Written and collected by Zia H Shah MD, Chief Editor of the Muslim Times

Historical Context: Knowledge Synthesis in the Islamic Golden Age

The Islamic Golden Age (8th–14th century) fostered a vibrant intellectual environment that combined and expanded the mathematical knowledge of earlier civilizations. Under the Abbasid Caliphate, centers of learning like the House of Wisdom in Baghdad became crucibles for scholarship and translation​ britannica.com. Caliph al-Ma’mun (reigned 813–833) founded the House of Wisdom as a library, academy, and translation bureau, attracting scholars from various faiths to translate Greek, Indian, and Persian works into Arabic​ britannica.com. Through this translation movement, the medieval Islamic world gained access to Greek mathematics (e.g. Euclid’s Elements, Archimedes’ treatises) and Indian mathematics (such as Brahmagupta’s work and the concept of the zero-based place-value system with Hindu-Arabic numerals)​ en.wikipedia.org britannica.com. Persian scholarly traditions and texts were also incorporated, creating a rich, syncretic knowledge base.

Crucially, Muslim scholars did not only preserve earlier works – they expanded and innovated upon them. For example, almost all of Apollonius’s works on conic sections were translated, and even a few of Archimedes’ treatises (on the sphere, cylinder, and circle) were rendered into Arabic, stimulating original research from the 9th century onward​ britannica.com. Indian advancements like the decimal notation and trigonometric sine function were adopted and refined: by the 10th century, Islamic mathematicians had extended the place-value system to include decimal fractions (with a decimal point)​ britannica.com, and had fully integrated trigonometry (sine, cosine, etc.) into astronomy and mathematics. The Banū Mūsā brothers in 9th-century Baghdad, for instance, produced original works on geometry and mechanics​ britannica.com, and mathematicians such as Thābit ibn Qurra not only translated Greek texts but also contributed new ideas (Thābit discovered a rule for generating amicable numbers)​.

By synthesizing these diverse sources, the scholars of the Islamic Golden Age created an environment where algebra, geometry, and arithmetic could evolve in novel ways. The infusion of Greek rigor, Persian and Babylonian astronomical insight, and Indian computational techniques led to important developments: notably, the systematic study of algebra as an independent discipline and major advances in geometry and trigonometry en.wikipedia.org. Scholars collaborated and corresponded across the Islamic world, from Central Asia and Persia to North Africa and al-Andalus (Islamic Spain), ensuring a wide dissemination of ideas. This cosmopolitan scholarly network, supported by enlightened patrons and buoyed by the practical needs of astronomy, commerce, and land surveying, set the stage for breakthroughs that foreshadowed concepts of precalculus and calculus.

Key Mathematicians and Their Contributions

Al-Khwarizmi: Algebra and Numerical Methods

Muḥammad ibn Mūsā al-Khwarizmi (c. 780–850) was a pioneering mathematician whose work laid critical groundwork for later mathematical advances. Working in Baghdad’s House of Wisdom during the early 9th century, Al-Khwarizmi authored The Compendious Book on Calculation by Completion and Balancing (Kitāb al-jabr wa’l-muqābala) around 830 CE – the first systematic treatise on algebra britannica.com. In this book, he introduced algebra as a distinct field, separate from geometry and arithmetic, solving linear and quadratic equations by reduction (“al-jabr” or completion) and balancing of terms​ mathshistory.st-andrews.ac.uk. This approach was revolutionary: it provided general methods for solving equations and established a clear, logical framework for manipulating unknowns. Al-Khwarizmi solved equations both algebraically and geometrically (by completing the square)​ mathshistory.st-andrews.ac.uk​, illustrating an early link between algebraic operations and geometric interpretation.

Al-Khwarizmi’s algebra had enduring influence. He is often called “the father of algebra” in recognition of how he defined and solidified the subject. His algebraic approach, departing from earlier rhetorical problem-solving, “laid the groundwork for the arithmetization of algebra”, influencing mathematical thought for centuries​ en.wikipedia.org. Successors like al-Karaji and Abu Kamil built on his methods, extending algebra to higher powers and more abstract realms. Al-Khwarizmi also wrote a treatise on Hindu-Arabic numerals (explaining the decimal positional system and algorithms for calculation) which was translated into Latin as Algoritmi de Numero Indorum. Through this work, the West learned of the efficient decimal system; indeed, the word “algorithm” derives from a Latin form of his name (Algoritmi)​. Likewise, the term “algebra” comes from al-jabr in the title of his book​. Both his algebra text and arithmetic text were translated into Latin in the 12th century, becoming standard references in Europe​. By introducing general methods and a logical structure, Al-Khwarizmi provided future mathematicians—Islamic and European alike—with essential tools that would eventually feed into the development of calculus (for example, algebraic manipulation of equations and polynomials).

