ON THE FIGURES OF THE TRANSPARENT BODIES
Christiaan Huygens
CHAPTER VI
Which serve for Refraction and for Reflexion
Preface | Chapter 1 | Chaper 2 | Chapter 3 | Chapter 4 | Chapter 5 | Chapter 6
After having explained how the properties of reflexion and refraction
follow from what we have supposed concerning the nature of light, and
of opaque bodies, and of transparent media, I will here set forth a
very easy and natural way of deducing, from the same principles, the
true figures which serve, either by reflexion or by refraction, to
collect or disperse the rays of light, as may be desired. For though I
do not see yet that there are means of making use of these figures, so
far as relates to Refraction, not only because of the difficulty of
shaping the glasses of Telescopes with the requisite exactitude
according to these figures, but also because there exists in
refraction itself a property which hinders the perfect concurrence of
the rays, as Mr. Newton has very well proved by experiment, I will yet
not desist from relating the invention, since it offers itself, so to
speak, of itself, and because it further confirms our Theory of
refraction, by the agreement which here is found between the refracted
ray and the reflected ray. Besides, it may occur that some one in the
future will discover in it utilities which at present are not seen.
[Illustration]
To proceed then to these figures, let us suppose first that it is
desired to find a surface CDE which shall reassemble at a point B rays
coming from another point A; and that the summit of the surface shall
be the given point D in the straight line AB. I say that, whether by
reflexion or by refraction, it is only necessary to make this surface
such that the path of the light from the point A to all points of the
curved line CDE, and from these to the point of concurrence (as here
the path along the straight lines AC, CB, along AL, LB, and along AD,
DB), shall be everywhere traversed in equal times: by which principle
the finding of these curves becomes very easy.
[Illustration]
So far as relates to the reflecting surface, since the sum of the
lines AC, CB ought to be equal to that of AD, DB, it appears that DCE
ought to be an ellipse; and for refraction, the ratio of the
velocities of waves of light in the media A and B being supposed to be
known, for example that of 3 to 2 (which is the same, as we have
shown, as the ratio of the Sines in the refraction), it is only
necessary to make DH equal to 3/2 of DB; and having after that
described from the centre A some arc FC, cutting DB at F, then
describe another from centre B with its semi-diameter BX equal to 2/3
of FH; and the point of intersection of the two arcs will be one of
the points required, through which the curve should pass. For this
point, having been found in this fashion, it is easy forthwith to
demonstrate that the time along AC, CB, will be equal to the time
along AD, DB.
For assuming that the line AD represents the time which the light
takes to traverse this same distance AD in air, it is evident that DH,
equal to 3/2 of DB, will represent the time of the light along DB in
the medium, because it needs here more time in proportion as its speed
is slower. Therefore the whole line AH will represent the time along
AD, DB. Similarly the line AC or AF will represent the time along AC;
and FH being by construction equal to 3/2 of CB, it will represent the
time along CB in the medium; and in consequence the whole line AH will
represent also the time along AC, CB. Whence it appears that the time
along AC, CB, is equal to the time along AD, DB. And similarly it can
be shown if L and K are other points in the curve CDE, that the times
along AL, LB, and along AK, KB, are always represented by the line AH,
and therefore equal to the said time along AD, DB.
In order to show further that the surfaces, which these curves will
generate by revolution, will direct all the rays which reach them from
the point A in such wise that they tend towards B, let there be
supposed a point K in the curve, farther from D than C is, but such
that the straight line AK falls from outside upon the curve which
serves for the refraction; and from the centre B let the arc KS be
described, cutting BD at S, and the straight line CB at R; and from
the centre A describe the arc DN meeting AK at N.
