The Unknown, the Variable, and Islamic Mathematical Philosophy: Algebra, Abstraction, and Algorithmic Reasoning

Last Updated June 23, 2026

The Unknown, the Variable, and Islamic Mathematical Philosophy examines one of the most important conceptual shifts in the history of algorithmic reasoning: the treatment of an unknown quantity as something that can be named, classified, transformed, reasoned about, and eventually represented symbolically. Long before modern algebraic notation became familiar, mathematicians working in Arabic, Persian, Hebrew, Latin, and vernacular traditions developed ways to reason about unknowns through words, cases, proportions, geometric demonstrations, arithmetic examples, and procedural rules.

The unknown is not merely a blank space. It is a disciplined object of inquiry. A problem asks for something not yet known. Algebra gives that missing quantity a role. It may be called a thing, a root, a square, a number, an amount, a debt, a share, a side, or a magnitude. Once named, the unknown can be manipulated according to rules. It can be restored, balanced, compared, completed, reduced, checked, and interpreted.

This article treats Islamic mathematical philosophy broadly: not as a single doctrine, but as a set of questions about number, magnitude, proof, procedure, abstraction, demonstration, usefulness, certainty, and the relation between calculation and reality. In this setting, the unknown becomes more than a practical convenience. It becomes a philosophical threshold between what is given, what is sought, and what can be made knowable through method.

This article introduces the unknown, variable thinking, Islamic mathematical philosophy, Arabic algebra, shayʾ, root, square, number, magnitude, al-jabr, al-muqābalah, equation cases, geometric demonstration, verbal procedure, abstraction, proof, practical calculation, Latin res, Italian cosa, symbolic transition, and the philosophical status of mathematical objects. It argues that the unknown is one of the deepest bridges between algorithmic procedure and mathematical thought: it allows a method to operate on what is not yet known, while still preserving rules, constraints, and correctness.

Why the Unknown Matters

The unknown matters because it changes what mathematics can do. Arithmetic can compute with given numbers. Algebra can reason about a quantity before it is known. This is a major conceptual expansion. The unknown allows a problem to be represented as a structure rather than as an isolated numerical task.

Once an unknown is named, a procedure can act on it. The method can classify the problem, transform the expression, complete a square, balance terms, reduce complexity, derive a value, and test the result. The unknown becomes a placeholder for disciplined reasoning.

The unknown makes mathematics capable of reasoning toward knowledge rather than merely calculating what is already given.

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The Unknown as a Mathematical Object

The unknown becomes a mathematical object when it can be named and treated according to rules. It is not yet a modern variable in the full symbolic sense, but it is no longer only a vague absence. It has a role in the problem. It can stand in relation to numbers, squares, roots, debts, shares, sides, areas, and magnitudes.

This is philosophically significant. To reason about an unknown is to reason about something that is not immediately present as a known number. Algebra turns absence into structure. It allows the mind to operate on the form of a problem before the answer is available.

The unknown is a mathematical object because it can be governed by method.

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Shayʾ, Root, Square, and Number

Arabic algebraic traditions used terms such as thing, root, square, and number to organize unknown quantities and their powers. The “thing” is not simply a casual word. It helps name what is sought. The root can refer to the unknown quantity itself. The square refers to the product of the root with itself. The number is the known quantity.

These terms make algebraic structure verbal. They allow a problem to be expressed without modern symbols. Instead of writing a compact equation, the author can describe a relation among squares, roots, and numbers. The procedure then identifies the case and applies a rule.

This vocabulary is a verbal technology for reasoning about the not-yet-known.

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Algebra Before Modern Symbolism

Modern readers often imagine algebra as symbolic notation: x, y, equals signs, exponents, and formulas. But algebra existed before this notation became standard. In many Islamic-world and Latin reception contexts, algebra was verbal, geometric, arithmetic, and procedural. The absence of modern symbols did not prevent systematic reasoning.

A verbal algebraic rule can still be algorithmic. It tells the reader what kind of problem is present, what operation to perform, and how to check the result. Symbolism later made algebra more compact and general, but the procedural structure came earlier.

Algebra was procedural before it was symbolically modern.

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Islamic Mathematical Philosophy

Islamic mathematical philosophy should be understood broadly as inquiry into the nature, certainty, usefulness, and structure of mathematical knowledge in Islamic-world intellectual contexts. It includes questions inherited from Greek philosophy, developed through Arabic commentary, shaped by arithmetic and algebra, and tested in astronomy, optics, inheritance, trade, surveying, music, and mechanics.

