refraction alleged by Alhazen and Vitellion. They say that the light
seeks to be compensated for the loss sustained at the oblique impact; so
that in proportion as it is enfeebled by striking against the denser
medium, in the same degree does it restore its energy by approaching the
perpendicular, that it may strike the bottom of the denser medium with
greater force; for those impacts are most forcible which are direct. And
they add some subtle notions, I know not what, how the motion of
obliquely incident light is compounded of a motion perpendicular and a
motion parallel to the dense surface, and that this compound motion is
not destroyed, but only retarded by meeting the denser medium.
[Illustration]
"I tried another way of measuring the refraction, which should include
the density of the medium and the incidence: for, since a denser medium
is the cause of refraction, it seems to be the same thing as if we were
to prolong the depth of the medium in which the rays are refracted into
as much space as would be filled by the denser medium under the force of
the rarer one.
"Let A be the place of the light, BC the surface of the denser medium,
DE its bottom. Let AB, AG, AF be rays falling obliquely, which would
arrive at D, I, H, if the medium were uniform. But because it is denser,
suppose the bottom to be depressed to KL, determined by this that there
is as much of the denser matter contained in the space DC as of the
rarer in LC: and thus, on the sinking of the whole bottom DE, the points
D, I, H, E will descend vertically to L, M, N, K. Join the points BL,
GM, FN, cutting DE in O, P, Q; the refracted rays will be ABO, AGP,
AFQ."—"This method is refuted by experiment; it gives the refractions
near the perpendicular AC too great in respect of those near the
horizon. Whoever has leisure may verify this, either by calculation or
compasses. It may be added that the reasoning itself is not very
sure-footed, and, whilst seeking to measure other things, scarcely takes
in and comprehends itself." This reflection must not be mistaken for the
dawn of suspicion that his examination of philosophical questions began
not altogether at the right end: it is merely an acknowledgment that he
had not yet contrived a theory with which he was quite satisfied before
it was disproved by experiment.
After some experience of Kepler's miraculous good fortune in seizing
truths across the wildest and most absurd theories, it is not easy to
keep clear of the opposite feeling of surprise whenever any of his
extravagancies fail to discover to him some beautiful law of nature. But
we must follow him as he plunges deeper in this unsuccessful inquiry;
and the reader must remember, in order fully to appreciate this method
of philosophizing, that it is almost certain that Kepler laboured upon
every one of the gratuitous suppositions that he makes, until positive
experiment satisfied him of their incorrectness.
"I go on to other methods. Since density is clearly connected with the
cause of the refractions, and refraction itself seems a kind of
compression of light, as it were, towards the perpendicular, it occurred
to me to examine whether there was the same proportion between the
mediums in respect of density and the parts of the bottom illuminated by
the light, when let into a vessel, first empty, and afterwards filled
with water. This mode branches out into many: for the proportion may be
imagined, either in the straight lines, as if one should say that the
line EQ, illuminated by refraction, is to EH illuminated directly, as
the density of the one medium is to that of the other—Or another may
suppose the proportion to be between FC and FH—Or it may be conceived
to exist among surfaces, or so that some power of EQ should be to some
power of EH in this proportion, or the circles or similar figures
described on them. In this manner the proportion of EQ to EP would be
double that of EH to EI—Or the proportion may be conceived existing
among the solidities of the pyramidal frustums FHEC, FQEC—Or, since the
proportion of the mediums involves a threefold consideration, since they
have density in length, breadth, and thickness, I proceeded also to
examine the cubic proportions among the lines EQ, EH.
"I also considered other lines. From any of the points of refraction as
G, let a perpendicular GY be dropped upon the bottom. It may become a
question whether possibly the triangle IGY, that is, the base IY, is
divided by the refracted ray GP, in the proportion of the densities of
the mediums.
"I have put all these methods here together, because the same remark
disproves them all. For, in whatever manner, whether as line, plane, or
pyramid, EI observes a given proportion to EP, or the abbreviated line
YI to YP, namely, the proportion of the mediums, it is sure that EI, the
tangent of the distance of the point A from the vertex, will become
infinite, and will, therefore make EP or YP, also infinite. Therefore,
IGP, the angle of refraction, will be entirely lost; and, as it
approaches the horizon, will gradually become less and less, which is
contrary to experiment.
"I tried again whether the images are equally removed from their points
of refraction, and whether the ratio of the densities measures the least
distance. For instance, supposing E to be the image, C the surface of
the water, K the bottom, and CE to CK in the proportion of the densities
of the mediums. Now, let F, G, B, be three other points of refraction
and images at S, T, V, and let CE be equal to FS, GT, and BV. But
according to this rule an image E would still be somewhat raised in the
perpendicular AK, which is contrary to experiment, not to mention other
contradictions. Thirdly, whether the proportion of the mediums holds
between FH and FX, supposing H to be the place of the image? Not at all.
For so, CE would be in the same proportion to CK, so that the height of
the image would always be the same, which we have just refuted.
Fourthly, whether the raising of the image at E is to the raising at H,
as CE to FH? Not in the least; for so the images either would never
begin to be raised, or, having once begun, would at last be infinitely
raised, because FH at last becomes infinite. Fifthly, whether the images
rise in proportion to the sines of the inclinations? Not at all; for so
the proportion of ascent would be the same in all mediums. Sixthly, are
then the images raised at first, and in perpendicular radiation,
according to the proportion of the mediums, and do they subsequently
rise more and more according to the sines of the inclinations? For so
the proportion would be compound, and would become different in
different mediums. There is nothing in it: for the calculation disagreed
with experiment. And generally it is in vain to have regard to the image
or the place of the image, for that very reason, that it is imaginary.
For there is no connexion between the density of the medium or any real
quality or refraction of the light, and an accident of vision, by an
error of which the image happens.
"Up to this point, therefore, I had followed a nearly blind mode of
inquiry, and had trusted to good fortune; but now I opened the other
eye, and hit upon a sure method, for I pondered the fact, that the image
of a thing seen under water approaches closely to the true ratio of the
refraction, and almost measures it; that it is low if the thing is
viewed directly from above; that by degrees it rises as the eye passes
towards the horizon of the water. Yet, on the other hand, the reason
alleged above, proves that the measure is not to be sought in the image,
because the image is not a thing actually existing, but arises from a
deception of vision which is purely accidental. By a comparison of these
conflicting arguments, it occurred to me at length, to seek the causes
themselves of the existence of the image under water, and in these
causes the measure of the refractions. This opinion was strengthened in
me by seeing that opticians had not rightly pointed out the cause of the
image which appears both in mirrors and in water. And this was the
origin of that labour which I undertook in the third chapter. Nor,
indeed, was that labour trifling, whilst hunting down false opinions of
all sorts among the principles, in a matter rendered so intricate by the
false traditions of optical writers; whilst striking out half a dozen
different paths, and beginning anew the whole business. How often did it
happen that a rash confidence made me look upon that which I sought with
such ardour, as at length discovered!
"At length I cut this worse than Gordian knot of catoptrics by analogy
alone, by considering what happens in mirrors, and what must happen
analogically in water. In mirrors, the image appears at a distance from
the real place of the object, not being itself material, but produced
solely by reflection at the polished surface. Whence it followed in
water also, that the images rise and approach the surface, not according
to the law of the greater or less density in the water, as the view is
less or more oblique, but solely because of the refraction of the ray of
light passing from the object to the eye. On which assumption, it is
plain that every attempt I had hitherto made to measure refractions by
the image, and its elevation, must fall to the ground. And this became
more evident when I discovered the true reason why the image is in the
same perpendicular line with the object both in mirrors and in dense
mediums. When I had succeeded thus far by analogy in this most difficult
investigation, as to the place of the image, I began to follow out the
analogy further, led on by the strong desire of measuring refraction.
For I wished to get hold of some measure of some sort, no matter how
blindly, having no fear but that so soon as the measure should be
accurately known, the cause would plainly appear. I went to work as
follows. In convex mirrors the image is diminished, and just so in rarer
mediums; in denser mediums it is magnified, as in concave mirrors. In
convex mirrors the central parts of the image approach, and recede in
concave farther than towards the circumference; the same thing happens
in different mediums, so that in water the bottom appears depressed, and
the surrounding parts elevated. Hence it appears that a denser medium
corresponds with a concave reflecting surface, and a rarer one with a
convex one: it was clear, at the same time, that the plane surface of
the water affects a property of curvature. I was, therefore, to
excogitate causes consistent with its having this effect of curvature,
and to see if a reason could be given, why the parts of the water
surrounding the incident perpendicular should represent a greater
density than the parts just under the perpendicular. And so the thing
came round again to my former attempts, which being refuted by reason
and experiment, I was forced to abandon the search after a cause. I then
proceeded to measurements."
Kepler then endeavoured to connect his measurements of different
quantities of refraction with the conic sections, and was tolerably well
pleased with some of his results. They were however not entirely
satisfactory, on which he breaks off with the following sentence: "Now,
reader, you and I have been detained sufficiently long whilst I have
been attempting to collect into one faggot the measure of different
refractions: I acknowledge that the cause cannot be connected with this
mode of measurement: for what is there in common between refractions
made at the plane surfaces of transparent mediums, and mixtilinear conic
sections? Wherefore, _quod Deus bene vortat_, we will now have had
enough of the causes of this measure; and although, even now, we are
perhaps erring something from the truth, yet it is better, by working
on, to show our industry, than our laziness by neglect."
