I want to bring together in this paper two figures from the history of mathematics—one from the ancient world and one from the modern. The first is the Greek, Euclid, and the second, the 19th century Swiss, Jacob Steiner. I have no intention of giving a complete picture of either man’s work (although I do hope readers will end up with their curiosity piqued, especially about the lesser known Steiner). What I do want to consider is a small instance in which both, in some sense, looked at the same geometrical facts. The latter are the theorems concerning the product of the segments of intersecting chords of a circle.

I say “in some sense, looked at the same geometrical facts” because, coming from different worlds, Euclid and Steiner brought to mathematics different perspectives, and that makes speaking about their looking at “the same geometrical fact” at least problematic. Indeed, Euclid’s treatment of these theorems in the Elements and Steiner’s transformation of them in his ‘power of a point’ bring into relief ancient and modern points of view. This brings me to the second part of my title and my second, but really principal, goal in this paper.

For it often happens that in the attempt to combine mathematics education and history of mathematics, the main lesson of the history of mathematics is lost, namely, that mathematics itself is an historical entity (see Fried, 2001 ). When teachers bring problems and mathematical ideas from the past into the classroom, they tend to speak about Roberval’s solution to this or Apollonius’ approach to that, as if the problems and ideas are eternal and only the solutions and approaches change. But to say that mathematics is historical is to say not only that its problems and ideas change but also what mathematics is and what it means to be mathematical.

It is in view of this, I want to consider the often-heard definition of mathematics as the “science of patterns.” Specifically, I want to show, by comparing Euclid and Steiner, that while this is presented to students as a timeless—that is, non-historical—definition, in fact, it represents a modern view of mathematics. I shall show that Greek mathematics, for example, is not a search for patterns but for concrete properties of concrete mathematical objects; and I shall show, conversely, that it is when mathematics becomes symbolic that patterns, as such, are suggested to mathematicians and become objects of their thought.

The example of Euclid and Steiner for this purpose is actually a rather subtle one, and so, before we begin, I should say a word as to why I chose it. There are two main reasons. First, in thinking about how modern mathematics is a science of patterns high school teachers do well to think about mathematics at the level they teach; in this way, an example from elementary geometry is better than one from, say, group theory, which in other respects would be ideal. Second, a subtle example shows how pattern-thinking lurks even where one does not expect.

I want to bring together in this paper two figures from the history of mathematics—one from the ancient world and one from the modern. The first is the Greek, Euclid, and the second, the 19th century Swiss, Jacob Steiner. I have no intention of giving a complete picture of either man’s work (although I do hope readers will end up with their curiosity piqued, especially about the lesser known Steiner). What I do want to consider is a small instance in which both, in some sense, looked at the same geometrical facts. The latter are the theorems concerning the product of the segments of intersecting chords of a circle.

I say “in some sense, looked at the same geometrical facts” because, coming from different worlds, Euclid and Steiner brought to mathematics different perspectives, and that makes speaking about their looking at “the same geometrical fact” at least problematic. Indeed, Euclid’s treatment of these theorems in the Elements and Steiner’s transformation of them in his ‘power of a point’ bring into relief ancient and modern points of view. This brings me to the second part of my title and my second, but really principal, goal in this paper.

For it often happens that in the attempt to combine mathematics education and history of mathematics, the main lesson of the history of mathematics is lost, namely, that mathematics itself is an historical entity (see Fried, 2001 ). When teachers bring problems and mathematical ideas from the past into the classroom, they tend to speak about Roberval’s solution to this or Apollonius’ approach to that, as if the problems and ideas are eternal and only the solutions and approaches change. But to say that mathematics is historical is to say not only that its problems and ideas change but also what mathematics is and what it means to be mathematical.

It is in view of this, I want to consider the often-heard definition of mathematics as the “science of patterns.” Specifically, I want to show, by comparing Euclid and Steiner, that while this is presented to students as a timeless—that is, non-historical—definition, in fact, it represents a modern view of mathematics. I shall show that Greek mathematics, for example, is not a search for patterns but for concrete properties of concrete mathematical objects; and I shall show, conversely, that it is when mathematics becomes symbolic that patterns, as such, are suggested to mathematicians and become objects of their thought.

The example of Euclid and Steiner for this purpose is actually a rather subtle one, and so, before we begin, I should say a word as to why I chose it. There are two main reasons. First, in thinking about how modern mathematics is a science of patterns high school teachers do well to think about mathematics at the level they teach; in this way, an example from elementary geometry is better than one from, say, group theory, which in other respects would be ideal. Second, a subtle example shows how pattern-thinking lurks even where one does not expect.

