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 that trigonometry, during
the very best period of Greek thought. But the importance of
trigonometry, both to 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.
三角学的起源并非源自对自然周期性的普遍考察。在这一点上，它的历史类似于圆锥曲线的历史，后者同样起源于非常特定的概念。事实上，对比这两门科学的发展历程，可以得出一些非常有启发性的类比和对比。三角学与圆锥曲线一样，起源于希腊。它的发明者是希腊天文学家喜帕恰斯（约公元前160年出生），他在罗得岛进行天文观测。他对天文学的贡献极为重大，使其成为一门真正的科学，确立了重要的研究成果，并指明了正确的研究方法。或许，三角学的发明正是他对其主要研究领域——天文学——所作贡献中最重要的一项。接下来扩展三角学的是我们之前提到的伟大的亚历山大天文学家托勒密。至此，我们可以清楚地看到圆锥曲线与三角学之间的显著对比。三角学的起源是实用的——它的发明是天文研究的必然需求；而圆锥曲线的起源则是纯理论的，它最初受到研究的唯一原因是其中所蕴含的抽象数学思想的趣味。很有代表性的是，圆锥曲线大约在三角学发明前150年问世，正是在希腊思想最鼎盛的时期。然而，三角学在数学理论和应用中的重要性，仅仅是数学这门学科从实践应用中获得的无数富有成效的思想之一。
We will try to 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, of, 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 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 is the eye of the astronomer, and and are the directions in which the stars
are seen, it is easy to devise instruments which shall measure the angle
. 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, angle 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 , and at B to measure the
angle , and at C to measure the
angle . 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 error 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.
我们将尝试弄清楚三角学是什么，以及为什么它应该由天文学的科学研究所产生。首先，天文学家可以进行哪些测量？主要是时间测量和角度测量。天文学家可以调整一架望远镜（因为讨论现代天文学家熟悉的仪器更为方便），使其只能围绕东西方向的固定轴旋转；其结果是，望远镜只能指向南方，并具有不同程度的仰角，或者如果旋转超过天顶，则可以指向北方。这种仪器被称为子午仪（Transit
Instrument），它是精确测量恒星经过南方或北方时间的重要工具。然而，间接地，该仪器也能够测量角度。因为，当记录下两颗恒星经过子午线的时间间隔时，假设地球以匀速自转，我们就能计算出地球在这段时间内旋转的角度。此外，天文学家还可以使用其他仪器直接测量两颗恒星之间的角度。例如，假设E
代表天文学家的眼睛，EA 和 EB
分别是观察两颗恒星的方向，那么很容易设计出能够测量角度
AEB
的仪器。因此，当天文学家对星空进行测绘时，他实际上是在测量角度，以便确定某一时刻恒星和行星的相对方向。类似地，在陆地测量（land-surveying）中，角度测量也是最主要的测量内容。直接测量长度往往难以精准进行，因为河流、房屋、森林、山脉以及地形的各种不规则性都会成为障碍。因此，一个国家的测绘工作通常仅依赖一到两次在特定地点（如索尔兹伯里平原）进行的高精度长度测量，而测绘的主要工作则是角度测量。例如，在某个区域测量时，A、B
和 C
可能是一些醒目的参照点，比如教堂塔楼的顶端。这些点可以相互看见。那么，在
A 处测量角 BAC，在 B 处测量角 ABC，在
C 处测量角
BCA，就是一件非常简单的事情。从理论上讲，只需要测量其中的两个角即可，因为根据几何学中的著名定理，三角形的三个内角之和总是两个直角（180°），因此当两个角已知时，第三个角可以推算出来。然而，在实际操作中，测量所有三个角更加稳妥，因为这样可以检查和纠正可能存在的微小测量误差。在制图（map-making）过程中，整个国家都会以这种方式用三角形网格覆盖。这种方法被称为三角测量法（triangulation），是测量工作中的基础方法。
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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 (1) if to any point in one figure a point in the other figure
corresponds, so that to every angle a corresponding angle, and (2) 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 and 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, triangulations 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 states 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 similar
figures; they only differ in scale. The lengths of the circumferences of
two circles, such as and
in Fig.26, are in
proportion to the lengths of their radii. Furthermore, if the two
circles have the same centre , as
do the two circle in Fig.26, then the arcs and intercepted by the arms of any
angle , are also in proportion
to their radii. Hence the ratio of the length of the arc to the length of the radius , that is is a number
which is quite independent of the length , and is the same as the function . This
fraction of 'arc divided by radius' is 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 represents the magnitude of
the angle . Now draw perpendicularly to . The the Greek mathematicians called
the line the sine of the arc AP,
and the line the cosine of the
arc . They were well aware that
the importance of the relations of these various line 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 diagram. For us the case is
different: we wish to embody our more powerful ideas. Hence in modern
mathematics, instead of considering the arc , we consider the fraction , which is a number the same
for all lengths of ; and, instead
of considering the lines and
, we consider the fractions and , which again are numbers
not dependent on the length of ,
i.e. not dependent on the scale of our diagrams. Then we define the
number to be the sine
of the number , and
the number to be the
cosine of the number .
