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History of geometry

The history of geometry encompasses the development of mathematical principles concerning shapes, sizes, spatial configurations, and their properties, originating from practical applications in ancient societies and progressing to abstract, axiomatic systems that underpin modern science and technology. This evolution reflects humanity's quest to understand and manipulate , beginning with empirical methods for and around 3000 BCE in and , advancing through deductive proofs in , and culminating in diverse geometric frameworks in the 19th and 20th centuries that challenged orthodoxy. In its earliest phases, emerged as a tool for and building in ancient civilizations. developed empirical formulas for volumes, such as the correct method for a of a , as recorded in the Rhind around 1650 BCE, driven by the annual floods that necessitated remeasuring fields. Similarly, Babylonians applied geometric knowledge to astronomy and , demonstrating special cases of the on clay tablets like around 1800 BCE, though without general proofs. In , the Sulbasutras (circa 800–500 BCE) provided rules for altar , including approximations of √2 and the , attributed to figures like Baudhayana. These practical origins laid the groundwork for more theoretical pursuits. The transformative shift to axiomatic geometry occurred in , where it became a deductive . Thales of Miletus (circa 624–546 BCE) introduced proofs for basic propositions, such as the equality of base angles in isosceles triangles, marking the transition from empirical to logical methods. (circa 570–495 BCE) and his school emphasized geometric proofs, famously associating the hypotenuse theorem with right triangles, while exploring irrational numbers through constructions like the . The pinnacle of this era was Euclid's Elements (circa 300 BCE), a systematic compilation of definitions, axioms, postulates—including the controversial —and theorems, such as those on circles and solids, which standardized geometry for over two millennia and influenced fields from to . Later Greeks like (287–212 BCE) advanced applications in and conic sections, while Apollonius (circa 240–190 BCE) formalized the study of ellipses, parabolas, and hyperbolas. During the (8th–14th centuries CE), scholars preserved and expanded Greek geometry through translations at centers like Baghdad's , ensuring Euclid's Elements and Apollonius's Conics survived into the modern era. Original contributions included , as (1048–1131) solved cubic equations using intersections of conic sections, and optical applications by (965–1040), who modeled reflection with parabolas. Figures like (973–1050) applied to and astronomy, computing with high accuracy. This period bridged antiquity and the , transmitting knowledge via Arabic texts to . The and revitalized geometry through integration with and . (1596–1650) introduced coordinate geometry in (1637), enabling algebraic representation of curves and revolutionizing problem-solving. The 19th century brought profound changes with non-Euclidean geometries: (1792–1856) and (1802–1860) independently developed by rejecting Euclid's , while (1826–1866) proposed with variable , influencing Einstein's . 's (1872) unified geometries via , classifying them by transformations. In the , geometry diversified into , differential forms, and computational applications, solidifying its role in physics, computer science, and engineering.

Ancient Geometry

Egyptian and Mesopotamian Practical Geometry

The practical geometry of ancient and emerged from empirical needs in land management, monumental construction, and celestial observation, predating abstract theorization. In , annual floods necessitated resurveying fertile boundaries, leading to the development of simple tools for measurement around 3000 BCE during the Early Dynastic Period. Surveyors employed knotted s, approximately 100 cubits long and divided into segments, to form essential for delineating fields and structures. This technique utilized the 3-4-5 property, where a knotted at 3, 4, and 5 units created a when stretched taut, enabling accurate rectangular layouts without advanced instruments. Tomb depictions, such as those in the 18th Dynasty tomb of Menna (c. 1400–1350 BCE), illustrate this "harpedonaptai" (rope-stretchers) method in action, confirming its continuity from earlier practices. Key mathematical insights appear in the , composed around 1650 BCE and attributed to the scribe , which compiles problems for areas and volumes encountered in daily administration. For circular areas, problem 50 approximates the with diameter 9 units as 64 square units, employing a method equivalent to squaring a side of (d - \frac{1}{9}d), yielding \pi \approx \frac{256}{81} \approx 3.1605. This approximation, derived by inscribing an in a square and adjusting for the circle, supported practical tasks like granary design. The papyrus provides the volume for simple pyramids, V = \frac{1}{3} \times \text{base area} \times \text{height}, applied to irrigation basins and architectural models. A correct for the volume of truncated pyramids (frustums) appears in the contemporary (c. 1850 BCE): V = \frac{h}{3} (A_1 + A_2 + \sqrt{A_1 A_2}), where A_1 and A_2 are the areas of the two bases and h is the height. In pyramid construction, Egyptians quantified slopes using the seked, defined as the horizontal run in palms per vertical rise of one cubit (7 palms). The Great Pyramid of Giza (c. 2580–2560 BCE), built for Pharaoh Khufu, employed a seked of 5 palms and 2 digits (5.5 palms total), corresponding to a rise-to-run ratio of 14:11 and a face slope of approximately 51°50′. This measure ensured uniform inclination across layers, with tools like plumb bobs and set squares maintaining precision during masonry placement. Mesopotamian geometry, recorded on clay tablets from the Old Babylonian period (c. 1800 BCE), emphasized algebraic solutions for right triangles in agrarian and building contexts. Tablets like compute square diagonals using approximations of \sqrt{2} \approx 1;24,51,10 (in notation), facilitating field divisions and structural alignments without geometric proofs. Similarly, lists 15 Pythagorean triples (e.g., sides 119, 120, 169), representing integer-sided right triangles for scaling land plots or foundations. These applications reflect a focus on reciprocal pairs and quadratic equations for practical reciprocity in trade and . The Babylonian sexagesimal (base-60) system profoundly shaped geometric measurement, particularly for angles and astronomical computations. Originating in the second millennium BCE, it divided the full circle into 360 s—approximating the solar year— with each further subdivided into 60 minutes and 60 seconds, enabling precise tracking of celestial arcs. This framework supported tables for planetary positions and eclipses, integrating with observational data in records. These empirical traditions in and provided foundational techniques that influenced subsequent theoretical advancements in .