Since the sums of the times along AK, KB, and along AC, CB are equal,
if from the former sum one deducts the time along KB, and if from the
other one deducts the time along RB, there will remain the time along
AK as equal to the time along the two parts AC, CR. Consequently in
the time that the light has come along AK it will also have come along
AC and will in addition have made, in the medium from the centre C, a
partial spherical wave, having a semi-diameter equal to CR. And this
wave will necessarily touch the circumference KS at R, since CB cuts
this circumference at right angles. Similarly, having taken any other
point L in the curve, one can show that in the same time as the light
passes along AL it will also have come along AL and in addition will
have made a partial wave, from the centre L, which will touch the same
circumference KS. And so with all other points of the curve CDE. Then
at the moment that the light reaches K the arc KRS will be the
termination of the movement, which has spread from A through DCK. And
thus this same arc will constitute in the medium the propagation of
the wave emanating from A; which wave may be represented by the arc
DN, or by any other nearer the centre A. But all the pieces of the arc
KRS are propagated successively along straight lines which are
perpendicular to them, that is to say, which tend to the centre B (for
that can be demonstrated in the same way as we have proved above that
the pieces of spherical waves are propagated along the straight lines
coming from their centre), and these progressions of the pieces of the
waves constitute the rays themselves of light. It appears then that
all these rays tend here towards the point B.
One might also determine the point C, and all the others, in this
curve which serves for the refraction, by dividing DA at G in such a
way that DG is 2/3 of DA, and describing from the centre B any arc CX
which cuts BD at N, and another from the centre A with its
semi-diameter AF equal to 3/2 of GX; or rather, having described, as
before, the arc CX, it is only necessary to make DF equal to 3/2 of
DX, and from-the centre A to strike the arc FC; for these two
constructions, as may be easily known, come back to the first one
which was shown before. And it is manifest by the last method that
this curve is the same that Mr. Des Cartes has given in his Geometry,
and which he calls the first of his Ovals.
It is only a part of this oval which serves for the refraction,
namely, the part DK, ending at K, if AK is the tangent. As to the,
other part, Des Cartes has remarked that it could serve for
reflexions, if there were some material of a mirror of such a nature
that by its means the force of the rays (or, as we should say, the
velocity of the light, which he could not say, since he held that the
movement of light was instantaneous) could be augmented in the
proportion of 3 to 2. But we have shown that in our way of explaining
reflexion, such a thing could not arise from the matter of the mirror,
and it is entirely impossible.
[Illustration]
[Illustration]
From what has been demonstrated about this oval, it will be easy to
find the figure which serves to collect to a point incident parallel
rays. For by supposing just the same construction, but the point A
infinitely distant, giving parallel rays, our oval becomes a true
Ellipse, the construction of which differs in no way from that of the
oval, except that FC, which previously was an arc of a circle, is here
a straight line, perpendicular to DB. For the wave of light DN, being
likewise represented by a straight line, it will be seen that all the
points of this wave, travelling as far as the surface KD along lines
parallel to DB, will advance subsequently towards the point B, and
will arrive there at the same time. As for the Ellipse which served
for reflexion, it is evident that it will here become a parabola,
since its focus A may be regarded as infinitely distant from the
other, B, which is here the focus of the parabola, towards which all
the reflexions of rays parallel to AB tend. And the demonstration of
these effects is just the same as the preceding.
But that this curved line CDE which serves for refraction is an
Ellipse, and is such that its major diameter is to the distance
between its foci as 3 to 2, which is the proportion of the refraction,
can be easily found by the calculus of Algebra. For DB, which is
given, being called _a_; its undetermined perpendicular DT being
called _x_; and TC _y_; FB will be _a - y_; CB will be sqrt(_xx + aa
-2ay + yy_). But the nature of the curve is such that 2/3 of TC
together with CB is equal to DB, as was stated in the last
construction: then the equation will be between _(2/3)y + sqrt(xx + aa
- 2ay + yy)_ and _a_; which being reduced, gives _(6/5)ay - yy_ equal
to _(9/5)xx_; that is to say that having made DO equal to 6/5 of DB,
the rectangle DFO is equal to 9/5 of the square on FC. Whence it is
seen that DC is an ellipse, of which the axis DO is to the parameter
as 9 to 5; and therefore the square on DO is to the square of the
distance between the foci as 9 to 9 - 5, that is to say 4; and finally
the line DO will be to this distance as 3 to 2.
[Illustration]
Again, if one supposes the point B to be infinitely distant, in lieu
of our first oval we shall find that CDE is a true Hyperbola; which
will make those rays become parallel which come from the point A. And
in consequence also those which are parallel within the transparent
body will be collected outside at the point A. Now it must be remarked
that CX and KS become straight lines perpendicular to BA, because they
represent arcs of circles the centre of which is infinitely distant.
