by Wells, David
The greatest wasan practitioner was Seki Kowa (c.1642–1708), a genius whose achievements included finding Horner's method for solving algebraic equations with numerical coefficients long before Horner, the concept of the discriminant of an equation, developing the Chinese idea of determinants ten years before Leibniz, and discovering the Bernoulli numbers, the Pappus–Guldin theorem and using negative and imaginary roots. Either Seki or his disciples also created the enri calculus through their study of arcs of circles, which they applied to curves and curved surfaces generally. However, this is where differences with the West appear:
The enri calculus, which developed from problems concerning the arc, an important topic in astronomy, coincided in its results with Western-style calculus. The course it followed…was completely different from that of Western mathematics, which began with problems in dynamics…There were greater limits to the problem development possible in enri. Ultimately the absence of kinematic and dynamic problems in Japan's scientific tradition handicapped and retarded wasan's approach to analysis and proved decisive in its race with the Western tradition.
[Nayakama 1975: 752, 750]
The wasan mathematicians had their own aesthetic criteria:
The purer the mathematical character of a problem and the greater its detachment from practicality, the greater the enthusiasm with which it was received.
[Nakayama 1975: 749]
In bibliographies, wasan was listed with flower arranging and the tea ceremony, social activities with a strong aesthetic component, influenced by Zen Buddhism [Ravina 1993: 206]. The other side of this coin was that wasan practitioners looked down on the sciences:
The demise of wasan stemmed largely from its divorce from the natural sciences…The natural world was deemed unsuitable as a subject of mathematical study. Mathematics advanced no new theories of the physical world, developed no new mechanical laws, challenged no theological principles, and produced no machines. Tokugawa scientists on neighbouring fields, such as astronomy and surveying, considered wasan intriguing but essentially useless.
[Ravina 1993: 205]
As a result, wasan degenerated:
As time passed, the problems became more intricate and mulifaceted…There was an emphasis on solving problems by some unusual means or in presenting problems for which it was not known whether a satisfactory solution existed.
[Nakayama 1975: 750]
John von Neumann expected our mathematics to degenerate also if it strayed too far from applications:
As a mathematical discipline travels far from its empirical source…if it is…only indirectly inspired by ideas coming from ‘reality’…It becomes more and more purely aestheticizing, more and more l'art pour l'art…there is a grave danger…that the stream .. will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities.
[Neumann 1947] [Newman 1956 v.4: 2063]
The interpretation of ‘applications’ is a moot point. It does not have to be applications to the sciences, but could be to other branches of mathematics. The creativity devoted to Fermat's Last Theorem seems not to have degenerated over several hundred years, while Fermat's contribution to optics is also still alive as we shall see shortly. In all the many miniature worlds of mathematics, cross-fertilisation is endemic, and fortunately so, because it forces the mathematician to face up to new challenges and prevents the degeneration that might otherwise attack the mathematical ivory tower aesthete.
15 Minimum paths: elegant simplicity
A familiar puzzle
Mary, who is standing at S, wishes to walk to the river for a drink and then back to T, walking as short a distance as possible. To what point on the river bank should she walk?
The solution to this favourite puzzle is delightfully simple and elegant: reflect the point T in the river bank to Tʹ, and join STʹ. Mary walks on the line STʹ to the bank and then heads back to T (Figure 15.1).
Figure 15.1 To the river and back
This problem, but expressed more seriously, was solved by Heron of Alexandria (c.75 CE) in his Catoptrica. He asked how a ray of light is reflected off a mirror and answered that it takes the shortest path and that its angles of incidence and reflection are equal (Figure 15.2).
Figure 15.2 Reflection of light in a mirror
It is a delightful feature that a problem about reflection of light is also solved by reflection.
A related problem is: to find the smallest triangle, measured by perimeter, that can be drawn in a given triangle (Figure 15.3). According to Heron's argument, the two lines FE and ED must meet the base AB at the same angle, because if they did not then we could adjust E slightly to reduce the length FE + ED. Similarly, FD and DE must strike BC at the same angles, and likewise EF and FD on AB. Therefore, if the solution exists, it is a triangle which is also the path of a ray of light being reflected endlessly round the inside the triangle. But does it exist? (Yes, it's the triangle formed by the base of the altitudes, so the triangle must be acute angled.)
Figure 15.3 The shortest path in triangle
Other puzzles can also be solved by reflection: the next is a popular investigation for school pupils. The figure represents a billiard table. A ball shoots out of the corner at 45° to both sides, bounces round the table and ends up – where (Figure 15.4)? And how long does it take to get there?