Vedic Indian and Early Chinese Geometry

The Vedic Sulbasutras, composed circa 800–500 BCE as appendices to the Vedic texts, form the earliest systematic exposition of in ancient , centered on constructing precise altars for sacrificial rituals. These manuals, attributed to scholars like Baudhayana, Apastamba, and Katyayana, emphasize practical and transformations using simple tools such as ropes (sulba) stretched between stakes to form geometric shapes. The focus was ritualistic accuracy, ensuring altars like the falcon-shaped fire altar adhered to symbolic proportions while maintaining equal areas across different configurations. A cornerstone of Sulbasutra geometry is the , articulated explicitly for construction; Baudhayana states that "the rope which is stretched across the diagonal of a square produces an area the size of the original square," generalizing to a^2 + b^2 = c^2 for right triangles in building altars. The texts also address squaring approximations, such as deriving a square of nearly equal area to a given by taking the side as \frac{13}{16} d + \frac{1}{4} \frac{d}{\sqrt{2}} (where d is the ), yielding an effective \pi \approx 3.088 for practical use. Constructions for transforming a into a square of equivalent area involved iterative rope manipulations and diagonal stretches, enabling efficient reshaping of plots without altering ritual volumes. These algebraic-geometric methods, rooted in ritual needs, anticipated later Indian advancements in . In parallel, early Chinese geometry emerged in texts tied to astronomy, calendrics, and engineering, with the Zhoubi Suanjing (Arithmetical Classic of the Gnomon and the Circular Paths of Heaven), compiled between circa 100 BCE and 100 CE, serving as a foundational work. This treatise applies the Gougu theorem—equivalent to the Pythagorean theorem for right triangles (a^2 + b^2 = c^2)—to compute distances in astronomical contexts, such as using gnomon shadow lengths to determine the sun's height or the earth's circumference. For instance, it describes a method where a gnomon's shadow at noon forms a right triangle with the observer's height, allowing derivation of celestial parameters through proportional scaling. Complementing these theoretical applications, practical geometry in early Chinese works included volume computations for civil engineering; the Nine Chapters on the Mathematical Art (circa 100 CE) provides formulas for granary and well capacities, such as the volume of a cylindrical granary as \frac{3d^2 h}{4} (using \pi \approx 3) and truncated pyramidal wells, facilitating storage and irrigation assessments. These empirical approaches, embedded in problem-solving traditions, underscored geometry's role in governance and agriculture without axiomatic foundations.

Classical Greek Geometry

Thales, Pythagoras, and Pre-Socratic Developments

The transition from empirical practices to deductive reasoning in geometry began with early Ionian thinkers in the 6th century BCE, marking a pivotal shift in Greek mathematical thought. Thales of Miletus (c. 624–546 BCE), often regarded as the founder of Greek geometry, introduced systematic proofs inspired by Egyptian techniques but adapted into rigorous demonstrations. He proved that a circle is bisected by its diameter, establishing a foundational property of circular symmetry. Thales also demonstrated that an angle inscribed in a semicircle is a right angle, a theorem that relies on the proportionality of segments and laid groundwork for later circle geometry. For practical applications, Thales measured the height of Egyptian pyramids by observing shadows at noon, using the similarity of triangles formed by the pyramid, its shadow, and a vertical staff with its shadow, where proportional sides yield the height via the ratio of shadows. Building on Thales's innovations, the , founded around 530 BCE by of (c. 570–495 BCE), elevated geometry to a philosophical pursuit, viewing it as a key to understanding cosmic harmony. The Pythagoreans provided the first known Greek proof of the , which states that in a right-angled , the square of the hypotenuse equals the sum of the squares of the other two sides; their method involved rearranging areas of squares on the sides to show equality without algebraic notation. This proof underscored the deductive power of geometry, transforming empirical observations into logical necessities. The school also explored Pythagorean triples—primitive integer solutions like (3,4,5)—which generate right and illustrate commensurable in lengths. Furthermore, they discovered irrational numbers through the diagonal of a , realizing that \sqrt{2} cannot be expressed as a of integers, a revelation attributed to of that challenged their belief in all magnitudes being commensurable and deepened geometric inquiry. Pre-Socratic developments intertwined with , portraying the universe as governed by geometric order. of (c. 610–546 BCE), a contemporary of Thales, created the first known Greek , depicting the inhabited as a within a geometric framework to represent spatial relations and cosmic balance. This mapping reflected a broader Pre-Socratic emphasis on as a tool for discerning the rational structure of the cosmos, where proportions and shapes underpin natural processes. By the mid-5th century BCE, (c. 470–410 BCE) advanced geometric techniques, achieving the first known squaring of lunes—curved regions between circular arcs—by demonstrating that certain lune areas equal straight-edged triangles, using properties of circles and semicircles to bridge curved and figures. The Pythagoreans extended geometric principles into , identifying harmonic intervals through string lengths on a monochord, where ratios like 2:1 (), 3:2 (fifth), and 4:3 (fourth) correspond to sounds. They employed the , such as \sqrt{ab} between arithmetic means, to explain these proportions, linking musical to cosmic and reinforcing the idea of numerical order in nature. These ideas influenced later philosophers like , who incorporated geometric ideals into his .