The unknown belongs to this philosophical setting because it raises questions about abstraction. What kind of thing is an unknown quantity? Is it a number, magnitude, root, side, debt, share, or formal object? Can it be reasoned about before it is known? What gives the procedure certainty? How does algebra relate to geometry, arithmetic, and practical calculation?

The unknown sits at the intersection of metaphysics, method, and practical calculation.

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Number, Magnitude, and Abstraction

Islamic-world mathematics inherited and transformed traditions in which number and magnitude were sometimes treated differently. Arithmetic dealt with numbers. Geometry dealt with magnitudes. Algebra crossed boundaries. It could handle numerical problems, but it often relied on geometric demonstration. It could seek a root or square, but it could also solve practical problems involving goods, money, inheritance, land, or measurement.

The unknown helps connect number and magnitude. It may be a number in a calculation, a length in a geometric demonstration, a share in an inheritance problem, or an amount in a commercial problem. Algebra abstracts across these settings while still requiring interpretation.

Algebra turns many kinds of quantity into objects of method.

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Classification and Equation Cases

Early algebra often worked through classification. A problem was reduced to a recognized case involving squares, roots, and numbers. Different cases required different rules. This is algorithmic in a deep sense: classification determines procedure.

In modern notation, several cases can be expressed through a single general equation. Earlier algebra often kept cases separate because negative coefficients, symbolic abstraction, and formal generality were handled differently. Case classification preserved clarity and kept procedures tied to meaningful quantities.

Case classification shows how algebra can be procedural without universal symbolic notation.

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Restoration, Balancing, and Transformation

Al-jabr and al-muqābalah are often translated as restoration and balancing or reduction. These operations matter because they show algebra as transformation. A problem is not solved merely by observing it. It is converted into a more tractable form.

Restoration can remove deficiency by moving terms or completing what is missing. Balancing can compare like terms and reduce both sides. Together, these operations create a method for changing the structure of a problem without changing its truth.

Algebraic transformation makes the unknown accessible to procedure.

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Geometric Demonstration and Algebraic Procedure

Geometric demonstration gave algebraic procedures a form of certainty. Completing the square, for example, can be shown through areas and line segments. The procedure is arithmetic or algebraic, but the justification may be geometric. This relationship between method and proof is central to Islamic-world algebraic traditions.

Geometric demonstration also reveals why the unknown can be meaningful before it is known. A line segment can stand for a root. A square can stand for the square of the root. Added areas can complete a larger square. The unknown becomes visible through construction.

Geometry helped show why an algebraic procedure was not just a trick, but a reasoned method.

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From Unknown to Variable

The unknown and the variable are related but not identical. An unknown usually refers to a particular sought quantity in a problem. A variable can range over possible values, represent a general relation, or serve as a formal symbol in a structure. The transition from unknown to variable is gradual.

Islamic-world algebra contributed to this transition by stabilizing procedures for unknown quantities. Latin, Hebrew, and vernacular reception then carried terms and methods forward. Later symbolic algebra made variable thinking more general and compact. But the conceptual groundwork includes the earlier ability to name and manipulate an unknown.

The unknown is one of the historical ancestors of the modern variable, but it should not be collapsed into it.

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Translation: Res, Cosa, and Symbolic Transition

As algebra moved into Latin and vernacular European traditions, the unknown was translated and reinterpreted. Latin terms such as res and later Italian cosa helped carry the idea of the “thing” into new mathematical cultures. This is why early algebraists in some European contexts were associated with cossic algebra.

Translation matters because the unknown is not only a mathematical idea but also a linguistic object. A language must make room for it. A term must be stable enough for teaching and manipulation. Eventually, symbolic notation reduces reliance on words, but the earlier verbal terms remain part of the conceptual genealogy.

The history of the variable is also a history of translation.

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Certainty, Verification, and Proof

A procedure must be correct, not merely effective. Islamic-world algebra often linked rule, example, and demonstration. A rule tells the reader what to do. An example shows the rule in action. A geometric demonstration can justify why the procedure works. Verification checks the answer against the original problem.

This layering matters for algorithmic reasoning. A correct algorithm is not only a sequence of steps; it is a sequence whose output is justified under defined conditions. In historical algebra, correctness often depended on classification, transformation, proof, and checking.

The unknown becomes knowable through justified procedure.

