An Introduction to Mathematics · Alfred North Whitehead
Trigonometry did not take its rise from
the general consideration of the periodicity of nature. In this respect its history is analogous to that of conic sections, which also had their origin in very particular ideas. Indeed, a comparison of the histories of the two sciences yields some very instructive analogies and contrasts. Trigonometry, like conic sections, had its origin among the Greeks. Its inventor was Hipparchus (born about 160 B.C.),
a Greek astronomer, who made his observations at Rhodes. His services to astronomy were very great, and it left his
hands a truly scientific subject with important results established, and the right method of progress indicated. Perhaps the invention of trigonometry was not the least of these services to the main science of his study. The next man who extended trigonometry was Ptolemy, the great Alexandrian astronomer,
whom we have already mentioned. We now
see at once the great contrast between conic sections and trigonometry. The origin of trigonometry was practical; it was invented because it was necessary for astronomical research. The origin of conic sections was purely theoretical. The only reason for its initial study was the abstract interest of the ideas involved. Characteristically enough conic sections were invented about 150 years earlier than trigonometry, during the very best period of Greek thought. But the importance of trigonometry, both to the theory and the application of mathematics, is only one of innumerable instances of the fruitful ideas which the general science has gained from its practical applications.
We will try and make clear to ourselves what trigonometry is, and why it should be generated by the scientific study of astronomy.
In the first place: What are the measurements which can be made by an astronomer? They are measurements of time and measurements of angles. The astronomer may adjust a telescope (for it is easier to discuss the familiar instrument of modern astronomers) so that it can only turn about a fixed axis pointing east and west; the result is that the telescope can only point to the south, with a greater or less elevation of direction, or, if turned round beyond the zenith, point to the north. This is the transit instrument, the
great instrument for the exact measurement of the times at which stars are due south or due north. But indirectly this instrument measures angles. For when the time elapsed between the transits of two stars has been noted, by the assumption of the uniform rotation of the earth, we obtain the angle through which the earth has turned in that period of time. Again, by other instruments, the angle between two stars can be directly measured. For if E is the eye of the astronomer,
and EA and EB are the directions in which the stars are seen, it is easy to devise instruments which shall measure the angle AEB. Hence, when the astronomer is forming a survey of the heavens, he is, in fact, measuring angles so as to fix the relative directions of the stars and planets at any instant. Again, in the analogous problem of
land-surveying, angles are the chief subject of measurements. The direct measurements of length are only rarely possible with any accuracy; rivers, houses, forests, mountains, and general irregularities of ground all get in the way. The survey of a whole country will depend only on one or two direct measurements of length, made with the greatest elaboration in selected places like Salisbury Plain. The main work of a survey is the measurement of angles. For example, A , B , and C will be conspicuous points in the district
surveyed, say the tops of church towers. These points are visible each from the others. Then it is a very simple matter at A to measure the angle BAC , and at B to measure the angle ABC , and at C to measure the angle BCA. Theoretically, it is only necessary to measure two of these angles; for, by a well-known proposition in geometry, the sum of the three angles of a triangle amounts to two
right-angles, so that when two of the angles are known, the third can be deduced. It is better, however, in practice to measure all three, and then any small errors of observation can be checked. In the process of map-making a country is completely covered with triangles in this way. This process is called triangulation, and is the fundamental process
in a survey.
Now, when all the angles of a triangle are
known, the shape of the triangle is known--that is, the shape as distinguished from the size. We here come upon the great principle of geometrical similarity. The idea is very familiar to us in its practical applications. We are all familiar with the idea of a plan drawn to scale. Thus if the scale of a plan be an inch to a yard, a length of three inches in the plan means a length of three yards in the original. Also the shapes depicted in the plan are the shapes in the original, so that a right-angle in the original appears as a right-angle in the plan. Similarly in a map, which is only a plan of a country, the proportions of the lengths in the map are the proportions of the distances between the places indicated, and the directions in the map are the directions in the country. For example, if in the map one place is north-north-west of the other, so it is in reality; that is to say, in a map the angles are the same as in reality.
