A treatise on astronomy

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A S"/"/"' ^'OB 'a, «?' ^

HARVARD
COLLEGE
LIBRARY

r

o

TREATISE ON ASTRONOMY.

BY

SIR JOHN F. W. HERSCHEL, KNT. GlIBLP.

rJL8J:..ftE. BULLA. FJULS. lAA, M.C.U.P.8.

VOEEBSPOllDBlfT O/ TBI ftOTAl ACAOBIIT OF •CIBNCBS Or PAllll, AMD OTBIB
fOEBiaR ■CISMTIFIO IMSTITimOMI.

PHILADELPHIA^
CAREY, LEA & BLANCHARD.

1834«

^1

UNIVlRSITY
V LIBRARY J

'^ Et quoniam eaaem natura cupiditatem ingenuit hominibus Ten

inveniendi, quod facilUmi apparet, cum vacui curis, e^am quid in

coeio fiat, scire avemus : his initiis inducti omnia vera diligimus ; id

est, fidelia, simplieia, constantia ; tum vana, talsa, fallendia odimus."

Cicero, de Fin, Bon. et J^aL iL 14.

And forasmuch as nature itself has implanted in man a craving
after the discovery of truth (which appears most clearly from thia,
that, when unoppressed by cares, we delight to know even what is
going on in the hea\ens), — led by this instinct, we learn to love all
truth for its own sake ; liiat is to say, whateVer is faithful, simple,
and consistent ; while we hold in abhorrence whatever is empty,
deceptive^ or untrue.

. 2

\

I MICROFILMED 1
I AT HARVARD |

CONTENTS.

iHmODDCnON

CHAFTERL

General Notioiw— Forai and MMnitnrfe of the Eaitii-»HonnB md
itB iMp— The AtiiH)iiiliere~ftt>inciMO--7Vilisht--AppMn^
reeoltmg from diurnal MoCkm-oPknUax— Fint &ep lowaida fti»>
iog an Idea ofthe Distance of the Stuv—Definitioai - • • 14

CHAPTER IL

Of die Nature of astroooinical Intframenli and ObaKwwAom in fM^
ral— Of sidereal and solar Tinie--Of the Bfawnw m ent of Tone—
Clocks, ChroDometei*, the Transit Instnunenv-Of the i
mentof an|[iilar Intervals— Application of the Telescope lo In
ments destined to that Purpose— Of the Maral Cirele— Date
tion of polar and horiaontal Pbinl»— The LsTel— Fhimb J
Artificial Horizon — CoUimator— Of ooospoand InsCn
ooKndinate Curcles, the Eqoatorial— Altitode and Aai
ment— Of the Sextant and Reflecting Cirele— Principle of Repeti*
tion ' f6

CHAPTER m.

or GBOOKAPHT.

Of Ifae Figare of &e Earth— Its exact DinMBaooa— Its Fom dnt of
Eqnilinrium modified by Centrifugal Force— Variation of Gfavitf
•on its Sorftee— Statical and dynamical Measures of Gmvitir— Tli»
Pendulum— Gravity to a Spheroid— Otbsr Efiects of Earth^s Rota-
tion— Trade-winds— Determination of geographical Positions— Of
Latit udes O f Longitudes— Conduct of a tiignometrical Survey--
Of Maps— Phgectkmi of the Sphere— Measnwmnnr of Heigliit by
4he Baaometer ,.106

CHAPTER IV.

or tTKANOOKAPHT.

CoDstniction of celestial Maps and Globes by Observations of right
Ascension and J)eclination— Celestial Olnects disfingiiishcd into
fixed and emtio— Of the ConstelUtions— Natural Regioni in the
Heaven»-The Milky Way— The Zodiac— Of the EcUj)tie-Celea-
tial Latitudes and Longitudes— Preoesrion of the Equmoaas— Nn-
. tfttion— Abenatioa— Uranographical Probiems - - 151

3

4 CONfENTS.

CHAPTER V.

OF THE 8UN*S MOTION.

Page
Apparent Motion of the Sun not uniform— Its apparent Diameter also
variable — Variation of its Distance concluded— Its apparent Orbit
an Ellipse about the Focus— Law of the angular Velocity — Equap
ble Description of Areas — Parallax of the !Sun — Its Distance and
Magnitude— Copeniican Explanation of the Sun's apparent Motion
— Parallelism of the Earth's Axis— The Seasonft-rHeat received
from the Sun in different P&rts of the Orbit - . . . . 176

CHAPTER VI.

Of the Moon— Its sidereal Period— Its apparent Diameter— Its Paral-
lax, Distance, and real Diametei^-First Approximation to its Orbit
— An Ellipse about the Earth in the Focus^Its Eccentricity and
Inclination — Motion of the Modes of its Orbit^Occultations— Solar
Eclipses— Phases of the Moon — ^Its 'synod ical Period— Lunar
Eclipses— Motion of the Apsides of its ( >roit— Physical Constitution
of the 'Moon — Its Mountains — Atmosphere— 'Rotation on Axis —
libration — ^Appearance of the Earth from it • * • 903

CHAPTER Vn.

Of terreatrial Gravity*-Of the Law of universal Gravitation— Paths
of Projectiles; apparent, real — ^The Moon retained in her Orbit bj
Gravity— Its Law of Diminution — Laws of elliptic Moiion*-Orbit
of the Earth round the Sun in accordance with these Laws —
Masses of the Earth . and Sun compared — Density of the Sun —
Force of Gravi^ at its Surface — Disturbing Effect of the Sun on
the Moon*s Motion 221

CHAPTER VIII.

' or THE SOLAR SYSTEM.

Apparent Motions of the PUmets^Their Stations and RetroffradaF
tions— >The Sun their natural Centre of Motion — Inferior Planets
— ^Their Phases, Periods, &c. — Dimensions and Form of their Orbits
— ^Transits across the Sun — Superior Pli^iets, their Distances, Pe-
riods, &c. — Kepler's Laws and their Interpretation — Elliptic Ele-
ments of a Planet's Orbit— Its heliocentric and geocentric Place —
Bode's Law of Planetary Distances— The four aitra<«M]iaoal PI*-
nets— Physical Peculiarities observable in each of the Planels - 981

CHAPTER DL

.OF THE 8ATBLUTI8.

Of the Moon, as a Satellite of the Earth— General Proximity of Satel-
lites to their Primaries, and consequent Subordination of their
Motions— Masses of the Primaries concluded from the Periods of
their Satellites^Maintenance of KepleKs Laws in the secondary
SystomB— Of Jupiter's Satellites— Their Eclipses, &c.— Velocity
or tight discovered by their Meana^-Satelhtes of Saturn— Of
Uranus 972

GONTENTS. •

CHAFT£RX.

or comn.

Ftg.

Great Number of recorded Comet*— The mimber of vueeonM
probably mach greater — Detcriptkm of a Comet— Comets without
Tails— Increase and Decay of their Tails— Their Motions— 49ab-
iect to the generel Laws of planetaiv Motion — ^Elements of their
Orbits — Periodic Return of certain Comets — ^Halley's— £neke*»— >
Biela's — ^Dimensions of Comets— Heir Resistanoe by the Ether,
gradual Decay, and possible Dispernon in Space - • . . m

CHAPTER XI.

OF PUlTUKBATXONf.

Sutgect propounded— Superposition of small Motioo[i->Ph>bleoi of
three Bodies— Estimation of disturbing Forces— Motion of Nodes
—Changes of Inclination-Compensation operated in a whole
ReToluuon of the Node— Luminge's Theorem of the Stability of
the Inclinations— Change of Obliquity of the Ediptio— Precession
of the Equinoxes— Nutation— Theorem respectingjbrced Vibra>
tions— Of the 'Hdes— Variation of Elements of the Planet's Orbits
—Periodic and secular— Disturbing Forces oonsidesed as tangen-
tial and radial— EfiectB of tangential Force— 1st, in circular OrbitB ;
2d, in elliptic— Compensations eflected— Case of near Coounen-
surability of mean Motions — ^The great Inequality of Jupiter and
Saturn explained— The lodff Inequality of Venus and the Earth-
Lunar Variation— E&cts of the radial Force- Mean Effect of the
Period and Dimensions of the disturbed Orbit— Variable Put of
its Effect— Lunar Evection — Secular Acceleration of the Moon's
Motion- Permanence of the Axes and Periods— Theoiy of the secu-
lar Variations of the Eccentricities and Periheliar— Bibtion of tiie
lunar Apsides — ^Lsji^nffe's Theorem of the Stability of the Ec-
centricities—Nutation of the lunar Orbit— Perturbatioas of Jupi-
ter's Satellites 894

CHAPTER XIL

or SIOKRIAL ASTRONOMY.

Of the Stars generally— Their distribution into Clasus according to
their apparent Magnitudes— Their apparent Distribution over the
Heavens— Of the Milky Way— Annual Parallax— Real Distances,
jMobable Dimensions, and Nature of the Stars— Variable Stars —
Temporary Stars— Of double Stars— Their Reyolution about each
other in ouiptic OrbitB— Extension of the Law of Gravity ts such
Systems— Of coloured Stars — Proper Motion of ttie Sun and Stan
— Systematic Aberration and Parallax— Of compound sidereal
Systems— Clusters of Stars— Of Nebulas — Nebulous Stan— Annu-
hur and planetary Nebu]s&— Zodiacal Light - '- - - - 349
a2

d contents;

CHAPTJBR XIII.

Pago

Or THE Caucndar - - 381

Synoptic Tftbktoftfasfileraeiifei of the Solar S]rptem • - -389

Synoptic Table of ike £lementa of the Qrbita of the Satellites, eo far
aa they are known .......... 390

I. The Moon 390

II. SetelUtea of Jupiter 390

III. Satellites of Saturn - - -391

IV. SateUites of Uranus 391

Index • ...» 393

▲
TREATISE

OR .

ASTRONOMY.

INTRODUCTION.

(1.) In entering upon any scientific panait, one of the
student^s first endeavours ought to be, to prepare his
mind for the reception of truth, by dismissing, or at least
loosening his hold on, all such crude and hastOy adopted
notions respecting the objects and relations he is about
to examine as. may tend to embarrass or mislead him;
and to strengthen himsdf,hy something of an effort and
a resolve, for the unprejudiced admission of any con-
clusion which shall appear to be supported by carefiil
observation and logical argument, even should it prove
of a nature adverse to notions he may have previously
formed for himself, or taken up, without examination,
on the credit of others. Such an effort is, in fact, a
commencement of that intellectual discipline which
forms one of the most important ends of all science*
It is the first movement of approach towards that state of
mental purity 'which alone can fit us for a full and steady
perception of moral beauty as well as physical adaptation.
It is the ^* euphrasy and rue" with which we must '* purge
our sight" before we can receive and contemplate as they
are the lineaments of truth and nature.

(2.) There is no science which, more than astronomy,
Mands in need of such a preparation, or. draws more
largely on that intellectual liberality which is ready to
adopt whatever is demonstrated, or concede whatever is
rendered highly probable, however new and uncommon

7

8 A TREATISE ON ASTRONOMY.

the points of view may be in which objects the most
familiar may thereby become placed. Almost all its
conclusions stand in open and striking contradiction
with those of superficial and vulgar observation, and
with what appears to every one, until he has understood
and weighed the proofs to the contrary, the most posi-
tive evidence of his senses. Thus, the earth on which
he stands, and which has served for ages as the un-
shaken foundation of the firmest structures, either of art
or nature, is divested by the astronomer of its attribute
of fixity, and conceived by him as turning swiftly on its
centre, and at the same time moving onwards through
space with great rapidity. The sun and the moon,
which appear to. untaught eyes round bodies of no very
considerable size, become enlarged in his imagination
into vast globes,— the one approaching in magnitude to
the earth itself, the other immensely surpassing it. The
planets, which appear only as stars somewhat brighter
than the rest, are to him spacious, elaborate, and habit-
able worlds ; several of them vastly greater and far more
curiously furnished than the earth he- inhabits, as there
are also others less so ; and the stars themselves, properly
so called, which to ordinary apprehension present only
lucid sparks or brilliant atoms, are to him suns of various
and transcendent glory — effulgent centres of life and light
to myriads of unseen worlds : so that when, after dilat-
ing his thoughts to comprehend the grandeur of those
ideas his calculations have called up, and exhausting his
imagination and the powers of his language to devise
similes and metaphors illustrative of the immensity of
the scale on which his universe is constructed, he shrinks
back to his native sphere ; he finds it,, in comparison, a
mere pointy so last— even in the minute system to
which it belongs-^as to be invisible and unsuspected
from some of its principal and remoter members.

(3.) There is hsu-dly any thing which sets in a stronger
light the inherent power of truth over the mind of
man, when opposed by no motives of interest or passion,
than the perfect readiness with which all these conclu-
sions are assented to as soon as their evidence is clearly
apprehended, and the tenacious hold they acquire over

INTRODVCnON. f

(NET belief when once admitted. In the oondaet, thereforei
of this volume, we shall take it for granted that oar
reader is more desirous to learn the system which it is
its object to teach as it now stands, than to raise or re*
vive objections against it; and that,. in short, he eomea
to the task with a willing mind ; an assumption which
will not only save ourselves the trouble of piling argu-
ment on argument to convince the skeptical, but will
greatly facilitate his actual progress, inasmuch as he will
find it at once easier and more satisfactory to pursue from
the outset a straight and definite path, than to be con*
stantly stepping aside, involving himself in perplexities
and circuits, which, after all, can only terminate in
finding himself compelled to adopt our road.

(4.) The method, therefore, we propose to follow is.
neither strictly the analytic nor the synthetic ,> but rather
such a combination of both, with a leaning to the latter,
as may best suit with a diJactic composition. Our object
is not to convince or refute opponents, nor to inquire,
under the semblance of an assumed ignorance, for prin*
ciples of which we are all the timo in full possession*—
but simply to teach what we know. The moderate limit
of a single volume, and the necessity of being on every
point, within that limit, rather diffuse and copious in ex*
planation, as well as the eminently itiatured* and ascer*
tained character of the science itself, render this course
both practicable and eligible. Practicable, because there
is now ho danger of any revolution in astronomy, like
those which are daily changing the features of the less
advanced sciences, supervening, to destroy all our hypo-
theses, and throw our statements into confusion. Eligible,
because the space to be bestowed, either in combating
refuted systems, or in leading the reader forward* by
slow and measqred steps from the known to the un-
known, maybe more advantageously devoted to stfch ex-
planatory illustrations as will impress on him a familiar
and, as it were, a practical sense of the sequence of phe-
nomena, and the manner in which they are produced.
We shall not, then, reject the analytic course where it
leads more easily and directly to our ■ objects, or in any
way fetter ourselves by a rigid adherence to method.

14) A TREATISE. ON A8TR0K0MT,

Writing only to be understood, and to communicate as
much information in as little space as possible, consist-
ently with its distinct and c^«c/t«j/ communication, we
can afford to make no sacrifice to system, to ,form, or to
affectation.

(5.) We shall take for granted, from the outset, the
Copemican system of the world ; relying on the easy,
obvious, and natural explanation it affords of all the phe-
nomena as they come to be described, to impress the
student with a sense of its truth, without either the form-
ality of demonstration or the superfluouts tedium of
eulogy, calling to mixid that important remark of Bacon :
— <-** Theoriarum vires, arcta et quasi se mutuo sustinente
partium adaptatione, qurU, quasi in orbem cohserent, fir-
mantur;"* nor failing, however,, to point out to the
reader, as occasion offers, the contrast which its superior
simplicity offers to the complication of other hypotheses.

(6.) The preliminary knowledge which it is desirable
that the student sl^ould possess, in order for the more
advantageous perusal of the following pages, consists in
the familiar practice of decimal and sexagesimal arith-
metic ; some moderate acquaintance with geometi;y and
trigonometry, both plane and spherical; the elementary
principles of mechanics ; and enough of optics to under-
stand the construction and use of the telescope, and some
other of the simpler instruments. For the acquisition of
these we may refer him to those other parts of this Gy-
clopaBdia which profess to tres^ of the several subjects in.
question. Of course, the more .of such knowledge Hb
brings to the perusal, the easier will be his progress^ and
the more complete the information gained ; but we shall
endeavour in every case, as far as it can be done with*-
out«a sacrifice of clearness, and of that useful brevity
which consists in the absence of prolixity and episode,
to render what we have to say as independent of other
books as possible.

(7.) After all, we must distinctly caution such of our
readers as may commence and terminate their astronomi-

* The confirmation of theories relies on the compact adaptation of
Ifaeir parts, by which, like those of an arch or dome, they mutually
aqatain each other, and fimn a ooherant whole.

IMTSOOVCnON. 11

eal studies with the present wofk (though of i
least in the latter predicament,^— we inist the nuniber will
be few), that its utmost pretension is to place them oa
the threshold of this particular wing of the tem]^ of sci-
ence, or rather on an eminence OKterior to it, whence
they may obtain something like a gtneral notion of its
structure ; or, at most, to give those who may wish to
enter, a ground-plan of its accesses, and put them in poa-
session of the pass-word. Admission to its sanctuary,
and to the privileges and feelings of a votary, is only to
be gained by one means,r— ii stnmd and n^fieieni knouH
ledge of mathematics, the great instntment of ail exact m-
quiry, unthout which no man can ever make meh ad*
ponces in this or any other of the higher departmenig of
science, as can entitle him to form an independent apt'
nion on any subject of discussion within their range. It
is not without an effort that those who possess this know-
ledge can communicate on such subjects with those who
do not, and adapt their language and their illostrations to
the necessities pf such an intercourse. Propositions
which to. the one are almost identical, are theorems of
import and difficulty to the other ; nor is their evidence
presented in the same way to the mind of each. In
leaching such propositions, under such circumstanees,
the appeal has to be made, not to the pare and abstract rea-
son, but to the sense of analogy, — to practice and expe-
rience : principles and modes of action have to be esta-
blished, pot by direct argument from acknowledged
axioms, but by bringing forward and dwelling on simple
and familiar instances in which the same principles and
the same or shnOar modes of action take place ; thus
erecting, as it were, in each particular case, a separate
induction, and constructing at each step a little body ol
science to meet its exigencies. The difference is that
of pioneering a road through an untraversed country, and
advancing at ease along a broad and beaten highway ;
that is to say, if we are determined to make ourselves
distinctly understood, and will appeal to reason at all.
As for the method of assertion, or a direct demand on
ihe faith of the student (though in some complex cases
. indispensable, where illustrative explanation would defeat

12 A TREATISE ON ASTRONOMY.

iU own end by becoming tedious and burdensome to both
parties), it is one which we shall neither adopt ourselves
nor would recommend to others.

(8.) On the other hand, although it is something new
to abandon the road of mathematical demonstration in the
treatment of subjects susceptible of it, and teach any con-
siderable branch of science entirely or chiefly by the way
of illustration and familiar parallels, it is yet not impossi-
ble that those who are already well acquainted wiUi our
subject, and whose knowledge has been acquired by that
confessedly higher and better practice which is incompa-
tible with the avowed objects of the present work, may
yet find their account in its perusal,^-for this reason, that
it is always of advantage to present any given body of
knowledge to the mind in as great a variety of different
lights as possible. It is a property of illustrations of this
kind to strike no two minds in the same manner, or with
the same force ; because no two minds are stored with
the same images, or have acquir^ their notions of them
by similar habits. Accordingly, it may very well hap-
pen, that a proposition, even to one best acquainted with
it, may be placed not merely in a new and uncommoh»
but in a more impressive and satisfactory light by such
a course— some obscurity may Be dissipated, some inward
misgiving cleared up, or even some link supplied which
may lead to the perception of connexions and deductions
altogether unknown before. And the probability of this
is increased when, as in the present instance, the illustra-
tions chosen have not been, studiously selected from books,
but are such as have presented themselves freely to the
author's mind as being'most in harmony with his own,
views ; by which, of course, he means to lay no claim
to originality in all or any of them beyond what they
may really possess,

(9.) Besides,' there are cases in the application of me-
chanical principles with which the mathematical student
is but too familiar, where, when the data are before hinif
and the numerical and geometrical relations of his pro-
blems all clear to his conception,— *when his forces are
estimated andhisUnes measured,-— nay , when even he has
foUowedup the application of his technical processes, and

niTHODircTioir. It

fairly arrived at his conclusion, — there is still somethiiif
wanting in his mind — ^not in the eridence, for he has ex-
amined each link, and finds the chain complete-— not in
the principles, for those he well knows are too firmly es-
tablished to be shaken — but precisely in the mode rfaC'
tion. He has followed out a train of reasoning by logical
and technical rules, but the signs he has employed are not
pictures of nature, or have lost their original meaning u
such to his mind : he has not seen, as it were, the pnh>
cess of nature passing under his eye in an instant of time*
and presented as a whole to his imagination. A familiar
parallel, or an illustration drawn from some artificial or
natural process, of which he has that direct and individual
impression which gives. it a reality and associates it with
a name, will, in almost every such case, supply in a mo-
ment this deficient feature, wiU convert all his symbols
into real pictures, and infuse an animated meaning into
what was before a lifeless succession of words and signs.
We cannot, indeed, always promise ourselves to attain
this degree of vividness in our illustrations, nor are the
points to be elucidated themselves always capable of be-
ing so paraphraaed (if we may use the expression) by
any single instance adducible in the ordinary course «f
experience ; but the object will at least be kept in view ;
and, as we are very conscious of having, in making such
attempts, gained for ourselves much clearer views of seve-
ral of the raoT^ concealed effects of planetary pertnxbap
tion than we had acquired by their mathematical investi-
gation in detail, we may reasonably hope thai the endeavour
will not always be unattended with a similar success in
others.

(10.) From what has been said, it will be evident that
our aim is not to offer to the public a technical treatise,
in which the student of practical or theoretical astronomy
shall find consigned the minute description of methocb
of observation, or the formula he requires prepared to
his hand, or their demonstrations drawn out in detail. In
all these the present woik will be found meagre, and quite
inadequate to his wants. Its aim is entirely different f be*
ing to present in each ease the mere ultimate tationaie of
faetS) arguments, and processes ; and, in all cases of nwtti&
B

14 A TREATISE OR ASTRONOMY. [CHAP. I*

maiical application, avoiding whatever would tend to en-
cumber its pages with algebraic or geometrical symbolsy
to place under his inspection that central thread of com-
mon same on which &e pearls of analytical research are
invariably strung ; but which, by the attentios the kttter
claim for themselves, is often concealed from the eye of
the gazer, and not always disposed in the straightest and
most convenient form to follow by those who string them.
This is no fault of those who have conducted the inqui-
ries to which we allude. The contention of mind for
which they call is enormous ; and it may, perhaps, be
owing to their experience of how little can be accomplish-
ed in carrying such processes on to their conclusion, by
mere ordinary clearness of head$ and how necessary it
often is to pay more attention to the purely mathematical
conditions which insure success, — the hooks-and«eyes of
their equations and series, — than to those which enchain
causes with their effects, and both with ^e human rea-
son, — that we must attribute something of that indistinct-
ness of view which is often complained of as a grievance
by the earnest student, and still more commonly ascribed
ironically to the native cloudiness of an atmosphere too
sublime for vulgar compreh^sion. We think we shall
render good service to both classes of readers, by dissi-
. pating, so far as our power lies, that accidental obscurity,
and by showing ordinary untutored comprehension clearly
what it can, and what it cannot ^ hope to attain.

CHAPTER I.

General NotioDfr->Fonn and Magnitude of the Earth— Hofizon and ita
Dip— The Atmosphere— Refraction— Twiliffht—Appearancea leault-
ing from diurnal Notion — Parallax — ^First Step towards fi>rmlng an
Idea of <he Distance of the Stars— Definitions.

(11.) The magnitudes, distances, arrangement, and
motions of the great bodies which make up the visible
universe, their constitution and physical condition, so
far as they can be known to us, with their mutual in-
fluences and actions on each other, so far as they can be

CHAP. I.J GENERAL NOTEOm. 16

tneed by the effects produced, and estaUished by legi*
timate reasoning, form the assemblage of objects to
which the attentbn of the astronomer is direeted. The
term astronomy* itself, which denotes the law or mla
of the eutra (by which the ancients nnderslood not only
the stars properly so caUed, but the sm, the moon^ and
all the visible constituents of the heavens), sufficiently
indicates this ; and, although the term astrology, which
denotes the reaton^ theory^ or inierpreiaiian of the
stars,t has become degraded in its application, and con*
fined to superstitious and delusive attempts to divine
future events by their dependence on pretended i^ane* •
tary influences, the same meaning originally attached
itself to that epithet.

(12«) But, besides the stars and other celestial bodies,
the earth itself, regarded as an individual bodyt is one
principal object of the astronomer's consideration^ and,
indeed, the chief of all. It derives its importance, in a
practioal as well as theoretical sense, not only from its
proximity, and its relation to us as animated beincs,
who draw from it the supply of all our wants, but as ttie
station from which we see all the rest, and as the only
one among them to which we can, in the first instanee,
r^er for any determinate marks and measures by which
to recognise >their changes of situation, or with which to
compare their distances.-

(13.) To the reader who now for the first time takes
up a book on astronomy, it will no doubt seem strange
to class the earth with the heavenly bodies, and to as-
sume any community of nature among things apparently
so different. For what, in fact, can be more apparently
different than the vast and seemingly immeasurable ex-
tent of the earth, and the stars, whieh appear but as
points, and seem to have no size at all ? The earth is
dark and opaque, while the celestial bodies are brilliant.
We perceive in it no motion, while in them we observe
a continual change of place, as we view them at different

*Arrti(, a iUtr ; vojuor, a km ; or f'iuov, to tend, as a shepherd his flock ;
go that •.rYe«i'»i»«ff means '^shepbezd of the itan." The two etymobgiM
aie, however, coincident ^ .

t Aoyof , reasoHi or a word^ the vehicle of reason ; the interpreter of
thought

16 A TREATISE ON ASTRONOMY. [CHAP. I.

hpufs of the day or night, or at different sesusons of the
year. The ancients, accordingly, one or two of the more
enlightened of them only excepted, admitted no such
community of nature ; and, by thus placing the heavenly
bodies and their movements without the pale of analogy
and experience, effectually intercepted the progress of
all reasoning from what passes here below, to what is
going on in the regions where they exist and move.
Under such conventions, astronomy, as a science of
cause and effect, could not exist, but must be limited to
a mere registry of appearances, unconnected with any
attempt to account for them on reasonable principles.
To get rid of this prejudice, therefore, is the first step
towards acquiring a knowledge of what is really the
case ; and the student has made his first effort towards
the acquisition of sound knowledge, when he has learnt
to familiarize himself with the idea that the earth, after
all, may be nothing but a great star. How correct such
an idea may be, and with what limitations and modlfica-
tioiis it is to be admitted, we shall see presently.

(14.) It is evident, that, to form any just notions of
the arrangement, in space, of a number of objects which
we cannot approach and examine, but of which adl the
information we can gain is by sitting still and watching
their evolutions, it must be very important for us to
know, in the first instance, whether what we call sitting
still is rmlly such : whether the station from which we
view them, with ourselves, and all objects which im-
mediately surround us, be not itself in motion, unper-
ceived by us;, and. if so, of what nature that motion is.
The apparent places of a number of objects, and their
apparent arrangement with respect to each other, will
of course be materially dependent on the situation of the
spectator among them ; and if this situation be liable to
ehange, unknown to the spectator himself^ an appearance
of change in the respective situations of the objects will
arise, without the reality. If, then, such be actually
the case, it will follow that all the movements we think
we perceive among the stars will not be real movements,
but that some part, at least, of whatever changes of re-
lative place we perceive among them must be merely

CfiAF. I.] ORNKRAL N0TI0K8. 17

apparent, the results of the shifting of our own point of
view ; and that, if we would eyer arriye at a knowledge
of their real motions, it can only be by first investigating
our own, and making due allowance for its effects.
Thus, the question whether the earth is in motion or at
rest, and if in motion, what that motion is, is no idle in-
quiry, but one on which depends our only chance of
arriving at true conclusions respecting the constitution
of the universe.

(15.) Nor let it be thought strange that we should
speak of a motion existing in the earth, unperceived by
its inhabitants : we must remember that it is of the earth
as a whole, with all that it holds within its substance,
or sustains on its surface, that we are speaking ; of a
motion common to the solid mass beneath, to the ocean
which flows around it, the air that rests upon it, and the
clouds which float above it in the air. Such a motiOUt
which should displace no terrestrial object from its re-
lative place among others, interfere with no natural pro-
cesses, and produce no sensations of shocks or jerks,
might, it is very evident, subsist undetected by us.
There is no peculiar sensation which advertises us thai
Tve are in motion. We perceive jerks, or shocks, it is
true, because these are sudden changes of motion, pro-
duced, as the laws of mechanics teach us, by sudden
and powerful forces acting during short times ; and these
forces, applied to our bodies, are what wefeii. When,
for example, we are carried along in a carriage with the
blinds down, or with our eyes dosed (to keep us from
seeing external objects), we perceive a tremor arising
from inequalities in the road, over which the carriage is
successively lifted and let fall, but we have no sense of
progress. As the road is smoother, our sense of motion
is diminished, though our rate of travelling is accelerated.
■Those who have travelled on the celebrated rail-road
between Manchester and Liverpool testify that but for
the noise of the train, and the rapidity with which ex-
ternal objects seem to dart by them, the sensation is al-
most that of perfect rest.

