the Renaissance. 

Modern Geometry 

Seventeenth Century 

Two significant geometric developments took place during the seventeenth century: The development of analytic geometry and projective geometry (Ferrarela, Mammona, & Pennisi, 2016). Pierre de Fermat and Rene Descartes invented the concept of analytic geometry; this is geometry typified by equations and coordinates. Analytic geometry was a significant antecedent to the invention of calculus and accurate conception of quantitative physics. Girard Desargues developed projective geometry which entails the study of geometry with no measurements. The calculus concept was later developed independently during the seventeenth century by Gottfried Wilhelm Leibniz and Isaac Newton; this marked the significant beginning to a new mathematical field currently referred to as analysis. Although not a geometric branch, analysis concepts are applicable in various geometric computations. 

The Eighteenth and Nineteenth Centuries 

Non-Euclidean Geometry 

Many mathematicians presented different alternatives or substitutes to the parallel postulate by Euclid during the nineteenth century. Omar Khayyam, though unsuccessful in establishing significant evidence against the parallel postulate by Euclid, his censure towards Euclid’s parallel theorem and his substantial proof on the attributes of figures in various non-Euclidean geometries were instrumental in the ultimate non-Euclidean geometry developments (Gray, 2011). Some of the researchers who attempted to develop the non-Euclidean geometry include Legendre, Lambert, Gauss, Saccheri, Lobatchewsky, and Johann Bolvai. In 1868, Beltrami proved the self-consistency of the non-Euclidean geometry mathematically; this led to the establishment of the conception on an equivalent mathematical foundation with the Euclidean geometry. 

Topology 

Topology is the most recent and sophisticated geometry branch which focuses on the elements of geometric objects which often remain unchanged when subjected to constant deformation, for instance, folding, stretching, and shrinking (Libby, 2017). The continual topology advancements date back to 1911 when L.E.J Brouwer, a Dutch mathematician introduced techniques applicable to the subject. 

Differential Geometry 

Carl Friedrich Gauss, a German mathematician, developed the differential geometry field in connection with the practicable geodesy and surveying problems. Gauss categorized the intrinsic properties of surfaces and curves using differential calculus (Libby, 2017). For instance, Gauss demonstrated that a cylinder’s intrinsic curvature is similar to that of a plane; this can be observed by cutting a cylinder through its axis and flattening it. 

Twentieth Century 

Advancements in algebraic geometry during the twentieth century involved the study of surfaces and curves along finite fields; this is evidenced in research conducted by Jean-Pierre Serre, Alexander Grothendieck, and Andre Well (Gray, 2011). The studies have also been focused on complex or real numbers. Finite geometry is applicable in cryptography and coding. 

Applications of Geometry 

Geometric concepts are usually employed in various day-to-day life applications, for instance, construction, navigation, astronomy, and surveying. Some of the applications of geometry in real life are usually vivid in various fields of knowledge. Art and mathematics are significantly related in multiple ways. For example, the perspective theory (an image's graphical representation on a flat surface as viewed by the eyes) which bases its foundation on projective geometry reveals geometry entails more than just a figure's metric properties (Libby, 2017). Secondly, geometric conceptualizations are usually applied in various technical fields such as video games, computer, and robotics. Geometry often presents significant conceptions for both video game and computer programmers. The procedure of designing the characters in video games and their subsequent movements often require geometric computations to develop paths around the obstacles in the virtual world. Video game engines often the raycasting technique, an approach that simulates the 3-D world through a 2-D map (Libby, 2017). Thirdly, architects employ various geometric concepts to define a building’s spatial characteristics; this includes the structure, height, and shape of a building. Fourthly, geometry may be used in the astronomy field to facilitate the mapping of the position of planets and stars on the celestial sphere (Libby, 2017). Geometry may be used to delineate the relation amid the movement of celestial bodies. In the domain of Physics, there is a significant connection amid general relativity and the pseudo-Riemannian geometry. Lastly, geometric conceptions may be used in GPS systems, particularly satellites to calculate the satellite’s position and the GPS’ location gauged by the longitudes and latitudes. 

References 

Coolidge, J. L. (2013). History of Geometrical Methods . Dover Publications. 

Ferrarello, D., Mammana, M. F., & Pennisi, M. (2016). From geometry to algebra: the Euclidean way with technology. International Journal of Mathematical Education in Science & Technology , 47(4), 597–605. 

Gray, J. (2011). Worlds out of nothing: A course in the history of geometry in the 19th century . London: Springer. 

Leech, B. C. (2007). Geometry's great thinkers: The history of geometry . New York: Rosen Publishing Group's PowerKids Press. 

Libby, J. (2017). Math for real life: Teaching practical uses for algebra, geometry, and trigonometry . Jefferson, North Carolina: McFarland & Company, Inc. 

Scriba, C. J., & Schreiber, P. (2015). 5000 years of geometry: Mathematics in history and culture . Basel: Birkhäuser.