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Ancient Greek mathematics

Ancient Greek mathematics encompasses the systematic study and development of mathematical concepts by Greek-speaking scholars from approximately the 6th century BCE to the 4th century CE, emphasizing deductive reasoning, axiomatic proofs, and geometric abstraction over empirical calculation.^[1] This tradition marked a profound shift from earlier practical arithmetical systems in Mesopotamia and Egypt toward theoretical inquiry, where mathematics was pursued as a pure intellectual discipline intertwined with philosophy and cosmology.^[2] Flourishing primarily in Ionia, Athens, and Alexandria, it laid foundational principles for Western mathematics, influencing fields from geometry to astronomy until its integration into Roman and later Byzantine scholarship.^[1] The origins trace back to the Ionian school around 600 BCE, with Thales of Miletus (c. 624–546 BCE) credited as the first to apply geometric proofs to practical problems, such as measuring the height of pyramids using similar triangles, drawing from Egyptian techniques but introducing abstraction.^[2] His work exemplified the Greek pursuit of natural explanations through rational means, rejecting mythological accounts in favor of demonstrable theorems.^[2] By the 6th century BCE, Pythagoras of Samos (c. 570–495 BCE) and his followers elevated numbers to a mystical and structural essence of the universe, discovering the Pythagorean theorem and irrational numbers like √2, which challenged their belief in commensurable magnitudes.^[1] The Pythagorean school also advanced number theory, identifying perfect numbers and amicable pairs, while linking mathematics to music through harmonic ratios.^[1] In the 4th century BCE, the Academy of Plato in Athens fostered mathematical rigor, with Eudoxus of Cnidus (c. 408–355 BCE) developing the method of exhaustion to handle ratios and volumes of irrational figures, resolving paradoxes of infinitesimals.^[1] This culminated in Euclid's Elements (c. 300 BCE), a comprehensive treatise compiling 13 books on plane and solid geometry, arithmetic, and proportion, structured axiomatically with 465 theorems proved deductively from five postulates and common notions.^[2] Euclid's work standardized geometric constructions using only ruler and compass, influencing mathematical pedagogy for over two millennia.^[1] Hellenistic advancements in the 3rd century BCE, centered in Alexandria's Museum, saw Archimedes of Syracuse (c. 287–212 BCE) extend geometry to mechanics and calculus precursors, calculating areas under parabolas, volumes of spheres, and the value of π to high precision via inscribed polygons.^[1] Apollonius of Perga (c. 240–190 BCE) systematized conic sections—ellipses, parabolas, and hyperbolas—in his Conics, applying them to astronomy and engineering.^[1] These innovations, alongside works in trigonometry by Hipparchus (c. 190–120 BCE) and Ptolemy (c. 100–170 CE), demonstrated Greek mathematics' versatility, though it largely prioritized theory over algebraic notation or decimal systems.^[1]

Early Foundations

Archaic and Pre-Socratic Contributions

The Archaic period of Greek mathematics, spanning the 7th and 6th centuries BCE, saw the emergence of foundational geometric ideas among Ionian philosophers, marking a transition from empirical practices to more systematic inquiry. Thales of Miletus (c. 624–546 BCE), often regarded as the first Greek mathematician, is credited with several key geometric theorems derived from observations of practical problems, such as measuring distances at sea or the heights of pyramids.^[3]^[4] Among these, Thales demonstrated that the angle inscribed in a semicircle is a right angle, that a diameter of a circle bisects the circle into two equal parts, and that the base angles of an isosceles triangle are equal.^[5] These results, attributed to him by later sources like Eudemus of Rhodes, relied on visual intercepts and proportional reasoning rather than fully axiomatic proofs, yet they represented an early application of geometry to real-world measurements.^[3] Anaximander of Miletus (c. 610–546 BCE), a successor to Thales, extended these ideas into cosmological models using geometric principles. He employed the gnomon—a simple vertical rod used to cast shadows—for determining solstices and equinoxes, thereby introducing quantitative observations into astronomy and timekeeping.^[6]^[7] In his model of the universe, Anaximander envisioned the Earth as a short cylinder suspended at the center, with celestial bodies as rings or wheels whose sizes he specified geometrically, such as the Sun's ring being 27 or 28 times the Earth's diameter.^[8] This approach integrated geometry with speculative cosmology, positing a balanced, symmetrical structure to explain natural phenomena without mythological intervention. By the 5th century BCE, Ionian contributions evolved toward more advanced techniques for measuring volumes and areas, exemplified by Hippocrates of Chios (fl. c. 460 BCE). Hippocrates achieved the quadrature of certain lunes—crescent-shaped regions bounded by circular arcs—by showing that the area of a lune could equal that of a right-angled triangle or a segment of a circle, using properties of isosceles triangles and circle proportions.^[9] This work on curvilinear figures represented an early attempt to square non-polygonal shapes exactly, bridging practical mensuration with theoretical geometry. Collectively, these pre-Socratic efforts introduced deductive reasoning as a departure from the predominantly empirical, problem-solving methods of Egyptian and Babylonian mathematics, emphasizing general principles over ad hoc calculations.^[10]^[11]

Pythagorean Mathematics

The Pythagorean school, founded by Pythagoras (c. 570–495 BCE) in Croton, southern Italy, integrated arithmetic, geometry, and philosophy, viewing numbers as the fundamental essence of reality and the key to understanding the cosmos. This doctrine posited that all things are composed of numbers, with numerical relationships governing natural phenomena, ethics, and the soul's harmony. Central to this worldview was the tetractys, a symbolic arrangement of the first four natural numbers (1 + 2 + 3 + 4 = 10) forming a triangular figure, revered as the source of all creation, oaths, and musical attunement.^[12]^[13] A cornerstone of Pythagorean mathematics was the discovery of harmonic ratios in music, linking sound to numerical proportions through experiments with vibrating strings or pipes. The octave corresponds to the ratio 2:1, the perfect fifth to 3:2, and the perfect fourth to 4:3, demonstrating that musical consonance arises from simple whole-number relationships, which the school extended to cosmic order. These findings underscored the Pythagoreans' belief in numerical mysticism, where such ratios revealed the underlying structure of the universe. The school classified numbers into categories like even and odd, with even numbers associated with multiplicity and odd with unity, and identified "perfect" numbers like 6 and 28, equal to the sum of their proper divisors. Triangular numbers, formed by successive rows of dots (e.g., the nth triangular number is \frac{n(n+1)}{2}), exemplified geometric patterns in arithmetic, as seen in the tetractys itself.^[12]^[13] The Pythagorean theorem, stating that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (a^2 + b^2 = c^2), was a pivotal geometric achievement attributed to the school, with proofs involving the rearrangement of areas from squares on the legs to form the square on the hypotenuse or using properties of similar triangles. However, the discovery of irrational numbers shattered the Pythagorean conviction that all magnitudes are commensurable ratios of integers; the diagonal of a unit square, \sqrt{2}, proved incommensurable with the side, sparking a profound crisis that threatened the doctrine of numerical universality and led to secretive handling of such findings within the community.^[13]^[12] Geometric applications flourished in Pythagorean practice, notably the construction of the pentagram—a five-pointed star inscribed in a pentagon—symbolizing health and the golden ratio (\frac{1 + \sqrt{5}}{2}), whose properties in regular pentagons demonstrated self-similar proportions and irrational continuations. These constructions highlighted the school's emphasis on figures embodying numerical harmony, influencing later Greek geometry while maintaining a veil of communal secrecy.^[13]^[12]

