Al-Kashi's general numerical root extraction

In the Key to Arithmetic, Ghiyath al-Din al-Kashi treated numerical extraction of roots of general orders, not only square roots. MacTutor reports Luckey's comparison of his procedure with a special case of the later Ruffini/Horner methods. It also retains Rashed's qualification: earlier procedures existed in the school of al-Karaji, particularly al-Samawal. This is development and integration of inherited computation, not the first root known to humanity ¹.

The subject differs from the sine of one degree, with its particular cubic problem, and from circumference calculation. Here the question is how to recover a number from one of its powers. The museum's examples are independently written modern explanations, not numbers or table layouts copied from an unexamined manuscript ².

Root extraction reverses raising a number to a power. The positive cube root of eight is two because2³=8; many other inputs do not yield a whole-number answer. Then a numerical procedure needs an approximation and some account of what that approximation guarantees. Iranica describes general root extraction with the square-root procedure as a special case ³.

This is not automatically a solver for every algebraic equation. Writingxⁿ=A specifies a particular problem; a later comparison with Horner does not turn it into the entire general polynomial algorithm. Our secondary accounts also do not disclose every pen movement, column arrangement or historical precision setting. We distinguish a verified account of the contribution from a complete reconstruction we have not undertaken ¹.

MacTutor explicitly acknowledges the al-Karaji school and al-Samawal through its discussion of Rashed's research. Their original works were not independently read for this entry. We therefore do not invent a table from them. Earlier existence does not erase the value of organizing or extending a procedure, but a strong later textbook cannot erase predecessors either ¹.

Iranica also discusses a possible connection with an earlier Chinese technique, distinguishing the integration achieved in al-Kashi's work. That is an attributed possibility, not a manuscript travel route established here. Similar procedures can be compared without assuming that every resemblance proves borrowing or that every difference proves wholly independent invention ³.

MacTutor dates completion of the Key to2March1427. The record uses1427/830 at year precision, as in the existing book context, not as the discovery date of the root algorithm. A book's completion, an individual procedure's development, a manuscript copy and a modern edition are different historical events. A teaching book can gather knowledge accumulated across generations ^{4 5}.

The work belongs to the Samarkand teaching environment and addresses calculation useful for astronomy, surveying, architecture, accounting and trade. These are purposes reported for the book, not evidence that our modern example was used in a particular workshop. We do not invent a conversation between teacher and students to make the setting appear better documented ⁴.

MacTutor individually identifies al-Kashi as an Islamic mathematician. His person record retains that basis separately from the museum's declared Sunni frame; birthplace, patronage and algorithmic success do not establish an additional personal school affiliation. A specific achievement deserves explanation without the title “inventor of all arithmetic” ⁶.

His role here is developer within an earlier tradition. Luckey and Rashed are later historians, not co-authors of the medieval book. Ruffini and Horner belong to a retrospective comparison, not a demonstrated direct teaching chain. The conjunction plate and muqarnas mensuration also remain separate contributions: sharing an author does not make them steps in a root-extraction algorithm ⁷.

Here is an independent modern teaching example, not a literal recovery of the historical table. Seek the positive cube root of ten. Since2³=8 and3³=27, it lies between two and three. “Near two” is insufficient if we want a stated accuracy. We compare powers at closer bounds and report an interval rather than an unexplained string of digits.

A modern identity explains the effect of an increment:if a is an estimate and b an increase,(a+b)³=a³+3a²b+3ab²+b³. Keeping just the first term does not account for the change. Doubling a cube's edge also does not merely double its volume. A new decimal digit can be tested using integer powers after scaling; this illustrates the mathematics without attributing our notation to the author.

At three decimal places,2154³=9993948264 and2155³=10007873875. Compare both with10000000000,which is ten multiplied by1000³. Consequently2.154≤the positive cube root of ten<2.155. The interval width is0.001,and the truncated lower value2.154 falls short by less than that. This is not a claim that the lower value is the nearest-rounded answer or that a cube's error equals its root's error.

The bound works because cubing increases on positive magnitudes. We explicitly select the positive branch:an even-power equation with positive input can have two real solutions, while the principal-root convention selects the nonnegative one. These modern distinctions prevent confusion; we do not claim that each was worded identically in al-Kashi's chapter.

TDV reports the Key's use in Ottoman education and mentions Ibrahim Kami's translation of its fourth treatise. That does not independently establish a translation of every root-extraction chapter. Nor is the popularity of a large book a count of users of one specific algorithm. Transmission of one portion cannot silently become proof of transmission of every portion ⁵.

MacTutor explains that some of al-Kashi's achievements remained unknown in Europe until modern historical study. Thus “comparable to Horner” cannot become a story of hidden correspondence or direct instruction. Structural comparison and proof that a later author read a particular earlier text are distinct claims requiring distinct evidence ⁸.

The biographical account places integer calculation, fractions and numerical systems within a broader computational world. Al-Kashi was not the first user of decimal fractions; MacTutor retains earlier al-Karaji-school and al-Samawal work in that discussion too. Decimal notation in our teaching example is not an excuse to reintroduce a sweeping invention claim ⁹.