And the intersection of the perpendicular CX with the arc FC will give
the point C, one of those through which the curve ought to pass. And
this operates so that all the parts of the wave of light DN, coming to
meet the surface KDE, will advance thence along parallels to KS and
will arrive at this straight line at the same time; of which the proof
is again the same as that which served for the first oval. Besides one
finds by a calculation as easy as the preceding one, that CDE is here
a hyperbola of which the axis DO is 4/5 of AD, and the parameter
equal to AD. Whence it is easily proved that DO is to the distance
between the foci as 3 to 2.
[Illustration]
These are the two cases in which Conic sections serve for refraction,
and are the same which are explained, in his _Dioptrique_, by Des
Cartes, who first found out the use of these lines in relation to
refraction, as also that of the Ovals the first of which we have
already set forth. The second oval is that which serves for rays that
tend to a given point; in which oval, if the apex of the surface which
receives the rays is D, it will happen that the other apex will be
situated between B and A, or beyond A, according as the ratio of AD to
DB is given of greater or lesser value. And in this latter case it is
the same as that which Des Cartes calls his 3rd oval.
Now the finding and construction of this second oval is the same as
that of the first, and the demonstration of its effect likewise. But
it is worthy of remark that in one case this oval becomes a perfect
circle, namely when the ratio of AD to DB is the same as the ratio of
the refractions, here as 3 to 2, as I observed a long time ago. The
4th oval, serving only for impossible reflexions, there is no need to
set it forth.
[Illustration]
As for the manner in which Mr. Des Cartes discovered these lines,
since he has given no explanation of it, nor any one else since that I
know of, I will say here, in passing, what it seems to me it must have
been. Let it be proposed to find the surface generated by the
revolution of the curve KDE, which, receiving the incident rays coming
to it from the point A, shall deviate them toward the point B. Then
considering this other curve as already known, and that its apex D is
in the straight line AB, let us divide it up into an infinitude of
small pieces by the points G, C, F; and having drawn from each of
these points, straight lines towards A to represent the incident rays,
and other straight lines towards B, let there also be described with
centre A the arcs GL, CM, FN, DO, cutting the rays that come from A at
L, M, N, O; and from the points K, G, C, F, let there be described
the arcs KQ, GR, CS, FT cutting the rays towards B at Q, R, S, T; and
let us suppose that the straight line HKZ cuts the curve at K at
right-angles.
[Illustration]
Then AK being an incident ray, and KB its refraction within the
medium, it needs must be, according to the law of refraction which was
known to Mr. Des Cartes, that the sine of the angle ZKA should be to
the sine of the angle HKB as 3 to 2, supposing that this is the
proportion of the refraction of glass; or rather, that the sine of the
angle KGL should have this same ratio to the sine of the angle GKQ,
considering KG, GL, KQ as straight lines because of their smallness.
But these sines are the lines KL and GQ, if GK is taken as the radius
of the circle. Then LK ought to be to GQ as 3 to 2; and in the same
ratio MG to CR, NC to FS, OF to DT. Then also the sum of all the
antecedents to all the consequents would be as 3 to 2. Now by
prolonging the arc DO until it meets AK at X, KX is the sum of the
antecedents. And by prolonging the arc KQ till it meets AD at Y, the
sum of the consequents is DY. Then KX ought to be to DY as 3 to 2.
Whence it would appear that the curve KDE was of such a nature that
having drawn from some point which had been assumed, such as K, the
straight lines KA, KB, the excess by which AK surpasses AD should be
to the excess of DB over KB, as 3 to 2. For it can similarly be
demonstrated, by taking any other point in the curve, such as G, that
the excess of AG over AD, namely VG, is to the excess of BD over DG,
namely DP, in this same ratio of 3 to 2. And following this principle
Mr. Des Cartes constructed these curves in his _Geometric_; and he
easily recognized that in the case of parallel rays, these curves
became Hyperbolas and Ellipses.
