by

David W. Henderson^{1}

Department of Mathematics, Cornell University

Ithaca, NY, 14853-7901, USA^{1}

**Theorem 1.*** Every rectangle is equivalent by dissection to a square.*

*HK* = (*AE*)(*HD*)/*AD* = *s*(*a - s*)/*a* =* s - s*/*a* = *s - b*.

Therefore, ... we have *D **EFK* @ *D** RCD*, *D** EBR* @ *D** KHD*."

Here is a diagram for Sutra 54:

This last assertion follows from sutra 50:

Proof of Theorem 3 known to the ancient Chinese:

*x*=*c*, which needs no solution,*x*=*bx*, which is easily solved,*x*^{2}=*c*, which has root*x*= Ö*c*,*x*^{2}+*bx*=*c*, with root*x*= Ö[(*b*/2)^{2}+*c*] -*b*/2,*x*^{2}+*c*=*bx*, with roots*x*=*b*/2 ± Ö[(*b*/2)^{2}-*c*] , if*c*< (*b*/2)^{2}, and*x*^{2}=*bx*+*c*, with root*x*=*b*/2 + Ö[(*b*/2) +*c*] .

* But why did mathematicians avoid negative numbers?* The avoidance of negative numbers was widespread until a few hundred years ago. In the Sixteenth Century, European mathematicians called the negative numbers that appeared as roots of equations, "numeri fictici" - fictitious numbers (see Witmer (1968), page 11).

To get a feeling for why, think about the meaning of 2 x 3 as two 3's and 3 x 2 as three 2's and then try to find a meaning for 3 x (-2) and -2 x (+3). Another answer is found in the reliance on geometric justifications, as al'Khayyam wrote (see Amir-Moez (1963), page 329):

"Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras (jabbre and maqabeleh) are geometric facts which are proved by propositions five and six of Book two of [Euclid's] *Elements*".

Some historians have quoted this passage but have left out all the words appearing after "proved". In my opinion, this omission changes the meaning of the passage. Euclid's propositions that are mentioned by al'Khayyam are the basic ingredients of Euclid's proof of the square root construction and form a basis for the construction of conic sections - see below. Geometric justification when there are negative coefficients is at least very cumbersome if not impossible. (If you doubt this try to modify some of the geometric justifications below.) In any case, Euclid, upon which these mathematicians relied, did not allow negative quantities.

For the geometric justification of (III) and the finding of square roots, al'Khayyam refers to Euclid's construction of the square root in Proposition II 14.

For (IV) we have as geometric justification:

and thus, by "completing the square" on * x* +* b*/2, we have (*x* + *b*/2)^{2} = *c* + (*b*/2)^{2} . Note the similarity between this and Baudhayana's construction of the square root (see Section 2).

For (V), first assume *x* < (*b*/2) and draw the equation as:

and note that the square on *b*/2 is (*b*/2 - *x*)^{2} + *c*.

This leads to *x* = * b*/2 - Ö[(*b*/2)^{2} - *c*]. Note that if *c* > (*b*/2)^{2} then this geometric solution is impossible. When *x* > (*b*/2), use the drawings:

For the solution of (VI) use the drawing:

** Do the above solutions find the negative roots?** Well, first, the answer is clearly, No, if you mean: Did al'Khowarizmi and al'Khayyam (or the earlier Greeks and Babylonians) mention negative roots? But let us not be too hasty, suppose -

*4: Conic Sections and Cube Roots*

The Greeks noticed that, if *a/c = c/d = d/b*, then (*a/c*)^{2} = (*c/d*)(*d/b*) = (*c/b*) and thus *c*^{3} = *a*^{2}*b*. Now setting * a* = 1, we see that we can find the cube root of *b*, if we can find *c* and *d * such that *c*^{2} = *d* and *d*^{2} = *bc*. If we think of *c* and *d* as being variables and *b* a constant, then we see these equations as the equations of two parabolas with perpendicular axes and the same vertex. The Greeks also saw it this way but first they had to develop the concept of a parabola!

