Changing dimensions, and the volume of a cone

Draw the square with side one, two and three, and record the areas. Ask how the area changes when the side doubles, then think about what might happen to a volume when all the dimensions double.

In al-Khwarizmi's chapter on mensuration, as the record reports it, the volume of a pointed cone or pyramid is a third of its base times its height, and a truncated one is completed to a point and the added part subtracted ^2 3. The study of al-Kashi's muqarnas measurement carries a related lesson on scale. Its Figure 8 gives approximately 0.707107 for a 45-degree rhombus and 0.353553 for its half; these are areas in squared-module units, not centimetres or volumes of cavities ⁴. The record adds that doubling every length quadruples area, so scale and units are part of the mathematical answer ⁵.

Al-Khwarizmi says in his preface that the measurement of land is among the needs he wrote his book for ⁶. The record preserves no sentence in which he or al-Kashi explains why they treated solids or ceilings. For al-Kashi, Dold-Samplonius separates what the text says from what she infers: her explanation for his relatively careless treatment of simple muqarnas is an interpretation, not direct testimony about the author's intentions ⁷.

Many printed digits can suggest absolute accuracy, but the record says they do not promise absolute accuracy for every curved surface ⁸. For curved muqarnas the cell bases are summed and multiplied by a coefficient; the historian explains it as an average that unifies the calculation, not a magical constant valid for any chosen curvature ⁹.

Dold-Samplonius identifies a projection inconsistency: equating the roof area with the area on which the muqarnas is built conflicts with inclined roofs in the definition and the examples ¹⁰. She also does not know how some of the values given for the intermediate elements were obtained, and her reconstructions yield different numbers ⁹. The record says that reading should be neither a celebration that suppresses the inconsistency nor a wholesale dismissal because it exists ¹⁰.

Dold-Samplonius finds that one step of al-Kashi's method conflicts with the inclined roofs in his own definition and examples. Take a position: should a student still learn the method? What is the strongest objection to your position, and how do you answer it?

• Mensuration (ilm al-misaha): the rule for the cone and pyramid and the truncated case, as the record reports them.
• Al-Kashi's modular muqarnas mensuration: the Figure 8 values, the average coefficient, and what the historian does not know.
• The Key to Arithmetic: the book that carries the method, and what the museum has read of it.
• Al-Khwarizmi: the record of the author of the mensuration chapter.