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Practical Problems and Philosophical Objects

Many algebraic problems were practical: inheritance, trade, measurement, surveying, debts, shares, partnerships, and legal distributions. But practical origin does not mean philosophical shallowness. A method developed for inheritance can raise questions about equality, abstraction, proportion, unknowns, proof, and the relation between number and social obligation.

The unknown often enters through practice. Someone’s share is not known. A debt must be computed. A side length must be found. A division must be made. Algebra turns practical uncertainty into formal structure. This is why applied mathematics and philosophy are not separate in this history.

The unknown connects everyday uncertainty to formal method.

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Origin Stories and Careful Interpretation

The history of the unknown and the variable should not be simplified into a single origin story. Diophantine traditions, Indian algebraic and arithmetic traditions, Arabic algebra, geometric demonstration, Latin reception, abacus schools, cossic algebra, and early modern symbolic notation all matter. Islamic-world mathematics is central, but it is not isolated.

Careful interpretation also avoids projecting modern symbolic algebra backward. Al-Khwārizmī did not write equations in modern notation. That does not make his algebra primitive. It means algebraic reasoning had a different form: verbal, case-based, demonstrative, and procedural. The history is richer when we let each stage keep its own structure.

The unknown is best understood as a historical continuum of representation, abstraction, and method.

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Mathematics, Computation, and Modeling

The unknown can be modeled as a placeholder in a constrained relation:

A case-based algebraic procedure can be represented as:

A modern version of a common case is:

Completing the square can be summarized as:

These formulas use modern notation to make the structure visible. They are interpretive models, not claims that medieval authors used this symbolic notation.

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Python Workflow: Unknown and Variable Concept Map

The Python workflow below creates a dependency-light interpretive map of the unknown, variable thinking, and Islamic mathematical philosophy. It scores themes by unknown representation, procedural transformation, abstraction, proof relation, translation continuity, practical grounding, philosophical depth, historical significance, ethical caution, and modern resonance, then writes reproducible CSV and JSON outputs.

# unknown_variable_islamic_mathematical_philosophy_map.py
# Dependency-light workflow for mapping the unknown as a philosophical and procedural object.

from __future__ import annotations

from dataclasses import asdict, dataclass
from pathlib import Path
from statistics import mean
import csv
import json
from datetime import datetime, timezone

ARTICLE_ROOT = Path(__file__).resolve().parents[1]
TABLES = ARTICLE_ROOT / "outputs" / "tables"
JSON_DIR = ARTICLE_ROOT / "outputs" / "json"


@dataclass(frozen=True)
class UnknownVariableConfig:
    article: str = "the_unknown_the_variable_and_islamic_mathematical_philosophy"
    core_threshold: float = 0.80
    high_unknown_threshold: float = 0.86


def timestamp_utc() -> str:
    return datetime.now(timezone.utc).isoformat()


def write_csv(path: Path, rows: list[dict[str, object]]) -> None:
    path.parent.mkdir(parents=True, exist_ok=True)
    if not rows:
        path.write_text("", encoding="utf-8")
        return
    fieldnames = sorted({key for row in rows for key in row.keys()})
    with path.open("w", newline="", encoding="utf-8") as handle:
        writer = csv.DictWriter(handle, fieldnames=fieldnames, extrasaction="ignore")
        writer.writeheader()
        writer.writerows(rows)


def write_json(path: Path, payload: object) -> None:
    path.parent.mkdir(parents=True, exist_ok=True)
    path.write_text(json.dumps(payload, indent=2, sort_keys=True), encoding="utf-8")