These fractional forms are clumsy to print; so let us put for the fraction , which represents the
magnitude of the angle , and put
for the fraction , and for the fraction . Then , are numbers, and , since we are
talking of angle , they are variable numbers. But a
correlation exists between their magnitudes, so that when (i.e. the angle ) is given, the magnitudes of and are definitely determined. Hence and are functions of the argument . we have called the sine of , and the cosine of . We wish to adapt the general
functional notation to these
special cases: so in modern mathematics we write sin for '' when we want to indicate the special
function of 'sine', and cos for ''
when we want to indicate the special function of 'cosine'. Thus, with
the above meaning for , we
get
三角学产生的根本问题是：给定一个三角形的角度大小，如何描述各边的相对大小。请注意，我们说的是“各边的相对大小”，因为根据相似理论，已知的仅仅是各边的比例。为了解答这个问题，引入了角度大小的某些函数，将角度的大小视为自变量。在最初，这些函数是通过考虑一个直角三角形得到的，角度的大小是通过圆弧的长度来定义的。在现代的基础教材中，圆弧的定义作为角度大小的标准已被稍微淡化，但这对理论或解释的清晰性并无好处。首先必须指出的是，在相似性相关的情况下，圆在曲线图形中占据的地位，与三角形在直线图形中的地位是相同的。任何两个相似的图形仅在尺度上有所不同。例如，在图26中，两个圆和的圆周长度与它们的半径成比例。而且，如果两个圆的圆心相同，如图26中的两个圆，那么由任意角的两条边截取的弧和也与它们的半径成比例。因此，弧长与半径之比，即，是一个与长度无关的数值，并且与相同。这种“弧长除以半径”的比值是度量角度大小的正确理论方法；因为它不依赖于任何任意的长度单位，也不依赖于任何假定角度（如直角）的划分方式。因此，分数代表了角度的大小。现在，画一条线垂直于。希腊数学家将线称为弧的正弦，而线称为弧的余弦。他们深知这些线之间的关系的重要性，正是基于我们刚才阐述的相似性理论。但他们并未将他们的定义表达为这一理论所产生的性质。此外，他们也没有现代关于函数作为变量对之间的关联这一概念，也未曾理解现代代数及代数分析的任何观念。因此，他们自然只考虑图示中某些线之间的关系。对我们来说，情况有所不同：我们希望体现更强大的概念。因此，在现代数学中，我们不再考虑弧，而是考虑分数，这是一个对于所有长度相同的数值；而且，我们不再考虑线和，而是考虑分数和，这些也是与长度无关的数值，即不依赖于我们图形的尺度。然后我们定义数值为数值的正弦，数值为数值的余弦。这些分数形式印刷起来不方便；因此，我们用表示分数，它代表了角度的大小，用表示分数，用表示分数。因此，、、是数值，并且由于我们讨论的是“任何”角度，它们是变量。但它们之间存在关联关系，因此当（即角度）给定时，和的大小是确定的。因此，和是的函数。我们称为的正弦，为的余弦。我们希望将一般的函数表示法适应到这些特殊的情况下：因此，在现代数学中，我们写sin表示“正弦”这一特殊函数，写cos表示“余弦”这一特殊函数。因此，按照上述、、的意义，我们得到 where the brackets surrounding the and are omitted for the special
functions. The meaning of these functions and as correlating the pairs of numbers
and , and and is, that the functional relations are
to be found by constructing (cf. Fig.26) an angle , whose measure ' divided by ' is equal to , and that then is the number given by ' divided by ' and is the number given by ' dividedby '.
It is evident that without some further definitions we shall get into
difficulties when the number is
taken too large. For then the arc may be greater that one-quarter of the
circumference of the circle, and the point (cf. Figs.26 and 27) may fall between
and and not between and . Also may be below the line and not above it as in Fig.26.