The characterization of mathematics as the “study of patterns” seems to have been first made by the British mathematician, G. H. Hardy. Lamenting his waning mathematical powers, Hardy, perhaps as a curative for his despair, wrote a small book on his life as a mathematician. Although the book was, indeed, an account of what it is to be a mathematician, it naturally could not escape also being an account of mathematics itself. Thus, when Hardy wrote in *A Mathematician’s Apology*,

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas” (Hardy, 1992 , p. 84)

he gave us something like a definition of mathematics, and a beautiful one at that!

Hardy may or may not have been the first to use the metaphor of patterns to describe the heart of mathematics, but he certainly was not the last. In recent years the most well known and often quoted statement to this effect is that of Lynne Steen, who referred to mathematics as the ‘science of patterns’ (Steen, 1988 ). Since then, the metaphor has become almost commonplace. One finds it in key documents in mathematics education, such as the *NCTM Principles and Standards* (NCTM, 2000 ), in books such as K. Devlin’s *Mathematics: The Science of Patterns* (Devlin, 1994 ), and in the classroom as well.

That it has become commonplace to call mathematics a science of patterns is probably a sign that there is something right about it. But what does it mean? Certainly, patterns are often the explicit subject of mathematics—sometimes even in the perfectly ordinary sense of the word, as in the study of ‘tilings’ and ‘wall-paper’ symmetries. Of course, the case may be made that the study of symmetry comprises a greater part of mathematics than might seem on first sight, but one hesitates to say that this is the reason it is right to call mathematics, in general, the science of patterns.

Why does this word ‘pattern’ seem so apt? No doubt because it connotes order, regularity, and lawfulness. Moreover, as the pattern, say, for a shirt is not cloth but the plan, scheme, or idea for a shirt, the word ‘pattern’ calls up the fact that, as one writer puts it (in a book called again *Mathematics as a Science of Patterns* (Resnik 1999)!), “…in mathematics the primary subject-matter is not the individual mathematical objects but rather the structures in which they are arranged” (Resnik, 1999 , p.201).

The view of mathematics contained in the last quotation did not arise all at once. A mathematics that looks at patterns rather than individual properties of individual mathematical objects was what Descartes’ sought in mathesis universalis, ‘universal mathematics’, which he associated with the then new subject of algebra. This ‘general science’, he said, existed “…to explain that element as a whole which gives rise to problems about order and measurement, restricted as these are to no special subject matter” (Descartes, 1970 , p.13).[i] What Descartes was suggesting, in other words, was that when one writes an expression like x^{2}-y^{2}=k one may look at it as a purely symbolic expression, an abstract pattern, to be manipulated and studied; one should not have to tie it to square figures whose sides have lengths x and y. Descartes said his algebraic approach was only a rediscovery of a mathematics secretly practiced by the Greeks; in fact mathematics itself was being reborn in a new form For this reason Felix Browder rightly points out that “From the 17th century on…a broader vision of mathematics arose in the minds of such intellectual innovators as Leibniz and Descartes, a vision of mathematics as the total science of intellectual order, as the science of pattern and structure” (Browder, 1975 , p.14)

We do not always appreciate how far the symbolic character of modern mathematics, which began to take shape in Descartes’ time, distinguishes modern mathematics from, for example, Greek mathematics. Greek mathematicians typically began with specific mathematical objects, such as a circle or a section of a cone, and then proved that those objects possess certain properties. They did not begin with some property and then find an object possessing it or a set of objects that could be related by it. For Greek mathematics was a non-algebraic mathematics (Klein, 1968; Grattan-Guinness, 1996; Fried & Unguru, 2001 ), and to begin with a property abstracted from any particular object is precisely what symbolic algebra allows us to do supremely well, indeed, what it is made for. Such abstracted properties are what we are looking for when we are looking for patterns. And this is what Hardy had in mind, surely, when he said the mathematician’s patterns “are made with ideas.” The symbolic nature of modern mathematics, then, is what allows mathematics to be a science of patterns, and it is now, indeed, a science of patterns; but because mathematics was not always symbolic we ought to take care and say that mathematics is the science of patterns because it has grown to be so. With that, let us turn to Euclid and Steiner. First, Euclid.

[i] Just a few lines before the quotation above, Descartes also says that “…all those matters only were referred to Mathematics in which order and measurement are investigated, and it makes no difference whether it be in numbers, figures, stars, sounds or any other object that the question of measurement arises.” In many ways, that sentiment encapsulates the modern view. Thus it is no accident that one hears strong echoes of it in this statement by Whitehead: “The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours” (Whitehead, 1964 , p.9).

