Al-Mahani's cubic reduction of sphere division

Al-Mahani tried to move an Archimedean problem from solid geometry into algebra: where should a plane cut a sphere so that the two resulting volumes have a prescribed ratio? He reached an equation involving a cube, a square and a number, but did not solve it. MacTutor and Dold-Samplonius report Khayyam's account and al-Khazin's later solution using conic sections ^{2 3}.

The contribution recorded is formulation, not the solution he could not complete, the invention of sphere cutting, or all cubic equations. It remains distinct from Khayyam's systematic geometric solutions. Changing the language of a question can alter what later investigators can attempt, even when its proposer cannot finish the solution ^{4 1}.

The requested ratio concerns volumes, not heights or surface areas. A plane near a pole creates a small cap and a larger remainder; its position must match the specified proportion. The inherited problem is from On the Sphere and Cylinder, book II, proposition4. The museum does not attribute the original question to al-Mahani ³.

TDV writes the modern form x³+c=ax², while Dold-Samplonius uses different coefficient names. Different letters do not create different historical problems. We choose our own teaching notation below, without claiming that the manuscript contained these printed signs or the particular numbers in our examples ^{1 3}.

The sphere, the problem and its earlier geometrical treatment were not al-Mahani's inventions. What Khayyam highlights in the transmitted account is an attempt to address the question algebraically. The appearance of a cube does not establish the first cubic anywhere, and difficulty completing this approach does not mean every previous geometrical method had failed ⁵.

The work connects an Archimedean question with algebraic treatment in its new setting. TDV also describes al-Mahani's Euclidean commentaries, ratio theory and irrational magnitudes. Their coexistence is not a fully documented causal chain, nor evidence that every page of a Euclid commentary was a necessary step toward this particular equation ⁶.

Little is known about his life. MacTutor gives birth about820; more firmly dated evidence concerns observations between853 and866 reported by Ibn Yunus. TDV places him in Baghdad around860. An observation date is not a documented invention day, and a working city does not establish a particular room in which the sphere problem was formulated ^{7 8}.

The sources describe a Sphere and Cylinder commentary and related fragments or commentary witnesses in Leiden. The clearest report of the unfinished solution comes through Khayyam as presented by modern scholars. We distinguish a later author's report, a scholar's manuscript description and an original reading we have not undertaken ^{9 3}.

MacTutor explicitly identifies Abu Abd Allah Muhammad ibn Isa al-Mahani as an Islamic mathematician; TDV places him among early Islamic mathematician-astronomers. This provides individual evidence, not an inference from name or birthplace. The separately declared Sunni museum frame is not a personal denominational certificate supplied by the consulted biographies ^{10 8}.

Death remains about880, with TDV explicitly questioning the267/880 pairing. Uncertainty is not republished as an exact day. Three roles remain distinct: Archimedes supplied the inherited problem, al-Mahani its algebraic reduction, and al-Khazin a later conic solution. Al-Khazin must not be confused with al-Khazini of the Balance of Wisdom ¹¹.

This is a modern museum explanation, not a reconstruction of every original step. Let R be the sphere's radius and h the cap height measured from its pole to the cutting plane. The cap volume is V=πh²(3R−h)/3; the whole sphere has volume4πR³/3. The cut-circle radius satisfies a²=h(2R−h). These are three-dimensional volumes, not disk or dome surface areas ¹².

Set t=h/R and let q be the cap's share of the entire sphere volume. Cancelling units gives q=t²(3−t)/4, hence t³+4q=3t². A cube, squares and a number have appeared from geometry. The physical interval is0≤t≤2: endpoints are an empty cap and a whole sphere, while two positive pieces require0<q<1. The normalization is ours, not a transcription of al-Mahani ¹³.

Our independent example t=1/2 gives q=5/32. The cap-to-remainder ratio is5:27, not5:32; the latter compares cap to whole. Substitution gives t³+4q=1/8+5/8=3/4, also3t²=3/4. These numbers check the formulation, not a claim that al-Mahani used them or completed the historical solution.

For half the volume, q=1/2, the modern equation becomes t³−3t²+2=0, factorized as(t−1)(t²−2t−2)=0. Its roots are1,1+√3 and1−√3. Only1 lies in the physical cap interval. Not every algebraic root represents a possible cut, and choosing an easy central-section example does not solve the original problem for every prescribed ratio.

In the modern check, the derivative of q with respect to t is3t(2−t)/4, positive between0 and2. Increasing cap height therefore increases its volume share, with one interior height for each interior share. Other polynomial roots do not multiply the physical cuts. This proves a property of our modern model, not that al-Mahani used differential calculus or that his lost reasoning has been recovered ¹³.

Khayyam's report shows the attempt becoming part of the remembered development of algebra, and the sources describe al-Khazin's subsequent solution. This is a sequence of contributions, not proof that all named scholars met or belonged to one personally documented teaching chain. The accounts read here do not supply a dated geographical adoption map ¹⁴.

TDV lists commentary witnesses in several libraries. A manuscript's present location is not an invention site or an automatically dated stage of medieval circulation. Surviving fragments and commentaries do not imply that every stage of their transmission is known; a catalogue list cannot simply become an animated historical route ¹⁵.

