ON THE STRANGE REFRACTION OF ICELAND CRYSTAL
1.
There is brought from Iceland, which is an Island in the North Sea, in
the latitude of 66 degrees, a kind of Crystal or transparent stone,
very remarkable for its figure and other qualities, but above all for
its strange refractions. The causes of this have seemed to me to be
worthy of being carefully investigated, the more so because amongst
transparent bodies this one alone does not follow the ordinary rules
with respect to rays of light. I have even been under some necessity
to make this research, because the refractions of this Crystal seemed
to overturn our preceding explanation of regular refraction; which
explanation, on the contrary, they strongly confirm, as will be seen
after they have been brought under the same principle. In Iceland are
found great lumps of this Crystal, some of which I have seen of 4 or 5
pounds. But it occurs also in other countries, for I have had some of
the same sort which had been found in France near the town of Troyes
in Champagne, and some others which came from the Island of Corsica,
though both were less clear and only in little bits, scarcely capable
of letting any effect of refraction be observed.
2. The first knowledge which the public has had about it is due to Mr.
Erasmus Bartholinus, who has given a description of Iceland Crystal
and of its chief phenomena. But here I shall not desist from giving my
own, both for the instruction of those who may not have seen his book,
and because as respects some of these phenomena there is a slight
difference between his observations and those which I have made: for I
have applied myself with great exactitude to examine these properties
of refraction, in order to be quite sure before undertaking to explain
the causes of them.
3. As regards the hardness of this stone, and the property which it
has of being easily split, it must be considered rather as a species
of Talc than of Crystal. For an iron spike effects an entrance into it
as easily as into any other Talc or Alabaster, to which it is equal in
gravity.
[Illustration]
4. The pieces of it which are found have the figure of an oblique
parallelepiped; each of the six faces being a parallelogram; and it
admits of being split in three directions parallel to two of these
opposed faces. Even in such wise, if you will, that all the six faces
are equal and similar rhombuses. The figure here added represents a
piece of this Crystal. The obtuse angles of all the parallelograms, as
C, D, here, are angles of 101 degrees 52 minutes, and consequently
the acute angles, such as A and B, are of 78 degrees 8 minutes.
5. Of the solid angles there are two opposite to one another, such as
C and E, which are each composed of three equal obtuse plane angles.
The other six are composed of two acute angles and one obtuse. All
that I have just said has been likewise remarked by Mr. Bartholinus in
the aforesaid treatise; if we differ it is only slightly about the
values of the angles. He recounts moreover some other properties of
this Crystal; to wit, that when rubbed against cloth it attracts
straws and other light things as do amber, diamond, glass, and Spanish
wax. Let a piece be covered with water for a day or more, the surface
loses its natural polish. When aquafortis is poured on it it produces
ebullition, especially, as I have found, if the Crystal has been
pulverized. I have also found by experiment that it may be heated to
redness in the fire without being in anywise altered or rendered less
transparent; but a very violent fire calcines it nevertheless. Its
transparency is scarcely less than that of water or of Rock Crystal,
and devoid of colour. But rays of light pass through it in another
fashion and produce those marvellous refractions the causes of which I
am now going to try to explain; reserving for the end of this Treatise
the statement of my conjectures touching the formation and
extraordinary configuration of this Crystal.
6. In all other transparent bodies that we know there is but one sole
and simple refraction; but in this substance there are two different
ones. The effect is that objects seen through it, especially such as
are placed right against it, appear double; and that a ray of
sunlight, falling on one of its surfaces, parts itself into two rays
and traverses the Crystal thus.
7. It is again a general law in all other transparent bodies that the
ray which falls perpendicularly on their surface passes straight on
without suffering refraction, and that an oblique ray is always
refracted. But in this Crystal the perpendicular ray suffers
refraction, and there are oblique rays which pass through it quite
straight.
[Illustration]
8. But in order to explain these phenomena more particularly, let
there be, in the first place, a piece ABFE of the same Crystal, and
let the obtuse angle ACB, one of the three which constitute the
equilateral solid angle C, be divided into two equal parts by the
straight line CG, and let it be conceived that the Crystal is
intersected by a plane which passes through this line and through the
side CF, which plane will necessarily be perpendicular to the surface
AB; and its section in the Crystal will form a parallelogram GCFH. We
will call this section the principal section of the Crystal.
9. Now if one covers the surface AB, leaving there only a small
aperture at the point K, situated in the straight line CG, and if one
exposes it to the sun, so that his rays face it perpendicularly above,
then the ray IK will divide itself at the point K into two, one of
which will continue to go on straight by KL, and the other will
separate itself along the straight line KM, which is in the plane
GCFH, and which makes with KL an angle of about 6 degrees 40 minutes,
tending from the side of the solid angle C; and on emerging from the
other side of the Crystal it will turn again parallel to JK, along MZ.
And as, in this extraordinary refraction, the point M is seen by the
refracted ray MKI, which I consider as going to the eye at I, it
necessarily follows that the point L, by virtue of the same
refraction, will be seen by the refracted ray LRI, so that LR will be
parallel to MK if the distance from the eye KI is supposed very great.
The point L appears then as being in the straight line IRS; but the
same point appears also, by ordinary refraction, to be in the straight
line IK, hence it is necessarily judged to be double. And similarly if
L be a small hole in a sheet of paper or other substance which is laid
against the Crystal, it will appear when turned towards daylight as if
there were two holes, which will seem the wider apart from one another
the greater the thickness of the Crystal.
10. Again, if one turns the Crystal in such wise that an incident ray
NO, of sunlight, which I suppose to be in the plane continued from
GCFH, makes with GC an angle of 73 degrees and 20 minutes, and is
consequently nearly parallel to the edge CF, which makes with FH an
angle of 70 degrees 57 minutes, according to the calculation which I
shall put at the end, it will divide itself at the point O into two
rays, one of which will continue along OP in a straight line with NO,
and will similarly pass out of the other side of the crystal without
any refraction; but the other will be refracted and will go along OQ.
And it must be noted that it is special to the plane through GCF and
to those which are parallel to it, that all incident rays which are in
one of these planes continue to be in it after they have entered the
Crystal and have become double; for it is quite otherwise for rays in
all other planes which intersect the Crystal, as we shall see
afterwards.
11. I recognized at first by these experiments and by some others that
of the two refractions which the ray suffers in this Crystal, there is
one which follows the ordinary rules; and it is this to which the rays
KL and OQ belong. This is why I have distinguished this ordinary
refraction from the other; and having measured it by exact
observation, I found that its proportion, considered as to the Sines
of the angles which the incident and refracted rays make with the
perpendicular, was very precisely that of 5 to 3, as was found also by
Mr. Bartholinus, and consequently much greater than that of Rock
Crystal, or of glass, which is nearly 3 to 2.
[Illustration]
12. The mode of making these observations exactly is as follows. Upon
a leaf of paper fixed on a thoroughly flat table there is traced a
black line AB, and two others, CED and KML, which cut it at right
angles and are more or less distant from one another according as it
is desired to examine a ray that is more or less oblique. Then place
the Crystal upon the intersection E so that the line AB concurs with
that which bisects the obtuse angle of the lower surface, or with some
line parallel to it. Then by placing the eye directly above the line
AB it will appear single only; and one will see that the portion
viewed through the Crystal and the portions which appear outside it,
meet together in a straight line: but the line CD will appear double,
and one can distinguish the image which is due to regular refraction
by the circumstance that when one views it with both eyes it seems
raised up more than the other, or again by the circumstance that, when
the Crystal is turned around on the paper, this image remains
stationary, whereas the other image shifts and moves entirely around.
Afterwards let the eye be placed at I (remaining always in the plane
perpendicular through AB) so that it views the image which is formed
by regular refraction of the line CD making a straight line with the
remainder of that line which is outside the Crystal. And then, marking
on the surface of the Crystal the point H where the intersection E
appears, this point will be directly above E. Then draw back the eye
towards O, keeping always in the plane perpendicular through AB, so
that the image of the line CD, which is formed by ordinary refraction,
may appear in a straight line with the line KL viewed without
refraction; and then mark on the Crystal the point N where the point
of intersection E appears.