(16.) But it is on shipboard, where a great system is
maintained in motion, and where we are surrounded
b2

18 A TR£ATISE ON ASTRONOMY. []GHAP. I.

with a multitude of objects which participate with our-
selves and each other in the common progress of the
whole mass, that wc feel most satisfactorily the identity
of sensation between a state of motion and one of rest.
In the cabin of a large and heavy vessel, going smoothly
before the wind in still water, or drawn along a canal,
not the smallest indication acquaints us with the way it
is making. We read, sit, walk, and perform every cus-
tomary action as if we were on land. If we throw a
ball into the air, it falls back into our hand ; or, if we
drop it, it lights at our feet. Insects buzz around us a»
in the free air ; and smoke ascends in the same manner
as it would do in an apartment on shore. If, indeed,
we come on deck, the case is, in Some r^^speets, diifer-
ent ; the air, not being carried along with us, drifts away
smoke and other light bodies — such a3 feathers aban*
doned to it — apparently, in the opposite direction to that
of the ship's progress ; but, in reality, they remain at
rest, and we leave them behind in the. air. Still, the
illusion, so far as massive objects and our own move^
ments are concerned, remains complete ; said when we
look at the shore, we then perceive the effect of our own
motion transferred, in a contrary direction, to external
objects — external, that is, to the system of which we
form a part,

" Pfo^ehimur portu« teirseque urbeaqu^ recediint"

(17.) Not only do external objects at rest appear in
motion generally, with respect to ourselves when we
are in motion among them, but they appear to move one
among the other — they shift their relative apparent
places. Let any one travelling rapidly along a high
road fix his eye steadily on any object, but at the same
time not entirely withdraw his attention from the gene-
ral landscape, — he will see, or think he sees, the iprhole
landscape thrown into rotation, and moving round that
object as a centre ; all objects between it and himself
appearing to move backwards, or the contrary way to
his own motion ; and %ll beyond it, forwards, or in the
direction in which he moves : but let him wididraw his
eye from that object, and fix it on another, — a nearer

CHAP. 1.3 FORM or THK EAKTB, 1%

one, for instance, — immediately the appearance of ro-
tation shifts also, and the apparent centre about which
this illusive circulation is performed is transferred to the
new object, which, for the moment, appears to rest.
This apparent change of situation of objects with re-
spect to one another, arising from a motion of the spec-
tator, is called a parallactic motion ; and it is, therefore,
evident that, before we can ascertain whether external
objects are really in motion or not, or whal their mo-
tions are, we must snbduct, or allow for, any sach pth
rallaetic motion which may exist

(18.) In order, however, to conceiTe the earth m in
motion, we must form to ourselves a conception of it«
shape and size. Now, aki object camiot have shape and
sixe, unless it is limited on all sides by some definite
ou^ne,t^so as to admit of our imagining it, at least, di«-
connected from other bodies, and existfaig msolaled.in
' space. The first rude notion we form of the earth i*
that of a flat surface, of indefinite extent in all direction*
from the spot where we stand, a^mve which are the air
and shf ; below, to an indefinite profundity, solid mat-
ter. This is a prejudice to be got rid of, like that of the
earth's inimobility ; but it is one much easier to rid our-
selves of, inasmuch as it originates only in our own
mental inactivity, in not questioning ourselves where we
will place a limit to a thing we have "been accustomed
from infancy to regard as immensely large ; and does
not, like that, originate in the testimony of our sense*
unduly interpreted. On the contrary, the direct testi-
mony of our senses lies the other way. When we see
the sun set in the evening in the west, and rise again in
the east, as we cannot doubt that it is the same sun we
see after a temporary absence, we must do violence ta
all our notions' of solid matter, to suppose it to have
made its way through the substance of t&e earth. It
must, therefore, have gone tmder it, and that not by a
mere subterraneous channel; for if we notice the pointe
where it sets and rises for many suopessive days, or for
a whole year, we shall find them constantly shifting,
round a very large extent of the horizon ; and, besidesi
the moon and stars also^8et and rise again in all point*

20 ' A TREATISE ON ASTRONOMY. [CHAP. I.

of the visible horizon. The conclusion is plain: the
earth cannot extend indefinitely in depth downwards,
nor indefinitely in surface laterally ; it must have not
only bounds in a horizontal direction, but also an under
side, round which the sun, moon, and stars can pass ;
and that side must, at least, be so far like what we see,
that it must have a sky and sunshine, and a day when it
is nigbt to us, and vice versa ; where, in short,

•—** redit a nobia Aurora, diemque reducit
Nosque ubi primus equis oriens afflavit anbelis,
lUic sera rubens accendit lumina Vesper." Georg.

(Id.) As soon as we have familiarized ourselves with
the conception of an earth -wiihoMi foundcUiona or fixed
supports — existing insulated in space from contact of
every thing extethal, it becomes easy to imagine it in
motion— -or, rather, difficult to imagine it otherwise ; for,
since there is nothing to retain it in one place, should
any causes of motion exist, or sny forces act upon it, it
must obey their impulse. Let us next see what obvious
circumstances there are to help us to a knowledge of the
shape of the eatth.

(20.) Let us first examine what we can actually see
of its shape.. Now, it is not on land (unless, indeed, on
uncommonly level and extensive plains) that we can see
any thing of the genergl figure of the earth ;— the hills,
trees, and other objects which roughen its surface, and
break and elevate the line of the horizon, though, ob-
viously bearing a most minute proportion to the whole
earth, are yet too considerable, with respect to ourselves
and to that sniall portion of it which we can see at a sin-
gle view, to allow of our forming any judgment of the
form of the whole, from that of a part so disfigured.
But with the surface of the sea, or any vastly extended
level plain, the case is otherwise. If we sail out of sight
of land, whether we stand on the deck of the ship or
climb the mast, we see the surface of the sea — ^not losing
itself in distance and mist, but terminated by a sharp,
clear, well defined line, or offing as it is called, which
runs all round us in a circle, having our station for its
centre. That this line is really a circle, we conclude,
first, from the perfect apparent similarity of all its parts ;

CHAP. I.] HOKIZON AMD TTB MP. 91

and, secondly, from the fact of all its ptrts sp pc siin y at
the same distance from as, and that evideDtly a mode-
rate one; and, thirdly, from this, that its apparent
diameterj measured with an instrument called the dip
sector, is the same (except under some singular atmo-
spheric circumstances, which produce a temporary distor-
tion of the outline), in whatever direction the measure
is taken, — ^properties which belong only to the circle
among geometrical figures. If we ascend a high emi-
nence on a plain (for instance, one of the Egyptian py-
ramids), the same holds good.

(21.) Masts of ships, however, and the edifices erected
by man are trifling eminences compared to what nature
itself affords; JStna, Tenerifie, Mowna Roa, are emi-
nences from which no contemptible aliquot part of the
whole earth's surface can be seen ; but from these again
—in those few and rare occasions when the transparency
of the air will permit the real boundary of the horizon,
the true sea-line, to be seen — ^the very same appearances
are witnessed, but with this remarkable addition, viz,
that the angular diameter of the visible area, as mea-
sured by the dip sector, is materially less than at a lower
level, or, in other words, that the apparent size of the
earth has sensibly diminished as we nave receded from
its surface, while yet the absolute quantity oi it seen at
once has been increased.

(22.) The same appearances are observed universally,
in every part of the earth's surface visited by man.
Now, the figure of a body which, however seen, w^
pears always circular^ can be no other than a sphere or
globe.

(23.) A diagram will elucidate this. Suppose the
earth to be represented by the sphere LHNQ* whose
centre is C, and let A, 6, M be stations at different
elevations above various points of its surface, represent
ed by a, g, m, respectively. From each of them (as
from M) let a line be drawn, as MNn, a tangent to Uie
surface at N, then will this line represent the visual ray
along which the spectator at M will see the visible ho*
rizon ; and as this tangent sweeps round M, and comes
successively into the positions M 9, M P/>, M Qg, the

33 A TREATISE ON ASTRONOMY. [cHAP. I.

point of contact N will mark out on the surface the circle
NOP Q. The area of this circle is the portion pf the

earth's surface visible to a spectator at'M, and the angle
NMQ included between the two extreme visual rayis is
the measure of its apparent angular diameter. Leaving,
at present, out of consideration the effect of refraction in
the air below M, of which more hereafter, and which
always tends, in some degree, to increase that angle, or
render it more obtuse, this is the angle measured by the
dip sector. Now, it is evident, 1st, that as the point M
is more elevated above m, the point immediately below
it on the sphere, the visible area, i. e, the spherical seg-
ment or slice NOPQ, increases ; 2dly, that the distance
of the visible horizon* or boundary of our view from
the eye, viiz. the line MN, increases ; and, 3dly, that
the angle M NQ becomes less obtuse, or, in other words,
the apparent angular diameter of the earth diminishes,

* *0f i^w, to terminal.

CHAP. I.J HOBIZON AND ITS MF. 98

being nowhere so great as 180^, or two right aaglef,
but falling short of it by some seosibie qaantity, and that
more and more the higher we ascend. The figure ex-
hibits three states or stages of eleration, with^e hori-
zon, &c. corresponding to each, a glance at which will
explain our meaning ; or, limiting ourselves to the larger
and more distinct, MNOPQ, let the reader imagina
nNM, MQg to be the two* legs of a ruler joined at M,*
and kept extended by the globe NmQ between them*
It is clear, that as the joint M is urged home towards
the surface, the legs will open, and the ruler will become
more nearly straight, but will not attain ^erfec/ straight
ness till M is brought fairly up to contact with the sur^
face at m, in which case its whole length will become a
tangent to the sphere at m, as is the line xy*

(24.) This explains what is meant by the dip of the
horizon. Mm, which is perpendicular to the gtnenl
surface of the sphere at m, is also the direction in which
a plumb-line* would hang ; for it is an observed (JM^t,
that in all situations, in every part of the earth, the di-
rection of a plumb-line is exactly perpendicular to the
surface of still-water; and moreover, that it is also ex-
actly perpendiculsA: to a line or surface truly adjusted by
a spirit-kveU * Suppose, then, that at our station M we
were to adjust a lliie (a wooden ruler for instance) by a
spirit-level, with perfect exactness ; then, if we sttj^ae
the direction of this line indefinitely prolonged both
ways, as XMY, the line so drawn wiU be at right
angles to Mm, and therefore parallel to xmu^ the tan-
gent to the sphere at m% A iqpectator placed at M will
tiierefore see not only all the vault of the sky obofH thia
line^ as XZY, but also that portion or zone of it which
lies between XN and YQ ; in other words, his sky will
be more than a hemisphere by the zone YQXN. It ia
the angular breadth of this redundant zone—the an|^
YMQ, by which the vinbh horizon appears depiesMd
below the direction of a spirit4evel'— that is called the
Up of the horizon. It is a correction of constant use in
nautical astronomy. «

*See thia instniment dMcribed in dup^ Q.

34 A HlXATISE ON ASTRONOMY. [CHAP. f •

(26.) From flie foregoing explanations it appears,
Ist, That the general figure of the earth (so far as it can
be gathered from this kind of observation) is that of a
sphere or globe. In this we also include that of the sea»
which, wherever it extends, covers and fills in those in-
equalities and local irregularities which exist on land,
but which can of course only be regarded as trifling de*
•Tiations from the general outline of the whole mass, as
we consider an orarige not the less round for the rough-
nesses on its, rind. 2dly, That the appearance of a visi^
ble horizon, or sea oiling, is a consequence of the cur-
vature of the surface, and does not arise from the inability
of the eye to follow objects to a greater distance, or
from atmospheric indistinctness. It will be. worth while
to pursue the general notion thus acquired uito some of
its consequences, by which its consistency with obser-
vations of a different kind^ and on a larger scale, will be
put to the test, and a clear conception be formed of the
manner in which the parts of the earth are related ta
each oth«r, and held together as a whole.

•(5J6.). In the first place, then, every one who Has passed
a little while at the sea side is aware that objects may be
seen perfectly well beyond the offing or visible horizon
—but not the whole of them. We only see their upper
partJB. Their bases where they rest On, or rise ou* of
the Water, are hid from view by the spherical surface of
the sea, which protrudes between them and ourselves^
Suppose a ship, for instance, to sail directly away from
our station ;— at first, when the distance ci the ship is
small, a spectator, S, situated at some certain height
above the sea, sees^ the whole of the ship, even to the
Ufofer Hne where it rests on the sea, as- at A. As it re-
cedes it diminishes, it is true, in apparent size, but still
the wlwle is seen down to the water line, till it reachea
the visible horizon at B. But as soon as it has passed
this distance, not only does the visible portion still con-
tinue to diminish in apparent size, but the hirll begins to*
disappear bodily, as if sunk below the surface. When
it has reached a certain distance, as at O, its hull
has entirely vanished, but the masts &nd sails remain, ^
presenting the appearance c. But if, in this state of

CHAP.

i-J

SIZE OF THE EARTH.

things, the spectator quickly ascends to a higher i
T, whose visible horizon is at D, the hall comes again
in sight ; and when he descends again he loses it. The
ship still receding, the lower sails seem to sink below
the water, as at <f, and at length the whole dis^peait:
while yet the distinctness with which the last portion of
the sail d is seen is such as to satisfy ns that were it not
for the interposed segment of the sea, ABCDE, the dis-
tance T£ is not so great as to have prevented an equally
perfect view of the whole.

(27.) In this manner, therefore, if we could measure
the heights and exact distance of two stations which
could barely be discerned from each other over the edge
of the horizon, we could ascertain the actual size of the
earth itself: and, in fact, were it not for the effect of re-
fraction, by which we are enabled to see in some small
degree round the interposed segment Tas will.be here-
after explained), this would be a tolerably good method
of ascertaining it. Suppose A and B to be two emi-

nences, whose perpendicular heights A a and B b (which,

for simplicity, we will suppose to be exactly equal) are

known, as well as their exact horizontal interval aDb^

C

26 A TREATISE ON ASTRONOMY. [cHAP. I.

by measurement; then is it clear that D, the visi-
ble horizon of both, will lie just half-way between
them, and if we suppose aBb to be the sphere of
the earth, and C its centre in the figure CD6B, we
know D &• the length of the arch of the circle between
D and 6, — viz. half the measured interval, and b B, the
excess of its secant above its radius — which is the height
of B,— Kiata which, by the solution of an easy geometrical
problem, enable us to find the length of the radius D C.
If, as is really the case, we suppose both the heights and
distance of the stations inconsiderable in comparison with
the size of the earth, the solution alluded to is contained
in the following proposition : —

Tlie eartVs diameter bears the same proportion to the
distance of the visible horizon from the eye as that dis'
tance does to the height of the eye above the sea level.

When the stations are unequal in height the problem
is a little more complicated.

(28.) Although, as we have observed, the effect of
refraction prevents this from being an exact method of
ascertaining the dimensions of the earth, yet it will suf-
fice to afford such an approximation to it as shall be of
use in the present stage of the reader's knowledge, and
help him to many just conceptions, on which account
we shall exemplify its application in numbers. Now, it
appears by observation, that two points, each ten -feet
• above the surface, cease to be visible from each otl^r
over still water, and in average atmospheric cireui»-
stances, at a distance of about 8 miles. But 10 feet is
the &28th part of a mile, so that half their distance, or
4 miks, is to the height of each as 4 x 528 or 2112 : 1^
ahd therefore in the same proportion to 4 miles is the
length of the earth's diameter. It must, therefore, be
equal to 4 X 2112 = 8448, or, in round numbers, about
8000 miles, which is not very far from the truth.

(29.) Such is the first rough result of an attempt to
ascertain the earth's magnitude ; and it will not be amiss
if we take advantage of it to compare it with objects we
have be^n accustomed to consider as of vast size, so as
to interpose a few steps between it and our ordinary ideas

CHAP. 1.3 TISIBL8 PORTION OF THS tURFACS. 97

of dimension. We have before likened the inequalities
on th« earth's surface, arising (xom mountains, vaUeys,
buildings, &>c, to the roughnesses on the rind of an
orange, compared with its general mass. The comp^
rtson is quite free from exaggeration. The highest moun-
tain known does not exceed five miles in perpendicular
elevation : this is only one 1600th part of die earth's
diameter ; consequenUy, on a ^be of sixteen inches in
diameter, such a mountain would be represented by a
protaberance of no more than one hundredth part of an
inch, which is about the thickness of ordinary drawing-
paper. Now as there is no entire continent, or even any
very extensive tract of land, known, whose general ele-
vation above the sea is any thing like half this quantity,
it follows, that if we would construct a correct model of
our earth, with its seas, continents, and mountains, on
a globe sixteen inches in diameter, the whole of the land,
with the exception of a few prominent points and ridges,
must be comprised on it within the thickness of diin
writing paper ; and the highest hills would be represented
by the smallest visible grains of sand.

(30.) The deepest mine existing does not penetrate
half a mile below the surface : a scratch, or pin-hole,
duly representing it, on the surface of such a globe as
our model, would be imperceptible without a magnifier.

(31.) The greatest depth of sea, probably, doe^ not
much exceed the greatest elevation of the continents;
and would, of course, be represented by an excavation,
in about the same proportion, ibto the substance of the
globe: so that the ocean comes to be conceived* as a
mere film of liquid, such as, on our model, would be left
by a brush dipped in colour and -drawn over those parts
intended to represent the sea : only in so conceiving it,
we must bear in mind that the resemblance extends no
farther than to proportion in point of quantity. The
mechanical laws which would regulate the distribution
and movements of such a filmj and its adhesion to the
surface, are altogether different from those which govern
the phenomena of the sea.

(32.) Lastly, the greatest extent of the earth's surface
which has ever been seen at once by -man, was that ex-

28 A TREATISE ON ASTRONOMY. [CHAP. I.

posed to the view of MM. Biot and Gay-Lussac, in their
celebrated aeronautic expedition to the enormous height
of 25,000 feet, or rather less than five miles. To esti-
mate the proportion of the area visible from this elevation
to the whole earth*s surface, we must have recourse to
the geometry of the sphere, which informs us that the
convex surface of a spherical segment is to the whole sur-
face of the sphere to which it belongs as the versed sine
or thickness of the segment is to the diameter of the
sphere ; and further, that this thickness, in the case we
are considering, is almost exactly equal to the perpen-
dicular elevation of the point of sight above the surface.
The proportion, therefore, of the visible area, in this
case, to the whole earth's surface, is that of five miles to.
8000, or 1 to 1600. The portion visible from iEtna, the
Peak of Teneriffe, or Mowna Roa, is about one 4000th.
(33.) When we ascend to any very considerable ele-
vation above the surface of the earth, either in a balloon,
or on mountains, we are made aware, by many uneasy
sensations, of an insufficient supply of air. The barome-
ter, an instrument which informs us of the weight of air
incumbent on a given horizontal surface, confirms this im-
pression, and affords a direct measure of the rate of dimi-
nution of the quantity of air which ^ given space includes
as we recede from the surface. From its indications we
learn, that when we have ascended to the height of 1000
feet, we have left below us about one thirtieth of the
whole mass of the atmosphere : — that at 10,600 feet of
perpendicular elevation (tirhich is rather less than that of
the summit of iEtna*) we have ascended through about
one third ; and at 18,000 feet (which is nearly that of Co-
topaxi) through one half the material, or, at least, ihe
ponderable, body of air incumbent on the earth's surface.
From the progression of these numbers, as well as, a jon-
ori, from the nature of the air itself, which, is compreasi"
ble, i. e. capable of being condensed, or crowded into a
smaller space in proportion to the incumbent pressure, it
is easy to see that, although by rising still higher we should

* The height of ^tna above the Mediterranean (as it results from a
barometrical measurement of my own, made in July, 1824, under very
fiivourable circumBtances) is 10,^ £nglish feetr-^Autftor,

I

CHiLP» I.J THS ATMOSPHERE. 29

continually get abore more and more of the air, and so re-
lieye ourselves more and more from the pressure with
which it weighs upon us, yet the amount of this additional
relief, or the ponderable quantity of dir surmounted, would
be by no means in proportion to the additional height as-
cended, but in a constantly decreasing ratio. An easy
calculation, however, founded on our experimental know-
ledge of the properties of air, and the mechanical laws
which regulate its dilation and compression, is sufficient
to show that, at an altitude above the surface of the earth
not exceeding the hundreth part of its diameter, the tenui-
ty, or rarefaction, of the air must be so excessive, that not
only animal life could not subsist, or combustion be main-
tained in it, but that the most delicate means we possess of
ascertaining the existence of any air at all would fail to
afford the slightest perceptible indications of its presence.

(34.) Laying out of consideration, therefore, at pre-
sent, all nice questions as to the probable esdstence of a
definite limit to the atmosphere, beyond which there is,
absolutely and rigorously speaking, no air, it is clear, that,
for all practical purposes, we may speak of those regions
which are more distant above the earth*8 surface than the
hundredth part of its diameter as void of air, and of course
of clouds (which are nothing but visible vapours, diffused
KSidJloating in the air, sustained by it, and rendering
it turbid as mud does water). It seems probable, from
many indications, that the greatest height at which visible
clouds ever exist does not exceed ten miles.; at which
height the density of the air is about an eighth part of
what it is at the level of the sea.

(35.)- We are thus led to regard the atmosphere of air,
with the clouds it supports, as constituting a coating of
equable or nearly equable thickness, enveloping our globe
on all sides ; or rather as an atrial ocean, of which the
surface of the sea and land constitutes the bed, and whose
inferior portions or strata, within a few miles of the
earth, contain by far the greater part of the whole mass,
the density diminishing with extreme rapidity as we re-
cede upwards, till, within a very moderate distance (such
as would be represented by the sixth of an inch on the mo-
del we have before spoken of, and which is not more in pro-
c2

30 A TREATISE ON ASTRONOMY. [CHAP. I.

portion to the globe on which it rests, than the downy-
skin of a peach in comparison with the fruit within it),
all sensible trace of the existence of air disappears.

(36.) Arguments, however, are not wanting to render
it, if not absolutely certain, at least in the highest degree
probable, that the surface of the aerial, like that of the
aqueous ocean, has a real and definite limit, as above hiirt'
ed at ; beyond which there is positively no air, and above
which a fresh quantity of air, could it be added from with-
out, or carried aloft from below, instead of dilating itself
indefinitely upwards, would, after a certain very enor-
mous but still finite enlargement of volume, sink and
merge, »3 water poured into the sea, and distribute itself
among the mass beneath. With the truth of this conclu-
sion,^ however, astronomy )ias little concern ; all the ef-
fects of the atmosphere in modifying astronomical phe-
nomena being the same, whether it be supposed of defi-
nite extent or not.

(37.) Moreover, whichever idea we adopt, it is equaUy
certain that, within those limits in which it possesses any
appreciable density, its constitution is the same over all
points on the earth's surface ; that is to say, on the great
scale, and leaving out of consideration temporary and local
causes of derangement, such as winds, and great fluc-
tuations, of the nature of waves, which prevail in it to an
immense extent : in other words, that the law of diminu-
tion of the air's density as we recede upwards from the
level of the sea is the same in every column into which
we may conceive it divided, or from whatever point of
the surface we* may set out. It may therefore be consi-
dered as consisting of successively superposed strata or
layers, each of the form of a spherical shell, concentric
with the general surface of the sea and land, and each of
which is rarer, or specific^y lighter, than Uiat immedi-
ately beneath it ; and denser 9 ct specifically heavier, than
that immediately above it. This kind of distribution of
its ponderable mass is necessitated by the laws of the
equilibrium of .fluids, whose results barometric observa-
tions demonstra^ to be in perfect accordance with expe-
rience.

It must be observed^ however, that with this distribu-

CHAP.

!•]

RBFRAOnON.

81

tion of its strata the inequalities of mountaiDS aad Tallajs
have no concern ; these exercise no more influence in
modifying their general spherical figure than the inequali-
ties at the bottom of the sea interfere with the general
sphericity of its surface.

(38.) It is the power which air possesses, in common
with all transparent media, of refracting the rays of lif ht,
or bending them out of their straight course, which renders
a knowledge of the constitution of the atmosphere import-
ant to the astronomer. Owing to this property, objects
seen obliquely through it appear oAerwise situated thao
they would to the same spectator, had the atmosphere no
existence ; it thus produces a fabe impression respecting
their places, which must be rectified by ascertaining the
amount and direction of the displacement so apparently
produced on each, before we can come at a knowledge
of the true directions in which they are sitaated from ns
at any assigned moment.

(39.) Suppose a spectator placed at A, any point of the
earth*8 surface KAit, and let li. Mm, Nn, represent

the successive strata or layers, of deereading density, into
which we may conceive the atoiosphere to be divided,
and which are spherical surfaces concentric with KA?, the
earth's surface. Let S represent a star, or other heavenly
body, beyond the utmost limit of the atmosphere ; then,
if the air were away, the spectator would see it in the di-

32 REFRACTION. [cHAP. I.

rection of the straight line AS. But, in reality, wheh
the ray of light SA reaches the atmosphere, suppose at dj
it will, by the laws of optics, begin to bend downwards,
and take a more inclined direction, as <f c. This bending
will at first be imperceptible, owing to the extreme tenu-
ity of the uppermost strata ; but as it advances downwards,
the strata continually increasing in density, it will continu-
ally undergo greater and greater refraction in the same di-
rection ; and thus, instead of pursuing the straight line
S(2A, it will describe a curve S(/c6 a, continually more
and more concave downwards, and will reach the earth,
not at A, but at a certain point a, nearer to S. This ray,
consequently, will not reach the spectator's eye. The ray
by which he will see the star is, therefore, not Sc^A, but
another ray which, had there been no atmosphere would,
have struck the earth at K, a point behind the spectator ;
but which, being bent by the air into the curve SDCBA,
actually strikes on A. Now, it is a law of optics, that an
object is seen in the direction which the visual ray has at
the instant of arriving at the eye^ without regard to what
may have been otherwise its course between the object and
the eye. Hence the star S will be seen, not in the di-
rection AS, but in that of Aa, a tangent to the curve
SDCBA, at A. But because the curve described by the
refracted ray is concave downward^!, the tangent A«, will
lie above As, the unrefracted ray : consequently the object
S will appear more elevated above the horizon AH, when
seen through the refracting atmosphere, than it would ap-
pear were there no such atmosphere. Since, however, the
disposition of the strata is the same in all directions around
A, the visual ray will not be made to deviate laterally, but
will remain constantly in the same vertical plane SAC,
passing through the eye, the object, and the earth's centre.
(40.) The effect of the air's refraction, then, is to raise
all the heavenly bodies higher above the horizon in ap-
pearance than they are in reality. Any such body, situ-
ated actually in the true horizon, will appear above it, or
will have some certain apparent altitude (as it is called).
Nay, even /some of those actually below the horizon, and
which would therefore be invisible but for the effect of
refraction, are, by that effect^ raised above it and brought

CHAP. I.j SSTRACnOK. 8)

into sight. Thus, the sun, when situated at P below the
trae horizon, AH, of the spectator, becomes visible to hinit
as if it stood at p, by the refracted ray Pgr^A, to which
Ap is a tangent.

(41.) The exact estimation of the amount of atm<^
spheric refraction, or the strict determination of the ancle
SA^, by which a celestial object at any assigned altitadev
HAS, is raised in appearance above its tme place, is, im«
fortunately, a very difficult subject of physical inquiry,
and one on which geometers (from whom alone we can
look for any information on the subject) are not yet en*
tirely agreed. The difficulty arises from this, that the
density of any stratum of air (on which its refracting
power depends) is affected not merely by the superincum-
bent pressure, but also by its temperature or degree of
heat. Now> although we know that as we recede iirom the
earth's surface the temperature of the air is constantly
diminishing, yet the law^ or amount of this diminution
at different heights, is not yet fully ascertained. More-
over, the refracting power of air is perceptibly affiected by
its moisture ; and &is; too, is not the same in every part of
an aerial column ; neither are we acquainted with the laws
of its distribution. 'The consequence of our ignorance on
these points is to introduce a corresponding degree of
uncertainty into the determination of the amount of refrac-
tion which affects, to a certain appreciable extent, our
knowledge of several of the most important data of as-
tronomy. The uncertainty thus induced is, however,
confined within such very narrow limits as to be no cause
of embarassment, except in the most delicate inquiries,
and to call for no further .allusion in a treatise like the
present.

(42.) A *< Table of Refractions," as it is called, or a
statement of the amount of apparent displacement aris-
ing from this cause, at all altitudes, or in every situation
of a heavenly body, from the horizon to the zenith^* or
point of the sky vertically above the spectator, and, under
all the circumstances in which astronomical observations
. are usually performed which may influence the result, is

* From aa Arabic word of this ngnification.

34 A TREATISE ON ASTRONOMY. [CHAP; 1.

one of the most important and indispensable of all astro-
nomical tables, since it is only by the use of such a table
we are enabled to. get rid of an illusion which must
otherwise pervert all our notions respecting the celestial
motions. Such have been, accordingly, constructed with
great care, and are to be found in every collection of
astronomical tables.* Our design, in the present treatise,
will not admi^ of the introduction of tables; and we
must, therefore, content ourselves here, and in similar
cases < with referring the reader to works especially des-
tined to furnish these useful aids to calculation. It is,
however, desirable that he should bear in mind the
following general notions of its amount, and law of
variation.

(43.) 1st. In the zenith there is no refraction ; a ce-
lestial object, situated vertically over head, is jseen in its
true direction, as if there were no atmosphere.

2dly. In descending from the zenith to the horizon,
the refraction continually increases ; objects, near the
horizon appearing more elevated by it above their true
directions than those at a high altitude.

3dly. The rate of its increase is nearly in proportion
to the tangent of the apparent angular distance of the
object from the zenith. But this rule, which is not far
from the truth, at moderate zenith distances ^ ce;isea U>
give correct results in the vicinity of the horizon, where
the law becomes much more complicated in its ex-
pression.

