by Bruce Director

Among the most interesting and provocative investigations of the thinkers of the ancient Greek speaking world, were problems concerning the construction, with straight edge and compass, of certain geometrical figures; specifically, the doubling of the cube, the trisection of the angle, the construction of the regular heptagon, and the quadrature of the circle. In most of the modern English language sources on the subject, these problems are generally portrayed as a certain type of puzzle, or brain teaser. Lacking in virtually all of this scholarship, is any conception of what these ancient Greek scientists were actually investigating. To answer the latter question, we need not hunt for some long-lost text, in which the deeper implications of these investigations are explicated. Rather, we need only to relive the discoveries ourselves, and, in the mirror of our own mind, those deeper implications will be reflected.

Instead of wasting time with today's academics, let us take as our guide Johannes Kepler, whose new and original discoveries arose from his own re-working of these investigations of ancient Greece. In the first book of the Harmony of the World, "The Construction of Regular Figures," Kepler presents some of the results of his re-discovery. Pertinent to this discussion, he provides the following definitions:

VII. "In geometrical matters, to know is to measure by a known measure, which known measure in our present concern, the inscription of figures in a circle, is the diameter of the circle."

VIII. "A quantity is said to be knowable if it is either itself immediately measurable by the diameter, if it is a line; or by its [the diameter's] square if a surface: or the quantity in question is at least formed from quantities such that by some definite geometrical connection, in some series [of operations] however long, they at last depend upon the diameter or its square. The Greek word for this is `gnorimon.'

IX. "The construction of a quantity which is either to be described or to be known is its deduction from the diameter, by permitted means, in Greek [these are called] `porima.'

"So construction generally yields either description or knowledge. But description declares mere quantity, whereas knowledge also in addition declares quality or a definite quantity. Now a line can be geometrically determined, in Greek, `takah,' even though its quality is not yet known intellectually. On the other hand, a line or lines may be known qualitatively, but that does not yet determine them or make them determinate, that is to say if their quality is common to many other things which are different in quantity. So for such lines description is easy, knowledge very difficult. Finally, many things can be described by some Geometrical means or other; but cannot be knowable by their nature: as knowledge has been defined above."

With these concepts in mind, take a first look at one of the classical Greek problems, the trisection of the angle. In proposition #46 of the same book Kepler restates this problem as:

"The division of any arc of a circle into three, five, seven, and so on, equal parts, and in any ratio which is not obtainable by repeated doubling from the ones which have been shown above, cannot be carried out in a Geometrical manner which produces knowledge."

His demonstration goes like this: To bisect an arc of a circle, we first bisect the chord drawn between the two ends of the arc. A line drawn from the center of the circle, through that point, will also bisect the arc. On the other hand, if we want to trisect the arc, the situation becomes much more ambiguous. Draw an arc and its chord, and label the end points A and B. Construct points P and Q on the chord, such that A-P = P-Q = Q-B. This trisects the chord. If we now draw lines from the center of the circle through P and Q, that intersect the arc at P' and Q', the arcs A-P' and Q'-B will be smaller than the middle arc P'-Q'. On the other hand, if we draw lines perpendicular to the chord, through P and Q, the result will be that arc P'-Q' will be larger than the arcs A-P' and Q'-B. Therefore, to trisect the arc, we have to draw lines through P and Q from a point that is somewhere between the center of the circle and infinity. Kepler shows that the position of this point gets farther from the center, as the arc gets smaller. But, this relationship is not proportional. That is, decreasing the arc by a given amount, does not change the position of the point by a proportional distance.

This is another manifestation of the phenomenon of non- linearity of circular action demonstrated three weeks ago with respect to the sine and cosine. One can illustrate that principle with the following experiment:

Draw a large circle on a black board. Draw two perpendicular diameters, one vertical and one horizontal. Get a string with a weight on it. Hold one end the string with your finger at the intersection of the horizontal diameter and the circumference, and let the weight hang down towards the floor. Now, move the end of the string with your finger along the circumference of the circle. The weight will rise, and the string will form a chord of the circle that intersects the horizontal diameter. Now watch the intersection of the string and the diameter, as you move the end of the string around the circumference of the circle. What is the relationship between the circular action of your finger, and the rectilinear movement of the point of intersection of the string with the diameter? The constant curvature of the circle, produces a non-constant motion of this point.

Back to the trisection of the angle. What Kepler's demonstration reveals, is a kind of boundary condition with respect to the divisions of a circular arc. Dividing a circular arc in half, or into powers of two, does not produce an immediate discontinuity between the circular arc and the straight line chord defined by it. But when we try to divide by three, division of the line and the arc diverge.