Classical Developments in Plato's Era

In the 4th century BCE, Plato (c. 428–348 BCE) elevated mathematics as a philosophical tool for training the mind in dialectics and apprehending eternal forms, viewing geometric figures not as physical objects but as ideal archetypes that foster intellectual ascent toward truth. In his dialogue Meno, Plato illustrates this through Socrates guiding an uneducated slave boy to rediscover a geometric proof for doubling the area of a square, demonstrating innate recollection (anamnesis) of eternal truths rather than empirical learning. This philosophical idealization distinguished Platonic mathematics from earlier Pythagorean numerical mysticism, emphasizing geometry's role in purifying the soul for higher reasoning.^[14]^[15] Plato's Academy in Athens institutionalized this vision through a structured curriculum of five mathematical disciplines, progressing from sensory to abstract contemplation: arithmetic to grasp unchanging numbers, plane geometry for intelligible forms, solid geometry for three-dimensional ideals, astronomy to model celestial harmonies mathematically, and harmonics to discern proportional ratios in sound. Arithmetic sharpened dialectical skills by abstracting beyond visible objects, while plane geometry, as Plato urged, should avoid practical applications like land measurement to focus on eternal truths. Solid geometry, though underdeveloped, was prioritized for state patronage to advance understanding of forms like the cube. Astronomy and harmonics completed the ascent, treating motions and sounds as visible/audible manifestations of numerical order, preparing philosophers for the Good.^[16]^[17] Eudoxus of Cnidus (c. 408–355 BCE), a prominent Academy member, advanced theoretical geometry with his method of exhaustion, a rigorous technique for determining areas and volumes of curvilinear figures by approximating them with inscribed polygons whose differences could be made arbitrarily small, serving as a precursor to integral calculus. Applied to circles, this method proved that their areas are proportional to the squares of their diameters, enabling approximations of π through successive polygonal inscriptions. For volumes, Eudoxus established that pyramids and cones equal one-third the volume of prisms and cylinders with the same base and height, laying groundwork for later Hellenistic syntheses like Euclid's Elements.^[18]^[19] Theaetetus of Athens (c. 417–369 BCE), another young Academy scholar praised in Plato's dialogue Theaetetus, systematized the theory of irrational numbers by classifying surds such as \sqrt{2}, \sqrt{3}, and \sqrt{5} as incommensurable magnitudes sharing a common "power" or form, extending Theodorus' demonstrations and resolving Pythagorean crises over non-rational ratios. This classification, preserved in Euclid's Elements Book X, grouped irrationals by their generative roots, distinguishing them from rationals while unifying them conceptually for geometric constructions. Theaetetus also contributed to regular polyhedra, linking number theory to solid geometry in the Academy's curriculum.^[20] Speusippus (c. 407–339 BCE), Plato's nephew and Academy successor, alongside Menaechmus (c. 380–320 BCE), pioneered conic sections as geometric tools to address the Delian problem of doubling the cube, constructing two mean proportionals to solve for

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. Menaechmus generated conics by slicing cones at angles, using intersections like a parabola defined by y^2 = 4ax and a hyperbola to find the required length, marking an early theoretical exploration beyond straightedge-and-compass methods. These innovations, born from Academy collaborations, bridged philosophical geometry to practical challenges, influencing later Hellenistic advancements.^[21]^[22]^[23]

Hellenistic Advancements

Euclid and the Elements

Euclid, active around 300 BCE in Alexandria, compiled existing mathematical knowledge into the Elements, a systematic treatise that synthesized and formalized geometric and arithmetic principles from earlier Greek traditions. This work established mathematics on a rigorous axiomatic basis, influencing scientific thought for over two millennia. The Elements consists of 13 books, beginning with foundational elements such as definitions (e.g., a point as "that which has no part" and a line as "breadthless length"), five postulates (including Postulate 1 on drawing a straight line between points and Postulate 5, the parallel postulate stating that a line intersecting two others to form interior angles less than two right angles will meet them on the side of the smaller angles), and five common notions (axioms like "things equal to the same thing are equal to one another"). These components provide the deductive framework for all subsequent propositions, ensuring proofs rely solely on prior established truths.^[24]^[25]^[26] Book I focuses on plane geometry, developing basic constructions and theorems, including criteria for triangle congruence (side-angle-side, angle-side-angle, and side-side-side) through propositions like I.4 (SAS), I.8 (SSS), and I.26 (ASA). It culminates in Proposition I.47, offering a proof of the Pythagorean theorem: in a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides, demonstrated by rearranging areas of squares on the sides to show equality. Books III and IV extend these ideas to circles and polygons; Book III explores circle properties, such as tangents being perpendicular to radii (III.18) and the angle in a semicircle being a right angle (III.31), while Book IV addresses constructions like inscribing and circumscribing regular polygons (triangle, square, pentagon, hexagon, and decagon) in or about circles, solving problems of inscription and circumscription systematically.^[26]^[27]^[28] Books VII through IX shift to number theory, treating numbers as magnitudes and proving properties deductively. Book VII introduces the Euclidean algorithm for finding the greatest common divisor (gcd) of two numbers via repeated subtraction or division (antenaresis), as outlined in Propositions VII.1 and VII.2, which establish the procedure to determine if numbers are relatively prime or to find their greatest common measure. Propositions build to relatively prime numbers and primes, with Book IX Proposition 20 providing a proof by contradiction of the infinitude of primes: assume finitely many primes p_1, \dots, p_k; form N = p_1 \cdots p_k + 1; N exceeds each p_i and is not divisible by any, so it must have a prime factor not among them, yielding a contradiction. Books X through XIII address more advanced topics; Book X classifies irrational magnitudes (e.g., distinguishing commensurable and incommensurable lines, building on earlier work with quadratic irrationals), while Books XI and XII introduce solid geometry, including parallels in space (XI.31) and the method of exhaustion for volumes. Book XII, influenced by Eudoxus of Cnidus, employs the exhaustion method—approximating curvilinear figures with inscribed polygons to bound areas and volumes rigorously—to prove results like the volume of a pyramid being one-third that of a prism with the same base and height (XII.5) and the volume of a cone being one-third that of a cylinder with the same base and height (XII.10). Book XIII constructs the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron) inscribed in spheres, culminating in Proposition XIII.17, which shows the icosahedron's side square relates to the sphere's diameter by a specific ratio involving the golden section, and compares their relations to prove the dodecahedron's "greatness" among them.^[29]^[30]^[31]^[18]^[32]

