Geometric Solutions of Quadratic and Cubic Equations


David W. Henderson1

Department of Mathematics, Cornell University

Ithaca, NY, 14853-7901, USA1

I am ready to lead you, the reader, on a path through part of the forest of mathematics - a path that has delighted me many times - and surprised me. Every time I walk along it I see something I had not seen before. We will bring with us the question: What are square roots? We will find what is one of the oldest written mathematical proofs, still very much alive, right along side some new results never before published.

These will be combined to solve quadratic equations by "completing the square" - a real square. These in turn lead to conic sections and cube roots and culminating in the beautiful general method from Omar al'Khayyam, the Persian geometer, philosopher, poet, which can be used to find all the real roots of cubic equations. Along the way we shall clearly see some of the ancestral forms of our modern Cartesian coordinates and analytic geometry. I will point our several inaccuracies and misconceptions that have crept in to the modern historical accounts of these matters. But I urge you to not look at this only for its historical interest but rather look for the meaning it has in our current-day understanding of mathematics. This path is not through a dead museum or petrified forest, this path passes through ideas which are very much alive and which have something to say to our modern technological, increasingly numerical, world.

1. The Beginning of the Path

For me the path started in eighth grade when I asked my teacher - "What is the square root?" I knew that the square root of  N was a number whose square was equal to N but where can I find it? (Hidden in that question is "How do I know it always exists?") I knew what the square roots of 4 and 9 were - no problem there.

I even knew that Ö2 was the length of the diagonal of a unit square, but what of Ö2.5 or Öp  ? At first the teacher showed me a Square Root Table (a table of numerical square roots), but I soon discovered that if I took the number listed in the table as Ö2 and squared it I got 1.999396 not 2. (Modern-day pocket calculators give rise to the same problem.) So I persisted asking my question - What is the square root? Then the teacher answered by giving me THE ANSWER - the Square Root Algorithm. Do you remember the Square Root Algorithm - that procedure, similar to long division, by which it is possible to calculate the square root? Or perhaps more recently you were taught the "Divide and Average" Method which goes like this: If A1 is an approximation of ÖN then the average of A1 and N/A1 is an even better approximation which we could call A2 . And then the next approximation A3 is the average of A2 and N/A2 . In equation form this becomes An+1 = (1/2)(An+(N/An)) . For example, if A1 = 1.5 is an approximation of Ö2 , then A2 = 1.417··· , A3 = 1.414216··· and so forth are better and better approximation. But wait! Most of the time these algorithms do not calculate the square root - they only calculate approximations to the square root. The algorithms have an advantage over the tables because I could, at least in theory, calculate approximations as close as I wished. However they are still only approximations and my question still remained - What is this square root which these algorithms approximate?

My eighth grade teacher then gave up, but later in college I found out that modern mathematics answers: "We make an assumption (The Completeness Axiom) which implies that the sequence of approximations from the Square Root Algorithm must converge to some real number." And, when I continued to ask my question, I found that in modern mathematics the square root is a certain equivalence class of Cauchy sequences of rational numbers or a certain Dedekind cut. I then let go of my question and forgot it in the turmoil of graduate school, writing my thesis and beginning my mathematical career.

Later, I started teaching a geometry course for mathematics majors and one of the topics was Dissection Theory which leads (among other things) to the result that every polygonal region in the plane can be cut up (dissected) into a finite number of pieces which can then be rearranged to form a square. In this case we say that the polygonal region is equivalent by dissection to a square. A preliminary step to the general result is the:

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

I presented to the class the following proof which I found slightly modified in a standard geometry text book, Eves (1963):

"Let s = Öab be the side of the square equivalent to the rectangle with sides a and b. Place the square, AEFH, on the rectangle, ABCD, as shown in [the figure].Draw ED to cut BC in R and HF in K. Let BC cut HF in G. From the similar triangles KDH and EDA we have HK/AE = HD/AD, or

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

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

(In case that ABCD is so long and skinny that K ends up between G and F we can, by cutting ABCD in half and stacking the halves, reduce the proof to the above case.)

I was satisfied with the proof until in the second year of the course when I started sensing student uneasiness with the proof. As I listened to their uneasiness there started to come up the question - What is Öab ? How do you find it ? -- Oh, yes, I remember -- that used to be my question!

The students and I also noticed that the facts used about similar triangles in the above proof are usually proved using the theory of areas of triangles and thus that this proof could not be used as part of a concrete theory of areas of polygons, which was our purpose in studying Dissection Theory in the first place.

That started me off on an exploration which continued on and off over many years. Some of what I found I will now show you (but in a different order from the order I first saw then).

2. What is a Square Root?

While reading an article about something else I ran across an item that said that the problem of changing a rectangle into a square appeared in the Sulbasutram by Baudhayana (see Prakash (1968)). "Sulbasutram" means "Rules of the cord" and is an ancient (at least 600 BC) book written in Sanskrit as a handbook for people who were building altars and temples. Most of the book gives detailed instructions on temple construction and design, but the first chapter is a geometry textbook which contains geometric statements called "Sutra". Sutra 54 is: (Here "oblong" means "rectangle".)