def unknown_variable_themes() -> list[dict[str, object]]:
    return [
        {"theme_id": "unknown_as_named_object", "unknown_representation": 0.98, "procedural_transformation": 0.94, "abstraction": 0.96, "proof_relation": 0.88, "translation_continuity": 0.90, "practical_grounding": 0.92, "philosophical_depth": 0.96, "historical_significance": 0.98, "ethical_caution": 0.86, "modern_resonance": 0.98},
        {"theme_id": "shay_root_square_number", "unknown_representation": 0.98, "procedural_transformation": 0.96, "abstraction": 0.94, "proof_relation": 0.90, "translation_continuity": 0.94, "practical_grounding": 0.90, "philosophical_depth": 0.94, "historical_significance": 0.98, "ethical_caution": 0.84, "modern_resonance": 0.96},
        {"theme_id": "case_classification_and_rule", "unknown_representation": 0.92, "procedural_transformation": 0.98, "abstraction": 0.92, "proof_relation": 0.90, "translation_continuity": 0.88, "practical_grounding": 0.92, "philosophical_depth": 0.90, "historical_significance": 0.96, "ethical_caution": 0.84, "modern_resonance": 0.96},
        {"theme_id": "geometric_demonstration", "unknown_representation": 0.90, "procedural_transformation": 0.92, "abstraction": 0.94, "proof_relation": 0.98, "translation_continuity": 0.86, "practical_grounding": 0.88, "philosophical_depth": 0.96, "historical_significance": 0.96, "ethical_caution": 0.84, "modern_resonance": 0.94},
        {"theme_id": "from_unknown_to_variable", "unknown_representation": 0.96, "procedural_transformation": 0.94, "abstraction": 0.98, "proof_relation": 0.88, "translation_continuity": 0.94, "practical_grounding": 0.86, "philosophical_depth": 0.96, "historical_significance": 0.98, "ethical_caution": 0.88, "modern_resonance": 0.98},
        {"theme_id": "res_cosa_symbolic_transition", "unknown_representation": 0.94, "procedural_transformation": 0.90, "abstraction": 0.94, "proof_relation": 0.84, "translation_continuity": 0.98, "practical_grounding": 0.86, "philosophical_depth": 0.90, "historical_significance": 0.94, "ethical_caution": 0.86, "modern_resonance": 0.96},
        {"theme_id": "origin_story_caution", "unknown_representation": 0.86, "procedural_transformation": 0.86, "abstraction": 0.90, "proof_relation": 0.86, "translation_continuity": 0.92, "practical_grounding": 0.84, "philosophical_depth": 0.94, "historical_significance": 0.94, "ethical_caution": 0.98, "modern_resonance": 0.94},
    ]


def score_theme(row: dict[str, object], config: UnknownVariableConfig) -> dict[str, object]:
    unknown_variable_score = mean([
        float(row["unknown_representation"]),
        float(row["procedural_transformation"]),
        float(row["abstraction"]),
        float(row["proof_relation"]),
        float(row["translation_continuity"]),
        float(row["practical_grounding"]),
        float(row["philosophical_depth"]),
        float(row["historical_significance"]),
        float(row["ethical_caution"]),
        float(row["modern_resonance"]),
    ])

    if unknown_variable_score >= config.core_threshold and float(row["unknown_representation"]) >= config.high_unknown_threshold:
        interpretive_status = "core_unknown_variable_philosophy_thread"
    elif unknown_variable_score >= config.core_threshold:
        interpretive_status = "major_unknown_variable_philosophy_thread"
    else:
        interpretive_status = "supporting_unknown_variable_philosophy_thread"

    return {
        "theme_id": row["theme_id"],
        "unknown_representation": round(float(row["unknown_representation"]), 6),
        "procedural_transformation": round(float(row["procedural_transformation"]), 6),
        "abstraction": round(float(row["abstraction"]), 6),
        "proof_relation": round(float(row["proof_relation"]), 6),
        "translation_continuity": round(float(row["translation_continuity"]), 6),
        "practical_grounding": round(float(row["practical_grounding"]), 6),
        "philosophical_depth": round(float(row["philosophical_depth"]), 6),
        "historical_significance": round(float(row["historical_significance"]), 6),
        "ethical_caution": round(float(row["ethical_caution"]), 6),
        "modern_resonance": round(float(row["modern_resonance"]), 6),
        "unknown_variable_score": round(unknown_variable_score, 6),
        "interpretive_status": interpretive_status,
    }


def interpretation_cautions() -> list[dict[str, str]]:
    return [
        {"caution": "do_not_project_modern_symbolism_backward", "meaning": "Verbal algebra can be abstract and rigorous without modern notation."},
        {"caution": "do_not_equate_unknown_and_variable_too_quickly", "meaning": "An unknown is usually problem-specific; a variable is a broader formal object."},
        {"caution": "do_not_treat_practical_problems_as_non_philosophical", "meaning": "Inheritance, trade, and measurement can raise deep questions about quantity and method."},
        {"caution": "do_not_reduce_algebra_to_etymology", "meaning": "Terms such as al-jabr, res, and cosa matter, but procedure and proof matter too."},
        {"caution": "do_not_create_single_origin_myths", "meaning": "The variable has a layered history across Greek, Indian, Arabic, Latin, vernacular, and symbolic traditions."},
    ]


def main() -> None:
    config = UnknownVariableConfig()
    themes = unknown_variable_themes()
    scored = [score_theme(row, config) for row in themes]
    cautions = interpretation_cautions()