Archimedes' Innovations

Archimedes of Syracuse (c. 287–212 BCE) made groundbreaking contributions to mathematics during the Hellenistic period, particularly in geometry, mechanics, and hydrostatics, by developing rigorous methods to compute areas, volumes, and physical principles that extended beyond Euclidean foundations. His innovative approach often combined theoretical proofs with practical mechanical insights, influencing later developments in calculus and engineering.^[33] One of Archimedes' key innovations was his "Method of Mechanical Theorems," a heuristic technique for discovering areas and volumes by balancing figures on levers, treating them as if composed of infinitesimally thin slices. In this method, he envisioned plane figures as assemblages of lines and solids as stacks of planes, using the principle of the lever to equate moments and infer geometric properties; for instance, he determined that the area of a parabolic segment is \frac{4}{3} times the area of the inscribed triangle with the same base and vertex. This mechanical approach, detailed in his treatise The Method, provided intuitive discoveries that he later verified through exhaustive geometric proofs, marking a precursor to integral calculus.^[33]^[34] Archimedes applied similar rigor to the quadrature of curved figures, notably in Quadrature of the Parabola, where he proved that the area of a segment bounded by a parabola and its chord equals \frac{4}{3} the area of the triangle formed by that chord and the parabola's vertex, using an iterative exhaustion process that doubled inscribed triangles until convergence. He extended this to the Archimedean spiral, a curve defined in polar coordinates by r = a\theta, where a is a constant and \theta is the angle from the origin; in On Spirals, he quadratured the area under the first turn of this spiral, showing it equals \frac{1}{3} the area of the circle with radius equal to the spiral's endpoint. These results demonstrated his mastery of handling non-linear curves through approximation and summation.^[34]^[35]^[36] In Measurement of a Circle, Archimedes approximated the value of \pi by inscribing and circumscribing regular polygons with up to 96 sides around a unit circle, establishing that \frac{223}{71} < \pi < \frac{22}{7}, which provided bounds accurate to about three decimal places (3.1408 to 3.1429) and remained the standard for centuries. His work on three-dimensional figures culminated in On the Sphere and Cylinder, where he derived the volume of a sphere as \frac{4}{3}\pi r^3 and its surface area as $4\pi r^2, showing that a sphere's volume is \frac{2}{3} that of the circumscribing cylinder of the same height and radius; he also computed the cone's volume as \frac{1}{3}\pi r^2 h and proved the sphere-cylinder relation through mechanical balancing and exhaustion. Archimedes requested that a sphere inscribed in a cylinder be engraved on his tombstone to symbolize this discovery.^[37]^[38] Archimedes pioneered hydrostatics in On Floating Bodies, formulating the principle that a body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced, enabling analysis of floating and submerged objects' equilibrium. This law, derived from considerations of fluid pressure and displacement, applies to irregular shapes and underpins modern fluid mechanics. To address astronomical scales, Archimedes developed a numeral system in The Sand-Reckoner capable of expressing numbers up to $10^{63}, using orders of magnitude based on multiples of 10,000 myriad (myriad-myriad = $10^8) to estimate the grains of sand needed to fill the universe, thereby demonstrating the potential for arbitrarily large finite quantities in a prepositional framework. This system highlighted his interest in infinity and combinatorics, countering the notion that the universe's size defied numerical reckoning.

Apollonius and Conic Sections

Apollonius of Perga (c. 240–190 BCE), often called the "Great Geometer," was a prominent mathematician in the Hellenistic period who advanced the study of conic sections through his seminal work Conics. Living primarily in Alexandria, he built upon earlier geometric traditions to provide a comprehensive, systematic analysis of these curves, emphasizing their properties derived from intersections with cones. His treatise, composed in eight books, survives partially: the first four in the original Greek, and the fifth through seventh in Arabic translations, with the eighth lost. This work not only classified and named the conic sections but also explored their intricate geometric behaviors, laying foundational principles for later mathematics.^[39] Apollonius defined conic sections—parabola, ellipse, and hyperbola—as the curves generated by a plane intersecting a right circular cone, distinguishing them based on the angle of the cutting plane relative to the cone's axis. He introduced the terminology that persists today: ellipse (from the Greek for "deficiency" or "compressed circle," referring to the shortfall in a geometric "application" of areas), parabola (meaning "application," for the equal application), and hyperbola (meaning "excess," for the surplus in application). Central to his analysis were properties akin to the focus-directrix definition, where a conic is the locus of points such that the ratio of the distance to a fixed point (focus) to the distance to a fixed line (directrix) is constant, known as the eccentricity e. For the ellipse, e < 1; for the parabola, e = 1; and for the hyperbola, e > 1. These properties allowed Apollonius to derive key characteristics without relying solely on coordinate methods.^[39]^[40] In the first two books of Conics, Apollonius established fundamental theorems on diameters, tangents, and asymptotes, treating conics as affine transformations of circles to simplify proofs. He demonstrated how to construct tangents from external points and analyzed intersections between conics and lines or other conics, including cases where they touch or cross. For instance, his work on the ellipse includes propositions equivalent to the modern equation \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, derived geometrically through parameters and ordinates along diameters. Books three and four extended these to more complex loci and applications, while the later books (five through seven) delved into normals, the evolute (the locus of centers of curvature), and maximum/minimum distances from the center—topics that anticipated differential geometry. Asymptotes were rigorously defined as lines approached by the hyperbola branches but never intersected, with constructions for their determination.^[39]^[41] Apollonius generated conic sections geometrically by specifying cone types (right-angled, obtuse-angled, or acute-angled) and plane orientations, using projective properties to unify their study. For example, he showed that all conics could be obtained from sections of a single cone under affine transformations, emphasizing their interrelations. This approach enabled precise constructions, such as drawing the parabola as the path where the plane is parallel to a cone's generator. His methods relied on Euclidean tools—compass and straightedge—for verifying theorems, avoiding algebraic notation entirely.^[39]^[40] Beyond pure geometry, Apollonius applied conics to classical problems, demonstrating how parabolic and hyperbolic sections could trisect an arbitrary angle and duplicate the cube—tasks impossible with straightedge and compass alone under Euclid's restrictions. In trisection, he used a marked ruler intersecting a circle and hyperbola to divide the angle into three equal parts; for cube duplication, conic intersections scaled volumes geometrically. These solutions highlighted the power of conics in resolving "impossible" constructions, influencing astronomical models like epicycles, though Apollonius focused on theoretical rigor over practical computation. His work thus bridged geometry and problem-solving, inspiring subsequent scholars in the Byzantine and Islamic traditions.^[39]^[40]

Specialized Fields

Arithmetic and Number Theory

Ancient Greek arithmetic and number theory built upon Pythagorean principles, emphasizing the properties of odd and even numbers, multiples, and means. The Pythagoreans classified numbers based on parity, viewing odd numbers as indivisible and masculine, while even numbers were divisible and feminine; products were categorized as even-even, odd-odd, even-odd, and odd-even, reflecting their structural qualities. They also explored arithmetic, geometric, and harmonic means, with the harmonic mean defined as the reciprocal of the arithmetic mean of reciprocals, applied in contexts like musical intervals to achieve proportional harmony. These ideas laid the groundwork for later systematic treatments, prioritizing numerical relationships over computational algorithms..pdf) Euclid's Elements, particularly Books VII, VIII, and IX, formalized much of this into rigorous number theory, extending Pythagorean insights. In Book VII, Euclid introduced the Euclidean algorithm (antenaresis), a subtractive process to find the greatest common divisor of two numbers, enabling solutions to linear indeterminate equations of the form ax + by = c where a, b, and c are integers, provided \gcd(a, b) divides c; particular solutions are obtained via back-substitution from the algorithm, with general solutions differing by multiples of b/\gcd(a, b) and -a/\gcd(a, b). Euclid defined perfect numbers as those equal to the sum of their proper divisors excluding themselves, exemplified by 6 (since $6 = 1 + 2 + 3) and 28 (since $28 = 1 + 2 + 4 + 7 + 14); in Proposition IX.36, he proved that if $2^p - 1 is prime (a Mersenne prime), then $2^{p-1}(2^p - 1) is even and perfect. These propositions established foundational results on primes, divisibility, and abundance, influencing subsequent classifications.^[29]^[42] Nicomachus of Gerasa, in his Introduction to Arithmetic (c. 100 CE), advanced these concepts by classifying numbers into perfect (sum of proper divisors equals the number), deficient (sum less than the number), and abundant (sum greater than the number), listing the first four perfect numbers: 6, 28, 496, and 8128. He also described amicable pairs, such as 220 and 284, where the proper divisors of each sum to the other number, extending the perfect number idea to relational pairs. Nicomachus further categorized figurate numbers, including polygonal numbers like triangles and squares, noting properties such as every square number beyond 1 being the sum of two consecutive triangular numbers, which underscored the geometric-arithmetic interplay in discrete forms.^[43]^[44]^[45] Iamblichus of Chalcis (c. 245–325 CE) preserved and expanded Nicomachus' work through his commentary on the Introduction to Arithmetic, integrating Neoplatonic mysticism with mathematical exposition. He elaborated on figurate numbers, providing formulas for triangular numbers as the sum of the first n naturals, \frac{n(n+1)}{2}, and square numbers as n^2, and proved relations like eight times any triangular number plus one yielding a square: $8 \cdot \frac{n(n+1)}{2} + 1 = (2n+1)^2. These commentaries emphasized the philosophical significance of such sequences, viewing them as manifestations of cosmic order, and transmitted Pythagorean-Nicoma chan ideas into late antiquity.^[46].pdf)