Book III of Euclid’s *Elements* concerns the basic properties of circles, for example, that one can always find the center of a given circle (proposition 1); that a line through the center is perpendicular to a chord if and only if it bisects the chord (proposition 3); that two circles can intersect one another in at most two points (proposition 10); that the diameter is the greatest chord (proposition 15); that a line is tangent to a circle if and only if it is perpendicular to a radius through the point of contact (propositions 18, 19); that the sum of the opposite angles of an inscribed quadrilateral is equal to two right angles (proposition 22); that the angle in a semicircle is right (proposition 31). Proposition 35 is the proposition stated above, namely:

If in a circle two straight lines cut each other, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.

Proposition 36 tells us, in addition:

If a point is taken outside the circle, and from it two lines fall on the circle, one cutting the circle and the other tangent, then the rectangle contained by whole of the line cutting the circle and the part of it intercepted outside the circle between the point and the convex circumference will be equal to the square on the tangent.

Proposition 37, which ends the book, is the converse of Proposition 36, providing a criterion for concluding when a line from a point outside a circle will be tangent to the circle.

The demonstration of proposition 35, which I shall present in a moment, is well worth seeing since Euclid’s approach is different than the usual classroom approach via similarity; indeed, Euclid does not treat similarity at all until the sixth book of the Elements. Before that, though, the reader ought to know why I go to pains to avoid the usual “product of the lengths of two segments” and insist on saying “the rectangle contained by two segments.” First of all, this is the way Euclid says it. And if one is dealing with history, one ought to be sensitive to the way things are put. Second, Euclid really means “the rectangle contained by two segments”; for Euclid, multiplication (*pollaplēsios* in Greek) is reserved for numbers, and ‘numbers’, for him, means only natural numbers. That a rectangle, for Euclid, is a rectangle and square a square is crucial when one considers propositions such as this from Book II of the Elements:

If a straight line be cut into equal and unequal segments the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half (Book II, proposition 5).

Thus if AB is bisected at C and divided again at D (see fig. 1), then Euclid says the rectangle contained by AD and DB (which I shall abbreviate hereafter as rect.AD,DB) together with the square on CD (which I shall abbreviate hereafter sq.CD) is equal to sq. CB. With CZ being the square built on CB, BE being joined, and KM and DH being drawn parallel to AB and BZ, respectively, Euclid must show that rectangle AQ together with the square LH is equal to the square CZ; once one realizes that AQ is equal to the figure CBZHQL, the proof becomes, with all the squares and rectangles in plain view, almost a ‘proof-without-words’.

Taking AC=a and CD=b, this proposition has been understood in the past to show, in geometric language, the algebraic identity (a+b)(a-b)+b^{2}=a^{2}, or, in its more familiar form, (a+b)(a-b)=a^{2}-b^{2}. For someone who already knows algebra and knows its importance in modern mathematics, this is a very seductive interpretation. The problem is that while the interpretation makes sense mathematically it really does not hold water historically, as I described above (see Fried & Unguru, 2001 for a discussion of this proposition in particular). But let us leave that issue and return to Euclid’s proof (slightly paraphrased) of III.35, which, incidentally, relies on the proposition just cited!

Let circle ABCD be given and let chords AC and BD meet at E (see fig. 2). We need to show that the rect.DE,EB equals rect.AE,EC. From the center Z, draw ZH and ZQ perpendicular to AC and BD (thus, also, H bisects AC and Q bisects BD), respectively, and let ZE, ZB, and ZC be joined.

Thus, by the theorem quoted from Book II, rect.AE,EC together with sq.HE equals sq.HC. Therefore, also, rect.AE,EC together with sq.HE and sq.ZH equals sq.HC and sq.ZH. But, sq.HE and sq.ZH equals sq.ZE, while sq.HC and sq.ZH equals sq.ZC, by the ‘Pythagorean Theorem’ (proposition 47 in Book I of the Elements). So, rect.AE,EC together with sq.ZE equals sq.ZC. Similarly, rect.DE,EB together with sq.ZE equals sq.ZB. But, ZC is equal to ZB because they are radii. Therefore rect.AE,EC together with sq.ZE equals rect.DE,EB together with sq.ZE, so that, rect.AE,EC equals rect.DE,EB.

