- UMAR AL-KHAYYAMI

Chapter 1. Definitions

ALGEBRA

By the help of God and with his precious assistance. I say that Algebra is a scientific art. The objects with which it deals are absolute numbers and measurable quantities which, though themselves unknown, are related to 'things' which are known, whereby the determination of the unknown quantities is possible. Such a thing is either a quantity or a unique relation, which is only determined by careful examination. What one searches for in the algebraic art are the relations which lead from the known to the unknown, to discover which is the object of Algebra as stated above. The perfection of this art consists in knowledge of the scientific method by which one determines numerical and geometric unknowns.

MEASURABLE QUANTITIES

By measurable quantities I mean continuous quantities of which there are four kinds, viz., line, surface, solid, and time, according to the custom ary terminology of the Categories² and what is expounded in metaphysics.³ Some consider space a subdivision of surface, subordinate to the division of continuous quantities, but investigation has disproved this claim. The truth is that space is a surface only under circumstances the determination of which is outside the scope of the present field of investigation. It is not customary to include ''time' among the objects of our algebraic studies, but if it were mentioned it would be quite admissible.

THE UNKNOWN

It is a practice among algebraists in connection with their art to call the unknown which is to be determined a ''thing," ⁴ the product obtained by multiplying it by itself a "square,"5 and the product of the square and the "thing" itself a ''cube.'' The product of the square multiplied by itself is ''the square of the square,'' the product of its cube multiplied by its square ''the cube of the square,'' and the product of a cube into itself ''a cube of the cube,'' and so on. as far as the succession is carried out. It is known from Euclid's book, the Elements, - that all the steps are in continuous proportion,' i.e.. that the ratio of one to the root is as the ratio of the root to the square and as the ratio of the square to the cube.' Therefore, the ratio of a number to a root is as the ratio of roots to squares, and squares to cubes, and cubes to the squares of the squares, and so on after this manner.
 

SOURCES

It should be understood that this treatise cannot be comprehended except by those who know  thoroughly Euclid's books, the Elements and the Data, as well as the first two books from Apollonius' work on Conics. Whoever lacks knowledge of any one of these books cannot possibly understand my work, as I have taken pains to limit myself to these three books only.

ALGEBRAIC SOLUTIONS

Algebraic solutions are accomplished by the aid of equations;  that is to say, by the well-known method of equating these degrees one with the other. If the algebraist were to use the square of the square in measuring areas, his result would be figurative and not real, because it is impossible to consider the square of the square as a magnitude of a measurable nature. What we get in measurable quantities is first one dimension, which is the "root" or the "side" in relation to its square; then two dimensions, which represent the surface and the (algebraic) square representing the square surface, and finally three dimensions, which represent the solid. The cube in quantities is the solid bounded by six squares, and since there is no other dimension, the square of the square does not fall under measurable quantities. This is even more true in the case of higher powers. If it is said that the square of the square is among measurable quantities, this is said with reference to its reciprocal  value in problems of measurement and not because it in itself is measurable. This is an important distinction to make.

The square of the square is, therefore, neither essentially nor accidentally a measurable quantity and is not as even and odd numbers, which are accidentally included in measurable quantities, depending on the way in which they represent continuous measurable quantities as discontinuous.

What is found in the books of algebra relative to these four geometric quantities - namely, the absolute numbers, the 'sides," the squares, and the cubes-are three equations containing numbers. sides. and squares. We, however, shall present methods by which one is able to determine the unknown quantities in equations including four degrees concerning which we have just said that they are the only ones that can he included in the category of measurable quantities, namely, the number, the thing, the square, and the cube.

The demonstration (of solutions) depending on the properties of the circle-that is to say, as in the two works of Euclid, the EIements and the Data - is easily effected; but what we can demonstrate only by the properties of conic sections should be referred to the first two books on conics. When, however, the object of the problem  is an absolute number, neither we, nor any of those who are concerned with algebra. have been able to prove this equation-perhaps others who follow us will be able to fill the gap-except when it contains only the three first degrees, namely, the number, the thing, and the square. For the numerical demonstration given in cases that could also be proved by Euclid's book, one should know that the geometric proof of such procedure does not take the place of its demonstration by number, if the object of the problem is a number and not a measurable quantity. Do you not see how Euclid proved certain theorems relative to proportions of geometric quantities in his fifth book and then in the seventh book gave a demonstration of the same theorems for the case when their object is a number?

Chapter II. Table of Equations

The equations among those four quantities are either simple or compound. The simple equations are of six species:

I. A number equals a root.

2. A number equals a square.

3. A number equals a cube.

4. Roots equal a square.

5. Squares equal a cube.

6. Roots equal a cube.

Three of these six species are mentioned in the books of the algebraists. The latter said a thing is to a square as a square is to a cube. Therefore, the equality between the square and the cube is equivalent to the equality of the thing to the square. Again they said that a number is to a square as a root is to a cube. And they did not prove by geometry. As for the number which is equal to a cube there is no way of determining its side when the problem is numerical except by previous knowledge of the order of cubic numbers. When the problem is geometrical it cannot be solved except by conic sections.

Chapter V. Preliminary Theorems for the Construction of Cubic Equations

After presenting those species of equations which it has been possible to prove by means of the properties of the circle. i.e.. by means of Euclid'sbook, we take up now the discussion of the species which cannot be proved except by means of the properties of conics. These include fourteen species: one simple equation, namely, that in which a number is equal to a cube; six trinomial equations: and seven tetranomial equations.

Let us precede this discussion by some propositions based on the book of Conics so that they may serve as a sort of introduction to the student and so that our treatise will not require familiarity with more than the three books already mentioned, namely, the two books of Euclid on the Elements and the Data, and the first two parts of the book on Conics.

Between two given lines it is required to find two other lines such that all four will form a continuous proportion.

Let there be two straight lines (given) AB, BC, and let them enclose the right angle B. Construct a parabola the vertex of which is the point B, the axis BC, and the parameter BC. Then the position of the conic BDE is known because the positions of its vertex and axis are known, and its parameter is given. It is tangent to the line BA, because the angle B is a right angle and it is equal to the angle of the ordinate, as was shown in the figure of the thirty-third proposition in the first book on Conics.²¹

In the same manner construct another parabola, with vertex B, axis AB, and parameter AB. This will be the conic BDZ, as was shown also by Apollonius in the fifty-sixth proposition of the first book. The conic BDZ is tangent to the line BC. Therefore, the two (parabolas) necessarily intersect. Let D be the point of intersection. Then the position of point D is known because the position of the two conics is known.

Let fall from the point D two perpendiculars, DH and DT, on AB and BC respectively. These are known in magnitude, as was shown in the Data.²³ I say that the four lines AB, BH, BT, BC are in continuous proportion.

DEMONSTRATION

The square of HD is equal to the product of BH and BC, because the line DH is the ordinate of the parabola BDE. Consequently BC is to MD, which is equal to BT as BT to MB. The line DT is the ordinate of the parabola BDZ.

The square of DT (which is equal to BH) is equal to the product of BA and BT. Consequently BT is to BH as BM to BA. Then the four lines are in continuous proportion and the line DH is of known magnitude, as it is drawn from the point the position of which is known, to a line whose position is known, at an angle whose magnitude is known. Similarly, the length of DT is known.

Therefore, the two lines, BM and BT, are known and are the means of the proportion between the two lines, AB and BC; that is to say, AB to BH is as BM to BT and is as BT to BC. That is what we wanted to demonstrate.