Geometry Beyond Euclid

While Euclid's Elements provided a foundational axiomatic framework for plane and solid geometry, later Greek mathematicians extended these principles through innovative techniques that approximated limits and addressed optimization problems in curves and solids. A pivotal advancement was the method of exhaustion, developed by Eudoxus of Cnidus around 370 BCE and refined by Archimedes in the 3rd century BCE. This approach involved inscribing and circumscribing polygons or polyhedra around curved figures, progressively increasing the number of sides to "exhaust" the difference between the approximations and the true area or volume, effectively foreshadowing integral calculus. For instance, Archimedes applied it to compute the area under a parabolic segment as equivalent to four-thirds the area of a triangle with the same base and height, demonstrating how the method could rigorously bound irrational quantities without invoking infinitesimals.^[47]^[48] Building on such techniques, Zenodorus (c. 200–140 BCE) tackled isoperimetric problems, seeking figures of equal perimeter that maximize enclosed area or, in three dimensions, solids of equal surface area that maximize volume. He proved that among plane figures, the circle encloses the greatest area for a given perimeter, surpassing regular polygons as the number of sides increases. Extending this to solids, Zenodorus demonstrated that the sphere possesses the maximum volume for a fixed surface area, using comparisons with cylinders and polyhedra derived from Archimedes' prior results on spheres and cylinders. These findings highlighted the optimality of curved surfaces in geometric optimization, influencing later variational problems.^[49] In spherical geometry, Hipparchus (c. 190–120 BCE) introduced chord tables that served as a precursor to trigonometry, listing chord lengths in a unit circle for central angles from 0° to 180° in increments of 7.5°. These tables enabled computations in spherical triangles, essential for astronomical applications, by providing a systematic way to relate arcs and angles on a sphere beyond Euclidean plane constructions. Hipparchus' work laid groundwork for solving problems in non-Euclidean contexts, such as determining distances on celestial spheres.^[50] Menelaus of Alexandria (c. 70–130 CE) further advanced transversal theorems in both plane and spherical geometry. His theorem states that for a transversal line intersecting the sides of a triangle ABC at points D, E, F on extensions if necessary, the product of the directed segment ratios satisfies \frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1. This relation, proved using areas and similar triangles, generalized Euclidean proportionality and proved invaluable for concurrency and collinearity proofs. Menelaus extended it to spherical triangles in his Sphaerica, adapting it for great circles on a sphere to compute angular distances.^[51]^[52] Developments in solid geometry progressed beyond Euclid's basic polyhedra by exploring relations between curved surfaces, notably in Archimedes' On the Sphere and Cylinder. He established that the surface area of a sphere is four times the area of its great circle, or equal to the lateral surface area of the circumscribing cylinder (with height equal to the diameter), while its volume is two-thirds that of the circumscribing cylinder. These ratios, derived via the method of exhaustion, revealed intrinsic properties of spheres not deducible from Euclid's postulates alone, such as the sphere's equilibrium within a cylinder and applications to levers and balances in three dimensions. Later geometers, including those in late antiquity, built on these to investigate sections and inscriptions of spheres in other solids.^[48]

Astronomy, Trigonometry, and Applications

Ancient Greek mathematicians made significant contributions to astronomy by applying geometric principles to model celestial motions and determine distances, with Aristarchus of Samos (c. 310–230 BCE) proposing an early heliocentric model where the Earth and other planets orbit the Sun, in contrast to the prevailing geocentric views.^[53] In his surviving treatise On the Sizes and Distances of the Sun and Moon, Aristarchus employed geometric arguments based on the geometry of lunar eclipses—where the Earth's shadow's diameter during totality provides a ratio—to estimate the relative sizes of the Sun, Moon, and Earth.^[54] Specifically, he calculated that the Sun's diameter is about 19 times that of the Earth, though his method relied on geometrically observed angles during lunar eclipses and quarter phases, which were imprecise, leading to an underestimate of the actual ratio of approximately 109.^[55] Trigonometry emerged as a key tool in Hellenistic astronomy for computing positions and distances on the celestial sphere, with Hipparchus of Nicaea (c. 190–120 BCE) credited as its founder through his development of the chord function, defined as the straight-line distance subtended by a central angle θ in a circle of radius r, given by \text{chord}(\theta) = 2r \sin(\theta/2).^[56] Hipparchus compiled the first known trigonometric table of chords for a circle divided into 360 parts, providing values for arcs in increments related to 24 equal divisions (corresponding to 15° steps as a foundational grid, with finer interpolations), enabling precise calculations of spherical distances and planetary motions without relying solely on geometric constructions.^[57] This table, reconstructed from later sources like Ptolemy, facilitated astronomical predictions, such as precession of the equinoxes, by allowing computation of chord lengths for angles up to 180°.^[58] Ptolemy of Alexandria (c. 100–170 CE) advanced these techniques in his Almagest, a comprehensive geocentric model using deferents (large circular orbits around Earth) and epicycles (smaller circles on deferents) to account for irregular planetary paths, with trigonometric tables essential for parameter calculations.^[59] Ptolemy's chord table, extending Hipparchus's work, listed values for every half-degree from 0° to 180° in a circle of radius 60 parts, achieving high precision through iterative methods like the Ptolemic theorem for chord differences.^[60] He converted chords to sines via the relation \sin(\theta) = \frac{1}{2} \text{[chord](/page/Chord)}(2\theta), producing an equivalent sine table that supported computations for eccentricities, longitudes, and latitudes in his epicycle model, influencing astronomical modeling for over a millennium.^[61] In practical applications, Heron of Alexandria (c. 10–70 CE) integrated geometry into mechanics and surveying, notably deriving the area formula for a triangle with sides a, b, c and semiperimeter s = (a+b+c)/2 as \sqrt{s(s-a)(s-b)(s-c)} in Book I of his Metrica, a result obtained through geometric dissection without trigonometry.^[62] This formula found use in Heron's engineering works, such as Pneumatica, where it aided in calculating areas for designing pneumatic devices like automated theaters and siphons, ensuring precise fluid volumes and structural balances in steam-powered mechanisms. Surveying benefited from Heron's innovations in instrumentation, including the dioptra, a precision sighting device akin to a theodolite with adjustable sights and a water level for measuring horizontal and vertical angles over long distances, as detailed in his treatise Dioptra.^[63] Complementing this, Heron described the odometer, a wheeled cart mechanism with geared axles that incremented counters based on wheel rotations to measure traveled distances accurately, applicable to road mapping and aqueduct alignment during the Hellenistic and Roman periods.^[64] These tools combined geometric principles with mechanical ingenuity, enabling large-scale infrastructure projects like the division of irregular lands into measurable plots.