4thly. The average amount of refraction, for an ohr
ject half-way between the zenith and horizon, or at an
apparent altitude of 45°, is about V (more exactly 57")^
a quantity hardly sensible to the naked eye ; but at the
visible horizon it amounts to no less a quantity than 33',
which is rather more than the greatest apparent diameter
of either the sun or the moon. Hence it follows, that
when we see the lower edge of the sun or moon just ap-
parently resting on the horizon, its whole disk is in
reality below it, and would be entirely out of sight and

*Vide " Requisite Tables to he used with the Nautical Almanac.'*
See also Nautical Almanac for 1833, Dr Pearson's Astronomical Tables,
and Mr. Baily's Astronomical Tables and Formuloe.

CHAP. 1.3 tWILIGUT. SO

concealed by the convexity of the earth but for the bend-
ing round it, which the rays of light have undeigone in
their passage through the air, as Eluded to in art. 40.

(44.) It follows from this, that one obvious effect of
refraction must be to shorten the duration of night and
darkness, by actually prolonging the stay of the sun and
moon above the horizon. But even after they are set,
the influence -of the atmosphere still continues to send
us a portion of their light ; not, indeed, by direct trans-
mission, but by reflection upon the vapours, and minute
solid particles, which float in it, and, perhaps, also on
the actual material atoms of the air itself. To understand
how this takes place, we must recollect, that it is not
only by the direct light of a luminous object that we
see, but that whatever portion of its light which would
not otherwise reach our eyes, is intercepted in its course,
and thrown back, or laterally, upon us, becomes to us a
means of illumination. Such reflective obstacles always
exist floating in the air. The whole course of a sun-
beam penetrating through the chink of a window-shutter
into a dark room, is viMle as a bright line in the air;
and even if it be stifled, or let out through an opposite
crevice, the light scattered through the apartment from
this source is sufficient to prevent entire darkness in the
room. The luminous lines occasionally seen in the air,
in a sky full of partially bsoken clouds, which the Vulgar
term " the sun drawing water," are similarly caused.
They are sunbeams, through apertures in clouds, par-
tially intercepted and reflected on the dust and vapours
of the air below. Thus it is with those solar rays which,
after the sun is itself concealed by the convexity of the
earth, continue to traverse the higher regions of the at-
mosphere above our heads, and pass through and out of
it, without directly striking on the earth at all. Some '
portion of them is intercepted, and reflected by the float-
ing particles above mentioned, and thrown back, or la-
terally, so as to reach us, and aflbrd us that secondary
illumination, which is twilight. The course of such rays
will be immediately understood from the annexed figure,
in which ABOD is the earth ; A a point on its surface,
where the sun S is in the act of setting ; its last lower

36

A TREATISE ON ASTRONOMY.

[chap.

ray SAM just grazing the surface at A, while its superior
rays SN, SO, traverse the atmosphere above A without
striking the earth, leaving it finally at the points PQR,
after being more or les» bent in passing through it, the

lower most, the higher less, and that which, like SRO,
merely grazes the exterior limit of the atmosphere, not
at all. Let us consider several points, A, B, C, D, each
more remote than the last from A, and each more deeply
involved in the earlKs shadow, which occupies the whole
space from A beneath the line AM. Now, A just receives
the sun's last direct ray, and, besides, is illuminated by
the whole reflective atmosphere PQRT. It therefore
receives twilight from the whole jsky. The point B, to
which the sun has set, receives no direct solar light, nor
any, direct or reflected, from all that part of its visible
atmosphere which is below APM ; but from the lenti-
cular portion FBo?, which is traversed by the sun's rays,
and which lies above the risible horizon BR of B, it re-
ceives a twilight, which is strongest at R, the point im-
medbttely below which the sun is, and fades away gradu-
ally towards P, as the luminous part of the atmosphere
thins off. At C, only the last or thinnest portion, PQz
of the lenticular segment, thus illuminated, lies above
the horizon, CQ, of that place : here, then, the twilight
is fdiebie, and confined to a small space in and near the

CHAP. 1.3 T&RRB8TRIAI. RBFRACTIDK^ 97

horizon, which the sun has quitted, while at D the twi«
light has ceased altogether.

(45.) When the sun is above the horizon, it illumi-
nates the atmosphere and clouds, and these again dis-
perse and scatter a portion of its light in all directions,
so as to send some of its rays to every exposed point,
from every point of the sky. The generally difiused
light, therefore, which we enjoy in the daytime, is a phe-
nomenon originating in the very same causes as the twi-
light. Were it not for the reflective and scattering power
of the atmosphere, no objects would be visible to us out
of direct sunshine ; every shadow of a passing cloud
would be pitchy datkness; the stars would be visible all
day, and every apartment, into which the sun had not di-
rect admission, would be involved in nocturnal obscurity.
This scattering action of the atmosphere on the solar
light, it should be observed, is greatly increased by the
irregularity of temperature caused by the same luminary
in its different parts, which, during the daytime, throws'
it into a Constant state of undulation, and, by thus brkig-
ing together masses of air of very unequal temperatures,
produces partial reflections and refractions at Uieir com-
mon boundaries, by which much light is turned aside
from the direct course, and diverted to the purposes of
general illumination.

(46.) From the explanation We have given, in arts*
39 and 40, of the nature of atmospheric refraction,
and the mode in which it is produced in the progress
of a ray of light through successive strata, or layeiB of
the atmosphere, it will be evident, that whenevei' a ray
passes obliquely from a higher level to a lower one, or
vice versa, its course is not rectilinear, but concave
downwards ; and of course any object seen by means of
such a ray, must appear deviated from its true place,
whether that object be, like the celestial bodies, entirely
beyond the atmosphere, or, like the summits of moiili-
tains, seen from the plains, or other terrestrial stations,
at different levels, seen from each othei*, immersed in it.
Every difference of level, accompanied, as it must be,
with a difference of density in the atrial strata, must also
have, corresponding to it, a certain amount of refraction ;
D

I

8S A TREATISE ON ASTRONOHT. [cHAP. I**

less, kttleed, than what would be produced by the whole
fttmosphere, but still oflen of very appreciable, and even*
considerable, amount* This refraction between terres-
trial stations i» termed terreatrial refraction^ to distin-
guish it from thaft total effect which is only produced on
celestial objects, or svck as are beyond the atmosphere,
and which is called eelestial or astronomical refraction.

(47.) Another effect of refraction is to distort the visi-
ble forms and proportions of objects seen near the hori-
zon. The sun, for instance, which, at a considerable
altitude, always appears round, assumes,, as it approac^hes
the horizon, a flattened or oval outline f ks horizontal
diameter being visibly greater than- that m a^ vertical di*
rectioR. When very near the horizon, this flattening ib
evidently more considerable on the lower side than on
the upper 'f so that the apparent form is neither circulai
nor elliptic, but a species of ovalr which deviates more
from a circle below than above. This singular effect,
which any one may notice hi a fine sunset, arises from the
rapid rate at which the lefiraiction increases in approach-
ing the horizon. Were every visible point in the sun's
circumference equally raised by refraction, k would still
i^pear circular, though displaced : but the lower portions
being more raised than the upper, the vertical diameter is
therrfiy shortened, while the two extremities^ of its hori-
zontal diameter are equally raised, and in parallel direc-
tions, so that its apparent length remains the samev The
dilated size (genenlly) of the sun or moon, when seea
near the horizon, beyond what they appear to have
when high up in the sky, has nothing to do with refrac-
tion. It is an iMusion of the judgment arising from the
terrestrial objeetfr interposed, or placed in close compari-
son with them. In that situation we view and judge of
them as we do of terrestrial objects — in detail, and with
an acquired habit of attention to parts. Aloft we have
no associations to guide us, and Uieir insulation in the
expanse of sky leads u» rather to undervalue than to
overrate their apparent magnitudes. Actual measure-
ment with a proper instrument corrects our error, with-
out, however, dispelling our illusion. By this we learn,
that the sun, when just on the horizon^ subtends at our

CHAP. 1.3 OF THE SPHERE OF THE HEAVENS. 39

eyes almost exactly tlie same, and the noon a materially
less angle, than when seen at a great altitude in the skyi^
owing to the effect of what is cadled parallax, to \m ex-
plained presently.

(48.) After what has been said of the small extent of
the atmosphere ia comparison of the mass of the earth,
we shall have little hesitation in admitting those lumina-
ries which people and adorn the sky, and which, while
Ihey obviously form no part of the eaurth, and receive no
support from it, are yet not borne along at random like
clouds upon the air, nor drifted by the winds, to be ex-
ternal to our atmosphere. As such we have considered
^em while speaking of their refractions — as existing in
tMe immensity of space beyond, and situated, perhaps,
for any thing we ean perceive to the contrafy, at enor*
mous distances from us and from each other.

(49.) Coi:dd a spectator exist unsustained by <he earthy
•r any solid support, he would see around him at one
view the whole contents of spac^-— the visible consti-
tuents of the universe : and, in the absence of any means
ef judging of their distances from him, would refer them,
in the directions in which they were seen from his sta^
tion, to the concave surface of an imaginary sphere,
having his eye for a centre, and its surface at some vast
indeterminate distance. Perhaps he might judge those
which appear to him large and bright, to be nearer to
him than the smaller and less brilliant ; but, independent
of other means of judging he would have no warrant for
this opinion, any more &aa for tiie idea that all were
equidistant from hkn^ and really arranged on such a
spherical surface. Nevertheless, there would be no
impropriety in his referring their places, geometrically
speaking, to those points of such a purely imaginary
sphere, which their respective visual rays intersect ; and
there would be much advantage in so doing, as by that
means .their appearance and relative situation could be
accurately measured, recorded, and mapped down. The
objects in a landscape are at every variety of distance
from the eye, yet we lay them all down in a picture on
one plane, and at one distance, in their actual apparent
proportions J and the likeness is not taxed with incorrec^r

40 A TREATISB ON ASTRONOMY. [CHAP. !•

aess, diough a man in the foreground should be repre-
eented larger than a mountain in the distance. So it is
to a spectator of the heavenly bodies pictured, pro; cc/crf,
or mapped down on that imaginary sphere we call the
sky or heaven. Thus, we may easily conceive that the
moon, which appears to us as large as the sun, though
less bright, may owe that apparent equality to its greater
proximity, and may be really much less ; while both the
moon and sun may only appear larger and brighter than
the stars, on account of the remoteness of the latter.

(50.) A spectator on the earth's surface is prevented,
by the great mass on which he stands, from seeing into
aU that portion of space which is below him, or to see
which he must look in any degree downwards. It is
true that, if his place of observation be at a great eleva-
tion, the dip of die horizon will bring within the scope
of vlsicm a little more than a hemisphere, and refraction,
wheFsver he may be situated, will enable him to look,
as it were, a little round the comer ; but the zone thus
added to his visual range can hardly ever, unless in very
extraordinary circumstances,* exceed a couple of degrees
in breadth, and is always ill seen on account of Uie va-
pouis near the horizon. Unless, then, by a change of
his geographical situation, he should shift his horizon
(which is always a plane touching the spherical con-
vexity of the earth at his station) ; or unless, by some
movements proper to the heavenly bodies, they should
of themselves come above his horizon ; or, lastly, vltlt
less, by some rotation of the earth itself on its centre,
the point of its surface which he occupies should be
carried round, and presented towards a di0erent region
(^ space ; he would Qever obtain a sight of almost one

*8kich as the Mowing, for iiMtonce: The late Mr. Sadler, the eele*
farated aeronaut, aacended in a balloon from Dublin at about 2 o'clock iiy
the afternoon, and was wafted across the channel. About sunset he ap^
proached the itngiish coast, when the balloon descended near the snrfitce
of the sea. By this time the sun was set, and the shades of evening began
to close in. He threw out nearly all his ballast, and suddenly sprung
upwards to a great height, and by so doing witnessed, the whole pheno-
menon of a western suprise. He subsequentl)r descended in Wales, and
witnessed a secflod sunset on the same evening. I have this anecdote
from Dr. Lardner, who was present at his asi^ent, and read his own ac-
count of the voyage.— AtitAor.

CHAP. 1.3 CHANGE OV LOCAL SITUATION. 41

half the objects external to 6ur atmosphere. But tf any
of these cases be supposed, more, or all, may come into
view according to the circumstances.

(51.) A traveller, for example, shifting his locality on
our globe, will obtain a view of celestial objects invisible
from his original station, in a way which may be not in-
aptly illustrated by comparing hun to a person standing
in a park close to a large iree. The massive obstacle
presented 'by its iamnk cuts off his view of all those parts
ef the landscape which it occupies as an object; but by
walking ronnd it a complete successive view of the
whole panorama may be obtained. Just in Ahe same
way, if we set off from any station, as LoBda«« and
travel southwards, we shall not fail to notice that many
celestial objects which are never seen from London
come successively into view, as if rising up above the
horizon, night after night, from the south» although it is
in reality our horizon, which, Jtorirlling vWi^ ns south-
wards round the sphere, iikiks in succession beneath
them. The novelty and splendotttr of fresh consteUar
tions thus gradually ^brought iate view in the clear calm
nights of tropical climates, in long voyaces to the, south,
is dwelt upon by aU who ha»e enjoyed this spec(&iDl9»

*

and never fails to impress itself on the recollection
among the most deligfaitfid and interesting of the asso-
ciations connected with extensive travel. A glance at
d2

43 A TREATISS ON ASTRONOMY. [cHAP. U

the accompanying figure, exhibiting three successive
stations of a traveller, A, B, C, with the horizon cor-
responding to each, will place this process in dearer
evidence than any description.

(52.) Again : suppose the earth itself to have a mo-
tion of rotatio|i on its centre. It is evident that a spec-
tator at resi (as it appears to him) on any part of it will*
unperceived by himself, be carried round with it : un-
perceived, we say, because his horizon will constantly
contain, and be limited by, the same terrestrial objects »
He will have the same landscape constantly before his
eyes, in which all the lamiliar objects in it, which serve
him for landmarl^ and directions, retain, witib respect
to himself or to each other, tlie same invariable situa-^
tions. The perfect smoothness and equality of the
motion of so vast a mass, in which every object he seea
around him participates alike, will (art. 1 5) prevent his
entertaining any suspicion of his actual change of place*
Yet, with respect to external objects, — that is to say,
all celestial ones which do not participate in the sup-
posed rotation of the earth, — his horizon will have been
all the.while shifting in its relation to them, precisely as
in the case of our traveller in the foregoing article. R&r
eurrin^ to the $gure of that article, it is evidently the
same thing, so far as their visibility is concerned,
whether he has been carried by the earth's rotation suc-
cessively into the situations A, B, O ; or whether, the
earth remaining at rest, he has transferred himself per^
son.aLly along its surface to those stations. Our spectator
in the park will obtain precisely the same view of the
landscape, whether he walk round the tree, or whether
we suppose it sawed off, and made to turn on an upright
pivot, while he stands on a projecting step attached to it,
and allows himself to be carried round by its motion.
The only difference vill be in his view of the tree it-
self, of which, in the former case, he will see every part,
but, in the latter, only that portion of it which remains
constantly opposite to him> and immediately imder his
eye, ■

(53.) By such a irotation of the earth, then, as we
have supposed, the horizon of a stationary spectator will

CHAP. X.} DIURNAL ROTATION 07 THB BASTH. 4$

be constantly depressing itself below those objects wbieh
lie in that region of space towards which the rotation is
carrying him, and elevating itself above those in the op-
posite quarter ; admitting into view the former, and snc*
cessively hiding the latter. As the horizon of every snch
spectator, however, appears to him motionless, idl snch
changes will be referred by him to a motion in the objects
themselves so successively disclosed and concealed. In
place of his horizon approaching the stars, therefore, he
will judge the stars to approach his horizon ; and when it
passes over and hides any of them, he will consider
them as having sunk below it, or »H^ while those it has
just disclosed, and itom which it is receding, will seem
to be rising above it.

(54.) If we suppose this rotation of the earth to con-
iinne in one and &e same direction, — ^that is to say, to be
performed round one and the same axis^ till it has com-
pleted an entire revolution, and come back to the position
firom which it set out when the spectator began his obser-
vations^— it is manifest that every thing w^l then be in
precisely the same relative position as at the outset : all
the heavenly bodies will appear to occupy the same
places in the concave of the sky which they did at that
instant, except such as may have actually moved in the
interim; and if the rotation still continue, the same phe-
nomena of their successive rising and setting, and return
to the same places, will continue to be repeated in the
same order, and (if the velocity of rotation be uniform)
in equal intervals of time, ad infinitum.

(55.) Now, in this we have a lively picture of that
grand phenomenon, ^e most important beyond all com-
parison which nature presents, the daily rising and setting
of the sun and stars, their progress through the vault of
the heavens, and their return to the same apparent places
at the same hours of the day and night. The accom-
plishment of this restoration in the regular interval of
twenty-four hours, is the first instance we encounter of
ihat great law of periodieityf* which, as we shall see,
pervades all astronomy ; by which expression \re under*

* nif <»to(, A going romd, a drcidatSflii or rey^lutkii

44 A IIUSATISE ON ASTRONOMY. [cHAP. I.

Stand the continual reproduction of the same phenomena,
in the same order, at equal intervals of time.

(56.) A free rotation of the earth round its centre, if it
exist and be performed in consonance with the same me-
chanical laws which obtain in the motions of masses of
matter under our immediate control, and within our ordi-
nary experience, must be such as to satisfy two essential
conditions. It must be invariable in its direction toith
respect to the sphere itself ^ and uniform in its velocity.
The rotation must be performed round an axis or diame-
ter of the sphere, whose poles^ or extremities, where it
meets the surface, correspond always to the same points
on the sphere^ Modes of rotation of a solid body under
ithe influence of external agency are conceivable, in which
the pol^s of the imaginary line or axis about which it is
at any moment revolving shall hold no fixed places on the
surface, but shift upon it every moment. Such changes,
however, are inconsistent with the idea of a rotation of
a body of regular figure about its axis of symmetry, per-
formed in free space, and without resistance or obstruc-
. tion from ^y surrounding medium. The complete ab-
sence of such obstructicms draws with it, of necessity,
the strict fulfilment of the two conditions above men-
tioned.

(57.) Now, these conditions are in perfect accordance
with what we observe, and what recorded observation
teaches us in respect of the diurnal motions of the hea-
venly bodies. We have no reason to believe, from his-
tory, that any sensible change has taken place since the
earliest ages in the interval of time elapsing between two
successive returns of the same star to the same point of
the sky ; or, rather, it is demonstrable from astronomical
records that no such change has taken place. And with
respect to the other condition,— ^^6 permanence i^ the
axis of rotation^^-^e appearances which any alteration
in that respect must produce, would be marked, as we
shall presently show, by a corresponding change of a
very obvious kind in the apparent motions of the stars ;
which, again, history decidedly declares them not to have
undergone.

(58.) But, before we proceed to examine more in de-

CBAP. I.J APPARENT DIUHNAL MOTION. 40

tail how the hypothesis of the rotation of the earth ahout
an axis accords with the phenomena which the dinmal
motion of the heavenly bodies offers to our notice, it will
be proper to describe, with precision, in what that diup-
nai motion consists, and how far it is participated in
by them all ; or whether any of them form exceptions,
wholly or partially, to the common analogy of the rest.
We will, therefore, suppose the reader to station himself,
on a clear evening, just after sunset, when the ^rst stars
begin to appear, in some open situation whence a good
general view of the heavens can be obtained. He will
then perceive, above and around him, as it were, a vast
concave hemispherical vault, beset with stars of various
magnitudes, of which the brightest only will first cateh
his attention in the twilight ; and more and more will
appear as the darkness increases, till the whole sky is
overspangled with them. When he has awhile adoured
the calm magnificence of this glorious spectacle, the
4heme of so jnueh song, and of so much thought, — a
spectacle ^hich no one can view without emotion, and
without « longing desire to know something of its na-
ture and purport,— 4et him fix his attention more particu-
larly on a few of the most brilliant stars, such as he can-
not fail to recognise again without mistake after looking
away from them for some time, and let him refer their ap-
parent situations to some surrounding objects, as build-
ings, trees, &c., selecting purposely such as are in dif-
ferent quarters of his horizon. On comparing them again
with their respective points of reference, after a moderate
interval, as the mght advances, he will not fail to per-
ceive that they have changed their places, and advanced,
as by a general movement, in a westward direction;
those towards the eastern quarter appearing to rise or re-
cede from the horizon, while those which lie towards the
west will be seen to approach it; and, if watched long
enough, will, for the most part, finally sink beneath it,
.and disappear ; while others, in the eastern quarter, will
be seen to rise as if out of the earth, and, joining in the
general procession, will take their course with the rest
towards the opposite quarter. ^

(59.) If he persists for a considerable time in watch-

46 A TREATISE ON ASTRONOMY. [cHAP* I.

ing their motions, on the same or on several successive
nights, he will perceive that each star appears to describe,
as far as its course lies above the horizon, a circle in the
sky ; that the circles so described are not of the same
magnitude for all the stars ; and that those described by
different stars differ greatly in respect of the parts of
them which lie above the horizon, some, which lie to-
wards the quarter of the horizon which is denominated
the South,* only remain for a short time above it, and dis-
appear, after describing in sight only the small upper seg-
ment of their diurnal circle ; others, which rise between
the south and east, describe larger segments of their cir-
cles above the horizon, remain proportionally longer in
sight, and ^et precisely as far to the westward of south
as.tjiey rose to the eastward ; while such as rise exactly
in the east remain just twelve hours visible, describe a
9emicircle, and set exactly in the west. With those,
again, which rise between the east and north, the same
law obtains ; at least, as far as regards the time of their
remaining above the horizon, and the proportion of the
visible segment of their diurnal circles to their whole cir-
cumferences. Both go on increasing; they remain in
view more than twelve hours, and their visible diurnal
a^s are more than semicircles. But the magnijudes of
the circles themselves diminish, as we go from the east,
northward ; the greatest of all Ithe circles being described
by those which rise exactly in the east point. Carrying
his eye farther northwards, he will notice, at length, stars
which, in their diurnal motion, just graze the horizon at
its north point, or only dip below it for a moment ; while
others never reach it all, but continue always above it,
revolving in entire circles round one point, called the
POLE, which appears to be the common centre of a]l
their motions, and which alone, in the whole heavens,
may be considered immovable. Not that this point is
marked by any star. It is a purely imaginary centre;
but there is near it one considerably brighjt star, called
the Pole Star, which is easily recognised by the very

* We sup^oM our obienrer to be stationed in wme Qorthem latitude ,*
^pmew^pre in Europe, for example.

CHAP, k.3 AFPAXEMT KVWNAL MOTimr. 47

smaU circle it describes : so smaU, indeed, thily WiAmU
paying particular attention, and referring its poatkm ^eiy
nicely to some fixed mark, it may easily be soppoeed at
rest, and be, itself, mistaken for the common eentre aboat
which all the others in that region describe thor cnrdes ;
or it may be known by its configoratioa with a very
splencUd and remarkable coruieilaHan or group of stars,
called by astronomers the Gsbat Beak.

(60.) He will further obsenre that the apparent id**
live situations of all the stars among one another is nol
changed hy their diurnal motion. £1 whalerer pails of
their circles they are observed, or at whatever hoar of die
night, they form with each other the same identical groepe
or configurations, to which the name of oomsteixatioiw
has been given. It is true, that, in different parts of their
course, these groups stand differently with respect to die
horizon ; and those towards the north, when m the course
of their diurnal movement they pass idtemately above and
below that common centre of motion described in the last
article, become actually inverted with respect to the hori-
zon, while, on the other hand, they always turn the sasM
points towards the pole. In short, he vrill perceive that
the whole assembhige of stars visiUe at once, or in sne-
cession, in the heavens, may be regarded as one great
constellation, which seems to revolve with a nnifoim mo-
tion, as if it formed one coherent mass ; or as if it were at-
tached to the mtemal surface of a vast hdlow sphere,
having the earth, or rather the spectator in the centre, and
turning round an axis inclined to his horizon, so as to pass
through that fixed point or pole already mentioBed«

(61.) Lastly, he will notice, if he have patienee to
outwatch a long wintei^s night, commencing at the earli-
est moment when the stars appear, and continuing till
morning tvrilight, that those stars which he observed set-
ting in the west hare again risen in the east, while those
wMdi were rising when he first began to notice them
have completed thehr coune, and are now set; and that
thus the hemisphere, or a great part of it, which was then
above, is now beneath him, and its place supplied by that
which was at first under his feet, which he will thus disco-
ver to be no less copiously finrnished with stars than the

48 A TftSATlSE ON ASTRONOMY. [CHAF. I*

Other, and bespangled with groups no less 'permanent and
distinctly recognisable. Thus he will learn that the great
constellation we have above spoken of as revolving round
the pole is coextensive with the whole surface of the
sphere, being iu reality nothing less than a universe of
luminaries surrounding the earth on all side^^-and brought
in succession before his view, and referred (each lumina-
ry according to its own visual ray or direction from his
eye) to the imaginary spherical surface, of which he him-
self occupies the centre. (See art 49.)

(62.) There is, however, one portion or segment of
this sphere of which he will not thus obtain a view. As
there is a segment towards the north, adjacent to the pole
above his horizon, in which the stars never set, so there
is a corresponding segment, about which the smaller cir-
cles of the more southern stars are described, in which
they never rise. The stars which border upon the extreme
circumference of this segment just graze the southern point
of his horizon, and show themselves for a few moments
above it, precisely as those near the circumference of the
northern segment graze his northern horizon, and dip for a
moment below it, to reappear immediately. Every point
in a spherical surface has, of course, another diametrically
opposite to it; and as the spectator's horizon divides his
sphere into two hemispheres— a superior and inferior-
there must of necessity exist a depressed pole to the south,
corresponding to the elevated one to the north, and a por^
tion surrounding it, perpetually beneath, as there is an«
other surrounding the north pole^ perpetually above it.

'* Hie vertex nobia semper sublimis ; at ilium
Sub pedibtts noz atra videt, manesque profundi."— Virgil.
Obe pole rfdee hi|^, one, plunged beneath the main,
Seeks the deep mght, and Pluto^a dusky lejgn.

(63.) To get sight of this segment, he must travel south-
wards. In so doing, a new set of phenomena come for-
ward. In proportion as he advances fo the south, some
of those constellations which, at his original station^ barely
grazed the northern horizon, will be observed to sink be-
low it and set ; at first remaining hid only for a very short
time,but gradually for a longer part of the twenty-four hoursr
They will continiie» however, to circulate about the i

CHAP. 1.3 EFFECT OF CHAKOE OF ttATITITBE. 40

point — that is, holding the same invariable position unih
respect to them in the concave of the heavens among ihe
stars ; but this point itself will become gradnally depress-
ed with respect to the spectator's horizon. The axis, in
short, about which the diurnal motion is performed, will
appear to have become continually less and less inclined
to the horizon ; and by the same degrees as the northern
pole is depressed the southern will rise, and oonstellatioiiB
surrounding it will come into view ; at first momentarily,
but by degrees for longer and longer times in each diur-
nal revolution — ^realizing, in short, what we have already
stated in art. 51. *

(64.) If he travel continually southwards, he will at
length reach a line on the earth's surface, callml the eqtiO'
tor J at any point of which, indifferently, if he take up his
station and recommence his observations, he will find that
he has both the. centres of diurnal motion in his horizon,
occupying opposite points, the northern pole having been
depressed, and the southern raised ; so that, in this geo-
graphical position, the diurnal rotation of the heavens
will appear to him to be performed about a horizontal
axis, every star describing half its diurnal circle above and.
half beneath his horizon, remaining alternately visible for
twelve hours, and concealed during the same interval.
In this situation, no part of the heavens is concealed from
his Hiccessive view. In a night of twelve hours (suppo-
sing such a continuance of darkness possible at the equa-
tor) the whole sphere will have passed in review orer
him — the whole hemisphere with which he began his
night's observation will have been carried down beneadi
him, and the entire opposite one brought up from belowr
(65.) If he pass the equator, and travel still iartfaer
southwards, the southern pole of the heavens will become
elevated above his horizon, and* the northern will sink
below it ; and the more, the farther he advances south-"
wards ; and when arrived at a station as far to the south
of the equator as that from which he started was to the
north, he will find the whole phenomena of the heavens
reversed. The stars which at his origin^ station de-
scribed their whole diurnal circles above his horizon, and
never set, now describe them entirdy below it, and never
E

60 A TREATISE ON ASTRONOMY. [cHAP. I.

me, but remain constantly invisible to him ; and vice
versa, those stars which at his former station he never
saw, he will now never cease to see.

(66.) Finally, if instead of advancing southwards from
his first station, he travel northwards, he will observe the
northern pole of the heavens to become more elevated
above his horizon, and the southern more depressed be-
low it. In consequence, his hemisphere will present a
less variety of stars, because a greater proportion of the
whole surface of the heavens remains constantly visible
or constantly invisible : the circle described by each star,
too, becomes more nearly parallel to the horizon; and,
in short; every appearance leads to suppose that could he
travel far enough to the north, he would at length attain
a point verticMy under the northern pole of the heavens,
at which none of the stars would either rise or set, but
each would circulate round the* horizon in circles parallel
to it. Many endeavours have been made to reach this
point, which is called the north pole of the earth, but
hitherto without success ; a barrier of almost insurmount-
able difficulty being presented by the increasing rigour
of the climate : but a very near approach to it has been
made ; and the phenomena of those regions, though not
precisely such as we have described as what must subsist o^
the pole itself, have proved to be in exact correspondence
with its near proximity. A similar remark applies to the
south pole of the eaith, which, however, is more unap-
proachable, or, at least, has been less nearly approached,
than the north.