Let us now return to our method and let us see how it leads without
difficulty to the finding of the curves which one side of the glass
requires when the other side is of a given figure; a figure not only
plane or spherical, or made by one of the conic sections (which is the
restriction with which Des Cartes proposed this problem, leaving the
solution to those who should come after him) but generally any figure
whatever: that is to say, one made by the revolution of any given
curved line to which one must merely know how to draw straight lines
as tangents.
Let the given figure be that made by the revolution of some curve such
as AK about the axis AV, and that this side of the glass receives rays
coming from the point L. Furthermore, let the thickness AB of the
middle of the glass be given, and the point F at which one desires the
rays to be all perfectly reunited, whatever be the first refraction
occurring at the surface AK.
I say that for this the sole requirement is that the outline BDK which
constitutes the other surface shall be such that the path of the
light from the point L to the surface AK, and from thence to the
surface BDK, and from thence to the point F, shall be traversed
everywhere in equal times, and in each case in a time equal to that
which the light employs, to pass along the straight line LF of which
the part AB is within the glass.
[Illustration]
Let LG be a ray falling on the arc AK. Its refraction GV will be given
by means of the tangent which will be drawn at the point G. Now in GV
the point D must be found such that FD together with 3/2 of DG and the
straight line GL, may be equal to FB together with 3/2 of BA and the
straight line AL; which, as is clear, make up a given length. Or
rather, by deducting from each the length of LG, which is also given,
it will merely be needful to adjust FD up to the straight line VG in
such a way that FD together with 3/2 of DG is equal to a given
straight line, which is a quite easy plane problem: and the point D
will be one of those through which the curve BDK ought to pass. And
similarly, having drawn another ray LM, and found its refraction MO,
the point N will be found in this line, and so on as many times as one
desires.
To demonstrate the effect of the curve, let there be described about
the centre L the circular arc AH, cutting LG at H; and about the
centre F the arc BP; and in AB let AS be taken equal to 2/3 of HG; and
SE equal to GD. Then considering AH as a wave of light emanating from
the point L, it is certain that during the time in which its piece H
arrives at G the piece A will have advanced within the transparent
body only along AS; for I suppose, as above, the proportion of the
refraction to be as 3 to 2. Now we know that the piece of wave which
is incident on G, advances thence along the line GD, since GV is the
refraction of the ray LG. Then during the time that this piece of wave
has taken from G to D, the other piece which was at S has reached E,
since GD, SE are equal. But while the latter will advance from E to B,
the piece of wave which was at D will have spread into the air its
partial wave, the semi-diameter of which, DC (supposing this wave to
cut the line DF at C), will be 3/2 of EB, since the velocity of light
outside the medium is to that inside as 3 to 2. Now it is easy to show
that this wave will touch the arc BP at this point C. For since, by
construction, FD + 3/2 DG + GL are equal to FB + 3/2 BA + AL; on
deducting the equals LH, LA, there will remain FD + 3/2 DG + GH equal
to FB + 3/2 BA. And, again, deducting from one side GH, and from the
other side 3/2 of AS, which are equal, there will remain FD with 3/2
DG equal to FB with 3/2 of BS. But 3/2 of DG are equal to 3/2 of ES;
then FD is equal to FB with 3/2 of BE. But DC was equal to 3/2 of EB;
then deducting these equal lengths from one side and from the other,
there will remain CF equal to FB. And thus it appears that the wave,
the semi-diameter of which is DC, touches the arc BP at the moment
when the light coming from the point L has arrived at B along the line
LB. It can be demonstrated similarly that at this same moment the
light that has come along any other ray, such as LM, MN, will have
propagated the movement which is terminated at the arc BP. Whence it
follows, as has been often said, that the propagation of the wave AH,
after it has passed through the thickness of the glass, will be the
spherical wave BP, all the pieces of which ought to advance along
straight lines, which are the rays of light, to the centre F. Which
was to be proved. Similarly these curved lines can be found in all the
cases which can be proposed, as will be sufficiently shown by one or
two examples which I will add.
Let there be given the surface of the glass AK, made by the revolution
about the axis BA of the line AK, which may be straight or curved. Let
there be also given in the axis the point L and the thickness BA of
the glass; and let it be required to find the other surface KDB, which
receiving rays that are parallel to AB will direct them in such wise
that after being again refracted at the given surface AK they will all
be reassembled at the point L.