"If you wish to turn an oblong into a square, take the tiryanmani, i.e. the shorter side of the oblong for the side of square. Divide the remainder (that part of the oblong which remains after the square has been cut off) into two parts and inverting (their places) join those two parts to two sides of the square. (We get thus a large square out of one corner of which a small square is cut out as it were.) Fill the empty place (in the corner) by adding a piece (a small square). It has been taught how to deduct it (the added piece).

"By adding the small square in the corner we get a large square which is equal to the oblong plus the small square, therefore we must deduct the small square from the large square (see Sutra 51) and then we have as remainder a square which is equal to the oblong."

Here is a diagram for Sutra 54:

So our rectangle has been changed into a large square from which a small square has been removed (or deducted). Now Sutra 51:

"If you wish to deduct one square from another Square, cut off a piece from the larger square by making a mark on the ground with the side of the smaller square which you wish to deduct (the process is the same as that described in Sutra 50; an oblong is cut off, the sides of which are equal to the sides of the two given squares); draw one of the sides (THE CORD REPRESENTING one OF THE longer SIDES of the oblong) across the oblong so that it touches the other side; where it touches (the other side), by this line which has been cut off the small square is deducted from the large one (i.e. the cutoff line is the side of a square the area of which is equal to the difference of the two squares.)"

This last assertion follows from sutra 50:

"If you wish to combine two squares of different size into one, scratch up with the side of the smaller square a piece cutoff from the larger one (i.e. cut off a piece from the larger square by scratching up the ground - or making a mark upon the ground - at a distance from one end of a side of the large square, which is equal to the length of the side of the smaller square; and by repeating this process on the opposite side of the larger square and joining the two marks on the ground by a line or cord, an oblong is cut off, of which the two longer sides are equal to the side of the large square). The diagonal of this cutoff piece is the side of the combined squares (of the square which combines the two squares)."

Does sutra 50 sound familiar? It should - it is a clear statement of what we call the Pythagorean Theorem, written before Pythagoras was born!

A. Seidenberg (1961) in an article entitled The Ritual Origin of Geometry gives a detailed discussion of the significance of the Sulbasutram. He argues that it was written before 600 BC (Pythagoras lived about 500 BC and Euclid about 300 BC) He gives evidence to support his claim that it contains codification of knowledge going "far back of 1700 BC" and that this knowledge was the common source of Indian, Egyptian, Babylonian and Greek mathematics. Combined together sutras 50, 51 and 54 describe a construction of a square with the same area as a given rectangle (oblong) and a proof (based on the Pythagorean Theorem) that this construction is correct. You can find stated in many books and articles that the ancient Hindus, in general, and the Sulbasutram in particular, did not have proofs or demonstrations or they are dismissed as being "rare". I suggest you decide for yourself.

Baudhayana avoids the Completeness Axiom by giving an explicit construction of the side of the square. The construction can be summarized in the diagram:

This is the same as Euclid's construction in Proposition II - 14 (see Heath (1956), page 409). But Euclid's proof is much more complicated.

Note that neither Baudhayana nor Euclid give a proof of Theorem 1 because the use of the Pythagorean Theorem obscures the dissection. However, they do give a concrete construction and a proof that the construction works. In addition, if supplemented with a dissection proof of the Pythagorean Theorem such as in Theorem 3, below, both Baudhayana's and Euclid methods prove (without using completeness):

Theorem 2: For every rectangle R there are squares S1 and S2 such that R + S2 is equivalent by dissection to S1 + S2 and thus R and S1 have the same area.

Theorem 3: (Dissection version of Pythagorean Theorem). In any right triangle, the union of the squares on the two sides is equivalent by dissection to the square on the hypotenuse.

Proof of Theorem 3 known to the ancient Chinese:

About a dozen different (and correct) proofs of Theorems 1 have been found by the students in my geometry course. One particularly clear one follows: (As far as I know this proof has never before been published.)

Let ABGH be the rectangle and extend the line AB to C so that BC @ BH. Draw the semicircle S on AC and let D be the intersection of S with the extension of BH.

Then D ADC is a right angle and the angles are congruent as indicated in the diagram. Construct the square DBEF. Then D DBC @ D IGA and D DBC @ D  DFJ both by Angle-Side-Angle. Easily D AEJ @ D IHD. Thus the rectangle ABHG is equivalent by dissection to the square DBEF.

Notice that this proof avoids assuming that the square root exists (and thus avoids the Completeness Axiom) and avoids using any facts about similar triangles. The proof explicitly constructs the square and shows in an elementary way that its area is the same as the area of the rectangle. There is no need for the area or the sides of the rectangle to be expressed in numbers. Also given a real number, b, the square root of b can be constructed by using a rectangle with sides b and 1.

So, finally, I have an answer to my question - What is a square root? I say "an answer" because every year I see more or see it from a different point of view.

