Al-Khwarizmi's "al-jabr" book: inheritance, trade and land-measurement calculations
"Algebra" brings to mind the x's and y's of school. But the author of the book that gave the subject its name said himself that he wrote it for people's everyday practical calculations. We look at the book's structure and at the operations of "al-jabr" and "al-muqabala" using a market example.
At the bazaar, two partners, Sardor and Jahongir, want to divide the profit from their dried fruit trade. Sardor put in the larger part of the capital, while Jahongir did more of the work. The question "who gets how much?" has not changed in a thousand years. Al-Khwarizmi openly stated that he wrote his book for exactly such questions.
The book's title and the word algebra
The full title of the book is roughly this: "Al-kitab al-mukhtasar fi hisab al-jabr wa-l-muqabala", "The concise book on calculation by completion and balancing". The word al-jabr in the title passed into Latin and became the name of an entire subject, algebra. Britannica also notes that the word "algebra" derives from the title of this book.
Why did the author write the book?
According to the introduction quoted in the scholarly biography on the University of St Andrews (MacTutor) website and in F. Rosen's 1831 translation, al-Khwarizmi says that he wrote the book for the practical calculations people constantly need. The cases listed in the introduction:
- inheritance;
- legacies;
- partition of property;
- lawsuits (court cases);
- trade;
- all other dealings of people with one another;
- land measurement;
- digging of channels (canals);
- geometrical computations;
- various other matters of a similar kind.
In other words, the book was conceived first of all not as theory but as a manual of practical calculation.
How is the book structured?
According to the sources, the book consists of three parts:
| Part | What it is about | Today's counterpart |
|---|---|---|
| 1. Algebra | Equations of the first and second degree and the rules for solving them | Finding an unknown amount |
| 2. Mensuration | Calculations of area and volume | Calculating the area of land or a room, or warehouse capacity |
| 3. Legacies | The largest part, consisting of fully solved problems | Dividing an inheritance into shares |
After the solved examples on algebra comes a short section devoted to commercial transactions: it explains how to find the fourth term of a proportion (the "rule of three"), a method still used today in price and quantity calculations. The Arabic manuscript of the book is kept in the Bodleian Library in Oxford (MS. Huntington 214), and in 1831 F. Rosen translated it into English and published it together with the Arabic text. Much earlier, in the 12th century, the book was translated into Latin and spread across Europe.
What do "al-jabr" and "al-muqabala" mean?
These two words are two operations for simplifying an equation:
- Al-jabr (completion, restoration): moving a subtracted term from one side of the equation to the other by adding it, so that all terms become "whole".
- Al-muqabala (balancing): cancelling identical terms on both sides.
Let us see this in Sardor and Jahongir's problem (prepared by the author, hypothetical figures). The agreement: Jahongir first receives 2 000 000 soum for his work, and the remaining profit is divided equally. It is known that Sardor received a total of 7 000 000 soum. What was the total profit (x)?
| Step | Notation | Comment |
|---|---|---|
| Condition | (x − 2 000 000) ÷ 2 = 7 000 000 | Sardor's share |
| Multiply by two | x − 2 000 000 = 14 000 000 | — |
| Al-jabr | x = 14 000 000 + 2 000 000 | The subtracted term was added to the other side |
| Answer | x = 16 000 000 | Jahongir 2 + 7 = 9 mln, Sardor 7 mln |
Now an example of balancing: "The price of the goods plus another 3 000 000 soum equals twice the price plus 1 000 000 soum", that is, x + 3 000 000 = 2x + 1 000 000. We cancel one x and 1 000 000 from both sides (al-muqabala): 2 000 000 = x.
A lesson for the finance professional
Today, when calculating profit distribution, loan repayments or shares, a finance professional uses an Excel formula or software. But behind every formula stands the same logic: gather the unknown on one side, cancel what is superfluous, check the result. Writing out the distribution as an equation before signing a contract is the cheapest way to prevent later disputes.
Note: the inheritance problems in the book were drawn up on the basis of the rules of that era. This article does not pass judgement on them; we have only shown the historical significance of the method of calculation.
See also
Sources
- MacTutor History of Mathematics (University of St Andrews): al-Khwarizmi, Dictionary of Scientific Biography article
- Encyclopaedia Britannica: al-Khwarizmi
- F. Rosen (translator): "The Algebra of Mohammed ben Musa", London, 1831 (Arabic text and English translation, Internet Archive)
- Bodleian Library (Oxford) catalogue: MS. Huntington 214, Arabic manuscript of al-Khwarizmi's algebra
- Critical edition of Gerard of Cremona's Latin translation of "al-jabr" (Internet Archive)
- Examples and amounts · prepared by the author, hypothetical figures