13. Then one will know the length and position of the lines NH, EM,
and of HE, which is the thickness of the Crystal: which lines being
traced separately upon a plan, and then joining NE and NM which cuts
HE at P, the proportion of the refraction will be that of EN to NP,
because these lines are to one another as the sines of the angles NPH,
NEP, which are equal to those which the incident ray ON and its
refraction NE make with the perpendicular to the surface. This
proportion, as I have said, is sufficiently precisely as 5 to 3, and
is always the same for all inclinations of the incident ray.
14. The same mode of observation has also served me for examining the
extraordinary or irregular refraction of this Crystal. For, the point
H having been found and marked, as aforesaid, directly above the point
E, I observed the appearance of the line CD, which is made by the
extraordinary refraction; and having placed the eye at Q, so that this
appearance made a straight line with the line KL viewed without
refraction, I ascertained the triangles REH, RES, and consequently the
angles RSH, RES, which the incident and the refracted ray make with
the perpendicular.
15. But I found in this refraction that the ratio of FR to RS was not
constant, like the ordinary refraction, but that it varied with the
varying obliquity of the incident ray.
16. I found also that when QRE made a straight line, that is, when the
incident ray entered the Crystal without being refracted (as I
ascertained by the circumstance that then the point E viewed by the
extraordinary refraction appeared in the line CD, as seen without
refraction) I found, I say, then that the angle QRG was 73 degrees 20
minutes, as has been already remarked; and so it is not the ray
parallel to the edge of the Crystal, which crosses it in a straight
line without being refracted, as Mr. Bartholinus believed, since that
inclination is only 70 degrees 57 minutes, as was stated above. And
this is to be noted, in order that no one may search in vain for the
cause of the singular property of this ray in its parallelism to the
edges mentioned.
[Illustration]
17. Finally, continuing my observations to discover the nature of
this refraction, I learned that it obeyed the following remarkable
rule. Let the parallelogram GCFH, made by the principal section of the
Crystal, as previously determined, be traced separately. I found then
that always, when the inclinations of two rays which come from
opposite sides, as VK, SK here, are equal, their refractions KX and KT
meet the bottom line HF in such wise that points X and T are equally
distant from the point M, where the refraction of the perpendicular
ray IK falls; and this occurs also for refractions in other sections
of this Crystal. But before speaking of those, which have also other
particular properties, we will investigate the causes of the phenomena
which I have already reported.
It was after having explained the refraction of ordinary transparent
bodies by means of the spherical emanations of light, as above, that I
resumed my examination of the nature of this Crystal, wherein I had
previously been unable to discover anything.
18. As there were two different refractions, I conceived that there
were also two different emanations of waves of light, and that one
could occur in the ethereal matter extending through the body of the
Crystal. Which matter, being present in much larger quantity than is
that of the particles which compose it, was alone capable of causing
transparency, according to what has been explained heretofore. I
attributed to this emanation of waves the regular refraction which is
observed in this stone, by supposing these waves to be ordinarily of
spherical form, and having a slower progression within the Crystal
than they have outside it; whence proceeds refraction as I have
demonstrated.
19. As to the other emanation which should produce the irregular
refraction, I wished to try what Elliptical waves, or rather
spheroidal waves, would do; and these I supposed would spread
indifferently both in the ethereal matter diffused throughout the
crystal and in the particles of which it is composed, according to the
last mode in which I have explained transparency. It seemed to me that
the disposition or regular arrangement of these particles could
contribute to form spheroidal waves (nothing more being required for
this than that the successive movement of light should spread a little
more quickly in one direction than in the other) and I scarcely
doubted that there were in this crystal such an arrangement of equal
and similar particles, because of its figure and of its angles with
their determinate and invariable measure. Touching which particles,
and their form and disposition, I shall, at the end of this Treatise,
propound my conjectures and some experiments which confirm them.
20. The double emission of waves of light, which I had imagined,
became more probable to me after I had observed a certain phenomenon
in the ordinary [Rock] Crystal, which occurs in hexagonal form, and
which, because of this regularity, seems also to be composed of
particles, of definite figure, and ranged in order. This was, that
this crystal, as well as that from Iceland, has a double refraction,
though less evident. For having had cut from it some well polished
Prisms of different sections, I remarked in all, in viewing through
them the flame of a candle or the lead of window panes, that
everything appeared double, though with images not very distant from
one another. Whence I understood the reason why this substance, though
so transparent, is useless for Telescopes, when they have ever so
little length.
21. Now this double refraction, according to my Theory hereinbefore
established, seemed to demand a double emission of waves of light,
both of them spherical (for both the refractions are regular) and
those of one series a little slower only than the others. For thus the
phenomenon is quite naturally explained, by postulating substances
which serve as vehicle for these waves, as I have done in the case of
Iceland Crystal. I had then less trouble after that in admitting two
emissions of waves in one and the same body. And since it might have
been objected that in composing these two kinds of crystal of equal
particles of a certain figure, regularly piled, the interstices which
these particles leave and which contain the ethereal matter would
scarcely suffice to transmit the waves of light which I have localized
there, I removed this difficulty by regarding these particles as being
of a very rare texture, or rather as composed of other much smaller
particles, between which the ethereal matter passes quite freely.
This, moreover, necessarily follows from that which has been already
demonstrated touching the small quantity of matter of which the bodies
are built up.
22. Supposing then these spheroidal waves besides the spherical ones,
I began to examine whether they could serve to explain the phenomena
of the irregular refraction, and how by these same phenomena I could
determine the figure and position of the spheroids: as to which I
obtained at last the desired success, by proceeding as follows.
[Illustration]
23. I considered first the effect of waves so formed, as respects the
ray which falls perpendicularly on the flat surface of a transparent
body in which they should spread in this manner. I took AB for the
exposed region of the surface. And, since a ray perpendicular to a
plane, and coming from a very distant source of light, is nothing
else, according to the precedent Theory, than the incidence of a
portion of the wave parallel to that plane, I supposed the straight
line RC, parallel and equal to AB, to be a portion of a wave of light,
in which an infinitude of points such as RH_h_C come to meet the
surface AB at the points AK_k_B. Then instead of the hemispherical
partial waves which in a body of ordinary refraction would spread from
each of these last points, as we have above explained in treating of
refraction, these must here be hemi-spheroids. The axes (or rather the
major diameters) of these I supposed to be oblique to the plane AB, as
is AV the semi-axis or semi-major diameter of the spheroid SVT, which
represents the partial wave coming from the point A, after the wave RC
has reached AB. I say axis or major diameter, because the same ellipse
SVT may be considered as the section of a spheroid of which the axis
is AZ perpendicular to AV. But, for the present, without yet deciding
one or other, we will consider these spheroids only in those sections
of them which make ellipses in the plane of this figure. Now taking a
certain space of time during which the wave SVT has spread from A, it
would needs be that from all the other points K_k_B there should
proceed, in the same time, waves similar to SVT and similarly
situated. And the common tangent NQ of all these semi-ellipses would
be the propagation of the wave RC which fell on AB, and would be the
place where this movement occurs in much greater amount than anywhere
else, being made up of arcs of an infinity of ellipses, the centres of
which are along the line AB.
24. Now it appeared that this common tangent NQ was parallel to AB,
and of the same length, but that it was not directly opposite to it,
since it was comprised between the lines AN, BQ, which are diameters
of ellipses having A and B for centres, conjugate with respect to
diameters which are not in the straight line AB. And in this way I
comprehended, a matter which had seemed to me very difficult, how a
ray perpendicular to a surface could suffer refraction on entering a
transparent body; seeing that the wave RC, having come to the aperture
AB, went on forward thence, spreading between the parallel lines AN,
BQ, yet itself remaining always parallel to AB, so that here the light
does not spread along lines perpendicular to its waves, as in ordinary
refraction, but along lines cutting the waves obliquely.
[Illustration]
25. Inquiring subsequently what might be the position and form of
these spheroids in the crystal, I considered that all the six faces
produced precisely the same refractions. Taking, then, the
parallelopiped AFB, of which the obtuse solid angle C is contained
between the three equal plane angles, and imagining in it the three
principal sections, one of which is perpendicular to the face DC and
passes through the edge CF, another perpendicular to the face BF
passing through the edge CA, and the third perpendicular to the face
AF passing through the edge BC; I knew that the refractions of the
incident rays belonging to these three planes were all similar. But
there could be no position of the spheroid which would have the same
relation to these three sections except that in which the axis was
also the axis of the solid angle C. Consequently I saw that the axis
of this angle, that is to say the straight line which traversed the
crystal from the point C with equal inclination to the edges CF, CA,
CB was the line which determined the position of the axis of all the
spheroidal waves which one imagined to originate from some point,
taken within or on the surface of the crystal, since all these
spheroids ought to be alike, and have their axes parallel to one
another.