Figure 15.4 The billiard ball problem
A solution well-known among teachers starts by reflecting the table several times, so that the path of the ball as it is reflected off the cushes becomes a straight line: the start of this solution is illustrated in Figure 15.5. Jolly ingenious! But there is another solution, discovered by a 12-year-old pupil after hours of drawing and thinking, which uses an entirely different argument and which you might think was simpler and superior. It goes like this: to end up in any of the corners, the ball must have travelled the length of the table a whole number of times and so it will have travelled one of the distances in the first row: 7
14
21
28
35
42
49
56…
5
10
15
20
25
30
35
40…
Figure 15.5 A solution by reflection
But it will also have travelled the width of the table a whole number of times, represented by the numbers in the second row. So the solution is found by comparing these lists to find a common number, which is 35.
This is impressive, but we can learn more. It travels the length of the table 5 times and so must end up at the righ-hand side of the table and it travels the width 7 times, so it ends up on the top edge. So it ends up in the top right corner.
We have not referred to 35 as the lowest common multiple of 5 and 7 because the pupil who discovered this method was too young to know the expression, though you could say that he half-discovered it in solving the puzzle.
Billiards is an enjoyable game but the puzzle might seem to be mathematically trivial. Not at all! If we generalise it and study other shapes of table we get physically interesting results. For example, in the circular table in Figure 15.6 the path of a ball will cover, eventually, the whole of an annulus, but will never enter a central circle. (To be more precise, the path will eventually pass arbitrarily close to any point in the annulus.)
Figure 15.6 Billiards between two circles
This motion is very regular when compared to Figure 15.7, made of semi-circles joined by straight lines and called the Bunimovich stadium after the Russian mathematician who studied its properties. The path is quite irregular, in fact it's chaotic. Such models are used in mechanics.
Figure 15.7 Billiards in a hippodrome
Developing Heron's theorem
Let's return to Mary's puzzle and to optics, one of the earliest sciences. Heron's conclusion that rays of light reflecting off a plane mirror tak
e the shortest possible path suggests Aristotle's claim that ‘Nature does nothing in vain’, [Aristotle De Anima III:12] which remained a popular idea among early modern scientists such as Newton who wrote:
To this purpose the philosophers say, that Nature does nothing in vain, and more is in vain, when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes.
[Newton 1687: Preface]
Heron's argument successfully linked this very attractive but frankly metaphysical axiom to mathematics and to physics. However, Heron's reference to shortest distance proved mistaken. Everyone has noticed that objects, such as a stick, appear ‘bent’ when seen standing in water.
The Greeks certainly did, but not until Kepler did anyone try to explain why [Kepler 1611]. Ten years later, Willebrord Snell did some careful experiments and concluded that the angles made by the incident and refracted rays with the normals to the surface fit Snell's Law (Figure 15.8).
Figure 15.8 Snell's law
Snell's law says that if a is the angle between the incident light and the normal and b is the angle between the normal and the refracted beam, then sin a/sin b is constant.
This is an elegant result but it contradicts Aristotle's claim, which upset Fermat (1601–1665). His own calculations suggested that Heron was wrong and that it was not the distance that was minimised, but the time taken. (In Heron's original problem, it makes no difference whether he minimises time or distance because the problem is reflection in one medium not refraction in two. Actually, the time can in some cases be a maximum: it is the fact that it is stationary that is crucial.) Using his corrected principle, Fermat then proved, rather to his disappointment, that Snell's law was indeed correct, giving further support to Aristotle's metaphysics.
Pierre de Maupertuis (1698–1759) was the next natural philosopher to build on Aristotle, Heron and Fermat. He gave a Christian spin to the idea that God as the creator of everything in nature and its prime mover always acted to minimise the ‘quantity of action’.
The laws of movement and of rest deduced from this principle being precisely the same as those observed in nature, we can admire the application of it to all phenomena. The movement of animals, the vegetative growth of plants…are only its consequences; and the spectacle of the universe becomes so much the grander, so much more beautiful, the worthier of its Author, when one knows that a small number of laws, most wisely established, suffice for all movements.
[Maupertuis 1746: 267]
Maupertuis came to his principle while working, appropriately, on the theory of light, but he believed that it applied to all the laws of nature. In physics, he suggested that the quantity to be minimised ought to be the product of the time taken by the event and the vis viva which was twice what we would call the kinetic energy of the system. The modern principle of least action minimises the difference between the kinetic energy and the potential energy, summed along the path of the action.
Ironically, recalling Newton's enthusiasm, the principle of least action provides a foundation for mechanics which is independent of Newton's laws as well as being used in the general theory of relativity and quantum mechanics. A principle that once seemed an attractive speculation has had a lasting influence on science and maths across more than 2000 years.