[Illustration]
From the point L let there be drawn to some point of the given line
AK the straight line LG, which, being considered as a ray of light,
its refraction GD will then be found. And this line being then
prolonged at one side or the other will meet the straight line BL, as
here at V. Let there then be erected on AB the perpendicular BC, which
will represent a wave of light coming from the infinitely distant
point F, since we have supposed the rays to be parallel. Then all the
parts of this wave BC must arrive at the same time at the point L; or
rather all the parts of a wave emanating from the point L must arrive
at the same time at the straight line BC. And for that, it is
necessary to find in the line VGD the point D such that having drawn
DC parallel to AB, the sum of CD, plus 3/2 of DG, plus GL may be equal
to 3/2 of AB, plus AL: or rather, on deducting from both sides GL,
which is given, CD plus 3/2 of DG must be equal to a given length;
which is a still easier problem than the preceding construction. The
point D thus found will be one of those through which the curve ought
to pass; and the proof will be the same as before. And by this it will
be proved that the waves which come from the point L, after having
passed through the glass KAKB, will take the form of straight lines,
as BC; which is the same thing as saying that the rays will become
parallel. Whence it follows reciprocally that parallel rays falling on
the surface KDB will be reassembled at the point L.
[Illustration]
Again, let there be given the surface AK, of any desired form,
generated by revolution about the axis AB, and let the thickness of
the glass at the middle be AB. Also let the point L be given in the
axis behind the glass; and let it be supposed that the rays which fall
on the surface AK tend to this point, and that it is required to find
the surface BD, which on their emergence from the glass turns them as
if they came from the point F in front of the glass.
Having taken any point G in the line AK, and drawing the straight line
IGL, its part GI will represent one of the incident rays, the
refraction of which, GV, will then be found: and it is in this line
that we must find the point D, one of those through which the curve DG
ought to pass. Let us suppose that it has been found: and about L as
centre let there be described GT, the arc of a circle cutting the
straight line AB at T, in case the distance LG is greater than LA; for
otherwise the arc AH must be described about the same centre, cutting
the straight line LG at H. This arc GT (or AH, in the other case) will
represent an incident wave of light, the rays of which tend towards
L. Similarly, about the centre F let there be described the circular
arc DQ, which will represent a wave emanating from the point F.
Then the wave TG, after having passed through the glass, must form the
wave QD; and for this I observe that the time taken by the light along
GD in the glass must be equal to that taken along the three, TA, AB,
and BQ, of which AB alone is within the glass. Or rather, having taken
AS equal to 2/3 of AT, I observe that 3/2 of GD ought to be equal to
3/2 of SB, plus BQ; and, deducting both of them from FD or FQ, that FD
less 3/2 of GD ought to be equal to FB less 3/2 of SB. And this last
difference is a given length: and all that is required is to draw the
straight line FD from the given point F to meet VG so that it may be
thus. Which is a problem quite similar to that which served for the
first of these constructions, where FD plus 3/2 of GD had to be equal
to a given length.
In the demonstration it is to be observed that, since the arc BC falls
within the glass, there must be conceived an arc RX, concentric with
it and on the other side of QD. Then after it shall have been shown
that the piece G of the wave GT arrives at D at the same time that the
piece T arrives at Q, which is easily deduced from the construction,
it will be evident as a consequence that the partial wave generated at
the point D will touch the arc RX at the moment when the piece Q shall
have come to R, and that thus this arc will at the same moment be the
termination of the movement that comes from the wave TG; whence all
the rest may be concluded.
Having shown the method of finding these curved lines which serve for
the perfect concurrence of the rays, there remains to be explained a
notable thing touching the uncoordinated refraction of spherical,
plane, and other surfaces: an effect which if ignored might cause some
doubt concerning what we have several times said, that rays of light
are straight lines which intersect at right angles the waves which
travel along them.
[Illustration]
For in the case of rays which, for example, fall parallel upon a
spherical surface AFE, intersecting one another, after refraction, at
different points, as this figure represents; what can the waves of
light be, in this transparent body, which are cut at right angles by
the converging rays? For they can not be spherical. And what will
these waves become after the said rays begin to intersect one another?