Notwithstanding the great length of this extract, we must add the
concluding paragraph of the Chapter, directed, as we are told in the
margin, against the "Tychonomasticks:"—
"I know how many blind men at this day dispute about colours, and how
they long for some one to give some assistance by argument to their rash
insults of Tycho, and attacks upon this whole matter of refractions;
who, if they had kept to themselves their puerile errors and naked
ignorance, might have escaped censure; for that may happen to many great
men. But since they venture forth publicly, and with thick books and
sounding titles, lay baits for the applause of the unwary, (for
now-a-days there is more danger from the abundance of bad books, than
heretofore from the lack of good ones,) therefore let them know that a
time is set for them publicly to amend their own errors. If they longer
delay doing this, it shall be open, either to me or any other, to do to
these unhappy meddlers in geometry as they have taken upon themselves to
do with respect to men of the highest reputation. And although this
labour will be despicable, from the vile nature of the follies against
which it will be directed, yet so much more necessary than that which
they have undertaken against others, as he is a greater public nuisance,
who endeavours to slander good and necessary inventions, than he who
fancies he has found what is impossible to discover. Meanwhile, let them
cease to plume themselves on the silence which is another word for their
own obscurity."
Although Kepler failed, as we have seen, to detect the true law of
refraction, (which was discovered some years later by Willibrord Snell,
a Flemish mathematician,) there are many things well deserving notice in
his investigations. He remarked, that the quantity of refraction would
alter, if the height of the atmosphere should vary; and also, that it
would be different at different temperatures. Both these sources of
variation are now constantly taken into account, the barometer and
thermometer giving exact indications of these changes. There is also a
very curious passage in one of his letters to Bregger, written in 1605,
on the subject of the colours in the rainbow. It is in these
words:—"Since every one sees a different rainbow, it is possible that
some one may see a rainbow in the very place of my sight. In this case,
the medium is coloured at the place of my vision, to which the solar ray
comes to me through water, rain, or aqueous vapours. For the rainbow is
seen when the sun is shining between rain, that is to say, when the sun
also is visible. Why then do I not see the sun green, yellow, red, and
blue, if vision takes place according to the mode of illumination? I
will say something for you to attack or examine. The sun's rays are not
coloured, except with a definite quantity of refraction. Whether you are
in the optical chamber, or standing opposite glass globes, or walking in
the morning dew, everywhere it is obvious that a certain and definite
angle is observed, under which, when seen in dew, in glass, in water,
the sun's splendour appears coloured, and under no other angle. There is
no colouring by mere reflexion, without the refraction of a denser
medium." How closely does Kepler appear, in this passage, to approach
the discovery which forms not the least part of Newton's fame!
We also find in this work a defence of the opinion that the planets are
luminous of themselves; on the ground that the inferior planets would,
on the contrary supposition, display phases like those of the moon when
passing between us and the sun. The use of the telescope was not then
known; and, when some years later the form of the disk of the planets
was more clearly defined with their assistance, Kepler had the
satisfaction of finding his assertions verified by the discoveries of
Galileo, that these changes do actually take place. In another of his
speculations, connected with the same subject, he was less fortunate. In
1607 a black spot appeared on the face of sun, such as may almost always
be seen with the assistance of the telescope, although they are seldom
large enough to be visible to the unassisted eye. Kepler saw it for a
short time, and mistook it for the planet Mercury, and with his usual
precipitancy hastened to publish an account of his observation of this
rare phenomenon. A few years later, Galileo discovered with his glasses,
a great number of similar spots; and Kepler immediately retracted the
opinion announced in his treatise, and acknowledged his belief that
previous accounts of the same occurrence which he had seen in old
authors, and which he had found great difficulty in reconciling with his
more accurate knowledge of the motions of Mercury, were to be referred
to a like mistake. On this occasion of the invention of the telescope,
Kepler's candour and real love of truth appeared in a most favourable
light. Disregarding entirely the disagreeable necessity, in consequence
of the discoveries of this new instrument, of retracting several
opinions which he had maintained with considerable warmth, he ranged
himself at once on the side of Galileo, in opposition to the bitter and
determined hostility evinced by most of those whose theories were
endangered by the new views thus offered of the heavens. Kepler's
quarrel with his pupil, Horky, on this account, has been mentioned in
the "Life of Galileo;" and this is only a selected instance from the
numerous occasions on which he espoused the same unpopular side of the
argument. He published a dissertation to accompany Galileo's
"Intelligencer of the Stars," in which he warmly expressed his
admiration of that illustrious inquirer into nature. His conduct in this
respect was the more remarkable, as some of his most intimate friends
had taken a very opposite view of Galileo's merit, and seem to have
laboured much to disturb their mutual regard; Mästlin especially,
Kepler's early instructor, seldom mentioned to him the name of Galileo,
without some contemptuous expression of dislike. These statements have
rather disturbed the chronological order of the account of Kepler's
works. We now return to the year 1609, in which he published his great
and extraordinary book, "On the Motions of Mars;" a work which holds the
intermediate place, and is in truth the connecting link, between the
discoveries of Copernicus and Newton.
CHAPTER IV.
_Sketch of the Astronomical Theories before Kepler._
KEPLER had begun to labour upon these commentaries from the moment when
he first made Tycho's acquaintance; and it is on this work that his
reputation should be made mainly to rest. It is marked in many places
with his characteristic precipitancy, and indeed one of the most
important discoveries announced in it (famous among astronomers by the
name of the Equable Description of Areas) was blundered upon by a lucky
compensation of errors, of the nature of which Kepler remained ignorant
to the very last. Yet there is more of the inductive method in this than
in any of his other publications; and the unwearied perseverance with
which he exhausted years in hunting down his often renewed theories,
till at length he seemed to arrive at the true one, almost by having
previously disproved every other, excites a feeling of astonishment
nearly approaching to awe. It is wonderful how he contrived to retain
his vivacity and creative fancy amongst the clouds of figures which he
conjured up round him; for the slightest hint or shade of probability
was sufficient to plunge him into the midst of the most laborious
computations. He was by no means an accurate calculator, according to
the following character which he has given of himself:—"Something of
these delays must be attributed to my own temper, for _non omnia
possumus omnes_, and I am totally unable to observe any order; what I do
suddenly, I do confusedly, and if I produce any thing well arranged, it
has been done ten times over. Sometimes an error of calculation
committed by hurry, delays me a great length of time. I could indeed
publish an infinity of things, for though my reading is confined, my
imagination is abundant, but I grow dissatisfied with such confusion: I
get disgusted and out of humour, and either throw them away, or put them
aside to be looked at again; or, in other words, to be written again,
for that is generally the end of it. I entreat you, my friends, not to
condemn me for ever to grind in the mill of mathematical calculations:
allow me some time for philosophical speculations, my only delight."
He was very seldom able to afford the expense of maintaining an
assistant, and was forced to go through most of the drudgery of his
calculations by himself; and the most confirmed and merest arithmetician
could not have toiled more doggedly than Kepler did in the work of which
we are about to speak.
In order that the language of his astronomy may be understood, it is
necessary to mention briefly some of the older theories. When it had
been discovered that the planets did not move regularly round the earth,
which was supposed to be fixed in the centre of the world, a mechanism
was contrived by which it was thought that the apparent irregularity
could be represented, and yet the principle of uniform motion, which was
adhered to with superstitious reverence, might be preserved. This, in
its simplest form, consisted in supposing the planet to move uniformly
in a small circle, called an _epicycle_, the centre of which moved with
an equal angular motion in the opposite direction round the earth.[187]
The circle D_d_, described by D, the centre of the epicycle, was called
the _deferent_. For instance, if the planet was supposed to be at A when
the centre of the epicycle was at D, its position, when the centre of
the epicycle had removed to _d_, would be at _p_, found by drawing _dp_
parallel to DA. Thus, the angle _adp_, measuring the motion of the
planet in its epicycle, would be equal to DE_d_, the angle described by
the centre of the epicycle in the deferent. The angle _p_E_d_ between
E_p_, the direction in which a planet so moving would be seen from the
earth, supposed to be at E, and E_d_ the direction in which it would
have been seen had it been moving in the centre of the deferent, was
called the equation of the orbit, the word equation, in the language of
astronomy, signifying what must be added or taken from an irregularly
varying quantity to make it vary uniformly.
[Illustration]
As the accuracy of observations increased, minor irregularities were
discovered, which were attempted to be accounted for by making a second
deferent of the epicycle, and making the centre of a second epicycle
revolve in the circumference of the first, and so on, or else by
supposing the revolution in the epicycle not to be completed in exactly
the time in which its centre is carried round the deferent. Hipparchus
was the first to make a remark by which the geometrical representation
of these inequalities was considerably simplified. In fact, if EC be
taken equal to _pd_, C_d_ will be a parallelogram, and consequently
C_p_ equal to E_d_, so that the machinery of the first deferent and
epicycle amounts to supposing that the planet revolves uniformly in a
circle round the point C, not coincident with the place of the earth.
This was consequently called the excentric theory, in opposition to the
former or concentric one, and was received as a great improvement. As
the point _d_ is not represented by this construction, the equation to
the orbit was measured by the angle C_p_E, which is equal to _p_E_d_. It
is not necessary to give any account of the manner in which the old
astronomers determined the magnitudes and positions of these orbits,
either in the concentric or excentric theory, the present object being
little more than to explain the meaning of the terms it will be
necessary to use in describing Kepler's investigations.