(67.) The above is an account of the phenomena of
the diurnal motion of the stars, as modified by different
geographical situations, not grounded on any specula-
tion, but actually observed and recorded by travellers
and voyagers. It is, however, in complete accordance
with the hypothesis of a rotation of the earth round a
fixed axis. In order to show this, however, it will be
. necessary to premise a few observations on the appear-
ances presented by an assemblage of remote objects,
when viewed from different parts of a small and circum-
scribed station.

f 68.) Imagine a landscape, in which a great multitude

DISTANCE 07 THE 8TABS.

51

OHAP. I.J

of objects are placed at every variety of distance from the
beholder. If he shift his point of view, though but for
a few paces, he will perceive a very great change in the
apparent positions of the nearer objects* both with re-
spect to himself and to each other. If he advance north-
wards, for instance, near objects on his riglU and left,
which were, therefore, to the east and west of his
original station, will be left behind him, and appear to
have receded southwards; some, which covered each
other at first, will appear to separate, and others to ap-
proach, and perhaps conceal each other. Remote objects,
■on the contrary, will exhibit no such great and remarka-
ble changes of relative position* An object to the east
of his original station, at a mile or two distance, will

still be referred by hii^ to the east point of his horizon,
with hardly any perceptible deviation. The reason of
this is, that the position of every object is referred by us
to the surface of an imaginary sphere of an indefinite ra-
dius, having our eye for its centre; and, as we advance
in any direction, AB, carrying this imaginary sphere
along with us, the visual rays AP, AQ, by which ob-
jects are referred to its surface (at C, for instance), shift
their positions with respect to the line in which we
move, AB, which serves as an a^is or line of reference,
and assume new positions, BPp, BQ^, revolving round
their respective objects as centres. Their intersections,
therefore, p^ q, with our visual sphere, will appear to
recede on its surface, but with different degrees of an-
gular velocity in proportion to their proximity; the
same, distance of advance AB subtending a greater an-
gle, APBscPp, at the near object P than at the remote
one Q.

53 A TBEATISS ON ASTRONOlfT. []gHAF. I.

(69.) This apparent angular motion of an object on
our sphere of vision,* arising from a change of our point
of view, is called parallax, and it is always expressed
by the angle BAP subtended at the object P by a line
joining the two points of view AB under consideration.
For it is evident that the difference of angular position
of P, with respect to the invariable direction ABD,
when viewed from A and from B, is the difference of
the two angles DBP and DAP ; now, DBP being the
exterior angle of the triangle, ABP is equal to the sum
of the interior and opposite, DBPasDAP-jf-APB, whence
DBP— DAP==:APB.

(70.) It follows from this, that the amount of paral-
lactic motion arising from any given change of Our point
of view is, ceteris paribus, less, as the distance of an
object viewed is greater ; and when that distance is ex-
tremely great in comparison with the change in our point
of vieWy the parallax becomes insensible; or, in other
words, objects do not appear to vary in situation at all.
It is on this principle, that in alpine regions visited for
'the first time we are surprised and confounded at the
little progress we appear to make by a considerable
change of place. An hour's walk, for instance, produces
but a small parallactic change in the relative situations
of the vast and distant masses which surround us.
Whether we walk round a circle of a hundred yards in
diamfeter, or merely turn ourselves round in its centre,
the distant panorama presents almost exactly the same as-
pect, — we hardly seem to have changed our point of view.

* The ideal sphere without us, to which we refer the plac66 of ohjects,
and which we carry along wiUi us wherever we eo, is no doubt inti-
mately connected by association, if not entirely dependent oq that ob-
scure perception of sensation in the retinse of our eyes, of which, even
when closed and unexcited, we cannot entirely divest them. We have
a real spherical surface within our eyes, the seat of sensation and vision,
corresponding, point for point, to the external sphere. On this the stars,
&jb. are really mapped down, as we have supposed them in the text to
be, on the imaginary concave of the heavens. When the whole surface
of the ietinsB is excited by light, habit leads us to aasociate it with the
idea of a real surface existing without us. Thus we become impressed
with the notion of a tky and a heaven, but the concave surface of the
retuuB itself is the true seat of all vitibte angular dimension and angular
motion. The substitution of the retina for the heavens would be awkwand
and inconvenient in language, but it may always be mentally made.
(See Schiller's pretty enigma on the eye in tiis Turandot)

DISTANCE OF TUB STASS.

61

OBAP. I.]

(71.) Whatever notion, in other respects^ we may
form of the stars, it is quite clear they most be im-
mensely distant. Were it not so, the apparent angolar
interval between any two of them seen over head would
be much greater than when seen near the horizon, and
the constellations, instead of preserving the same ap-
pearances and dimensions during their whole diornal
course, would appear to enlarge as they rise higher in
the sky, as we see a small cloud in the horizon swell
into a great overshadowing canopy when drifted by the
wind across our zenith, or as may be seen in the annex-
ed figure, where a 6, AB, a 6, are three different positions of

the same stais, as they would, if near the earth, be seen
from a spectator S, under the visual angles OSbf ABB.
No such change of apparent dimension, however, is ob-
8erved« The nicest measurements of the apparent an-
gular distance of any two stars inter se, taken in any
parts of their (Jhumad course, (after allowing for the un-
equal effectB of refraction, or when taken at such times
tluit this cause of distortion shall act equally on both,)
manifest not the slightest perceptible variation. Not
only this, but at whatever point of the earth's surfaoe the
measurement is performed, the results are absohUdy
identical. No instruments ever yet invented by man
are delicate enough to indicate, by an increase or dimi-
nution of the angle subtended, ^t one point of the
earth is nearer to or iurther from the stars than another.
(72.) • The necessary conclusion from this is, that the
dimensions of the earth, large as it is, are comparatively
nothings absolutely imperceptible, when compared with
b2

54 A TREATISE ON ASTROKOMT. []cHAP* I.

the interval which separates the stars from the earth. If
an observer walX round a circle not more than a few
vBrds in diameter, and from different points in its cir-
cumference measure with a sextant, or other more exact
instrument adapted for the purpose, the angles PAQ,
PBQ, PCQ, subtended at those stations by two well
defined points in his visible horizon, PQ, he will at once

be advertised, ^yy the difference of the results, of his
change of distance &om them arising from his change
of place, although that difference may be so small as to
produce no change in their general aspect to his unas-
sisted sight This is one of the innumerable instancea
where accurate measurement obtained by instrumental
means places us in a totally different situation in respect
to matters of fact, and conclusions thence deducible,
from what we should hold, were we to rely in all oases
on the mere judgment of the eye. To so great a nicety
have such observations been carried by 3ie aid of an
instrument called a theodolite, that a circle of the dia-
meter above mentioned may thus be rendered aenaiblet
may thus be detected to have a size^ and an ascertainable
place, by reference to objects distant by fiiUy 100>000
times its own dimensions. Observations, differing, it is
true, somewhat in method, but identical in principlei
and executed with nearly as much exactness, have been
applied to the stars, and with a result such as has been
already stated. Hence it follows, incontrovertibly, that
^e distanee of the stats from the earth cannot be 90

[

CRAF* 1.3 DISTANCE OF THE StARS. 55

small as 100,000 of the earth's diameters. It is, indeed,
incomparably greater ; for we shall hereafter find it fully
demonstrated that the distance just named, immense as it
may appear, is yet much underrated.

(73.) From snch a distance, to a spectator with onr
faculties, and furnished with our instruments, the earth
would be imperceptible ; and, reciprocally, an object of
the earth's size, placed at the distance of the stars, would
be equally undiscemible. If, therefore, at the point on
which a spectator stands, we draw a plane touching the
globe, and prolong it in imagination till it attain the
region of the stars, and through the centre of the earth
conceive another plane parallel to the former, and co-
extensive with it, to pass; these, although separated
throughout their whole extent by the same interval, viz.
a semi-diameter of the earth, will yet, on account of the
vast distance at which that interval is seen, be confound-
ed together, and undistinguishable from each other in the
region of the stars^ when viewed by a spectator on the
earth. The zone they there include -^ill be of evanescent
breadth to his eye, and will only mark out a great circle in
the heavens, which, like the vaniahirig point in perspec-
tive to which all parallel lines in a picture appear to
convene, is, in fad, the vanishing line to which all
planes parallel to the horizon offer a similar appearance
of ultimate convergence in the great panorama of nature.
' ("74.) The two planes just described are termed, in
astronomy, the sensible and rational horizon of the ob-
server's station ; and the great circle in theheaveps which
marks their vanishing line, is also spoken of as a circle
of the sphere, under the name of the celestial horizon^
or simply the horizon.

From what has been said (art. 72) of the distance
of the stars, it follows, that if we suppose a spectator
at the centre of the earth to have his vi^w bounded by
^e ratiorwl horizon, in the same manner as that of a
corresponding spectator on the surface is by his sensible
horizon, the two observers will see the same stars in the
same relative situations, each beholding that entire he-
misphere of the heavens which is above the celestial
korizoB, eoirespondiag to their common zenith.

M A TREATISE ON ASTRONOMY. [cHAP. I.

(75.) Now, so far as appearances go, it is clearly the
same thing whether the heavens, that is, all space, with
its contents, revolve round a spectator attest in the earth's
centre, or whether that spectator simply turn round in the
opposite direction in his place, and view them in suc-
cession. The aspect of the heavens, at every instant, as
referred to his hofrizon (which must be supposed to turn
with him), will be the same in both suppositions. And
since, as has been shown, appearances are also, so far as
the stars are concerned^ th^ same to a spectator on the sur-
face as to one at the centre, it follows that, whether we sup-
pose the heavens to revolve without the earth, or the earth
within the heavens, in the opposite direction^ the diurnal
phenomena, to all its inhabitants, will be no way different.

(76.) The Copemican astronomy adopts the latter as
the true explanation of these phenomena, avoiding there-
by the necessity of otherwise resorting tq the cumbrous
mechanism of a solid but invisible sphere, to which the
stars must be supposed attached, in order that they may
be carried round die earth without derangement of their
relative situations inter se. Such a oontrivance would,
indeed, suffice to explain the diurnal revolution of the
stars, so as to ** save appearances ;' ■ but the movements of
the sun and moon, as well as those of the planets, are in-
compatible with suclti a supposition, as wUl appear when
we come to treat of these bodies. On the other hand, that
a spherical mass of. moderate dimensions (or, rather,
when compared widi the surrounding and visible universe,
of evanescent magnitude), held by no tie, and free to move
and to revolve, :Should do so, in conformity with thode
general laws which, so far as we know, regulate the mo-
tions of all material bodies, is «o far from being a postu-
late difficult to be conceded, that the wonder would rather
be should the fact prove otherwise. As a postulate, there-
fore, we shall henceforth regard it; and as, in the pro-
gress of our work, analogies offier themselves in its sup-
port from what we observe of other celestial bodies, we
shall not fail to point them out to the reader's notice.
Meanwhile, it will be proper to define a variety of terms
which will be continually employed hereafter.

(77.) Definition 1. The axis of the earth is that di-

CHAP. !•]] DEFINITIONS. 67

ameter about which it revolves, with a uniform inotioii»
frora west to east ; performing one revolution in the in-
terval which elapses between any star leaving a certain
point in the heavens, and returning to the same point
again.

(78.) Def. 2. The poles of the earth are the points
where its axis meets its surface. The North Pole is that
nearest to Europe ; the South Pole that most remote from it.

(79.) Dbf. 3. The sphere of the heavens, or the sphere
of the stars, is an imaginary spherical surface of infinite
radius, and having the centre of the earth, or, which
comes to the very same thing, the eye of any spectator
on its surface, for its centre. Every point in this sphere
may be regarded as the vanishing point of a system of
lines parallel to that radius of the sphere which passes
through it, seen in. perspective from the earth ; and any
great circle on it, as the vanishing line of a system of
planes parallel to its own. This mode of conceiving such
points and circles has great advantages in a variety of cases.

(80.) Def. 4. The zenith and nadir* are thfe two points
of the sphere of the heavens, vertically over the specta-
tor's head, and vertically under his feet ; they are, there-
fore, the vanishing points of all lines mathematically par
xallel to the direction of a plumb-line at his station. The
plumb-line itself is, at every point of the earth, perpen-
dicular to its spherical surface : at no two stations, there-
fore, can the actual directions of two plumb-lines be re-
garded as mathematically parallel. They converge to-
wards the centre of the earth : but for very small intervals
(as in the area of a building — in one and the same town,
&c.) the difference from exact parallelism is so small, that
it may be practically disregarded. An interval of a mile
corresponds to a convergence of plumb-lines amounting
to about 1 minute. The zenith and nadir are the poles
of the celestial horizon ; that is to say, points 90^ distant
ftom every point in it. The celestial horizon itself is
the vanishing line of a system of planes parallel to the
sensible and rational horizon.

*From AralHC words. Nadir oomspo&dt eyidenUy to the Geimaii
(down)

58 A TREATISE ON ASTRONOMY. [cHAP. I.

(81.) Def. 5. Vertical circles of the sphere are great
circles passing through the zenith and nadir, or great cir-
cles perpendicular to the horizon. On these are mea-
sured the altitudes of objects above the horizon — the
complements to which are their zenith distances.

(82.) Def. 6. The poles of the heavens are the points
of the sphere to which the earth's axis is directed ; or
the vanishing points of all lines parallel thereto.

(83.) Def. 7. The earth^s equator is a great circle on
its surface, equidistant from its poles, dividing it into
two hemispheres — a northern and a southern; in the
midst of which are situated the respective poles of the
earth of those names. The plane of the equator is,
therefore, a plane perpendicular to the earth's axis, and
passing through its centre. The celestial equator is a
great circle of the heavens, marked out by the indefinite
extension of the plane of the terrestrial, and is the vanish-
ing line of all planes parallel to it. This circle is called
by astronomers the equinoctial.

(84.) Def. 8. The terrestrial meridian of a station
on the earth's surface is a great circle passing through
both the poles and through the place. When its plane
is prolonged to the sphere of the heavens, it marks out
the ce/e«^ta/men(/ian of a spectator stationed at that place.
When we speak of the meridian of a spectator, we intend
the celestial meridian, which is a vertii?al circle passing
through the poles of the heavens.

The plane of the meridian is the plane of this circle,
and its intersection with the sensible horizon of the spec-
tator is called a meridian line, and marks the north and
south points of hli horizon.

(85.) Def. 9. Azimuth is the angular distance of a
celestial object from the north or south point of the hori-
zon (according as it is the north or south pole which is
elevated), when the object is referred to the horizon by
a vertical circle ; or it is the angle comprised between
two vertical planes— -one passing through the elevated
pole, the other through the object. The altitude and
azimuth of an object being known, therefore its place in
the visible heavens is determined. For their simultane-
ous measurement, a peculiar instrument has been ima-

CHAP. I«3 LATITUDE AND LONenUDB. &•

gined, called an altitude and azimuth instrnment, wbicli
win be described in the next chapter.

(86.) Def. 10. The latitude of a place on the earth's
surface is its angular distance from the equator, measured
on its own terrestrial meridian : it is reckoned in degiees,
minutes, and seconds, from up to 90®, and northwards
or southwards according to the hemisphere the place lies
in. Thus, the observatory at Greenwich is situated in
bl"" 28' 40" north latitude. This definition of latitude, it
will be observed, is to be considered as only temporary.
A more exact knowledge of the physical structure and
figure of the earth, and a better acquaintance with the
niceties of astronomy, will render some modification of its
terms, or a different manner of considering it, necessary.

(87.) Def. 11. Parallels of latitude are small circles
on the earth's surface parallel *to the equator. Every
point in such a circle has the same latitude. Thus, Green-
wich is said to be situated in the parallel rfbl^2S' 40".

(88.) Def. 12. The ^gi^ude of a place on the earth's
surface is the inclination of its meridian to that of some
fixed station referred to as a point to reckon from. Gng-
lish astronomers and geographers use the observatory at
Greenwich for this station ; foreigners, the principal ob-
servatories of their respective nations. Some geographers
have adopted the island of Ferro. Hereafler, when we
speak of longitude, we reckon from Greenwich. The
longitude of a place is, therefore, measured by the are of
the equator intercepted between the meridian of a place
and that of Greenwich ; or, which is the same thing, by
the spherical angle at the pole included between these
meridians.

As latitude is reckoned north or south, so longitude
is usually said to be reckoned west or east. It would
add greatly, however, to systematic regularity, and tend
much to avoid confusion and ambiguity in computations,
were this mode of expression abandoned, and longitudes
reckoned invariably westward from their origin round
the whole circle from to 360''. Thus the longitude
of Paris is, in common parlance, either 2® 20' 22" east,
or 357"" 39' 38" west of Greenwich. But, in the sense
on which we shall henceforth use and recommend others

60 A TREATISE ON ASTRONOMY. [CHAF. I*

to use the term, the latter is its proper designation.
Longitude is also reckoned in time at the rate of 24 h.
for 360°, or 15® per hour. In this system the longitude
of Paris is 23h. 50m. 38^s.

(89.) Knowing the longitude and latitude of a place,
it may be laid down on an artificial globe ; and thus a
map of the earth may be constructed. Maps of particu-
lar countries are detached portions of this general map,
extended into planes ; or, rather, they are representations
on planes of such portions, executed according to certain
conventional systems of rules, called projections, the
object of which is either to distort as little as possible
the outlines of countries from what they are on the globe
—or to establish easy means of ascertaining, by inspec-
tion or graphical measurement, the latitudes and lo];^gi-
tudes of places which occur in them, without refernng
to the globe or to books — or for other peculiar uses. See
chap. III.

(90.) A globe, or general map of the heavens, as well
as charts of particular parts, may also be constructed,
and the stars laid down in their proper situations rela*
tive to each other, and to the poles of the heavens and
the celestial equator. Such a representation, once made,
will exhibit a true appearance of the stars as they pre-
sent themselves in succession to every spectator on the
surface, or as ,they tmay be conceived to be seen at once
by one at the centre of the globe. \i is, therefore, in-
dependent of all geographical localities. There will
occur in such a representation neither zenith, nadir, nor
horizon — neither east nor west points; and although
great circles may be drawn on jt from pole to pole, cor-
responding to terrestrial meridians, they can no longer,
in this point of view, be regarded as the celestial meri-
dians of fixed points on the earth's surface, since, in
the course of one diurnal revolution, every point in it
passes beneath each of them. It is on account of this
change of conception, and with a view to establish a
complete distinction between the two branches of Oeo-
graphy and Uranography,* that astronomers have
adopted different terms (viz. declination, and right
• ^^, the earth; y«»?»»», to describe or repreient: »«««»•*, the heavaiM.

COUP. I.J LATIIUDE AND LONOITUDK. 61

ofcension) to represent thosp arcs in the heavens which
correspond to latitudes and longitudes on the earth. It
is for this reason that they tenn the equator of the hea-
vens the equinoctial ; that what are meridians on the
earth are called hour circles in the heavens, and the
angles they include between them at the poles are called
hour angles, AU this is convenient and intelligible;
and had they been content with this nomenclature, no
confusion could ever have arisen. Unluckily, the early
astronomers have employed also the words latitude and
longitude in their uranography, in speaking of arcs of
eireles not corresponding to those meant by the same
words on the earth, but having reference to the motion
of the sun and planets among the stars. It is now too
late to remedy this confusion, which is ingrafted into
every existing work on astronomy : we can only regret,
and warn the reader of it, that he may be on his guard
when, at a more, advanced period of our work, we
shall have occasion to define and use the terms in their
celestial sense^ at the same time urgently recommending
to future writers the adoption of others in their places.

(91.) As terrestrial longitudes reckon from an assumed
fixed meridian, or from a determinate point on the equa*
tor ; so right ascensions in the heavens require some
determinate hour circle, or some known point in the
equinoctial, as the commencement of their reckoning, or
their zero point. The hour circle passing through some
remarkably bright star might have been chosen ; but there
would have been no particular advantage in this ; and
astronomers have adopted, in preierence, a point in the
equinoctial called the eqtdnoXj through which they sup-
pose the hour circle to pass, from which all others are
reckoned, and which point is itself the zero point of all
right ascensions, counted on the equinoctial.

The right ascensions of celestial objects are always
reckoned eastward from the equinox, and are estimated
either in degrees, minutes, and seconds, as in the case
of terrestrial longitudes, from 0° to 360^, which com-
pletes the circle ; or, in time, in hours, minutes, and
seconds, from h. to 24 h. The appattnt diurnal motiim
of the heavens being contrary to ^e real motion of the
F

02 A TREATISE ON ASTRONOMY. [CHAP. I.

earth, this is in conformity with the westward reckon-
ing of longitudes. (Art. 87.)

(92.) Sidereal time is reckoned by the diurnal motion
of the stars, or rather of that point in the equinoctial
from which right ascensions are reckoned. This point
may be considered as a star, though no star is, in fact,
there ; and, moreover, the point itself is liable to a cer-
tain slow variation, — so slow, however, as not to affect,
perceptibly, the interval of any two of its successive
returns to the meridian. This interval is called a side-
real day, and is divided into 24 sidereal hours, and these
again into minutes and seconds. A clock which marks
sidereal time, i. e. which goes uniformly at such a rate
as always to show h. Om. Os. when the equinox comes
on the meridian, is called a sidereal clock, and is an in-
dispensable piece of furniture in every observatory.

(93.) It remains to illustrate these descriptions by
reference to a figure. Let C be the centre of the earth.

NCS its axis ; then are N and S itspole$s EQ its equtp-
tor; AB the parallel of latitude of the station A on its
surface ; AP parallel to SON, the direction in whieb an
observer at A will see the elevated pole of the heavens ;
and AZ, the prolongation of the terrestrial radius CA,

CHAP. lO DEFINITIONS EXEMFUFIBD. 68

that of his zenith. NAES will be his meridian ; NOS
that of some dxed station, as -Greenwich ; and GE, or
the spherical angie GNE, his longitude, and EA his lar
titade. Moreover, if fi« be a plane touching the surface
in A, this wiH be his sensible horizon ; nk$ marked on
that plane by its intersection with his meridian will be
his meiidian line, and n and $ the north and soulh points
of his horizon.

(94.) Again, neglecting the size of the earth, or con-
ceiving him stationed at its centre, and referring jevery
thing to his rational horizon; let the annexed figure
represent the sphere of the heavens ; C the spectator ;
Z his zenith ; and N his nadir ; then will HAO a great
circle of the sphere, whose poles are ZN, be his eeU$*
Hal horizon ; Vp this devaftd and depressed poles .of

B z

the heavens ; HP the altitude of the pole, and HPZEO
his meridian; ETQ, a great circle perpendicular to P/i,
will be the equinoctial ; and if T represent the equinox,
*y» T will be ihe right ascension^ TS the declination, and
PS the polar distance of any star or objec;t S, referred to
the equinoctial by the hour circle VSTp; and BSD will
be the diurnal circle it will appear to describe about the
pole. Again* if we refer it to the horizon by the vertical
circle ZSA, HA will be its j^zimuth, AS its altitude, -and
ZS its zenith distance^ H and O are the north and
south, and ew the east and west points of his horizon,

64 A, TKIATISE ON A8TR0N0MT. [cHAP. I.

or of the heavens. Moreoyer, if Hh, Oo, be small cir-
cles, Of parallels of declination, touching the horizon in
its north and south points, Kh will be the circle of per-
petual apparition^ between which and the elevated pole
the stars nepver set ; Oo that of perpetual occultation^
between which and the depressed pole they never rise.
In all the zone of the heavens between HA and Oo,
they rise and set, any one of them, as S, remaining above
the horizon, in that part of its diurnal circle represented
by ABA, and below it throughout all the part represented
by AD a. It will exercise the reader to construct this
figure for several different elevations of the pole, and for
a variety of positions of the star S in each. The fol-
lowing consequences result from these definitions, and
are propositions which the reader will readily bear in
mind : —

(95.) The altitude of the elevated pole is equal to the
latitude of the spectator's geographical station. . For,
comparing the figures of arts. 93 and 94, it appears that
the angle PAZ, between the pole and zenith, in the one
figure, which is the co-altitude (complement to 90° of the
altitude) of the pole, is equal to the angle NCA in the
other ; CN and AP being parallels whose vanishing point
is the pole. Now, NCA is the co-latitude of the plane A.

(96.) The same stars, in their diurnal revolution, come
to the meridian, successively, of every place on^the globe
once in twenty-four sidereal hours. And, since the di-
urnal rotation is uniform, the interval, in sidereal time,
which elapses between the same star coming upon the
meridians of two difierent places is measured by the dif-
ference of longitudes of the places.

(97.) Vice versa — the interval elapsing between two
different stars coming on the meridian of one and the
same place, expressed in sidereal time, is tfie measure of
the difference of right ascensions of the stars.

This explains the reason of the double division of the
equator and equinoctial into degrees and hours.

(9B.) The equinoctial intersects the horizon in the east
and west points, and the meridian in a point whose alti-
tude is equal to the co-latitude of the place. Thus, at

CHAP. I.] 8TAS9 VI9IBLB BT DAT. W

Greenwich, the altitude of the intenectioii of the equi-
noctial and meridian is 38'' 31' 20".

(99.) All the heavenly bodies culminate (t. e. come to
their greatest altitudes) on the meridian ; which is, there-
fore, Qie best situation to observe them, being least oon-
lused by the inequalities and vapowrs of the atmosphere,
as well as least displaced hj refraction.

(100.) All celestial objects within the cirde of perpe-
tual apparition come twice on the meridian, above the hori-
zon, in every diurnal revolution ; once abiwe and once
bdow the pole. These are called their tqtper and hufer
eulminatiaru.

(101.) We shall conclude this chapter by calling the
reaider's attention to a fact, which, if he now learn for the
first time, will not fail to surprise him, viz. that the stars
continue visible through telescopes during the day as WjcU
as the night ; and that, in proportion to Sie power of the
instrument, not only the largest and brightest of them,
but even those of inferior lustre, such as scarcely strike
the eye at night as at all conspicuous, are readily found
and followed even at &ooiiday,--^4}nle8s in that part of the
sky which is very near the 8ua,-^by those who possess the
means of polluting a telescope accurately to the proper
places. Indeed, from the bottonts of deep narrow pits, such
as a well, or the shaft of a mine, such bright stars as pass
the zenith may even be discerned by the naked eye ; and
we have otfrselvQs heard it stated by a celebrated optician,
that the earliest circumstan^ which drew his attention
to astronomy was the regular appearance, at a certain
hour, for several successive days, of a considerable star,
through the shaft of a chimney.

f2

66 A TREATISE ON ASTRONOMY. [CBAP. li

CHAPTER II.

Of the Nature of astronomical iDStramentB and ObservationB in general —
'0£ sitlereal and Bolar Time — Of the Measurement of ^me^-CIocks,
Chronometers, the Transit Instrument^-Of the Measurement of angular
Intervals — AppUcationof the Telescope to Instruments destmed to that
Purpose — Of the Mural Circle — ^Fixation of polar and horizontal points
— ^The Level— Plumb-line — ^Artificial Horizon — Collimator— Of com*
pound Instruments with co-ordinate Circles, the Equatorial — ^Altitude
and Azimuth Instrument— Of the Sextant and leflectmg Gttde— Princi-
ple of Repetition.

(102.) Our first chapter has heen devoted to the
acquisition i&hiefly of preliminary notions respecting the
globe we inhabit, its relation to the celestial objects which
surround it, and the physical circumstances under which
all astronomical obsenrations must be made, as well as to
provide ourselves with a stock of technical words of most
frequent and familiar use in the sequel. . We might now
proceed to a more exact and detailed statement oif the
racts and theories of astronomy ; but in order to do this
with full effect, it will be desirable that the reader be
made acquainted with the principal means which astrono-
mers possess, of determining, with the degree of nicety
their theories require, the data on which they ground their
Cfmclusions ; in other words, of ascertaining by measure-
ment the apparent and real magnitudes with which they
aare conversant. It is only when in possession of this
knowledge that he can fully appreciate either the truth of
the theories themselves, or the degree of reliance to be
placed on any of their conclusions antecedent ''to trial ;
since it is only by knowing what amount of erro?r can
certainly be perceived and distinctly measured, that he
call satisfy himself whether any theory offers so close an
approximation, in its numerical results, to actual phe-
nomena, as will justify him in receiving it as a true repre*
mentation, of nature.