26. Considering after this the plane of one of these three sections,
namely that through GCF, the angle of which is 109 degrees 3 minutes,
since the angle F was shown above to be 70 degrees 57 minutes; and,
imagining a spheroidal wave about the centre C, I knew, because I have
just explained it, that its axis must be in the same plane, the half
of which axis I have marked CS in the next figure: and seeking by
calculation (which will be given with others at the end of this
discourse) the value of the angle CGS, I found it 45 degrees 20
minutes.
[Illustration]
27. To know from this the form of this spheroid, that is to say the
proportion of the semi-diameters CS, CP, of its elliptical section,
which are perpendicular to one another, I considered that the point M
where the ellipse is touched by the straight line FH, parallel to CG,
ought to be so situated that CM makes with the perpendicular CL an
angle of 6 degrees 40 minutes; since, this being so, this ellipse
satisfies what has been said about the refraction of the ray
perpendicular to the surface CG, which is inclined to the
perpendicular CL by the same angle. This, then, being thus disposed,
and taking CM at 100,000 parts, I found by the calculation which will
be given at the end, the semi-major diameter CP to be 105,032, and the
semi-axis CS to be 93,410, the ratio of which numbers is very nearly 9
to 8; so that the spheroid was of the kind which resembles a
compressed sphere, being generated by the revolution of an ellipse
about its smaller diameter. I found also the value of CG the
semi-diameter parallel to the tangent ML to be 98,779.
[Illustration]
28. Now passing to the investigation of the refractions which
obliquely incident rays must undergo, according to our hypothesis of
spheroidal waves, I saw that these refractions depended on the ratio
between the velocity of movement of the light outside the crystal in
the ether, and that within the crystal. For supposing, for example,
this proportion to be such that while the light in the crystal forms
the spheroid GSP, as I have just said, it forms outside a sphere the
semi-diameter of which is equal to the line N which will be determined
hereafter, the following is the way of finding the refraction of the
incident rays. Let there be such a ray RC falling upon the surface
CK. Make CO perpendicular to RC, and across the angle KCO adjust OK,
equal to N and perpendicular to CO; then draw KI, which touches the
Ellipse GSP, and from the point of contact I join IC, which will be
the required refraction of the ray RC. The demonstration of this is,
it will be seen, entirely similar to that of which we made use in
explaining ordinary refraction. For the refraction of the ray RC is
nothing else than the progression of the portion C of the wave CO,
continued in the crystal. Now the portions H of this wave, during the
time that O came to K, will have arrived at the surface CK along the
straight lines H_x_, and will moreover have produced in the crystal
around the centres _x_ some hemi-spheroidal partial waves similar to
the hemi-spheroidal GSP_g_, and similarly disposed, and of which the
major and minor diameters will bear the same proportions to the lines
_xv_ (the continuations of the lines H_x_ up to KB parallel to CO)
that the diameters of the spheroid GSP_g_ bear to the line CB, or N.
And it is quite easy to see that the common tangent of all these
spheroids, which are here represented by Ellipses, will be the
straight line IK, which consequently will be the propagation of the
wave CO; and the point I will be that of the point C, conformably with
that which has been demonstrated in ordinary refraction.
Now as to finding the point of contact I, it is known that one must
find CD a third proportional to the lines CK, CG, and draw DI parallel
to CM, previously determined, which is the conjugate diameter to CG;
for then, by drawing KI it touches the Ellipse at I.
29. Now as we have found CI the refraction of the ray RC, similarly
one will find C_i_ the refraction of the ray _r_C, which comes from
the opposite side, by making C_o_ perpendicular to _r_C and following
out the rest of the construction as before. Whence one sees that if
the ray _r_C is inclined equally with RC, the line C_d_ will
necessarily be equal to CD, because C_k_ is equal to CK, and C_g_ to
CG. And in consequence I_i_ will be cut at E into equal parts by the
line CM, to which DI and _di_ are parallel. And because CM is the
conjugate diameter to CG, it follows that _i_I will be parallel to
_g_G. Therefore if one prolongs the refracted rays CI, C_i_, until
they meet the tangent ML at T and _t_, the distances MT, M_t_, will
also be equal. And so, by our hypothesis, we explain perfectly the
phenomenon mentioned above; to wit, that when there are two rays
equally inclined, but coming from opposite sides, as here the rays RC,
_rc_, their refractions diverge equally from the line followed by the
refraction of the ray perpendicular to the surface, by considering
these divergences in the direction parallel to the surface of the
crystal.
30. To find the length of the line N, in proportion to CP, CS, CG, it
must be determined by observations of the irregular refraction which
occurs in this section of the crystal; and I find thus that the ratio
of N to GC is just a little less than 8 to 5. And having regard to
some other observations and phenomena of which I shall speak
afterwards, I put N at 156,962 parts, of which the semi-diameter CG is
found to contain 98,779, making this ratio 8 to 5-1/29. Now this
proportion, which there is between the line N and CG, may be called
the Proportion of the Refraction; similarly as in glass that of 3 to
2, as will be manifest when I shall have explained a short process in
the preceding way to find the irregular refractions.
31. Supposing then, in the next figure, as previously, the surface of
the crystal _g_G, the Ellipse GP_g_, and the line N; and CM the
refraction of the perpendicular ray FC, from which it diverges by 6
degrees 40 minutes. Now let there be some other ray RC, the refraction
of which must be found.
About the centre C, with semi-diameter CG, let the circumference _g_RG
be described, cutting the ray RC at R; and let RV be the perpendicular
on CG. Then as the line N is to CG let CV be to CD, and let DI be
drawn parallel to CM, cutting the Ellipse _g_MG at I; then joining CI,
this will be the required refraction of the ray RC. Which is
demonstrated thus.
[Illustration]
Let CO be perpendicular to CR, and across the angle OCG let OK be
adjusted, equal to N and perpendicular to CO, and let there be drawn
the straight line KI, which if it is demonstrated to be a tangent to
the Ellipse at I, it will be evident by the things heretofore
explained that CI is the refraction of the ray RC. Now since the angle
RCO is a right angle, it is easy to see that the right-angled
triangles RCV, KCO, are similar. As then, CK is to KO, so also is RC
to CV. But KO is equal to N, and RC to CG: then as CK is to N so will
CG be to CV. But as N is to CG, so, by construction, is CV to CD. Then
as CK is to CG so is CG to CD. And because DI is parallel to CM, the
conjugate diameter to CG, it follows that KI touches the Ellipse at I;
which remained to be shown.
32. One sees then that as there is in the refraction of ordinary
media a certain constant proportion between the sines of the angles
which the incident ray and the refracted ray make with the
perpendicular, so here there is such a proportion between CV and CD or
IE; that is to say between the Sine of the angle which the incident
ray makes with the perpendicular, and the horizontal intercept, in the
Ellipse, between the refraction of this ray and the diameter CM. For
the ratio of CV to CD is, as has been said, the same as that of N to
the semi-diameter CG.
33. I will add here, before passing away, that in comparing together
the regular and irregular refraction of this crystal, there is this
remarkable fact, that if ABPS be the spheroid by which light spreads
in the Crystal in a certain space of time (which spreading, as has
been said, serves for the irregular refraction), then the inscribed
sphere BVST is the extension in the same space of time of the light
which serves for the regular refraction.
[Illustration]
For we have stated before this, that the line N being the radius of a
spherical wave of light in air, while in the crystal it spread through
the spheroid ABPS, the ratio of N to CS will be 156,962 to 93,410. But
it has also been stated that the proportion of the regular refraction
was 5 to 3; that is to say, that N being the radius of a spherical
wave of light in air, its extension in the crystal would, in the same
space of time, form a sphere the radius of which would be to N as 3 to
5. Now 156,962 is to 93,410 as 5 to 3 less 1/41. So that it is
sufficiently nearly, and may be exactly, the sphere BVST, which the
light describes for the regular refraction in the crystal, while it
describes the spheroid BPSA for the irregular refraction, and while it
describes the sphere of radius N in air outside the crystal.
Although then there are, according to what we have supposed, two
different propagations of light within the crystal, it appears that it
is only in directions perpendicular to the axis BS of the spheroid
that one of these propagations occurs more rapidly than the other; but
that they have an equal velocity in the other direction, namely, in
that parallel to the same axis BS, which is also the axis of the
obtuse angle of the crystal.