To explain the irregularities observed in the other planets, it became
necessary to introduce another hypothesis, in adopting which the
severity of the principle of uniform motion was somewhat relaxed. The
machinery consisted partly of an excentric deferent round E, the earth,
and on it an epicycle, in which the planet revolved uniformly; but the
centre of the epicycle, instead of revolving uniformly round C, the
centre of the deferent, as it had hitherto been made to do, was
supposed to move in its circumference with an uniform angular motion
round a third point, Q; the necessary effect of which supposition was,
that the linear motion of the centre of the epicycle ceased to be
uniform. There were thus three points to be considered within the
deferent; E, the place of the earth; C, the centre of the deferent, and
sometimes called the centre of the orbit; and Q, called the centre of
the equant, because, if any circle were described round Q, the planet
would appear to a spectator at Q, to be moving equably in it. It was
long uncertain what situation should be assigned to the centre of the
equant, so as best to represent the irregularities to a spectator on the
earth, until Ptolemy decided on placing it (in every case but that of
Mercury, the observations on which were very doubtful) so that C, the
centre of the orbit, lay just half way in the straight line, joining Q,
the centre of equable motion, and E, the place of the earth. This is the
famous principle, known by the name of the bisection of the
excentricity.
[Illustration]
The first equation required for the planet's motion was thus supposed to
be due to the displacement of E, the earth, from Q, the centre of
uniform motion, which was called the excentricity of the equant: it
might be represented by the angle _d_EM, drawing EM parallel to Q_d_;
for clearly M would have been the place of the centre of the epicycle at
the end of a time proportional to D_d_, had it moved with an equable
angular motion round E instead of Q. This angle _d_EM, or its equal
E_d_Q, was called the equation of the centre (_i.e._ of the centre of
the epicycle); and is clearly greater than if EQ, the excentricity of
the equant, had been no greater than EC, called the excentricity of the
orbit. The second equation was measured by the angle subtended at E by
_d_, the centre of the epicycle, and _p_ the planet's place in its
circumference: it was called indifferently the equation of the orbit, or
of the argument. In order to account for the apparent stations and
retrogradations of the planets, it became necessary to suppose that many
revolutions in the latter were completed during one of the former. The
variations of latitude of the planets were exhibited by supposing not
only that the planes of their deferents were oblique to the plane of the
ecliptic, and that the plane of the epicycle was also oblique to that of
the deferent, but that the inclination of the two latter was continually
changing, although Kepler doubts whether this latter complication was
admitted by Ptolemy. In the inferior planets, it was even thought
necessary to give to the plane of the epicycle two oscillatory motions
on axes at right angles to each other.
The astronomers at this period were much struck with a remarkable
connexion between the revolutions of the superior planets in their
epicycles, and the apparent motion of the sun; for when in conjunction
with the sun, as seen from the earth, they were always found to be in
the apogee, or point of greatest distance from the earth, of their
epicycle; and when in opposition to the Sun, they were as regularly in
the perigee, or point of nearest approach of the epicycle. This
correspondence between two phenomena, which, according to the old
astronomy, were entirely unconnected, was very perplexing, and it seems
to have been one of the facts which led Copernicus to substitute the
theory of the earth's motion round the sun.
As time wore on, the superstructure of excentrics and epicycles, which
had been strained into representing the appearances of the heavens at a
particular moment, grew out of shape, and the natural consequence of
such an artificial system was, that it became next to impossible to
foresee what ruin might be produced in a remote part of it by any
attempt to repair the derangements and refit the parts to the changes,
as they began to be remarked in any particular point. In the ninth
century of our era, Ptolemy's tables were already useless, and all those
that were contrived with unceasing toil to supply their place, rapidly
became as unserviceable as they. Still the triumph of genius was seen in
the veneration that continued to be paid to the assumptions of Ptolemy
and Hipparchus; and even when the great reformer, Copernicus, appeared,
he did not for a long time intend to do more than slightly modify their
principles. That which he found difficult in the Ptolemaic system, was
none of the inconveniences by which, since the establishment of the new
system, it has become common to demonstrate the inferiority of the old
one; it was the displacement of the centre of the equant from the centre
of the orbit that principally indisposed him against it, and led him to
endeavour to represent the appearances by some other combinations of
really uniform circular motions.
There was an old system, called the Egyptian, according to which Saturn,
Jupiter, Mars, and the Sun circulated round the earth, the sun carrying
with it, as two moons or satellites, the other two planets, Venus and
Mercury. This system had never entirely lost credit: it had been
maintained in the fifth century by Martianus Capella[188], and indeed it
was almost sanctioned, though not formally taught, by Ptolemy himself,
when he made the mean motion of the sun the same as that of the centres
of the epicycles of both these planets. The remark which had also been
made by the old astronomers, of the connexion between the motion of the
sun and the revolutions of the superior planets in their epicycles, led
him straight to the expectation that he might, perhaps, produce the
uniformity he sought by extending the Egyptian system to these also, and
this appears to have been the shape in which his reform was originally
projected. It was already allowed that the centre of the orbits of all
the planets was not coincident with the earth, but removed from it by
the space EC. This first change merely made EC the same for all the
planets, and equal to the mean distance of the earth from the sun. This
system afterwards acquired great celebrity through its adoption by Tycho
Brahe, who believed it originated with himself. It might perhaps have
been at this period of his researches, that Copernicus was struck with
the passages in the Latin and Greek authors, to which he refers as
testifying the existence of an old belief in the motion of the earth
round the sun. He immediately recognised how much this alteration would
further his principles of uniformity, by referring all the planetary
motions to one centre, and did not hesitate to embrace it. The idea of
explaining the daily and principal apparent motions of the heavenly
bodies by the revolution of the earth on its axis, would be the
concluding change, and became almost a necessary consequence of his
previous improvements, as it was manifestly at variance with his
principles to give to all the planets and starry worlds a rapid daily
motion round the centre of the earth, now that the latter was removed
from its former supposed post in the centre of the universe, and was
itself carried with an annual motion round another fixed point.
[Illustration]
The reader would, however, form an inaccurate notion of the system of
Copernicus, if he supposed that it comprised no more than the theory
that each planet, including the earth among them, revolved in a simple
circular orbit round the sun. Copernicus was too well acquainted with
the motions of the heavenly bodies, not to be aware that such orbits
would not accurately represent them; the motion he attributed to the
earth round the sun, was at first merely intended to account for those
which were called the second inequalities of the planets, according to
which they appear one while to move forwards, then backwards, and at
intermediate periods, stationary, and which thenceforward were also
called the optical equations, as being merely an optical illusion. With
regard to what were called the first inequalities, or physical
equations, arising from a real inequality of motion, he still retained
the machinery of the deferent and epicycle; and all the alteration he
attempted in the orbits of the superior planets was an extension of the
concentric theory to supply the place of the equant, which he considered
the blot of the system. His theory for this purpose is shown in the
accompanying diagram, where S represents the sun, D_d_, the deferent or
mean orbit of the planet, on which revolves the centre of the great
epicycle, whose radius, DF, was taken at ¾ of Ptolemy's excentricity of
the equant; and round the circumference of this revolved, in the
opposite direction, the centre of the little epicycle, whose radius, FP,
was made equal to the remaining ¼ of the excentricity of the equant.
The planet P revolved in the circumference of the little epicycle, in
the same direction with the centre of the great epicycle in the
circumference of the deferent, but with a double angular velocity. The
planet was supposed to be in the perigee of the little epicycle, when
its centre was in the apogee of the greater; and whilst, for instance, D
moved equably though the angle DS_d_, F moved through _hdf_ = DS_d_, and
P through _rfp_ = 2 DS_d_.
It is easy to show that this construction gives nearly the same result
as Ptolemy's; for the deferent and great epicycle have been already
shown exactly equivalent to an excentric circle round S, and indeed
Copernicus latterly so represented it: the effect of his construction,
as given above, may therefore be reproduced in the following simpler
form, in which only the smaller epicycle is retained:
[Illustration]
In this construction, the place of the planet is found at the end of any
time proportional to F _f_ by drawing _fr_ parallel to SF, and taking
_rfp_ = 2F _of_. Hence it is plain, if we take OQ, equal to FP, (already
assumed equal to ¼ of Ptolemy's excentricity of the equant,) since SO is
equal to ¾ of the same, that SQ is the whole of Ptolemy's excentricity
of the equant; and therefore, that Q is the position of the centre of
his equant. It is also plain if we join Q_p_, since _rfp_ = 2F _of_, and
_o_Q = _fp_, that _p_Q is parallel to _fo_, and, therefore, _p_QP is
proportional to the time; so that the planet moves uniformly about the
same point Q, as in Ptolemy's theory; and if we bisect SQ in C, which is
the position of the centre of Ptolemy's deferent, the planet will,
according to Copernicus, move very nearly, though not exactly, in the
same circle, whose radius is CP, as that given by the simple excentric
theory.