To the Greeks, and later al'Khayyam, if * AB* is a line segment, then *the parabola with vertex B and parameter AB* is the curve *P* such that, if *C* is on *P*, then the rectangle *BDCE* (see the drawing) has the property that (*BE*)^{2} = *DB · AB* . Since in Cartesian coordinates the coordinates of

Points of the parabola may be constructed by using the construction for the square root given in Section 2. In particular, *E* is the intersection of the semicircle on *AD* with the line perpendicular to *AB* at * B*. (The construction can also be done by finding *D' * such that *AB* = *DD'*, then the semicircle on *BD'* intersects *P* at *C*.) I encourage you to try this construction yourself; it is very easy to do if you use a compass and graph paper.

** Now we can find the cube root.** Let

Construct a parabola with vertex * B* and parameter *AB * and construct another parabola with vertex *B* and parameter *CB*. Let *E* be the intersection of the two parabolas. Draw the rectangle *BGEF*. Then (*EF*)^{2} = *BF·AB* and (*GE*)^{2} = *GB·CB*. But, setting *c* = *GE* = *BF* and *d* = *GB* = *EF*, we have *d*^{2} = *cb* and * c*^{2} = *d*. Thus *c*^{3} = *b*. If you use a fine graph paper it is easy to get three digit accuracy in this construction.

The Greeks did a thorough study of conic sections and their properties which culminated in Appolonius's book ** Conics** which appeared in 200 BC. You can read this book in English translation, see Heath (1961).

To find roots of cubic equations in the next section we shall also need to know *the (rectangular) hyperbola with vertex* *B* * and parameter* *AB*. This is the curve *H*, such that if *E * is on *H* and *ACED* is the determined rectangle (see drawing), then (*EC*)^{2} = *BC·AC*.

The point *E* can be constructed using Section 2. Let * F* be the bisector of *AB*. Then the circle with center *F* and radius *FC* will intersect at *D* the line perpendicular to *AB* at * A*. From the drawing it is clear how these circles also construct the other branch of the hyperbola (with vertex *A*.)

Notice how these descriptions and constructions of the parabola and hyperbola look very much like they were done in Cartesian coordinates. *The ancestral forms of Cartesian coordinates and analytic geometry are evident here.* Also they are evident in the solutions of cubic equations in the next section. The ideas of Cartesian coordinates did not come to Descartes out of nowhere. The underlying concepts were developing in Greek and Muslim mathematics. One of the apparent reasons that full development did not occur until Descartes is that, as we have seen, negative numbers were not accepted. The full use of negative numbers is essential for the realization of Cartesian coordinates.

*5: Roots of Cubic Equations*

In his *Al-Jabr wa'l muqabalah* Omar al'Khayyam also gave geometric solution to cubic equations. We shall see that his methods are sufficient to find geometrically all real (positive or negative) roots of cubic equations; however; in his first chapter al'Khayyam says: (see Kasir (1931), page 49.)

"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."

By "absolute number", al'Khayyam is referring to, what we call, algebraic solution as opposed to geometric one. This quotation suggests, contrary to what many historical accounts say, that al'Khayyam expected that algebraic solutions would be found.

Al'Khayyam found 19 types of cubic equations (when expressed with only positive coefficients). (See Kasir (1931), page 51). Of these 19, five reduce to quadratic equations (e.g.,* x*^{3 }*+ ax = bx * reduces to * x*^{2}* + ax = b*). The remaining 14 types al'Khayyam solves by using conic sections. His methods find all the positive roots of each type although he failed to mention some of the roots in a few cases; and, of course, he ignores the negative roots. Instead of going through his 14 types, I will show how a simple reduction will reduce all the types to only 3 types in addition to types already solved such as, * x*^{3}* = b*. I will then give al'Khayyam's solutions to these types.

In the cubic *y*^{3}* + py*^{2} * + gy + r = 0* (where, here, * p, g, r*, are positive, negative, or zero) set * y = x - (p/*3*)*. Try it! The resulting equation in *x* will have the form *x*^{3}* + sx + t = 0*, (where, here, *s * and * t* are positive, negative or zero). If we rearrange this equation so all the coefficients are positive then we get four types that have not been previously solved:

(I) *x*^{3}* + ax = b*, (II) *x*^{3}* + b = ax*, (III) *x*^{3}* = ax + b*, and (IV) *x*^{3}* + ax + b = *0,

where * a* and * b* are positive, in addition, to types previously solved. Now (IV) has no positive roots and the absolute value of its negative roots are the (positive) roots of (I). Also, the absolute value of the negative roots of (II) are the roots of (III) and vice - versa. Thus, we need only find the positive roots of types (I), (II), and (III).