Geometrical similarity may be defined thus: Two figures are similar (i) if to any point in one figure a point in the other figure corresponds, so that to every line there is a corresponding line, and to every angle a corresponding angle, and (ii) if the lengths of corresponding lines are in a fixed proportion, and the magnitudes of corresponding angles are the same. The fixed proportion of the lengths of corresponding lines in a map (or plan) and in the original is called the scale of the map. The scale should always be indicated on the margin of every map and plan. It has already been pointed out that two triangles whose angles are respectively equal are similar. Thus, if the two triangles
ABC and DEF have the angles at A and D equal, and those at B and E , and those at C and F , then DE is to AB in the same proportion
as EF is to BC , and as FD is to CA. But it is not true of other figures that similarity is guaranteed by the mere equality of angles. Take for example, the familiar cases of a rectangle and a square. Let ABCD be a square, and ABEF be a rectangle. Then all the corresponding angles are equal. But
whereas the side AB of the square is equal to the side AB of the rectangle, the side BC of the square is about half the size of the side BE of the rectangle. Hence it is not true that the square ABCD is similar to the rectangle ABEF. This peculiar property of the triangle, which is not shared by other rectilinear figures, makes it the fundamental figure in the theory of similarity. Hence in surveys, triangulation is the fundamental process; and hence also arises the word "trigonometry,"
derived from the two Greek words trigonon a triangle and metria measurement. The fundamental question from which trigonometry arose is this: Given the magnitudes of the angles of a triangle, what can be stated as to the relative magnitudes of the sides. Note that we say "relative magnitudes of the sides," since by the theory of similarity it is only the proportions of the sides which are known. In order to answer this question, certain functions of the magnitudes of an angle, considered as the argument, are introduced. In their origin these functions were got at by considering a right-angled triangle, and the magnitude of the angle was defined by the length of the arc of a circle. In modern elementary books, the fundamental position of the arc of the circle as defining the magnitude of the angle has been pushed somewhat to the background, not to the advantage either of theory or clearness of explanation. It must first be noticed that, in relation to similarity, the circle holds the same fundamental position among curvilinear figures, as does the triangle among rectilinear figures. Any two circles are similar figures; they only differ in scale. The lengths of the circumferences of two circles, such as APA' and A_1 P_1 A_1' in the fig. 26 are in proportion to the lengths of their radii. Furthermore, if the two circles have the same
centre O , as do the two circles in fig. 26, then the arcs AP and A_1 P_1 intercepted by the arms of any angle AOP , are also in proportion to their radii. Hence the ratio of the
length of the arc AP to the length of the radius OP , that is arc AP/radius OP is a number which is quite independent of the length OP , and is the same as the fraction arc A_1 P_1/radius OP_1. This fraction of "arc divided by radius" is the proper theoretical way to measure the magnitude of
an angle; for it is dependent on no arbitrary unit of length, and on no arbitrary way of dividing up any arbitrarily assumed angle, such as a right-angle. Thus the fraction AP/OA represents the magnitude of the angle AOP. Now draw PM perpendicularly to OA. Then the Greek mathematicians called the line PM the sine of the arc AP , and the line OM the cosine of the arc AP. They were well aware that the importance of the relations of these various lines to each other was dependent on the theory of similarity which we have just expounded. But they did not make their definitions express the properties which arise from this theory. Also they had not in their heads the modern general ideas respecting functions as correlating pairs of variable numbers, nor in fact were they aware of any modern conception of algebra and algebraic analysis. Accordingly, it was natural to them to think merely of the relations between certain lines in a diagram. For us the case is different: we wish to embody our more powerful ideas.
Hence, in modern mathematics, instead of considering the arc AP , we consider the fraction AP/OP , which is a number the same for all lengths of OP ; and, instead of considering the lines PM and OM , we consider
the fractions PM/OP and OM/OP , which again are numbers not dependent on the length of OP , i.e. not dependent on the scale of our diagrams. Then we define the number PM/OP to be the sine of the number PA/OP , and the number OM/OP to be the cosine of the number PA/OP. These fractional forms are clumsy to print; so let us put u for the fraction AP/OP , which represents the magnitude of the angle AOP , and put v for the fraction PM/OM , and w for the fraction OM/OP. Then u , v , w , are numbers, and, since we are talking of any angle AOP , they are variable numbers. But a correlation exists between their magnitudes, so that when u (i.e. the angle AOP ) is given the magnitudes of v and w are definitely determined. Hence v and w are functions of the argument u. We have called v the sine of u , and w the cosine of u. We wish to adapt the general functional notation y = f(x) to these special cases: so in modern mathematics
we write " sin " for " f " when we want to
indicate the special function of "sine," and " cos " for " f " when we want to indicate the special function of "cosine." Thus, with the above meanings for u , v , w , we get v = sin u,and w = cos u, where the brackets surrounding the x in f(x) are omitted for the special functions. The meaning of these functions sin and cos as correlating the pairs of numbers u and v , and u and w is, that the functional relations are to be found by constructing (cf. fig. 26) an angle AOP , whose measure " AP divided by OP " is equal to u , and that then v is the number given by " PM divided by OP " and w is the number given by " OM divided by OP."