Nasir al-Din al-Tusi: Trigonometry and Mathematical Innovation

Nasīr al-Dīn al-Tūsī (1201–1274) was a Persian polymath who made outstanding contributions in mathematics and astronomy. He is especially renowned for elevating trigonometry to an independent discipline, distinct from astronomy​. In his treatise “On the Sector Figure” (and other works), al-Tusi systematically presented the plane and spherical trigonometric functions and laws of triangles. He stated and proved the law of sines for plane triangles and extended it to spherical triangles, providing the first general proof of this law​ en.wikipedia.org. In fact, Tusi’s work advanced the development of trigonometry as an independent field, comparable to what Regiomontanus would accomplish in Europe 200 years later​

early-astronomy.classics.lsa.umich.edu. He formulated the six fundamental trigonometric relationships for right-angled spherical triangles (analogous to modern spherical law of cosines, etc.) and clarified the definitions of the trigonometric functions sine, cosine, tangent, and their reciprocals​ science.howstuffworks.com. By doing so, Tusi freed trigonometry from its purely astronomical context and made it a general mathematical tool – a crucial step for precalculus.

Beyond trigonometry, al-Tusi made other contributions that resonate with early calculus ideas. He developed techniques for binomial expansions and binomial coefficients in his study of numerical relationships​ science.howstuffworks.com. (Notably, Pascal’s triangle is known in Persia as the “Khayyam-Tusi triangle,” since both Omar Khayyam and al-Tusi discussed binomial coefficients centuries before Pascal​ science.howstuffworks.com.) Al-Tusi is also famous for inventing the Tusi couple, a mathematical device involving a small circle rotating inside a larger circle twice its radius. This mechanism produces a back-and-forth linear motion from the sum of two circular motions​ commons.wikimedia.org – effectively an early study of harmonic motion. The Tusi couple was devised to improve planetary models, but its geometric ingenuity foreshadows concepts of fourier decomposition or kinematic motion used in later mathematics. (Copernicus, in the 16th century, employed an identical couple in his heliocentric theory, likely informed by knowledge of Tusi’s work​ beforenewton.blog.) In addition, al-Tusi wrote on the philosophy of mathematics: he asserted, for instance, that the product of ratios is commutative and that any rational number can be considered a “real” number in a general sense​. Such insights reflect a mature understanding of number and magnitude, concepts underpinning continuous quantities in calculus.

Al-Tusi’s legacy was profound. He led the Maragha observatory in the 13th century and trained a generation of scientists. Through his works (many written in Arabic and Persian), he transmitted both Greek and Islamic mathematical knowledge forward. His clear formulations in trigonometry provided the tools for precise astronomical tables and navigation. Moreover, by emphasizing rigorous proof (in geometry and trig) and by exploring compounded motions (the Tusi couple), Nasir al-Din al-Tusi touched on ideas of functional relationships and kinematics that are cornerstones of calculus. His influence would eventually reach Europe via translations and the work of later astronomers.

Ibn al-Haytham: Optical Analysis and Early Integration Ideas

Abū ‘Alī al-Ḥasan ibn al-Haytham (965–c.1040), Latinized as Alhazen, was an Arab mathematician, astronomer, and physicist often hailed as the “father of modern optics.” While Ibn al-Haytham is best known for his groundbreaking work Kitāb al-Manāẓir (Book of Optics), he also developed mathematical techniques that foreshadow integration and analysis. In attempting to solve geometric problems related to optics and astronomy, Ibn al-Haytham employed sophisticated methods of summation. Most famously, he derived a formula for the sum of fourth powers: 14+24+⋯+n41^4 + 2^4 + \cdots + n^414+24+⋯+n4​ en.wikipedia.org. He did this by devising a clever recursive method that could, in principle, be generalized to sum any integral power​. In essence, Alhazen found a way to compute ∑k=1nk4\sum_{k=1}^n k^4∑k=1n​k4 (and by extension k3,k2,k^3, k^2,k3,k2, etc.) and used this result to determine the volume of a paraboloid, treating the problem much like a continuous integration​. This is a striking instance of integral calculus in embryo: he was summing infinitesimal cross-sectional areas (via summing powers) to find a volume, much as one would integrate x4x^4×4 to get x55\frac{x^5}{5}5×5​ in modern calculus. Ibn al-Haytham’s achievement here is sometimes credited as the first formulation of integration beyond the method of exhaustion. He effectively could find the formula for the integral (area/volume) of any polynomial up to fourth degree without a general closed-form notation​.

In developing these summation formulas, Ibn al-Haytham pioneered an early form of mathematical induction or finite difference method to extrapolate patterns​ pmc.ncbi.nlm.nih.gov. His work on sums of powers was later known to European mathematicians (it’s cited by Pietro Cataldi in the 17th century) and is a direct precursor to what would become known as Faulhaber’s formulas for sums of powers. In addition to this algebraic approach, Ibn al-Haytham made conceptual leaps in treating continuous phenomena. In his Book of Optics, he argued that a light ray emanates from every point on an object in straight lines in all directions, effectively modeling a continuum of rays​. He considered objects as “a compilation of an infinite number of points, from which rays of light are projected”​ – an idea that implies breaking a continuum (the object or a light source) into infinitely many infinitesimal pieces. This conceptualization mirrors the integral calculus view of continuous matter or area composed of infinitely many infinitesimal elements.