    summary = {
        "article": config.article,
        "timestamp_utc": timestamp_utc(),
        "themes_reviewed": len(scored),
        "core_threads": sum(1 for row in scored if row["interpretive_status"] == "core_unknown_variable_philosophy_thread"),
        "major_threads": sum(1 for row in scored if row["interpretive_status"] == "major_unknown_variable_philosophy_thread"),
        "supporting_threads": sum(1 for row in scored if row["interpretive_status"] == "supporting_unknown_variable_philosophy_thread"),
        "mean_unknown_variable_score": round(mean(float(row["unknown_variable_score"]) for row in scored), 6),
        "cautions": len(cautions),
        "interpretation": "The unknown should be studied as a disciplined mathematical object: named, classified, transformed, demonstrated, translated, verified, and gradually generalized toward variable thinking.",
    }

    write_csv(TABLES / "unknown_variable_themes.csv", themes)
    write_csv(TABLES / "unknown_variable_map.csv", scored)
    write_csv(TABLES / "interpretation_cautions.csv", cautions)
    write_csv(TABLES / "unknown_variable_summary.csv", [summary])

    write_json(JSON_DIR / "unknown_variable_config.json", asdict(config))
    write_json(JSON_DIR / "unknown_variable_map.json", scored)
    write_json(JSON_DIR / "interpretation_cautions.json", cautions)
    write_json(JSON_DIR / "unknown_variable_summary.json", summary)

    print("Unknown, variable, and Islamic mathematical philosophy map complete.")
    print(TABLES / "unknown_variable_summary.csv")


if __name__ == "__main__":
    main()

This workflow turns the unknown into a reproducible interpretive artifact: thing, root, square, number, case classification, transformation, demonstration, translation, variable transition, and caution are documented together.

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R Workflow: Unknown, Variable, and Philosophy Diagnostics

The R workflow reads the generated CSV outputs, summarizes unknown-variable themes, visualizes theme dimensions, and writes an additional diagnostic table.

# unknown_variable_islamic_mathematical_philosophy_summary.R
args <- commandArgs(trailingOnly = FALSE)
file_arg <- grep("^--file=", args, value = TRUE)

if (length(file_arg) > 0) {
  script_path <- normalizePath(sub("^--file=", "", file_arg[1]), mustWork = TRUE)
  article_root <- normalizePath(file.path(dirname(script_path), ".."), mustWork = TRUE)
} else {
  article_root <- getwd()
}

setwd(article_root)

tables_dir <- file.path(article_root, "outputs", "tables")
figures_dir <- file.path(article_root, "outputs", "figures")
dir.create(tables_dir, recursive = TRUE, showWarnings = FALSE)
dir.create(figures_dir, recursive = TRUE, showWarnings = FALSE)

map_path <- file.path(tables_dir, "unknown_variable_map.csv")
summary_path <- file.path(tables_dir, "unknown_variable_summary.csv")

if (!file.exists(map_path)) {
  stop(paste("Missing", map_path, "Run the Python workflow first."))
}

unknown_map <- read.csv(map_path, stringsAsFactors = FALSE)
summary <- read.csv(summary_path, stringsAsFactors = FALSE)

png(file.path(figures_dir, "unknown_variable_dimensions.png"), width = 1200, height = 850)
score_matrix <- t(as.matrix(unknown_map[, c("unknown_representation", "procedural_transformation", "abstraction", "proof_relation", "translation_continuity", "practical_grounding", "philosophical_depth", "historical_significance", "ethical_caution", "modern_resonance")]))
barplot(score_matrix,
        beside = TRUE,
        names.arg = unknown_map$theme_id,
        las = 2,
        ylim = c(0, 1),
        ylab = "Interpretive Score",
        main = "Unknown, Variable, and Islamic Mathematical Philosophy Dimensions")
legend("bottomright",
       legend = rownames(score_matrix),
       cex = 0.68,
       bty = "n")
grid()
dev.off()

png(file.path(figures_dir, "unknown_variable_score_by_theme.png"), width = 1000, height = 750)
barplot(unknown_map$unknown_variable_score,
        names.arg = unknown_map$theme_id,
        las = 2,
        ylab = "Unknown-Variable Score",
        main = "Unknown and Variable Score by Theme")
grid()
dev.off()

r_summary <- data.frame(
  themes_reviewed = summary$themes_reviewed[1],
  core_threads = summary$core_threads[1],
  major_threads = summary$major_threads[1],
  supporting_threads = summary$supporting_threads[1],
  mean_unknown_variable_score = summary$mean_unknown_variable_score[1],
  cautions = summary$cautions[1],
  diagnostic_note = "The unknown should be studied as a disciplined mathematical object: named, classified, transformed, demonstrated, translated, verified, and gradually generalized toward variable thinking."
)

write.csv(r_summary, file.path(tables_dir, "r_unknown_variable_diagnostic_summary.csv"), row.names = FALSE)
print(r_summary)

The R layer makes the interpretive structure visible: unknown representation, procedural transformation, abstraction, proof, translation, practical grounding, philosophy, historical significance, caution, and modern resonance can be examined as related but distinct dimensions.

