An Introduction to Mathematics · Alfred North Whitehead
The methods and ideas of coordinate geometry
have already been employed in the previous chapters. It is now time for us to consider them more closely for their own sake; and in doing so we shall strengthen our hold on other ideas to which we have attained. In the present and succeeding chapters we will go back to the idea of the positive and negative real numbers and will ignore the imaginaries which were introduced in the last two chapters.
We have been perpetually using the idea that, by taking two axes, XOX' and YOY' , in a plane, any point P in that plane can be determined in position by a pair of positive or negative numbers x and y , where (cf. fig. 13) x is the length OM and y is the length PM. This conception, simple as it looks, is the main idea of the great subject of coordinate geometry. Its discovery marks a momentous epoch in the history of mathematical thought. It is due (as has been
already said) to the philosopher Descartes,
and occurred to him as an important mathematical method one morning as he lay in bed. Philosophers, when they have possessed a thorough knowledge of mathematics, have been among those who have enriched the
science with some of its best ideas. On the other hand it must be said that, with hardly an exception, all the remarks on mathematics made by those philosophers who have possessed but a slight or hasty and late-acquired knowledge of it are entirely worthless, being either trivial or wrong. The fact is a curious one; since the ultimate ideas of mathematics
seem, after all, to be very simple, almost childishly so, and to lie well within the province of philosophical thought. Probably their very simplicity is the cause of error; we are not used to think about such simple abstract things, and a long training is necessary to secure even a partial immunity from error as soon as we diverge from the beaten track of thought.
The discovery of coordinate geometry, and also that of projective geometry about the same time, illustrate another fact which is being continually verified in the history of knowledge, namely, that some of the greatest discoveries are to be made among the most well-known topics. By the time that the seventeenth century had arrived, geometry had already been studied for over two thousand years, even if we date its rise with the Greeks. Euclid, taught in the University of Alexandria,
being born about 330 B.C.; and he only systematized and extended the work of a long series of predecessors, some of them men of genius. After him generation after generation of mathematicians laboured at the improvement of the subject. Nor did the subject suffer from that fatal bar to progress, namely, that its study was confined to a narrow group of men of similar origin and outlook--quite the contrary was the case; by the seventeenth century it had passed
through the minds of Egyptians and Greeks, of Arabs and of Germans. And yet, after all this labour devoted to it through so many ages by such diverse minds its most important secrets were yet to be discovered.
No one can have studied even the elements of elementary geometry without feeling the lack of some guiding method. Every proposition has to be proved by a fresh display of ingenuity; and a science for which this is true lacks the great requisite of scientific thought, namely, method. Now the especial point of coordinate geometry is that for the first time it introduced method. The remote deductions of a mathematical science are not of primary theoretical importance. The science has not been perfected, until it consists in essence of the exhibition of great allied methods by which information, on any desired topic which falls within its scope, can easily be obtained. The growth of a science is not primarily in bulk, but in ideas; and the more the ideas grow, the fewer are the deductions which it is worth while to write down. Unfortunately, mathematics is always encumbered by the repetition in text-books of numberless subsidiary propositions, whose importance has been lost by their absorption into the role of particular cases of more general truths--and, as we have already insisted, generality is the soul of mathematics.
Again, coordinate geometry illustrates another feature of mathematics which has already been pointed out, namely, that mathematical sciences as they develop dovetail into each other, and share the same ideas in common. It is not too much to say that the various branches of mathematics undergo a perpetual process of generalization, and that as they become generalized, they coalesce. Here again the reason springs from the very nature of the science, its generality, that is to say, from the fact that the science deals with the general truths which apply to all things in virtue of their very existence as things. In this connection the interest of coordinate geometry lies in the fact that it relates together geometry, which started as the science of space, and algebra, which has its origin in the science of number.
Let us now recall the main ideas of the two sciences, and then see how they are related by Descartes' method of coordinates. Take
algebra in the first place. We will not trouble ourselves about the imaginaries and will think merely of the real numbers with positive or negative signs. The fundamental idea is that of any number, the variable number, which is denoted by a letter and not by any definite numeral. We then proceed to the consideration of correlations between variables. For example, if x and y are two variables,
we may conceive them as correlated by the equations x + y = 1 , or by x - y = 1 , or in any one of an indefinite number of other ways. This at once leads to the application of the
idea of algebraic form. We think, in fact, of any correlation of some interesting type, thus rising from the initial conception of variable numbers to the secondary conception of variable correlations of numbers. Thus we generalize the correlation x + y = 1 , into the correlation ax + by = c. Here a and b and c , being letters, stand for any numbers and are in fact themselves variables. But they are the variables which determine the variable correlation; and the correlation, when determined, correlates the variable numbers x and y. Variables, like a , b , and c above, which are used to determine the correlation are called "constants," or parameters. The use
of the term "constant" in this connection for what is really a variable may seem at first sight to be odd; but it is really very natural. For the mathematical investigation is concerned with the relation between the variables x and y , after a , b , c are supposed to have been determined. So in a sense, relatively to x and y , the "constants" a , b , and c are constants. Thus ax + by = c stands for the general example of a certain algebraic form, that is, for a variable correlation belonging to a certain class.