* Al'Khayyam's solution for type (I):** x*^{3}* + ax = b*.

"A cube and sides are equal to a number. Let the line *AB* [see figure] be the side of a square equal to the given number of roots, [that is, (*AB*)^{2}=*a*, the coefficient.] Construct a solid whose base is equal to the square on *AB*, equal in volume to the given number, [ *b *]. The construction has been shown previously. Let *BC* be the height of the solid. [I.e. *BC·(AB) ^{2} = b*.] Let

The root is *EB*. Al'Khayyam's proof (using a more compact notation) is: From the properties of the parabola (Section 4) and circle (Section 2) we have

(*DZ*)^{2}* *=* *(*EB*)^{2} * *=* BZ·AB *and* *(*ED*)^{2}* *= (*BZ*)^{2} =* EC·EB *,

thus

*EB*·(*BZ*)^{2} = (*EB*)^{2}·*EC = BZ*·*AB*·*EC*

and therefore

*AB*·*EC = EB*·*BZ *and (*EB*)^{3} = *EB*·(*BZ*·*AB*) = (*AB*·*EC*)·*AB = *(*AB*)^{2}·*EC*;

So

(*EB*)^{3} + *a*(*EB*) = (*AB*)^{2}·*EC* + (AB)^{2}·(*EB*) = (*AB*)^{2}·*CB = b*.

Thus *EB* is a root of * x*^{3} + *ax* = * b*. Since *x*^{2} + *ax* increases as *x* increases, there can be only this one root.

*Al'Khayyam's solutions for types (II) and (III): ** x*^{3}* + b = ax *and* x*^{3} *= ax + b*.

Al'Khayyam treated these equations separately but by allowing negative horizontal lengths we can combine his two solutions into one solution of *x*^{3} ± *b = ax*. Let *AB* be perpendicular to *BC *and as before let (*AB*)^{2} = *a* and (*AB*)^{2}·*BC* = *b*. Place *BC* to the left if the sign in front of *b* is negative (type (III)) and place *BC* to the right is the sign in front of *b* is positive (type (II)). Construct a parabola with vertex *B *and parameter * AB*. Construct both branches of the hyperbola with vertices *B* and *C* and parameter *BC*.

Each intersection of the hyperbola and the parabola (except for *B* ) gives a root of the cubic. Suppose they meet at *D*. Then drop perpendiculars *DE * and *DZ*. The root is *BE* (negative if to the left and positive if to the right). Again, if you use fine graph paper it is easy to get three digit accuracy here. I leave it for you, the reader, to provide the proof which is very similar to type (I).

** A little more history:** Most historical accounts assert correctly that al'Khayyam did not find the negative roots of cubics. However, they are misleading in that they all fail to mention that his methods are fully sufficient to find the negative roots as we have seen above. This is in contrast to the common assertion (see, for example, Davis & Hersch (1981)) that Girolamo Cardano (16th century Italian) was the first to publish the general solution of cubic equations when in fact, as we shall see, he himself admitted that his methods are insufficient to find the real roots of many cubics.

Cardano published his algebraic solutions in his book, *Artis Magnae* (The Great Art) which was published in 1545. For a readable English translation and historical summary, see Witmer (1968). Cardano used only positive coefficients and thus divided the cubic equations into the same 13 types (excluding * x*^{3} = *c* and equations reducible to quadratics) used earlier by al'Khayyam. Cardano also used geometry to prove his solutions for each type. As we did above we can make a substitution to reduce these to the same types as above:

(I) *x*^{3}* + ax = b*, (II) *x*^{3}* + b = ax*, (III) *x*^{3}* = ax + b*, and (IV) *x*^{3}* + ax + b = *0.

If we allow ourselves the convenience of using negative numbers and lengths then we can reduce these to one type: *x*^{3}* + ax + b = *0, where now we allow *a* and *b* to be either negative or positive.

The main "trick" that Cardano used was to assume that there is a solution of *x*^{3}* + ax + b = *0 of the form *x* =* t*^{1/3} + *u*^{1/3} . Plugging this into the cubic we get

(*t*^{1/3} + *u*^{1/3})^{3}* + a*(*t*^{1/3} + *u*^{1/3})* + b = *0.