Platonic Influences and the Academy

Plato (c. 428–348 BCE) integrated geometry into his philosophical framework, viewing it as a means to access the eternal, ideal forms that underpin reality. In his dialogue Timaeus, he describes the fundamental building blocks of the universe as geometric atoms composed of the five regular Platonic solids, each associated with a classical element to explain the cosmos's construction by the Demiurge. Specifically, the tetrahedron represents fire due to its sharp points, the cube earth for its stability, the octahedron air, the icosahedron water, and the dodecahedron the overarching universe, inscribed in a sphere. This cosmological model underscores geometry's role in revealing the mathematical harmony of ideal forms, contrasting with empirical observation. At Plato's , founded around 387 BCE in , geometry held a central place in the , prioritized over as a tool for intellectual ascent toward philosophical truth. The emphasized dialectical inquiry and mathematical training, with serving as a bridge from sensory perception to abstract reasoning about universals. This educational focus is exemplified in the dialogue, where guides an uneducated slave boy through a geometric problem—doubling the area of a square—using only questions to elicit the solution, demonstrating that geometric knowledge is innate and recollected rather than learned anew. Such exercises illustrated the Academy's conviction that geometry fosters the soul's purification and prepares students for dialectic, as outlined in the 's educational program where plane geometry follows arithmetic to train the mind in visualizing immaterial forms. Under Platonic influence at the Academy, Eudoxus of Cnidus (c. 408–355 BCE) advanced the method of exhaustion, a rigorous technique for determining volumes and areas by approximating curved figures with inscribed polygons of increasing sides, avoiding infinitesimals. Eudoxus, who studied directly with Plato, refined this approach to handle irrational quantities precisely, applying it to problems like the volumes of pyramids and cones, which later informed Euclidean geometry. His work exemplified the Academy's commitment to geometry as a pathway to demonstrating universal truths without reliance on sensory deception. Aristotle, Plato's student, offered critiques of this geometric emphasis in his Metaphysics, questioning the separation of mathematical objects like geometric forms from sensible reality. In Books XIII and XIV, he argues that geometry deals with attributes of physical things rather than independent universals or substances, rejecting Plato's view that such forms exist apart from particulars. For instance, Aristotle contends that "the objects of mathematics are not substances... but are attributes and affections of sensible things," thus limiting geometry's metaphysical scope to descriptive rather than causative roles in understanding universals. These critiques highlighted tensions within the Academy's tradition, paving the way for more empirical integrations in later Hellenistic mathematics.

Euclidean Synthesis

Euclid, active around 300 BCE in , , compiled the , a comprehensive treatise that synthesized and standardized the geometric knowledge accumulated by earlier Greek mathematicians. This work, consisting of thirteen books, established a rigorous axiomatic framework for geometry, drawing on contributions from figures such as Thales and the Pythagoreans to create a deductive system beginning from primitive definitions, postulates, and common notions. The opens with twenty-three definitions, five postulates, and five common notions in Book I, providing the foundational assumptions for all subsequent deductions. The postulates include the ability to draw a straight line between any two points, to extend a finite straight line continuously, to describe a with any and , that all right angles are equal, and the parallel postulate stating that if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines will meet on that side. The common notions, or axioms, encompass principles of equality such as things equal to the same thing are equal to one another, if equals are added to equals the wholes are equal, if equals are subtracted from equals the remainders are equal, things that coincide are equal to one another, and the whole is greater than the part. These elements enabled to derive theorems through logical proof, including criteria for congruence such as side-angle-side () and angle-side-angle () in Propositions I.4 and I.8, as well as the in Proposition I.47, which states that in a right-angled , the square on the equals the sum of the squares on the other two sides. Books I through VI focus on plane geometry, building from basic constructions to advanced properties. Book I establishes principles of triangles, parallels, and area, including proofs of congruence and the . Book II introduces , applying areas and ratios to problems like the transformation of rectangles. Book III examines circles, proving theorems on tangents, inscribed angles, and the power of a point. Book IV addresses constructions of inscribed and circumscribed polygons, such as regular pentagons. Book V, drawing on Eudoxus's theory, develops proportions for magnitudes, while Book VI applies these to similar figures, including similarity of triangles and parallelograms. This sequence culminates in a systematic treatment of plane figures, emphasizing deductive rigor over empirical measurement. Books VII through X shift to arithmetic and the theory of irrationals, treating numbers as magnitudes to bridge and . Books VII and VIII cover fundamentals of , including divisibility, primes, the for greatest common divisors, and continued proportions. Book IX explores properties like perfect numbers and the infinitude of primes. Book X, the longest with 115 propositions, classifies irrational magnitudes such as medials and apotomes, using the to handle incommensurables without resolving foundational paradoxes outright. These books demonstrate 's synthesis of into geometric deduction, highlighting irrationals discovered by earlier scholars like Theaetetus. Books XI through XIII extend the axiomatic approach to solid geometry. Book XI introduces planes, solid angles, and parallels in three dimensions. Book XII employs the to measure volumes of , cones, cylinders, and , proving results like the volume of a pyramid being one-third the base area times height. Book XIII constructs and investigates the five solids—, , , , and —inscribed in a , relating their edges to the . This culminates the in a geometric exploration of regular polyhedra, tying back to ideals. Although containing no major original discoveries, the achieved enduring influence through its methodical organization and proof-based exposition, serving as the primary geometry textbook for over two thousand years across cultures, from the to , with more than a thousand editions printed since 1482. Its axiomatic model shaped mathematical and until the advent of non-Euclidean geometries in the nineteenth century.