It will be seen in the solution of this difficulty that something very
remarkable comes to pass herein, and that the waves do not cease to
persist though they do not continue entire, as when they cross the
glasses designed according to the construction we have seen.
According to what has been shown above, the straight line AD, which
has been drawn at the summit of the sphere, at right angles to the
axis parallel to which the rays come, represents the wave of light;
and in the time taken by its piece D to reach the spherical surface
AGE at E, its other parts will have met the same surface at F, G, H,
etc., and will have also formed spherical partial waves of which these
points are the centres. And the surface EK which all those waves will
touch, will be the continuation of the wave AD in the sphere at the
moment when the piece D has reached E. Now the line EK is not an arc
of a circle, but is a curved line formed as the evolute of another
curve ENC, which touches all the rays HL, GM, FO, etc., that are the
refractions of the parallel rays, if we imagine laid over the
convexity ENC a thread which in unwinding describes at its end E the
said curve EK. For, supposing that this curve has been thus described,
we will show that the said waves formed from the centres F, G, H,
etc., will all touch it.
It is certain that the curve EK and all the others described by the
evolution of the curve ENC, with different lengths of thread, will cut
all the rays HL, GM, FO, etc., at right angles, and in such wise that
the parts of them intercepted between two such curves will all be
equal; for this follows from what has been demonstrated in our
treatise _de Motu Pendulorum_. Now imagining the incident rays as
being infinitely near to one another, if we consider two of them, as
RG, TF, and draw GQ perpendicular to RG, and if we suppose the curve
FS which intersects GM at P to have been described by evolution from
the curve NC, beginning at F, as far as which the thread is supposed
to extend, we may assume the small piece FP as a straight line
perpendicular to the ray GM, and similarly the arc GF as a straight
line. But GM being the refraction of the ray RG, and FP being
perpendicular to it, QF must be to GP as 3 to 2, that is to say in the
proportion of the refraction; as was shown above in explaining the
discovery of Des Cartes. And the same thing occurs in all the small
arcs GH, HA, etc., namely that in the quadrilaterals which enclose
them the side parallel to the axis is to the opposite side as 3 to 2.
Then also as 3 to 2 will the sum of the one set be to the sum of the
other; that is to say, TF to AS, and DE to AK, and BE to SK or DV,
supposing V to be the intersection of the curve EK and the ray FO.
But, making FB perpendicular to DE, the ratio of 3 to 2 is also that
of BE to the semi-diameter of the spherical wave which emanated from
the point F while the light outside the transparent body traversed the
space BE. Then it appears that this wave will intersect the ray FM at
the same point V where it is intersected at right angles by the curve
EK, and consequently that the wave will touch this curve. In the same
way it can be proved that the same will apply to all the other waves
above mentioned, originating at the points G, H, etc.; to wit, that
they will touch the curve EK at the moment when the piece D of the
wave ED shall have reached E.
Now to say what these waves become after the rays have begun to cross
one another: it is that from thence they fold back and are composed of
two contiguous parts, one being a curve formed as evolute of the curve
ENC in one sense, and the other as evolute of the same curve in the
opposite sense. Thus the wave KE, while advancing toward the meeting
place becomes _abc_, whereof the part _ab_ is made by the evolute
_b_C, a portion of the curve ENC, while the end C remains attached;
and the part _bc_ by the evolute of the portion _b_E while the end E
remains attached. Consequently the same wave becomes _def_, then
_ghk_, and finally CY, from whence it subsequently spreads without any
fold, but always along curved lines which are evolutes of the curve
ENC, increased by some straight line at the end C.
There is even, in this curve, a part EN which is straight, N being the
point where the perpendicular from the centre X of the sphere falls
upon the refraction of the ray DE, which I now suppose to touch the
sphere. The folding of the waves of light begins from the point N up
to the end of the curve C, which point is formed by taking AC to CX in
the proportion of the refraction, as here 3 to 2.
As many other points as may be desired in the curve NC are found by a
Theorem which Mr. Barrow has demonstrated in section 12 of his
_Lectiones Opticae_, though for another purpose. And it is to be noted
that a straight line equal in length to this curve can be given. For
since it together with the line NE is equal to the line CK, which is
known, since DE is to AK in the proportion of the refraction, it
appears that by deducting EN from CK the remainder will be equal to
the curve NC.
