Units and converting between them

This lesson asks you to solve problems with different units within a measurement system and between measurement systems, including those of other cultures. The museum offers the conversion of a module to the cubit; other systems and technology are for your class to bring. ¹

Suppose a large unit equals some number of small ones. Convert a length and then a square area using both units, and compare the two factors.

Al-Kashi's calculation starts with a module: a standard length associated with a facet base and assigned the value one. Lengths and areas are first expressed through it, then converted to cubits or another unit. If one cubit equals b modules, the numerical area in module-squared units is divided by b² to obtain square cubits; dividing by b only would confuse length conversion with area conversion ². The study also distinguishes simple, clay-plastered, curved and Shirazi muqarnas: the inventory of elements and the measurement procedure change with the type, and a convenient method does not make every real ceiling identical in geometry or curvature coefficient ³.

The record preserves no sentence in which al-Kashi explains why he chose a module. It reports that the addendum attributes panel construction to masons, and that the author calculates from existing craft knowledge ⁴. For curved panels, a mean value avoids following every different cell diagonal, and the historian explains why this makes practical calculation manageable; writing the method as a calculation preserves a connection between craft and mathematics ⁴.

The reported 1427 (830 AH) dates the completion of the book, not the first building measured with it or the invention of every muqarnas vault ^{5 6}. In the same way, the record says a modern photograph of a ceiling in Qom is not the endpoint of an arrow claiming that all its calculations came from Samarkand; it illustrates the problem's subject ^{7 8}.

The witness examined is the selected pages of the 1992 study and the translated statements embedded in its analysis, not al-Kashi's manuscript or the complete Arabic text and translation printed in the article. Figure 8 on printed page 213 was visually inspected to confirm the plane forms and numbers, and it is not republished ⁹. The Cairo 1967 and Damascus 1977 editions are bibliographical information through Unat, not editions claimed to have been read ¹⁰.

In al-Kashi's method the module is a length taken from a facet base and given the value one, and it is converted to cubits by dividing lengths by b and areas by b². Take a position: is the module a unit of measurement like the cubit, or only a convenience for one ceiling? What is the strongest objection to your position, and how do you answer it?

• Al-Kashi's modular muqarnas mensuration: the module, the division of an area by b², and the types of muqarnas.
• The Key to Arithmetic: the book that carries the method, and what the museum has not read of it.
• Ghiyath al-Din al-Kashi: the record of the method's author.

Read Al-Kashi's modular muqarnas mensuration. Measure the length of your desk in your own forearms and then in centimetres, and convert the area of its top between the two units.