Archimedean Innovations and Conics

During the , the study of conic sections originated with Menaechmus around 350 BCE, who discovered these curves while attempting to solve the classical problem of duplicating the cube using geometric constructions. Menaechmus intersected cones to generate what would later be identified as parabolas, ellipses, and hyperbolas, providing an early framework for their properties in . Archimedes of Syracuse (c. 287–212 BCE) advanced geometric methods beyond axioms by employing the , an iterative akin to , to determine areas and volumes with rigorous . In his work , proved that the volume of a is \frac{4}{3} \pi r^3 and its surface area is $4 \pi r^2, using inscribed and circumscribed polyhedra to bound the and converge to these values. He further demonstrated that the volume of a equals two-thirds that of the circumscribing cylinder, a result he considered so significant that it was inscribed on his tombstone. also applied the in to show that the area of a parabolic segment is \frac{4}{3} times the area of the inscribed triangle with the same base and height, establishing key properties of parabolas through successive triangular approximations. Archimedes bridged geometry and physics in On Floating Bodies, where he formulated the hydrostatic principle stating that a body immersed in a experiences an upward equal to the weight of the displaced , laying foundational principles for . This work included practical applications, such as the , a helical device for raising water by rotating it within a , demonstrating early integration of with . (c. 240–190 BCE) systematized conic sections in his eight-volume treatise Conics, defining the , , and as plane sections of right circular cones at varying angles to the axis. He introduced the -directrix property, where a conic is the locus of points such that the ratio of the distance to a fixed point () and a fixed line (directrix) is constant (the : less than 1 for , equal to 1 for parabola, greater than 1 for ), enabling algebraic derivations of their equations.

Medieval Geometry

Classical Indian Advances

In the classical period of Indian mathematics, spanning roughly from the 5th to the 12th century CE, geometers built upon earlier synthetic traditions to develop sophisticated algebraic and trigonometric tools, often applied to astronomical problems. These advances emphasized precise calculations for circles, triangles, and quadrilaterals, independent of Greek influences, and integrated geometric constructions with numerical approximations. Key texts like the Aryabhatiya and Brahmasphutasiddhanta introduced methods that facilitated solutions to practical and theoretical issues in measurement and celestial modeling. Aryabhata (c. 476–550 CE), in his seminal work (499 CE), constructed the earliest known sine table, using the term jya to denote approximations of the half- (sine) values for in a of 3438 units, enabling efficient computation of circular arcs and shadows. He derived these through chord theorems, such as relations between chords subtended by arcs and central , which allowed for geometric solutions to problems in . Additionally, employed geometric models to explain Earth's axial rotation, positing that the planet spins daily on its axis, producing the apparent motion of stars relative to an observer, a concept supported by calculations of sidereal periods using circular geometry. Brahmagupta (c. 598–668 ), in his Brahmasphutasiddhanta (628 ), advanced geometry with a formula for the area of a , given by \sqrt{(s-a)(s-b)(s-c)(s-d)}, where s is the semiperimeter and a, b, c, d are the side lengths; this generalization of was derived using properties of inscribed figures and similarity in right triangles. He also advanced geometric methods for right-angled triangles, leveraging restricted similarity transformations to determine segments, altitudes, and areas, which extended Pythagorean relations to more complex configurations within cyclic frameworks. These contributions underscored a synthetic approach, combining visual proofs with algebraic verification for astronomical and applications. Bhaskara II (c. 1114–1185 ), in his treatise Lilavati (1150 ), offered elegant geometric proofs of variants, notably a method rearranging four right triangles around a square to visually demonstrate a^2 + b^2 = c^2, emphasizing intuitive spatial equality over numerical computation. He further contributed a rational \pi \approx 3.1416 ($3927/1250), derived from polygonal inscriptions and arc divisions, which supported precise measurements in and calendars. A hallmark of classical geometry was the use of to solve equations visually, representing unknowns as line segments and products as areas of or similar figures; for instance, geometrically transformed x^2 + px = q into a constructed , revealing through side lengths without symbolic manipulation. This method, evident in works from to Bhaskara, bridged arithmetic and geometry, allowing solutions to indeterminate equations and conic sections via diagrammatic proofs, and influenced later algebraic developments.