(103.) Astronomical instrument-making may be justly
regard^ as the most refined of the mechanical arts, and
that in which the nearest approach to geometrical preci-
sion is required, and has been attained* It may be thought

CSAP. n*] PRACTICAL DIlPFICUlTUBS. 67

an easy thing, by one unacquainted with the niceties re-
quired, to turn a circle in metal, to divide its circumfe-
rence into 360 equal parts, and these again into smaller sub-
divisions, — ^to place it accurately on Its centre, land to ad-
just it in a given position ; but practically it is found to be
one of the most difficult. Nor will this appear extraordina-
ry, when it is considered that, owing to liie application of
telesd^pes to the purposes of angular measurement, every
imperfection of structure or division becomes magnified
by the whole optical power of that instrument ; and that
thus, not only direct errors of workmaniship, arising from
unsteadiness of hand or imperfection of tools, but those
inaccuracies which originate in far more uncontrollable
causes, such as the unequal expansion and contraction of
metallic masses, by a change of temperature, and their
unavoidable flexure or bending by their own weight, be»
come perceptible and measurable. An angle of oa» mjr
nute occupies, on the circumference of a cirde of 10
inehes in radius, only about Tf^fth part of an inch, a quan-
tity iAo small to be certcdmy dealt with without the use
of magnifying glasses ; yet one minute is a gross quan-
tity in the astronomical measurement of an angle. With
the instruments nbw emplo^d in observatories, a single
secuftd, or the 00th part of a minute, is rendered a <JUs-
tinctly visible and appreciable quantity. Now, the arc
of a circle, subtended by one second, is leas than the
200,000th part of the radius, so that on a circle of 6 feet
in diameter it would occupy no greater linear extent than
y^th part of an inch ; a quantity requiring a powerful
microscope to be discerned at all. , Let any one flgui^ jlo
himself, therefore, the difficulty of placing on the circum-
ference of a metallic circle of such dimensions (supposing
the difficulty of its construction surmounted) 360 marks«
dots, or cognizable divisions, which shall be true to their
places within such minute limits ; to say nothing of the
siibdivision of the degrees so marked off into minutes, and
of 4iese again into seconds. Such^a work has probably
baffled, and will probably for ever contisue to baffle, the
utmost stretch of human sjkffl and industry ; nor, if exe-
cuted, could it endure. The ever varying fluctuations of
heal and cold have a t^dency to produce notmerely tem*'

68 A TREATISE ON ASTRONOMV. [CHAF. II,

porary and transient, but permanent, uncompensated
changes of form in all considerable masses of those metals
whi(m alone are applicable to such uses ; and their own
weight, however symmetrically formed, must always be
unequally sustained, since it is impossible to apply the
sustaining power to every part separately ; even could
this be done, at all events force must be used to move and
to fix them ; which can never be done without producing
temporary and risking permanent change of form. It is
true, by dividing them on their centres^ and in the identi-
cal places they are destined to occupy, and by a tfiousand
ingenious and delicate contrivances, wonders have been
accomplished in this department of art, and a degree of
perfection has been given, not merely to chefs d'ceuore^
but to instruments of moderate prices and dimensions, and
in ordinary use, which, on due consideration, must ap-
pear very surprising. But though we are entitled to look
for wonders at the hands of scientific artists, we are not
to expect miracles. The demands of the astronomer
will adways surpass the power of the artist ; and it must,
therefore, be constantly the aim of the former to make
himself, as far as possible, independent of the imperfec-
tions incident to every work the latter can place in his
bands. He must, therefore,* endeavour so to combine his
observations, so to choose his opportunities, and so to
fandUarize him«(elf with all the causes which may pro-
duce instrumental derangement, and with all the pecu-
liarities of structure and material of esLch instrument he
possesses, ajs not to. alloir himself to be misled by their
errors, but to extract from their indications, as far as possi-
ble, all that is trucj and reject all that 19 erroneous. It
is in this that the art of the practical astronomer consists,
—an art of itself of a curious and intricate nature, and of
which we can here only notice some of the leading and
general features.

(104.) The great aim of the practical astronomer be-
ing numerical correctness in the results of instrumental
meajsurement, his constant care and vigilance must be
directed to the detection and compensation of errors,
either by annihilating, or by taking account of, and al«
lowing for them. Now, if we examine the sources, from

CaXAP. II.3 CLASSIFICATION OF SOURCES OF BSSOR. 00

which errors may arise in any instrumental determina-
tion, we shall find them chiefly reducible to three prin-
cipal heads : —

(105.) 1st, External or incidental causes of error;
comprehending such as depend on external, uncontrol-
lable circumstances : such as, fluctuations of weather,
which disturb the amount of refraction from its tabu-
lated value, and, being reducible to no fixed law, induce
uncertainty to the extent of their own possible magni-
tude ; such as, by varying the temperature of the air,
vary also the form and position of the instruments used,
by altering relative magnitude and the tension of diear
parts ; and others of the like nature.

(106.) 2dly, Errorg of observation : such as tunse, for
example, from inexpertness, defective vision, slownesf
in seizing the exact instant of occurrence of a pheno^
menon, or precipitancy in anticipating it, 'dsc. ; from at-
mospheric indistinctness; insufficient optical power in
the instrument, and the like. Under this head may also
be classed all errors arising from momentary instrumental
derangement, — ^slips in clamping, looseness of screws, iit.,

(107.) 3dly, The third, and by far the most numerous
class of errors to which astronomical measurements are
liable, arise from causes which may be deemed instru-
mental, and which may be subdivided into two principal
classes. The^ir^^ comprehends those which arise from
an instrument not being what it professes to be, which
is error of workmanship. Thus,' if a pivot or axis, in-
stead of being, as it ought, exact^ylindrical, be slightly
flattened, or elliptical, — if it be sot exactly (as it is in-
tended it should) concentric with the circle it carries ;— -
if this circle (so called) be in reality not exactly circular,
or not in one plane ;— -if its divisions, intended to be
precisely equidistant, should be placed in reality at un-
equal intervals, — ^and a hundred other things of the same
sort. These are not mere speculative sources of error,
but practical annoyances, which every observer has to
contend with.

(108.) The other subdivision of instrumental errors
comprehends such as arise from an instrument not being
placed in the position it ought to have ; and from those

70 A TREATISK ON ASTRONOMY. [CHAF. II.

of its parts, which ^re made purposely inoveahle, not
being properly disposed inter se. These are errors of
adjustment. Some are unavoidable, as they arise from
a general unsteadiness of the soil or building in which
the instruments are placed ; which, though too minute
to be noticed in any oUier way, become appreciable in
delicate astronomical observations: others, again^ are
consequences of imperfect workmanship, as where an
instrument once weU adjusted will not remain so, but
keeps deviating and shifting. But the most important
of Uiis class of errors arise from the non-e;Kistence of
natural indications, other than those afforded by astrono*
mical observations themselves^ whether an instrument
has or has not the ex^acjt position, with respect to the
horizon and its cardinal points, the axis of the ^arth^ or
to other principal astronomical lines and circles, which
it ought to have to fulfil properly its objects^

(109.) Now, with respect to the first two classes of
error, it must be observed* that, in so far as they cannot
be reduced to known laws, and thereby become subjects
of calculation and due allowance, Uiey actually vitiate, to
theii^full extent, the results of any observations in which
they subsist. Being, however^ in their nature casual
and accidental, their effecits necessarily lie sometimes
one way, sometimes the other ; sometime^ diminishing,
sometimes tending to increase the results^ Hen£e, by
greatly multiplying observations, under varied circum-
stances, and taking the mean or average of their results
this class of errors may be so far subdued, by setting
them to destroy one j»nother, as no longer sensibly to
vitiate any theoretical or practical coneiusion. This is
the great and indeed only resource against.such errors not
merely to the astronomer, but to the investigator of nu-
merical results in every department of physical research.

(110.) With regard to errors of adjustment €md work-
manship, not only the possibility, but the certainty, of
their existence, in every imaginable form, in all instru-
ments, must be contemplated. Human hands or ma-
chines never formed a circle, drew a straight line, or
erected a perpendicular, nor ever placed an instrument
m perfect adjustment, unless accidentally $ and then only

CHAP, n.]] MUTUAL DESTRUCTION OF ERRORS. 71

during an instant of time. This does not ptevent, how*
ever, that a great approximation to all these desiderata
should he attained. But it is the peculiarity of astronO'
mical observation to be the ultimate means of detection
of all mechanical defects which elude by their minute-
ness every other mode of detection* What the eye can-
not discern, nor the touch perceive, a coumse of astrono-
mical observations wiU make distinctly evident The
imperfect products of man^9 hands are here tested by
being brought into comparison with the perfect work-
manship of nature ; and there is none which will bear
the trial. Now, it may seem like arguing in a vicious
circle, to deduce theoretics^ conclusions ami laws from
observation, and then to turn round upon the instruments
with which those observations were made, accuse them
of imperfection, and attempt to detect and rectify their
errors by means of the very laws and theories which
they have helped us to a knowledge of. A little consi-
deration, however, will suffice to show that such a course
of proceeding is perfectly legitimate.

(111.) The steps by which we arrive at the laws of
natural phenomena, and especially those which depend
for their verification on numerical determinations, are
necessarily successive. Gross results and palpable law9
are arrived at by rude observation with coarse instru-
ments, or without any instruments at all ; and these are
corrected and refined upon by nicer scrutiny with more
delicate means. In the progress of this, subordinate
laws are brought into view, which modify both the verbal
statement and numerical results of those which first of-
fered themselves to our notice ; and when these are traced
out, and reduced to certainty, others, again, subordinate
to Ihem, make their appearand;^, and become subjects of
further inquiry. Now, it invariably happens (and the
reason is evident) that tiie first glimpse we catch of such
subordinate laws^the first form in which they arc
dimly shadowed out to our minds — is that of errors.
We perceive a discordance betv^een what we expect
and what we find. The first occurrence of such a dis-
cordance we attribute to accident. It happens again and
again ; and we begin to suspect our instruments. We

It JL TREATISE ON ASTRONOHtT. [CHAP. II*

tiien inquire, to what amount of error their determina-
tiona can, bt/ possibility, be liable. If their limit of pos-
sible error excee^ the observed deviation, we at once
eondemn the instrument, and set about improving its
construction or adjustments. Still the same deviations
occur, and, so far from being palliated, are more marked
and better dei&ned than before. We are now sure that
we are on the traces? of a law of nature, and we pursue
it till we have reduc^ it to a definite statement, and
verified it by repeated observation, under every variety
of circumstances.

(112.) Now, in the eourse of this inquiry, it will
not fail to happen that other discordances will strike us..
Taught by experience, we suspect the existence of some
Batural law, before unknown ; we tabulate (i. e. draw out
in order) the results of our observations ; and we per*
eeive, in this synoptic statement of them, distinct indi-
cations of a r^ular progression. Again we improve or
vary our instruments, and we now lose sight of this sup*
posed new law of nature altogether, or find it replaced
By some other, of a totally different character. Thus
we are led to suspect an instrumental cause for what
we have noticed* We examine, therefore, the theonf
of our instrument; we suppose delects in its struc*
tuie, and, by the aid of geometry, we trace their in*
flttsnce in introducing actual errors into its indications.
These errors have their lawMv whic)), so long as we
have no knowledg^^ of cautses to guide us, may be con*
founded with laws of nature, and are mixed up with
them in &eir effects. They are not fortuitous, like
errors of observation, but, as they arise from sources
inherent in tiite mstorument, and unchangeable while it
and its adjustments remain unchanged, they are reduci«
ble to .fixed and ascertainable forms ; each particular
defect, whether of structure or adjustment, producing its
own appropriate form of error. When these are tho-
roughfy investigated, we recognise among them one
which coincides in its nature and progression with that
of our observed discordances. The mystery is at once
solved : we have, detected^ by direct observation, an in-
strumental defect.

CHAP. II.3 DETECTION OF INSTRUMENTAL ERRORfl. 78

(11'3.) It is, therefore, a chief requisite for the practi-
cal astronomer to make himself completely familiar with
the theory of his instruments, so as to be able at once to
decide what effect on his observations any given imperfec-
tion of structure or adjustment will produce in any given
circumstances under which an observation can be made.
Suppose, for example, that the principle of an instrument
required that a circle should be exactly concentric with
the axis on which it is made to turn. As this is a condi-
tion which no workmanship can fulfil, it becomes neces-
sary to inquire what errors will be produced in observa-
tions made and registered on the faith of such an instm-
ment, by any assigned deviation in this respect ; that is
to say, what would be the disagreement between obser-
vations made with it and with one absolutely perfect,
could such be obtained. Now, a simple theorem in geo-
metry shows that, whatever be the extent of this d^ia*
tion, it may be annihilated in its effect on the result of
observations depending on the graduation of the limb«
by the very easy method of reading off the divisions on
two diametrically opposite points of the circle, and tak-^
ing a mean ; for the effect of eccentricity is always to
increase one such reading by just the same quantity by
which it diminishes the other. Again, suppose that the
proper use of the instrument required that this axis should
be exacfly parallel to that of the earth. As it never can
be placed or remain so, it becomes a question, what
amount of error will arise in its use ftoni any assigned
deviation, whether in a horizontal or vertical plane, from
this precise position. Such inquiries constitute the theory
of instrumental errors ; a theory of the uUnost import*
ance to practice, and one of which a complete knowledge
will enable an observer, with very moderate instnimenw
means, to attain a degree of precision which might seeni
to belong only to the most refined and costly. In the
present work, however, we have no further concern witli
it. The few astronomical instraments we propose to de^
scribe in this chapter will be considered as perfect both in
constraction and adjustment.

(114.) As the above remarks are very essential to a
right understanding of the philosophy of our subject and
G

74 A TREATISE ON A8TllONOHr. [cHAP. II.

the spirit of astronomical methods, we shall elucidate
them by taking a case. Observant persons, before the
invention of astronomical instruments, had already con-
cluded the apparent diurnal motions of the stars to be
performed in circles about fixed poles in the heavens, as
shown in the foregoing chapter. In drawing this con-
clusion, however, refraction was entirely overlooked, or,
if fofced on their notice by its great magnitude in the
immediate neighbourhood of the horizon, was regarded
as a local irregularity, and, as such, neglected or slurred
over. As soon, however, as the diurnal paths of the stars
were attempted to be traced by instruments, even of the
coarsest kind, it became evident that the notion of exact
circles described about one and the same pole would not
represent the phenomena correctly, but that, owing to
some cause or others the apparent diurnal orbit of every
star is distorted from a circular into an oval form, its
lower segment being ^flatter tlian its upper ; and the de-
viation being greater the nearer the star approached the
horizon, the effect being the same as if the circle had
been 8(|aeezed upwards from below, and the lower parts
more than the higher. For such an effect, as it was soon
found to arise from no casual or instrumental cause, it
became necessary to seek a natural one ; and refraction
readily occurred to solve the difHculty. In fact, it is a
case precisely analagous to what we have already (art.
47) noticed, of the apparent distortion of the sun near
the horizon, only on a larger scale, and traced up to greater
altitudes. This new law once established, it became ne-
cessary to modify the expression of that anciently re-
ceived, by inserting in iJt^4(dvo for the effect of refraction,
or by msJLing a diQtinction between tlie apparent diurnal
orbits, as affected by refraction, and the true ones cleared
of that effect.

(115.) Again: The first impression produced by a
view of the diurnal movement of the heavens is, that all
the heavenly bodies perform this revolution in one com-
mon period, viz. a day, or 24 hours. But no sooner do
we come to examine the matter instrumentally, i. e. by
noting, by timekeepers, their successive arrivals on the
lAeridian, than we find differences which cannot be ac-

CHAP. II.3 LAWS TRACED BY OBSERVATION. 75

counted for by any error of observation. All the itarSt
it is true, occupy the same interval of time between their
successive appulses to the meridian, or to any vertical
circle ; hnt this is a very different one from that occupied
by the sun. It is palpably shorter : being, in fact, only
23*" 56' 4*09'', instead of 24 hours, such hours as our
common clocks mark. Here, then, we have already tufo
diff^erent day 8^ a sidereal and a salary and if, instead of
the sun, we observe the moon, we find a third, much
longer than either, — a lunar day, whose average dura-
tion is 24** 54" of our ordinary time, which last is iolar
time, being of necessity conformable to the stints succes-
sive reappearances, on which all the business of life de-
pends.

(116.) Now, all the stars are found to be unanimous
in giving the «ame exact duration of 23** 66' 4"*09, for
the sidereal day ; which, therefore, we cannot hesitate to
receive as the period in which the earth makes one revo-
lution on its axis. We are, therefore, compelled to look
on the sun and moon as exceptions to the general law ;
as having a different nature, or at least a different relation
to us, from the stars ; and as having motions, real or ap-
parent, of "^eir own, independent of the rotation of the
earth on its axis. Thus a great and most important dis-
tinction is disclosed to us.

(117.) To establish these facts, almost no apparatus is
required. An observer need only station himself to the
north of some well defined vertical object, as the angle
of a building, and placing his eyes exactly at a certain
fixed point (such as a smsdl hole in a plate of metal nail-
ed to some immoveable support), notice the successive
disappearances of any star behind the building, by a
watch.* When he observes the sun, he must shade his
eye with a dark-coloured or smoked glass, and notice the
moments when its western and eastern edges successively

* This is an excellent practical method of ascertaining the rate of a
dock or watch, being exceedingly accurate if a few precautions are a^
tended to ; the chief of which is, to take care that that part of the edge
behind which the star (a bright one, not a pland) disappears shall be

amte smooth ; as otherwise variable reflection may transfer the point of
isappearance from a protuberance to a notch, and. thus vary the moment
of observation unduly: this is easily secured, by nailing up a smooth
edged board.

70 A TREATISE ON ASTRONOMY. [CHAP. H.

come up to the wall, from which, by taking half the in*
terval he wiU ascertain (what he cannot directly observe)
the moment of disappearance of its centre.

(118.) When, in pursuing and establishing this gene-
ral fact, we are led to attend more nicely to £e times of
the daily arrival of the sun on the meridian, irregularir
ties (so they first seem) begin to be observed. The inter-
vals between two successive arrivals are not the same at
all times of the year. They are sometimes greater,
sometimes less, than 24 hours, as shown by the clock |
that is to say, the solar day is not always of tfae same
length. About the 22st of December, for example, it is
half a minute longer, and about the same day of Septem*
ber nearly as much shorter, than its average duration.
And thus a distinction is again prei^ed upon our notice
between the acttiol solar day, which is never two days in
succession alike ; and the mean solar day of 24 hours ,
which is an average of all the solar days throughout the
year. Here, then, a new source of inquiry opens u]>oa
us^ The sun's apparent motion is not only not the same
with that of the stars, but it is not (as the latter is) uni^
form. It is subject to fluctuatiims, whose laws become
matter of investigation. But to pursue these laws, we
require nicer means of observation than what we have
described, and are obliged to call into our aid an instru-
ment called the trtmMit instrument, especially destined
for such observations, and to attend minutely to all the
causes of irregularity in the going of clocks and watches
which may affect our reckoning of time. Thus we be-
come involved by degrees in more and more delicate in-
strumental inquiries ; and we speedily find that, in pro*
portion as we ascertain the amount and law of one great
or leading fluctuation, or inequality, as it is called, of the
sun's diurnal motion, we bring into view others cocttinur
ally smaller and smaller, which were before obscured, or
mixed up with errors of observation and instrumental inir
perfections. In short, we may hot inaptly compare the
mean length of the solar day to the mean or average
height of water in a harbour, or the general level of the
sea unagitated by tide or waves. The great annual fluc-
tuation above noticed may be compared to the daily varia-

CBAP. 11.3 OF TIMS AND IT8 MEASUREaiEMT. 77

lions of level produced by the tides, which are nothing
but enormous waves extending over the whole ocean,
while the smaller subordinate inequalities may be assi-
milated to waves ordinarily so called, on which, when
large, we perceive lesser undulations to ride, and on these
again, minuter ripplings, to the series of whose subordi-
nation we can perceive no end.

(119.) With the causes of these irregularities in the
solar motion we have no concern at present; their expla-
nation belongs to a more advanced part of our subject ;
but the distinction between the solar and sidereal days, as
it pervades every part of astronomy, requires to be early
introduced, and never lost sight of. It is, as already ob-
served, the mean or average length of the solar day,
which is used in the civil reckoning of time. It com-
mences at midnight, but astronomers (at least those of
tills country), even when they use mean solar time, de-
part from the civil reckoning, commencing their day at
noon, and reckoning tiie hours from round to 24.
Thus, 1 1 o'clock in the forenoon of the second of Janu-
ary, in the civil reckoning of time, corresponds to January
1 day 23 hours in the astronomical reckoning ; and one
o'clock in the afternoon of the former, to January 2 days
1 hour of the latter reckoning. This usage has its ad-
vantages and disadvantages, but the latter seem to pre-
ponderate ; and it would be well if, in consequence, it
could be broken through, and tiie eivil reckoning substi-
tuted.

(120.) Both' astronomers and civilians, however, who
inhabit different points of the earth's surface, differ from
each other in their reckoning of time ; as it is obvious
tiiey must, if we consider tiiat, when it is noon at one
place, it is midnight at a place diametrically opposite ;
sunrise at another ; and sunset» again, at a fourth. Hence
arises considerable inconvenience, especially as respects
places differing very widely in situatioif^ and which may
even in some critical cases involve the mistake of a whole
day^ To obviate this inconvenience, there has lately
been introduced a system of reckoning time by mean so-
lar days and parts of a day counted from a fixed instant,
common to all the world, and determined by no local cir-
o2

78 A TREATISE ON ASTRONOMY. [cHAP. H.

cumstaace, such as noon or midnight, but by the motion
of the sun among the stars. Time, so reckoned, is called
equinoctial time, and is numerically the same, at the same
instant, in efery part of the globe. Its origin will be ex-
plained i^iore fully at a more advanced stage of our work.

(121.) Time is an essential element in astronomical
observation, in a twofold point of view : — 1st, As the
representative of angular motion. The earth's diurnal
motion being uniform, every star describes its diurnal cir-
cle uniformly ; and the time elapsing between the pas-
sage of the stars in succession across the meridian of any
observer becomes, therefore, a direct measure of their dif-
ferences of right ascension. 2dly, As the fundamental
element (or, independent variable, to use the language of
geometers) in all dynamical theories. The great object of
astronomy is the determination of the laws of the celestial
motions, and their reference to their proximate or remote
causes. Now, the statement of the law of any observed
motion in a^elestial object can be no other than a propo-
sition declaring what has been, is, and will be, the real
or apparent situation of that object at any time past, pre-
sent, or future. To cov^stre such laws, therefore, with
observation, we must possess a register of the observed
situations of the object in question, and of the times when
they were observed.

(122.) The measurement of time is performed by
clocks, chronometers, clepsydras, and hour-glasses : the
two former are alone used in modem astronomy. The
hour-glass is a coarse and rude contrivance for measuring,
or radier counting out, fixed portions of time, and is en*
tirely disused. The clepsydra, which measured time by
the gradual emptying of a large vessel of water through a
detenninate orifice, is suj^peptible of considerable exact-
ness, and was the only dependence of astronomers before
the invention of clocks and watches At present it is
abandoned, owing to the greater convenience and exact-
ness of the latter instruments^ In one case only has the
revival of its use been proposed; viz. for, the accurate
measurement of very small portions of time, by the flow-
ing out of mercury from a small orifice in ihe bottom of
a vessel, kept constantly full to a fixed height The

CEAP. n.] CLOCKS— CHRONOMSTSKS. . 79

Stream is intercepted at the moment oi noting any event,
and directed aside into a receiver, into which itcontinnei
to ran, till the moment of noting any otiier event, when
the intercepting canse is snddendy removed, the stream
flows in its original course, and ceases to m% into the
receiver. The weight of mercury received, compared
with the weight received in an interval of time ohserved
by the dock, gives the interval between the events ob-
served. This ingenious and simple method of resolving,
with all possible precision, a problem which has of late
been much agitated, is due to Captain Kater.

(123.) The pendulum clock, however, and the balance
watch, with those improvements and refinements in its
structure which constitute it emphatically a thm/nomtier^
are the instruments on which the astronomer depends
for his knowledge of the lapse of time. These instru-
ments are now brought to such perfection, that an irrega-
larity in the rait of going, to the extent of a single se-
cond in twenty-four hours in two consecutive days, is not
tolerated in one of good character ; so that any interval
of time less than twenty-four hours may be certainly
ascertainedwithin a few tentht of a second, by their use.
In proportion as intervals are longer, the risk of error, as
well as the amount*of error risk^, becomes greater, be-
cause the accidental errors of many days may aceunra-
late ; and causes produdng a slow progressive change in
the rate of going may subsist unpereeived. It is not safe,
therefore, to tost the determination of time to docks, or
watches, for many days in succession, without checking
them, and ascertaining their errors by reference to natu-
ral events which we know to happen, day after day, at
equal intervals. Bui if this be done, the longest intervals
may be fixed with the sama precision as the shortest;
since, in fact, it is then only the times intervening be-
tween the first and last moments of such long intervals,
and such of those periodically recurring events adopted
for our points of reckoning, as occur within twenty-four
hours respectively of either, that we measure by artifi-
cial means. The whole days are counted out for us by
nature ; tiie fractional parts only, at either end, ate mea^
* xfOMf, time ; ^«re**»»to meMure.

80 A TREATISE ON ASTRONOMT. [cHAP. II*

siired by our clocks. To keep the reckoning of the inte-
ger days correct, so that none shall be lost or counted
twice, is the object of the calendar. Chronology marks
out the order of succession of events, and refers them to
their proper years and days ; while chronometry, ground-
ing its determinations on the precise observation of such
regularly periodical events as can be conveniently and
exactly subdivided, enables us to fix the moments in
which phenomena occur, with the last degree of preci-
sion.

(124.) In' the culmination^ or transit («. e, the pas-
sage across the meridian of an observer) of every star in
the heavens, he is furnished with such a regularly pe-
riodical natural event as we allude to. Accordingly, it is
to the transits of the brightest and most conveniently
situated fixed stars that astronomers resort to ascertain
their exact time^ or, which comes to the same thing, to
determine the exact amount of error of their clocks.

(125.) The instrument with which the culminations of
celestial objects are observed is called a transit instrU'
ment. It consists of a telescope firmly fastened on a hori-
zontal axis directed to the east and west points of the
horizon, or at right angles to the plane of the meridian of
the place of observation. The exti'emities of the axis
are formed into cylindrical pivots of exactly equal diame-
ters, which rest in notches fofmed in metailie supports,
bedded (in the case of large instruments) on strong piers
of stone, and susceptible of nice adjustment by screws,
both in a vertical and horizontal direction. By the for-

mer adjustment, the axis can be rendered precisely hori-
siontal, hy levelling it with a level made to rest on the

CHAP.

II.J

TRANSIT INSTRUMSMT.

81

piTots. By the latter adjustment the axis is brought pr»-
cisely into the east and west direction, the criterion of
which is furnished by the observations themselres made
with the instrument, or by a well-defined abject caUed a
meridian mark, originally determined by such obsenra-
tions, and then, for convenience of ready reference, per-
manently established, at a great distance, exactly in a
meridian line passing through the central point of the
whole instrument. It is evident, from this description,
that, if the central line of the telescope (that which joins
the centres of its object-glass and eye-glass, and which
is called in astronomy its Une of coUimation) be once well
adjusted at right angles to the axis of the transit, it will
never quit the plane of the meridian, when the instmment
is turned round on its axis.

(126.) In the focus of the eye-piece, and at ri|^t an-
gles to tiie length of the telescope, is placed a system oi
one horizontal and five equidistant vertical threads or
wires, as rejuresented in the annexed figure, which always
appear in the fidd of vieWj, when properly illuminated,

by ilay by the light of the sky, by night by that of a lamp
introduced by a contrivance not necessary here to explain.
The place of this system of wires may be altered by ad-
justing screws, giving it alateral (horizontal) motion ; and
it is by this means brought to such a position, that the
middle one of the vertical wires shall intersect the line of
eoUimittion of the telescope, where it is arrested and
permanently fastened. In thii^ situation it is evident
that the middle thread will be a visible representation of
that portion of the celestial meridian to which the tele-
scope is pointed ; and when a star is seen to cross this
wire in the telescope, it is in the act of culminating, or
passing the celestial meridian. The instant of this event is

8!^ A TREATISE ON ASTRONOMV. [CBAP. H.

noted by the clock or chronometer, which forms an in--
dispensable accompaniment of the transit instrument.
For greater precision, the moments of its crossing all the
five vertical threads is noted, and a mean taken, which
(since the threads are equidistant) would give exactly the
same result, were all the observations perfect, and wiU,
of course, tend to subdivide and destroy their errors in
an average of the whole.

(127.) For the mode of executing the adjustments,
and allowing for the errors unavoidable in the use of this
simple and elegant instrument, the reader must consult
works especially devoted to this department of practical
astronomy.* We shall here only mention one import-
ant verification of its correctness, which consists in re"
versing the ends of the axis, or turning it east for west.
If this be done, and it continue to give the same results*
and intersect the same point on the meridian mark, we
may be sure that the line of eollimation of the telescope
is truly at right angles to the axis, and describes strictly
a plane, i, e, marks out in the heavens a great circle. In
good transit observations, q» error of two or three tenths
of a second of time in the moment of a star's culmination
is the utmost which need be apprehended, exclusive of
the error of the clock : in other words, a clock may be
compared with the earth's diurnal motion by a single
observation, without risk of greater error. By multiply-
ing observations, of course, a yet greater degree of pre-
cision may be obtained.

(128.) The angular intervals measured by means of
. the transit instrument and clock are arcs of the equinoc-
tial, intercepted between circles of declination passing
through the objects observed ; and their measurement,
in this case, is performed by no artificial graduation of
circles, but by Uie help of the earth's diurnal motion,
which carries equal arcs of the equinoctial across the
meridian, in equal times, «t the rate of 15° per sidereal
hour. In all other cases, when we would measure an-
gular intervals, it is necessary to have recourse to cir-
cles, or portions of circles, constructed of metal or other

* See Dr. PearBon's Treatise on Practical Astronomy. Also Bianchi
fk>pra lo Stromento de' Paasagi. Ephem. di Milano, 1884.