Late Antiquity Developments

Diophantus' Arithmetica

Diophantus of Alexandria (c. 200–284 CE), often called the father of algebra, composed the Arithmetica around the mid-3rd century CE as a comprehensive treatise on solving arithmetic problems through algebraic methods. Originally comprising 13 books, only the first six survive in the original Greek (containing 189 problems), while an Arabic translation discovered in 1973 preserves books IV–VII, adding content from book VII. These problems primarily involve indeterminate equations seeking positive rational or integer solutions, marking a shift from geometric to more abstract arithmetic approaches in Greek mathematics. The work's emphasis on systematic problem-solving laid foundational techniques for later number theory.^[65]^[66] A key innovation in the Arithmetica is Diophantus's use of syncopated notation, an early form of symbolic algebra that abbreviated terms for efficiency. The unknown quantity, termed "the number" (arithmos), is represented by the sigma symbol ς. Powers are denoted by specific abbreviations: ∆υ for the square (second power), Kυ for the cube (third power), ∆∆υ for the fourth power, and so forth up to the sixth power (Tυ), with reciprocals indicated by overbars or suffixes like ςχ for 1/ς. Constants are prefixed with ˚M, addition achieved by simple juxtaposition, subtraction by symbols like an inverted delta (∧), and equality by ις. This notation, while not fully symbolic like modern algebra, allowed concise expression of quadratic and higher-degree equations, limited primarily to powers up to the quadratic in practice for most problems.^[65]^[66] The problems in the Arithmetica are classified into definite and indefinite categories, further subdivided by the number of conditions imposed. Definite problems seek unique or finitely many solutions under strict constraints, such as dividing a given number into parts with specified ratios or products (primarily in Book I). Indefinite problems, the core of the work, admit infinitely many solutions and require additional conditions to yield specific instances, categorized as having single or multiple parameters. Books II–III focus on representing numbers as sums or differences of squares, while later books explore more complex indeterminate analysis, including equations involving cubes and polygonal numbers. Diophantus's methods often involve trial assumptions, parameter introduction, and the "false hypothesis" technique to adjust solutions.^[65]^[66] Representative examples illustrate the depth of indeterminate analysis in the Arithmetica. In Book II, Problem 8, Diophantus solves for dividing 16 into two squares, yielding rational solutions such as \frac{256}{25} + \frac{144}{25} = 16, demonstrating techniques for sum-of-squares representations. Pell-like equations of the form x^2 - D y^2 = [1](/page/1) appear in various guises, with Diophantus providing solutions for specific D values; for instance, problems in Books III and V lead to forms equivalent to solving such equations for small D, and his methods influenced later solutions for challenging cases like D=61, known for its exceptionally large minimal solution (x=1766319049, y=226153980). A notable applied problem is the cannonball stacking in Book III, Problem 19, where Diophantus finds three squares such that the differences between them satisfy a given ratio (3:1), analogous to arranging cannonballs in triangular formations where pyramidal stacks form triangular numbers. These examples highlight Diophantus's focus on constructive solutions over general proofs.^[65]^[67] The Arithmetica exerted profound influence on subsequent mathematics, particularly through marginal notes in a 17th-century copy owned by Pierre de Fermat. On Book II, Problem 8 (the sum-of-squares division), Fermat scribbled that he had discovered a proof that no four positive integers satisfy x^4 + y^4 = z^4, challenging others to match it without revelation—a claim sparking Fermat's Last Theorem and centuries of research in Diophantine equations. This annotation underscores how Diophantus's problem-solving framework inspired modern number theory.^[65]^[66]

Pappus' Collection

Pappus of Alexandria, active in the early fourth century CE, authored the Synagoge (often translated as Collection or Mathematical Collection), an eight-book compendium that served as a vital synthesis and commentary on Hellenistic mathematics, preserving knowledge from earlier figures amid the declining classical tradition.^[68] Composed around 340 CE, the work aimed to guide readers through key geometric and arithmetic advancements, often referencing lost treatises and providing lemmas for problem-solving, though only parts survive in Greek, with others known through Arabic translations or fragments.^[69] Unlike focused monographs, Pappus' Collection emphasized interconnections across topics, influencing later Byzantine and Islamic scholars by cataloging methods for analysis and synthesis in geometry.^[70] Book I of the Collection, now lost, focused on arithmetic topics, including recreations and paradoxes that explored conceptual challenges such as impossible divisions of magnitudes, highlighting the limits of numerical manipulation in Greek thought.^[68] These elements drew from earlier traditions, using arithmetic to illustrate counterintuitive results, though direct contents are inferred from Pappus' prefaces and cross-references in later books.^[70] In contrast, surviving portions of Book II extend into handling large numbers via methods attributed to Apollonius, underscoring Pappus' role in bridging arithmetic with geometric applications.^[68] Book VII stands out as the "Treasury of Analysis," a catalog of lemmas and techniques for solving geometric problems, particularly those involving conic sections and centers of gravity, building on lost works by Euclid, Apollonius, and Aristaeus.^[68] Pappus outlines analytical methods—reducing problems to known constructions through reverse reasoning (analysis) and forward verification (synthesis)—with applications to conics for loci and intersections, such as determining points on parabolas or hyperbolas satisfying given conditions.^[70] Central to this book are centroid theorems for plane figures, including the position of the center of gravity; for instance, the centroid of a parabolic segment lies at \frac{3}{5} the height from the vertex to the base, enabling computations of stability and equilibrium in curved regions.^[71] In Book V, Pappus addresses practical geometry through the honeycomb theorem, proving that a regular hexagonal tiling minimizes the total perimeter for partitioning the plane into regions of equal area, outperforming triangular or square grids.^[72] He demonstrates this by comparing perimeters: for a given boundary length, hexagons enclose the maximum area among tilable polygons, explaining the efficiency of bee honeycombs, though he notes the circle achieves the absolute maximum but cannot tile without gaps.^[73] This result, derived from isoperimetric considerations, prefigures modern optimization problems and highlights Pappus' integration of natural observation with rigorous proof.^[68] Book VI references lost works on spherical geometry, including those by Theodosius of Tripoli on the Sphaerica and related astronomical treatises, providing lemmas for great circles, poles, and day-night divisions on the sphere.^[68] Pappus comments on Theodosius' theoretical approach to spherical loci and intersections, contrasting it with more applied texts, and preserves propositions on spherical triangles and equators that would otherwise be unknown.^[74] These discussions underscore the Collection's value as a repository, linking plane geometry to three-dimensional extensions in astronomy.^[68]

Neoplatonic Commentaries and Compilations

In the late antique period, Neoplatonic scholars in the 4th to 6th centuries CE produced extensive commentaries that preserved, interpreted, and philosophically enriched earlier Greek mathematical texts, integrating them into a broader metaphysical framework. These works, often emerging from lecture traditions in Alexandria and Athens, emphasized mathematics as a bridge to understanding divine order and the soul's ascent, while providing technical clarifications and alternative proofs for classical results.^[75] Proclus (412–485 CE), a leading Neoplatonist in Athens, authored a detailed Commentary on the First Book of Euclid's Elements, which explores the hypotheses and axioms of Euclidean geometry through a philosophical lens. In this work, Proclus examines the ontological status of mathematical objects, arguing that they exist as intermediate realities between the sensible world and intelligible forms, serving as paradigms for demonstrating Platonic ideas. He structures his analysis around the division of mathematical propositions into enunciation, definition, and demonstration, linking geometric truths to Neoplatonic cosmology.^[76]^[77] Simplicius (c. 490–560 CE), active in Athens and later Persia, contributed commentaries on Aristotle's Physics and other works that engaged with mathematical concepts, particularly Archimedean mechanics and the notion of infinite divisibility. In his Commentary on Aristotle's Physics, Simplicius references Archimedes' mechanical principles, such as the lever and equilibrium, to illustrate Aristotelian natural philosophy, while defending the infinite divisibility of continua against atomist critiques through Neoplatonic arguments for the unity of magnitudes. These discussions preserve fragments of earlier mechanical traditions and reconcile them with Platonic and Aristotelian views on motion and infinity.^[78] Eutocius (c. 480–540 CE), associated with the Alexandrian school, wrote influential commentaries on Archimedes' treatises (On the Sphere and Cylinder, On the Measurement of a Circle) and the first four books of Apollonius' Conics. His annotations supply detailed proofs for methods that were likely lost or mechanical in origin, including neuseis constructions and solutions to problems like angle trisection using conic sections, thereby elucidating Archimedes' and Apollonius' approaches to quadrature and sectioning. Eutocius' work demonstrates a practical focus on recovering and verifying classical techniques within a scholarly tradition.^[79]^[80] Olympiodorus the Younger (c. 495–570 CE), a prominent figure in Alexandria's Neoplatonic circle, integrated harmonic theory into philosophical exegesis, particularly in his commentaries on Plato's dialogues. He viewed harmonics as a mathematical discipline revealing cosmic harmony and the soul's attunement to divine principles, drawing on Ptolemy's Harmonics to connect musical intervals with Neoplatonic emanations. This synthesis extended the quadrivium's role in spiritual pedagogy, blending quantitative analysis with metaphysical symbolism.^[81] These commentaries emerged from Alexandria's Neoplatonic school, the final major hub of Greek mathematical and philosophical inquiry before its decline amid rising Christian dominance in the 6th century CE. Under figures like Ammonius Hermiae, the school fostered interdisciplinary scholarship that safeguarded texts through interpretive layers, ensuring their transmission despite political upheavals.^[82]^[75]