Islamic Golden Age Contributions

During the , the (Bayt al-Hikma) in , established around 800 CE under Caliph and expanded by , became a pivotal center for synthesizing , , and knowledge in mathematics and science. Scholars there systematically translated key texts into Arabic, including Euclid's Elements—first rendered by al-Hajjaj ibn Yusuf al-Matar around 810 CE—and Ptolemy's , preserving and adapting these works for further study. Muhammad ibn Musa al-Khwarizmi (c. 780–850 CE), a prominent figure at the House, contributed to these efforts by revising Euclid's Elements and integrating algebraic techniques to address geometric problems, such as inheritance divisions and land measurement, thereby linking symbolic with visual proofs. Al-Khwarizmi's seminal Kitab al-Jabr wa'l-Muqabala (c. 820 CE), often regarded as the foundational text of , demonstrated how to solve equations through geometric constructions, such as to find areas of rectangles and squares. For instance, he illustrated the to equations like x^2 + 10x = 39 by drawing a square of side x and attaching rectangles to its sides, then adjusting to form a larger square whose area could be calculated. This approach not only resolved practical geometric issues but also established as a tool for , influencing subsequent Islamic mathematicians. His works on polyhedral , including commentaries on figures from Euclid's Book XI, explored volumes and surfaces of regular polyhedra, building on Greek foundations to support applications in and astronomy. Omar Khayyam (c. 1048–1131 CE), a , advanced the algebra-geometry synthesis in his on Demonstration of Problems of Algebra (c. 1070 CE), where he classified and solved all 25 types of cubic equations geometrically. Unlike al-Khwarizmi's quadratics, Khayyam used intersections of conic sections—such as a and a —to find roots, as in solving x^3 + a^2 x = b x^2 by constructing perpendiculars and hyperbolas whose intersection points yielded the solution. This method highlighted the power of conics for higher-degree problems, though it required graphical construction rather than purely algebraic manipulation, and it laid groundwork for later . Ibn al-Haytham (c. 965–1040 CE), known in the West as Alhazen, made profound geometric contributions through optics in his monumental Kitab al-Manazir (Book of Optics, c. 1011–1021 CE), a seven-volume treatise that revolutionized the field. In Books IV and V on catoptrics, he analyzed reflection in plane, spherical, and parabolic mirrors, solving Alhazen's problem: finding the point on a spherical mirror where light from two points reflects to the eye, using conic intersections for precise calculations. For dioptrics in Book VII, he examined refraction through lenses and interfaces, deriving laws for curved surfaces. Additionally, in his Doubts Concerning Ptolemy and commentaries on Euclid, al-Haytham attempted to prove the parallel postulate via contradiction, assuming an alternative and deriving inconsistencies, but ultimately recognized its independence from the other axioms— an early insight into its foundational status. His work on parabolic mirrors, detailed in a separate treatise On Burning Mirrors (c. 1015 CE), geometrically demonstrated their focal properties for concentrating sunlight, advancing applications in engineering. Islamic geometric innovations also manifested in polyhedral constructions and tessellations, integral to and . Al-Khwarizmi's revisions to Euclid extended to , enabling explorations of polyhedra like tetrahedra and octahedra for architectural modeling. Tessellations in , employing regular polygons, evolved from the 9th century onward, with —a using decagonal and pentagonal motifs to create intricate patterns—emerging around the and reflecting deep understanding of and principles without figural representation, as guided by aniconic traditions. These contributed to the transmission of advanced to via translations in and during the .

Chinese and Byzantine Continuations

In medieval China, mathematical traditions rooted in earlier texts like The Nine Chapters on the Mathematical Art (compiled around 100 CE but expanded through commentaries in subsequent centuries) continued to emphasize algorithmic and practical geometry. This foundational work, which included problems on areas, volumes, and constructions, was extensively annotated by around 263 CE, providing rigorous proofs and innovative methods for geometric computations. employed a precursor to the —dividing shapes into thin slices to approximate areas and volumes—yielding an approximation of \pi \approx 3.1416 for the via a 192-sided inscribed and accurate formulas for volumes. Building on these approaches, (429–500 CE) advanced polygonal approximations to further refine \pi, establishing the bounds $3.1415926 < \pi < 3.1415927 using inscribed and circumscribed polygons up to 24,576 sides, a precision unmatched until the . These developments reflected a focus on computational accuracy rather than axiomatic theory, with Zu's work also informing practical reforms in the to align lunar and solar cycles more precisely through geometric modeling of celestial motions. Chinese geometry found direct applications in and during this period, particularly in methods for converting between circular and square forms to design structures like granaries and observatories. For instance, techniques from The Nine Chapters and Hui's annotations enabled the computation of areas for circular bases inscribed in squares or vice versa, essential for and site planning in imperial projects. In the , geometric knowledge from classical Greek sources was primarily preserved through scholarly recopying and compilation rather than significant original innovation, ensuring the transmission of texts to later European traditions. Manuscripts of 's Elements were meticulously reproduced in monastic scriptoria from the 9th century onward, with scholars like Maximus Planudes (c. 1260–1305 CE) playing a key role in editing and disseminating these works during the Palaeologan Renaissance. Planudes, a and , recopied alongside other classical authors, compiling anthologies that included geometric propositions and facilitating their eventual movement to via diplomatic exchanges in the . This custodial approach maintained the integrity of Greek geometry amid political upheavals, though Byzantine contributions remained largely commentary-based and oriented toward theology or rather than novel geometric theory.