CHAP. II.] MEASUREMENT OF ANOI.E8. 83

firm and durable material, and mechanically subdivided
into equal parts, such as degrees, minutes, &c. Let
ABCD be such a circle, divided into 360 degrees (num-

bered in order from any point 0° in the circumference,
round to the same point again), and connected with its
centre by spokes or rays, xyz^ firmly united to its cir-
cumference or limb. At the centre let a circular hole be
pierced, in which shall move a pivot exactly fitting it,
carrying a tube, whose axis^ a 6, is exactly parallel to
the plane of the circlcr or perpendicular to the pivot ; and
also the two arms m n, at right angles to it, and forming
one piece with the tube and the axis ; so that the motion
of the axis on the centre shall carry the tube and arms
smoothly round the circle, to be arrested and fixed at any
point we please, by a contrivance called a clamp. Sup-
pose, now, we would measure the angular interval be-
tween two fixed objects, ST. The plane of the circle
must first be adjusted so as to pass through them both.
This done, let the axis ab of the tube be directed to
one of them, S, and clamped. Then will a mark on the
ann m point either exactly to some one of the divisions
on the limb, or between two of them adjacent. In the
former case, the division must be noted, as the reading
of the arm. m. In the latter, the fractional part of one
whole interval between the consecutive divisions by
which the mark on m surpasses the last inferior division
must be estimated or measured by some mechanical or
optical means. (See art. 130.) The divisiortirand frac«
tional part thus noted, and reduced into degrees, minutes,,
and seconds, is to be set down as the reading of the limb

84 A TREATISE ON ASTRONOMY. [CHAP* lb

oorresponding to that position of the tube ab, where it
points to the object 8. The same must then be done for
tiie object T ; die tube pointed to it, and the limb '* read
o^." It is manifest, then, that, if the lesser of these
readings be subtracted from the greater, their difference
will be the angular interval between S and T, as seen
from the centre of the circle, at whatever point of the
limb the commencement of the graduations on the point
0^ be situated.

(129.) The very same result will be obtained, if, in-
stead of making the tube moveable upon the circle, we
connect it invariably with the latter, and make both re-
volve together on an axis concentric with the circle, and
forming one piece with it, working in a hollow formed
to receive and fit it in some fixed support. Such a com-
bination is represented in section in the annexed sketch.
T is the tube or sight, fastened, ^ipp, on the circle AB,

fW

whose axis, D, works in the solid metallic centring E,
from which originates an arm, P, carrying at its ex-
tremity an index, or other proper mark, to point out and
read off the exact division of the circle at B, the point
close to iu It is evident that, as the telescope and circle
revolve through any angle, the part of the limb of the
latter, which by such revolution is carried past the index
F, will measure the angle described. This is the most
usual mode of applying divided circles in astronomy.

(130.) The index F may either be a simple pointer,
like a clock hand (fig. a) ; or a vernier ^g, b) ; or.

CHAP, n.3 APPLICATION OP' THE TB1E8C0PE. 85

lastly, a compound microscope (jig, c), represented in
section (in Jig, d), and furnished with a cross in the
common focus, of its object and eye-glass, moveable by
a fine threaded screw, by which the intersection of the
cross may be brought to exact coincidence with the
image of the nearest of the divisions of the circle ; and by
the turns and parts of a turn of the screw required for this
purpose the distance of that division from the original
or zero point of the microscope may be estimated. This
simple but delicate contrivance gives to the reading off
of a circle a degree of accuracy only limited by the power
of the microscope, and the perfection with which a screw
can be executed, and places the subdivision of angles on
the same footing of optical certainty which is introduced
into their measurement by the use of the telescope.

(131.) The exactness of the result thus obtained must
depend, 1st, on the precision with which the tube a b
can be pointed to the objects ; 2dly, on the accuracy of
graduation of the limb ; 3dly, on the accuracy with
which the subdivision of the intervals between any two
consecutive graduations can be accomplished. The
mode of accomplishing the latter object with any re-
quired exactness has been explained in the last article*
With regard to the graduation pf the limb, being merely
of a mechanical nature, we shall pass it without Iremark*
further than this, that, in the present state of instrument
making, the amount of error from this source of inacca-
racy is reduced within very narrow limits indeed. With
regard to the first, it must be obvious that, if the sights
a 6 be nothing more than what they are represented in
the figure (art. 128), simple crosses or pin-holes at the,
ends of a hollow tube, or an eye-hole at one end, and a
cross at the other, no greater nicety in pointing can be
expected than what simple visi<m with the naked eye
can command. But if, in place of these simple but
coarse contrivances, the tube itself be converted into a
tdeseope, having an object-glass at b, and an eye-piece
at a ; and if the motion of the tube on the limb of the
circle be arrested when the object is brought just into
the centre of the field of view, it is evident tifiat a great^^
degree of exactness may be attained in the pointing oi
H

86 A TREATISE ON ASTRONOMY. [CHAP. II.

the tube than by the unassisted eye, in proportion to the
magnifying power and distinctness of the telescope used.
The last attainable degree of exactness is secured by
stretching in the common focus of the object and eye-
glasses two delicate fibres, such as fine hairs or spider-
lines, intersecting each other at right angles in the centre
of the field of view. Their points of intersection afiford
a permanent mark with which the image of the object
can be brought to exact coincidence by a proper degree
of caution (aided by mechanical contrivances), in bringing
the telescope to its final situation on t^e limb of the circle,
and retaining it there till the '< reading off" is finished.
(132.) This application of the telescope may be con-
sidered as completely annihilating that part of the error
of observation which might otherwise arise from errone-
ous estimation of the direction in which an object lies
from the observer's eye, or from the centre of the in-
strument. It is, in fact, the grand source of all the pre-
cision of modem astronomy, without which all other re-
finements in instrumental workmanship would be thrown
away; the errors capable of being committed in point-
ing to an object, without such assistance, being far greater
than what could arise from any but the very coarsest
graduation.* In fact, the telescope thus applied becomes,

* Tbe honour of this capital improvement has been successfully vin-
dicated by Derham (Pliil. Trans, xxx. 603) to our young, talentea, and
unfortunate countryman Gascoisne, from lus correspondence with Crab-
tree and Hprrockes, in his (Derham's) possession. The passages cited
by Derham from these letters leave no doubt that, so early as 1640,
Gascoigne had applied telescopes to his quadrants and sextants, wUk

threads in the common foctu (f the glasses ; and had even carried the in-
vention so far as to illuminate the field of view by artificial light, which
he found " very helpful when the moon appearefh not, or it is not otherwise
light enoughJ* These inventions were treely communicated by him to
Crabtree, and through him to his jQriend Horrockes, the pride and boast
of British astronomy ; both of whom expressed their unbounded admira-
tion of this and many other of his delicate and admirable improvements
in the art of observation. Gascoigne, however, perished at the age of
twenty-three at the batde of Marston Moor ; and the premature and
sudden death of Horrockes, at a yet earlier age, will account for the
temporary oblivion of the invention. It vras revived, or re-invented, in
1667, by Picard and Auzout (Lalande, Astron. 2310), after which its vue
became universal. Morin, even earlier than Gascoiffne (in 1635), had
proposed to substitute the telescope for plain sights ; but it is the thread
or wire stretched in the focus with which the image of a star can be
^ brought to exact coincidence, which gives the telescope its advantage in
practice ; and the idea of this does not seem to have occurred to I^rin.
<3ee Lalande, vH supra.)

CHAP, n.] INTERVALS IN DECLINATION MSA8URBD. 87

with respect to angular, what the microscope is with
respect to linear dimension. By concentrating attention
on its smallest points, and magnifying into palpable in-
tervals the minutest differences, it enables us not only to
scrutinize the form and structure of the objects to which
it is pointed, but to refer their apparent places, with all
but geometrical precision, to the parts of any scale with
which we propose to compare them.

(133.) The simplest mode in which the measurement
of an angular interval can be executed, is what we have
just described ; but, in strictness, this mode is applicable
only to terrestrial angles, such as those occupied on the
sensible horizon by the objects which surround our sta-
tion, — because these only remain stationary during the
interval while the telescope is shifted on the limb from
one object to the other. But the diurnal motion of the
heavens, by destroying this essential condition, renders
the direct measurement of angular distance from object
to object by this means impossible. The same objection,
however, does not apply if we seek only to determine
the interval between the diurrud circles described by any
two celestial objects. Suppose every star, in its diurnal
revolution, were to leave behind it a visible trace in the
heavens, — a fine line of light, for instance,-— then a teles-
cope once pointed to a star, so as to have its image
brought to coincidence with the intersection of the wires,
would constantly remain pointed to some portion or other
of this line, which would therefore continue to appear
in its field as a luminoifs line, permanently intersecting
the same point, till the star came round again. From
one such line to another the telescope might be shifted,
at leisure, without error ; and then the angular interval
between the two diurnal circles, in the plane of the tele-
scope^ 8 rotation, might be measured. Now, though we
cannot see the path of a star in the heavens, we can wait
till the star itself crosses the field of view, and seize the
moment of its passage to place the intersection of its
wires so that the star shall traverse it ; by which, when
the telescope is well clamped, we equally well secure the
position of its diurnal circle as if we continued to see it
ever so long. The reading off of die limb may then be

88 A TREATISE ON ASTRONOKT. [CHAP. 11.

perfonned at leisure; and when another star comes
round into the plane of the circle, we may unclamp the
tdescope, and a similar ohservation will enable us to as-
sign the place of its diurnal circle on the limb : and the
enervations may be repeated alternately, every day, as
the stars pass, till we are satisfied with their result.

(134.) This is the principle of the mural circle, which
is nothing more than such a circle as we have described
in art. 129, firmly supported, in the plane of the meri-
dian, on a long and powerful horizontal axis. This axis
is let into a massive pier, or wall, of stone (whence the
name of the instrument), and so secured by screws as to
be capable of adjustment both in a vertical and horizon-
tal direction ; so that, like the axis of the transit, it can
be maintained in the exact direction of the east and west
points of the horizon, the plane of the circle being con-
sequently truly meridional.

(135.) The meridian, being at right angles to all the
diurnal circles described by the stars, its arc intercepted
between any two of them will measure the least distance
between these circles, and will be equal to the difference
of the declinations, as also to the difference of the meri"
dian aUitudea of the objects — at least when corrected
for refraction. These differences, then, are the angular
intervals directly measured by the mural circle. But
from these, supposing the law of refraction known, it is
easy to conclude, not their differences only, but the
quantities themselves, as we shall now explain.

(136.) The declination of a heavenly body is the com-
plement of its distance from the pole. The pole, being
a point in the meridian, might be directly observed on the
limb of the circle, if any star stood exactly therein ; and
thence the polar distances, and of course, the declina-
tions of all the rest, might be at once determined. But
this not being the case, a bright star as near the pole as
can be found is selected, and observed in its upper and
lower culminations ; that is, when it passes the meridian
above and below the pole. Now, as its distance from
the pole remains the same, the difference of reading off
the circle in the two cases is, of course (when corrected
for refraction), equal to twice the polar distance of the

(SAP. n.] MURAL CIRCLE. 89

«tar; the arc intercepted on the limb of the circle being,
in this case, equal to the angular diameter of the star's
diurnal circle. In the annexed diagram, HPO represents
the celestial meridian, P the pole, BR, AQ, CD, the di-
ttmal circles of stars which arrive on the meridian — at
BA and C in their upper, and at RQD in their lower cul-

minations, of which D happens above the horizon HO.
P b the pole ; and if we suppose hpo to be the mural
circle, having S for its centre, bacpd will be the points
on its circumference corresponding to BACPD in the
heavens. Now, the arcs ba^hc^hd^ and c ^ are given
immediately by observation ; and since CP=sPD, we
have also cp^^p d, and each of them a=ic rf, consequently
the place of the polar point, as it is called, upon the limb
of the circle becomes known, and the arcs p b,p a, p c,
vhich represent on the circle the polar distances re-
quired become also known.

(137.) The situation of the pole star, which is a very
brilliant one, is eminently favourable for this purpose,
being only about a degree and a half from the pole ; it
w» therefore, the star usually and almost solely chosen
for this important purpose ; tfie more especially because,
both its culminations taking place at great and not very
different altitudes, the refractions by which they are
affected are of small amount, and differ but slightly from
«ach other, so that their correction is easily and safely
*PpUed. The brightness of the pole star, too, allows
«< to be easily observed in the daytime. In consequence

r9

00 A TREATISE ON ASTRONOMY. [cRAP* II»

of these peculiarities, this star is one of constant resort
with astronomers for the adjustment and verification of
instruments of almost every description. In the case of
the transit, for example, it furnishes a ready means of
ascertaining whether the plane of the telescope's motion
is coincident with the meridian. For since this latter
plane bisects its diurnal circle, the eastern and western
portion of it require equal times for their description.
Let, therefore, the moments of its transit above and be-
low the pole be noted ; and if they are found to follow
at equal intervals of 12 sidereal hours, we may conclude
with certainty that the plane of the telescope's motion is
meridional, or the position of its horizontal axis exactly
east and west. But if it pasd from one to the other ap-
parent culmination in unequal intervals of time, it is ^
equally certain that an extra-meridional error must exist,
the deviation lying towards that side on which the least
interval is occupied. And the axis must be moved in
azimuth accordingly, till the difference in question dis-
appears on repeating the observations.

(138.) The place of the polar paint on the limb of
the muial circle once determined, becomes an origm, or
zero point, from which the polar distances of all objects,
referred to other points on the same lines, reckon. It
matters not whether the actual commencement 0*^ of the
graduations stand there, or not; since it is only by
the difference of the readings that the arcs on the
limb are determined ; and hence a great advantage is
obtained in the power of commencing anew a fresh series
of observations, in which a different part of the circum-
jEbrence of the circle shall be employed, and different
graduations brought into use, by which inequalities of
division may be detected and neutralized. Thia is ae-
complished practically by detaching the telescope from
its old bearings on the circle, and fixing it afresh on a
different part of the circumference.

(139.) A point on the limb of the mural circle, not
less important than the polar point j is the horizmtd
pai$Uf which, being once known, becomes in like muh
net an origin, or zero point, from which altitudes an
reckoned. The principle of its determinatioii is Mf>

CfiAP* n. J POLAR AND HORIZONTAL POIirTS. 91

mately nearly the same with that of the polar poinU
As no star exists in the celestial horizon, the observer
must seek to determine two points on the limb, the one
of which shall be precisely as far b^aw the horizontal
point as the other is above it. For this purpose, a star
is observed at its culmination on one night, by pointing
the telescope directly to it, and the next, by pointing to
the image of the same star r^Ucted in the stiU, unruffled
surface of a fluid at perfect rest. Mercury, as the most
reflective fluid known, is generally chosen for that use.
As the surface of a fluid at rest is necessarily horizontal,
and as the angle of reflection, by the laws of optics, is
equal to that of incidence, this image will be just as
much depressed below the horizon, as the star itself is
elevated above it (allowing for the diflerence of refrac-
tion at the moments of observation). The arc inter-
eepted on the limb of the circle between the star and its
reflective image thus consecutively observed, when cor-
rected for refraction, is the double altitude of the star,
and its point of bisection the horizontal point. The re-
flecting surface of a fluid so used for the determination
of the altitudes of objects is called an artificial horizon,

(140.) The mural circle is, in fact, at the same time, a
transit instrument ; and, if furnished with a proper sys-
tem of vertical wires in the focus of its telescope, may
be used as such. As tlie axis, however, is only support-
ed at one end, it has not the strength and permanence ne-
cessary for the more delicate purposes of a transit ; nor
can it be vended, as a transit may, by the reversal of the
two ends of its axis, east for west. Nothing, however,
prevents a divided circle being pennanently fastened on
the axis of a transit instrument, near to one of its extre-
mities, so as to revolve with it, the reading ofi* being per-
formed by a microscope fixed on one of its piers. Such
an instrument is called a transit circle, or a mxridian
CIRCLE, and serves for the simultaneous determination of
the right ascensions and polar distances of objects ob-
served with it ; the time of transit being noted by the clock,
and the circle being read ofl* by the lateral microscope.

(HI.) The detennination of the horizontal point on
the limb of an instrument is of such essential importance
in astronomy, that the student should be made acquaint-

92 A TREATISE ON ASTROKOHT. [cHAP. 11.

ed with every means employed for this purpose. These
are the artificial horizon, the plumb-line, the level, and the
floating collimator. The artificial horizon has been al*
ready explained. The plumb-line is a fine thread or wire,
to which is suspended a weight, whose oscillations are
impeded and quickly reduced to rest by plunging it in
water. The direction ultimately assumed by such a line,
admitting its perfect flexibility, is that of gravity, or per-
pendicular to the surface of still water. Its application
to the purposes of astronomy is, however, so delicate, and
difficult, and liable to error, unless extraordinary precau-
tions are taken in its use, that it is at present almost uni-
versally abandoned, for ^e more convenient and equally-
exact mstrument the level,

(142.3 '^^^ ^^^®l ^^ nothing more than a glass tube
nearly filled with a liquid (spirit of wine being that now
generally used, on account of its extreme mobility^ and
not being liable to freeze), the bubble in which, when the

tube is placed horizontally, would rest indifferently in any
part if the tube could be mathematically straight. But
that being impossible to execute, and every tube having
some slight curvature, if the convex side be placed up-
wards, the bubble will occupy the higher part, as in the
figure (where the curvature is purposely exaggerated);
Suppose such a tube as AB firmly fastened on a straight
bar, CD, and marked at a 6, two points distant by the
length of the bubble ; then, if the instrument be so placed
iJiat the bubble shall occupy this interval, it is clear that
CD can have no other than one definite inclination to the
horizon ; because, were it ever so little moved one way
or other, the bubble would shift its plaee, and run towards
the elevated side. Suppose, now, that we would ascer-
tdn whether any given line PQ be horizontal ; let the
base of the level CD be set upon it, and note the points

CHAP, n.3 OF THE LEVEL. 9S

abf between which the bubble is exactly contained ; .then
torn the level end for end, so that C shall rest on Q, and
D on P. If then the bubble continue to occupy ^e same
place between a and 6, it is evident that PQ can be no
otherwise than horizontal. If not, the side towards which
the bubble runs is highest, and must be lowered. Astro-
Bomical levels are furnished with a divided scale, by
which ihe places of the ends of the bubble can be nicely
marked ; and it is said that they can be executed with
nieh delicacy, as to indicate a single second of angular
deviation from exact horizontality.

(143.) The mode in which a level may be applied to
find the horizontal point on the limb of a vertical divided
circle may be thus explained : Let AB be a telescope
fiimly fixed to such a circle, DEF, and moveable in one

with it on a horizontal axis C, which must be like that of
a transit, susceptible of reversal (see art. 127), and with
which the circle is inseparably connected. Direct the
telescope on some distant well-defined object S, and bi-
Mct it by its horizontal wire, and in this position clamp
it fast. Let L be a level fastened at right angles to an
Mm, LEP, furnished with a microscope, or vernier at F,
«id, if we please, another at E. Let this arm be fitted by
gnnding on the axis C, but capable of moving smoothly
on it without carrying it round, and also of being clamped
fi»t on it, so as to prevent it from moving until required,
While the telescope is kept fixed on the obiect S, let the

94 A TREATISE ON ASTRONOKT. [CHAP.II.

level be set so as to bring its bubble to the marks a by and
clamp it there. Then will the arm LCF have some cer-
tain determinate inclination (no matter what) to the hori-
zon. In this position let the circle be read off at F, and
then let the whole apparatus be reversed by turning its
horizontal axis end for end, without unclamping the level
arm from the axis. This done, by the motion of the
whole instrument (level and all) on its axis, restore the
level to its horizontal position with the bubble at a 6.
* Then we are sure that the telescope has now the same
inclination to the horizon the other way, that it had when
pointed to S, and the reading off at F will not have been
changed. Now, unclamp the level, and, keeping it nearly
horizontal, turn round the circle on the axis, so as to car-
ry back the telescope through the zenith to 8, and in
that position clamp the circle and telescope fast. Then it
is evident that an angle equal to twice the zenith distance
of S has been moved over by the axis of the telescope
from its last position. Lastly, without unclamping the
telescope and circle, let the level be once more rectified.
Then will the arm LEF once more assume the same de-
finite position with respect to the horizon ; and, conse-
quently, if the circle be again read off, the difference be-
tween this and the previous reading must measure the
arc of its circumference which has passed under the
point F, which may be considered as having all the
while retained an invariable position. This difference,
then, will be the double zenith distance of S, and its half
the zenith distance simply, the complement of which is
its altitude. Thus the altitude corresponding to a given
reading of the limb becomes known, or, in other words,
the horizontal point on the limb is ascertained. Circuit-
ous as this process may appear, there is no other mode
of employing the level for this purpose which does not
in the end come to the same thing. Most commonly,
however, the level is used as a mere fiducial reference,
to preserve a horizontal point once well determined by
other means, which is done by adjusting it so as to stand
level when the telescope is truly horizontal, and thus
leaving it depending on the permanence of its adjustment.
(144.) The last, but probably not the least exact, as it

CHAP. II.3, THE FLOATING COLLIHATOR. 95

certainly is, in innumerable cases, the most convenient
means of ascertaining the horizontal point, is that af-
forded by the floating collimator, a recent invention of
Captain Kater. This elegant instrument is nothing more
than a small telescope furnished with a cross-wire in iti
focus, and fastened horizontally, or as nearly so as may
be, on a flat iron float, which is made to swim on mer-
cury, and which, of course, will, when left to itself, as-
sume always one and the same invariable inclination to
the horizon. If the cross-wires of the collimator be illu-

minated by a lamp, being in the focus of its object-glass,
the rays from them will issue parallel, and will therefore
be in a fit state to be brought to a focus by the object-
glass of any other telescope, in which they will form an
image as if they came from a celestial object in their di-
rection, i. e. at an altitude equal to their inclination.
Thus the intersection of the cross of the collimator may
be observed as if it were a star, and that, however near
the two telescopes are to each other. By transferring then,
the collimator still floating on a vessel of mercury from
the one side to the other of a circle, we are furnished with
two quasi-celestial objects, at precisely equal altitudes,
on opposite sides of the centre ; and if these be observed
in succession with the telescope of the circle, bringing its
cross to bisect the image of the cross of the collimator (for
which end the wires of the the latter cross
are purposely set 45° inclined to the hori-
zon) the diflference of the readings on its limb
■^ill be twice the zenith distance of either ;
whence, as in the last article, the horizontal
or zenith point is immediately determined.*

* Another, ami in many respects, jveferable fiirm of the floating oolU-
"wtor.in wUch the telescope is vertuxUt and whereby the zenith point is
directly ascertained, is described in the Phil. Trans. 1828, p. 257 Inr tti«
■«me author.

96 A TREATISE ON ASTRONOlTir. [CHAT. n.

^145.) The transit and mural circle are essentially me-
ridian instruments, being used only to observe the stars
at the moment of their meridian passage. Independent
of this being the most favourable moment for seeing them,
it is that in which their diurnal motion is parallel to the
horizon. It is therefore easier at this time than it could
be at any other, to place the telescope exactly in their
true direction ; since their apparent course in the field of
view being pauraUel to the horizontal thread of the system
of wires therein, they may, by giving a fine motkm to
the telescope, be brought to exact coincidence with it,
and time may be allowed to examine and correct this co-
incidence, if not at first accurately hit, which is the case
in no other situation. Generally speaking, all angular
magnitudes, which it is of importance to ascertain ex-
actly, should, if possible, be observed at their maxima or
minima of increase or diminution; because at these
points they remain not perceptibly changed during a time
long enough to complete,* and even, in many cases, to re-
peat and verify our observations in a careful and leisurely
manner. The angle which, in the case before us, is in
this predicament, is the altitude of the star, which attains
its maximum or minimum on the meridian, and which is
measured on the limb of the mural circle.

(146.) The purposes of astronomy, however, require
that an observer should possess the means of observing
any object not directly on the meridian, but at any point
of its diurnal course, or wherever it may present itself
in the heavens. Now, a point in the sphere is determined
by reference to two great circles at right angles to each
other ; or of two circles one of which passes through the
pole of the other. These, in the language of geometry,
are co-ordinates by which its situation is ascertained :
for instance,^-on the earth, a place is known if we know
its longitude and latitude ;— in the starry heavens, if we
know its right ascension and declination ; — in the visible
hemisphere, if we know its azimuth and altitude, 4m5.

(147.) To observe an object at any point of its diurnal
course, we must possess the means of directing a tele-
scope to it; which, therefore, must be capable of motion
in two planes at right angles to each other ; and the

CHAP, n.] CO-ORDINATE CIRCLES.

07

amount of its angular motion in each must be measured
on two circles co-ordinate to each other, whose planes
must be parallel to those in which the telescope moves.
The practical accomplishment of this condition is effect-
ed by making the axis of one of the circles penetrate that
of the other at right angles. The pierced axis turns on
fixed supports, while the other has no connexion with
any external support, but is sustained entirely by that
which it penetrates, which is strengthened and enlarged
at the point of penetration to receive it. The annexed
figure exhibits the simplest form of such a combination,
though by no means the best in point of mechanism.
The two circles are read off by verniers, or microscopes ;
the one attached to the fixed support which carries the
principal axis, the other to an arm projecting from that
axis. ' Both circles also are susceptible of being clamped,
the clamps being attached to the same ultimate bearing
with which the apparatus for reading off is connected.

(148.) It is manifest that such a combination, however
its principal axis be pointed (provided that its direction
be invariable), will enable us to ascertain the situation of

any object with respect to
the observer's station, by
angles reckoned upon two
great circles in the visible
hemisphere, one of which
has for its poles the pro-
longations of the principal
axis or the vanishing points
of a system of lined parallel
to it, and the other passes
always through these poles ;
for ihe former great circle
is the vanishing line of all
planes parallel to the circle
AB, while the latter, in any
position of the instrument,
is the vanishing line of all
the planes parallel to the
circle GH ; and these two planes being, by the construc-
tion of the instrument, at right angles, the great circles,

98 A TREATISE ON ASTRONOMIT. [CHAF. H^

which are their vanishing lines, must he so too. Now,
if two great circles of a sphere be at right angles 40 each
other, the one will always pass through the other's
poles.

( 149.) There are, however, but two positions in which
such an apparatus can be mounted so as to be of any
practical utility in astronomy. The first is, when the
principal axis CD is parallel to the earth's axis, and
therefore points to the poles of the heavens which are the
vanishing points of all lines in his system of parallels :
and when, of course, the plane of the circle AB is paral-
lel to the earth's equator, and therefore, has the equi-
noctial for its vanishing circle, and measures, by its arcs
read off, hour angles, or differences of right ascension.
In this case, the great circles in the heavens, correspond-
ing to the various positions,- which the circle GH can be
made to assume, by the rotation of the instrument round
its axis CD, are all hour-circles : and the arcs read off
on this circle will be declinations, or polar distances, or
their differences.

(150.) In this position the apparatus assumes the name
of an equatorial J or, as it was formerly called, a paralltictic
instrument. It is one of the most convenient instruments
for all such observations as require an object to be kept
long in view, because, being once set upon the object,
it can be followed as long as we please by a single motion^
i. e. by merely turning the whole apparatus round on its
polar axis. For since, when the telescope is set on a
star, the angle between its direction and that of Jthe polar
axis is equal to the polar distance of the star, it follows,
that when turned about its axis, without altering the posi-
tion of the telescope on the circle 6H, the point to which
it is directed will always lie in the smaH circle of the
heavens coincident with the star's diurnal path. In many
observations this is an inestimable advantage,^ and one
which belongs to no other instrument. The equatorial
is also used for determining the place of an unknown by
comparison with that of a known object, in a manner to
be described in the fourth chapter. The adjustments of
the equatorial are somewhat complicated and difficult
They are best performed by following the pole-star round

CHAP. II.] AZIIITJTH AND ALTITUDE INSTRUMENT. 99

the entire diurnal circle, and by observing, at proper in-
tervals, other considerable stars v^hose places are well
ascertained.*

(151.) The other position in which such a compound
apparatus as we have described in art. 147 may be advan-
tageously mounted, is that in which the principal axis
occupies a vertical position, and the one circle, AB, con-
sequently corresponds to the celestial horizon, and the
other, GA, to a vertical circle of the heavens. The an-
gles measured on the former are therefore azimuths, or
differences of azimuth, and those on the latter zenith dis-
tances, or altitudes, according as the graduation com-
mences from the upper point of its limb, or from one 90**
distant from it. It is therefore known by the name of *
an azimuth and altitude instrument. The vertical posi-
tion of its principal axis is secured either by a plumb-
line suspended from the upper end, which, however it
be turned round, should continue always to intersect one
and the same fiducial mark near its lower extremity, or
by a level fixed directly across it, whose bubble ought
not to shift its place, on moving the instrument in azi-
muth. The north or south point on the horizontal cir-
cle is ascertained by bringing the vertical circle to coin-
cide with the plane of the meridian, by the same criterion
by which the azimiithal adjustment of the transit is per-
formed (art. 137), and noting, in this position, the read-
ing off of the lower circle, or by the following process.

(152.) Let a bright star be observed at a considerable
distance to the east of the meridian, by bringing it on
the cross wires of the telescope. In this position let the
horizontal circle be read off, and the telescope securely
clamped on the vertical one. When the star has passed
the meridian, and is in the descending point of its daily
course, let it be followed by moving the whole instrument
round to the west, without, however, unclamping the
telescope, until it comes into the field of view ; and, until,
by continuing the horizontal motion, the star, and the
cross of the wires come once more to coincide. In this
position it is evident the star must have the same precise

♦See Uttrowonth« A^jiwtmentof the Equatorial— J&w. AMrm, Sac

100 A TRBATISE ON ASTRONOMY. |^CHAP« U.

altitude about the western horizon, that it had at the mo-
meut of the first observation above the eastern. At this
point let the motion be arrested, and the horizontal circle
be again read off. The difference of the readings will be
the azimuthal arc described in the interval. Now, it is
evident that when the altitudes of any star are equal on
either side of the meridian, its azimuths^ whether reckon-
ed both from the north or both from the south point of the
horizon, must also be equal,— consequently the north or
south point of the horizon must bisect the azimuthal arc
thus determined, and will therefore become known.