Transmission and Legacy

Preservation in the Byzantine Empire

The preservation of ancient Greek mathematical texts in the Byzantine Empire relied heavily on systematic copying efforts in Constantinople and monastic scriptoria, ensuring the continuity of works in their original Greek language from late antiquity through the early Middle Ages. Major treatises such as Euclid's Elements were meticulously reproduced, with a notable parchment manuscript dated to 888 CE produced in Constantinople by the scribe Stephanos, exemplifying the ongoing scribal tradition that maintained geometric proofs and axioms without significant alteration.^[83] These copying practices, centered in imperial libraries and remote monasteries, safeguarded core texts on arithmetic, geometry, and astronomy against the ravages of time and political instability, allowing Byzantine scholars to engage directly with foundational works like those of Euclid and Ptolemy.^[84] The period of iconoclasm from the 8th to 9th centuries CE contributed to a general decline in scholarly production amid broader cultural upheavals.^[85] However, the 9th-century Macedonian Renaissance marked a revival, bolstered by diplomatic and trade contacts with the Arab world that facilitated the exchange of knowledge and introduction of new computational methods from the Arab world, enabling scholars to rebuild and expand upon preserved materials.^[86] This resurgence drew briefly on late antique commentaries as foundational source texts, integrating them into renewed copying initiatives.^[87] Astronomical knowledge, rooted in Ptolemaic models, persisted through practical applications in Byzantine chronology and calendrical computations, as seen in the 9th-century Chronographia of Theophanes the Confessor, which incorporated epact tables derived from Ptolemy's Handy Tables to align solar and lunar cycles for ecclesiastical purposes.^[88] These tables, adapted for Byzantine use, reflected the empire's reliance on Greek astronomical frameworks for timekeeping and prediction, underscoring mathematics' role in supporting imperial administration and religious observance. Meanwhile, Nicomachus of Gerasa's Introduction to Arithmetic was integrated into Christian theological discourse, influencing Byzantine quadrivium treatises that explored numbers' mystical properties—such as the symbolism of perfect numbers—in harmony with Neoplatonic and scriptural interpretations, echoing Boethius' earlier Latin adaptations but preserved in Greek originals.^[89] In the late 13th and early 14th centuries, scholars like Maximus Planudes (c. 1260–1305 CE) further adapted and disseminated Greek mathematics, producing a translation and commentary on Diophantus' Arithmetica that clarified indeterminate equations and algebraic methods for contemporary audiences.^[90] Planudes also briefly introduced eastern Arabic numerals through his treatise The Great Calculation According to the Indians, promoting their use alongside traditional Greek notation in Byzantine computational practices, though adoption remained limited.^[91] These efforts highlighted the empire's role as a bridge for Greek mathematical heritage, sustaining it amid evolving cultural influences until the fall of Constantinople in 1453 CE.

Adoption in the Islamic World

During the Islamic Golden Age, from the 8th to the 12th centuries CE, Greek mathematical texts were systematically translated into Arabic, primarily at the House of Wisdom (Bayt al-Hikma) in Baghdad, where scholars integrated and expanded upon these works, fostering innovations in algebra, geometry, and trigonometry.^[92] Established under the Abbasid caliphs, this intellectual center served as a hub for translation efforts that preserved and adapted ancient knowledge, with key figures like al-Kindi (c. 801–873) overseeing renditions of Euclid's Elements and Ptolemy's Almagest, while Hunayn ibn Ishaq (809–873) and his school contributed to translating Archimedes' treatises on geometry and mechanics, often via intermediate Syriac versions to ensure accuracy.^[93] These translations not only safeguarded Greek texts from loss but also enabled Muslim scholars to critique and refine them, blending Hellenistic methods with local and Indian influences.^[94] A pivotal advancement came through Muhammad ibn Musa al-Khwarizmi (c. 780–850), whose Kitab al-Jabr wa al-Muqabala (The Compendious Book on Calculation by Completion and Balancing) formalized algebra as a distinct discipline, addressing linear and quadratic equations through geometric proofs and rhetorical descriptions.^[95] Drawing indirect inspiration from Diophantus' Arithmetica, al-Khwarizmi tackled indeterminate problems—equations with multiple solutions—by classifying cases and emphasizing practical applications in inheritance and commerce, while his separate work on Hindu numerals introduced the positional decimal system to the Islamic world, revolutionizing computation.^[96] This synthesis marked a shift from Greek geometric algebra toward a more systematic, problem-solving approach.^[97] In the 11th century, Omar Khayyam (1048–1131) extended these foundations in his Treatise on the Demonstration of Problems of Algebra, devising geometric methods to solve cubic equations by intersecting conic sections, such as hyperbolas and circles, thereby addressing equations previously intractable by quadratic techniques.^[98] For instance, to solve the general cubic

x^3 + a x^2 + b x + c = 0,

Khayyam constructed a parabola or hyperbola intersecting a circle, where the abscissa of the intersection point yielded the root, building on Apollonian conics while avoiding numerical approximation.^[99] This approach highlighted the power of synthetic geometry for higher-degree polynomials and influenced later Persian and Islamic mathematicians.^[100] Ibn al-Haytham (965–1040), known as Alhazen, further advanced mathematical optics in his monumental Kitab al-Manazir (Book of Optics), employing infinitesimal methods reminiscent of Archimedes' exhaustion technique to model light propagation and calculate volumes of rotation for lenses and mirrors.^[101] By dividing rays into infinitesimally small segments and integrating their paths, he quantified refraction and reflection with unprecedented precision, extending Archimedean statics to dynamic visual phenomena and laying groundwork for later calculus-like analyses in physics.^[102] His rigorous experimental validation of these models distinguished his work from purely speculative Greek optics.^[103] Trigonometry also saw refinement through Abu Abd Allah Muhammad ibn Jabir ibn Sinan al-Battani (c. 858–929), whose Zij (astronomical tables) improved Ptolemy's chord-based calculations by adopting sine functions, producing more accurate tables for sines, tangents, and cosines up to two decimal places.^[104] Al-Battani's enhancements, derived from direct observations, corrected Ptolemaic errors in solar and lunar parameters, facilitating precise astronomical predictions and influencing subsequent zij compilations across the Islamic world.^[105] This trigonometric framework proved essential for navigation, surveying, and spherical astronomy, bridging Greek theory with empirical practice.^[106]

