按键盘上方向键 ← 或 → 可快速上下翻页，按键盘上的 Enter 键可回到本书目录页，按键盘上方向键 ↑ 可回到本页顶部！
————未阅读完？加入书签已便下次继续阅读！
principle。 Let B be the outer rainbow； A the inner one； let R stand
for the red colour； G for green； V for violet； yellow appears at the
point Y。 Three rainbows or more are not found because even the
second is fainter； so that the third reflection can have no strength
whatever and cannot reach the sun at all。 （See diagram。）
5
The rainbow can never be a circle nor a segment of a circle
greater than a semicircle。 The consideration of the diagram will prove
this and the other properties of the rainbow。 （See diagram。）
Let A be a hemisphere resting on the circle of the horizon； let
its centre be K and let H be another point appearing on the horizon。
Then； if the lines that fall in a cone from K have HK as their axis；
and； K and M being joined； the lines KM are reflected from the
hemisphere to H over the greater angle； the lines from K will fall
on the circumference of a circle。 If the reflection takes place when
the luminous body is rising or setting the segment of the circle above
the earth which is cut off by the horizon will be a semi…circle； if
the luminous body is above the horizon it will always be less than a
semicircle； and it will be smallest when the luminous body culminates。
First let the luminous body be appearing on the horizon at the point
H； and let KM be reflected to H； and let the plane in which A is；
determined by the triangle HKM； be produced。 Then the section of the
sphere will be a great circle。 Let it be A （for it makes no difference
which of the planes passing through the line HK and determined by
the triangle KMH is produced）。 Now the lines drawn from H and K to a
point on the semicircle A are in a certain ratio to one another； and
no lines drawn from the same points to another point on that
semicircle can have the same ratio。 For since both the points H and
K and the line KH are given； the line MH will be given too；
consequently the ratio of the line MH to the line MK will be given
too。 So M will touch a given circumference。 Let this be NM。 Then the
intersection of the circumferences is given； and the same ratio cannot
hold between lines in the same plane drawn from the same points to any
other circumference but MN。
Draw a line DB outside of the figure and divide it so that
D：B=MH：MK。 But MH is greater than MK since the reflection of the
cone is over the greater angle （for it subtends the greater angle of
the triangle KMH）。 Therefore D is greater than B。 Then add to B a line
Z such that B＋Z：D=D：B。 Then make another line having the same ratio to
B as KH has to Z； and join MI。
Then I is the pole of the circle on which the lines from K fall。 For
the ratio of D to IM is the same as that of Z to KH and of B to KI。 If
not； let D be in the same ratio to a line indifferently lesser or
greater than IM； and let this line be IP。 Then HK and KI and IP will
have the same ratios to one another as Z； B； and D。 But the ratios
between Z； B； and D were such that Z＋B：D=D： B。 Therefore
IH：IP=IP：IK。 Now； if the points K； H be joined with the point P by the
lines HP； KP； these lines will be to one another as IH is to IP； for
the sides of the triangles HIP； KPI about the angle I are
homologous。 Therefore； HP too will be to KP as HI is to IP。 But this
is also the ratio of MH to MK； for the ratio both of HI to IP and of
MH to MK is the same as that of D to B。 Therefore； from the points
H； K there will have been drawn lines with the same ratio to one
another； not only to the circumference MN but to another point as
well； which is impossible。 Since then D cannot bear that ratio to
any line either lesser or greater than IM （the proof being in either
case the same）； it follows that it must stand in that ratio to MI
itself。 Therefore as MI is to IK so IH will be to MI and finally MH to
MK。
If； then； a circle be described with I as pole at the distance MI it
will touch all the angles which the lines from H and K make by their
reflection。 If not； it can be shown； as before； that lines drawn to
different points in the semicircle will have the same ratio to one
another； which was impossible。 If； then； the semicircle A be
revolved about the diameter HKI； the lines reflected from the points
H； K at the point M will have the same ratio； and will make the
angle KMH equal； in every plane。 Further； the angle which HM and MI
make with HI will always be the same。 So there are a number of
triangles on HI and KI equal to the triangles HMI and KMI。 Their
perpendiculars will fall on HI at the same point and will be equal。
Let O be the point on which they fall。 Then O is the centre of the
circle； half of which； MN； is cut off by the horizon。 （See diagram。）
Next let the horizon be ABG but let H have risen above the
horizon。 Let the axis now be HI。 The proof will be the same for the
rest as before； but the pole I of the circle will be below the horizon
AG since the point H has risen above the horizon。 But the pole； and
the centre of the circle； and the centre of that circle （namely HI）
which now determines the position of the sun are on the same line。 But
since KH lies above the diameter AG； the centre will be at O on the
line KI below the plane of the circle AG determined the position of
the sun before。 So the segment YX which is above the horizon will be
less than a semicircle。 For YXM was a semicircle and it has now been
cut off by the horizon AG。 So part of it； YM； will be invisible when
the sun has risen above the horizon； and the segment visible will be
smallest when the sun is on the meridian； for the higher H is the
lower the pole and the centre of the circle will be。
In the shorter days after the autumn equinox there may be a
rainbow at any time of the day； but in the longer days from the spring
to the autumn equinox there cannot be a rainbow about midday。 The
reason for this is that when the sun is north of the equator the
visible arcs of its course are all greater than a semicircle； and go
on increasing； while the invisible arc is small； but when the sun is
south of the equator the visible arc is small and the invisible arc
great； and the farther the sun moves south of the equator the
greater is the invisible arc。 Consequently； in the days near the
summer solstice； the size of the visible arc is such that before the
point H reaches the middle of that arc； that is its point of
culmination； the point is well below the horizon； the reason for
this being the great size of the visible arc； and the consequent
distance of the point of culmination from the earth。 But in the days
near the winter solstice the visible arcs are small； and the
contrary is necessarily the case： for the sun is on the meridian
before the point H has risen far。
6
M