[Illustration]
34. The proportion of the refraction being what we have just seen, I
will now show that there necessarily follows thence that notable
property of the ray which falling obliquely on the surface of the
crystal enters it without suffering refraction. For supposing the same
things as before, and that the ray makes with the same surface _g_G
the angle RCG of 73 degrees 20 minutes, inclining to the same side as
the crystal (of which ray mention has been made above); if one
investigates, by the process above explained, the refraction CI, one
will find that it makes exactly a straight line with RC, and that thus
this ray is not deviated at all, conformably with experiment. This is
proved as follows by calculation.
CG or CR being, as precedently, 98,779; CM being 100,000; and the
angle RCV 73 degrees 20 minutes, CV will be 28,330. But because CI is
the refraction of the ray RC, the proportion of CV to CD is 156,962 to
98,779, namely, that of N to CG; then CD is 17,828.
Now the rectangle _g_DC is to the square of DI as the square of CG is
to the square of CM; hence DI or CE will be 98,353. But as CE is to
EI, so will CM be to MT, which will then be 18,127. And being added to
ML, which is 11,609 (namely the sine of the angle LCM, which is 6
degrees 40 minutes, taking CM 100,000 as radius) we get LT 27,936; and
this is to LC 99,324 as CV to VR, that is to say, as 29,938, the
tangent of the complement of the angle RCV, which is 73 degrees 20
minutes, is to the radius of the Tables. Whence it appears that RCIT
is a straight line; which was to be proved.
35. Further it will be seen that the ray CI in emerging through the
opposite surface of the crystal, ought to pass out quite straight,
according to the following demonstration, which proves that the
reciprocal relation of refraction obtains in this crystal the same as
in other transparent bodies; that is to say, that if a ray RC in
meeting the surface of the crystal CG is refracted as CI, the ray CI
emerging through the opposite parallel surface of the crystal, which
I suppose to be IB, will have its refraction IA parallel to the ray
RC.
[Illustration]
Let the same things be supposed as before; that is to say, let CO,
perpendicular to CR, represent a portion of a wave the continuation of
which in the crystal is IK, so that the piece C will be continued on
along the straight line CI, while O comes to K. Now if one takes a
second period of time equal to the first, the piece K of the wave IK
will, in this second period, have advanced along the straight line KB,
equal and parallel to CI, because every piece of the wave CO, on
arriving at the surface CK, ought to go on in the crystal the same as
the piece C; and in this same time there will be formed in the air
from the point I a partial spherical wave having a semi-diameter IA
equal to KO, since KO has been traversed in an equal time. Similarly,
if one considers some other point of the wave IK, such as _h_, it will
go along _hm_, parallel to CI, to meet the surface IB, while the point
K traverses K_l_ equal to _hm_; and while this accomplishes the
remainder _l_B, there will start from the point _m_ a partial wave the
semi-diameter of which, _mn_, will have the same ratio to _l_B as IA
to KB. Whence it is evident that this wave of semi-diameter _mn_, and
the other of semi-diameter IA will have the same tangent BA. And
similarly for all the partial spherical waves which will be formed
outside the crystal by the impact of all the points of the wave IK
against the surface of the Ether IB. It is then precisely the tangent
BA which will be the continuation of the wave IK, outside the crystal,
when the piece K has reached B. And in consequence IA, which is
perpendicular to BA, will be the refraction of the ray CI on emerging
from the crystal. Now it is clear that IA is parallel to the incident
ray RC, since IB is equal to CK, and IA equal to KO, and the angles A
and O are right angles.
It is seen then that, according to our hypothesis, the reciprocal
relation of refraction holds good in this crystal as well as in
ordinary transparent bodies; as is thus in fact found by observation.
36. I pass now to the consideration of other sections of the crystal,
and of the refractions there produced, on which, as will be seen, some
other very remarkable phenomena depend.
Let ABH be a parallelepiped of crystal, and let the top surface AEHF
be a perfect rhombus, the obtuse angles of which are equally divided
by the straight line EF, and the acute angles by the straight line AH
perpendicular to FE.
The section which we have hitherto considered is that which passes
through the lines EF, EB, and which at the same time cuts the plane
AEHF at right angles. Refractions in this section have this in common
with the refractions in ordinary media that the plane which is drawn
through the incident ray and which also intersects the surface of the
crystal at right angles, is that in which the refracted ray also is
found. But the refractions which appertain to every other section of
this crystal have this strange property that the refracted ray always
quits the plane of the incident ray perpendicular to the surface, and
turns away towards the side of the slope of the crystal. For which
fact we shall show the reason, in the first place, for the section
through AH; and we shall show at the same time how one can determine
the refraction, according to our hypothesis. Let there be, then, in
the plane which passes through AH, and which is perpendicular to the
plane AFHE, the incident ray RC; it is required to find its refraction
in the crystal.
[Illustration]
37. About the centre C, which I suppose to be in the intersection of
AH and FE, let there be imagined a hemi-spheroid QG_qg_M, such as the
light would form in spreading in the crystal, and let its section by
the plane AEHF form the Ellipse QG_qg_, the major diameter of which
Q_q_, which is in the line AH, will necessarily be one of the major
diameters of the spheroid; because the axis of the spheroid being in
the plane through FEB, to which QC is perpendicular, it follows that
QC is also perpendicular to the axis of the spheroid, and consequently
QC_q_ one of its major diameters. But the minor diameter of this
Ellipse, G_g_, will bear to Q_q_ the proportion which has been defined
previously, Article 27, between CG and the major semi-diameter of the
spheroid, CP, namely, that of 98,779 to 105,032.
Let the line N be the length of the travel of light in air during the
time in which, within the crystal, it makes, from the centre C, the
spheroid QC_qg_M. Then having drawn CO perpendicular to the ray CR and
situate in the plane through CR and AH, let there be adjusted, across
the angle ACO, the straight line OK equal to N and perpendicular to
CO, and let it meet the straight line AH at K. Supposing consequently
that CL is perpendicular to the surface of the crystal AEHF, and that
CM is the refraction of the ray which falls perpendicularly on this
same surface, let there be drawn a plane through the line CM and
through KCH, making in the spheroid the semi-ellipse QM_q_, which will
be given, since the angle MCL is given of value 6 degrees 40 minutes.
And it is certain, according to what has been explained above, Article
27, that a plane which would touch the spheroid at the point M, where
I suppose the straight line CM to meet the surface, would be parallel
to the plane QG_q_. If then through the point K one now draws KS
parallel to G_g_, which will be parallel also to QX, the tangent to
the Ellipse QG_q_ at Q; and if one conceives a plane passing through
KS and touching the spheroid, the point of contact will necessarily be
in the Ellipse QM_q_, because this plane through KS, as well as the
plane which touches the spheroid at the point M, are parallel to QX,
the tangent of the spheroid: for this consequence will be demonstrated
at the end of this Treatise. Let this point of contact be at I, then
making KC, QC, DC proportionals, draw DI parallel to CM; also join CI.
I say that CI will be the required refraction of the ray RC. This will
be manifest if, in considering CO, which is perpendicular to the ray
RC, as a portion of the wave of light, we can demonstrate that the
continuation of its piece C will be found in the crystal at I, when O
has arrived at K.
38. Now as in the Chapter on Reflexion, in demonstrating that the
incident and reflected rays are always in the same plane perpendicular
to the reflecting surface, we considered the breadth of the wave of
light, so, similarly, we must here consider the breadth of the wave CO
in the diameter G_g_. Taking then the breadth C_c_ on the side toward
the angle E, let the parallelogram CO_oc_ be taken as a portion of a
wave, and let us complete the parallelograms CK_kc_, CI_ic_, Kl_ik_,
OK_ko_. In the time then that the line O_o_ arrives at the surface of
the crystal at K_k_, all the points of the wave CO_oc_ will have
arrived at the rectangle K_c_ along lines parallel to OK; and from the
points of their incidences there will originate, beyond that, in the
crystal partial hemi-spheroids, similar to the hemi-spheroid QM_q_,
and similarly disposed. These hemi-spheroids will necessarily all
touch the plane of the parallelogram KI_ik_ at the same instant that
O_o_ has reached K_k_. Which is easy to comprehend, since, of these
hemi-spheroids, all those which have their centres along the line CK,
touch this plane in the line KI (for this is to be shown in the same
way as we have demonstrated the refraction of the oblique ray in the
principal section through EF) and all those which have their centres
in the line C_c_ will touch the same plane KI in the line I_i_; all
these being similar to the hemi-spheroid QM_q_. Since then the
parallelogram K_i_ is that which touches all these spheroids, this
same parallelogram will be precisely the continuation of the wave
CO_oc_ in the crystal, when O_o_ has arrived at K_k_, because it forms
the termination of the movement and because of the quantity of
movement which occurs more there than anywhere else: and thus it
appears that the piece C of the wave CO_oc_ has its continuation at I;
that is to say, that the ray RC is refracted as CI.