The explanation offered by Copernicus, of the motions of the inferior
planets, differed again in form from that of the others. He here
introduced what was called a _hypocycle_, which, in fact, was nothing
but a deferent not including the sun, round which the centre of the
orbit revolved. An epicycle in addition to the hypocycle was introduced
into Mercury's orbit. In this epicycle he was not supposed to revolve,
but to librate, or move up and down in its diameter. Copernicus had
recourse to this complication to satisfy an erroneous assertion of
Ptolemy with regard to some of Mercury's inequalities. He also retained
the oscillatory motions ascribed by Ptolemy to the planes of the
epicycles, in order to explain the unequal latitudes observed at the
same distance from the nodes, or intersections of the orbit of the
planet with the ecliptic. Into this intricacy, also, he was led by
placing too much confidence in Ptolemy's observations, which he was
unable to satisfy by an unvarying obliquity. Other very important
errors, such as his belief that the line of nodes always coincided with
the line of apsides, or places of greatest and least distance from the
central body, (whereas, at that time, in the case of Mars, for instance,
they were nearly 90° asunder,) prevented him from accurately
representing many of the celestial phenomena.
These brief details may serve to show that the adoption or rejection of
the theory of Copernicus was not altogether so simple a question as
sometimes it may have been considered. It is, however, not a little
remarkable, while it is strongly illustrative of the spirit of the
times, that these very intricacies, with which Kepler's theories have
enabled us to dispense, were the only parts of the system of Copernicus
that were at first received with approbation. His theory of Mercury,
especially, was considered a masterpiece of subtle invention. Owing to
his dread of the unfavourable judgment he anticipated on the main
principles of his system, his work remained unpublished during forty
years, and was at last given to the world only just in time to allow
Copernicus to receive the first copy of it a few hours before his
death.
FOOTNOTES:
[187] By "the opposite direction" is meant, that while the motion in the
circumference of one circle appeared, as viewed from its centre, to be
from left to right, the other, viewed from its centre, appeared from
right to left. This must be understood whenever these or similar
expressions are repeated.
[188] Venus Mercuriusque, licet ortus occasusque quotidianos ostendunt,
tamen eorum circuli terras omnino non ambiunt, sed circa solem laxiore
ambitu circulantur. Denique circulorum suorum centron in sole
constituunt.—De Nuptiis Philologiæ et Mercurii. Vicentiæ. 1499.
CHAPTER V.
_Account of the Commentaries on the motions of Mars—Discovery of
the Law of the equable description of Areas, and of Elliptic
Orbits._
WE may now proceed to examine Kepler's innovations, but it would be
doing injustice to one of the brightest points of his character, not to
preface them by his own animated exhortation to his readers. "If any one
be too dull to comprehend the science of astronomy, or too feeble-minded
to believe in Copernicus without prejudice to his piety, my advice to
such a one is, that he should quit the astronomical schools, and
condemning, if he has a mind, any or all of the theories of
philosophers, let him look to his own affairs, and leaving this worldly
travail, let him go home and plough his fields: and as often as he lifts
up to this goodly heaven those eyes with which alone he is able to see,
let him pour out his heart in praises and thanksgiving to God the
Creator; and let him not fear but he is offering a worship not less
acceptable than his to whom God has granted to see yet more clearly with
the eyes of his mind, and who both can and will praise his God for what
he has so discovered."
Kepler did not by any means underrate the importance of his labours, as
is sufficiently shewn by the sort of colloquial motto which he prefixed
to his work. It consists in the first instance of an extract from the
writings of the celebrated and unfortunate Peter Ramus. This
distinguished philosopher was professor of mathematics in Paris, and in
the passage in question, after calling on his contemporaries to turn
their thoughts towards the establishment of a system of Astronomy
unassisted by any hypothesis, he promised as an additional inducement to
vacate his own chair in favour of any one who should succeed in this
object. Ramus perished in the massacre of St. Bartholomew, and Kepler
apostrophizes him as follows:—"It is well, Ramus, that you have
forfeited your pledge, by quitting your life and professorship together:
for if you still held it, I would certainly claim it as of right
belonging to me on account of this work, as I could convince you even
with your own logic." It was rather bold in Kepler to assert his claim
to a reward held out for a theory resting on no hypothesis, by right of
a work filled with hypotheses of the most startling description; but of
the vast importance of this book there can be no doubt; and throughout
the many wild and eccentric ideas to which we are introduced in the
course of it, it is fit always to bear in mind that they form part of a
work which is almost the basis of modern Astronomy.
The introduction contains a curious criticism of the commonly-received
theory of gravity, accompanied with a declaration of Kepler's own
opinions on the same subject. Some of the most remarkable passages in it
have been already quoted in the life of Galileo; but, nevertheless, they
are too important to Kepler's reputation to be omitted here, containing
as they do a distinct and positive enunciation of the law of universal
gravitation. It does not appear, however, that Kepler estimated rightly
the importance of the theory here traced out by him, since on every
other occasion he advocated principles with which it is scarcely
reconcileable. The discussion is introduced in the following terms:—
"The motion of heavy bodies hinders many from believing that the earth
is moved by an animal motion, or rather a magnetic one. Let such
consider the following propositions. A mathematical point, whether the
centre of the universe or not, has no power, either effectively or
objectively, to move heavy bodies to approach it. Let physicians prove
if they can, that such power can be possessed by a point, which, neither
is a body, nor is conceived unless by relation alone. It is impossible
that the form[189] of a stone should, by moving its own body, seek a
mathematical point, or in other words, the centre of the universe,
without regard of the body in which that point exists. Let physicians
prove if they can, that natural things have any sympathy with that which
is nothing. Neither do heavy bodies tend to the centre of the universe
by reason that they are avoiding the extremities of the round universe;
for their distance from the centre is insensible, in proportion to their
distance from the extremities of the universe. And what reason could
there be for this hatred? How strong, how wise must those heavy bodies
be, to be able to escape so carefully from an enemy lying on all sides
of them: what activity in the extremities of the world to press their
enemy so closely! Neither are heavy bodies driven into the centre by the
whirling of the first moveable, as happens in revolving water. For if we
assume such a motion, either it would not be continued down to us, or
otherwise we should feel it, and be carried away with it, and the earth
also with us; nay, rather, we should be hurried away first, and the
earth would follow; all which conclusions are allowed by our opponents
to be absurd. It is therefore plain that the vulgar theory of gravity is
erroneous.
"The true theory of gravity is founded on the following axioms:—Every
corporeal substance, so far forth as it is corporeal, has a natural
fitness for resting in every place where it may be situated by itself
beyond the sphere of influence of a body cognate with it. Gravity is a
mutual affection between cognate bodies towards union or conjunction
(similar in kind to the magnetic virtue), so that the earth attracts a
stone much rather than the stone seeks the earth. Heavy bodies (if we
begin by assuming the earth to be in the centre of the world) are not
carried to the centre of the world in its quality of centre of the
world, but as to the centre of a cognate round body, namely, the earth;
so that wheresoever the earth may be placed, or whithersoever it may be
carried by its animal faculty, heavy bodies will always be carried
towards it. If the earth were not round, heavy bodies would not tend
from every side in a straight line towards the centre of the earth, but
to different points from different sides. If two stones were placed in
any part of the world near each other, and beyond the sphere of
influence of a third cognate body, these stones, like two magnetic
needles, would come together in the intermediate point, each approaching
the other by a space proportional to the comparative mass of the other.
If the moon and earth were not retained in their orbits by their animal
force or some other equivalent, the earth would mount to the moon by a
fifty-fourth part of their distance, and the moon fall towards the earth
through the other fifty-three parts and they would there meet; assuming
however that the substance of both is of the same density. If the earth
should cease to attract its waters to itself, all the waters of the sea
would be raised and would flow to the body of the moon. The sphere of
the attractive virtue which is in the moon extends as far as the earth,
and entices up the waters; but as the moon flies rapidly across the
zenith, and the waters cannot follow so quickly, a flow of the ocean is
occasioned in the torrid zone towards the westward. If the attractive
virtue of the moon extends as far as the earth, it follows with greater
reason that the attractive virtue of the earth extends as far as the
moon, and much farther; and in short, nothing which consists of earthly
substance any how constituted, although thrown up to any height, can
ever escape the powerful operation of this attractive virtue. Nothing
which consists of corporeal matter is absolutely light, but that is
comparatively lighter which is rarer, either by its own nature, or by
accidental heat. And it is not to be thought that light bodies are
escaping to the surface of the universe while they are carried upwards,
or that they are not attracted by the earth. They are attracted, but in
a less degree, and so are driven outwards by the heavy bodies; which
being done, they stop, and are kept by the earth in their own place. But
although the attractive virtue of the earth extends upwards, as has been
said, so very far, yet if any stone should be at a distance great enough
to become sensible, compared with the earth's diameter, it is true that
on the motion of the earth such a stone would not follow altogether; its
own force of resistance would be combined with the attractive force of
the earth, and thus it would extricate itself in some degree from the
motion of the earth."
Who, after perusing such passages in the works of an author, whose
writings were in the hands of every student of astronomy, can believe
that Newton waited for the fall of an apple to set him thinking for the
first time on the theory which has immortalized his name? An apple may
have fallen, and Newton may have seen it; but such speculations as those
which it is asserted to have been the cause of originating in him had
been long familiar to the thoughts of every one in Europe pretending to
the name of natural philosopher.
As Kepler always professed to have derived his notion of a magnetic
attraction among the planetary bodies from the writings of Gilbert, it
may be worth while to insert here an extract from the "New Philosophy"
of that author, to show in what form he presented a similar theory of
the tides, which affords the most striking illustration of that
attraction. This work was not published till the middle of the
seventeenth century, but a knowledge of its contents may, in several
instances, be traced back to the period in which it was written:—
"There are two primary causes of the motion of the seas—the moon, and
the diurnal revolution. The moon does not act on the seas by its rays or
its light. How then? Certainly by the common effort of the bodies, and
(to explain it by something similar) by their magnetic attraction. It
should be known, in the first place, that the whole quantity of water is
not contained in the sea and rivers, but that the mass of earth (I mean
this globe) contains moisture and spirit much deeper even than the sea.