Similarly the waves that are folded back in reflexion by a concave
spherical mirror can be found. Let ABC be the section, through the
axis, of a hollow hemisphere, the centre of which is D, its axis being
DB, parallel to which I suppose the rays of light to come. All the
reflexions of those rays which fall upon the quarter-circle AB will
touch a curved line AFE, of which line the end E is at the focus of
the hemisphere, that is to say, at the point which divides the
semi-diameter BD into two equal parts. The points through which this
curve ought to pass are found by taking, beyond A, some arc AO, and
making the arc OP double the length of it; then dividing the chord OP
at F in such wise that the part FP is three times the part FO; for
then F is one of the required points.
[Illustration]
And as the parallel rays are merely perpendiculars to the waves which
fall on the concave surface, which waves are parallel to AD, it will
be found that as they come successively to encounter the surface AB,
they form on reflexion folded waves composed of two curves which
originate from two opposite evolutions of the parts of the curve AFE.
So, taking AD as an incident wave, when the part AG shall have met the
surface AI, that is to say when the piece G shall have reached I, it
will be the curves HF, FI, generated as evolutes of the curves FA, FE,
both beginning at F, which together constitute the propagation of the
part AG. And a little afterwards, when the part AK has met the surface
AM, the piece K having come to M, then the curves LN, NM, will
together constitute the propagation of that part. And thus this folded
wave will continue to advance until the point N has reached the focus
E. The curve AFE can be seen in smoke, or in flying dust, when a
concave mirror is held opposite the sun. And it should be known that
it is none other than that curve which is described by the point E on
the circumference of the circle EB, when that circle is made to roll
within another whose semi-diameter is ED and whose centre is D. So
that it is a kind of Cycloid, of which, however, the points can be
found geometrically.
Its length is exactly equal to 3/4 of the diameter of the sphere, as
can be found and demonstrated by means of these waves, nearly in the
same way as the mensuration of the preceding curve; though it may also
be demonstrated in other ways, which I omit as outside the subject.
The area AOBEFA, comprised between the arc of the quarter-circle, the
straight line BE, and the curve EFA, is equal to the fourth part of
the quadrant DAB.
INDEX
Archimedes, 104.
Atmospheric refraction, 45.
Barrow, Isaac, 126.
Bartholinus, Erasmus, 53, 54, 57, 60, 97, 99.
Boyle, Hon. Robert, 11.
Cassini, Jacques, iii.
Caustic Curves, 123.
Crystals, see Iceland Crystal, Rock Crystal.
Crystals, configuration of, 95.
Descartes, Rénê, 3, 5, 7, 14, 22, 42, 43, 109, 113.
Double Refraction, discovery of, 54, 81, 93.
Elasticity, 12, 14.
Ether, the, or Ethereal matter, 11, 14, 16, 28.
Extraordinary refraction, 55, 56.
Fermat, principle of, 42.
Figures of transparent bodies, 105.
Hooke, Robert, 20.
Iceland Crystal, 2, 52 sqq.
Iceland Crystal, Cutting and Polishing of, 91, 92, 98.
Leibnitz, G.W., vi.
Light, nature of, 3.
Light, velocity of, 4, 15.
Molecular texture of bodies, 27, 95.
Newton, Sir Isaac, vi, 106.
Opacity, 34.
Ovals, Cartesian, 107, 113.
Pardies, Rev. Father, 20.
Rays, definition of, 38, 49.
Reflexion, 22.
Refraction, 28, 34.
Rock Crystal, 54, 57, 62, 95.
Römer, Olaf, v, 7.
Roughness of surfaces, 27.
Sines, law of, 1, 35, 38, 43.
Spheres, elasticity of, 15.
Spheroidal waves in crystals, 63.
Spheroids, lemma about, 103.
Sound, speed of, 7, 10, 12.
Telescopes, lenses for, 62, 105.
Torricelli's experiment, 12, 30.
Transparency, explanation of, 28, 31, 32.
Waves, no regular succession of, 17.
Waves, principle of wave envelopes, 19, 24.
Waves, principle of elementary wave fronts, 19.
Waves, propagation of light as, 16, 63. |