Beyond optics and summation, Ibn al-Haytham solved equations involving higher-degree polynomials and tackled what is now known as Alhazen’s problem (finding the point of reflection on a spherical mirror – an early max/min problem). His method for the reflection problem led him to solve a difficult quartic equation, arguably the first instance of someone solving a fourth-degree equation numerically. In doing so, he had to approximate solutions – a step into numerical methods. He also made early strides in understanding limits: examining what happens to an expression as it approaches some extreme value (this occurs implicitly in his geometric proofs and analysis of vision). Ibn al-Haytham’s blending of geometry, algebra, and physical reasoning exemplifies how Islamic scholars began moving toward an analytic mode of thinking, carving the path towards calculus. His works were highly influential; the Book of Optics was translated into Latin by the late 12th century and became a core text in Europe, directly influencing scholars like Roger Bacon and Johannes Kepler​ en.wikipedia.org. Through these translations, his mathematical ideas (like his integration technique) also percolated into European knowledge.

Other Notable Scholars: Al-Samaw’al, Omar Khayyam, and Sharaf al-Din al-Tusi

In addition to the figures above, several other scholars of the Islamic Golden Age made significant contributions to precalculus and proto-calculus concepts:

• Al-Samaw’al ibn Yaḥyā al-Maghribī (c.1130–1180) was a mathematician who built on the algebraic legacy of Al-Khwarizmi and al-Karaji. In his treatise Al-Bāhir fī’l-Jabr (“The Brilliant in Algebra”), written when he was only nineteen, Al-Samaw’al pushed algebra to new heights. He is credited with fully arithmetizing algebra, treating polynomial expressions with the same rigor as numeric quantities ​mathshistory.st-andrews.ac.uk. Al-Samaw’al explicitly defined powers of the unknown x not just for positive integers, but also zero and negative exponents – i.e. he allowed terms like x−1,x−2x^{-1}, x^{-2}x−1,x−2 in algebraic expressions ​mathshistory.st-andrews.ac.uk​mathshistory.st-andrews.ac.uk. This was a remarkable leap; it amounts to working with what we today call rational functions or power series. He described rules for polynomial addition, subtraction, multiplication, and division in general, even cases like dividing by xxx (yielding negative powers)​ mathshistory.st-andrews.ac.uk​. In doing so, Al-Samaw’al had to embrace the concept of negative numbers. He provided rules for operations with negative quantities, stating for example that subtracting a negative is equivalent to adding a positive​ mathshistory.st-andrews.ac.uk – rules fundamental to modern algebra and crucial for calculus (where negative exponents represent reciprocals and facilitate series expansions). This concept of handling negative and zero exponents would reappear much later in Europe; Al-Samaw’al’s algebraic techniques predated by several centuries the algebraic notation of Viète and Descartes that made calculus calculations feasible.
• Omar Khayyam (1048–1131), better known in the West as a poet, was also a brilliant mathematician who advanced algebra and its geometric applications. In his work Treatise on the Demonstration of Problems of Algebra, Khayyam made a systematic study of cubic equations. He classified cubic equations into various standard forms and, recognizing the limitations of algebraic methods of his time, he solved them by geometric means – specifically by intersecting conic sections​ encyclopedia.com. For example, to solve a cubic of the form x3+ax=bx^3 + a x = bx3+ax=b, Khayyam would construct a circle and a hyperbola such that their intersection’s coordinates yield the solution to the equation​ digitaleditions.sheridan.com​sfu.ca. This approach effectively created a dictionary between algebraic equations and geometric curves, a step towards analytic geometry. Khayyam’s use of conic intersections to solve cubics can be seen as an early implicit use of a continuum of solutions (the curves) to find a specific numeric solution – akin to solving equations by finding function intersections, a routine concept in calculus and precalculus. Moreover, Khayyam was aware of