From this it is to be noted that the proportion of the refraction for
this section of the crystal is that of the line N to the semi-diameter
CQ; by which one will easily find the refractions of all incident
rays, in the same way as we have shown previously for the case of the
section through FE; and the demonstration will be the same. But it
appears that the said proportion of the refraction is less here than
in the section through FEB; for it was there the same as the ratio of
N to CG, that is to say, as 156,962 to 98,779, very nearly as 8 to 5;
and here it is the ratio of N to CQ the major semi-diameter of the
spheroid, that is to say, as 156,962 to 105,032, very nearly as 3 to
2, but just a little less. Which still agrees perfectly with what one
finds by observation.
39. For the rest, this diversity of proportion of refraction produces
a very singular effect in this Crystal; which is that when it is
placed upon a sheet of paper on which there are letters or anything
else marked, if one views it from above with the two eyes situated in
the plane of the section through EF, one sees the letters raised up by
this irregular refraction more than when one puts one's eyes in the
plane of section through AH: and the difference of these elevations
appears by comparison with the other ordinary refraction of the
crystal, the proportion of which is as 5 to 3, and which always raises
the letters equally, and higher than the irregular refraction does.
For one sees the letters and the paper on which they are written, as
on two different stages at the same time; and in the first position of
the eyes, namely, when they are in the plane through AH these two
stages are four times more distant from one another than when the eyes
are in the plane through EF.
We will show that this effect follows from the refractions; and it
will enable us at the same time to ascertain the apparent place of a
point of an object placed immediately under the crystal, according to
the different situation of the eyes.
40. Let us see first by how much the irregular refraction of the plane
through AH ought to lift the bottom of the crystal. Let the plane of
this figure represent separately the section through Q_q_ and CL, in
which section there is also the ray RC, and let the semi-elliptic
plane through Q_q_ and CM be inclined to the former, as previously, by
an angle of 6 degrees 40 minutes; and in this plane CI is then the
refraction of the ray RC.
[Illustration]
If now one considers the point I as at the bottom of the crystal, and
that it is viewed by the rays ICR, _Icr_, refracted equally at the
points C_c_, which should be equally distant from D, and that these
rays meet the two eyes at R_r_; it is certain that the point I will
appear raised to S where the straight lines RC, _rc_, meet; which
point S is in DP, perpendicular to Q_q_. And if upon DP there is drawn
the perpendicular IP, which will lie at the bottom of the crystal, the
length SP will be the apparent elevation of the point I above the
bottom.
Let there be described on Q_q_ a semicircle cutting the ray CR at B,
from which BV is drawn perpendicular to Q_q_; and let the proportion
of the refraction for this section be, as before, that of the line N
to the semi-diameter CQ.
Then as N is to CQ so is VC to CD, as appears by the method of finding
the refraction which we have shown above, Article 31; but as VC is to
CD, so is VB to DS. Then as N is to CQ, so is VB to DS. Let ML be
perpendicular to CL. And because I suppose the eyes R_r_ to be distant
about a foot or so from the crystal, and consequently the angle RS_r_
very small, VB may be considered as equal to the semi-diameter CQ, and
DP as equal to CL; then as N is to CQ so is CQ to DS. But N is valued
at 156,962 parts, of which CM contains 100,000 and CQ 105,032. Then DS
will have 70,283. But CL is 99,324, being the sine of the complement
of the angle MCL which is 6 degrees 40 minutes; CM being supposed as
radius. Then DP, considered as equal to CL, will be to DS as 99,324 to
70,283. And so the elevation of the point I by the refraction of this
section is known.
[Illustration]
41. Now let there be represented the other section through EF in the
figure before the preceding one; and let CM_g_ be the semi-ellipse,
considered in Articles 27 and 28, which is made by cutting a
spheroidal wave having centre C. Let the point I, taken in this
ellipse, be imagined again at the bottom of the Crystal; and let it be
viewed by the refracted rays ICR, I_cr_, which go to the two eyes; CR
and _cr_ being equally inclined to the surface of the crystal G_g_.
This being so, if one draws ID parallel to CM, which I suppose to be
the refraction of the perpendicular ray incident at the point C, the
distances DC, D_c_, will be equal, as is easy to see by that which has
been demonstrated in Article 28. Now it is certain that the point I
should appear at S where the straight lines RC, _rc_, meet when
prolonged; and that this point will fall in the line DP perpendicular
to G_g_. If one draws IP perpendicular to this DP, it will be the
distance PS which will mark the apparent elevation of the point I. Let
there be described on G_g_ a semicircle cutting CR at B, from which
let BV be drawn perpendicular to G_g_; and let N to GC be the
proportion of the refraction in this section, as in Article 28. Since
then CI is the refraction of the radius BC, and DI is parallel to CM,
VC must be to CD as N to GC, according to what has been demonstrated
in Article 31. But as VC is to CD so is BV to DS. Let ML be drawn
perpendicular to CL. And because I consider, again, the eyes to be
distant above the crystal, BV is deemed equal to the semi-diameter CG;
and hence DS will be a third proportional to the lines N and CG: also
DP will be deemed equal to CL. Now CG consisting of 98,778 parts, of
which CM contains 100,000, N is taken as 156,962. Then DS will be
62,163. But CL is also determined, and contains 99,324 parts, as has
been said in Articles 34 and 40. Then the ratio of PD to DS will be as
99,324 to 62,163. And thus one knows the elevation of the point at the
bottom I by the refraction of this section; and it appears that this
elevation is greater than that by the refraction of the preceding
section, since the ratio of PD to DS was there as 99,324 to 70,283.
[Illustration]
But by the regular refraction of the crystal, of which we have above
said that the proportion is 5 to 3, the elevation of the point I, or
P, from the bottom, will be 2/5 of the height DP; as appears by this
figure, where the point P being viewed by the rays PCR, P_cr_,
refracted equally at the surface C_c_, this point must needs appear
to be at S, in the perpendicular PD where the lines RC, _rc_, meet
when prolonged: and one knows that the line PC is to CS as 5 to 3,
since they are to one another as the sine of the angle CSP or DSC is
to the sine of the angle SPC. And because the ratio of PD to DS is
deemed the same as that of PC to CS, the two eyes Rr being supposed
very far above the crystal, the elevation PS will thus be 2/5 of PD.
[Illustration]
42. If one takes a straight line AB for the thickness of the crystal,
its point B being at the bottom, and if one divides it at the points
C, D, E, according to the proportions of the elevations found, making
AE 3/5 of AB, AB to AC as 99,324 to 70,283, and AB to AD as 99,324 to
62,163, these points will divide AB as in this figure. And it will be
found that this agrees perfectly with experiment; that is to say by
placing the eyes above in the plane which cuts the crystal according
to the shorter diameter of the rhombus, the regular refraction will
lift up the letters to E; and one will see the bottom, and the letters
over which it is placed, lifted up to D by the irregular refraction.
But by placing the eyes above in the plane which cuts the crystal
according to the longer diameter of the rhombus, the regular
refraction will lift the letters to E as before; but the irregular
refraction will make them, at the same time, appear lifted up only to
C; and in such a way that the interval CE will be quadruple the
interval ED, which one previously saw.
43. I have only to make the remark here that in both the positions of
the eyes the images caused by the irregular refraction do not appear
directly below those which proceed from the regular refraction, but
they are separated from them by being more distant from the
equilateral solid angle of the Crystal. That follows, indeed, from all
that has been hitherto demonstrated about the irregular refraction;
and it is particularly shown by these last demonstrations, from which
one sees that the point I appears by irregular refraction at S in the
perpendicular line DP, in which line also the image of the point P
ought to appear by regular refraction, but not the image of the point
I, which will be almost directly above the same point, and higher than
S.
