An Introduction to Mathematics · Alfred North Whitehead
When the Greek geometers had exhausted,
as they thought, the more obvious and interesting properties of figures made up of straight lines and circles, they turned to the study of other curves; and, with their almost infallible instinct for hitting upon things worth thinking about, they chiefly devoted themselves to conic sections, that is, to the curves in which planes would cut the surfaces of circular cones. The man who must have the credit of inventing the study is Menaechmus (born 375 B.C. and
died 325 B.C.); he was a pupil of Plato and one of the tutors of Alexander the Great. Alexander, by the by, is a conspicuous
example of the advantages of good tuition, for another of his tutors was the philosopher Aristotle. We may suspect that
Alexander found Menaechmus rather a dull teacher, for it is related that he asked for the
proofs to be made shorter. It was to this request that Menaechmus replied: "In the
country there are private and even royal roads, but in geometry there is only one road for all." This reply no doubt was true enough in the sense in which it would have been immediately understood by Alexander. But if Menaechmus thought that his proofs could not be shortened, he was grievously mistaken; and most modern mathematicians would be horribly bored, if they were compelled to study the Greek proofs of the properties of conic sections. Nothing illustrates better the gain in power which is obtained by the introduction of relevant ideas into a science than to observe the progressive shortening of proofs which accompanies the growth of richness in idea. There is a certain type of mathematician who is always rather impatient at delaying over the ideas of a subject: he is anxious at once to get on to the proofs of "important" problems. The history of the science is entirely against him. There are royal roads in science; but those who first tread them are men of genius and
not kings.
The way in which conic sections first presented themselves to mathematicians was as follows: think of a cone (cf. fig. 15), whose vertex (or point) is V , standing on a circular base STU. For example, a conical shade to
an electric light is often an example of such a surface. Now let the "generating" lines which pass through V and lie on the surface be all produced backwards; the result is a double cone, and PQR is another circular cross section on the opposite side of V to the cross section STU. The axis of the cone CVC' passes through all the centres of these circles and is perpendicular to their planes, which are parallel to each other. In the diagram the parts of the curves which are supposed to lie behind the plane of the paper are dotted lines, and the parts on the plane or in front of it are continuous lines. Now suppose this double cone is cut by a plane not perpendicular to the axis CVC' , or at least not necessarily perpendicular to it. Then three cases can arise:--
(1) The plane may cut the cone in a closed
oval curve, such as ABA'B' which lies entirely on one of the two half-cones. In this case the plane will not meet the other half-cone at all. Such a curve is called an ellipse; it is an oval curve. A particular case of such a section of the cone is when the plane is perpendicular to the axis CVC' , then the section, such as STU or PQR , is a circle. Hence a
circle is a particular case of the ellipse.
(2) The plane may be parallelled to a tangent plane touching the cone along one of its "generating" lines as for example the plane of the
curve D_1A_1D_1' in the diagram is parallel to the tangent plane touching the cone along the generating line VS ; the curve is still confined to one of the half-cones, but it is now not a closed oval curve, it goes on endlessly as long as the generating lines of the half-cone are produced away from the vertex. Such a conic section is called parabola.
(3) The plane may cut both the half-cones,
so that the complete curve consists of two detached portions, or "branches" as they are called, this case is illustrated by the two branches G_2A_2G_2' and L_2A_2'L_2' which together make up the curve. Neither branch is closed, each of them spreading out endlessly as the two half-cones are prolonged away from the vertex. Such a conic section is called a hyperbola.
There are accordingly three types of conic sections, namely, ellipses, parabolas, and hyperbolas. It is easy to see that, in a sense, parabolas are limiting cases lying between ellipses and hyperbolas. They form a more special sort and have to satisfy a more particular condition. These three names are apparently due to Apollonius of Perga (born
about 260 B.C., and died about 200 B.C.), who wrote a systematic treatise on conic sections which remained the standard work till the sixteenth century.
It must at once be apparent how awkward
and difficult the investigation of the properties of these curves must have been to the Greek geometers. The curves are plane curves, and yet their investigation involves the drawing in perspective of a solid figure. Thus in the diagram given above we have practically drawn no subsidiary lines and yet the figure is sufficiently complicated. The
curves are plane curves, and it seems obvious that we should be able to define them without
going beyond the plane into a solid figure. At the same time, just as in the "solid"
definition there is one uniform method of definition--namely, the section of a cone by
a plane--which yields three cases, so in any "plane" definition there also should be one uniform method of procedure which falls into three cases. Their shapes when drawn on their planes are those of the curved lines in the three figures 16, 17, and 18. The points A and A' in the figures are called
the vertices and the line AA' the major axis. It will be noted that a parabola (cf. fig. 17)
has only one vertex. Apollonius proved (Cf. Ball, loc. cit., for this account of Apollonius and Pappus.) that the ratio of PM^2 to AM·MA' (PM^2/AM·MAMA') remains constant both for the ellipse and the hyperbola (figs. 16 and 18), and that the ratio
of PM^2 to AM is constant for the parabola of fig. 17; and he bases most of his work on this fact. We are evidently advancing towards the desired uniform definition which does not go out of the plane; but have not yet quite attained to uniformity.