Again we generalize x^2 + y^2 = 1 into ax^2 + by^2 = c , or still further into ax^2 + 2hxy + by^2 = c , or, still further, into ax^2 + 2hxy + by^2 + 2gx + 2fy = c.
Here again we are led to variable correlations which are indicated by their various algebraic forms.
Now let us turn to geometry. The name of the science at once recalls to our minds the thought of figures and diagrams exhibiting triangles and rectangles and squares and circles, all in special relations to each other. The study of the simple properties of these figures is the subject matter of elementary geometry, as it is rightly presented to the beginner. Yet a moment's thought will show that this is not the true conception of the subject. It may be right for a child to commence his geometrical reasoning on shapes, like triangles and squares, which he has cut out with scissors. What, however, is a triangle? It is a figure marked out and bounded by three bits of three straight lines.
Now the boundary of spaces by bits of lines is a very complicated idea, and not at all one which gives any hope of exhibiting the simple general conceptions which should form the bones of the subject. We want something more simple and more general. It is this obsession with the wrong initial ideas--very natural and good ideas for the creation
of first thoughts on the subject--which was the cause of the comparative sterility of the study of the science during so many centuries. Coordinate geometry, and Descartes its inventor, must have the credit of disclosing the true simple objects for geometrical thought.
In the place of a bit of a straight line, let us think of the whole of a straight line throughout its unending length in both directions. This is the sort of general idea from which to start our geometrical investigations. The Greeks never seem to have found any use for this conception which is now fundamental in all modern geometrical thought. Euclid always contemplates a straight line as drawn between two definite points, and is very careful to mention when it is to be produced beyond this segment. He never thinks of the line as an entity given once for all as a whole. This careful definition and limitation, so as to exclude an infinity not immediately apparent to the senses, was very characteristic of the Greeks in all their many activities. It is enshrined in the difference between Greek architecture and Gothic architecture, and between the Greek religion and the modern religion. The spire on a Gothic cathedral and the importance of the unbounded straight line in modern geometry are both emblematic of the transformation of the modern world.
The straight line, considered as a whole, is accordingly the root idea from which modern geometry starts. But then other sorts of lines occur to us, and we arrive at the conception of the complete curve which at every point of it exhibits some uniform characteristic, just as the straight line exhibits at all points the characteristic of straightness. For example, there is the circle which
at all points exhibits the characteristic of being at a given distance from its centre, and again there is the ellipse, which is an oval
curve, such that the sum of the two distances of any point on it from two fixed points, called
its foci, is constant for all points on the curve. It is evident that a circle is merely a particular case of an ellipse when the two foci are superposed in the same point; for then the sum of the two distances is merely twice the radius of the circle. The ancients knew the properties of the ellipse and the circle and, of course, considered them as wholes. For example, Euclid never starts with mere segments (i.e., bits) of circles, which are then prolonged. He always considers the whole circle as described. It is unfortunate that the circle is not the true fundamental line in geometry, so that his defective consideration of the straight line might have been of less consequence.
This general idea of a curve which at any
point of it exhibits some uniform property is expressed in geometry by the term "locus." A locus is the curve (or surface, if we do not confine ourselves to a plane) formed by points, all of which possess some given property. To every property in relation to each other which points can have, there corresponds some locus, which consists of all the points possessing the property. In investigating the properties of a locus considered as a whole, we consider any point or points on the locus. Thus in geometry we again meet with the fundamental idea of the variable. Furthermore, in classifying loci under such headings as straight lines, circles, ellipses, etc., we again find the idea of form.
Accordingly, as in algebra we are concerned with variable numbers, correlations between variable numbers, and the classification of correlations into types by the idea of algebraic form; so in geometry we are concerned with variable points, variable points satisfying some condition so as form to a locus, and the classification of loci into types by the idea of conditions of the same form.
Now, the essence of coordinate geometry is the identification of the algebraic correlation with the geometrical locus. The point on a plane is represented in algebra by its two coordinates, x and y , and the condition satisfied by any point on the locus is represented
by the corresponding correlation between x and y. Finally to correlations expressible in some general algebraic form, such as ax + by = c , there correspond loci of some general type, whose geometrical conditions are all of the same form. We have thus arrived at a position where we can effect a complete interchange in ideas and results between the two sciences. Each science throws light on the other, and itself gains immeasurably in power. It is impossible not to feel stirred at the thought of the emotions of men at certain historic moments of adventure and discovery--Columbus
when he first saw the Western shore, Pizarro when he stared at the Pacific
Ocean, Franklin when the electric spark came
from the string of his kite, Galileo when he
first turned his telescope to the heavens. Such moments are also granted to students in the abstract regions of thought, and high among them must be placed the morning when Descartes lay in bed and invented the method
of coordinate geometry.