But as to the apparent elevation of the point I in other positions of
the eyes above the crystal, besides the two positions which we have
just examined, the image of that point by the irregular refraction
will always appear between the two heights of D and C, passing from
one to the other as one turns one's self around about the immovable
crystal, while looking down from above. And all this is still found
conformable to our hypothesis, as any one can assure himself after I
shall have shown here the way of finding the irregular refractions
which appear in all other sections of the crystal, besides the two
which we have considered. Let us suppose one of the faces of the
crystal, in which let there be the Ellipse HDE, the centre C of which
is also the centre of the spheroid HME in which the light spreads, and
of which the said Ellipse is the section. And let the incident ray be
RC, the refraction of which it is required to find.
Let there be taken a plane passing through the ray RC and which is
perpendicular to the plane of the ellipse HDE, cutting it along the
straight line BCK; and having in the same plane through RC made CO
perpendicular to CR, let OK be adjusted across the angle OCK, so as
to be perpendicular to OC and equal to the line N, which I suppose to
measure the travel of the light in air during the time that it spreads
in the crystal through the spheroid HDEM. Then in the plane of the
Ellipse HDE let KT be drawn, through the point K, perpendicular to
BCK. Now if one conceives a plane drawn through the straight line KT
and touching the spheroid HME at I, the straight line CI will be the
refraction of the ray RC, as is easy to deduce from that which has
been demonstrated in Article 36.
[Illustration]
But it must be shown how one can determine the point of contact I. Let
there be drawn parallel to the line KT a line HF which touches the
Ellipse HDE, and let this point of contact be at H. And having drawn a
straight line along CH to meet KT at T, let there be imagined a plane
passing through the same CH and through CM (which I suppose to be the
refraction of the perpendicular ray), which makes in the spheroid the
elliptical section HME. It is certain that the plane which will pass
through the straight line KT, and which will touch the spheroid, will
touch it at a point in the Ellipse HME, according to the Lemma which
will be demonstrated at the end of the Chapter. Now this point is
necessarily the point I which is sought, since the plane drawn through
TK can touch the spheroid at one point only. And this point I is easy
to determine, since it is needful only to draw from the point T, which
is in the plane of this Ellipse, the tangent TI, in the way shown
previously. For the Ellipse HME is given, and its conjugate
semi-diameters are CH and CM; because a straight line drawn through M,
parallel to HE, touches the Ellipse HME, as follows from the fact that
a plane taken through M, and parallel to the plane HDE, touches the
spheroid at that point M, as is seen from Articles 27 and 23. For the
rest, the position of this ellipse, with respect to the plane through
the ray RC and through CK, is also given; from which it will be easy
to find the position of CI, the refraction corresponding to the ray
RC.
Now it must be noted that the same ellipse HME serves to find the
refractions of any other ray which may be in the plane through RC and
CK. Because every plane, parallel to the straight line HF, or TK,
which will touch the spheroid, will touch it in this ellipse,
according to the Lemma quoted a little before.
I have investigated thus, in minute detail, the properties of the
irregular refraction of this Crystal, in order to see whether each
phenomenon that is deduced from our hypothesis accords with that which
is observed in fact. And this being so it affords no slight proof of
the truth of our suppositions and principles. But what I am going to
add here confirms them again marvellously. It is this: that there are
different sections of this Crystal, the surfaces of which, thereby
produced, give rise to refractions precisely such as they ought to be,
and as I had foreseen them, according to the preceding Theory.
In order to explain what these sections are, let ABKF _be_ the
principal section through the axis of the crystal ACK, in which there
will also be the axis SS of a spheroidal wave of light spreading in
the crystal from the centre C; and the straight line which cuts SS
through the middle and at right angles, namely PP, will be one of the
major diameters.
[Illustration: {Section ABKF}]
Now as in the natural section of the crystal, made by a plane parallel
to two opposite faces, which plane is here represented by the line GG,
the refraction of the surfaces which are produced by it will be
governed by the hemi-spheroids GNG, according to what has been
explained in the preceding Theory. Similarly, cutting the Crystal
through NN, by a plane perpendicular to the parallelogram ABKF, the
refraction of the surfaces will be governed by the hemi-spheroids NGN.
And if one cuts it through PP, perpendicularly to the said
parallelogram, the refraction of the surfaces ought to be governed by
the hemi-spheroids PSP, and so for others. But I saw that if the plane
NN was almost perpendicular to the plane GG, making the angle NCG,
which is on the side A, an angle of 90 degrees 40 minutes, the
hemi-spheroids NGN would become similar to the hemi-spheroids GNG,
since the planes NN and GG were equally inclined by an angle of 45
degrees 20 minutes to the axis SS. In consequence it must needs be, if
our theory is true, that the surfaces which the section through NN
produces should effect the same refractions as the surfaces of the
section through GG. And not only the surfaces of the section NN but
all other sections produced by planes which might be inclined to the
axis at an angle equal to 45 degrees 20 minutes. So that there are an
infinitude of planes which ought to produce precisely the same
refractions as the natural surfaces of the crystal, or as the section
parallel to any one of those surfaces which are made by cleavage.
I saw also that by cutting it by a plane taken through PP, and
perpendicular to the axis SS, the refraction of the surfaces ought to
be such that the perpendicular ray should suffer thereby no deviation;
and that for oblique rays there would always be an irregular
refraction, differing from the regular, and by which objects placed
beneath the crystal would be less elevated than by that other
refraction.
That, similarly, by cutting the crystal by any plane through the axis
SS, such as the plane of the figure is, the perpendicular ray ought to
suffer no refraction; and that for oblique rays there were different
measures for the irregular refraction according to the situation of
the plane in which the incident ray was.
Now these things were found in fact so; and, after that, I could not
doubt that a similar success could be met with everywhere. Whence I
concluded that one might form from this crystal solids similar to
those which are its natural forms, which should produce, at all their
surfaces, the same regular and irregular refractions as the natural
surfaces, and which nevertheless would cleave in quite other ways, and
not in directions parallel to any of their faces. That out of it one
would be able to fashion pyramids, having their base square,
pentagonal, hexagonal, or with as many sides as one desired, all the
surfaces of which should have the same refractions as the natural
surfaces of the crystal, except the base, which will not refract the
perpendicular ray. These surfaces will each make an angle of 45
degrees 20 minutes with the axis of the crystal, and the base will be
the section perpendicular to the axis.
That, finally, one could also fashion out of it triangular prisms, or
prisms with as many sides as one would, of which neither the sides nor
the bases would refract the perpendicular ray, although they would yet
all cause double refraction for oblique rays. The cube is included
amongst these prisms, the bases of which are sections perpendicular to
the axis of the crystal, and the sides are sections parallel to the
same axis.
From all this it further appears that it is not at all in the
disposition of the layers of which this crystal seems to be composed,
and according to which it splits in three different senses, that the
cause resides of its irregular refraction; and that it would be in
vain to wish to seek it there.
But in order that any one who has some of this stone may be able to
find, by his own experience, the truth of what I have just advanced, I
will state here the process of which I have made use to cut it, and to
polish it. Cutting is easy by the slicing wheels of lapidaries, or in
the way in which marble is sawn: but polishing is very difficult, and
by employing the ordinary means one more often depolishes the surfaces
than makes them lucent.
After many trials, I have at last found that for this service no plate
of metal must be used, but a piece of mirror glass made matt and
depolished. Upon this, with fine sand and water, one smoothes the
crystal little by little, in the same way as spectacle glasses, and
polishes it simply by continuing the work, but ever reducing the
material. I have not, however, been able to give it perfect clarity
and transparency; but the evenness which the surfaces acquire enables
one to observe in them the effects of refraction better than in those
made by cleaving the stone, which always have some inequality.
Even when the surface is only moderately smoothed, if one rubs it over
with a little oil or white of egg, it becomes quite transparent, so
that the refraction is discerned in it quite distinctly. And this aid
is specially necessary when it is wished to polish the natural
surfaces to remove the inequalities; because one cannot render them
lucent equally with the surfaces of other sections, which take a
polish so much the better the less nearly they approximate to these
natural planes.
Before finishing the treatise on this Crystal, I will add one more
marvellous phenomenon which I discovered after having written all the
foregoing. For though I have not been able till now to find its cause,
I do not for that reason wish to desist from describing it, in order
to give opportunity to others to investigate it. It seems that it will
be necessary to make still further suppositions besides those which I
have made; but these will not for all that cease to keep their
probability after having been confirmed by so many tests.