Renaissance and Early Modern Geometry

European Revival and Perspective

The European revival of geometry during the Renaissance began with the recovery and translation of ancient Greek texts, often transmitted through Islamic scholars, which fueled advancements in astronomy, art, and architecture. A key figure in this resurgence was Johannes Regiomontanus (c. 1436–1476 CE), who produced influential trigonometry tables essential for astronomical computations and completed a critical Latin translation of Ptolemy's Almagest, making Ptolemaic geometry accessible to Western scholars. His De triangulis omnimodis (1464) systematized plane trigonometry, introducing a sine table with a radius of 60,000 units for precise calculations of celestial positions. Perspective geometry emerged as a cornerstone of , transforming artistic representation by applying geometric principles to create illusions of depth. (1377–1446 CE) pioneered around 1415, demonstrating the through experimental panels depicting Florence's and Palazzo dei Signori, where parallel lines converged to a single point on the horizon. This innovation, initially shared orally with artists like and , emphasized the proportional decrease in object size with distance. formalized these ideas in Della pittura (1435), describing via the costruzione legittima method, which used a of orthogonals converging at the and transversals to divide the picture plane. Central to Alberti's approach was the regula, a rule for dividing lines harmonically to ensure balanced proportions in depictions of and figures, aligning visual harmony with geometric precision. Leonardo da Vinci (1452–1519 CE) further bridged geometry and art through his illustrations for Luca Pacioli's De divina proportione (1509), showcasing polyhedra as embodiments of divine proportion. Da Vinci provided 60 woodcut plates depicting the five Platonic solids, including detailed views of the in both solid and "vacua" (unfolded) forms to highlight edges, vertices, and spatial relationships. These illustrations, rooted in principles, exemplified geometry's role in exploring form and perspective, influencing later architectural and artistic designs.

Analytic Geometry and Coordinates

Analytic geometry emerged in the early as a transformative fusion of algebraic methods with geometric representation, primarily through the independent efforts of and . Fermat, around the 1630s, developed an algebraic approach to by associating coordinates with points on , enabling the solution of problems involving maxima and minima as well as the determination of tangents to curved lines. In his unpublished treatise Method for Determining Maxima and Minima and Tangents to Curved Lines, Fermat employed a technique known as "adequation," which involved setting up equations in two variables—typically an abscissa (horizontal distance) and ordinate (vertical distance)—to approximate tangents by considering small increments along the curve. This method allowed him to find points of tangency without relying solely on classical geometric constructions, marking an early step toward analysis. Descartes formalized and expanded these ideas in his 1637 appendix , part of , where he introduced the Cartesian coordinate system to systematically link algebraic equations to geometric loci. By assigning perpendicular axes and using variables such as x and y to denote distances from an origin, Descartes represented straight lines as equations of the form y = mx + c, where m is the slope and c the y-intercept, allowing geometric problems to be reduced to algebraic manipulation. He further classified conic sections—circles, ellipses, parabolas, and hyperbolas—as curves defined by second-degree equations of the general form ax^2 + bxy + cy^2 + dx + ey + f = 0, demonstrating how such equations could be solved through systematic factorization and intersection with lines. This "geometrical calculus," as Descartes termed it, shifted geometry from ruler-and-compass constructions to symbolic computation, resolving longstanding debates on the constructibility of curves by degree. The introduction of coordinates facilitated applications to physical phenomena, particularly in describing projectile motion and planetary orbits. Galileo's investigations into parabolic trajectories during the 1600s, detailed in Two New Sciences (1638), had established through geometric means that projectiles under constant horizontal velocity and uniform vertical acceleration follow parabolic paths, but analytic geometry provided an algebraic framework to express these trajectories precisely as y = ax^2 + bx + c, where coefficients relate to initial conditions and gravitational acceleration. This algebraic representation simplified calculations for range and height, influencing later ballistic studies. Similarly, Kepler's laws of planetary motion (1609–1619), originally derived geometrically from Tycho Brahe's observations, found rigorous proofs through coordinate methods; Isaac Newton, in his Principia (1687), used polar coordinates centered at the sun to demonstrate that elliptical orbits (Kepler's first law) arise from inverse-square gravitational forces, with the area-sweeping property (second law) following from conservation of angular momentum expressed as r^2 \dot{\theta} = \constant. These applications underscored analytic geometry's power in modeling dynamic systems. Analytic geometry laid essential groundwork for the development of by enabling the study of curves through equations amenable to analysis. Both and Leibniz, building on Fermat's and Descartes' foundations, employed —quantities smaller than any finite length but nonzero—to compute areas under curves and tangents at points. 's fluxional calculus (circa 1660s), applied to curves like parabolas and ellipses, treated variables as flowing quantities whose rates (fluxions) yielded tangents, while integrals recovered areas as sums of elements; for instance, the area under y = f(x) was approximated by summing rectangles of width dx and height f(x). Leibniz, independently in the 1670s, refined this with his differential notation dy/dx for tangents and \int y \, dx for areas, directly leveraging coordinate equations to generalize methods for arbitrary algebraic curves. This transition from static coordinate descriptions to dynamic processes formalized , revolutionizing the analysis of geometric forms in motion and variation.