When one has once grasped the idea of coordinate geometry, the immediate question which starts to the mind is, What sort of loci correspond to the well-known algebraic forms? For example, the simplest among the general types of algebraic forms is ax + by = c. The sort of locus which corresponds
to this is a straight line, and conversely to every straight line there corresponds an equation of this form. It is fortunate that the simplest among the geometrical loci should correspond to the simplest among the algebraic forms. Indeed, it is this general correspondence of geometrical and algebraic simplicity which gives to the whole subject its power. It springs from the fact that the connection between geometry and algebra is not casual and artificial, but deep-seated and essential. The equation which corresponds to a locus is called the equation "of" (or "to") the locus. Some examples of equations of straight lines will illustrate the subject.
Consider y - x = 0 ; here the a , b , and c , of the general form have been replaced by -1 , 1 , and 0 respectively. This line passes through the "origin," O , in the diagram and bisects the angle XOY. It is the line L'OL of the diagram. The fact that it passes through the origin, O , is easily seen by observing that the equation is satisfied by putting x = 0 and y = 0 simultaneously, but 0 and 0 are the coordinates of O. In fact it is easy to generalize and to see by the same method that the equation of any line through the origin is of the form ax + by = 0. The locus of the equation y + x = 0 also passes through the origin and bisects the angle X'OY : it is the line L_1OL_1' of the diagram.
Consider y - x = 1 : the corresponding locus does not pass through the origin. We therefore seek where it cuts the axes. It must cut the axis of x at some point of coordinates x and 0. But putting y = 0 in the equation, we get x = -1 ; so the coordinates of this point (A) are 1 and 0. Similarly the point (B) where the line cuts the axis OY are 0 and 1. The locus is the line AB in the figure and is parallel to LOL'. Similarly y + x = 1 is the equation of line A_1B of the figure; and the locus is parallel to L_1OL_1'. It is easy to prove the general theorem that two lines represented by equations of the forms ax + by = 0 and ax + by = c are parallel.
The group of loci which we next come upon are sufficiently important to deserve a chapter to themselves. But before going on to them we will dwell a little longer on the main ideas of the subject.
The position of any point P is determined by arbitrarily choosing an origin, O , two axes,
OX and OY , at right-angles, and then by noting its coordinates x and y , i.e. OM and PM (cf. fig. 13). Also, as we have seen in the last chapter, P can be determined by the "vector" OP , where the idea of the vector includes a determinate direction as well as a determinate length. From an abstract mathematical point of view the idea of an arbitrary origin may appear artificial and clumsy, and similarly for the arbitrarily drawn axes, OX and OY. But in relation to the application of mathematics to the event of the Universe we are here symbolizing with direct simplicity the most fundamental fact respecting the outlook on the world afforded to us by our senses. We each of us refer our sensible perceptions of things to an origin which we call "here": our location in a particular part of space round which we group the whole Universe is the essential fact of our bodily existence. We can imagine beings who observe all phenomena in all space with an equal eye, unbiassed in favour of any part. With us it is otherwise, a cat at our
feet claims more attention than an earthquake at Cape Horn, or than the destruction of a world in the Milky Way. It is true that in making a common stock of our knowledge with our fellowmen, we have to waive something of the strict egoism of our own individual "here." We substitute "nearly here" for "here"; thus we measure miles from the town hall of the nearest town, or from the capital of the country. In measuring the earth, men of science will put the origin at the earth's centre; astronomers
even rise to the extreme altruism of putting their origin inside the sun. But, far as this last origin may be, and even if we go further to some convenient point amid the nearer fixed stars, yet, compared to the immeasurable infinities of space, it remains true that our first procedure in exploring the Universe is to fix upon an origin "nearly here."
Again the relation of the coordinates OM and MP (i.e. x and y ) to the vector OP is an instance of the famous parallelogram law, as
can easily be seen (cf. fig. 8) by completing the parallelogram OMPN. The idea of the "vector" OP , that is, of a directed magnitude, is the root-idea of physical science. Any moving body has a certain magnitude of velocity in a certain direction, that is to say, its velocity is a directed magnitude, a vector. Again a force has a certain magnitude
and has a definite direction. Thus, when in analytical geometry the ideas of the "origin," of "coordinates," and of "vectors" are introduced, we are studying the abstract conceptions which correspond to the fundamental facts of the physical world.
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End of part 9 of 19
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