It is evident that without some further definitions we shall get into difficulties when the number u is taken too large. For then the arc AP may be greater than one-quarter of the circumference of the circle, and the point M (cf. figs. 26 and 27) may fall between O and A' and not between O and A. Also P may be below the line AOA' and not above it as in fig. 26. In order to get over this difficulty we have recourse to the ideas and conventions of coordinate geometry in making our complete definitions of the sine and cosine. Let one arm OA of the angle be the axis OX , and produce the axis backwards to obtain its negative part OX'. Draw the
other axis YOY' perpendicular to it. Let any point P at a distance r from O have coordinates x and y. These coordinates are both positive in the first "quadrant" of the plan, e.g. the coordinates x and y of P
in fig. 27. In the other quadrants, either one or both of the coordinates are negative, for example, x' and y for P' , and x' and y' for P" , and x and y' for P"' in fig. 27, where x' and y' are both negative numbers. The positive angle POA is the arc AP divided by r , its sine is y/r and its cosine is x/r ; the positive
angle AOP' is the arc ABP' divided by r , its sine is y/r and cosine x'/r ; the positive angle AOP" is the arc ABA'P" divided by r , its sine is y'/r and its cosine is x'/r ; the positive angle AOP"' is the arc ABA'B'P"' divided by r , its sine is y'/r and its cosine is x/r.
But even now we have not gone far enough. For suppose we choose u to be a number greater than the ratio of the whole circumference of the circle to its radius. Owing to the similarity of all circles this ratio is the same for all circles. It is always denoted in mathematics by the symbol 2π , where π is the Greek form of the letter p and its name in the Greek alphabet is "pi." It can be proved that π is an incommensurable number, and that therefore its value cannot be expressed by any fraction, or by any terminating or recurring decimal. Its value to a few decimal places is 3.14159 ; for many purposes a sufficiently accurate approximate value is 22/7. Mathematicians can easily calculate π to any degree of accuracy required, just as sqrt(2) can be so calculated. Its value has been actually given to 707 places of
decimals. Such elaboration of calculation is merely a curiosity, and of no practical or theoretical interest. The accurate determination of π is one of the two parts of the famous problem of squaring the circle.
The other part of the problem is, by the theoretical methods of pure geometry to describe a straight line equal in length to the circumference. Both parts of the problem are now known to be impossible; and the insoluble problem has now lost all special practical or theoretical interest, having become absorbed in wider ideas.
After this digression on the value of π , we now return to the question of the general definition of the magnitude of an angle, so as to be able to produce an angle corresponding to any value u. Suppose a moving point, Q , to start from A on OX (cf. fig. 27), and to rotate in the positive direction (anti-clockwise, in the figure considered) round the circumference of the circle for any number of times, finally resting at any point, e.g. at P or P' or P" or P"'. Then the total length of the curvilinear circular path traversed, divided by the radius of the circle, r , is the generalized definition of a positive angle of any size. Let x , y be the coordinates of the point in which the point Q rests, i.e. in one of the four alternative positions mentioned in fig. 27; x and y (as here used) will either be x and y , or x' and y , or x' and y' , or x and y'.
Then the sign of this generalized angle is y/r and its cosine is x/r. With these definitions the functional relations v = sin u and w = cos u , are at last defined for all positive real values of u. For negative values of u we simply take rotation of Q in the opposite (clockwise) direction; but it is not worth our while to elaborate further on this point, now that the general method of procedure has been explained.