the number of solutions a cubic could have and even speculated about the existence of a general algebraic solution (which would not be found until the 16th century in Europe). Aside from cubics, Omar Khayyam is credited with studying the binomial expansion (Pascal’s triangle) for general powers. He understood the triangular arrangement of binomial coefficients and used it to extract roots and powers ​en.wikipedia.org​en.wikipedia.org. In 11th-century Persia, this array is known as Khayyam’s triangle, and Khayyam “popularised its use” for binomial coefficients​ mapleprimes.com. There are hints that he considered cases of the binomial theorem beyond integer exponents – possibly exploring series expansions for irrational exponents ​iranchamber.com. This would be a direct precursor to the binomial series expansion that Newton famously generalized. Khayyam also made a significant contribution to the foundations of geometry by addressing Euclid’s parallel postulate, but in the context of calculus development, his fusion of algebra and geometry in solving problems is most noteworthy. His work on cubics “greatly influenced future Islamic mathematicians, and through them the mathematicians of Renaissance Europe”​ encyclopedia.com.
• Sharaf al-Dīn al-Tūsī (c.1135–1213) was another Persian mathematician (not to be confused with Nasir al-Din al-Tusi) who wrote a penetrating treatise on algebraic equations, especially cubics. Sharaf al-Dīn’s work Al-Muʿādalāt (“On Equations”) marks a leap toward the concept of a mathematical function and its analysis ​en.wikipedia.org. He investigated cubic equations of the form f(x)=cf(x) = cf(x)=c (using modern notation) where f(x)f(x)f(x) is a polynomial, and systematically found conditions for these equations to have one, two, or no positive solutions ​en.wikipedia.org​en.wikipedia.org. In particular, Sharaf al-Dīn considered an equation like x3+ax2+bx+c=0x^3 + ax^2 + bx + c = 0x3+ax2+bx+c=0 (with one side zero) and examined the cubic expression f(x)f(x)f(x) on the other side. He discovered that such an equation can have at most two positive roots, and to determine how many roots exist, one should find the maximum value of the function f(x)f(x)f(x) on the positive real line ​en.wikipedia.org​en.wikipedia.org. Sharaf al-Dīn actually computed the location mmm where f(x)f(x)f(x) attains its maximum (for various cubic forms) and then compared f(m)f(m)f(m) to the target value ccc ​en.wikipedia.org​en.wikipedia.org. He concluded: if c<f(m)c < f(m)c<f(m), the equation f(x)=cf(x)=cf(x)=c has two solutions; if c=f(m)c = f(m)c=f(m), it has one solution (a double root); and if c>f(m)c > f(m)c>f(m), it has no positive solution ​en.wikipedia.org​en.wikipedia.org. This reasoning is essentially an analysis of the shape of a cubic curve, identifying its turning point (local maximum) and using it to infer solution counts – exactly what we do with derivatives and the discriminant in modern polynomial analysis. In fact, Sharaf al-Dīn’s method implies finding where f’(x)=0f’(x) = 0f’(x)=0 for the cubic (even though he did not write an explicit derivative). Some historians argue that Sharaf al-Dīn “systematically” took the derivative of the cubic to find the critical point​ en.wikipedia.org, while others think he may have used geometric reasoning without formalizing differentiation. Either way, he clearly understood the concept of a function’s extremum. He was effectively using an early form of differential calculus: determining the maxima of functions and employing them to solve problems. Additionally, Sharaf al-Dīn devised what is now known as the Ruffini-Horner method to numerically approximate roots of equations ​en.wikipedia.org – a technique to successively hone in on a solution. This is an iterative algorithm similar in spirit to the later Newton’s method for finding roots. Sharaf al-Dīn’s work was ahead of its time: he came very close to linking algebra and calculus. However, his insights did not immediately spread to Europe (his manuscript remained relatively unknown until modern times). Nonetheless, they indicate a clear trajectory in Islamic mathematics toward the key ideas of calculus: functions, continuous change, maxima and minima, and numerical solution of equations.

