** - **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 Categories ^{2
}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.

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

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.**