In the diagrams 16 and 18, two points, S and S' , will be seen marked, and in diagram 17 one point, S. These are the foci of the curves, and are points of the greatest importance. Apollonius knew that for an ellipse the sum of SP and S'P (i.e. SP + S'P ) is constant as P moves on the curve, and is equal to AA'. Similarly for a hyperbola the difference S'P - SP is constant, and equal to AA' when P is on one branch, and the difference SP' - S'P' is constant and equal to AA' when P' is on the other branch. But no corresponding point seemed to exist for the parabola.
Finally 500 years later the last great Greek geometer, Pappus of Alexandria, discovered
the final secret which completed this line of thought. In the diagrams 16 and 18 will be seen two lines, XN and X'N' , and in diagram 17 the single line, XN. These are the directrices of the curves, two each for the ellipse and the hyperbola, and one for the parabola. Each directrix corresponds to its nearer focus.
The characteristic property of a focus, S , and its corresponding directrix, XN , for any one of the three types of curve, is that the ratio
SP to PN (SP/PN) is constant, where PN is the perpendicular on the directrix from P , and P is any point on the curve. Here we have finally found the desired property of the curves which does not require us to leave the plane, and is stated uniformly for all three curves. For ellipses the ratio (Cf. Note B, p. noteB.) is less than 1 , for parabolas it is equal to 1 , and for hyperbolas it is greater than 1.
When Pappus had finished his investigations,
he must have felt that, apart from minor extensions, the subject was practically exhausted; and if he could have foreseen the history of science for more than a thousand years, it would have confirmed his belief. Yet in truth the really fruitful ideas in connection with this branch of mathematics had not yet been even touched on, and no one had guessed their supremely important applications in nature. No more impressive warning can be given to those who would confine knowledge and research to what is apparently useful, than the reflection that conic sections were studied for eighteen hundred years merely as an abstract science, without a thought of any utility other than to satisfy the craving for knowledge on the part of mathematicians, and that then at the end of this long period of abstract study, they
were found to be the necessary key with which to attain the knowledge of one of the most important laws of nature.
Meanwhile the entirely distinct study of astronomy had been going forward. The
great Greek astronomer Ptolemy (died 168 A.D.)
published his standard treatise on the subject in the University of Alexandria, explaining the apparent motions among the fixed stars of the sun and planets by the conception of the earth at rest and the sun and the planets circling round it. During the next thirteen hundred years the number and the accuracy of the astronomical observations increased, with the result that the description of the motions of the planets on Ptolemy's hypothesis had to be made more and more complicated. Copernicus (born
1473 A.D. and died 1543 A.D.) pointed out that the motions of these heavenly bodies could be explained in a simpler manner if the sun were supposed to rest, and the earth and planets were conceived as moving round it. However, he still thought of these motions as essentially circular, though modified by a set of small corrections arbitrarily superimposed on the primary circular motions. So the matter stood when Kepler was born at Stuttgart
in Germany in 1571 A.D. There were two sciences, that of the geometry of conic sections and that of astronomy, both of which
had been studied from a remote antiquity without a suspicion of any connection between the two. Kepler was an astronomer,
but he was also an able geometer, and on the subject of conic sections had arrived at ideas in advance of his time. He is only one of many examples of the falsity of the idea that success in scientific research demands an exclusive absorption in one narrow line of study. Novel ideas are more apt to spring from an unusual assortment of knowledge--not necessarily from vast knowledge, but from a thorough conception of the methods and ideas of distinct lines of thought. It will be remembered that Charles Darwin was helped
to arrive at his conception of the law of evolution by reading Malthus' famous Essay
on Population, a work dealing with a different subject--at least, as it was then thought.
Kepler enunciated three laws of planetary
motion, the first two in 1609, and the third ten years later. They are as follows:
(1) The orbits of the planets are ellipses, the sun being in the focus.
(2) As a planet moves in its orbit, the radius vector from the sun to the planet sweeps out equal areas in equal times.
(3) The squares of the periodic times of the several planets are proportional to the cubes of their major axes.
These laws proved to be only a stage towards a more fundamental development of ideas. Newton (born 1642 A.D. and died
1727 A.D.) conceived the idea of universal gravitation, namely, that any two pieces of
matter attract each other with a force proportional to the product of their masses and inversely proportional to the square of their distance from each other. This sweeping general law, coupled with the three laws of motion which he put into their final general shape, proved adequate to explain all astronomical phenomena, including Kepler's laws, and has formed the basis of modern physics. Among other things he proved that comets might move in very elongated ellipses, or in parabolas, or in hyperbolas, which are nearly parabolas. The comets which return--such as Halley's comet--must, of course, move in
ellipses. But the essential step in the proof of the law of gravitation, and even in the suggestion of its initial conception, was the verification of Kepler's laws connecting the motions of the planets with the theory of conic sections.