(153.) This method of determining the north and
south points of a horizontal circle (by which, when
known, we may draw a meridian line) is called the
** method of equal altitudes," and is of great and constant
use in practical astronomy. If we note, at the moments
of the two observations, the time, by a clock or chrono-
meter, the instant halfway between them will be die
moment of the star's meridian passage, which may thus
be determined without a transit ; and, vice versa^ the
error of a clock or chronwneter may by this process be
discovered. For this last purpose, it is not necessary
that our instrument should be provided with a horizontal
circle at all. Any means by which altitudes can be mea-
sured will enable us to determine the moments v(rhen the
same star arrives at equal altitudes in the eastern and
western halves of its diurnal course ; and, these once
known, the instant of meridian passage and the error of
the clock become also known.

(154.) One of the chief purposes to which the altitude
and azimuth circle is applicable is the investigation of
the amount and laws of refraction. For, by following
with it a circumpolar star which passes the zenith, and
another which grazes the horizon, through their whole
diurnal course, the exact apparent form of their diurnal
orbits, or the ovals into which their circles are distorted
by refraction, can be traced ; and their deviation from
circles, being at every moment given by the nature of
the observation in the direction in which the refraction
itself takes place (i. e. in altitude), is made a matter of
direct observation.

CHAP, n.] hadlet's sextant. 101

(155.) The zenith sector and the theodolite are pecu-
liar modifications of the altitude and azimuth instrument.
The former is adapted for the very exact observation of
stars in or near the zenith, by giving a great length to
the vertical axis, and suppressing all the circumference of
the vertical circle, except a few degrees of its lower
part, by which a great length of radius, and a consequent
proportional enlargement of the divisions of its arc, is
obtained. The latter is especially devoted to the mea-
sure of horizontal angles between terrestrial objects, in
which the telescope never requires to* be elevated more
than a few degrees, and in which, therefore, the vertical
circle is either dispensed wiUi, or executed on a smaller
scale, and with less delicacy ; while, on the other hand,
great care is bestowed on securing the exact perpendieu^
larity of the plane of the telescope's motion, by resting
its horizontal axis on two supports like the piers of a
transit-instrument, while themselves are firmly bedded on
the spokes of the horizohtal circle, and turn with it.

(156.) The last instrument we shall describe is one
by whose aid the direct angular distance of any two ob^
jects may be measured, or the altitude of a single ono
determined, either by measuring its distance from the
visible horizon (such as the sea-ofiing, allowing for its
dip), or from its own reflection on the surface of mercury.
It is the sextant, or quadrant, commonly called Hadley^Sy
from its reputed inventor, though the priority of invention
belongs undoubtedly to Newton, whose claims to the
gratitude of the navigator are thus doubled, by his having
furnished at once the only theory by which his vessel
can be securely guided, and the only instrument which
has ever been found to avail, in applying that theory to
its nautical uses.*

(157.) The principle of this instrument is the optical
property of reflected rays, thus announced : — "The

* Newton comiminicated it to Dr. Halley, who suppressed it The
description of the instrument was found, after the aeath of Halley,
among li^ papers, in Newton*s own handwriting, by his executor, who
communicated the papers to toe Royal Society, twenty-five jreais after
Newton's death, and eleven after the publicatian of Hadley's invention,
which might be, and probably was, independent of any knowledge of
Newton's, though Hutton immuates the contraiy.
i2

lOS A TREATISB^ ON ASTRONOMY. [CHAP. II.

angle between the first and last directions of a ray which
has suffered two reflections in one plane is equal to twice

the inclina^n of the reflecting suirfaces to each other.'*
Let AB be the limb, or graduated arc, of a portion of a
circle 60° in extent, but divided 4ntQ*120 equal parts*
On the radius CB let a silvered plane glass D be fixed,
at right angles to the plane of the circle, and on the
moveable radius C£ let another such silvered glass, C,
be fixed. The glass D is permanently fixed parallel to
AC, and only one half of it is silvered, the other half
allowing oi^ects to be seen through it. The glasf C is
wholly silvered, and its plme is parallel to 2ie length
of the moveable radius C£, at the extremity E, of which
a vernier is placed to read off the divisions of the limb.
On the radius AC is set a telescope F, through which
any object, Q, may be seen by direct rays which pass
throuf^ the unsilvered portion of the glass D, while
another object, P, is seen through the s^^e telescope
by rays, which, after reflection at C, have been thrown
upon the silveied part of D, and are thence directed by
a second reflection into the telescope. The two images
so formed will both be seen in the field of view at once,
aad by moving the radius CE will (if the reflectors be
traly perpendicular to the plane of ihe circle) meet and
pass over, without obliters^g each other. The motion,
however, is arrested when they meet, and at this point
the angle included between die direction CP of one
object, and FQ of the other, is twice the angle ECB in-
cluded between the fixed and moveable radii CB, CE.
Now the graduations of the limb being purposely made

GHAP. n.] FRINCIFLB OF RBPBTITIOll, IQt

only half as distant as would correspond to degi/beBf the
arc BE, when read off, as< if the graduations were whole
degrees, will, in fact, read double its real amountt and
therefore the numbers to read off will express not the
angle BOB, but its double, the angle subtended by the
objects.

(158.) To determine the exact distances between the
sttffs by direct observation is comparatively of little ser-
vice ; but in nautical astronomy the measurement of
their distances from the moon, and of tlieir altitudes, is
of e^/sential imftf>rtance ; and as the sextant requires no
fixed support, but can be held in the hand, and used on
ship-board, the utility of the instrument becomes at once
obvious. For altitudes at sea, as no level, plumb-line,
or artificial horizon can be used, the sea-offing affords
the only resource p- and the image of the star observed^
seen by reflection, is brought to coincide with the bowi^
dary of the sea seen by (Srect rays. Thus the altkude
above the sea-line is found ; and this corrected for the
dip i^the honzon (art. 24) gives the true altitude of the
star. On land, an artificial horizon may be used (art. 180),
and tiie consideration of dip is rendered unnecessary.

(159.) The reflecting circle is an instrument destined
for the same uses as the sextant, but more complete, th#
circle being en^, and the divisions carried all round.
It is usually furnished with three verniers, so as to admit
of three distinct readings off, by the average ff whick
the error of graduation and ef reading is reduced. This
is altogether a very r^ned and elegant instrumeiit

{KM) We must not conclude this chapter without
mention ef the *« principle of repetition T* an ioventioii
ef Borda, by which the error of graduation may be di*
minished to any degree, and, praciioally speaking, ennn
hilated. Let PQ be two o^eets which we may suppose
fixed, for- purposes ef mere explanation, and let KL be a
telescope moveable on O, the common axis of two cir-
cles, AML and a 6 e, of which the former, AML» is ab-
solutely fixed in the plane of the objects, and earries the
graduations, and the latter is freely nteveaUe on the axie*
The telescope is attached permanentf y to the latter drcle,
^d moves with it. An arm OoA carries the index, or

104 A TREATISE ON ASTRONOMY. ([CHAP, n.

vernier,' which reads off the graduated limb of the &xei.
circle. This arm is provided with two clamps, by which
it can be temporarily connected with either circle, an4

detached at pleasure. Suppose, now, the telescope du
rected to P. Clamp the index arm OA to the inner
circle, and unclamp it from the outer, and read off. Then
carry the telescope round to the other object Q. In so
doing, the inner circle, and the index-arm which is
clamped to it, will also be carried round, over an arc AB,
on the graduated limb of the outer, equal to the angle
POQ. Now clamp the index to the outer circle, and
unclamp the inner, and read off: the difference of readings
will of course measure the angle POQ ; but the result
will be liable to two sources of error — that of graduatior^
and that of observation, both which it is our object to
get rid of. To this end transfer the telescope back to P^
without unclamping the arm from the out^r circle; then^
having made the bisection of P, clamp the arin to 6, an4
unclamp it from B, and again transfer the telescope to Q»
by which the arm will now be carried with it to C, over
a second arc, IBC, equal to the angle POQ. Now agaiiji
read off ; then will the difference between this reading
and the original one measure twice the angle POQ,
affected with both errors of observation, but only wiUi
the same error of graduation as before. Let this pro-
eess be repeated as often as we please (suppose ten
times) ; then will the final arc ABCD read off on the
eirele be ten times the required angle, affected by the

CHAP. III.J GEOGRAPHY. 106

joint errors of all the ten obserrations, but only bj the
same constant error of graduation, which depends on tlit
initial and final readings off alone. Now the errors o£
observation, when numerous, tend to balance and destroj
one another ; so that, if sufficiently multiplied, their; in-
fluence will disappear from the result. There remaimiy
then, only the constant error of graduation, which comes
to be divided in the iinal result by the number of obser-
vations, and is therefore diminished in its influence to
one tenth of its possible amount, or to less if need be.
The abstract beauty and advantage of this principle seem
to be counterbalanced in practice by some unlmwm
cause, which, probably, must be sought tax in imperfect
clamping*

CHAPTER in.

OF OBOGRAPHT.

or the Figure of the £arthr-Ito exact DimeiMioni— Ito Fona duU ef fiqu-
libriam modified hy centriftif^ Force — ^Variation of Gnvifar on iti
Surface— Statical and Dynamical Measures of Gravity— TheFendu-
lum— Gravity to a S^roid— Other E£E6c1m of Earth's RDtatioi^— Trade
Winds— Determination of geo^phical Positions— Of TatitndiW' ■ Of
Longitudes— Conduct of a trigonometrical Survey— Of Maps — Tifh
jections of the Sphere— Measurement ef Hei|^ts by the Baionieter.

(161.) Geography is not only the most impcnrtant of
the practical branches of knowledge to which astronomy
is applied, but is also, theoretically speaking, an essen*
tial part of the latter science. The earth being the ge-
ner^ station from which we view the heavens, a know*
ledge of the local situation of particular stations on its
surface is of great consequence, when we come to inquire
the distances of the nearer heavenly bodies from ns, as
concluded from observations of their parallax as well as
on all other occasions, where a difference of locality can
be supposed to influence astronomical results. We pro*
pose, therefore, in this chapter, to explain the principles
by which astronomical observation is applied to geo»
graphicaSl determinations, and to give at the same timt

106 A TREATISE ON ASTRONOMY. [cHAP. III.

an outline of geography so far as it is to be considered a
part of astronomy.

(162.) Geography, as the word imports, is a delinea-
tion or description of the earth. In its widest sense, this
comprehends not only the delineation of the form of its
continents and seas, its rivers and mountains, but their
physical condition, climates, and products, and their
appropriation by communities oi men. With physical
and political geography, however, we have no concern
here. Astronomical geography has for its objects the
exact knowledge of the form and dimensions of the earth,
the parts of its surface occupied by sea and land, and the
configuration of the surface of the latter, regarded as pro-
tuberant above the ocean, and broken into the various
forms of mountain, table land, and valley ; neither should
the form of the bed of the ocean, regarded as a continual
tion of the surface of the land beneath the water, be left
out of consideration ; we know, it is true, very little of
it ; but this is an ignorance rather to be lamented, and,
if possible, remedied, than jocquiesced in, inasmuch as there
are many very important branches of inquiry which would
be greatly advanced by a better acquaintance with it.

(163.) With regard to the figure of the earth as a
whole, we have already shown that, speaking loosely, it
may be regarded as spherical ; but the reader who has
duly appreciated the remarks in art. 23 will not be at a
loss to perceive that this result, concluded from observa-
tions not susceptible of much exactness, and embracing
very small portions of the eurface at once, can only be
regarded as a first approximation, and may require to be
materially modified by entering into minutiae before neg-
lected, or by increasing the delicacy of our observations,
or by including in their extent larger areas of its surface.
For instance, if it should turn out (as it will), on minuter
inquiry, that the true figure is somewhat elliptical, or
fiattened, in the manner of an orange, having the diame-
ter which coincides with the axis about 3 J^yth part shorter
than tlie diameter of its equatorial circle ; this is so
trifling a deviation from the spherical form that, if a mo-
del of such proportions were turned in wood, and laid
before us on 'a table, the nicest eve or hand would not

CHAP. III.3 FIGUBS OF THE BARTH. 107

detect the flattening, since the difference of diameterst in
a globe of sixteen inches would amount only to ^^^th of
an inch. In all common pailance, and for all oroinary
purposes, then, it would still be called a ^obe ; while»
nevertheless, by careful measurement, the diffisrence
would not fail to be noticed, and, speaking strictly, it
would be termed, not a globe, but an oblate ellipsoid, or
spheroid, which is the name appropriated by gecHUeters
to the form above described.

(164.) The sections of such a figure by a plane are not
circles, but ellipses ; so that, on such a shaped earth, the
horizon of a spectator would nowhere (except at the
poles) be exactly circular, but somewhat elliptical. It is
easy to demonstrate, however, that its deviation from the
circular form, arising from so very slight an ** eUipticity**
as above supposed, would be quite imperceptible, not
only to our eyesight, but to the test of the dipsector ; so
that by that mode of observation we should never be 1^
to notice so small a deviation from perfect sphericity.
How we are led to this conclusion, as a practical resiut,
will appear, when we have explained the means of de-
termining with accuracy the dimensions of the whole, or
any part of the earth.

(165.) As we cannot grasp the earth, nor recede from
it far enough to view it at once as a whole, and compare
it with a known standard of measure in any degree com-
mensurate to its own size, but can only creep about upon
it, and apply our diminutive measures to comparativ^y
small parts of its vast surface in succession, it becomes
necessary to supply, by geometrical reasoning, the defect
of our physical powers, and fnmi a delicate aiid careful
measurement of such small parts to conclude the form
and dimensions of the whole mass. This would present
little difficulty, if we were sure the earth were strictly a
sphere, for the proportion of the circumference of a circle
to its diameter being known (viz. that of 3*1415026 to
l-OOOOOOO), we have only to ascertain the length of the
entire circumference of any great circle, such as a meri-
dian, in mileSf feet, or any oQier standard units, to know
the diameter in units of the same kind. Now the cir-
cumference of the whole eireic is known as soon as we

lOH A TftEATUE OM ASTB0N0H7. [cBAF. Ill*

know the exact length of any aliquot part of it, such as
1^ or ^j^th part ; and, this heing not more than about
seventy miles in length, is not beyond the limits of very
exact measurement, and could in fact, be measured (if
we knew its exact termination at each extremity) within
a very few feet, or, indeed, inches, by methods presently
to be particularized.

(166.) Supposing, then, we were to begin measuring
with all due nicety from any station, in the exact direc^
fion of a meridian, and go measuring on, till by some in-
dication we were informed that we had accomplished an
exact degree from the point we set out from, our problem
would then be at once resolved. It only remains, there-
fore, to inquire by what indications we can be sure, 1st,
that we heme advanced an exact degree ; and, 2dly , that we
have been measuring in the exact direction of a great circle*

(X67.) Now, the earth has no landmarks on it to in*
dicate degrees, nor traces inscribed on its surface to guide
US in snch a course. The compass, though it affords a
tolerable guide to the mariner or the traveller, is far too
uncertain in its indications, and too litde known in its
laws, to be of any use in such an operation. We must,
therefore, look outwards and refer our situation on the
surface of our globe to natural marks, external to it,
and which are of equal permanence and stability with the
earth itself. Such maiks are afforded by the stars. By
observations of their meridian altitudes, performed at any
station, and from their known polar distances, we con-
clude the height of the pole ; and since the altitude of the
pole is equal to the latitude of the place (art* 95), the
same observations give the latitudes of any stations where
we may establish the requisite instruments. When our
latitude, then, is found to have diminished a degree, we
know that, provided we have kept to the meridian^ we
have described one three hundred and sixtieth part of the
earth^s circumference. ''

(168.) The direction of the meridian may be secured
at every instant by the observations described in art. 187t
and although local difficulties may oblige us to deviate in
our measurement from this exact direction, yet if we
keep a strict account of the amount of this deviationi a

CHAP. UI.] LENGTH OF ▲ DEORBE OF LATtTUDIt. 109

very simple calculation will enable us to reduce our ob-
served measure to its meridionid value.

(160.) Such is the principle of that most important
geographical operation, the measurement of an arc of
3ie meridian. In its detail, however, a s<Hnewhat modi**
fied course piust be followed. An observatory cannot be
mounted and dismounted at every step ; so that we can-
not identify and measure an exact degree neither more
nor less. But this is of no consequence, provided we
know with equal precision how much, more or less, we
have measured. In place, then, of measuring this pre-
cise aliquot part, we take the moie convenient meuod
of measuring from one good observing station to another,
aboiU a degree, or two or three degrees, as the case may
be, apait, and determining by astronomical observation
the precise difference of latitudes between the stations.

(170.) Again, it is of great consequence to avoid in
this operation every source of uncertainty, because an
error committed in the length of a single degree will be
multiplied 360 times in the circumference, and nearly
115 times in the diameter of the earth concluded from it.
Any error which may afiect the astronomical determination
of a star's altitude will be especially influential* Now
there is still too much uncertainty and fluctuation in the
amount of refraction at moderate altitudes, not to make it
especially desirable to avoid this source of error. To
eflfect this, we take care to select for observation, at the
extreme stations, some star which passes through or near
the zeniihs of both. The amount of refraction, Within a
few degrees of the zenith, is very small, and its fluctnaF
tions and uncertainty, in point of quantity, so excessively
minute as to be utterly inappreciable. Now, it is the
same thing whether we observe the pole to be raised or
depressed a degree, or the zenith distance of a star when
on the meridian to have changed by the same quantity.
If at one station we observe any star to pass through the
zenith, smd at the other to pass one degree south or north
of the zenith, we are sure that the geographical latitudes,
or the akitudes of the pole at the two stations, must dif-
fer by the same amount.

(171.) Granting that the terminjal points of one degree
K

110 A TREATISE ON ASTRONOMY. [cHAP. HI.

can be ascertained, its length may be measured by the
methods which will be presently described, as we have
before remarked, to within a very few feet Now, the
error which may be committed in fixing each of these
terminal points cannot exceed that which may be com-
mitted in the observation of the zenith distance of a star,
properly situated for the purpose in question. This erfor,
with proper care, can hardly exceed a single second.
Supposing we grant the possibility of ten feet of error io
the measured length of one degree, and of one second in
each of the zenith distances of one star, observed at the
northern and southern stations, and, lastly, suppose all
these errors to conspire, so as to tend all of them to give
a result greater or all less than the truth, it will appear,
by a very easy proportion, that the whole amount of
error which would be thus entailed on an estimate of the
earth's diameter, as concluded from such a measure,
would not exceed 544 yards, or about the third part of a
mile, and this would be large allowance.

(172.) This, however, supposes that the form of the
earth is that of a perfect sphere, and, in consequence, the
lengths of its degrees in all parts precisely equal. But
when we come to compare the measures of meridional
arcs made in various parts of the globe, the results ob-
tained, although they agree sufficiently to show that the
supposition of a spheri^ figure is not very remote from
the tiiith, yet exhibit discordances far greater than what
we have shown to be attributable to error of observation,
and which render it evident that the hypothesis, in strict-
ness of its wording, is untenable. The following table
exhibits the lengths of a degree of the meridian (astro-
nomically determined as above described), expressed in
British standard feet, as resulting from actual measure-
ment, made with alt possible care and precision, by com-
missioners of various nations, men of the first eminence,
supplied by their respective governments with the best
instruments, and furnished with every facility which
could tend to insure a successful result of their import-
ant labours.*

* The first three columns of this table are estiscted from aram the
data giv^ to Proiessor Airjr's excellent paper " On the Figure of the
Earth." in the Encyclopeedia MetropoUtana.

CHAP, m.] DEGRESS IN DIFFERENT LATITUDES. Ill

Country.

Latitude
of Middle
of the Arc

Arc
okeasured.

Lensth

of the

Degree

eoDcluded.

Obaerrefs^

Sweden

66 90 10

. I«»3ri9"

365783

Svanberg.

Russia - '. '

58 17 37

335 5

365368 .

Struve.

Eofland

53 35 45

3 57 13

364971

Roy. Kater.

France

4653 2

8 90

364872

Lacailie, Caaaini.

Fraa«9 .

44 51 3

13 S3 13

364535

Delambre, M ecbain.

Borne - -

4350

2 9 47

364362

Boacovich.

America, U. 8. •

26) 13

1 38 45

363786

Mason. Dixon.

Cape of Good Hope

33 18 30

1 13 17i

364713

Lacailie.

India . .

16 8 82

15 57 40

363044

Lanbton, Evereat.

India . .

13 33 21

1 34 56

363013

Lambton.

Peru

1 31

3 7 3

363808

Condamioe, ice.

It is evident from a mere inspection of the second and
fourth columns of this tahle that the mecLSured length of
a degree increases tvith the latitude, being greatest near
the poles, and least near the equator. LiCt us now con-
sider what interpretation is to be put upon this conclusion,
as regards the form of the earth.

(173.) Suppose we held in our hands a model of the
earih smoothly turned in wood, it woi^ld be, as already
observed, so nearly spherical, that neither by the eye nor
the touch, unassisted by ins^mepts, could we detect any
deviation from that form. SupposCj too, we were debar-
red from measuring directly across from surface to surface
in different directions with any instrument, by which we
might at once ascertain whether one diameter were longer
than another ; how, then, we may ask, are we to ascer-
tain whether it is a true sphere or not ? It is clear that
we have no resource, but to endeavour to discover, by

some nicer means than simple inspection or feeling,
whether the convexity of its surface is the same in
every part ; and if not, where it is greatest, and where
least Suppose, then, a thin plate of metal to be cut into

112 A TREATISE ON ASTRONOMY. [cHAP. III.

a concavity at its edge, so as exactly to fit the surface at
A ; let this now be removed from A, and applied succes-
sively to several other parts of the surface, taking care to
keep its plane always on a great circle of the globe, as
here represented. If, then, we find any position, &, in
which the light can enter in the middle between the globe
and plate> or any other, C, where the latter tilts by pres-
sure, or admits the light under its edges, we are sure that
the curvcUure of the surface at B is less^ and at O greater
than at A.

(174.) What we here do by the applicalion of a metal
plate of determinate length and curvature, we do on the
earth by the measurement of a degree of variation in the
altitude of the pole. Curvature of a surface is nothing
but the continual deflection of its tangent from one fixed
direction as we advance along it. When, in the same
measured distance of advance^ w6 find the tangent
(which answers to our horizon) to have shifted its posi-
tion with respect to a fixed direction in space (such as
the axis of the heavens, or the line joining the earth's
centre and some given star), more in one part of the
earth's meridian than in another, we conclude, of ne-
cessity, that the curvature of the surface at the former
spot is greater than at the latter ; and, vice versa, when,
in order to produce the same change of horizon with
respect to the pole (suppose P), we require to travel
over a longer measured space at one point than at an-
other, we assign to that point a less curvature. Hence
we conclude that the curvature of a meridional section
of the earth is sensibly greater at the equator than to-
wards the poles ; or, in «ther words, that the earth is
not spherical, but flattened at the poles, or, which comes
to the same, protuberant at the equator.

(175.) Let NABDEF represent a meridional section
of the earth, C its centre, and NA, BD, GE, arcs of a
meridian, each corresponding to one degree of difference
of latitude,. or to one degree of variation in the meridian
altitude of a star, as referred to the horizon of a spectator
travelling along the meridian. Let nN, cA, 6B, rfD, gG,
«E, be the respective directions of the plumb-line at the
stations N, A, B, D, G, E, of which we will suppose N

I

jCHAF. ni.3 MSIXDIONAL SECTION OF THS BASTH. 118

to be at the pole and E at the equator ; then will the tan-
gents to the surface at these points respectively be per-
pendicular to these directions ; and, consequently, if each
pair, viz. nN and aA, 6B and dD, gG and eE, be pro-
longed till they intersect each other (at the points x, y • z)^
the angles NorA, ByD, OzE, will each be one degree,
tmd, therefore, all equal ; so that the small curvilinear
arcs NA, BD, GE, may be regarded «s arcs of circles
of one degree each, described about a?, y, z, as centres.
These are what in geometry arc called centres ofcurva-
iurt, and the radii arN or xA, yB or yD, zQ or zE, fe-
present radii of curvature^ by which the ciurvatues at
those points are determined and measured. Now, as the
arcs of different circles, which subtend equal angles at
their respective centres, are in the direct proportion of
their radu, and as the arc NA is greater than BD, and
that again than 6E, it follows that the radius N:r must
be greater than By, and By than Ej. Thus it appears
that the mutual intersections of the plumb-lines will not,
as in the sphere, all coincide in one point G, the centre,
but will be arranged along a certain curve, ocyz (virhich
will be rendered more evident by considering a number
of intermediate stations). To this curve geometers have
given the name of the evolute of the curve NABDGE,
from whose centres of curvature it is constructed.
k2

114 A TREATISE ON ASTRONOMY. [CBAP. III.

(176.) In' the flattening of around fignre at two op-
posite points, and its protuberance at points rectangularly
situated to the former, we recognise the distinguishing
feature of the elliptic torm. Accordingly, the next and
simplest supposition that we can make respecting the
nature of the meridian, since il; is proved not to be a
eirde, is, that it is an ellipse, or nearly so, having NS,
the axis of the earth, for its shorter, and EF, the equa-
torial diameter, for its longer axis ; and that the form of
the earth's surface is that which would arise from making
such a curve rev6lve about its shorter axis NS. Thi«
agrees well with the general course of the increase of
the degree in going from the equator to the pole. In the
ellipse, the radius of curvature at £, the extremity of the
longer B.tii» is.lhe leaM, and at that of the shorter axis,
the greatest it admits, and the form of its evoJufe agrees
with that here represented.* Assuming, then, that it is
an ellipse, the geometrical properties of that curve enar
ble us to assign the proportion between the lengths of its
•axes which shall correspond to any proposed rate of var
nation in lis curvature, as well as to fix upon their ab-
solute lengths, corresponding to any assigned length of
the degree in a given latitu(te. Without troubling th«
reader with the investigation (which may be found in
any work on the conic seetions), it will be suffi.cient to
atate that the lengths which agree on the whole best with
the entire series of meridional arcs which have been
0i9.tisfactorily measured, are as follow :t —

Feet. Miles.

Greater or equatorial diameter :;= 41 ^7,486 =7925-648

Lesser or polar diameter = 41,707,680= 7899170

Diflbmice of diameters, or polar com-) ^ 139306= 26*478
pression )

J he proportion of the diameters is very neatly that of
: 299, and their difference -^^ of the greater, or a
very little greater than ^ij^.

Q77.)« Thus we see that the rough diameter of 8000
miles w6 have hitherto used is rather too great, the ex-
cess being about 100 miles, or ^th part. We consider
it extremely improbable that ah error to the extent of

* The dotted lines are the portions of the evolute belonging to the other
quadrants,
t See FroleM. Aiiy^s Emay before cited.

CHAP, in.] EXACT DIMENSIONS OF THE EARTH. 115

five miles can subsist in the diameters, or aa uncertainty
to that of a tenth of its whole quantity in the com-
pression just stated. As convenient numbers to remem-
'ber, the reader may bear in mind, that in our latitude
there are just as many thousands of feet in a degree of
the meridian as there are days in the year (365) : that,
speaking loosely^ a degree is about 70 British statute
miles, and a second about 100 feet ; and that the equa*
torial circumference of the earth is a little less than
25,000 miles (24,890).

(178.) The supposition of an elliptic form of the
earth's ^section through the axis is recommended by its
simplicity, and confirmed by comparing Uie numeiieai
results we have just set down with those of actual mea-
surement. When this comparison is executed, discord-
ances, it is true, are observed, which, although still too
great to be referred to error of measurement are yet so
Bmall, compared to the errors which would result from
the spherical hypothesis, as completely to justify our
regarding the earth as an ellipsoid, and referring the
lObserved deviations to either local or, if general, to com-
paratively small eauses.

(170.) Now, it is highly satisfactory to find that the
general elliptical figure thus practicaily proved to exist,
is precisely what ought theoretically to result from the
rotation of the earth on its axis. For, let us suppose .
the earth a sphere,, at rest, of uniform materials through-
out, and externally covered with an ocean of equal depth
in every part. Under such circumstances it would ob-
viously be in a state of equilibrium j and the water on
1*8 surface would have no tendency to run one way or
the other. Suppose, now, a quantity of its materials
were taken from the polar regions, and piled up all
around the equator, so as to produce that difference of
the polar and equatorial diameters of 96 miles which we
know to exist. It is not less evident that a mouft^ain
ridge or equatorial continent^ only, would be thus form-
ed, from which the water would run down to the ex-
cavated part at the poles. However solid matter might
rest where it was placed, the liquid part, at least, would
not remain there, any more than if it were thrown oa
the side of a hilL The consequence* therefore, would

116

A TRSATIBE ON ASTRONOMIT. [CHAP. Ill*

be the fonnation of two great polar seas, hemmed in all
round by equatorial land. Now, this is by no means
the case in nature. The ocean occupies, indifferentlyy
all latitudes, with no more partiality to the polar than to
the equatorial. Since, then, as we see, the water oc-
cupies an elevation above the centre no less than 13
, miles greater at the equator than at the poles, and yet
manifests no tendency to leave the former and run to-
wards the latter, it is evident that it must be retained in
that situation by some adequate power. No such power,
however, would exist in the case we have supposed,
which is therefore not conformable to nature. In other
words, the spherical form is not the figure of equili'
brium ; and therefore the earth is either not at rest or
is so internally constituted as to attract the water to its
equatorial regions, and retain it there. For the latter
supposition .there is no prima fade probability, nor any
analogy to lead us to such an idea. The former is in
accordance with all the phenomena of the apparent
diurnal motion of the heavens ; and, therefore, if it will
furnish us with the power in question, we can have no
hesitation in adopting it as the true one.