Revival in Medieval and Renaissance Europe

The revival of ancient Greek mathematics in medieval and Renaissance Europe began with the 12th-century translations from Arabic sources, facilitated by scholars in Toledo who bridged Islamic and Latin traditions. Gerard of Cremona (c. 1114–1187), a key figure in this movement, produced Latin translations of Euclid's Elements and Ptolemy's Almagest, making these foundational Greek texts accessible to Western scholars for the first time in complete form. These translations, drawn from Arabic editions that preserved and expanded upon the originals, spurred interest in geometry and astronomy across European monasteries and universities.^[107] In the early 13th century, Leonardo Fibonacci (c. 1170–1250) further integrated Greek mathematical ideas into European practice through his Liber Abaci (1202), which introduced Hindu-Arabic numerals—ultimately rooted in Greek numeral systems via Islamic intermediaries—and addressed Diophantine problems involving integer solutions to equations. Fibonacci's work demonstrated practical applications of these numerals for commerce and calculation, drawing on Greek arithmetic traditions preserved in Arabic texts, and helped supplant Roman numerals in Europe. By solving problems akin to those in Diophantus's Arithmetica, such as finding numbers satisfying multiple conditions, Fibonacci highlighted the enduring relevance of ancient Greek number theory.^[108]^[109] The 15th century saw advancements in trigonometry, with Johannes Regiomontanus (1436–1476) compiling detailed trigonometric tables in works like De triangulis omnimodis (c. 1464), building directly on Ptolemy's chord tables from the Almagest. These tables, emphasizing sines and spherical trigonometry, were instrumental for navigation, as evidenced by their use in Christopher Columbus's voyages via Regiomontanus's ephemerides. His efforts systematized Greek trigonometric methods for practical astronomy and surveying, marking a shift toward more precise computational tools in Europe.^[110]^[111] The invention of printing accelerated the dissemination of Greek mathematics during the Renaissance. Erhard Ratdolt's 1482 edition of Euclid's Elements, the first printed version in Latin, featured innovative geometric diagrams and made the text widely available to scholars and artisans. Similarly, Federico Commandino (1509–1575) translated and edited Archimedes's works, publishing Archimedis opera non nulla in 1558, which included Latin versions of treatises on spheres, cylinders, and conoids, complete with commentaries that clarified Greek proofs. These printed editions preserved and interpreted ancient Greek innovations in geometry and mechanics.^[112]^[113] This revival culminated in transitions to modern mathematics, as seen in René Descartes's La Géométrie (1637), which developed analytic geometry by applying algebraic coordinates to geometric problems, explicitly inspired by Apollonius's Conics and its Greek roots in synthetic geometry. Descartes's method unified algebra and geometry, transforming ancient Greek conic sections into a foundational tool for calculus and physics, while acknowledging the classical heritage that informed his innovations.^[114]^[115]