Our explanatory model uses multiplication, powers and bounds. An exact value such as the cube root of eight remains different from an approximation to another value, however many digits are printed. An equality can establish exactness; two comparisons can establish a guaranteed interval. A long displayed number is not reliable merely because its appearance suggests precision.

The historical comparison places al-Kashi within development between inherited procedures and later forms that historians can compare. It does not demonstrate ancestry for every computer program or modern calculator. Nor should any numerical iteration be named after him without checking the problem's domain and the structure of the actual procedure ¹.

Modern teaching can separate three aims:an integer part, decimal places and an error bound. We answer those questions in our own examples rather than putting the answers into the medieval author's mouth. Knowing how to find an nth root also does not solve every real or complex branch of every equation. Clear limits make the contribution more intelligible, not less valuable.

Astronomy, surveying, architecture, accounting and trading are purposes of the Key reported by MacTutor. Roots can arise when recovering lengths from other magnitudes, but the account read here contains no record of a ten-unit cube made in Samarkand using this procedure. Our cube is a teaching comparison, not a measurement of a surviving artefact ⁴.

We report no time savings or reduction in builders' errors without evidence. What the sources support is a computational tool within a book teaching varied practical calculation. That is a meaningful connection, not permission to attribute every bridge, dome or transaction to one chapter. The book's intended range and a particular documented use must remain distinct ^{4 3}.

This account moves beyond the choice between solitary first inventor and passive transmitter. MacTutor qualifies two expansive priority claims—decimal fractions and root methods—while retaining al-Kashi's achievement. Earlier material does not imply that later organization or explanation adds nothing; acknowledging improvement does not require deleting earlier work ⁹.

Iranica emphasizes computational integration and development while distinguishing it from novel theoretical planetary astronomy. The entry follows the evidenced action:calculation, teaching and synthesis, not a general title making one scholar the owner of every science. An attributed historian's interpretation remains visible as interpretation, not the museum's final verdict on all possible precedence ³.

**Sunni—declared frame:**the individual Muslim basis is retained; the museum's frame does not settle algorithmic priority. **Academic account:**Luckey's Ruffini/Horner comparison is accompanied by Rashed's al-Karaji/al-Samawal precedents, as reported by MacTutor. The record therefore treats improvement within a tradition, not a birth certificate for roots in human history ¹⁰.

Iranica's possible Chinese connection remains attributed possibility. This entry has not compared original texts on both sides or traced their routes. It neither rules out exchange nor pretends to prove it. It also does not count the sine-related cubic problem a second time:general extraction is a separately bounded subject ^{3 11}.

Our example can be checked with large integer powers instead of relying solely on a calculator's rounded root display. Comparing2154³ and2155³ with the reference integer proves an interval without printing every root digit. Its guarantee is describable:the lower truncation differs by less than one thousandth, not the promotional phrase “without error.”

The image is a modern cube showing why volume can raise a question about edge length. It is not al-Kashi's table and proves neither his algorithm nor a medieval building's measurement. Three labelsd identify equal edges, not a circle's diameter or original fifteenth-century dimensions ¹².

The biographical, Iranica and TDV passages on roots and the Key were read. The full Arabic book and Luckey's or Rashed's independent studies were not read for this research. A2000summary paper was blocked by a security check in both retrieval and browser; Kenyon's2020resource metadata was seen, but the animation was not watched. Blocked content is not counted as reading.

The image by HMcKenna is based on Marcelo Reis's code, published on Commons in2006underCC BY-SA2.5. We inspected its rights and actual preview, including a white-background inspection composite to interpret transparency. The published original SVG remains unmodified. We neither reproduce a scholarly page nor treat an inaccessible teaching video as freely reusable media ¹³.

The title Key to Arithmetic links knowledge with teaching access. MacTutor treats clarity and breadth of purpose as part of the book's value. Not every beneficiary needs to be a new inventor, and not every teacher needs a separate worldwide-priority title. Exposition can matter without inflating totals or turning a book into a collection of trophies ⁴.

The root story invites a question about answers:an exact equality, a nearby number or a guaranteed range? Those are the museum's teaching questions today, not quotations from a recovered medieval lesson. Separating evidence from explanation prevents current mathematical knowledge from silently replacing the past we are trying to understand.

If a cube has volume eight cubic units, its edge is two length units because2×2×2=8. Do not mix the volume unit with the edge unit, or assume doubling an edge doubles volume only once. This is a modern exercise about roots, not an artefact we measured or an experiment attributed to al-Kashi.

Now ask about volume ten. Its edge is not exactly two, since two cubed is eight. The interval between2.154and2.155narrows the answer beyond a broad guess. Check the printed powers, then ask which part a historical source supports and which part the museum explains today. Understanding does not require a fabricated medieval table.

• MacTutor:Key,roots and predecessors ¹⁴.
• Iranica:computational synthesis and possible connection ³.
• TDV:book dating and circulation ⁵.
• Commons:modern cube/rights,not a manuscript image ¹³.