From the seventeenth century onwards the abstract theory of the curves has shared in the double renaissance of geometry due to the introduction of coordinate geometry and of projective geometry. In projective geometry
the fundamental ideas cluster round
the consideration of sets (or pencils, as they
are called) of lines passing through a common point (the vertex of the "pencil"). Now (cf. fig. 19) if A , B , C , D , be any four fixed points on a conic section and P be a variable point on the curve, the pencil of lines PA ,
PB , PC , and PD , has a special property, known as the constancy of its cross ratio. It
will suffice here to say that cross ratio is a fundamental idea in projective geometry. For projective geometry this is really the definition of the curves, or some analogous property which is really equivalent to it. It
will be seen how far in the course of ages of study we have drifted away from the old original idea of the sections of a circular cone. We know now that the Greeks had got hold of a minor property of comparatively slight importance; though by some divine good fortune the curves themselves deserved all the attention which was paid to them. This unimportance of the "section" idea is now marked in ordinary mathematical phraseology by dropping the word from their names. As often as not, they are now named merely "conics" instead of "conic sections."
Finally, we come back to the point at
which we left coordinate geometry in the last chapter. We had asked what was the type of loci corresponding to the general algebraic form ax + by = c , and had found that it was the class of straight lines in the plane. We had seen that every straight line possesses an equation of this form, and that every equation of this form corresponds to a straight line. We now wish to go on to the next general type of algebraic forms. This is evidently to be obtained by introducing terms involving x^2 and xy and y^2. Thus the new general form must be written:-- ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0. What does this represent? The answer is
that (when it represents any locus) it always represents a conic section, and, furthermore, that the equation of every conic section can always be put into this shape. The discrimination of the particular sorts of conics as given by this form of equation is very easy. It entirely depends upon the consideration of ab - h^2 , where a , b , and h , are the "constants" as written above. If ab - h^2 is a positive number, the curve is an ellipse; if ab - h^2 = 0 , the curve is a parabola: and if ab - h^2 is a negative number, the curve is a hyperbola.
For example, put a = b = 1 , h = g = f = 0 , c = -4. We then get the equation x^2 + y^2 - 4 = 0. It is easy to prove that this is the equation of a circle, whose centre is at the origin, and radius is 2 units of length. Now ab - h^2 becomes 1 × 1 - 0^2 , that is, 1 , and is therefore positive. Hence the circle is a particular case of an ellipse, as it ought to be. Generalising, the equation of any circle can be put into the form a(x^2 + y^2) + 2gx + 2fy + c = 0. Hence ab - h^2 becomes a^2 - 0 , that is, a^2 , which is necessarily positive. Accordingly all circles satisfy the condition for ellipses. The general form of the equation of a parabola is (dx + ey)^2 + 2gx + 2fy + c = 0, so that the terms of the second degree, as
they are called, can be written as a perfect square. For squaring out, we get d^2 x^2 + 2dexy + e^2 y^2 + 2gx + 2fy + c; so that by comparison a = d^2 , h = de , b = e^2 , and therefore ab - h^2 = d^2 e^2 - (de)^2 = 0. Hence the necessary condition is automatically satisfied. The equation 2xy - 4 = 0 , where a = b = g = f = 0 , h = 1 , c = -4 , represents a hyperbola. For the condition ab - h^2 becomes 0 - 1^2 , that is, -1 , which is negative.
The limitation, introduced by saying that,
when the general equation represents any locus, it represents a conic section, is necessary, because some particular cases of the general equation represent no real locus. For example x^2 + y^2 + 1 = 0 can be satisfied by no real values of x and y. It is usual to say that the locus is now one composed of imaginary points. But this idea of imaginary points in geometry is really one of great complexity, which we will not now enter into.
Some exceptional cases are included in the general form of the equation which may not be immediately recognized as conic sections. By properly choosing the constants the equation can be made to represent two straight lines. Now two intersecting straight lines may fairly be said to come under the Greek idea of a conic section. For, by referring to
the picture of the double cone above, it will be seen that some planes through the vertex, V , will cut the cone in a pair of straight lines intersecting at V. The case of two parallel straight lines can be included by considering a circular cylinder as a particular case of a cone. Then a plane, which cuts it and is parallel to its axis, will cut it in two parallel straight lines. Anyhow, whether or no the
ancient Greek would have allowed these special cases to be called conic sections, they are certainly included among the curves represented by the general algebraic form of the second degree. This fact is worth noting; for it is characteristic of modern mathematics to include among general forms all sorts of particular cases which would formerly have received special treatment. This is due to its pursuit of generality.
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End of part 10 of 19
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