by Wells, David
Yes, there is: Ceva proved that AZ/ZB · BX/XC · CY/YA = 1. The ratio AZ/ZB is the ratio of the areas of the triangles AZC and BZC, and also, similarly, of the areas of triangles AZO and BZO. But in that case, AZ/ZB is also the ratio of their differences, AOC and BOC.
Similarly, BX/XC is the ratio of BOC to COA and CY/YA is the ratio of COB to COA. So the product AZ/ZB · BX/XC · CY/YA is 1.
Curiously, although Ceva's theorem is relatively modern, it strongly resembles the theorem of Menelaus (c.70–130 CE) which says that if a line crosses the sides of a triangle (Figure 7.4), then
AY/YC · CX/XB · BZ/ZA = 1
Ceva's theorem is about concurrent lines, while Menelaus's theorem is about collinear points. Menelaus’ theorem also has an elegant proof. We draw a line WC parallel to YXZ (Figure 7.5).
Figure 7.4 Menelaus' theorem
Figure 7.5 Proof of Menelaus
We now change the first two ratios into ratios on the side AB: By similar triangles,
AY/YC = AZ/ZW
and,
CX/XB = WZ/ZB
We now multiply the three ratios: AZ/ZW · WZ/ZB · BZ/ZA = 1.
Simson's line
Another elegant theorem about transversals concerns a triangle within a circle and a point P on its circumference (Figure 7.6).
Figure 7.6 Simson's line
We drop perpendiculars from P to each side of the triangle. The feet of the perpendiculars lie on a straight line. The simplest proof uses two basic circle properties, that angles in the same segment are equal and that if a quadrilateral has opposite angles which sum to 180o, then its vertices lie on a circle.
The converse of Simson's theorem is also true: if a line traverses the sides of a triangle and the perpendiculars at the points of intersection are concurrent at P, then P lies on the circumcircle. We shall meet this theorem again.
The parabola and its geometrical properties
The only curve that Euclid considered was the ‘perfect’ circle which because it is so simple and symmetrical has many equally simple and elegant properties. The parabola has good claims to be the next simplest curve and we can deduce several of its properties using basic Euclidean properties of triangles, (and sometimes of circles also), plus its own definition and starting with an intuitively convincing but imperfectly logical dynamic argument.
The parabola is the path of a point which moves so that it is always the same distance from a point F, called the focus, and a line d, called the directrix: Figure 7.7 shows a basic parabola.
Figure 7.7 Parabola, focus and directrix
Now let's draw a tangent (Figure 7.8). Think of the point P actually moving along the tangent in the direction of the arrow: since it is equidistant from F and d and moving momentarily so as to remain so, the tangent bisects the angle
Figure 7.8 Parabola with constructions
FPA. This physical argument, of a kind Archimedes favoured, gives us our first basic angle property. It means that the triangles FPX and APX are congruent, and X is the mid-point of FA and AO = FO, and the perpendicular from F to the tangent at P meets the directrix at A. Since the vertex of the parabola (its lowest point) is halfway between F and d, it follows that X, the foot of the perpendicular from F to the tangent at P, lies on the tangent at the vertex (Figure 7.9).
Figure 7.9 Parabola with further constructions
Now complete the chord through F, add the tangent at Q, and mark B so that FO = AO = OB. Then since FQ = QB by definition, FOBQ is a kite, OQ is a line of symmetry and so OQ is the tangent at Q.
Also, since AO = OF = OB, AFB is a right angle. Since APF + BQF = 180, OPF + OQF = 90° and so POQ is a right angle also.
It turns out, typically, that the figure formed by the parabola, the two tangents and lines of equal length from F and to d is extremely simple. It illustrates a basic structure that is forced by the definition and basic Euclidean properties – but this is only the start: any number of other properties can be discovered, and proved, starting from the parabola's extremely simple definition. One more is shown in Figure 7.10.
Figure 7.10 Parabola, three tangents and circumcircle
We first add a third tangent, so that the three tangents at A, B and C form a triangle. Next we draw the circumcircle of this triangle. The perpendiculars from the focus F will all lie a straight line, the tangent at the vertex, and so by the converse of Simson's theorem, F lies on the circle too.