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GitHub Repository

The companion repository contains reproducible workflows, synthetic interpretive data, outputs, calculators, documentation, and multilingual examples for this article.

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A Practical Method for Studying the Unknown

A careful study of the unknown asks how the not-yet-known becomes a structured object of method.

This method treats the unknown as a disciplined object of procedure, proof, translation, and abstraction.

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Common Pitfalls

The first pitfall is thinking that abstraction requires modern symbols. The second is treating the unknown and the modern variable as identical. The third is treating practical problems as philosophically unimportant. The fourth is reducing algebra to etymology or origin stories.

The unknown becomes clearer when it is studied as a historical object, not forced into modern categories too quickly.

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Why the Unknown Belongs in Algorithmic Reasoning

The unknown, the variable, and Islamic mathematical philosophy belong in algorithmic reasoning because algorithms often operate on what is not yet known. A problem begins with uncertainty. A method gives that uncertainty structure. Algebra names the missing quantity, classifies the relation, applies a transformation, proves or justifies the method, and verifies the result.

This history expands the meaning of computation. Computation is not only calculation with known values. It is also disciplined inference from givens toward what is sought. The unknown is the doorway into that process.

The lesson for modern systems is direct. Variables, placeholders, model parameters, missing data, latent states, optimization targets, and inferred quantities all depend on the same deep idea: we can reason systematically about what we do not yet know. The Islamic-world history of algebra helps show how powerful and philosophically serious that idea is. AI belongs in the toolkit, not in control.

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Related Articles

• From Baghdad to Latin Europe: Algorism, Algebra, and Reception
• Historiography and Origin Stories of Algorithms
• Al-Jabr wa’l-Muqābalah: Algebra as Rule-Governed Problem Solving
• Al-Khwārizmī, Algorism, and the Procedural Imagination
• Hindu-Arabic Numerals and the Transmission of Positional Calculation

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Further Reading

• Oaks, J.A. (2015) The Algebra of Mohammed ben Musa. Cham: Springer.
• Rashed, R. (2009) Al-Khwārizmī: The Beginnings of Algebra. London: Saqi.
• Rashed, R. (1994) The Development of Arabic Mathematics: Between Arithmetic and Algebra. Dordrecht: Kluwer.
• Berggren, J.L. (2007) Mathematics in Medieval Islam. Princeton: Princeton University Press.
• Høyrup, J. (1994) In Measure, Number, and Weight: Studies in Mathematics and Culture. Albany: SUNY Press.
• Netz, R. (2004) The Transformation of Mathematics in the Early Mediterranean World. Cambridge: Cambridge University Press.
• National Academies (2006) ‘A Real and Imaginary History of Algebra’. Washington, DC: National Academies Press.
• MacTutor History of Mathematics (n.d.) ‘Al-Khwarizmi’. University of St Andrews.

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References

• Berggren, J.L. (2007) Mathematics in Medieval Islam. Princeton: Princeton University Press.
• Høyrup, J. (1994) In Measure, Number, and Weight: Studies in Mathematics and Culture. Albany: SUNY Press.
• MacTutor History of Mathematics (n.d.) ‘Al-Khwarizmi’. University of St Andrews. Available at: https://mathshistory.st-andrews.ac.uk/Biographies/Al-Khwarizmi/.
• National Academies (2006) ‘A Real and Imaginary History of Algebra’. Available at: https://www.nationalacademies.org/read/11540/chapter/5.
• Netz, R. (2004) The Transformation of Mathematics in the Early Mediterranean World. Cambridge: Cambridge University Press.
• Oaks, J.A. (2015) The Algebra of Mohammed ben Musa. Cham: Springer.
• Rashed, R. (1994) The Development of Arabic Mathematics: Between Arithmetic and Algebra. Dordrecht: Kluwer.
• Rashed, R. (2009) Al-Khwārizmī: The Beginnings of Algebra. London: Saqi.

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