Extremal problems
Heron's problem and all its descendants are extremal problems: find the greatest or the least distance, time, area, volume, action…The first person on record to tackle an extremal problem was Queen Dido of Carthage. According to Virgil's Aeneid, she arrived by ship on the coast of north Africa as a refugee and asked the native inhabitants for as much land as could be surrounded by a bull's hide. They agreed and she cut it into fine strips, tied them together and surrounded a local hill. Had she been a mathematician she could have enclosed a larger area by constructing a semicircle with the sea as one side, if you accept that a circle surrounds the greatest area possible for its circumference (Figure 15.9).
Figure 15.9 The area enclosed by a rope against a shoreline
Suppose that Dido chose the shape on the left: we reflect it in the sea shore. The total circumference is now double the length of her hide rope, but it is not surrounding the maximum area, which would be larger if it were a circle. Therefore, she should have chosen to make a semicircle.
Pappus and the honeycomb
The Greek mathematician Pappus (290–350 CE) discussed an extremal problem that continues to delight us today:
It is of course to men that God has given the best and the most perfect notion of wisdom in general and of mathematical science in particular…to the other animals, while denying them reason, he granted that each of them should, by virtue of a certain natural instinct, obtain just so much as is needful to support life.
[Pappus: Heath 1921]
This instinct, says Pappus, is seen in all animals but most of all in bees. He then refers to the way they store their honey, assuming they would naturally choose a way of dividing the plane which leaves no spaces between the cells through which ‘foreign material could enter…and so defile the purity of their produce’. There are three such tessellations of regular figures, composed of equilateral triangles, squares, and regular hexagons and the bees, ‘by reason of their instinctive wisdom chose for the construction of the honeycomb the figure which has the most angles because they conceived that it would contain more honey than either of the two others’.
Their choice is therefore the regular hexagon, because as Pappus is now going on to demonstrate,
of all equilateral and equiangular plane figures having an equal perimeter, that which has the greater number of angles is always greater, and the greatest plane figure of all those which have a perimeter equal to that of the polygon is the circle.
[Heath 1921]
This problem is quite unlike anything in Euclid but it is mathematical and it led in due course to a novel branch of mathematics called the calculus of variations after a paper by that universal mathematician, Euler. So natural history, physics and mathematics, not forgetting chemistry, have all contributed to an extraordinary symbiosis in which an attractive but strange metaphysical principle has turned out to be a goldmine of powerful theories, concepts and methods. An enigma indeed.
16 The foundations: perception, imagination, insight
Perception is a puzzle. We think of visual perception because we use our eyes so much though we perceive through all our senses. But we also say, ‘Can you see the next move?’ or ‘Can you spot the next move?’ which often means ‘Can you work it out?’ We also ask, ‘Do you see what I mean?’ which happens to be the commonest use of ‘see’ in the English language – and it's not about seeing, but about understanding!
Psychologists tell us that perception is an active and neurologically complex process. We see nothing instantly, only in real time. As we look at a geometrical diagram our eyes pick out particular features that are already there on the paper (or the computer screen) but we may also ‘see’ lines or circles or points that are not yet there. They are potential – we could add them if we chose to do so. The Greeks were keen on such constructions and often used them in their proofs. Indeed, many proofs would be impossible without them.
So we see what is actually there, and we ‘see’ or imagine what might be, emphasising how active mathematics is. Indeed, we can sometimes ‘see’ what would happen if the parts of the diagram started moving around. We see, in our mind's eye, the effect of a transformation. Computer graphics can create this movement for you but this can be dangerous: a machine which replaces your brain may stop your brain working, a disaster because maths is an imaginative activity that needs an active brain.
Strangely, very much the same processes occur when you look at an algebraic equation. Equations don't look like a picture and you wouldn't frame one and hang it on your wall (that is not quite true – you can purchase beautifully printed mathematical and scientific equations over the Internet) but you do scan it to pick out fea
tures, to make sense of it. In looking at this series,
you might notice, in no particular order, that the variable is always x, that it contains only odd powers of x, and that the coefficients are rising rapidly. Look harder, and you might ‘see’ that the differences between the coefficients seem to be the rising powers of 3: 3, 9, 27…and so on.
You might also ‘see’ possible transformations of the equation. As always, this takes imagination. As in a chess position, there are many possible ‘moves’ and it is often not obvious which one is best. Here's an example. This quadratic equation can be transformed by dividing throughout by x:
Fair enough, but what's the point? What does it mean? Well, it means that now you have two numbers, x and 15/x, whose product is 15, and whose sum is 8. So the original quadratic seems to be equivalent to the problem: find a and b if,