Each of these scholars extended the mathematical toolkit in a way that later became crucial for calculus. By the 13th century, Islamic mathematicians had conceptually defined polynomials and their arithmetic (al-Samaw’al), solved higher-degree equations via curves (Khayyam), and analyzed the variation of functions (Sharaf al-Dīn). These achievements formed a natural bridge from classical Greek geometry to the analytic methods of the early modern period.

Mathematical Innovations Paving the Way for Calculus

The Golden Age of Islam yielded a number of technical contributions and innovations that collectively paved the way for the emergence of calculus. These can be grouped into several key areas:

• Development of Algebraic Techniques: Islamic mathematicians transformed algebra from a problem-solving art into a general theory of equations, which is fundamental for calculus. Al-Khwarizmi’s introduction of symbolic operations for solving equations provided a language to describe general curves (like parabolas and cubics) algebraically​ britannica.com. Later, algebraists like Al-Karaji and Al-Samaw’al extended these techniques to high-degree polynomials and even to infinite processes. They introduced the idea of mathematical induction and recursive formulas to prove statements for all integers – Al-Karaji (c. 1000) used a rudimentary inductive argument to prove the formula for the sum of cubes​ apcentral.collegeboard.org, and Ibn al-Haytham later did similar for fourth powers. Al-Samaw’al’s negatives and zero exponents allowed what we now recognize as polynomial division and power series expansion ​mathshistory.st-andrews.ac.uk​. This algebraic groundwork was essential for the symbolic manipulation in calculus, where one expands functions in series or factors polynomials to find roots and critical points. By treating symbols and unknowns in a general way, Islamic algebra made it possible to conceive of an arbitrary function f(x)f(x)f(x) and reason about its properties abstractly.
• Early Uses of Infinitesimals and Limits: Several Islamic scholars displayed an intuitive understanding of infinitesimal quantities and the idea of letting something tend to zero or infinity. The method of exhaustion (a Greek technique to find areas by inscribing polygons) was known to Islamic mathematicians through Archimedes’ works and was applied in various forms. For example, the astronomer al-Battani in the 9th century used very large polygons to approximate a circle’s circumference (akin to a limiting process). Ibn al-Haytham’s reasoning in optics considered objects as made of an “infinite” number of light-emitting points ​en.wikipedia.org, a conceptual leap treating the continuum via infinitely small pieces. Also, in determining the qibla direction, some mathematicians considered geometrical projections that in effect involve limiting cases. While a rigorous infinitesimal calculus didn’t yet exist, we see proto-limit concepts: notions like “arbitrarily large”, “arbitrarily small”, or approaching a value. Importantly, Islamic mathematicians were comfortable summing large sequences and sometimes even summing diverging series for practical approximations (e.g., in astronomical computations, although convergence was not formally addressed). This tentative handling of the infinite and infinitesimal set the stage for the later formal limit concepts of calculus.
• Trigonometric Identities and Equations: Trigonometry was massively advanced in the Islamic Golden Age, and this had direct relevance to calculus. Precise trigonometric identities (such as sin⁡(A±B)\sin(A\pm B)sin(A±B) formulas, double-angle formulas, etc.) were derived by scholars like Abu’l-Wafā’ and al-Tusi​ en.wikipedia.org ​science.howstuffworks.com. They introduced the law of sines and law of cosines for spherical geometry, and defined the six trig functions clearly. Trigonometry provided methods to deal with periodic phenomena and angular motion, which are central to calculus in physics (e.g., orbital motion analysis). Moreover, Islamic astronomers compiled accurate sine and tangent tables to high precision. Solving equations using trig identities became common – for instance, finding an angle that satisfies a certain sine value (a precursor to inverse functions) was essential in astronomy. The concept of a function was implicitly present in trigonometry: for example, treating sin⁡x\sin xsinx as a magnitude depending on the angle xxx. Sharaf al-Dīn al-Tusi went further and discussed equations in a functional form f(x)=cf(x) = cf(x)=c, essentially treating f(x)f(x)f(x) as an object of study itself​ en.wikipedia.org. This gradual shift from static equations to an idea of a varying quantity f(x)f(x)f(x) is a prerequisite for the function concept in calculus. Additionally, Islamic mathematicians’ work on trig helped develop the idea of series approximations for sine and cosine. While the full power series (like Madhava’s series in India or later Gregory’s series) were not yet known in the Islamic world, the groundwork of understanding how to approximate nonlinear functions was laid with interpolation techniques on trig tables.
• Geometric Methods for Integration and Area/Volume: Many problems tackled by Islamic scholars were essentially integration problems – finding areas, volumes, or arc lengths. They often approached these with classical geometry or ingenious new methods. For example, the Banu Musa (9th century) devised methods to find the area under curves (like a parabola segment) by geometric dissection. Ibn al-Haytham’s determination of the volume of a paraboloid by summing fourth powers was effectively an integration of a polynomial function by converting it to a summation problem ​en.wikipedia.org. Astronomers like Al-Biruni (11th century) measured the Earth’s circumference by what we can interpret as integration-like reasoning: summing small angles over a long distance. There was also work on determining the area of spherical surfaces (spherical integration) in connection with map projection problems. The Islamic tradition preserved Archimedes’ idea of the balance of small slices: for instance, the equilibrium of the lever was used as an analogy to derive centroids (which involves integrating x dmx\,dmxdm). While these geometric methods were presented in static proofs, they contain the seeds of calculus: considering shapes composed of indivisible elements and summing those elements. In some cases, scholars even discussed pseudointegrals: Abū Sahl al-Qūhī in the 10th century studied volumes of revolution (a prelude to Pappus’s centroid theorem) and essentially integrated to find volumes of solids like paraboloids and hyperboloids (using conic sections and balancing arguments). These efforts indicate an understanding that a whole can be obtained by accumulating infinitesimal parts, a central idea of integration.
• Contributions to Understanding Rates of Change: A true conception of derivative (instantaneous rate of change) did not fully emerge in medieval Islam, but some glimmers were present. The science of motion (in physics) was discussed by thinkers like Ibn Sina (Avicenna) and Ibn Bājja (Avempace), who considered uniform vs. accelerated motion. Avempace in the 12th century even talked about something analogous to velocity increment during non-uniform motion, which hints at the derivative concept. More concretely, Sharaf al-Dīn al-Tusi’s work directly touched on the rate of change of a polynomial function. By finding where a cubic polynomial increases or decreases, he was examining how f(x+Δx)−f(x)f(x+\Delta x) – f(x)f(x+Δx)−f(x) changes sign – effectively a difference quotient approach to a derivative​ en.wikipedia.org. Some historians believe he computed derivative-like expressions (though without modern notation) to locate extrema​en.wikipedia.org. Additionally, the development of astronomical models introduced the need to understand rates: the motion of planets in the Islamic Ptolemaic tradition (Maragha school) involved variable speeds along epicycles, requiring knowledge of when motion was fastest or slowest (again, an extremum problem). Nasir al-Din al-Tusi’s Tusi couple generated linear oscillation – analyzing this motion means implicitly dealing with velocity (the linear speed of the point) and how it relates to the constant speeds on the circles. Such analyses inch towards a conception of periodic rates of change. Furthermore, in the study of optics, Ibn al-Haytham considered the path of quickest time vs. shortest path (a rudimentary form of Fermat’s principle), essentially comparing rates of light in different media. All these are precursors to the idea that one can have a quantity that changes with time or with some parameter, and that one can analyze that change. By the 14th century, even as the Islamic Golden Age waned, thinkers like Ibn Khaldun recognized that later scholars “had achieved a level of precision in the study of motion” unknown to the Ancients, suggesting a growing proto-kinematics mindset.

