Measuring areas and surfaces

Draw the shape on a grid and cut it into rectangles and triangles. Add the areas of the parts, then check by cutting it another way.

In the Key to Arithmetic, al-Kashi organised the measurement of muqarnas: he selects a module, counts elements by shape, and adds the surfaces of facets, roofs and connecting pieces ². In the definition the study discusses, a muqarnas is a staircase-like ceiling: a cell consists of two facets and a roof, and cells whose bases lie in one horizontal plane form a tier ³. Some visible pieces belong to cells and others fill the gaps between them, so counting the hollows in a photograph is not enough to determine their exposed surface area ³. For plane surfaces the facet bases are summed and multiplied by the height, the roof pieces are added, and then the tiers are combined ⁴.

The museum's records preserve no sentence in which al-Kashi says why he measured muqarnas. They show that his book places arches, vaults and buildings among the matters a calculator needs to understand ^5 6. In an earlier book, al-Khwarizmi says in his preface that the measurement of land is among the needs he wrote his book for ⁷.

Al-Kashi can be taken for the inventor of muqarnas, but builders already knew how to assemble the cells before his mathematical account, and the historian Dold-Samplonius explains that he did not need to teach his practical readers the whole craft from its beginning ⁸. The record describes his method as a calculation within an older architectural practice, not an attribution of muqarnas itself to one inventor ².

In class you split a composite shape two ways and got the same area. Al-Kashi's method sorts the pieces of a ceiling by shape and adds their areas. Why might a measurement of a ceiling go wrong if some pieces are counted as cells when they only fill the gaps between cells?

• Al-Kashi's modular muqarnas mensuration: the module, the classes of pieces, and how their areas are added.
• The Key to Arithmetic: the book that carries the method, and where its author placed it.
• Mensuration (ilm al-misaha): the rules for the triangle, the rhombus and the circle in al-Khwarizmi's chapter.
• Ghiyath al-Din al-Kashi: the record of the book's author.