These functions of sine and cosine, as thus defined, enable us to deal with the problems concerning the triangle from which Trigonometry took its rise. But we are now in a position to relate Trigonometry to the wider idea of Periodicity of which the importance
was explained in the last chapter. It is easy to see that the functions sin u and cos u are periodic functions of u. For consider the position, P (in fig. 27), of a moving point, Q , which has started from A and revolved round the circle. This position, P , marks the angles arc AP/r , and 2π + arc AP/r , and 4π + arc AP/r , and 6π + arc AP/r , and so on indefinitely. Now, all these angles have the same sine and cosine, namely, y/r and x/r. Hence it is easy to see that,
if u be chosen to have any value, the arguments u and 2π + u , and 4π + u , and 6π + u , and 8π + u and so on indefinitely, have all the same values for the corresponding sines and cosines. In other words, sin u = sin (2π + u) = sin (4π + u) = sin (6π + u) = etc.; ; cos u = cos (2π + u) = cos (4π + u) = cos (6π + u) = etc. This fact is expressed by saying that sin u and cos u are periodic functions with their period equal to 2π.
The graph of the function y = sin x (notice that we now abandon v and u for the more familiar y and x ) is shown in fig. 28. We take on the axis of x any arbitrary length at pleasure to represent the number π , and on the axis of y any arbitrary length at pleasure to represent the number 1. The numerical values of the sine and cosine can never exceed unity. The recurrence of the figure after periods of 2π will be noticed. This graph represents the simplest style of periodic function, out of which all others are constructed. The cosine gives nothing fundamentally different from the sine. For it is easy to prove that cos x = sin (x + π/2) ; hence it can be seen that the graph of cos x is simply fig. 28 modified by
drawing the axis of OY through the point on OX marked π/2 , instead of drawing it in its actual position on the figure.
It is easy to construct a 'sine' function in
which the period has any assigned value a. For we have only to write y = sin 2π x/a, and then sin 2π (x + a)/a = sin (2π x/a + 2π) = sin 2π x/a. Thus the period of this new function is now a. Let us now give a general definition of what
we mean by a periodic function. The function f(x) is periodic, with the period a , if (i) for any value of x we have f(x) = f(x + a) , and (ii) there is no number b smaller than a such that for any value of x , f(x) = f(x + b).
The second clause is put into the definition because when we have sin 2π x/a , it is not only periodic in the period a , but also in the periods 2a and 3a , and so on; this arises since sin 2π (x + 3a)/a = sin (2π x/a + 6π) = sin 2π x/a. So it is the smallest period which we want to get hold of and call the period of the function. The greater part of the abstract theory of periodic functions and the whole of the applications of the theory to Physical Science are dominated by an important theorem called Fourier's Theorem; namely that, if f(x) be a
periodic function with the period a and if f(x) also satisfies certain conditions, which practically are always presupposed in functions suggested by natural phenomena, then f(x) can be written as the sum of a set of terms in the form c_0 + c_1 sin (2π x/a + e_1) + c_2 sin (4π x/a + e_2) ; + c_3 sin (6π x/a + e_3) + etc.
In this formula c_0 , c_1 , c_2 , c_3 , etc., and also e_1 , e_2 , e_3 , etc., are constants, chosen so as to suit the particular function. Again we have to ask, How many terms have to be chosen? And here a new difficulty arises: for we can prove that, though in some particular cases a definite number will do, yet in general all we can do is to approximate as closely as we like to the value of the function by taking more and more terms. This process of gradual approximation brings us to the consideration of the theory of infinite series, an essential part of mathematical theory which we will consider in the next chapter.
The above method of expressing a periodic
function as a sum of sines is called the "harmonic analysis" of the function. For example, at any point on the sea coast the tides rise and fall periodically. Thus at a point near the Straits of Dover there will be two daily tides due to the rotation of the earth. The daily rise and fall of the tides are complicated by the fact that there are two tidal waves, one coming up the English Channel, and the other which has swept round the North of Scotland, and has then come southward down the North Sea. Again some high tides are higher than others: this is due to the fact that the Sun has also a tide-generating influence as well as the Moon. In this way monthly and other periods are introduced.
We leave out of account the exceptional influence of winds which cannot be foreseen. The general problem of the harmonic analysis of the tides is to find sets of terms like those in the expression on page 191 above, such that each set will give with approximate accuracy the contribution of the tide-generating influences of one "period" to the height of the tide at any instant. The argument x will therefore be the time reckoned from any convenient commencement.
Again, the motion of vibration of a violin string is submitted to a similar harmonic analysis, and so are the vibrations of the ether and the air, corresponding respectively to waves of light and waves of sound. We are here in the presence of one of the fundamental processes of mathematical physics--namely, nothing less than its general method of dealing with the great natural fact of Periodicity.
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End of part 13 of 19
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