In summary, the mathematical advances in the medieval Islamic world – from robust algebraic symbolism and polynomial theory to precise trigonometry, inductive summation, and geometric quadrature – collectively formed an essential prelude to calculus. The language of algebra allowed general curves and rates to be written down; the tentative use of infinitesimals and integrals allowed areas and volumes to be calculated; and the consideration of functional relationships and extremes prepared the way for the formal concepts of derivative and integral. These contributions were transmitted to later ages, ensuring that when calculus was formally developed in the 17th century, the underlying ideas were already “in the air,” so to speak, thanks in no small part to the Islamic scholars of many centuries earlier.

Transmission to Europe and Influence on Later Mathematics

The achievements of Islamic mathematicians eventually found their way to Europe, where they had a profound impact on the Renaissance and early modern mathematics, including the work of Newton and Leibniz. This transmission occurred through multiple channels:

Translation into Latin: The 12th and 13th centuries in Europe saw a concerted effort to translate Arabic scientific works into Latin, especially in centers like Toledo (Spain) and Sicily. Key Arabic texts on mathematics and astronomy were rendered accessible to European scholars. For example, Al-Khwarizmi’s algebra was translated into Latin in the 12th century (by Robert of Chester in 1145) as Liber algebrae et almucabala, which introduced the very concept of algebra to Europe​ britannica.com. His work on arithmetic (Algoritmi) was also translated, bringing the Hindu-Arabic numeral system to the Latin-reading world – a development that revolutionized European calculation and commerce. The word “algorithm” in European languages is a direct homage to Al-Khwarizmi​. Furthermore, Euclid’s Elements entered medieval Europe mainly via Arabic-to-Latin translations (often through translations by Adelard of Bath and others), since the original Greek was less accessible; these included the Islamic scholars’ commentaries and enhancements.

Crucially, advanced Arabic works beyond Euclid also made their way west: Ibn al-Haytham’s Book of Optics was translated into Latin (circa late 12th or early 13th century) as Opticae Thesaurus. This text became hugely influential in Europe – it shaped the field of optics for centuries and was studied by the likes of Roger Bacon, Witelo, and later Kepler. Through it, European thinkers absorbed not only optical theories but also Ibn al-Haytham’s mathematical methods (such as his approach to summing powers and his rigorous experimental geometry)​ en.wikipedia.org. In astronomy, works by al-Battani (his Sabian Tables) and al-Zarqālī were translated and used by Copernicus. Al-Battani’s sine and tangent tables and his use of trigonometry helped European astronomers move away from Ptolemy’s chord-based trigonometry to the sine-based system – an essential improvement that carried into modern astronomy and calculus-based physics. Nasir al-Din al-Tusi’s astronomical handbook Zīj-i Īlkhānī was known to later astronomers (and some have argued that Copernicus was indirectly influenced by the geometrical devices like the Tusi couple from the Maragha school​ beforenewton.blog). Whether directly or through later works, Tusi’s trigonometry treatise also had echoes in Europe: by the 15th century, the systematic treatment of trigonometry was taken up by Regiomontanus in his De Triangulis (1464), and he cites Arabic authors like al-Battani. Thus, the sine law and other trig identities proven by al-Tusi​ en.wikipedia.org were part of the mathematical arsenal of Renaissance scholars.

Influence on the Renaissance: By 1400, the cumulative effect of two centuries of translation was evident. European mathematics in the Renaissance blossomed by building on the imported knowledge. For instance, the Italian mathematician Leonardo of Pisa (Fibonacci) traveled to North Africa and learned from Muslim teachers; in 1202 he published Liber Abaci, which introduced Arabic numerals and algebraic methods to Europe, directly citing algebraic problems from Al-Khwarizmi and others. Fibonacci also learned of Islamic contributions in geometry and number theory during his travels. Later, during the 14th century, scholars at Merton College (Oxford) and in Paris (Oresme, etc.) began examining infinite series and motion—ideas that likely trickled down from the translated works of thinkers like al-Haytham and Avempace. Nicole Oresme, for example, in the 14th century discussed the sum of infinite series and even graphed a time-speed diagram (a proto-graph of a function), possibly influenced by the general atmosphere of combining algebra with geometry, something done by Khayyam and others.

By the time of Newton and Leibniz in the 17th century, many of the algebraic and technical tools they employed had roots in Islamic scholarship. The very ability to write equations in a general polynomial form was enabled by the algebraic tradition that came via Al-Khwarizmi and was later refined by Viète. The concept of the decimal fraction (e.g., 0.5, 0.125) which is indispensable for calculus in measurement, was popularized in Europe by Simon Stevin in 1585, but Stevin’s work was preceded by the innovations of al-Uqlīdīsī and al-Kāshī in base-10 fractions a few centuries earlier​ britannica.com. Trigonometric tables and identities were at Newton’s disposal, courtesy of Islamic and Greek advances transmitted by translations. When Newton derived the general binomial series and infinite series for sine and cosine, he was solving problems that mathematicians from Omar Khayyam to Madhava (in India) had set the stage for. In fact, the binomial coefficients that Newton used (Pascal’s triangle) had been described by al-Karaji and Omar Khayyam​ en.wikipedia.org. It’s telling that Leibniz, a great synthesizer of knowledge, acknowledged the importance of both ancient and non-European mathematics. There is evidence that by the 17th century, European scholars were increasingly aware that “the Arabs” had preserved and augmented Greek learning – for example, the historian John Wallis wrote about the Arabic contribution in his preface to Mathesis Universalis.