The moon draws this out by sympathy, so that they burst forth on the
arrival of the moon, in consequence of the attraction of that star; and
for the same reason, the quicksands which are in the sea open themselves
more, and perspire their moisture and spirits during the flow of the
tide, and the whirlpools in the sea disgorge copious waters; and as the
star retires, they devour the same again, and attract the spirits and
moisture of the terrestrial globe. Hence the moon attracts, not so much
the sea as the subterranean spirits and humours; and the interposed
earth has no more power of resistance than a table or any other dense
body has to resist the force of a magnet. The sea rises from the
greatest depths, in consequence of the ascending humours and spirits;
and when it is raised up, it necessarily flows on to the shores, and
from the shores it enters the rivers."[190]
This passage sets in the strongest light one of the most notorious
errors of the older philosophy, to which Kepler himself was remarkably
addicted. If Gilbert had asserted, in direct terms, that the moon
attracted the water, it is certain that the notion would have been
stigmatized (as it was for a long time in Newton's hands) as arbitrary,
occult, and unphilosophical: the idea of these subterranean humours was
likely to be treated with much more indulgence. A simple statement, that
when the moon was over the water the latter had a tendency to rise
towards it, was thought to convey no instruction; but the assertion that
the moon draws out subterranean spirits by sympathy, carried with it a
more imposing appearance of theory. The farther removed these humours
were from common experience, the easier it became to discuss them in
vague and general language; and those who called themselves philosophers
could endure to hear attributes bestowed on these fictitious elements
which revolted their imaginations when applied to things of whose
reality at least some evidence existed.
It is not necessary to dwell upon the system of Tycho Brahe, which was
identical, as we have said, with one rejected by Copernicus, and
consisted in making the sun revolve about the earth, carrying with it
all the other planets revolving about him. Tycho went so far as to deny
the rotation of the earth to explain the vicissitudes of day and night,
but even his favourite assistant Longomontanus differed from him in this
part of his theory. The great merit of Tycho Brahe, and the service he
rendered to astronomy, was entirely independent of any theory;
consisting in the vast accumulation of observations made by him during a
residence of fifteen years at Uraniburg, with the assistance of
instruments, and with a degree of care, very far superior to anything
known before his time in practical astronomy. Kepler is careful
repeatedly to remind us, that without Tycho's observations he could have
done nothing. The degree of reliance that might be placed on the results
obtained by observers who acknowledged their inferiority to Tycho Brahe,
maybe gathered from an incidental remark of Kepler to Longomontanus. He
had been examining Tycho's registers, and had occasionally found a
difference amounting sometimes to 4´ in the right ascensions of the same
planet, deduced from different stars on the same night. Longomontanus
could not deny the fact, but declared that it was impossible to be
always correct within such limits. The reader should never lose sight of
this uncertainty in the observations, when endeavouring to estimate the
difficulty of finding a theory that would properly represent them.
When Kepler first joined Tycho Brahe at Prague, he found him and
Longomontanus very busily engaged in correcting the theory of Mars, and
accordingly it was this planet to which he also first directed his
attention. They had formed a catalogue of the mean oppositions of Mars
during twenty years, and had discovered a position of the equant, which
(as they said) represented them with tolerable exactness. On the other
hand, they were much embarrassed by the unexpected difficulties they met
in applying a system which seemed on the one hand so accurate, to the
determination of the latitudes, with which it could in no way be made to
agree. Kepler had already suspected the cause of this imperfection, and
was confirmed in the view he took of their theory, when, on a more
careful examination, he found that they overrated the accuracy even of
their longitudes. The errors in these, instead of amounting as they
said, nearly to 2´, rose sometimes above 21´. In fact they had reasoned
ill on their own principles, and even if the foundations of their theory
had been correctly laid, could not have arrived at true results. But
Kepler had satisfied himself of the contrary, and the following diagram
shews the nature of the first alteration he introduced, not perhaps so
celebrated as some of his later discoveries, but at least of equal
consequence to astronomy, which could never have been extricated from
the confusion into which it had fallen, till this important change had
been effected.
[Illustration]
The practice of Tycho Brahe, indeed of all astronomers till the time of
Kepler, had been to fix the position of the planet's orbit and equant
from observations on its mean oppositions, that is to say, on the times
when it was precisely six signs or half a circle distant from the mean
place of the sun. In the annexed figure, let S represent the sun, C the
centre of the earth's orbit, T_t_. Tycho Brahe's practice amounted to
this, that if Q were supposed the place of the centre of the planet's
equant, the centre of P_p_ its orbit was taken in QC, and not in QS, as
Kepler suggested that it ought to be taken. The consequence of this
erroneous practice was, that the observations were deprived of the
character for which oppositions were selected, of being entirely free
from the second inequalities. It followed therefore that as part of the
second inequalities were made conducive towards fixing the relative
position of the orbit and equant, to which they did not naturally
belong, there was an additional perplexity in accounting for the
remainder of them by the size and motion of the epicycle. As the line of
nodes of every planet was also made to pass through C instead of S,
there could not fail to be corresponding errors in the latitudes. It
would only be in the rare case of an opposition of the planet in the
line CS, that the time of its taking place would be the same, whether O,
the centre of the orbit, was placed in CQ or SQ. Every other opposition
would involve an error, so much the greater as it was observed at a
greater distance from the line CS.
It was long however before Tycho Brahe could be made to acquiesce in the
propriety of the proposed alteration; and, in order to remove his doubts
as to the possibility that a method could be erroneous which, as he
still thought, had given him such accurate longitudes, Kepler undertook
the ungrateful labour of the first part of his "Commentaries." He there
shewed, in the three systems of Copernicus, Tycho Brahe, and Ptolemy,
and in both the concentric and excentric theories, that though a false
position were given to the orbit, the longitudes of a planet might be so
represented, by a proper position of the centre of the equant, as never
to err in oppositions above 5´ from those given by observation; though
the second inequalities and the latitudes would thereby be very greatly
deranged.
The change Kepler introduced, of observing apparent instead of mean
oppositions, made it necessary to be very accurate in his reductions of
the planet's place to the ecliptic; and in order to be able to do this,
a previous knowledge of the parallax of Mars became indispensable. His
next labour was therefore directed to this point; and finding that the
assistants to whom Tycho Brahe had previously committed this labour had
performed it in a negligent and imperfect manner, he began afresh with
Tycho's original observations. Having satisfied himself as to the
probable limits of his errors in the parallax on which he finally fixed,
he proceeded to determine the inclination of the orbit and the position
of the line of nodes. In all these operations his talent for
astronomical inquiries appeared pre-eminent in a variety of new methods
by which he combined and availed himself of the observations; but it
must be sufficient merely to mention this fact, without entering into
any detail. One important result may be mentioned, at which he arrived
in the course of them, the constancy of the inclination of the planet's
orbit, which naturally strengthened him in his new theory.
Having gone through these preliminary inquiries, he came at last to fix
the proportions of the orbit; and, in doing so, he determined, in the
first instance, not to assume, as Ptolemy appeared to have done
arbitrarily, the bisection of the excentricity, but to investigate its
proportion along with the other elements of the orbit, which resolution
involved him in much more laborious calculations. After he had gone over
all the steps of his theory no less than seventy times—an appalling
labour, especially if we remember that logarithms were not then
invented—his final result was, that in 1587, on the 6th of March, at 7ʰ
23´, the longitude of the aphelion of Mars was 4ˢ 28° 48´ 55´´; that the
planet's mean longitude was 6ˢ 0° 51´ 35´´; that if the semidiameter of
the orbit was taken at 100000, the excentricity was 11332; and the
excentricity of the equant 18564. He fixed the radius of the greater
epicycle at 14988, and that of the smaller at 3628.
When he came to compare the longitudes as given by this, which he
afterwards called the _vicarious_ theory, with the observations at
opposition, the result seemed to promise him the most brilliant success.
His greatest error did not exceed 2´; but, notwithstanding these
flattering anticipations, he soon found by a comparison of longitudes
out of opposition and of latitudes, that it was yet far from being so
complete as he had imagined, and to his infinite vexation he soon found
that the labour of four years, which he had expended on this theory,
must be considered almost entirely fruitless. Even his favourite
principle of dividing the excentricity in a different ratio from
Ptolemy, was found to lead him into greater error than if he had
retained the old bisection. By restoring that, he made his latitudes
more accurate, but produced a corresponding change for the worse in his
longitudes; and although the errors of 8´, to which they now amounted,
would probably have been disregarded by former theorists, Kepler could
not remain satisfied till they were accounted for. Accordingly he found
himself forced to the conclusion that one of the two principles on which
this theory rested must be erroneous; either the orbit of the planet is
not a perfect circle, or there is no fixed point within it round which
it moves with an uniform angular motion. He had once before admitted the
possibility of the former of these facts, conceiving it possible that
the motion of the planets is not at all curvilinear, but that they move
in polygons round the sun, a notion to which he probably inclined in
consequence of his favourite harmonics and geometrical figures.