Modern Geometry

Non-Euclidean Revolutions

In the early 19th century, mathematicians began systematically challenging Euclid's fifth postulate, known as the parallel postulate, which states that through a point not on a given line, exactly one parallel line can be drawn. This effort built on earlier attempts, such as that of Girolamo Saccheri in his 1733 work Euclides ab omni nævo vindicatus, where he sought to prove the postulate via by assuming its and deriving consequences. Saccheri explored what are now recognized as the hypotheses of the acute and obtuse angles, arriving at propositions that led to contradictions he described as "repugnant to the nature of the straight line," though his rejection of these outcomes relied on unproven Euclidean intuitions, ultimately failing to resolve the issue but highlighting the postulate's independence. Nikolai Lobachevsky advanced this inquiry decisively in 1829 by constructing a consistent that negates the postulate, allowing infinitely many lines through a point to a given line. In this , the sum of angles in a is less than 180 degrees, with the angular defect (the difference from 180 degrees) directly proportional to the triangle's area, providing a measure of . Lobachevsky's system, detailed in his 1837 publication "Géométrie imaginaire," demonstrated that such a geometry could be developed without , challenging the long-held view of as the sole foundation for space. Independently, developed a similar framework in 1832, formulating what he termed by excising the parallel postulate from Euclid's axioms, creating a system to which either the or version could be appended. Complementing these efforts, in his 1854 lecture "On the Hypotheses Which Lie at the Foundations of Geometry" introduced , where no exist—every pair of lines intersects—and the sum of a triangle's angles exceeds 180 degrees. Riemann modeled this on , where great circles serve as "straight lines," and extended the concept to spaces of constant positive curvature, laying groundwork for more general manifolds. These non-Euclidean geometries profoundly influenced physics, particularly Albert Einstein's formulation of in , which describes through the of modeled on Riemann's differential geometry. In Einstein's theory, deviates from Euclidean flatness, with paths of light and matter following geodesics in curved manifolds, as elaborated in his 1916 , thus integrating non-Euclidean principles into the description of the physical .

Rigorous Foundations and Projective Geometry

In the early , mathematicians sought to establish on firmer analytical foundations by introducing rigorous concepts from , particularly limits and , to address ambiguities in earlier treatments of curves and series. played a pivotal role with his 1821 publication Cours d'analyse de l'École Royale Polytechnique, where he provided the first systematic definitions of limits, , and , applying them to to prove their summation for ratios less than unity using what became known as the Cauchy criterion. This work clarified the behavior of infinite series underlying geometric progressions, ensuring precise handling of limits in geometric contexts like approximating areas and volumes. Building on Cauchy's advances, further refined these ideas in his lectures at the University of Berlin during the 1850s and 1860s, formalizing the epsilon-delta definition of limits and for real-valued functions, which extended to representations of curves in the . 's approach eliminated reliance on intuitive notions of infinitesimals, providing a strict analytical basis for studying the differentiability and smoothness of geometric curves, thus bridging with geometric rigor. These developments transformed geometry from a synthetic discipline into one amenable to precise , influencing subsequent work in function theory and spatial configurations. Parallel to these analytical efforts, emerged as a unifying framework for conic sections and , emphasizing properties invariant under projection. laid groundwork in the 1790s through his development of descriptive geometry, a method for projecting three-dimensional designs onto two planes to facilitate mechanical drawing and . , initially devised for fortifications, enabled accurate of complex solids without metrics, proving essential for industrial applications. Jean-Victor Poncelet advanced this in his 1822 Traité des propriétés projectives des figures, formalizing projective principles where the —a measure of four collinear points—remains invariant under central projections, allowing conics to be studied uniformly regardless of viewpoint. This invariance unified disparate conic forms like ellipses and hyperbolas as projections of a single circle, simplifying proofs in . Central to projective geometry is the projective plane, an extension of the Euclidean plane incorporating points at infinity, where all parallels in a given direction intersect, thereby eliminating the special status of parallel lines and achieving uniformity in incidence properties. This construction, popularized by Poncelet and contemporaries, resolved inconsistencies in handling asymptotic behaviors of conics and lines. Karl Georg Christian von Staudt culminated these ideas in his 1847 Geometrie der Lage, constructing a purely synthetic free from Euclidean metrics or coordinates, defining harmonic divisions and cross-ratios via incidence alone to derive all projective theorems. Von Staudt's embedded the as a foundational , influencing later abstract geometries by prioritizing relational properties over measurement.