Although Newton and Leibniz developed calculus largely independently, the intellectual soil had been fertilized by Islamic mathematics. Concepts like algorithmic computation, algebraic solving of equations, functional trigonometry, and summation of series were all readily available in the mathematical culture they inherited​. Indeed, Newton’s approach to calculus (especially his Method of Fluxions) heavily relies on algebraic manipulation of infinite series and polynomials – a style of mathematics that would have been impossible without the prior “arithmetization of algebra” initiated by Al-Khwarizmi and advanced by Al-Samaw’al​ en.wikipedia.org mathshistory.st-andrews.ac.uk. Leibniz’s notation and his idea of a function y=f(x)y = f(x)y=f(x) likewise have precedents in the function-like thinking of Sharaf al-Dīn​ en.wikipedia.org. Even the mechanical conception of orbits as combinations of circular motions (central to Newton’s physics) harks back to devices like the Tusi couple and Ibn al-Shatir’s planetary models from the Islamic world.

In summary, the work of Islamic scholars was transmitted to Europe through translations and scholarly exchange (some facilitated by events such as the Crusades and the reconquista of Spain)​ en.wikipedia.org. This imported knowledge “contributed substantially to the evolution of Western mathematics” en.wikipedia.org. The Renaissance saw European scholars digest and build upon these ideas, culminating in the groundbreaking work of the 17th century. The continuum from Al-Khwarizmi’s algebra to Euler’s analysis is a chain in which the Islamic Golden Age is a vital link.

Primary Sources and Legacy

Original Arabic Manuscripts: Many of the Islamic advancements in precalculus and calculus are documented in manuscripts that have survived (at least in part) to the present day. For instance, Al-Khwarizmi’s Kitāb al-jabr wa-l-muqābala (c. 830) is extant in several Arabic manuscripts and was summarized in Latin in the 12th century​ britannica.com. The title literally means “The Compendious Book on Calculation by Completion and Balancing,” and it systematically lays out algebraic rules and solutions. Another foundational text is Abu Kamil’s 10th-century algebra book, which built on Al-Khwarizmi and was used by later mathematicians like Fibonacci. The works of Al-Karaji (e.g. al-Fakhri fī’l-jabr, c. 1000) survive only partially, but through quotations in Al-Samaw’al’s Al-Bāhir. In that text, Al-Samaw’al recorded and extended Al-Karaji’s now-lost results, giving us insight into the development of algebraic induction and polynomial division​ mathshistory.st-andrews.ac.uk.

Nasir al-Din al-Tusi’s influential text on trigonometry, Kitāb al-Shakl al-qattāʿ (“Book on the Complete Quadrilateral”), exists in manuscript form and was studied in the Islamic East for centuries. His astronomical works, like the Zīj-i Īlkhānī, were also preserved and eventually studied by later astronomers. Ibn al-Haytham’s seven-volume Kitāb al-Manāẓir (Book of Optics, written 1011–1021) is preserved in Arabic (with a famous copy in the Süleymaniye Library, Istanbul)​ en.wikipedia.org. That work not only includes optical experiments but also mathematical discussions like the summation of series. Ibn al-Haytham wrote other treatises, including Opuscula on analysis and number theory; one of these contains his solution of Wilson’s theorem in congruences​ en.wikipedia.org. Omar Khayyam’s algebra book (written in 1070) was relatively unknown in the West until modern times, but manuscripts in Arabic and Persian exist – for example, one copy is kept at the University of Tehran. In it, Khayyam details the geometric solutions of cubics and even remarks on the possibility of a general algebraic method, showing a deep understanding of equations​ encyclopedia.com. Sharaf al-Dīn al-Tusi’s manuscript on cubic equations (Risāla fi al‐ʿAml bi’l‐Jabr wa’l-Muqābala) was rediscovered in the 20th century and analyzed by historians; it contains numerical examples of his derivative-like method and iterative root-finding​ en.wikipedia.org.

Key Latin Translations: The conduit of knowledge to Europe can be traced through specific translated works. In 1145, Robert of Chester’s translation of Al-Khwarizmi’s Algebra introduced Europe to the very idea of algebraic equations beyond the simple arithmetic of the Romans​ britannica.com. In the late 12th century, Gerard of Cremona translated dozens of scientific books from Arabic to Latin in Toledo – among them Ptolemy’s Almagest (from Arabic) and possibly Alhazen’s Optics (some sources credit Gerard for the Latin Perspectiva). By around 1200, as noted, an anonymous scholar had translated Ibn al-Haytham’s Optics en.wikipedia.org. This Latin Opticae Thesaurus circulated widely; the English scholar Roger Bacon cited Alhazen repeatedly in his own optical compendium, and the Polish scholar Witelo based his influential work Perspectiva (1270s) largely on Ibn al-Haytham’s findings. These optical works, with their geometric-mechanical approach, contributed indirectly to the development of the scientific method and the quantitative analysis of nature, which is closely tied to calculus.