It is typical that the apparently idiosyncratic Simson's theorem can be used to prove a theorem about parabolas. Euclidean geometry and all its relations are intimately linked together and repeatedly refer to each other. (We can also add that our original dynamic argument that the tangent bisects FPA must be correct in some sense, to produce such a simple, rich structure.)
We conclude with a new perspective on the parabola and the other conics that was found 2000 years after Euclid in 1822, after being missed first by Euclid and then by Apollonius and a host of other geometers.
Dandelin's spheres
The conics were originally studied by the ancient Greeks as slices of a circular cone which is the meaning of conic (Figure 7.11).
Figure 7.11 A slice of a cone
An immediate puzzle is why this asymmetrical slice should produce a symmetrical ellipse. At first sight you might think that the slice will be wider at the bottom end B, because the cone is ‘larger’ further down. On second thoughts, you might think that it would be wider at the top, at T, because there the slice cuts the edge at a larger angle. Neither of these speculations is correct: the elliptical cross-section is completely symmetrical.
A further puzzle is: what is the connection between the foci and directrices of a conic and the slice of the cone which creates the conic?
This was answered, rather late in the day, by Germinal Pierre Dandelin (1794–1847) who discovered a surprising way of finding the focus and directrix by constructing the two spheres that touch the cone along a circle and the plane of the conic (one sphere on either side of the cutting plane for the ellipse) in two single points (Figure 7.12). The figure, once again, looks asymmetrical but no, the two points of contact turn out to be the foci, and the directrices are the lines in which the plane of the elliptical slice meets the planes through the circles along which each sphere touches the cone.
Figure 7.12 Dandelin's spheres
The line through A is a generator of the cone touching both spheres on their circles of contact. The line from A to the focus on either sphere is equal in length to that part of the segment, from which it follows that the total distance from A to both foci is constant.
Truly, the consequences of Euclid's geometrical game are endless!
8 New concepts and new objects
Mathematical explorers continually discover and name new objects in the mathematical landscape, and their features. Here are just some nouns and adjectives – there are plenty of mathematical verbs also – from the hundreds in any mathematical dictionary.
Abelian group, abundant number, affine space, area, automorphism, axis, binomial coefficient, braid, Cantor set, cardinal, Cauchy sequence, characteristic, Chinese remainder theorem, closed interval, combination, complex integration, continued fraction, cross ratio, deltoid, Desargues’ theorem, discriminant, duality, eccentricity, equation, equivalence, Euclidean, evolute, extremal, factor, factorial, Fagnano's problem, field, finite, free group, function, Galois theory, Gaussian integers, golden section, the greedy algorithm, group, Hamiltonian, harmonic, height, heptagon, hypercube, identity, imaginary number, incidence matrix, index, infinitesimal, infinite series, integer, integration, invariant, inversion, isogonal, isometry, isomorphism, iteration, Jacobian, Jordan curve, Julia set, Klein bottle, knot, Kurschak tile, Latin square, lattice, linear function, locus, logarithm, magic square, Mandelbrot set, matrix, maximum, mean value theorem, median, metric space, Möbius strip, model, modular arithmetic, nephroid, node, normal, orbit, ordinate, origin, orthogonal, packing, parabola, parallel, Pascal's triangle, Peano curve, pencil, perfe
ct number, permutation, pigeonhole principle, pole, polygon, power, prime, probability, projection, quadrilateral, quadratic, quartile, radical axis, radius, rational, reflection, ring, rotation, ruled surface, saddle, sample, scalar, set, similarity, Simpson line, sphere, square, Stirling's formula, subtraction, symmetry, tangent, Taylor series, tessellation, topological space, torus, trace, transitive, tree, triangle, unduloid, uniform, union, value, variable, vector, vertex, Viviani's theorem, volume, weighted average, word, x-axis, zero, zeta function, Zorn's lemma.
Some of these are, as it were, very small objects, others are large-scale features. Many of the nouns are associated with adjectives and verbs, as in any colloquial language.