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Heron of Alexandria - Biography - MacTutor - University of St Andrews
His best known mathematical work is the formula for the area of a triangle in terms of the lengths of its sides. Thumbnail of Heron of Alexandria View two ...<|separator|>
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Surveying Instruments of Greece and Rome - ResearchGate
It explores the history of surveying instruments, notably the Greek dioptra and the Roman libra, and with the help of tests with reconstructions explains ...Missing: odometer | Show results with:odometer
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(PDF) Heron of Alexandria (c. 10–85 AD) - ResearchGate
PDF | Heron of Alexandria was a mathematician, physicist and engineer who lived around 10–85 AD. He taught at Alexandria's Musaeum and wrote many books.Missing: triangle | Show results with:triangle
[65]
[PDF] The Symbolic and Mathematical Influence of Diophantus's Arithmetica
Jan 1, 2015 · Diophantus to use a syncopated style, and the notation of the Algebra closely resembles that of the Arithmetica [2]. For instance, for x6 ...Missing: sources | Show results with:sources
[66]
[PDF] Diophantus and the Arithmetica Spencer Neff
Nesselmann calls this intermediate stage Syncopated Algebra. Diophantus uses the Greek letter ς' to represent the unknown quantity. “This symbol in verbal.Missing: sources | Show results with:sources
[67]
[PDF] Diophantine Equations An Introduction
Fermat's challenge of 1657 to find an integral solution for d = 61 brought this equation, attributed to Pell, to prominence. However, three centuries ...<|control11|><|separator|>
[68]
Pappus (290 - 350) - Biography - MacTutor History of Mathematics
Pappus of Alexandria is the last of the great Greek geometers and one of his theorems is cited as the basis of modern projective geometry.
[69]
[PDF] Pappus of Alexandria, Book VIII of the Mathematical Collection
Pappus of Alexandria was one of the final figures in the main line of ancient Greek mathematics. Since he mentions works by Claudius Ptolemy (ca 100–170 CE) ...
[70]
[PDF] Pappus of Alexandria, (fl. c. 300-c. 350)!
Aug 27, 2000 · Summary of Contents: • Book I and first 13 (of 26) propositions of Book II. Book II was concerned with very large numbers { powers of ...Missing: Synagoge | Show results with:Synagoge
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Pappus's Centroid Theorem -- from Wolfram MathWorld
The first theorem of Pappus states that the surface area S of a surface of revolution generated by the revolution of a curve about an external axis is equal ...Missing: parabola 3/5 height
[72]
[math/9906042] The Honeycomb Conjecture - arXiv
Jun 8, 1999 · Pappus discusses this problem in his preface to Book V. This paper gives the first general proof of the conjecture.
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[PDF] Book V of the Mathematical Collection of Pappus of Alexandria ...
In Hultsch's Proposition 18, Pappus shows that the sphere is greater than any regular polyhedron of the same surface area. The remainder of Book V then ...<|control11|><|separator|>
[74]
[PDF] Book VI of the Mathematical Collection of Pappus of Alexandria ...
For example, they say of the sixth theorem of the third book of Theodosius' Spherica that the two great circles must be cut by the great circle through the ...
[75]
https://brill.com/view/journals/jpt/18/1/article-p1_1.xml
[76]
Proclus: A Commentary on the First Book of Euclid's Elements - jstor
Proclus' commentary on book I of Euclid's Elements is almost certainly a written version of lectures which he presented to students and associates in Athens ...
[77]
Proclus' division of the mathematical proposition into parts: how and ...
Feb 11, 2009 · ... Proclus: A Commentary on the First Book of Euclid's Elements, trans. Morrow, G. (Princeton, 1992), pp. xlviii–1.Google Scholar. 41. 41 I ...
[78]
Full article: The problems of exceptionality: The case of Archimedes ...
Dec 6, 2022 · The very first piece of extended Greek mathematical reasoning that we have is the account in Simplicius (Commentary on Aristotle's Physics 54.12 ...
[79]
Eutocius (480 - 540) - Biography - MacTutor History of Mathematics
Eutocius's commentary on Apollonius's "Conics" is extant for the first four Books, and it is probably owing to their having been commented on by Eutocius, as ...
[80]
Translated into English, together with Eutocius' Commentaries, with ...
Jul 14, 2004 · The Works of Archimedes: Translated into English, together with Eutocius' Commentaries, with Commentary, and Critical Edition of the Diagrams.
[81]
[PDF] ON THE ROLE OF HARMONIA IN PLATO'S PHILOSOPHY
Mathematics and harmonics in Republic VII ... Neoplatonist Olympiodorus suggests that temperance (sophrosynē) is the 'order' (kosmos) of the parts of ...
[82]
THE LATE NEOPLATONIC SCHOOL OF ALEXANDRIA
Aug 6, 2025 · This essay will provide a survey of the major philosophical schools in ancient Greek and Roman ethics, from the time of Plato (429–347 bc) until ...<|control11|><|separator|>
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Early History - Euclid's Elements
The first is a manuscript written in Greek, dated to 888 AD, and written in Constantinople on parchment by the scribe (clerk) Stephanos. It is a copy of Euclid ...
[84]
Greek Mathematics and its Modern Heirs - Ibiblio
In Byzantium, the capital of the Greek-speaking Eastern empire, the original Greek texts were copied and preserved. In the Islamic world, in locales that ...
[85]
Mathematics - Rome Reborn: The Vatican Library & Renaissance ...
In Byzantium, the capital of the Greek-speaking Eastern empire, the original Greek texts were copied and preserved. In the Islamic world, in locales that ...
[86]
The Great Myths 8: The Loss of Ancient Learning - History for Atheists
Mar 28, 2020 · Leo the Mathematician and his fellow scholars led a revival after the long centuries of loss, infighting and turmoil following the Arab ...
[87]
[PDF] 5. The late Greek period - UCR Math Department
During the period between 400 B.C.E. and 150 B.C.E., Greek mathematical knowledge had increased very substantially. Over the next few centuries, ...
[88]
[PDF] The Chronographia of George the Synkellos and Theophanes
Epact table in the Vatican copy of the Handy Tables of Ptolemy. The diagram's outer ring indicates the epact for each year from the 30th year of the ...
[89]
8. How the Early Christian Theology of Arithmetic Shaped Neo ...
After the early third century, the controversy over the theology of arithmetic disappeared from the Church. The dispute need not have died down.Missing: Boethius | Show results with:Boethius<|separator|>
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[PDF] Untitled
Between this mention of Diophantus and the intensive study of him by the late Byzantine authors George Pachymeres and Maximus Planudes there is another ...
[91]
The Great Calculation According to the Indians, of Maximus Planudes
The present work, The Great Calculation According to the Indians, introduces (i) the (eastern) Arabic form of the Indian numerals, as used in Persia, along with ...
[92]
(PDF) "Greco-Arabic Translation Movement in the Muslim World, an ...
Aug 22, 2025 · Bayt al-Hikmah in Baghdad translated Greek and Persian works into Arabic, with scholars like Hunain bin Ishaq and al-Kindi leading the way.
[93]
Greek Sources in Arabic and Islamic Philosophy
Feb 23, 2009 · The compendium of Plato's Timaeus has been translated into Syriac by Hunayn ibn Ishaq and from Syriac into Arabic by Hunayn's pupil ʿIsa ibn ...
[94]
[PDF] THE TRANSMLSSION OF GREEK GEOMETRY TO MEDIEVAL ISLAM
They included Euclid's Data and his Phaenornena, Theodosius's Sphaerica, Menelaus's work of the same title. (translated by Hunayn ibn lshaq (b. 809)), Hypsicles ...
[95]
The Art of Algebra from Al-Khwārizmī to Viète: A Study in the Natural ...
... Hindu-Arabic numerals. It is also important to note ... indeterminate problems which may have been inspired indirectly by Diophantus through al-Karajī.
[96]
[PDF] Islamic Mathematics - University of Illinois
Modern numeral notation certainly has its roots in al-Khwarizmi and other. Arab mathematicians; though influenced by Hindu numerals, al-Khwarizmi and his ...
[97]
Islamic Mathematics (Chapter 2) - The Cambridge History of Science
Islamic algebraists contributed importantly to the study of Diophantine equations, which demand integer or fractional solutions to a single equation in more ...
[98]
(PDF) Omar Khayyam: Geometric Algebra and Cubic Equations
Sep 3, 2020 · Khayyam presented methods to nd solutions of cubic equations using geometric constructions by identifying the intersection of hyperbolas, parabolas, circles, ...
[99]
A Geometric Solution of a Cubic by Omar Khayyam… in Which ...
Dec 13, 2017 · This graphical presentation removes the modern reliance on algebraic notation and focuses instead on a visualization that emphasizes Khayyam's ...Missing: scholarly | Show results with:scholarly
[100]
[PDF] The Works of Omar Khayyam in the History of Mathematics
The preceding form of a cubic equation was only one type of cubic equation Omar Khayyam developed a method of finding a solution to. In fact, he found solutions.
[101]
[PDF] Ibn Al‐Haytham (Alhazen) - CFCUL
He composed 12 treatises on infinitesimal mathematics and then on conic theory. To those can be added a third area in which Ibn al‐Haytham takes up several ...Missing: Archimedean | Show results with:Archimedean<|separator|>
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(PDF) Ibn Al-Haytham : Father of modern optics - Academia.edu
Alhazen's problem involved deriving the fourth-degree equation from geometric principles, contributing to the foundational developments of infinitesimal ...
[103]
Setting up the Empire of Mathematics - ResearchGate
Feb 9, 2024 · geometrical objects. In the early mathematizations of Euclid's optics and Archimedes' statics, the theorems. express something necessarily true ...
[104]
[PDF] Trigonometry.pdf - Adelphi University
The main difference was that al-Bāttānī relied on trigonometric calculations rather than the geometrical models of Ptolemy. For some of his computations he ...
[105]
(PDF) E. S. Kennedy : Survey of Islamic Astronomical Tables 1956
This groundbreaking study from 1956 identified some 125 Islamic astronomical handbooks with tables and explanatory text prepared in Muslim lands between 750 ...
[106]
Mathematical Science - Contributions of Islamic Scholars to the ...
Sep 2, 2025 · Thus, the Muslims not only developed the methods of solving quadratic equations they also produced tables containing sine, cosine, cotangent and ...<|control11|><|separator|>
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Gerard of Cremona's Latin translation of the Almagest and the ...
Feb 4, 2023 · The first complete, printed edition of Ptolemy's Almagest appeared in 1515 and is based on Gerard of Cremona's translation from Arabic into ...
[108]
Fibonacci - Evansville
In the Liber Abbaci, Fibonacci explains (sometimes tediously) the nature of the Hindu-Arabic numerals, their use in calculations with integers and fractions, ...
[109]
[PDF] Leonardo Pisano Fibonacci By Susmita Paruchuri Born in 1170 ...
Fibonacci solves. Diophantine problems involving second degree equations with two or more variables, with solutions required to be given as integers or rational ...
[110]
Applications of trigonometry - cs.clarku.edu
Ptolemy (100–178) used trigonometry in his Geography and used trigonometric tables in his works. Columbus carried a copy of Regiomontanus' Ephemerides ...
[111]
Trigonometry for the heavens - Physics Today
Dec 1, 2017 · Trigonometry, both plane and spherical, was intended for astronomers; 15th-century astronomer Regiomontanus called it “the foot of the ladder to ...Trigonometry For The Heavens · Triangles In Curved Space · From The Heavens To Earth
[112]
Euclid's Elementa Geometriae Printed by Ratdolt
It is in Latin, published by Erhard Ratdolt on May 25, 1482, in Venice, based on Campanus' translation of Euclid's Elements from Arabic, and contains 15 books ...
[113]
Mathematical Treasure: Commandino's Archimedes
Archimedis opera non nulla (The Complete Archimedes, 1558) was a translation from Greek into Latin by Federico Commandino. Commandino (1509–1575) was an ...
[114]
Descartes' Mathematics - Stanford Encyclopedia of Philosophy
Nov 28, 2011 · In La Géométrie, Descartes details a groundbreaking program for geometrical problem-solving—what he refers to as a “geometrical calculus” ( ...
[115]
DESCARTES AND THE BIRTH OF ANALYTIC GEOMETRY
In the field of geometry, a very clear application of the coordinate principle is to be found in the first book of Apollonius's. Conies. Hero of Alexandria used ...