Another important translation was the Toledo translators’ rendition of Al-Battani’s astronomical treatise (De motu stellarum), which included his refined sine tables and use of trig functions. This text gave Europe improved tools for astronomy, which were essential for the later work of Copernicus, Galileo, and Kepler (all of whom needed trigonometry to calculate orbits and trajectories, calculations we now do with calculus). In the 13th century, Fibonacci’s interactions with Islamic knowledge led him to incorporate problems from Al-Khwarizmi and others into Flos and other works, seeding European mathematics with advanced problem types (including algebraic equations that would later be tackled with algebraic geometry and calculus).

By the 15th and 16th centuries, the process came full circle. As Europe became the leader in mathematical innovation, scholars like Regiomontanus and later Galileo and Fermat were aware of some of the historic sources. Regiomontanus’s book on trigonometry mentions earlier authorities including Islamic ones. The humanist scholar Grosseteste had earlier commented on Alhazen, and by 1600 the first printed edition of Alhazen’s Optics was published (1572 by Friedrich Risner)​ en.wikipedia.org, showing continued interest. When calculus was finally formalized by Newton (in Principia, 1687, and Methodus Fluxionum) and Leibniz (in his 1684 paper), they did not cite medieval sources, but they worked in a mathematical universe richly furnished by their predecessors.

To give a concrete example of direct lines of influence: the concept of the derivative as a tool to find maxima/minima was described by Sharaf al-Dīn in 12th-century Baghdad​ en.wikipedia.org, and a similar idea appears in Fermat’s method of adequality (1630s) for finding maxima/minima, which then evolved into calculus’s derivative test. We don’t have evidence Fermat knew of Sharaf al-Dīn, but both drew on a common algebraic tradition. Likewise, integration as summation of infinitesimals was practiced by Ibn al-Haytham​ en.wikipedia.org and later by Cavalieri (1630s) in Italy before Newton/Leibniz. Cavalieri’s method of indivisibles was a direct stepping stone to calculus, and while it was mostly inspired by Greek Archimedean ideas, the Islamic preservation and extension of Archimedes (like the work of al-Tusi’s student al-Farisi on the rainbow, which included ideas on sums of series) helped keep such ideas alive.

In conclusion, the primary sources from the Golden Age of Islam – both the original Arabic manuscripts and their Latin translations – form a vital part of the historical tapestry of calculus. They show a continuous progression of ideas: from solving concrete problems to abstracting general techniques. The Islamic mathematicians’ legacy is preserved in these works and was transmitted through translation to the West, where it became the foundation upon which modern mathematics was built​ en.wikipedia.org encyclopedia.com. Newton once famously said, “If I have seen further, it is by standing on the shoulders of giants.” Those giants were not only Greek but also Persian, Arab, and other Muslim scholars whose shoulders proved broad enough to support the edifice of calculus.

Sources:

1. R. Rashed & A. Djebbar, Encyclopedia of the History of Arabic Science, Routledge, 1996. (Discusses the translation movement and development of algebra and analysis in Islam) ​en.wikipedia.org​en.wikipedia.org
2. Britannica, “Mathematics in the Islamic world, 8th–15th century.” (Overview of major projects including algebra and trigonometry) ​britannica.com​early-astronomy.classics.lsa.umich.edu
3. Wikipedia, “Mathematics in medieval Islam”. (Al-Khwarizmi’s role and spread of Arabic mathematics to the West) ​en.wikipedia.org​en.wikipedia.org
4. Wikipedia, “Ibn al-Haytham”. (Details his sum-of-powers formula and volume of paraboloid as early calculus) ​en.wikipedia.org
5. Wikipedia, “Sharaf al-Dīn al-Tūsī”. (Describes his method of finding maxima of cubic polynomials and the derivative debate)​ en.wikipedia.org​
6. “Sharaf al-Dīn al-Tūsī and the Number of Positive Roots of Cubic Equations,” Historia Mathematica 16 (1989) 69–85 (J.P. Hogendijk). (Academic paper analyzing Sharaf’s cubic equation work and its significance for calculus concepts).
7. Encyclopedia.com, “Omar Khayyam and the Solution of Cubic Equations.” (Highlights Khayyam’s synthesis of Greek geometry and Islamic algebra, and influence on later mathematicians)​encyclopedia.com
8. MacTutor History of Mathematics (St. Andrews University) – biographies of Al-Khwarizmi, Al-Samaw’al, Omar Khayyam, and Nasir al-Din al-Tusi​ britannica.com​mathshistory.st-andrews.ac.uk. (Provides summaries of their contributions: algebra, negative exponents, geometric solutions, and independent trigonometry respectively.)
9. HowStuffWorks, “Nasir al-Din al-Tusi.” (Mentions Tusi’s pioneering of spherical trig and binomial coefficients)​ science.howstuffworks.com.
10. Wikipedia, “Pascal’s Triangle”. (Notes that the triangle was known as Khayyam’s triangle in Iran and that Khayyam used binomial coefficients for extracting roots) ​en.wikipedia.org​en.wikipedia.org.