Mensuration (ilm al-misaha)

Mensuration (al-misaha) is the science of measuring areas and volumes. Al-Khwarizmi gave it a chapter of his Book of al-Jabr and al-Muqabala, which opens by saying that "one by one" means an area, that is, a cubit by a cubit. ¹ In the preface he lists the measurement of land among the needs he wrote the book for. ²

• An equal-sided square of side two is four times the cubit-by-cubit area, and a square of side half a cubit is a quarter of it. ³
• The taksir (area) of a triangle is its height times half its base, and of a rhombus one diagonal times half the other. ³
• The circumference of a circle is the diameter times three and a seventh; its area is half the diameter times half the circumference, or the square of the diameter less a seventh and half a seventh of it. ^{3 4}
• A segment of a circle shaped like a bow is a semicircle if its arrow (sagitta) equals half the chord, and less or more if the arrow is less or more; the diameter of its circle is the square of half the chord divided by the arrow, plus the arrow. ⁴
• Triangles are of three kinds, right, acute and obtuse; for an acute triangle of unequal sides (13, 14 and 15 cubits) the foot of the height and the height itself are found by al-jabr and al-muqabala, giving a height of twelve and an area of eighty-four. ^{5 6}
• The volume of a pointed cone or pyramid is a third of its base times its height; a truncated one is completed to a point and the added part subtracted. ^{7 8}
• For a square inside a triangle with sides of ten cubits and a base of twelve, the side of the square is made a "thing" and comes out at four and four-fifths cubits. ^{8 9}

Al-Khwarizmi says that taking the circumference as three and a seventh times the diameter is "a convention among people, not a necessity", and gives two other views of the geometers: multiply the diameter by itself and by ten and take the root; and the astronomers' rule, multiply the diameter by 62,832 and divide by 20,000; all of them, he says, are close to one another. ⁴ The editors of the Cairo edition comment that the astronomers' value is the closest to the truth and the root of ten the furthest, and quote the manuscript's gloss that this is "an approximation, not an exact determination", because the circumference is not a straight line. ³ In the problem of a circle of diameter seven cubits he takes the circumference as twenty-two cubits, which the editors note is approximate. ⁷