Six Rhetoric Quadratic: the Six Types of Quadratic Equations Presented by Abū Ja’far Muḥammad ibn Mūsā al-Khwārizmī in al-Kitāb al-Mukhtaṣor fī Ḥisāb al-Jabr wa al-Muqōbala

This work explains the six types of quadratic equations presented by Abū Ja’far Muḥammad ibn Mūsā al-Khwārizmī (Arabic: أبو جعفر محمد بن موسی الخوارزمی) in al-Kitāb al-Mukhtaṣor fī Ḥisāb al-Jabr wa al-Muqōbalah (Arabic: الكتاب المختصر في حساب الجبر والمقابلة). In this mathematical treatise written approximately 820 CE, equations are verbally described in terms of “squares” (Arabic: مربع الجذر; what would today be “x^2”), “roots” (Arabic: الجذور; what would today be “x”) and “numbers” (Arabic: الأعداد; “constants”: ordinary spelled out numbers, like ‘twenty-six’). The six types equations are:[1] <المربعات تساوي الجذور> or squares equal roots or with current notations ax^2=bx;[2] <المربعات تساوي الأعداد> or squares equal number or ax^2=c;[3] <الجذور تساوي الأعداد> or roots equal number or bx=c;[4] <المربعات والجذور تساوي الأعداد> or squares and roots equal number or ax^2+bx=c;[5] <المربعات والأعداد تساوي الجذور> or squares and number equal roots or ax^2+c=bx; and[6] <الجذور والأعداد تساوي المربعات> or roots and number equal squares or bx+c=ax^2.Al-Kitāb al-Mukhtaṣor fī Ḥisāb al-Jabr wa al-Muqōbala thoroughly rhetorical, with the syncopation that is the numbers were written out in words rather than symbols. However, in author’s day, most of this notation had not yet been invented, so he had to use ordinary text to present problems and their solutions. The types of problems which the book discusses reveals middle east mathematicians didn’t deal with negative numbers at all. Hence an equation like bx+c=0 doesn’t appear in the classification, because it has no positive solutions if all the coefficients are positive. Similarly equation types 4, 5 and 6, which look equivalent to our current view, were distinguished because the coefficients must all be positive.