In consequence of the failure of a theory conducted with such care in
all its practical details, Kepler determined that his next trial should
be of an entirely different complexion. Instead of first satisfying the
first inequalities of the planet, and then endeavouring to account for
the second inequalities, he resolved to reverse the process, or, in
other words, to ascertain as accurately as possible what part of the
planet's apparent motion should be referred solely to the optical
illusion produced by the motion of the earth, before proceeding to any
inquiry of the real inequality of the planet's proper motion. It had
been hitherto taken for granted, that the earth moved equably round the
centre of its orbit; but Kepler, on resuming the consideration of it,
recurred to an opinion he had entertained very early in his astronomical
career (rather from his conviction of the existence of general laws,
than that he had then felt the want of such a supposition), that it
required an equant distinct from its orbit no less than the other
planets. He now saw, that if this were admitted, the changes it would
everywhere introduce in the optical part of the planet's irregularities
might perhaps relieve him from the perplexity in which the vicarious
theory had involved him. Accordingly he applied himself with renewed
assiduity to the examination of this important question, and the result
of his calculations (founded principally on observations of Mars'
parallax) soon satisfied him not only that the earth's orbit does
require such an equant, but that its centre is placed according to the
general law of the bisection of the excentricity which he had previously
found indispensable in the other planets. This was an innovation of the
first magnitude, and accordingly Kepler did not venture to proceed
farther in his theory, till by evidence of the most varied and
satisfactory nature, he had established it beyond the possibility of
cavil.
It may be here remarked, that this principle of the bisection of the
eccentricity, so familiar to the Ptolemaic astronomers, is identical
with the theory afterwards known by the name of the simple elliptic
hypothesis, advocated by, Seth Ward and others. That hypothesis
consisted in supposing the sun to be placed in one focus of the elliptic
orbit of the planet, whose angular motion was uniform round the other
focus. In Ptolemaic phraseology, that other focus was the centre of the
equant, and it is well known that the centre of the ellipse lies in the
middle point between the two foci.
It was at this period also, that Kepler first ventured upon the new
method of representing inequalities which terminated in one of his most
celebrated discoveries. We have already seen, in the account of the
"Mysterium Cosmographicum," that he was speculating, even at that time,
on the effects of a whirling force exerted by the sun on the planets
with diminished energy at increased distances, and on the proportion
observed between the distances of the planets from the sun, and their
periods of revolution. He seems even then to have believed in the
possibility of discovering a relation between the times and distances in
different planets. Another analogous consequence of his theory of the
radiation of the whirling force would be, that if the same planet should
recede to a greater distance from the central body, it would be acted on
by a diminished energy of revolution, and consequently, a relation might
be found between the velocity at any point of its orbit, and its
distance at that point from the sun. Hence he expected to derive a more
direct and natural method of calculating the inequalities, than from the
imaginary equant. But these ingenious ideas had been checked in the
outset by the erroneous belief which Kepler, in common with other
astronomers, then entertained of the coincidence of the earth's equant
with its orbit; in other words, by the belief that the earth's linear
motion was uniform, though it was known not to remain constantly at the
same distance from the sun. As soon as this prejudice was removed, his
former ideas recurred to him with increased force, and he set himself
diligently to consider what relation could be found between the velocity
and distance of a planet from the sun. The method he adopted in the
beginning of this inquiry was to assume as approximately correct
Ptolemy's doctrine of the bisection of the excentricity, and to
investigate some simple relation nearly representing the same effect.
In the annexed figure, S is the place of the sun, C the centre of the
planet's orbit AB_ab_, Q the centre of the equant represented by the
equal circle DE_de_, AB, _ab_, two equal small arcs described by the
planet at the apsides of its orbit: then, according to Ptolemy's
principles, the arc DE of the equant would be proportional to the time
of passing along AB, on the same scale on which _de_ would represent the
time of passing through the equal arc _ab_.
[Illustration]
QD:QA :: DE:AB, nearly; and because QS is bisected in C, QA, CA or QD,
and SA, are in arithmetical proportion: and, therefore, since an
arithmetical mean, when the difference is small, does not differ much
from a geometrical mean, QD:QA :: SA:QD, nearly. Therefore, DE:AB :: S
A:QD, nearly, and in the same manner _de_:_ab_ :: S_a_:Q_d_ nearly; and
therefore DE:_de_ :: SA:S_a_ nearly. Therefore at the apsides, the
times of passing over equal spaces, on Ptolemy's theory, are nearly as
the distances from the sun, and Kepler, with his usual hastiness,
immediately concluded that this was the accurate and general law, and
that the errors of the old theory arose solely from having departed from
it.
It followed immediately from this assumption, that after leaving the
point A, the time in which the planet would arrive at any point P of
its orbit would be proportional to, and might be represented by, the
sums of all the lines that could be drawn from S to the arc AP, on the
same scale that the whole period of revolution would be denoted by the
sum of all the lines drawn to every point of the orbit. Kepler's first
attempt to verify this supposition approximately, was made by dividing
the whole circumference of the orbit into 360 equal parts, and
calculating the distances at every one of the points of division. Then
supposing the planet to move uniformly, and to remain at the same
distance from the sun during the time of passing each one of these
divisions, (a supposition which manifestly would not differ much from
the former one, and would coincide with it more nearly, the greater was
the number of divisions taken) he proceeded to add together these
calculated distances, and hoped to find that the time of arriving at any
one of the divisions bore the same ratio to the whole period, as the sum
of the corresponding set of distances did to the sum of the whole 360.
This theory was erroneous; but by almost miraculous good fortune, he was
led by it in the following manner to the true measure. The discovery was
a consequence of the tediousness of his first method, which required, in
order to know the time of arriving at any point, that the circle should
be subdivided, until one of the points of division fell exactly upon the
given place. Kepler therefore endeavoured to discover some shorter
method of representing these sums of the distances. The idea then
occurred to him of employing for that purpose the area inclosed between
the two distances, SA, SP, and the arc AP, in imitation of the manner in
which he remembered that Archimedes had found the area of the circle, by
dividing it into an infinite number of small triangles by lines drawn
from the centre. He hoped therefore to find, that the time of passing
from A to P bore nearly the same ratio to the whole period of revolution
that the area ASP bore to the whole circle.
This last proportion is in fact accurately observed in the revolution of
one body round another, in consequence of an attractive force in the
central body. Newton afterwards proved this, grounding his demonstration
upon laws of motion altogether irreconcileable with Kepler's opinions;
and it is impossible not to admire Kepler's singular good fortune in
arriving at this correct result in spite, or rather through the means,
of his erroneous principles. It is true that the labour which he
bestowed unsparingly upon every one of his successive guesses, joined
with his admirable candour, generally preserved him from long retaining
a theory altogether at variance with observations; and if any relation
subsisted between the times and distances which could any way be
expressed by any of the geometrical quantities under consideration, he
could scarcely have failed—it might be twenty years earlier or twenty
years later,—to light upon it at last, having once put his
indefatigable fancy upon this scent. But in order to prevent an
over-estimate of his merit in detecting this beautiful law of nature,
let us for a moment reflect what might have been his fate had he
endeavoured in the same manner, and with the same perseverance, to
discover a relation, where, in reality, none existed. Let us take for
example the inclinations or the excentricities of the planetary orbits,
among which no relation has yet been discovered; and if any exists, it
is probably of too complicated a nature to be hit at a venture. If
Kepler had exerted his ingenuity in this direction, he might have wasted
his life in fruitless labour, and whatever reputation he might have left
behind him as an industrious calculator, it would have been very far
inferior to that which has procured for him the proud title of the
"Legislator of the Heavens."
However this may be, the immediate consequence of thus lighting upon the
real law observed by the earth in its passage round the sun was, that he
found himself in possession of a much more accurate method of
representing its inequalities than had been reached by any of his
predecessors; and with renewed hopes he again attacked the planet Mars,
whose path he was now able to consider undistorted by the illusions
arising out of the motion of the earth. Had the path of Mars been
accurately circular, or even as nearly approaching a circle as that of
the earth, the method he chose of determining its position and size by
means of three distances carefully calculated from his observed
parallaxes, would have given a satisfactory result; but finding, as he
soon did, that almost every set of three distances led him to a
different result, he began to suspect another error in the long-received
opinion, that the orbits of the planets must consist of a combination
of circles; he therefore, determined, in the first instance, to fix the
distances of the planet at the apsides without any reference to the form
of the intermediate orbit. Half the difference between these would, of
course, be the excentricity of the orbit; and as this quantity came out
very nearly the same as had been determined on the vicarious theory, it
seemed clear that the error of that theory, whatever it might be, did
not lie in these elements.
Kepler also found that in the case of this planet likewise, the times of
describing equal arcs at the apsides were proportional to its distances
from the sun, and he naturally expected that the method of areas would
measure the planet's motion with as much accuracy as he had found in the
case of the earth. This hope was disappointed: when he calculated the
motion of the planet by this method, he obtained places too much
advanced when near the apsides, and too little advanced at the mean
distances. He did not, on that account, immediately reject the opinion
of circular orbits, but was rather inclined to suspect the principle of
measurement, at which he felt that he had arrived in rather a precarious
manner. He was fully sensible that his areas did not accurately
represent the sums of any distances except those measured from the
centre of the circle; and for some time he abandoned the hope of being
able to use this substitution, which he always considered merely as an
approximate representation of the true measure, the sum of the
distances. But on examination he found that the errors of this
substitution were nearly insensible, and those it did in fact produce,
were in the contrary direction of the errors he was at this time
combating. As soon as he had satisfied himself of this, he ventured once
more on the supposition, which by this time had, in his eyes, almost
acquired the force of demonstration, that the orbits of the planets are
not circular, but of an oval form, retiring within the circle at the
mean distances, and coinciding with it at the apsides.