The given condition is a ratio between two pieces; the required relation connects cap volume and cutting-plane position. The model assumes an ideal sphere and plane, not irregular fruit or a vessel of unknown thickness. The modern example shows why reformulation requires defining the quantity, not merely writing a cube beside something round ¹⁶.

For a prescribed ratio m:n, the cap share is q=m/(m+n), not m/n. Finding the corresponding height returns the algebra to geometry. This is our explanation of the reported ratio condition, not a literal translation of Archimedes or a claim that his original text was examined for this batch ¹⁷.

The sources say al-Khazin solved the equation with conics, and assess the attempt's importance in connecting algebra with geometry. Solving an earlier particular case remains different from Khayyam's later organization of multiple cubic forms and from modern radical formulas. These stages are not counted as interchangeable names for one record ¹⁴.

Today's notation allows comparisons of volumes and computational selection of the appropriate root. Those tools are not attributed to al-Mahani merely because a museum lesson uses them. Successful substitution also does not establish a direct transmission route from his commentary to every modern computer program for cubic equations ^{1 13}.

The readings provide no workshop register of sphere cutting based on al-Mahani, material savings attributed to the method, or an industry created by the equation. The supported impact concerns mathematical formulation and history. We do not fill the gap with a fictional craftsperson, or turn the solid-geometry model into instructions for physically cutting objects ⁴.

A useful distinction can nevertheless be learned: length, area and volume ratios are not interchangeable. Doubling both radius and cap height multiplies volume by eight while preserving q. That is a consequence of the modern formula, not a measurement made with an old instrument or a historically documented economic effect ¹³.

As MacTutor reads Khayyam, al-Mahani's inability to finish did not make formulation worthless. Expressing a problem in another language can expose approaches not apparent before. That does not make every unsolved equation a great invention: this case has a particular reported question and a particular later continuation ⁴.

The record makes failure part of the story rather than a footnote beneath a triumphal title. We do not write that al-Mahani solved sphere division when the sources say he stopped at a cubic. Distinguishing the result attained from the result still sought protects this history from exaggerated retrospective credit ³.

Sunni — declared frame: the individual Islamic identity is documented, separate from a universal-priority claim. Academic account: the sources report Khayyam's description of al-Mahani's attempt and al-Khazin's later solution. The attribution concerns a particular algebraic reduction, not every cubic known in every civilization or a museum verdict after examining all manuscripts ^{18 1}.

The sources also retain different confidence about witnesses: TDV attributes a later commentary to al-Quhi, while Dold-Samplonius writes “probably.” We preserve that qualification rather than manufacture certainty by combining accounts. A non-Muslim contributor to the inherited mathematical story is not omitted because this browser's maker criterion is narrower ^{9 3}.

Comparing t with2−t gives complementary caps whose volume shares sum to1. For t=3/2, q=27/32 and the cap-to-remainder ratio is27:5. This independently checked modern example reverses the viewpoint on the same cutting plane; it is not the discovery of a second historical example used by al-Mahani.

In the illustration, r is sphere radius, h cap height and a cut-circle radius; our explanation uses R for sphere radius. The picture's a is not the coefficient carrying that letter in TDV's general cubic. A violet cap and blue sphere are modern illustration, not evidence about the author's physical apparatus ¹⁹.

The actual readings are MacTutor's complete biography, İhsan Fazlıoğlu's TDV body and work list, and Dold-Samplonius's reproduced biography with bibliography in a single-page PDF. No Leiden fragment, complete al-Mahani original or listed Rashed study was inspected merely because it appeared in references ^{20 9 21}.

TDV's other work entries include lost revisions and catalogue notices; not every title represents a complete surviving book. Pbroks13's2008 illustration was released into the public domain, and its rights and current image were inspected. The unchanged original is used; it is not a page from al-Mahani's commentary ^{15 22}.

A history of knowledge can show a question opening a path without creating a hero who answered everything. The report helps distinguish the language of a problem from the language of its solution. No invented maxim or undocumented dramatic Baghdad scene is needed to make the distinction interesting ⁴.

“Cubic” here describes a power in an equation, not the shape of the piece cut from a sphere. “Division” means specifying a plane and two volumes, not just numerical division on paper. Explaining these terms helps a reader interpret the picture without converting modern terminology into a supposed quotation from the author.

Draw a circle standing for a cross-section of a sphere and a line near its top. The small area on paper suggests a three-dimensional cap, but is not itself that cap's volume. No real ball or sharp cutting tool is needed. You can draw different plane positions and ask how volume changes rather than measuring coloured area alone.

Choosing h=R/2 in our example gives5/32 of the entire volume, not half. Al-Mahani did not necessarily choose these numbers; we chose them to teach the question. Formulation was one step, solution another. When reading an invention title, ask what its maker achieved and what remained for later investigators ^{13 2}.

• MacTutor: biography and Khayyam's reported account ²³.
• Fazlıoğlu, TDV: equation and works ²⁴.
• Dold-Samplonius: biography, witnesses and qualifications ²⁵.
• MathWorld: modern formulas, not invention history ¹².
• Commons: modern illustration and rights ²².