Topology, Higher Dimensions, and Differential Geometry

In the mid-19th century, developed quaternions as a four-dimensional extension of numbers to handle rotations in . Discovered on October 16, 1843, while walking along the Royal Canal in , quaternions provided a non-commutative for representing oriented rotations, addressing limitations in earlier vector approaches. Hamilton's seminal paper, "On Quaternions, or on a New System of Imaginaries in ," published in installments from 1844 to 1850, formalized quaternions as elements of the form a + bi + cj + dk, where i, j, k satisfy i^2 = j^2 = k^2 = ijk = -1, enabling efficient computations for 4D transformations that underpin modern and . Building on this multidimensional perspective, advanced in his 1854 lecture, "Über die Hypothesen, welche der Geometrie zu Grunde liegen" (On the Hypotheses Which Lie at the Bases of Geometry), where he generalized spaces to arbitrary dimensions and introduced intrinsic metrics on manifolds. Riemann's framework allowed geometries to be defined independently of embedding in , treating n-dimensional manifolds as equipped with a that measures distances and angles locally. A contribution was his of for surfaces, exemplified by K, which for a surface parametrized by coordinates u, v with coefficients E, F, G and coefficients e, f, g is given by K = \frac{eg - f^2}{EG - F^2}. This invariant quantifies the intrinsic bending of a surface, influencing later developments in general relativity. Felix Klein's 1872 Erlangen Program, outlined in "Vergleichende Betrachtungen über neuere geometrische Forschungen," unified various geometries through group theory, classifying them by their underlying transformation groups that preserve specific invariants. Klein proposed that Euclidean geometry arises from the group of isometries (rigid motions), projective geometry from collineations, and affine geometry from affine transformations, providing a hierarchical structure where broader groups encompass narrower ones. This group-theoretic approach, delivered as an inaugural lecture at the University of Erlangen, emphasized studying geometric objects via symmetries, laying groundwork for modern abstract algebra in geometry. Toward the end of the century, pioneered in his 1895 memoir "Analysis Situs," introducing tools to classify spaces up to continuous deformation, independent of metrics or dimensions. defined the , a algebraic invariant capturing loops in a space that cannot be continuously shrunk to a point, and laid to theory by associating cycles and boundaries to multidimensional chains, enabling the study of holes in spaces. His work notably applied these ideas to knot theory, distinguishing tangled curves via group presentations, as in the trefoil knot's \langle x, y \mid x^2 = y^3 \rangle, which revolutionized the understanding of qualitative spatial properties.

20th-Century Expansions and Computational Approaches

In the early 20th century, saw profound advancements through David Hilbert's work, particularly his Nullstellensatz, which established a correspondence between algebraic varieties and radical ideals in polynomial rings, providing a foundational link between and . This , proved in Hilbert's on invariant systems, resolved key questions in by showing that if a system of polynomials has no common zeros in , then some power of the ideal they generate is the unit ideal, enabling rigorous algebraic descriptions of geometric objects. Building on this, in the 1940s extended the Riemann-Roch theorem to algebraic curves over finite fields, quantifying the dimension of spaces of differentials and functions with prescribed divisors, which bridged classical with arithmetic and influenced later developments in scheme theory. The integration of geometry with physics advanced significantly in the context of , where introduced the concept of affine connections on Riemannian manifolds in 1917, defining in a metric-compatible and torsion-free manner that specified the tensor geometrically. This framework, essential for interpreting as a curved manifold, allowed for the covariant of tensors and underpinned Einstein's field equations by providing a differential-geometric tool for describing motion. Complementing this, in the mid-20th century, developed by William V. D. Hodge and formalized through sheaf cohomology in the 1950s, decomposed the cohomology of compact Kähler manifolds into harmonic forms, pairing differential forms with topological invariants via the and Laplacian. This theory not only unified with complex structure but also yielded the Hodge decomposition theorem, where H^k(M, \mathbb{C}) = \bigoplus_{p+q=k} H^{p,q}(M), facilitating applications in mirror symmetry and . Computational geometry emerged as a distinct field in the mid-20th century, leveraging digital to solve geometric problems efficiently, with the of 1972 providing an O(n \log n) method for computing the of a of points in the by angularly around a lowest point and eliminating non-extreme candidates via cross-product tests. This scan, pivotal for applications in and , constructs the boundary by maintaining a of hull candidates and on left-turn violations. Voronoi diagrams, originally defined by Georgy Voronoi in 1908 as the of into cells of nearest sites based on , were efficiently computed starting in the 1970s (e.g., O(n log n) by Shamos and Hoey in 1975), with Fortune's sweep-line in 1986 enabling practical O(n log n) construction for nearest-neighbor queries in geographic information systems and . Fractal geometry, formalized by in the 1970s, expanded traditional measures by introducing self-similar structures prevalent in , such as coastlines and clouds, where the D quantifies irregularity via the scaling relation D = \frac{\log N}{\log (1/s)}, with N as the number of self-similar copies at scale factor s. In his book Fractals: Form, Chance, and , Mandelbrot demonstrated how this , exceeding the topological but below the embedding space, models phenomena like and , revolutionizing fields from chaos theory to by emphasizing scale-invariance over smoothness.