If you expand and simplify this you get to

*t + u + b + *(3*t*^{1/3}*u*^{1/3} + *a*)(*t*^{1/3} + *u*^{1/3}) = 0.

Thus *x* = *t*^{1/3} + *u*^{1/3} is a root if

*t + u = *- *b *and *t u* = -(*a*/3)^{3}.

Solving, we find that *t* and *u* are the roots of the quadratic equation *z*^{2} + *bz *- (*a*/3)^{3} = 0 which Cardano solved geometrically (and you can use the quadratic formula) to get

*t = **-**b*/2* + *Ö[(*b*/2)^{2} + (*a*/3)^{3}] and *u* = -*b*/2 - Ö[(*b*/2)^{2} + (*a*/3)^{3}] .

Thus the cubic has roots

*x* = *t*^{1/3} + *u*^{1/3} = {*-**b*/2* + *Ö[(*b*/2)^{2} + (*a*/3)^{3}] }^{1/3} + {-*b*/2 - Ö[(*b*/2)^{2} + (*a*/3)^{3}] }^{1/3}.

This is Cardano's cubic formula. But, a strange thing happened, Cardano noticed that the cubic *x*^{3 }= 15*x* + 4 has a positive real root 4 but, for this equation, *a* = -15 and *b* = -4, and if we put these values into his cubic formula we get that the roots of *x*^{3 }= 15*x* + 4 are

*x* = { 2 + Ö-121 }^{1/3} + { 2 - Ö-121 }^{1/3} .

In Cardano's time there was no theory of complex numbers and so he reasonably concluded that his method would not work for this equation; Cardano writes (Witmer (1968), page 103):

"When the cube of one-third the coefficient of x is greater than the square of one-half the constant of the equation ... then the solution of this can be found by the aliza problem which is discussed in the book of geometrical problems."

It is not clear what book he is referring to but the "aliza problem" presumably refers to al'Hazen, an Arab, who lived around 1000 AD and whose works were known in Europe in Cardano's time. Al'Hazen had used intersecting conics to solve specific cubic equations and the problem of describing the image seen in a spherical mirror - this later problem is in some books called "Alhazen's problem".

In addition, we know today that each complex number has three cube roots and so the formula *x = *{ 2 + Ö-121 }^{1/3} + { 2 - Ö-121 }^{1/3} is ambiguous. In fact, some choices for the two cube roots give roots of the cubic and some do not. (Experiment with *x*^{3} = 15*x* + 4.) Faced with Cardano's Formula and equations like *x*^{3} = 15*x* + 4, Cardano and other mathematicians of the time started exploring the possible meanings of these complex numbers and thus started the theory of complex numbers. This leads to another interesting path which we may take another day.

*6: So What Does This All Point To?*

It points to different things for each of us. I conclude that it is worthwhile paying attention to the meaning in mathematics. Often in our haste to get to the modern, powerful, analytic tools we ignore and trod upon the meanings and images that are there. Sometimes it is hard even to get a glimpse that some meaning is missing. One way to get this glimpse and find meaning is to listen to and follow questions of "What does it mean?" that come up in oneself and in one's students. We must listen creatively because we and our students often do not know how to express precisely what is bothering us.

Another way to find meaning is to read the mathematics of old and keep asking "Why did they do that?" or "Why didn't they do this?" Why did the early algebraists (up until at least 1600 and much later I think) insist on geometric proofs? I have suggested some reasons above. Today, we normally pass over geometric proofs in favor of analytic ones based on the 150 year old notion of Cauchy sequences and the Axiom of Completeness. However, for most students and, I think, most mathematicians, our intuitive understanding of the real numbers is based on the geometric real line. As an example, think about multiplication: What does *a* x *b* mean? Compare the geometric images of *a* x *b* with the multiplication of two infinite, nonrepeating, decimal fractions. What is Ö2 x p?