We start by glancing at the problem of naming objects. We all know that there are lots of numbers, but how can we refer to them? Next, a kind of dual problem, of objects that you can talk about easily although they don't exist! In mathematics as in real life, talk is cheap. Just because we can refer to an odd perfect number doesn't mean it exists. However, just because mathematicians are doing mathematics and not science they can draw firm conclusions about these ‘objects’ even if it turns out in time that their existence is impossible because it would be contradictory.
A different kind of object is constructed to fit a specific purpose – rather as a DIY enthusiast might make a special tool – but, because this is mathematics, the tool is forced by the situation, if only we can work out how. We realise in the long run that the concept-object ‘has to exist’ because all the evidence points inexorably to that conclusion: and when we discover it, we are rewarded by its simplicity and power.
A different problem arises when we take an informal idea such as infinity or tangent, and ask how they can be made game-like. It's easy to talk about a parabola ‘going off to infinity’ but what ought that to mean, precisely? It's easy to sketch a curve – such as a parabola – and then use a ruler to draw a straight line that (roughly) touches it, but how do we make that idea precise? While on the subject of the parabola, even the everyday idea of shape turns out to be surprisingly ambiguous.
Creating new objects
Once upon a time, the very numbers we now take for granted were a novelty. The ancient Greeks had no good method of naming numbers – they used the letters of their alphabet and soon ran short of possibilities – and the Romans as we know used the letters I, V, X, L, C, D, M for a selection of numbers from 1 to 1000. It is no surprise that calculation with these numbers was tricky, which is why for all practical purposes they used an abacus instead.
The Indian-Arabic counting system, with its notation for decimal fractions added, is superior because it lets us name far larger numbers according to a simple plan. The number 1034667 stands for,
Today, mathematicians couldn't do without series based on the same idea: the variable x in this series appears in powers from zero upwards:
However, even the Arabic system is stretched to express high numbers and so we use the notation of powers and powers of powers:
and
can be written briefly as 10256. This is more than merely an abbreviation – it avoids error by eliminating the need to count all those zeros. It is also greater by far than the number of particles in the universe, if cosmologists are correct, although minuscule, even negligible, when compared to numbers so large that we can hardly imagine them: infinity is a very long way away!
However, by inventing a new notation we can very easily go far beyond the customary notation for powers and powers of powers. Suppose we define 2* to mean 22 or 2∧2 and 3* to mean 333, and so on. Conventionally, mathematicians read these ‘towers of powers’ from the top downwards so that 3* = 3∧(3∧3) = 327 and so on.
Then 2* = 4, which is very small, and 3* = 327 = 7625597484987, which is much larger. How large is 4*? Well, 4∧4 = 256, so 4∧(4∧4) = 4256 which is approximately 1.3407796 × 10154 or a number of 155 digits. So 4* is approximately 4∧(1.3407796 × 10154) which is a number of about 8.072297 × 10∧153 digits.
5* will be vastly larger again and 100* will be unimaginably large, and yet we can write it down – we have just done so – and we can also write down very briefly the fantabulously ginormous number (100*)*.
Does it exist?
Can you draw four equidistant points which are, let's say, 10 cm from each other? Tricky. You might start by doing what Euclid did in his Elements Bk.1, 1–1, and construct an equilateral triangle but when you look for a fourth equidistant point it doesn't exist – in two dimensions. However, you could add a fourth point in three dimensions, to make this regular tetrahedron (Figure 8.1).
Figure 8.1 Four dots form a tetrahedron
In other words, in two dimensions there is a contradiction in the idea of four mutually equidistant points, but in three dimensions there isn't.
Other objects are less obviously fake, for example, the odd perfect numbers. Perfect numbers are the sum of all their factors excluding the number itself, so 6 = 1 + 2 + 3 is the first and 28 = 14 + 7 + 4 + 2 + 1 is the second. Euclid proved in his Elements, Book IX, that if 2n − 1 is prime then 2n−1(2n − 1) is perfect. Since 25 − 1 = 31 is prime, 16·31 = 496 is the third perfect number. However, Euclid's formula gives only even perfect numbers and says nothing about odd PNs. Are there any odd PNs? No one knows, but it has been proved that if an odd PN exists then it must be greater than 10200, must have at least eight prime factors and must be divisible by a prime power greater than 1018. And if an odd PN cannot exist? Then these statements will still be true – but irrelevant – curiosities.