Through Kepler we have now re-discovered this problem in the form confronted in 5th-4th Century B.C. One attempted solution was devised by Hippias of Elis<fn1>, who is credited with producing the first non-circular curve, today called the quadratrix. This curve was later investigated by Leibniz, Huygens, Bernoulli, et al., from the higher standpoint of that Leibniz developed out of Kepler's discoveries.

The quadratrix of Hippias is generated as follows. Draw a square. Label the corners clockwise from the upper left hand corner A, B, C, D. Now imagine side A-D rotating clockwise around point D. As this segment rotates, point A will trace a quarter of a circle from A to C. Now imagine that as this line is rotating, side A-B, moves down the square to side D-C, at the same rate as side A-D is rotating. Side A-B remains parallel to side D-C as it moves. The quadratrix is the curve traced out by the intersection of these two lines as they move. Thus, the quadratrix is the intersection of circular rotation and linear motion.

Because of the way the quadratrix is constructed, Hippias used it to trisect the angle. That is, since both sides move at the same rate, when A-D has rotated 1/2 way, side A-B has moved down 1/2 way. Similarly, when A-D has rotated 1/3 of the way, side A-B has moved 1/3 of the way down. And so on for any other division.

To trisect an angle, mark off any angle with vertex at D, such that one side of the angle will be D-C, and the other side of the angle will intersect the quadratrix at some point K and the arc A-C at L. Draw a line parallel to the side D-C through K. That line will intersect side A-D at some point E. Now, find the point E' that divides D-E by one third. Draw a line from E' that is parallel to side D-C. This line will intersect the quadratrix at some point K'. Connect K' to D and the angle formed with side D-C will be one third of the original angle.

But, has Hippias constructed a means for trisection that is a "knowable" quantity as Kepler re-stated the problem? We could construct an mechanical apparatus to draw a quadratrix, but is this curve "knowable," that is, constructable by the circle and it's diameter?

Ah, there's the rub! To construct the quadratrix by "knowable" means, Hippias proposed the following:

First draw the perpendicular bi-sector of side D-C. Then draw the bisector of the angle C-D-A. The intersection of these two lines is a point on the quadratrix. Then bisect the two new segments of side D-C and the two new angles formed by bisecting C-D-A. These intersections will define two more points on the quadratrix. This process can be repeated again and again, to fill in so many points on the quadratrix, that by connecting the dots, the quadratrix can be drawn.

But, wait! All these points were determined by division by two, and so they will only precisely determine points on the quadratrix that intersect lines that divide angle C-D-A by powers of two. None of these dots will precisely determine a point on the quadratrix that trisects an angle. Those points, will always lie on the indeterminate, "filled" in parts of the curve. The "non-knowable" parts.

We seem to have hit a stumbling block. The boundary between division of a circular arc by two and division by three has re- emerged. Think about this a while. The closed door we've seem to run into, is, perhaps, an open hallway, through which our predecessors have strolled.

1. This Hippias is the subject of Plato's dialogues Hippias Major and Hippias Minor. He is also mentioned in the Apology. He was apparently a traveling philosopher, with some facility in mathematics.

by Bruce Director

This week we look at another classical Greek problem as reported by Theon of Smyrna:

In his work entitled {Platonicus} Eratosthenes says that, when the god announced to the Delians by oracle that to get rid of a plague they must construct an altar double of the existing one, their craftsmen fell into great perplexity in trying to find how a solid could be made double of another solid, and they went to ask Plato about it. He told them that the god had given this oracle, not because he wanted an altar of double the size, but because he wished, in setting this task before them, to reproach the Greeks for their neglect of mathematics and their contempt for geometry.

In future weeks we will re-live this problem. For now think about it.

by Bruce Director

According to Eratosthenes, Plato took great pleasure at the prospect, that the cognitive capacity of his fellow Greeks might be improved, when, having asked the gods for help, the gods answered with a question, that required the Greeks to think. It might improve our own cognitive functioning, and make us smile, to find out why Plato was so delighted.

Before taking up this problem directly, let's first look at the Pythagorean investigation of the doubling of the square. Plato discussed these investigations in the famous Meno dialogue. We can reconstruct an essential feature of that discovery by the following means:

Draw a square. Double its side and draw the square on the doubled side. Repeat this process several times. If the area of the first square is considered 1, then the subsequent areas are 1, 4, 16, 32 ... etc. The sides of the corresponding squares are 1, 2, 4, 8, 16, ...