[Illustration]
The phenomenon is, that by taking two pieces of this crystal and
applying them one over the other, or rather holding them with a space
between the two, if all the sides of one are parallel to those of the
other, then a ray of light, such as AB, is divided into two in the
first piece, namely into BD and BC, following the two refractions,
regular and irregular. On penetrating thence into the other piece
each ray will pass there without further dividing itself in two; but
that one which underwent the regular refraction, as here DG, will
undergo again only a regular refraction at GH; and the other, CE, an
irregular refraction at EF. And the same thing occurs not only in this
disposition, but also in all those cases in which the principal
section of each of the pieces is situated in one and the same plane,
without it being needful for the two neighbouring surfaces to be
parallel. Now it is marvellous why the rays CE and DG, incident from
the air on the lower crystal, do not divide themselves the same as the
first ray AB. One would say that it must be that the ray DG in passing
through the upper piece has lost something which is necessary to move
the matter which serves for the irregular refraction; and that
likewise CE has lost that which was necessary to move the matter
which serves for regular refraction: but there is yet another thing
which upsets this reasoning. It is that when one disposes the two
crystals in such a way that the planes which constitute the principal
sections intersect one another at right angles, whether the
neighbouring surfaces are parallel or not, then the ray which has come
by the regular refraction, as DG, undergoes only an irregular
refraction in the lower piece; and on the contrary the ray which has
come by the irregular refraction, as CE, undergoes only a regular
refraction.
But in all the infinite other positions, besides those which I have
just stated, the rays DG, CE, divide themselves anew each one into
two, by refraction in the lower crystal so that from the single ray AB
there are four, sometimes of equal brightness, sometimes some much
less bright than others, according to the varying agreement in the
positions of the crystals: but they do not appear to have all together
more light than the single ray AB.
When one considers here how, while the rays CE, DG, remain the same,
it depends on the position that one gives to the lower piece, whether
it divides them both in two, or whether it does not divide them, and
yet how the ray AB above is always divided, it seems that one is
obliged to conclude that the waves of light, after having passed
through the first crystal, acquire a certain form or disposition in
virtue of which, when meeting the texture of the second crystal, in
certain positions, they can move the two different kinds of matter
which serve for the two species of refraction; and when meeting the
second crystal in another position are able to move only one of these
kinds of matter. But to tell how this occurs, I have hitherto found
nothing which satisfies me.
Leaving then to others this research, I pass to what I have to say
touching the cause of the extraordinary figure of this crystal, and
why it cleaves easily in three different senses, parallel to any one
of its surfaces.
There are many bodies, vegetable, mineral, and congealed salts, which
are formed with certain regular angles and figures. Thus among flowers
there are many which have their leaves disposed in ordered polygons,
to the number of 3, 4, 5, or 6 sides, but not more. This well deserves
to be investigated, both as to the polygonal figure, and as to why it
does not exceed the number 6.
Rock Crystal grows ordinarily in hexagonal bars, and diamonds are
found which occur with a square point and polished surfaces. There is
a species of small flat stones, piled up directly upon one another,
which are all of pentagonal figure with rounded angles, and the sides
a little folded inwards. The grains of gray salt which are formed from
sea water affect the figure, or at least the angle, of the cube; and
in the congelations of other salts, and in that of sugar, there are
found other solid angles with perfectly flat faces. Small snowflakes
almost always fall in little stars with 6 points, and sometimes in
hexagons with straight sides. And I have often observed, in water
which is beginning to freeze, a kind of flat and thin foliage of ice,
the middle ray of which throws out branches inclined at an angle of 60
degrees. All these things are worthy of being carefully investigated
to ascertain how and by what artifice nature there operates. But it is
not now my intention to treat fully of this matter. It seems that in
general the regularity which occurs in these productions comes from
the arrangement of the small invisible equal particles of which they
are composed. And, coming to our Iceland Crystal, I say that if there
were a pyramid such as ABCD, composed of small rounded corpuscles, not
spherical but flattened spheroids, such as would be made by the
rotation of the ellipse GH around its lesser diameter EF (of which the
ratio to the greater diameter is very nearly that of 1 to the square
root of 8)--I say that then the solid angle of the point D would be
equal to the obtuse and equilateral angle of this Crystal. I say,
further, that if these corpuscles were lightly stuck together, on
breaking this pyramid it would break along faces parallel to those
that make its point: and by this means, as it is easy to see, it would
produce prisms similar to those of the same crystal as this other
figure represents. The reason is that when broken in this fashion a
whole layer separates easily from its neighbouring layer since each
spheroid has to be detached only from the three spheroids of the next
layer; of which three there is but one which touches it on its
flattened surface, and the other two at the edges. And the reason why
the surfaces separate sharp and polished is that if any spheroid of
the neighbouring surface would come out by attaching itself to the
surface which is being separated, it would be needful for it to detach
itself from six other spheroids which hold it locked, and four of
which press it by these flattened surfaces. Since then not only the
angles of our crystal but also the manner in which it splits agree
precisely with what is observed in the assemblage composed of such
spheroids, there is great reason to believe that the particles are
shaped and ranged in the same way.
[Illustration: {Pyramid and section of spheroids}]
There is even probability enough that the prisms of this crystal are
produced by the breaking up of pyramids, since Mr. Bartholinus relates
that he occasionally found some pieces of triangularly pyramidal
figure. But when a mass is composed interiorly only of these little
spheroids thus piled up, whatever form it may have exteriorly, it is
certain, by the same reasoning which I have just explained, that if
broken it would produce similar prisms. It remains to be seen whether
there are other reasons which confirm our conjecture, and whether
there are none which are repugnant to it.
[Illustration: {paralleloid arrangement of spheroids with planes of
potential cleavage}]
It may be objected that this crystal, being so composed, might be
capable of cleavage in yet two more fashions; one of which would be
along planes parallel to the base of the pyramid, that is to say to
the triangle ABC; the other would be parallel to a plane the trace of
which is marked by the lines GH, HK, KL. To which I say that both the
one and the other, though practicable, are more difficult than those
which were parallel to any one of the three planes of the pyramid; and
that therefore, when striking on the crystal in order to break it, it
ought always to split rather along these three planes than along the
two others. When one has a number of spheroids of the form above
described, and ranges them in a pyramid, one sees why the two methods
of division are more difficult. For in the case of that division which
would be parallel to the base, each spheroid would be obliged to
detach itself from three others which it touches upon their flattened
surfaces, which hold more strongly than the contacts at the edges. And
besides that, this division will not occur along entire layers,
because each of the spheroids of a layer is scarcely held at all by
the 6 of the same layer that surround it, since they only touch it at
the edges; so that it adheres readily to the neighbouring layer, and
the others to it, for the same reason; and this causes uneven
surfaces. Also one sees by experiment that when grinding down the
crystal on a rather rough stone, directly on the equilateral solid
angle, one verily finds much facility in reducing it in this
direction, but much difficulty afterwards in polishing the surface
which has been flattened in this manner.
As for the other method of division along the plane GHKL, it will be
seen that each spheroid would have to detach itself from four of the
neighbouring layer, two of which touch it on the flattened surfaces,
and two at the edges. So that this division is likewise more difficult
than that which is made parallel to one of the surfaces of the
crystal; where, as we have said, each spheroid is detached from only
three of the neighbouring layer: of which three there is one only
which touches it on the flattened surface, and the other two at the
edges only.
However, that which has made me know that in the crystal there are
layers in this last fashion, is that in a piece weighing half a pound
which I possess, one sees that it is split along its length, as is the
above-mentioned prism by the plane GHKL; as appears by colours of the
Iris extending throughout this whole plane although the two pieces
still hold together. All this proves then that the composition of the
crystal is such as we have stated. To which I again add this
experiment; that if one passes a knife scraping along any one of the
natural surfaces, and downwards as it were from the equilateral obtuse
angle, that is to say from the apex of the pyramid, one finds it quite
hard; but by scraping in the opposite sense an incision is easily
made. This follows manifestly from the situation of the small
spheroids; over which, in the first manner, the knife glides; but in
the other manner it seizes them from beneath almost as if they were
the scales of a fish.