The force of circumstance
Mathematicians sometimes realise that an object exists but can't easily identify it or find the best, the most ‘natural’, definition. For example, it is obviously possible to measure angles by dividing a circle into a number of parts. Usually, we use 360 parts, or degrees, for reasons that go back to the ancient Babylonians and are highly arbitrary. This is very unsatisfactory, because the standard series for sin x,
and all the other usual trigonometrical series would have to be written with factors of 360 in them, if we chose any such arbitrary measure. To divide the circle into 100 parts would be just as arbitrary. So how ought angles to be measured most naturally?
Mysteriously, the answer is not to divide the circumference of the circle into any whole number of parts, but to divide it into 2π parts. The entire circumference of a unit circle is 2π by the usual formula, and so 180° becomes π, 90° becomes π/2, and so on.
(Some mathematicians seriously believe that it is an anomaly that it is 2π parts and that it should be divided into τ parts, where τ is the Greek letter tau, and τ = 2π or approximately 6.28. This argument will not be decided by pure logic, but by the collective judgements of many mathematicians – and the weight of history.)
The new measure of angle is called the radian and only if we use this measure and no other will the usual series for sin x be so simple. Curiously, the term radian was first used in 1873, but the idea goes back to Roger Coates (1682–1716), and Euler who explained in his Elements of Algebra that angles should be measured by the length of the arc cut off on the unit circle – so although we could say that the radian was invented, it was also discovered because this simplest possible angle measure is unique and so is forced on us. The old idea of angle measure was simple but naive and it was only with advances in mathematics that the ‘true’ measure was realised.
We can say the same for everyday ideas of infinity: they may be intriguing (and attractive to poets) but they are also crude and confusing for the mathematician who needs to formalise the informal.
Infinity and infinite series
What is infinity? It's a quintessentially mathematical idea, because the counting numbers go on for ever and we can imagine geometrical space extending for ever, even if that's physically impossible. For more than 300 years mathematicians have exploited infinite series, suggesting that they understand ‘infinity’, but it's not so simple.
Metaphors for infinity are not game-li
ke and not mathematical. To tame infinity we must bring it down from poetic heaven to solid ground. One way is to interpret it in finite terms, which we can illustrate with this series whose ‘sum’ is ‘obviously’ 1:
Its sum is ‘obvious’ for several reasons. Here are two, one visual (Figure 8.2), and one arithmetical.
Figure 8.2 Picture of sum of 2−n
We have cut off the top half of the square, then the bottom-left quarter, and so on. The terms of the series fill up the square so that the amount left unfilled is as small as we choose. We can reach the same conclusion by arithmetic by calculating the partial sums: the sums of the first 1, 2, 3, 4…terms are,
The pattern seems obvious, though perhaps not quite as obvious as the picture. More precisely, the sum of the first n terms of the series is (2n − 1)/2n and this does get closer and closer to 1 as n increases. However, that fact by itself does not quite pin down the idea of an infinite sum: after all, it never reaches 1, but is always definitely less – so how can 1 be its sum?
The answer lies, ironically, in changing the meaning of sum, in order to fit the idea of an infinite sum. We agree to use sum for a number that the series when summed term by term never actually reaches. That idea is not sharp enough by itself because the term-by-term sum will never reach 100 either, so we add the (imprecise) condition that it gets as ‘close as anything’ to 1. We then make the idea of ‘close as anything’ precise, like this. We suppose that you challenge me with a very small number, say, 1/1000000. To meet the challenge I must show that by adding up enough terms of the series I can make the difference between the partial sum and 1, less than 1/1 000 000. If I can always meet your challenge, (in this case I can,) then 1 is the sum of the infinite series.