3. Quadratic Equations

Finding square roots is the simplest case of solving quadratic equations. If you look in some history of mathematics books, you will find that quadratic equations were extensively solved by the Babylonians (numerically) and by the Greeks (geometrically). However, the earliest known general discussion of quadratic equations took place between 800 and 1100 AD in the Muslim Empire. Best known are Mohammed Ibn Musa al'Khowarizmi (who lived in Baghdad) and Omar al'Khayyam (who lived in Persia, now Iran, and is mostly known in the West for his poetry The Rubaiyat. Both wrote books entitled Al-jabr W'al mugabalah, al'Khowarizmi about 820 AD and al'Khayyam about 1100 AD. From al'Khowarizmi we get our word "algorithm" and from the title of their books our word "algebra". An English translation of both books is available in many libraries, if you can figure out whose name it is catalogued under (see References, Karpinski (1915) and Kasir (1931)).

In these books you find geometric and numerical solutions to quadratic equations and geometric proofs of these solutions. But the first thing that you notice is that there is not one general quadratic equation as we are used to it: 2ax + bx + c = 0. Rather, because the use of negative coefficients and negative roots was avoided, they list six types of quadratic equations (we follow al'Khayyam's lead and set the leading coefficient equal to 1):

  1. x = c, which needs no solution,
  2. x = bx, which is easily solved,
  3. x2 = c, which has root x = Öc ,
  4. x2 + bx = c, with root x = Ö[(b/2)2 + c] - b/2,
  5. x2 + c = bx, with roots x = b/2 ± Ö[(b/2)2 - c] , if c < (b/2)2 , and
  6. x2 = bx + c, with root x = b/2 + Ö[(b/2) + c] .
Here b and c are always positive numbers or a geometric length (b) and area (c). These types are the only possibilities with positive coefficients and positive roots. ( x2 + bx + c = 0 has no positive roots.)

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 -r (r, positive) is the negative root of x2 + bx = c. Then (-r)2 + b(-r) = c or r2 = br + c. Thus r is a positive root of x2 = bx + c ! The absolute value of the negative root of x2 + bx = c is the positive root of x2 = bx + c and vice versa. Also, the absolute values of the negative roots of x2 + bx + c = 0 are the positive roots of  x2 + c = bx. So, in this sense, Yes, the above geometric solutions do find all the real roots of all quadratic equations. Thus it is misleading to state, as most historical accounts do, that the geometric methods failed the find negative roots. The users of these methods did not find negative roots because they did not conceive of them. However, the methods can be easily and directly used to find all the negative roots.

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 c3 = a2b. Now setting a = 1, we see that we can find the cube root of b, if we can find c and d such that c2 = d and d2 = 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 C are (BE,BD) this last equation becomes a familiar equation for a parabola.

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 b be a positive number or length and let AB = b and construct C so that CB is perpendicular to AB and such that CB = 1.

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 d2 = cb and c2 = d. Thus c3 = 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+ ax = bx reduces to x2 + 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, x3 = b. I will then give al'Khayyam's solutions to these types.

In the cubic y3 + py2 + 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 x3 + 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) x3 + ax = b, (II) x3 + b = ax, (III) x3 = ax + b, and (IV) x3 + 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): x3 + 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, [ ]. The construction has been shown previously. Let BC be the height of the solid. [I.e. BC·(AB)2 = b.] Let BC be perpendicular to AB ... Construct a parabola whose vertex is the point B ... and parameter AB. Then the position of the conic HBD will be tangent to BC. Describe on BC a semicircle. It necessarily intersects the conic. Let the point of intersection be D; drop from D, whose position is known, two perpendiculars DZ and DE on BZ and BC. Both the position and magnitude of these lines are known."

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 ,


EB·(BZ)2 = (EB)2·EC = BZ·AB·EC

and therefore

AB·EC = EB·BZ and (EB)3 = EB·(BZ·AB) = (AB·ECAB = (AB)2·EC;


(EB)3 + a(EB) = (AB)2·EC + (AB)2·(EB) = (AB)2·CB = b.

Thus EB is a root of x3 + ax = b. Since x2 + ax increases as x increases, there can be only this one root.

Al'Khayyam's solutions for types (II) and (III): x3 + b = ax and x3 = ax + b.

Al'Khayyam treated these equations separately but by allowing negative horizontal lengths we can combine his two solutions into one solution of x3 ± 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 x3 = 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) x3 + ax = b, (II) x3 + b = ax, (III) x3 = ax + b, and (IV) x3 + ax + b = 0.

If we allow ourselves the convenience of using negative numbers and lengths then we can reduce these to one type: x3 + 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 x3 + ax + b = 0 of the form x = t1/3 + u1/3 . Plugging this into the cubic we get

(t1/3 + u1/3)3 + a(t1/3 + u1/3) + b = 0.

If you expand and simplify this you get to

t + u + b + (3t1/3u1/3 + a)(t1/3 + u1/3) = 0.

Thus x = t1/3 + u1/3 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 z2 + 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 = t1/3 + u1/3 = {-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= 15x + 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= 15x + 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 x3 = 15x + 4.) Faced with Cardano's Formula and equations like x3 = 15x + 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:

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.


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1 1 This paper was written while I was a visiting member of the faculty at Birzeit University, a Palestinian university in the Israeli-occupied West Bank. I appreciate the hospitality and support given me by the students, faculty and staff during my visit.