I will not undertake to say anything touching the way in which so many
corpuscles all equal and similar are generated, nor how they are set
in such beautiful order; whether they are formed first and then
assembled, or whether they arrange themselves thus in coming into
being and as fast as they are produced, which seems to me more
probable. To develop truths so recondite there would be needed a
knowledge of nature much greater than that which we have. I will add
only that these little spheroids could well contribute to form the
spheroids of the waves of light, here above supposed, these as well as
those being similarly situated, and with their axes parallel.
_Calculations which have been supposed in this Chapter_.
Mr. Bartholinus, in his treatise of this Crystal, puts at 101 degrees
the obtuse angles of the faces, which I have stated to be 101 degrees
52 minutes. He states that he measured these angles directly on the
crystal, which is difficult to do with ultimate exactitude, because
the edges such as CA, CB, in this figure, are generally worn, and not
quite straight. For more certainty, therefore, I preferred to measure
actually the obtuse angle by which the faces CBDA, CBVF, are inclined
to one another, namely the angle OCN formed by drawing CN
perpendicular to FV, and CO perpendicular to DA. This angle OCN I
found to be 105 degrees; and its supplement CNP, to be 75 degrees, as
it should be.
[Illustration]
To find from this the obtuse angle BCA, I imagined a sphere having its
centre at C, and on its surface a spherical triangle, formed by the
intersection of three planes which enclose the solid angle C. In this
equilateral triangle, which is ABF in this other figure, I see that
each of the angles should be 105 degrees, namely equal to the angle
OCN; and that each of the sides should be of as many degrees as the
angle ACB, or ACF, or BCF. Having then drawn the arc FQ perpendicular
to the side AB, which it divides equally at Q, the triangle FQA has a
right angle at Q, the angle A 105 degrees, and F half as much, namely
52 degrees 30 minutes; whence the hypotenuse AF is found to be 101
degrees 52 minutes. And this arc AF is the measure of the angle ACF in
the figure of the crystal.
[Illustration]
In the same figure, if the plane CGHF cuts the crystal so that it
divides the obtuse angles ACB, MHV, in the middle, it is stated, in
Article 10, that the angle CFH is 70 degrees 57 minutes. This again is
easily shown in the same spherical triangle ABF, in which it appears
that the arc FQ is as many degrees as the angle GCF in the crystal,
the supplement of which is the angle CFH. Now the arc FQ is found to
be 109 degrees 3 minutes. Then its supplement, 70 degrees 57 minutes,
is the angle CFH.
It was stated, in Article 26, that the straight line CS, which in the
preceding figure is CH, being the axis of the crystal, that is to say
being equally inclined to the three sides CA, CB, CF, the angle GCH is
45 degrees 20 minutes. This is also easily calculated by the same
spherical triangle. For by drawing the other arc AD which cuts BF
equally, and intersects FQ at S, this point will be the centre of the
triangle. And it is easy to see that the arc SQ is the measure of the
angle GCH in the figure which represents the crystal. Now in the
triangle QAS, which is right-angled, one knows also the angle A, which
is 52 degrees 30 minutes, and the side AQ 50 degrees 56 minutes;
whence the side SQ is found to be 45 degrees 20 minutes.
In Article 27 it was required to show that PMS being an ellipse the
centre of which is C, and which touches the straight line MD at M so
that the angle MCL which CM makes with CL, perpendicular on DM, is 6
degrees 40 minutes, and its semi-minor axis CS making with CG (which
is parallel to MD) an angle GCS of 45 degrees 20 minutes, it was
required to show, I say, that, CM being 100,000 parts, PC the
semi-major diameter of this ellipse is 105,032 parts, and CS, the
semi-minor diameter, 93,410.
Let CP and CS be prolonged and meet the tangent DM at D and Z; and
from the point of contact M let MN and MO be drawn as perpendiculars
to CP and CS. Now because the angles SCP, GCL, are right angles, the
angle PCL will be equal to GCS which was 45 degrees 20 minutes. And
deducting the angle LCM, which is 6 degrees 40 minutes, from LCP,
which is 45 degrees 20 minutes, there remains MCP, 38 degrees 40
minutes. Considering then CM as a radius of 100,000 parts, MN, the
sine of 38 degrees 40 minutes, will be 62,479. And in the right-angled
triangle MND, MN will be to ND as the radius of the Tables is to the
tangent of 45 degrees 20 minutes (because the angle NMD is equal to
DCL, or GCS); that is to say as 100,000 to 101,170: whence results ND
63,210. But NC is 78,079 of the same parts, CM being 100,000, because
NC is the sine of the complement of the angle MCP, which was 38
degrees 40 minutes. Then the whole line DC is 141,289; and CP, which
is a mean proportional between DC and CN, since MD touches the
Ellipse, will be 105,032.
[Illustration]
Similarly, because the angle OMZ is equal to CDZ, or LCZ, which is 44
degrees 40 minutes, being the complement of GCS, it follows that, as
the radius of the Tables is to the tangent of 44 degrees 40 minutes,
so will OM 78,079 be to OZ 77,176. But OC is 62,479 of these same
parts of which CM is 100,000, because it is equal to MN, the sine of
the angle MCP, which is 38 degrees 40 minutes. Then the whole line CZ
is 139,655; and CS, which is a mean proportional between CZ and CO
will be 93,410.
At the same place it was stated that GC was found to be 98,779 parts.
To prove this, let PE be drawn in the same figure parallel to DM, and
meeting CM at E. In the right-angled triangle CLD the side CL is
99,324 (CM being 100,000), because CL is the sine of the complement of
the angle LCM, which is 6 degrees 40 minutes. And since the angle LCD
is 45 degrees 20 minutes, being equal to GCS, the side LD is found to
be 100,486: whence deducting ML 11,609 there will remain MD 88,877.
Now as CD (which was 141,289) is to DM 88,877, so will CP 105,032 be
to PE 66,070. But as the rectangle MEH (or rather the difference of
the squares on CM and CE) is to the square on MC, so is the square on
PE to the square on C_g_; then also as the difference of the squares
on DC and CP to the square on CD, so also is the square on PE to the
square on _g_C. But DP, CP, and PE are known; hence also one knows GC,
which is 98,779.
_Lemma which has been supposed_.
If a spheroid is touched by a straight line, and also by two or more
planes which are parallel to this line, though not parallel to one
another, all the points of contact of the line, as well as of the
planes, will be in one and the same ellipse made by a plane which
passes through the centre of the spheroid.
Let LED be the spheroid touched by the line BM at the point B, and
also by the planes parallel to this line at the points O and A. It is
required to demonstrate that the points B, O, and A are in one and the
same Ellipse made in the spheroid by a plane which passes through its
centre.
[Illustration]
Through the line BM, and through the points O and A, let there be
drawn planes parallel to one another, which, in cutting the spheroid
make the ellipses LBD, POP, QAQ; which will all be similar and
similarly disposed, and will have their centres K, N, R, in one and
the same diameter of the spheroid, which will also be the diameter of
the ellipse made by the section of the plane that passes through the
centre of the spheroid, and which cuts the planes of the three said
Ellipses at right angles: for all this is manifest by proposition 15
of the book of Conoids and Spheroids of Archimedes. Further, the two
latter planes, which are drawn through the points O and A, will also,
by cutting the planes which touch the spheroid in these same points,
generate straight lines, as OH and AS, which will, as is easy to see,
be parallel to BM; and all three, BM, OH, AS, will touch the Ellipses
LBD, POP, QAQ in these points, B, O, A; since they are in the planes
of these ellipses, and at the same time in the planes which touch the
spheroid. If now from these points B, O, A, there are drawn the
straight lines BK, ON, AR, through the centres of the same ellipses,
and if through these centres there are drawn also the diameters LD,
PP, QQ, parallel to the tangents BM, OH, AS; these will be conjugate
to the aforesaid BK, ON, AR. And because the three ellipses are
similar and similarly disposed, and have their diameters LD, PP, QQ
parallel, it is certain that their conjugate diameters BK, ON, AR,
will also be parallel. And the centres K, N, R being, as has been
stated, in one and the same diameter of the spheroid, these parallels
BK, ON, AR will necessarily be in one and the same plane, which passes
through this diameter of the spheroid, and, in consequence, the points
R, O, A are in one and the same ellipse made by the intersection of
this plane. Which was to be proved. And it is manifest that the
demonstration would be the same if, besides the points O, A, there had
been others in which the spheroid had been touched by planes parallel
to the straight line BM.