Geometry lessons usually include also the complement to propositions 35 and 36, namely, that if two lines from a point outside a circle cut the circle then the rectangle contained by the whole of one of the lines and its exterior segment is equal to the rectangle contained by the whole of the other line and its exterior segment (call it 36*). This proposition, however, is not found in the Elements. The reason, presumably, is simply that the proposition follows from proposition 36 almost immediately; Euclid does not have to spell out explicitly every property of circles that can be easily deduced from the main propositions.

But is this not the important point? Euclid is interested in finding properties of geometrical objects, not patterns they manifest. A pattern in propositions 35, 36, 36* emerges particularly clearly when one gets over what Descartes’ called the “scruple that kept the ancients from using arithmetical terms in geometry” (*La Géométrie* , p.305) and writes these propositions, as we do now and Euclid did not do, in terms of products. Thus, if AC and BD are chords of a circle meeting at a point E (see fig. 3), it will always be the case that AExEC=BExED, even where B and D are the same (i.e. when EB is tangent to the circle).

One with an algebraic eye can spot a slightly different, and even more compelling, pattern in Euclid’s own proof. It is where, in the course of the proof, Euclid shows that rect.AE,EC together with sq.ZE equals sq.ZC (see fig. 2): let ZE, which is the distance between the center of the circle and the point E where the chords meet, be d, and let ZC, which is the radius of the circle, be r. Then, AExEC + d^{2}= r^{2} or AExEC = r^{2}- d^{2}. Similarly, had we gone over the proof for proposition 36* where E is outside the circle, we would have found that AExEC + r^{2}= d^{2} or. Thus, AExEC = r^{2}- d^{2} if E is inside the circle, AExEC= d^{2}- r^{2} if E is outside the circle, and, obviously, AE x EC=0 if E is on the circle (for then d=r). Put even more succinctly, if AC is any chord of a circle through a point E (possibly on the circumference of the circle so that C and E or A and E coincide), then AE x EC=|r^{2}-d^{2}|. This brings us to Jacob Steiner.

Jacob Steiner lived from 1796 until 1863. He was a fascinating figure in the history of mathematics not only because of the depth and originality of his geometrical work but also because of his unique educational background. For he was born to a poor peasant family that could hardly afford to send him to school; he could not even write before the age of fourteen! Luck came, however, in the form of the great Swiss educational reformer, Johann Heinrich Pestalozzi, who discovered Steiner, and, in 1814, enrolled him in his school at Yverdon. Later, in an application to the Prussian Ministry of Education written in 1826, Steiner credited Pestalozzi’s methods in forming his general approach to mathematics, his desire to find “the deeper bases” of mathematical theorems (see Burckhardt, 1970 ).

In 1826, the same year he wrote the application just mentioned, he also wrote a long article entitled “Einige geometrische Betrachtungen” (“A Few Geometrical Observations”) (Steiner, 1971 , I, pp.17-76). It is in this work that Steiner defines the "power of a point." To do this, Steiner refers to the Euclidean propositions discussed above but shifts the focus from the chords AC and BD to the point E (see fig. 4).[i] Since the product AExEC for any chord AC through E is the constant value |r^{2}-d^{2}|, Steiner defines the "power of a point (*Potenz des Puncts*) E with respect to a given circle" to be this invariant number. Incidentally, one ought to note that when the point E is outside the circle, the power of E is just the square of the tangent from E.

The shift from the chords in a circle to a point is more significant than it might seem at first. The power of a point is not a property of a point, for, unlike chords in a circle, points in geometry really have no properties; the power of a point is a relation at a point with respect to a circle and having the form A^{2}-B^{2}=constant; it is, indeed, the recognition of a certain pattern. Accordingly, Steiner precedes the definition of the power of a point with a geometrical locus having, ostensibly, nothing to do with circles—rather, a locus connected to the form A^{2}-B^{2}=constant. He notes that if Mm is a line segment containing the point G and PG is perpendicular to Mm at G, then if P is any point on PG, we have MP^{2}-mP^{2}=MG^{2}-mG^{2}, which is constant since M, m, and G are fixed. (Fig. 5) The proof follows immediately by the Pythagorean theorem: MP^{2}-MG^{2} =PG^{2} and PG^{2}=mP^{2}-mG^{2}, so MP^{2}-MG^{2}= mP^{2}-mG^{2}, or MP^{2}-mP^{2}=MG^{2}-mG^{2}.

Since the converse also follows easily, Steiner can state that the locus of points whose distances D and d from two fixed points M and m satisfy the relation D^{2}-d^{2}=constant lies along a straight line perpendicular to the line Mm. Following this and the definition of the power of a point, Steiner develops these ideas in a series of beautiful theorems and constructions that fully justify Hardy’s statement that “A mathematician, like a painter or a poet, is a maker of patterns.”