Now draw another square. Draw the diagonal. Draw a square on the diagonal. Draw the diagonal of the new square. Draw a third square on the diagonal of the second square. Continue this several more times.

You should see a series of squares and diagonals in a spiral formation. If the area of the first square is considered to be 1, then the subsequent squares have the areas, 1, 2, 4, 8 ..., respectively. The area of each square in this series is in the same proportion to the one proceeding it, as to the one succeeding it. That is, 1:2::2:4::4:8::8:16 ...

Notice that the squares produced by doubling the sides are every other one, of the series of squares produced from the diagonals. The squares of doubled area, are in between the squares of doubled sides. If we think of the curvature of the whole series, the squares of double areas are found in a smaller interval of action, than the squares of doubled sides. The proportionality between the terms of the series remain the same, but something completely new emerges in the smaller interval of action, to wit: doubled areas.

(It should be particularly thought provoking, to think of the curvature in the small, of a discontinuous series!)

Now look at an even smaller interval of action; the interval between two squares. That interval includes, the side of one of the squares, the diagonal, and the side of the succeeding square. Here again, the principle of proportionality remains the same, but a new singularity emerges -- the incommensurability of the side to the diagonal of the square. (For a more complete discussion of incommensurability, see "Incommensurability and Analysis Situs" Parts 1 and 2; NF Vol. XI #22 6/9/97 97237jbt101 and NF Vol. XI #23 6/16/97 (7267jbt101).

In Plato's Theatetus dialogue, Theatetus reports on an investigation of a still smaller interval of action of this same curvature, by Theodorus of Cyrene, a Pythagorean who was one of Plato's teachers. Theodorus produced a series of triangles whose hypotenuses were the square roots of 2, 3, 4, 5, etc. For now, we leave to the reader the fun of re-constructing this series.

Now think back on these series of squares. All are characterized by the same principle of proportionality. Each one is a smaller interval of action than the previous one. With each smaller interval, new singularities emerge. What is invariant in every smaller interval, however, is the principle of proportionality.

Now if a square is doubled through this principle of proportionality, how is the cube doubled? A generation or two before Plato, Hippocrates of Chios, (not the same Hippocrates of medicine fame) made a crucial discovery concerning this problem. This Hippocrates supposedly was a merchant who ended up broke in Athens around 500 B.C. and started to teach thinking to earn a living. His discovery can be re-created in the following way:

If we begin with a cube whose side is one, its volume will also be one. If we double the side, the new cube will have a volume of 8. Between 1 and 8 are two means, 2 and 4. That is, 1:2::2:4::4:8. So, the cube whose volume is double, is the lesser of the two means, between the cube whose side is doubled. Since the sides of the cubes are in the same proportion as the volumes, the side of the cube whose volume is double, is the lesser of two means between 1 and 2.

Eutocius reported this discovery in his commentaries on Archimedes as, "It became a subject of inquiry among geometers in what manner one might double the given solid, while it remained the same shape, and this problem was called the duplication of the cube; for, given a cube, they sought to double it. When all were for a long time at a loss, Hippocrates of Chios first conceived that, if two mean proportionals could be found in continued proportion between two straight lines, of which the greater was double the lesser, the cube would be doubled, so that the puzzle was by him turned into no less of a puzzle."

Plato reflected on this discovery in the Timeaus: "But it is not possible that two things alone should be conjoined without a third, for there must needs be some intermediary bond to connect the two. And the fairest of bonds is that which most perfectly unites into one both itself and the things which it binds together; and to effect this in the fairest manner is the natural property of proportion. For whenever the middle term of any three numbers, cubic or square, is such that the first term is to it, so is it to the last term, and again, conversely, as the last term is to the middle, so is the middle to the first,-- then the middle term becomes in turn the first and the last, while the first and last become in turn middle terms, and the necessary consequence will be that all the terms are interchangeable, and being interchangeable they all form a unity. Now if the body of the All had had to come into existence as a plane surface, having no depth, one middle term would have sufficed to bind together both itself and its fellow-terms, but now it is otherwise, for it behoved it to be solid of shape, and what brings solids into unison is never one middle term alone but always two."

Hippocrates' discovery amounts to investigating the geometric principle of proportionality, in an even smaller interval, trying to find two means between 1 and 2. His successors soon discovered a new boundary had to be crossed, when investigating this smaller interval. Those familiar with Gauss' Disquisitiones Arithmeticae and his theory of bi-quadratic residues, will recognize the seeds of those great investigations, in these ancient Greek inquiries.

In future weeks we will re-construct some of the ancient Greek studies of this problem. For now, keep thinking about it.