This notion was not altogether new; it had been suggested in the case of
Mercury, by Purbach, in his "Theories of the Planets." In the edition of
this work published by Reinhold, the pupil of Copernicus, we read the
following passage. "Sixthly, it appears from what has been said, that
the centre of Mercury's epicycle, by reason of the motions
above-mentioned, does not, as is the case with the other planets,
describe the circumference of a circular deferent, but rather the
periphery of a figure resembling a plane oval." To this is added the
following note by Reinhold. "The centre of the Moon's epicycle describes
a path of a lenticular shape; Mercury's on the contrary is egg-shaped,
the big end lying towards his apogee, and the little end towards his
perigee."[191] The excentricity of Mercury's orbit is, in fact, much
greater than that of any of the other planets, and the merit of making
this first step cannot reasonably be withheld from Purbach and his
commentator, although they did not pursue the inquiry so far as Kepler
found himself in a condition to do.
Before proceeding to the consideration of the particular oval which
Kepler fixed upon in the first instance, it will be necessary, in order
to render intelligible the source of many of his doubts and
difficulties, to make known something more of his theory of the moving
force by which he supposed the planets to be carried round in their
orbits. In conformity with the plan hitherto pursued, this shall be done
as much as possible in his own words.
"It is one of the commonest axioms in natural philosophy, that if two
things always happen together and in the same manner, and admit the same
measure, either the one is the cause of the other, or both are the
effect of a common cause. In the present case, the increase or languor
of motion invariably corresponds with an approach to or departure from
the centre of the universe. Therefore, either the languor is the cause
of the departure of the star, or the departure of the languor, or both
have a common cause. But no one can be of opinion that there is a
concurrence of any third thing to be a common cause of these two
effects, and in the following chapters it will be made clear that there
is no occasion to imagine any such third thing, since the two are of
themselves sufficient. Now, it is not agreeable to the nature of things
that activity or languor in linear motion should be the cause of
distance from the centre. For, distance from the centre is conceived
anteriorly to linear motion. In fact linear motion cannot exist without
distance from the centre, since it requires space for its
accomplishment, but distance from the centre can be conceived without
motion. Therefore distance is the cause of the activity of motion, and a
greater or less distance of a greater or less delay. And since distance
is of the kind of relative quantities, whose essence consists in
boundaries, (for there is no efficacy in relation _per se_ without
regard to bounds,) it follows that the cause of the varying activity of
motion rests in one of the boundaries. But the body of the planet
neither becomes heavier by receding, nor lighter by approaching.
Besides, it would perhaps be absurd on the very mention of it, that an
animal force residing in the moveable body of the planet for the purpose
of moving it, should exert and relax itself so often without weariness
or decay. It remains, therefore, that the cause of this activity and
languor resides at the other boundary, that is, in the very centre of
the world, from which the distances are computed.—Let us continue our
investigation of this moving virtue which resides in the sun, and we
shall presently recognize its very close analogy to light. And although
this moving virtue cannot be identical with the light of the sun, let
others look to it whether the light is employed as a sort of instrument,
or vehicle, to convey the moving virtue. There are these seeming
contradictions:—first, light is obstructed by opaque bodies, for which
reason if the moving virtue travelled on the light, darkness would be
followed by a stoppage of the moveable bodies. Again, light flows out in
right lines spherically, the moving virtue in right lines also, but
cylindrically; that is, it turns in one direction only, from west to
east; not in the opposite direction, not towards the poles, &c. But
perhaps we shall be able presently to reply to these objections. In
conclusion, since there is as much virtue in a large and remote circle
as in a narrow and close one, nothing of the virtue perishes in the
passage from its source, nothing is scattered between the source and the
moveable. Therefore the efflux, like that of light, is not material, and
is unlike that of odours, which are accompanied by a loss of substance,
unlike heat from a raging furnace, unlike every other emanation by which
mediums are filled. It remains, therefore, that as light which
illuminates all earthly things, is the immaterial species of that fire
which is in the body of the sun, so this virtue, embracing and moving
all the planetary bodies, is the immaterial species of that virtue which
resides in the sun itself, of incalculable energy, and so the primary
act of all mundane motion.—I should like to know who ever said that
there was anything material in light!—Guided by our notion of the
efflux of this species (or archetype), let us contemplate the more
intimate nature of the source itself. For it seems as if something
divine were latent in the body of the sun, and comparable to our own
soul, whence that species emanates which drives round the planets; just
as from the mind of a slinger the species of motion sticks to the
stones, and carries them forward, even after he who cast them has drawn
back his hand. But to those who wish to proceed soberly, reflections
differing a little from these will be offered."
Our readers will, perhaps, be satisfied with the assurance, that these
sober considerations will not enable them to form a much more accurate
notion of Kepler's meaning than the passages already cited. We shall
therefore proceed to the various opinions he entertained on the motion
of the planets.
He considered it as established by his theory, that the centre E of the
planet's epicycle (see fig. p. 33.) moved round the circumference of the
deferent D_d_, according to the law of the planet's distances; the point
remaining to be settled was the motion of the planet in the epicycle. If
it were made to move according to the same law, so that when the centre
of the epicycle reached E, the planet should be at F, taking the angle
BEF equal to BSA, it has been shewn (p. 19) that the path of F would
still be a circle, excentric from D_d_ by DA the radius of the epicycle.
But Kepler fancied that he saw many sound reasons why this could not be
the true law of motion in the epicycle, on which reasons he relied much
more firmly than on the indisputable fact, which he mentions as a
collateral proof, that it was contradicted by the observations. Some of
these reasons are subjoined: "In the beginning of the work it has been
declared to be most absurd, that a planet (even though we suppose it
endowed with mind) should form any notion of a centre, and a distance
from it, if there be no body in that centre to serve for a
distinguishing mark. And although you should say, that the planet has
respect to the sun, and knows beforehand, and remembers the order in
which the distances from the sun are comprised, so as to make a perfect
excentric; in the first place, this is rather far-fetched, and requires,
in any mind, means for connecting the effect of an accurately circular
path with the sign of an increasing and diminishing diameter of the sun.
But there are no such means, except the position of the centre of the
excentric at a given distance from the sun; and I have already said,
that this is beyond the power of a mere mind. I do not deny that a
centre may be imagined, and a circle round it; but this I do say, if the
circle exists only in imagination, with no external sign or division,
that it is not possible that the path of a moveable body should be
really ordered round it in an exact circle. Besides, if the planet
chooses from memory its just distances from the sun, so as exactly to
form a circle, it must also take from the same source, as if out of the
Prussian or Alphonsine tables, equal excentric arcs, to be described in
unequal times, and to be described by a force extraneous from the sun;
and thus would have, from its memory, a foreknowledge of what effects a
virtue, senseless and extraneous from the sun, was about to produce: all
these consequences are absurd.
"It is therefore more agreeable to reason that the planet takes no
thought, either of the excentric or epicycle; but that the work which it
accomplishes, or joins in effecting, is a libratory path in the diameter
B_b_ of the epicycle, in the direction towards the sun. The law is now
to be discovered, according to which the planet arrives at the proper
distances in any time. And indeed in this inquiry, it is easier to say
what the law is not than what it is."—Here, according to his custom,
Kepler enumerates several laws of motion by which the planet might
choose to regulate its energies, each of which is successively
condemned. Only one of them is here mentioned, as a specimen of the
rest. "What then if we were to say this? Although the motions of the
planet are not epicyclical, perhaps the libration is so arranged that
the distances from the sun are equal to what they would have been in a
real epicyclical motion.—This leads to more incredible consequences
than the former suppositions, and yet in the dearth of better opinions,
let us for the present content ourselves with this. The greater number
of absurd conclusions it will be found to involve, the more ready will a
physician be, when we come to the fifty-second chapter, to admit what
the observations testify, that the path of the planet is not circular."
The first oval path on which Kepler was induced to fix, by these and
many other similar considerations, was in the first instance very
different from the true elliptical form. Most authors would have thought
it unnecessary to detain their readers with a theory which they had once
entertained and rejected; but Kepler's work was written on a different
plan. He thus introduces an explanation of his first oval. "As soon as I
was thus taught by Brahe's very accurate observations that the orbit of
a planet is not circular, but more compressed at the sides, on the
instant I thought that I understood the natural cause of this
deflection. But the old proverb was verified in my case;—the more haste
the less speed.—For having violently laboured in the 39th chapter, in
consequence of my inability to find a sufficiently probable cause why
the orbit of the planet should be a perfect circle, (some absurdities
always remaining with respect to that virtue which resides in the body
of the planet,) and having now discovered from the observations, that
the orbit is not a perfect circle, I felt furiously inclined to believe
that if the theory which had been recognized as absurd, when employed in
the 39th chapter for the purpose of fabricating a circle, were modulated
into a more probable form, it would produce an accurate orbit agreeing
with the observations. If I had entered on this course a little more
warily, I might have detected the truth immediately. But, being blinded
by my eagerness, and not sufficiently regardful of every part of the
39th chapter, and clinging to my first opinion, which offered itself to
me with a wonderful show of probability, on account of the equable
motion in the epicycle, I got entangled in new perplexities, with which
we shall now have to struggle in this 45th chapter and the following
ones as far as the 50th chapter."