[i] It is interesting to note at this point that when Descartes’ younger contemporary, Spinoza, wanted to show that an infinity of things (existent and non-existent) are comprehended in the idea of God, he used Elements III.35 as an illustration of his argument (Ethics, Part 2, prop. viii). Spinoza supplied no diagram, but, it seems clear that a diagram like figure 4 is what he had in mind. (The connection between my discussion and this part of Spinoza’s Ethics was pointed out to me by Joseph Cohen of St. John’s College in Annapolis, Maryland).

To start, Steiner asks what is the locus of points having the same power with respect to two given circles? Let the centers and radii of the circles be M and m and R and r, respectively. Then we are looking for the set of points P satisfying the relation, MP^{2}-R^{2}=mp^{2}-r^{2 } or R^{2}-MP^{2}=r^{2}-mP^{2} (see fig. 6).

In either case, this means that the points P satisfy the relation MP^{2}-mP^{2}=R^{2}-r^{2}, which we have just seen is a line perpendicular to the line Mm, the line joining the centers of the circles! This ‘line of equal powers’, as Steiner called it, is also known as the ‘radical axis’ of the two circles. When the circles intersect, the radical axis is particularly easy to find, for the power of the points intersection are 0 with respect to both circles; therefore, the radical axis is the common chord of the two circles (and, of course, it follows immediately, that that line is perpendicular to the line joining the centers of the circles). Similarly, if the circles are tangent the radical axis has to be the tangent line. The various cases are shown in the figure below (fig. 7):

In the cases where the radical axis lies outside the two circles it is clear that the axis can be given another interpretation, namely, the locus of all points from which the tangents to the two circles are equal since the power of a point P with respect to a circle equals the square of the tangent to the circle from P.

From here, Steiner moves on to three circles. Let the centers of the circles, which we shall assume do not lie along a line, be M1, M2, M3, and let the radical axis of circles 1 and 2 be denoted *l*(12), of circles 2 and 3, *l*(23), and of circles 1 and 3, *l*(13) (these are all Steiner’s notations) (see fig. 8). Suppose *l*(12) and *l*(23) meet at point p(123). Then the power of *p*(123) is the same with respect to circles 1 and 2 and also with respect to 2 and 3; therefore, the power of *p*(123) with respect to circles 1 and 3 must be the same, so that *p* (123) must also lie on *l* (13).

In other words, given three circles whose centers do not all lie on a line, the radical axes all pass through one point. That point is also known as the radical center of the three circles. That there is a radical center means, among other things, that 1) if three circles intersect pair-wise then the three common chords intersect at a point (see fig. 9a), 2) if three circles are tangent pair-wise then the three tangents meet at a point and are equal (see fig. 9b), and, similarly, 3) if three circles are all non-intersecting then the three tangents from the radical center to the three circles are equal (see fig. 9c). It is clear, moreover, that a circle is orthogonal to three given circles, its center will be the radical center of the three circles and its radius the length of the equal tangents.

None of these theorems which Steiner demonstrates is immediate without the idea of the ‘power of a point’, but all are almost obvious with it. How it makes these things obvious is not by supplying some previously unknown property of some geometrical object, but by supplying a kind of organizational principle, a pattern to look for, something providing “scientific unity and coherence,” as Steiner says in another context. Thus, the comparison between Euclid and Steiner makes it clear that the difference between them is not so much knowledge as it is perspective and how they perceive what it is they are doing when they do mathematics. Both seem to be concerned with circles, but, in fact, while Euclid looks at circles as objects with properties, Steiner looks at circles as the carriers of patterns. The ability to take a pattern as a starting point, even if one has a definite object in view, placed Steiner and moderns like him in a conceptual camp quite different from that of Euclid—in fact, one might say that if truth is a great ocean, as Newton put it, surely Euclid and Steiner stood on opposite shores. But where should we mathematics teachers stand, especially those of us who consider the history of mathematics important? I am afraid we have no other choice—we are all moderns. Of course there is no shame in that: the approach to mathematics via patterns has proven very powerful, and mathematics teachers, therefore, do well by helping students see this strength of modern mathematics. But they ought to make it clear also how far the search for pattern is indeed distinct of modern mathematics, that mathematicians have not always approached their subject in this way. By making the effort—and it is not an easy one—to see how mathematics has been different in the past, teachers will not only show that mathematics can change its way of seeing things, but also that their students may be the ones to change it.

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