Al-Kashi's Key to Arithmetic

Al-Kashi wrote this teaching book,described by MacTutor as mathematics for astronomy,surveying,architecture,accounting and commerce. It reports completion1427;Dold-Samplonius also gives830AH. Diverse intended audiences do not prove that the author practised every occupation or that every group used the same copy ^{2 3}.

The study calls it an encyclopaedic practical treatment of elementary mathematics. It locates sexagesimal arithmetic in BookIIIand muqarnas at the end of chapter9,BookIV,on measuring structures/buildings. This connects compound forms to previously studied plane shapes and areas rather than reducing the book to one decorative account ⁴.

MacTutor discusses decimal fractions and root extraction,warning against calling al-Kashi the first decimal inventor or ignoring al-Karaji's school and al-Samawal. It also describes approximating dome surface/shell measurements,not interchangeable with muqarnas. These observations describe a wide work;each paragraph is not a new invention record or a claim that every original algorithm was read here ⁵.

The modular mensuration article explains a bounded procedure:one module,element classes,facet/roof/intermediate-area sums,and squared unit conversion. Treatment differs for simple,clay-plastered,curved and Shirazi forms. Curved calculation uses an average coefficient derived from masons' panels,not a promise of universal exactness ⁶.

Criticism remains visible:a projection/surface inconsistency,different recomputed values,and the historian's explicit uncertainty about some derivations. Modern recalculation is not retrospectively rewritten as al-Kashi's own text. This neither generates a judgement about belief nor erases the book's contribution;it describes evidence and its limits ⁷.

Unat reports Ottoman madrasa teaching,Ibrahim Kami's translation of BookIV,and editions including Cairo1967and Damascus1977. These are bibliographical notices,not copies claimed examined. Selected pages of the1992study and statements embedded in its analysis were read here,and figure8,p213,visually inspected;not the complete Arabic book or the whole text/translation printed later in that study ^{8 9}.

The module and elements show the book connecting arithmetic with craft. Area-based artisan payment in the Arab world remains the historian's possible application,not a verified payment ledger. The modern photograph on the invention record likewise illustrates a subject without proving that this book was used for that building ^{10 11}.

The general-root entry adds a subject distinct from muqarnas and the sine-one-degree problem. MacTutor reports Luckey's later Ruffini/Horner comparison while retaining Rashed's al-Karaji/al-Samawal school precedents. Iranica describes general extraction with square roots as a special case and an attributed possible Chinese connection,not a demonstrated transfer route. This is improvement/integration,not first roots or all algebraic algorithms ^{12 13}.

Those secondary accounts were read,not the original Arabic chapter or Luckey/Rashed independently;the security-blocked summary and unwatched animation are not reading. Integer cube bounds on the invention page are independent modern teaching,not a recovered manuscript table. The book's completion date does not date discovery of every procedure ¹⁴.