There is another reason for why a geometric solution may be more meaningful: Sometimes we want a geometric result instead of a numerical one. As an example, I shall describe an experience that I had while a friend and I were building a small house using wood. The roof of the house consists of 12 isosceles triangles which together form a 12-sided cone (or pyramid). It was necessary for us to determine the angle between two adjacent triangles in the roof so that we could appropriately cut the log rafters. I immediately started to calculate the angle using (numerical) trigonometry and algebra. But then I ran into a problem. For finding square roots and values of trigonometric functions I had only a slide rule with three-place accuracy. At one point in the calculation I had to subtract two numbers that differed only in the third place (e.g. 5.68 - 5.65) thus my result had little accuracy. As I started to figure out a different computational procedure that would avoid the subtraction, I suddenly realized - *I didn't want a number, I wanted a physical angle*. In fact, a numerical angle would be essentially useless - imagine taking two rough boards and putting them at a given numerical angle apart using only an ordinary protractor! What I needed was the physical angle, full size. So I constructed the angle on the floor of the house using a rope as a compass. Note the relationship between this and Baudhayana's descriptions of using cords. This geometric solution had the following advantages over a numerical solution:

- The geometric solution resulted in the desired physical angle, while the numerical solution resulted in a number.
- The geometric solution was quicker than the numerical solution.
- The geometric solution was immediately understood
by my friend (and follow builder), who had almost no mathematical training, while the numerical solution was beyond my friend's understanding because it involved trigonometry (such as the "Law of Cosines").*and trusted* - And, since the construction was done full-size, the solution automatically had the degree of accuracy appropriate for the application.

I close with the words written in 1934 by the "father of Formalism", David Hilbert, from the Preface to ** Geometry and the Imagination** (see Hilbert, Cohn-Vossen (1952), page iii):

"In mathematics, as in any scientific research, we find two tendencies present. On the one hand, the tendency toward *abstraction* seeks to crystallize the *logical* relations inherent in the maze of material that is being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency toward *intuitive understanding* fosters a more immediate grasp of the objects one studies, a live *rapport* with them, so to speak, which stresses the concrete meaning of their relations.

"As to geometry, in particular, the abstract tendency has here led to the magnificent systematic theories of Algebraic Geometry, of Riemannian Geometry, and of Topology; these theories make extensive use of abstract reasoning and symbolic calculation in the sense of algebra. Notwithstanding this, it is still as true today as it ever was that intuitive understanding plays a major role in geometry. And such concrete intuition is of great value not only for the research worker, but also for anyone who wishes to study and appreciate the results of research in geometry.

"In this book, it is our purpose to give a presentation of geometry, as it stands today, in its visual, intuitive aspects. With the aid of visual imagination we can illuminate the manifold facts and problems of geometry, ...

"In this manner, geometry being as many-faceted as it is and being related to the most diverse branches of mathematics, we may even obtain a summarizing survey of mathematics as a whole, and a valid idea of the variety of its problems and the wealth of ideas it contains."

Hilbert is emphasizing the point which I am trying to make in this paper: ** Meaning is important in mathematics and geometry is an important source of that meaning**.

**References:**

Amir-Moez, A.R. (1963). A Paper of Omar Khayyam, **Scripta Mathematica**, **26**, 323-337.

Davis, P.J. & Hersh, R. (1981). **The Mathematical Experience**. Boston: Birkhäuser.

Eves, H. (1963).** A Survey of Geometry, Vol. 1**. Boston: Allyn and Bacon.

Heath, T.L. (1956). **The Thirteen Books of Euclid's Elements**. New York: Dover.

Heath, T.L. (1961). **Appolonios of Perga, Treatise on Conic Sections**. New York: Dover.

Hilbert, David, & Cohn-Vossen (1952). **Geometry and the Imagination**. New York: Chelsea.

Karpinski, L.C., editor (1915). **Robert of Chester's Latin Translation of the Algebra of al-Khowarizmi**. New York: Macmillan. (This is an English translation.)

Kasir, D.S., editor (1931). **The Algebra of Omar Khayyam**. New York: Columbia Teachers College.

Prakash (1968). **Baudhayana-Sulbasutram. **Bombay.

Seidenberg, A. (1961). The Ritual Origin of Geometry, **Archive for the History of the Exact Sciences**, **1**, 488-527.

Valens, E.G. (1976). **The Number of Things: Pythagoras, Geometry and Humming Strings**. New York: Dutton.

Witmer, T.R., editor (1968). **The Great Art or the Rules of Algebra by Girolano Cardano**. Cambridge: The MIT Press.