In this theory, Kepler supposed that whilst the centre of the epicycle
was moving round a circular deferent according to the law of the
planets' distances (or areas) the planet itself moved equably in the
epicycle, with the mean angular velocity of its centre in the deferent.
In consequence of this supposition, since at D, when the planet is at A
the aphelion, the motion in the deferent is less than the mean motion,
the planet will have advanced through an angle BEP greater than BEF or
BSA, through which the centre of the epicycle has moved; and
consequently, the path will lie everywhere within the circle A_a_,
except at the apsides. Here was a new train of laborious calculations to
undergo for the purpose of drawing the curve AP_a_ according to this
law, and of measuring the area of any part of it. After a variety of
fruitless attempts, for this curve is one of singular complexity, he was
reduced, as a last resource, to suppose it insensibly different from an
ellipse on the same principal axes, as an approximate means of
estimating its area. Not content even with the results so obtained, and
not being able to see very clearly what might be the effect of his
alteration in substituting the ellipse for the oval, and in other
simplifications introduced by him, he had courage enough to obtain the
sums of the 360 distances by direct calculation, as he had done in the
old circular theory.
[Illustration]
In the preface to his book he had spoken of his labours under the
allegory of a war carried on by him against the planet; and when
exulting in the early prospects of success this calculation seemed to
offer, he did not omit once more to warn his readers, in his peculiar
strain, that this exultation was premature.
"Allow me, gentle reader, to enjoy so splendid a triumph for one little
day (I mean through the five next chapters), meantime be all rumours
suppressed of new rebellion, that our preparations may not perish,
yielding us no delight. Hereafter if anything shall come to pass, we
will go through it in its own time and season; now let us be merry, as
then we will be bold and vigorous." At the time foretold, that is to
say, at the end of the five merry chapters, the bad news could no longer
be kept a secret. It is announced in the following bulletin:—"While
thus triumphing over Mars, and preparing for him, as for one altogether
vanquished, tabular prisons, and equated eccentric fetters, it is buzzed
here and there that the victory is vain, and that the war is raging anew
as violently as before. For the enemy, left at home a despised captive,
has burst all the chains of the equations, and broken forth of the
prisons of the tables. For no method of geometrically administering the
theory of the 45th chapter was able to come near the accuracy of
approximation of the vicarious theory of the 16th chapter, which gave me
true equations derived from false principles. Skirmishers, disposed all
round the circuit of the excentric, (I mean the true distances,) routed
my forces of physical causes levied out of the 45th chapter, and shaking
off the yoke, regained their liberty. And now there was little to
prevent the fugitive enemy from effecting a junction with his rebellious
supporters, and reducing me to despair, had I not suddenly sent into the
field a reserve of new physical reasonings on the rout and dispersion of
the veterans, and diligently followed, without allowing him the
slightest respite, in the direction in which he had broken out."
In plainer terms, Kepler found, after this labour was completed, that
the errors in longitude he was still subject to were precisely of an
opposite nature to those he had found with the circle; instead of being
too quick at the apsides, the planet was now too slow there, and too
much accelerated in the mean distances; and the distances obtained from
direct observation were everywhere greater, except at the apsides, than
those furnished by this oval theory. It was in the course of these
tedious investigations that he established, still more satisfactorily
than he had before done, that the inclinations of the planets' orbits
are invariable, and that the lines of their nodes pass through the
centre of the Sun, and not, as before his time had been supposed,
through the centre of the ecliptic.
When Kepler found with certainty that this oval from which he expected
so much would not satisfy the observations, his vexation was extreme,
not merely from the mortification of finding a theory confuted on which
he had spent such excessive labour, for he was accustomed to
disappointments of that kind, but principally from many anxious and
fruitless speculations as to the real physical causes why the planet did
not move in the supposed epicycle, that being the point of view, as has
been already shewn, from which he always preferred to begin his
inquiries. One part of the reasoning by which he reconciled himself to
the failure exhibits much too curious a view of the state of his mind to
be passed over in silence. The argument is founded on the difficulty
which he met with, as above mentioned, in calculating the proportions of
the oval path he had imagined. "In order that you may see the cause of
the impracticability of this method which we have just gone through,
consider on what foundations it rests. The planet is supposed to move
equably in the epicycle, and to be carried by the Sun unequably in the
proportion of the distances. But by this method it is impossible to be
known how much of the oval path corresponds to any given time, although
the distance at that part is known, unless we first know the length of
the whole oval. But the length of the oval cannot be known, except from
the law of the entry of the planet within the sides of the circle. But
neither can the law of this entry be known before we know how much of
the oval path corresponds to any given time. Here you see that there is
a _petitio principii_; and in my operations I was assuming that of which
I was in search, namely, the length of the oval. This is at least not
the fault of my understanding, but it is also most alien to the primary
Ordainer of the planetary courses: I have never yet found so
ungeometrical a contrivance in his other works. Therefore we must either
hit upon some other method of reducing the theory of the 45th chapter to
calculation; or if that cannot be done, the theory itself, suspected on
account of this _petitio principii_, will totter." Whilst his mind was
thus occupied, one of those extraordinary accidents which it has been
said never occur but to those capable of deriving advantage from them
(but which, in fact, are never noticed when they occur to any one else),
fortunately put him once more upon the right path. Half the extreme
breadth between the oval and the circle nearly represented the errors of
his distances at the mean point, and he found that this half was 429
parts of a radius, consisting of 100000 parts; and happening to advert
to the greatest optical inequality of Mars, which amounts to about 5°
18´, it struck him that 429 was precisely the excess of the secant of 5°
18´ above the radius taken at 100000. This was a ray of light, and, to
use his own words, it roused him as out of sleep. In short, this single
observation was enough to produce conviction in his singularly
constituted mind, that instead of the distances SF, he should everywhere
substitute FV, determined by drawing SV perpendicular on the line FC,
since the excess of SF above FV is manifestly that of the secant above
the radius in the optical equation SFC at that point. It is still more
extraordinary that a substitution made for such a reason should have the
luck, as is again the case, to be the right one. This substitution in
fact amounted to supposing that the planet, instead of being at the
distance SP or SF, was at S_n_; or, in other words, that instead of
revolving in the circumference, it librated in the diameter of the
epicycle, which was to him an additional recommendation. Upon this new
supposition a fresh set of distances was rapidly calculated, and to
Kepler's inexpressible joy, they were found to agree with the
observations within the limits of the errors to which the latter were
necessarily subject. Notwithstanding this success, he had to undergo,
before arriving at the successful termination of his labours, one more
disappointment. Although the distance corresponding to a time from the
aphelion represented approximately by the area ASF, was thus found to be
accurately represented by the line S_n_, there was still an error with
regard to the direction in which that distance was to be measured.
Kepler's first idea was to set it off in the direction SF, but this he
found to lead to inaccurate longitudes; and it was not until after much
perplexity, driving him, as he tells us, "almost to insanity," that he
satisfied himself that the distance SQ equal to FV ought to be taken
terminating in F_m_, the line from F perpendicular to A_a_, the line of
apsides, and that the curve so traced out by Q would be an accurate
ellipse.
[Illustration]
He then found to his equal gratification and amazement, a small part of
which he endeavoured to express by a triumphant figure on the side of
his diagram, that the error he had committed in taking the area ASF to
represent the sums of the distances SF, was exactly counterbalanced; for
this area does accurately represent the sums of the distances FV or SQ.
This compensation, which seemed to Kepler the greatest confirmation of
his theory, is altogether accidental and immaterial, resulting from the
relation between the ellipse and circle. If the laws of planetary
attraction had chanced to have been any other than those which cause
them to describe ellipses, this last singular confirmation of an
erroneous theory could not have taken place, and Kepler would have been
forced either to abandon the theory of the areas, which even then would
have continued to measure and define their motions, or to renounce the
physical opinions from which he professed to have deduced it as an
approximative truth.
These are two of the three celebrated theorems called Kepler's laws: the
first is, that the planets move in ellipses round the sun, placed in the
focus; the second, that the time of describing any arc is proportional
in the same orbit to the area included between the arc and the two
bounding distances from the sun. The third will be mentioned on another
occasion, as it was not discovered till twelve years later. On the
establishment of these two theorems, it became important to discover a
method of measuring such elliptic areas, but this is a problem which
cannot be accurately solved. Kepler, in offering it to the attention of
geometricians, stated his belief that its solution was unattainable by
direct processes, on account of the incommensurability of the arc and
sine, on which the measurement of the two parts AQ_m_, SQ_m_ depends.
"This," says he in conclusion, "this is my belief, and whoever shall
shew my mistake, and point out the true solution,
_Is erit mihi magnus Apollonius._"
FOOTNOTES:
[189] It is not very easy to carry the understanding aright among these
Aristotelian ideas. Many at the present day might think they understood
better what is meant, if for "form" had been written "nature."
[190] De mundo nostro sublunari, Philosóphia Nova. Amstelodami, 1651.
[191] Theoricæ novæ planetarum. G. Purbachii, Parisiis, 1553.
CHAPTER VI.
_Kepler appointed Professor at Linz—His second marriage—Publishes
his new Method of Gauging—Refuses a Professorship at Bologna._