In order to get over this difficult we have recourse to the ideas and
conventions of co-ordinate geometry in making our complete definitions
of the sine and cosine. Let one arm of the angle be the axis , and produce the axis backwards to
obtain its negative part .
Draw the other axis
perpendicular to it. Let any point at a distance from have co-ordinates and . These co-ordinates are both positive
in the first 'quadrant' of the plan, e.g the co-ordinates and of in Fig.27. In the other quadrants,
either one or both of the co-ordinates are negative, for example, and for , and and for , and and for in Fig.27, where and are both negative numbers. The
positive angle is the arc divided by , its sine is and its cosine is ; the positive angle is the arc divided by , its sine is and cosine ; the positive angle
is the arc divided by , its sine is and its cosine is ; the positive angle
is the arc divided by
, its sine is and its cosine is .
But even now we have not gone far enough. For suppose we choose 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 , where is the Greek form of the letter and its name in the Greek alphabet is
'pi' . It can be proved that 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 . Mathematicians can easily
calculate to any degree of
accuracy required, just as
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
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 , 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 value . Suppose a moving point, , to start from on (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 or or or . Then the total length
of the curvilinear circular path traversed, divided by the radius of the
circle, , is the generalized
definition of a positive angle of size. Let be the co-ordinates of the point in
which the point rests, i.e. in
one of the four alternative positions mentioned in Fig.27; and (as here used) will either be and , or and , or and , or and . Then the sine of this generalized
angle is and its cosine
is . With these
definitions the functional relations and , are at last defined for
all positive real values of . For
negative values of we simply take
rotation of 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 and cos are periodic functions of . For consider the position, (in Fig.27), of a moving point, , which has started from and revolved round the circle. This
position marks the angles , and , and , and , and so on
indefinitely. Now, all these angles have the same sine and cosine,
namely , and . Hence it is easy to see that
, if be chosen to have any value,
the arguments and , and , and , and and so on indefinitely, have all
the same values for the corresponding sines and cosines. In other
words,
This fact is expressed by saying that sin and cos are periodic functions with their
period equal to .
这个事实可以通过说sin 和cos
是周期性函数，并且它们的周期等于来表达。
The graph of the function (notice that we now abandon and for the more familiar and ) is shown in Fig.28. We take on the
axis of any arbitrary length at
pleasure to represent the number , and on the axis of 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 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
form the sine. For it is easy to prove that ; hence
it can be seen that the graph of is simply Fig.28 modified by drawing the axis of through the point on marked , instead of drawing it in
its actual position on the figure.
函数的图像（注意，我们现在放弃了和，而使用更熟悉的和）如图28所示。我们在轴上任意选取一个长度来表示数字，在轴上也选取任意一个长度来表示数字1。正弦和余弦的数值永远不会超过1。可以注意到，图像每经过的周期就会重复一次。这个图像代表了最简单的周期性函数形式，所有其他周期性函数都可以由它构造出来。余弦与正弦没有本质上的不同。因为很容易证明$
x = (x + ) x OYOX$的点，而不是按照图中的实际位置绘制。
image-20250212003052256
It is easy to construct a 'sine' function in which the period has any
assigned value . For we have only
to write
构造一个周期可以赋值为任意值
的正弦函数是很容易的。我们只需写出 and then
然后 Thus the period of this new function is now . Let us now give a general definition
of what we mean by a periodic function. The function is periodic, with the period , if (1) for value of we have , and (2) there is no number
smaller than such that for value of , .
The second clause is put into the definition because when we have sin
, it is not only
periodic in the period , but also
in the periods and , and so on; this arises since
第二个条款被包含在定义中，因为当我们有 sin 时，它不仅在周期 内是周期性的，而且在周期 、 等周期内也是周期性的；这出现的原因是
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 whole of the applications of the theory to
physical science are dominated by an important theorem called Fourier's
Theorem [傅里叶定理]; namely that, if be a periodic function with the
period and if also satisfies certain conditions,
which practically are always presupposed in functions suggested by
natural phenomena, then can be
written as the sum of a set of terms in the form
因此，我们想要掌握的是最小的周期，并称之为函数的周期。周期函数抽象理论的大部分内容以及该理论在物理科学中的应用，都由一个重要定理主导，这个定理称为傅里叶定理；即如果
是一个周期函数，周期为 ，并且如果
还满足某些条件，这些条件在由自然现象提出的函数中实际上几乎总是被预设的，那么
可以写成一组项的和，形式为
In this formula , and , 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 sine
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 that 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 wind 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 142 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 will therefore be the
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 that its general method of
dealing with the great natural fact of Periodicity.