^ (180.) Now, every body knows that

when a weight is whirled round, it ac-
quires thereby a tendency to recede
from the centre of its motion ; which is
called the centrifugal force. A stone
whirled round in a sling is a common
illustration ; but a better, for our pre-
sent purpose, will be a pail of water, sus-
pended by a cord, and made to 9pin
round, while the cord hangs perpendi-
cularly. The surface of the water, in-
stead of remaining horizontal, will be-
come concave, as in the figure. The
centrifugal force generates a tendency in
ail the water to leave the axis, and
press towards the circumference ; it is,
therefore, urged against the pail, and
forced up its sides, till the excess of
^ height, and consequent increase of pres**
sure downwards, just counterbalances its

CHAP. Itl.] ACTION OF THE SEA ON THE LAND. 117

centrifugal force, and a sttUe of equilibrium is attained.
The experiment is a very easy and instructive one, and
is admirably calculated to show how the form of equiR'
brium accommodates itself to varying circumstances.
If, for example, we allow the rotation to cease by degrees,
as it becomes slower we shall see the concavity of the water
regularly diminish ; the elevated outward portion will de-
scend, and the depressed central rise, while all the time a
perfeedy smooth surface is maintained, till the rotation is
exhausted, when the water resumes its horizontal state.

(181.) Suppose, then, a globe, of the size of the earth,
at rest, and covered with a uniform ocean, were to be set
in rotation about a certain axis, at first very Slowly, but
by degrees more rapidly, till it turned round once in
twenty-four hours ; a centrifugal force would be thus gene-
rated, whose general tendency would be to urge the water
at every point of the surface to recede from the aoois,
A rotation might, indeed, be conceived so swift as to flirt
the whole ocean from the surface^ like water from a mop.
But this would Squire a far greater velocity than what
we now speak of. In the case supposed, the weight
of the water would still keep it on the earth : and the
tendency to recede from the axis cotdd only be satisfied,
therefore, by the water leaving the poles, and flowing
towards the equator; there heaping itself up in a ridge,
just as the water in our pail accuinulates against the side ;
and being retained in opposition to its weight, or natural
tendency towards the centre, by the pressure thus caused.
This, however, could not take place without laying dry
the polar portions of the land in the form of immensely
protuberant continents ; and the difference of our supposed
cases, therefore, is this:—- in the. former, a great equato-
rial continent and polar seas would be formed ; in the
latter, protuberant land would appear at the poles, and a
zone of ocean be disposed around the equator. This
would be the first or immediate effect. Let us now see
what would afterwards happen, in the two cases, if things
were allowed to take their natural course.

(182.) The sea is constantly beating on the land,
grinding it down, and scattering its worn off pslrticles and
fragments, in the st^te of mud and pebbles, over its bed.

118 A TREATISE ON ASTRONOMY. [cHAP. HI*

Geological facts afford abundant proof that the existing
continents have all of them undergone this process, even
more than once, and been entirely torn in fragments, or
reduced to powder, and submerged and reconstructed.
Land, in this view of the subject, loses its attribute of
fixity. As a mass it might hold together in opposition
to forces which the water freely obeys ; but in its state
of successive or simultaneous degradation, when dissemi-
nated through the water, in the state of sand or mud, it
is subject to all the impulses of that fluid. In the lapse
of time, then, the protuberant land in both cases would
be destroyed, and spread over the bottom of the ocean,
filling up the lower parts, and tending continually to re-
model the surface of the solid nucleus, in correspondence
with the /orm of equilibrium in both cases. Thus, after
a sufficient lapse of time, in the case of an earth at rest,
the equatorial continent, thus forcibly constructed, would
again be levelled and transferred to the polar excavations,
and the spherical figure be so at length restored. In
that of an earth in rotation, the poktr protuberances
would gradually be cut down and disappear, being trans-
ferred to the equator (as being then the deepest sea), till
the earth would assume by degrees the form we observe
it to have— that of a flattened or oblate ellipsoid.

(183.) We are far from meaning here to trace the pro-
cess by which the earth really assumed its actual form ;
all we intend is, to show that this is the form to which,
under the condition of a rotation on its axis, it must tendf
and which it would attain, even if originally xmd (so to
speak) perversely constituted otherwise.

(184.) But, further, the dimensions of the earth and
the time of its rotation being known, it is easy thence to
calculate the exact amount of the centrifugal force,*
which, at the equator, appears to . be -^ ^ ^^ of the
force or weight by which all bodies, whether solid or
liquid, tend to fall towards the earth. By this fraction
of its weight, then, the sea at the equator is lightened^
and thereby rendered susceptible of being supported at a
higher level, or more remote from the centre &an at the
poles, where no such counteracting force exists; and
* See Cab. Cjrc Mechaniob, c. viiL

GHAF. III. J LOCAL VARIATION OF ORAVfTT. 119

where, in consequence, the water may be considered as
8pecificaUy heavier. Taking this principle as a guide,
and combining it with the laws of gravi^ (as developed
by Newton, and as hereafter to be more fully explained),
mathematicians have been enabled to investigate, a pri»
ariy what would be the figure of equilibrium of such a
body, constituted internally as we have reason to believe
the earth to be ; covered wholly or partially with a fluid ;
and revolving uniformly in twenty-four hours ; and the
result of this inquiry is foutid to agree very satisfactorily
with what experience shows to be the case. From their
investigations it appears that the form of equilibrium is,
in fact, no other Uian an oblate ellipsoid, of a degree of
ellipti(»ty very nearly identical with what is observed,
and which would be no doubt accurately so, did we know
the internal constitution and materials of the earth.

(185.) The confirmation thus incidently furnished, of
the hypothesis of the earth's rotation on its axis, cannot
fail to strike the reader. A deviation of its figure from
that of a sphere was not contemplated among the original
reasons for adopting that hypothesis, which was asstimed
solely on account of the easy explanation it ofTers of the
apparent diurnal motion of the heaivens. Yet we see
that, once admitted, it draws with it, as a necessary con-
sequence, this other remarkable phenomenon, of which
no other satisfactory account could be rendered. Indeed,
so direct is their connexion, that the ellipticity of the
earth's figure wiis discovered and demonstrated by New-
ton to be a consequence of its rotation, and its amount
actually calculated by him, long before any measurements
had suggested such a conclusion. As we advance with
our. subject, we shall find the same simple principle
branching out into a whole train of singular and import-
ant consequences, some obvious enough, others which
at first seem entirely unconnected with it, and which,
until traced by Newton up to this their origin, had
ranked among the mest inscrutable arcana of astronomy,
as well as among its grandest phenomena.

(186.) Of its more obvious consequences, we may here
mention one which falls in naturally with our present
subject. If the earth really revolve on its axis, this rota-

120 A TREATISE ON ASTRONOMY. [cHAP. III.

tioa must generate a centrifugal force (see art. 184), ike
effect of which must of course be to counteract a certain,
portion of the weight of every body situated at the equa-
tor, as compared with its weight at the poles, or in any
intermediate latitudes. Now, this is fully confirmed by
experience. There is actually observed to exist a differ-
ence in the gravity, or downward tendency, of one and
the same body, when conveyed successively to stations
in different latitudes. Experiments made with the great-
est care, and in every accessible part of the globe, have
fully demonstrated the fact of a regular and progressive
increase in the weights of bodies corresponding to the
increase of latitude, and fixed its amount and the law of
its progression. From these it appears, that the extreme
amount of this variation of gravity, or the difference be-
tween the equatorial and polar weights of one and the
same mass of matter, is one part in 104 of its whole
weight, the rate of increase in travelling from the equa-
tor to the pole being as the square of the sine of the lati-
tilde.

(187.) The reader will here naturally inquire, what is
meant by speaking of the same body as having different
weights at different stations ; and, how such a fact, if
true, can be ascertained. When we weigh a body by a
balance or a steelyard we do but counteract its weight by
the equal weight of another body under the very same
circumstances; and if both the body weighed and its
counterpoise be removed to another station, their gravity,
if changed at all, will be changed equally, so tl^t they
will stiU continue to counterbalance each other. A dif-
ference in the intensity of gravity could, therefore, never
be detected by these means ; nor is it in this sense that
we assert that a body weighing 104 pounds at the equa-
tor will weigh 105 at the pole. If counterbalanced in a
scale or steelyard at the former station, an additional
pound placed In one or other scale at the latter would
inevitably sink the beam.

(188.) The meaning of the proposition may be thus ex-
plained :-— Conceive a weight x suspended at the equator
by a string without weight passing over a pulley, A, and
conducted (supposing such a thing possible) over other

CHAP. III.J STATICAL MEASURE OF GRAVITY.

131

pulleys, such as B, round the earth's convexity, till the
other end hung down at the pole, and there sustained the
weight y. If, then, the weights x
and y were such as, at any one sta-
tion, equatorial or polar, would ex-
actly counterpoise each other on a
balance or when suspended side
by side over a single pulley, they
would not counterbalance each
other in this supposed situation, but
the polar weighty would preponde-
rate ; and to restore the equipoise
the weight a? must be increased by^^th part of its quantity.
(189.) The means by which this variation of gravity
may be shown to exist, and its amount measured, are
twofold (like all estimations of mechanical power), stati-
cal and dynamical. The former consists in putting the
gravity of a weight in equilibrium, not with that of an-
other weight, but with a natural power of a different kind
not liable to be affected by local situation. Such a power
is the elastic force of a spring. Let ABC be a strong
support of brass standing on the foot AED cast in one
piece with it, into which is let a
smooth plate of agate, D, which can
be adjusted to perfect horizontality
by a level. At G let a spiral spring
6 be attached, which carries at its
lower end a weight F, 'polished and
convex below. The length and
strength of the spring must be so ad-
justed that the weight F shall be sus-
tained by it just to swing clear of
contact with the agate plate in the
highest latitude at which it is intend-
ed to use the instrument. Then, if
small weights be added cautiously, it
may be made to descend till it just
grazes the agate, a contact which can
be made with the utmost imaginable «
delicacy. Let these weights be noted ; the weight F de-
tached ; the spring G carefully lifted off ita hook, and
L

122 A lllEATISE ON ASTRONOMY. [^CHAP. lU*

secured, for travelling, from rust, strain, or disturbance,
and the whole apparatus conveyed to a station in a lower
latitude. It will then be found, on remounting it, that,
although loaded with the same additional weights as be-
fore, the weight F will no longer have power enough
to stretch the spring to the extent required for producing
a similar contact. More weights will require to be add-
ed ; and the additional quantity necessary will, it is evi-
dent, measure the difference of gravity between the two
stations, as exerted on the whole quantity of pendent
matter, i. e. the sum of the weig-ht of F and Iialf that
of the spiral spring itself. Granting that a spiral spring
can be constructed of such strength and dimensions
that a weight of 10,000 grains, including its own, shall
produce an elongation of 10 inches without permanently
straining it,* one additional grain will produce a further
extension of y^Virth of an inch, a quantity which cannot
possibly be mistaken in Buch a contact as that in questioii*
Thus we should be provided with the means of mea-
suring the power of gravity at any station to within
TUTTTT^h of its whole quantity.

(100.) The other, or dynamical process, by which the
force urging any given weight to the earth may be de-
termined, consists in ascertaining the velocity imparted
by it to the Weight when suffered to fall freely in a given
time, as one second. This velocity cannot, indeed, be
directly measured ; but indirectly, the principles of me-
chanics furnish an easy and certain means of deducing it,
and, consequently, the intensity of gravity, by observing
the oscillations of a pendulum. It is proved in mecha-
nics (see Cab. Cyc, Mechanics, 216), that, if one and
the same pendulum be made to oscillate at different sta-
tions, or under the influence of different forces, and the
numbers of oscillations made in the same time in each

* Whether the process above described could ever be so far perfected
and refined as to become a substitute fi>r the use of the pendulum most
depend on the degree of permanence and unifbrmily of action of springs,
on the constancy or variaoility of the efiect of temperature, on their elas-
tic force, on the possibility of transporting them, absolutely unaltered,
fh>m place to place, &c. The great advantages, however, which such
an apparatus and mode oT observation would possess, in point of cmxv^
iiience, cheiqness, portability, and expedition, over the present laborioiis,
tedious, and ezpenaivo process, render the attempt well worth making.

CHAP, in.3 ORAVITT ON A SPHEROID. 123

case be counted, the intensities of the forces will be to
each other inversely as the squares of the numbers of
oscillations made, and thus ^eir proportion becomes
known. For instance, it is found that, under the equa-
tor, a pendulum of a certain form and length makes
86,400 vibrations in a mean solar day ; and that, when
transported to London, the same pendulum makes 86,535
vibrations in the same time. Hence we conclude, that
the intensity of the force urging the pendulum down-
wards at the equator is to that at London as 86400 to
86535, or as 1 to 1*00315; or, in other words, that a
mass of matter at the equator weighing 10,000 pounds
exerts the same pressure on the ground, and the same
effort to crush a body placed below it, that 10,031^ of the
same pounds ^ transported to London, would exert there.

(191.) Experiments of this kind have been made, as
above stated, with the utmost care and minutest precaution
to insure exactness in all accessible latitudes ; and their
general and final result has been, to give j^ for the frac-
tion expressing the difference of gravity at the equator
and poles. Now, it will not fail to be noticed by the
the reader, and will, probably, occur to him as an objec-
tion against the explanation here given of the fact by the
earth's rotation, that this differs materially from the frac-
tion jij expressing the centrifugal force at the equator.
The difference by which the former fraction exceeds the
latter is j^^^, a small quantity in itself, but still far too
large, compared with the others in question, not to be
distinctly accounted for, and not to prove fatal to this ex-
planation if it will not render a strict account of it.

(192.) The mode in which this difference arises af-
fords a curious and instructive example of the indirect
influence which mechanical causes often exercise, and
of which astronomy furnishes innumerable instances.
The rotation of the earth gives rise to the centrifugal
force ; the centrifugal force produces an eUipticity in 3ie
form of the earth itself; and this very eUipticity of form
modifies its power of attraction on bodies placed at its
- surface, and thus gives rise to the difference in question.
Here, then, we have the same cause exercising at once a
direct and an indirect influence . The amount of the former

124 A TREATISE ON ASTRONOMT. [cHAP. III.

id easily calculated, that of the latter with far more diffi-
culty, by an intricate and profound application of geo-
metry, whose steps we cannot pretend to trace in a work
like the present, and can only state its nature and result.

(193.) The weight of a body (considered as undimi-
nished by a centrifugal force) is die effect of the earth's
attraction on it. This attraction, as Newton has demon-
strated, consists, not in a tendency of all matter to any
one particular centre, but in a disposition of every parti-
cle of matter m the universe to press towards, and if not
opposed to approach to, every other. The attraction of
the earth, then, on a body placed on its surface, is not a
simple but a complex force, resulting from the separate
attractions of all its parts. Now, it is evident, that if the
earth were a perfect sphere, the attraction exerted by it
on a body any where placed on its surface, whether at
its equator or pole, must be exactly alike, for the simple
reason of the exact symmetry of the sphere in every di-
rection. It is not less evident that, the earth being ellip-
tical, and this symmetry or similitude of all its parts not
existing, the same result cannot be expected. A body
placed at the equator, and a similar one at the pole of a
flattened ellipsoid, stand in a different geometrical rela-
tion to the mass as a whole. This difference, without
entering further into particulars, may be expected to
draw with it a difference in its forces of attraction on the
two bodies. Calculation confirms this idea. It is a
question of purely mathematical investigation, and has
been treatied with perfect clearness and precision by New-
ton, Maclaurin, Clairaut, and many other eminent geo-
meters ; and the result of their investigations is to show
that owing to the elliptic form of the earth alone, and in-
dependent of the centrifugal force, its attraction ought to
increase the weight of a body in going from the equator
to the pole by almost exactly y^th part ; which, toge-
ther with ^-J^th due to the centrifugal force, make up the
whole quantity, 777*^, observed.

(194.) Another great geographical phenomenon, which
owes its existence to the earth's rotation, is that of the
trade-winds. These mighty currents in our atmosphere,
on which so important a part of navigation depends.

CHAP. III.] THE TRADE-WINDS. 125

arise from, 1st, the unequal exposure of the earth's sur-
face to the sun's rays, by which it is unequally heated
in different latitudes ; and, 2dly, from that general law
in the constitution of all fluids, in virtue of which they
occupy a larger bulk, and become specifically lighter
when hot than when cold. These causes, combined with
the earth's rotation from west to east, afford an easy and
satisfactory explanation of the magnificent phenomena in
question.

(195.) It is a matter of observed fact, of which we
shall give the explanation farther on, that the sun is con-
stantly vertical over some one or other part of the earth
between two parallels of latitude, called the tropics, re-
spectively 23s^ north, and as much south of the equator;
and that the whole of that zone Or belt of the earth's sur-
£aice included between the tropics, and equally divided
by the equator, is, in consequence of the great altitude
attained by the sun in its diurnal course, maintained at a
much higher temperature than those regions to the north
and south which lie nearer the poles. Now, the heat thus
acquired by the earth's surface is communicated to the
incumbent air, which is thereby expanded, and rendered
specifically lighter than the air incumbent on the rest of
the globe. It is, therefore, in obedience to the general
laws of hydrostatics, displaced and buoyed up from the
surface, and its place occupied by colder, and therefore
heavier air, which glides in, on both sides, along the
surface, from the regions beyond the tropics ; while the
displaced air, thus raised above its due level, and unsus-
tained by any lateral pressure, fiows over, as it were,
and forms an upper current in the contrary direction, or
toward the poles ; which, being cooled in its course, and
also sucked down to supply the deficiency in the extra-
tropical regions, keeps us thus a continual circulation.

(196.) Since the earth revolves about an axis passing
through the poles, the equatorial portion of its surface
has the greatest velocity of rotation, and all other parts
less in the proportion of the radii of the circles of lati-
tude to which they correspond. But as the air, when
relatively and apparently at rest on any part of the earth's
surface* is only so because in reality it participates in the
l2

126 A TREATISE ON ASTRONOMY. [CHAP. Ill,

motion of rotation proper to that part, it follows that
when a mass of air near the poles is transferred to the
region near the equator by any impulse. urging it direct-
ly towards that circle, in every point of its progress to-
wards its new situation it must be found deficient in ro-
tatory velocity, and therefore unable to keep up with the
speed of the new surface over which it is brought.
Hence, the currents of air which set in towards the
equator from the north and south must, as they glide
along the surface, at the s^me time lag, or hang back,
and drag upon it in the direction opposite to the earth's
rotation, L e. from east to west. Thus these currents,
which but for the rotation would be simply northerly
and southerly winds, acquire, from this cause, a relative
direction towards the west, and assume the character of
permanent north-easterly and south-easterly winds.

(197.) Were any considerable niE^ss of air to be sud-
denly transferred from beyond the tropics to the equator,
the difference of the rotatory velocities proper to the two
situations would be so great as to produce not merely a
wind, but a tempest of the most destructive violence.
But this is not the case ; the advance of the air from the
north and south is gradual, and all the while the earth is
continually acting on, and by the friction of its surface
accelerating its rotatory velocity. Supposing its progress
towards the equator to cease at any point, this cause
would almost immediately communicate to it the defi-
cient motion of rotatioi;!, after which it would revolve
quietly with the earth, and be at relative rest. We have
only to call to mind the comparative thinness of the coat-
ing which the atmosphere forms around the globe (art.
34), and the immense mass of the latter, compared with
the former (which it exceeds at least 100,000,000 times),
to appreciate fully the absolute command of any exten-
sive territory of the earth over the atmosphere immedi-
ately incumbent on it, in point of motion.

(198.) It follows from this, then, that as the winds on
botii sides approach the equator, their easterly tendency
must diminish.* The lengths of the diurnal circles in-

♦ See Captain Hall's " Fragments of Voyages and Travels," 2d series,
vol. i. p. 162, where this is veiy distmctly, and, so far as I am aware, fi>r
the first time, reasoned (mt— Author.

CHAP. III.^ COMPENSATION OF THE TRADE-WINDS. 127

crease very slowly in the immediate vicinity of the equate
tor, and for several degrees on either side of it hardly
change at all. Thus the friction of the surface has more
time to act in accelerating the velocity of the air, bring-
ing it towards a state of relative rest, and diminishing
thereby the relative set of the currents from east to west,
which, on the other hand, is feebly, and, at length, not
at all reinforced by the cause which originally produced
it. Arrived, then, at the equator, the trades must be
expected to lose their eastedy character altogether. But
not only this but the northern and southern currents, here
meeting and opposing, will mutually destroy each other,
leaving only such preponderancy as may be due to a
difference of local causes acting in the two hemispheres,
which in some regions around the equator may lie one
way, in some ano&er.

(199.) The result, theu, must be the production of
two great tropical belts, in the northern of which a con-
stant north-easterly and in the southern a south-easterly,
wind must prevail,, while the winds in the equatorial
belt, which separates the two former, should be compa-
ratively calm and free from any steady prevalence of
easterly character. All these consequences are agreeable
to observed fact, and the system of aerial currents above
described constitutes in reality what is understood by
the regular trade-unnds.*

(20a.) The constant friction thus produced between
the earth and atmosphere in the regions near the equator
must (it may be objected) by degrees reduce and at
length destroy the rotation of the whole mass. The
laws of. dynamics, however, render such a coAsequence
generally impossible ; ^ind it is easy to see, in the pre-
sent case, where and how the compensation takes place.
The heated equatorial air, while it rises and flows over
towards the poles, carries with it the rotatory velocity
due to its equatorial situation into a higher latitude,
where the earth's surfax;e has less motion. Hence, as
it travels northward or southward, it will gain conti-
nually more and more on the surface of the earth in its
diurnal motion, and assume constantly more and more a
* See the work last cited.

128 A TREATISE ON ASTRONOIUY. [CfHAP. III.

westerly relative direction ; and when at length it returns
to the surface, in its circulation, which it must do more
\>x less in all the interval between the tropics and the
poles, it will act on it by its friction as a powerful south-
west wind in the northern hemisphere, and a north-west
in the southern, and restore to it the impulse taken up
from it at the equator. We have here the origin of the
south-west and westerly gales so prevalent in our lati-
tudes, and of the almost universal westerly winds in
the North Atlantic, which are, in fact, nothing else than
a part of the general system of the reaction of the
trades, and of the process by which the equilibrium of
the earth's motion is maintained under their action.*

(201.) In order to construct a map or model of the
earth, and obtain a knowledge of the distribution of sea
and land over its surface, the forms of the outlines of its
continents and islands, the courses of its rivers and
mountain chains, and the relative situations, with respect
to each other, of those points which chiefly interest us,
as centres of human habitation, or from other causes, it
is necessary to possess the means of determining correctly
the situation of any proposed station on its surface. For
this, two elements require to be known, the latitude and
longitude, the former assigning its distance from the
poles or the equator, the latter, the meridian on which
that distance is to be reckoned. To these, in strictness,
should be added, its height above the sea level ; but the

* As it is our object merely to illustrate the mode in which the earth's
Botation afibcts the atmosphere on the great scale, we omit all considera-
tion, of local periodical winds, such as monsoons, &c.

It seems worth inquiry, whether hurricanes in tropical climates may
not arise from portions of the upper currents prematurely diverted down-
wards before their relative velocity has been sufficiently reduced by fric-
tion (HI, and gradual mixing with, the lower strata ; and so dashing upon
the earth with that tremendous velocity which gives them their destruc-
tive character, and of which hardly any rational account has yet been
given. Their course, generally speaking, is in opposition to the regular
trade-wind, as it ought to be, in conformity with this; idea. (Young's
Lectures, i. 704.) But it by no means follows that this must always be
the case. In |^eneral, a rapid transfer, either way, in latitude, ot any
mass of air which local or temporary causes might carry abwe the im-
mediate reach of the friction of the earth's surface, would give a fearfiil
exan^eration to its velocity. Wherever such a mass should strike the
aarth^ a hurricane might arise ; and should two such masses encouQter
in mid-air, a tornado of any degree of intensity on record might easily
result fiom their combination.— Att(Aor.

CHAP, m.^ GEOGRAPHICAL LATITUDES DETERmNED. 120

consideration of this had better be deferred, to avoid
complicating the subject.

(202.) The latitude of a station on a sphere would be
merely the length of an arc of the meridian, intercepted
between the station and the nearest point of the equator,
reduced into degrees. (See art. 86.) But as the earth
is elliptic, this mode of conceiving latitudes becomes
inapplicable, and we are compelled to resort for our de-
finition of latitude to a generalization of that property
(art. 95), which affords the readiest means of determin-
ing it by observation, and which has the advantage of
being independent of the figure of the earth; which,
after all, is not exactly an ellipsoid, or any known geo-
metrical solid. The latitude of a station, then, is the
altitude of the elevated pole, and is, therefore, astrono-
mically determined by those methods already explained
for ascertaining that important element. In consequence,
it will be remembered that, to make a perfectly correct
map of the whole, or any part of the earth's surface^
equal differences of latitude are not represented by ex-
actly equal intervals of surface.

(203.) To determine the latitude of a station, then, is
easy. It is Otherwise with its longiti^de, whose exact de-
termination is a matter of more difficulty. The reason
is this: — ^ there are no meridians marked upon the
earth, any more than parallels of latitude, we are obliged
in this case, as in the case of the latitude, to resort to
marks external to the earth, L e. to the heavenly bodies,
for the objects of our measurement ; but with this dif-
ference in the two cases — to observers situated at sta-
tions on the same rtieridian (t. c. differing in latitude)
the heavens present different aspects at aU moments.
The portions of them which become visible in a com-
plete diurnal rotation are not the same, and stars which
are common to both describe circles differently inclined
to their horizons, and differently divided by them, and
attain different altitudes. On the other hand, to ob-
aervers situated on the same parallel (i. 6. differing only
^ longitude) the heavens present the same aspects.
Their visible portions are the same ; and the same stars
describe circles equally inclined, and similarly divided

130 A TAEATISE ON ASTHONOMY. fcHAP. TJLl,

by their honzons, and attain the same altitudes. In the
former case there is, in the latter there is not, any thing
in the appearance of the heavens, watched through a
whole diurnal rotation, which indicates a difference of
locality in the observer.

(204.) But no two observers, at different points of the
earth's surface, can have at the same instant the same
celestial hemisphere visible. Suppose, to fix our ideas,
an observer stationed at a given point of the equator,
and that at the moment when he noticed some bright
star to be in his zenith, and therefore on his meridian,
he should be suddenly transported, in an instant of time,
round one quarter of the globe in a westerly direction, it
is evident that he will no longer have the same star ver-
tically above him : it will now appear to him to be just
rising, and he will have to wait six hours before it again
comes to his zenith, t. e. before the earth's rotation from
west to east carries him back again to the line joining
the star and tfie earth's centre from which he set out.

(206.) The difference of the cases, then, may be thus
stated, so as to afford a key to the astronomical solution
of the problem of the longitude. In the case of stations
differing only in latitude, the same star comes to the
meridian at the same time, but at different altitudes. In
that of stations differing only in longitude, it comes to
the meridian at the same cdtitude, but at different times.
Supposing, then, that an observer is in possession of any
means by which he can certainly ascertain the time of a
known star's transit across his meridian, he knows his
longitude ; or if he knows the difference between its
times of transit across his meridian and across that of
any other station, he knows their difference of longitudes.
For instance, if the same star pass the meridian of a
place A at a certain moment, and that of B exactly one
hour of sidereal time, or one twenty-fourth part of the
earth's diurnal period, later, then the difference of longi-
tudes between A and B is one hour of time or 15^, and
B is so much west of A.

(206.) In order to a perfectly clear understanding of
the principle on which the problem of finding the longi-
tude by astronomical observations is resolved, the reader

CHAP. III.] DETERMINATION OF LONGITUDES. 131

must learn to distinguish between time, in the abstract,
as common to the whole universe, and therefore reckoned
fix)m an epoch independent of local situation, and local
timet which reckons, at each particular place, from an
epoch, or initial instant, determined by local convenience.
Of time reckoned in the former, or abstract maimer, we
have an example in what we have before defined as equi-
noctial time, which dates from an epoch determined by
the sun's motion among the stars. Of the latter, or locM
reckoning, we have instances in every sidereal clock in
an observatory, and in every town clock for common use.
Every astronomer regulates, or aims at regulating, his
sidereal clock, so that it shall indicate 0** 0" ()•, when a
certain point in the heavens, called the equinox, is on the
meridian of his station. This is the epoch of his side-
real time ; which is, therefore, entirely a local reckoning.
It gives no information to say that an event happened at
such and such an hour of sidereal time, unless we parti-
cularize the station to which the sidereal time meant,
appertains. Just so it is with mean or common time.
This is also a local reckoning, having for its epoch mean
noon, or the average of all the times throughout the year,
when the sun is on the meridian of that particular
place to which it belongs; and, therefore, in like man-
ner, when we date any event by mean time, it is nece*s-
sary to name the place, or particularize what mean time
we mtend. On the- other hand, a date by equinoctial
time is absolute, and requires no such explanatory ad-
dition. ■

(207.) The astronomer sets and regulates his sidereal
clock by observing the meridian passages of the more
conspicuous and well known stars. Each of these holds
in the heavens a certain determinate and known place
with respect to that imaginary point called the equinox,
and by noting the times of their passage in succession by
Us clock he knows when the equinox passed. At that
?ioment his clock ought to have marked 0^ 0" 0* ; and if
it did not, he knows and can correct its error, and by the
^pcement or disagreement of the errors assigned by each
star he can ascertain whether his clock is correctly regu-
lated to go twenty-four hours in one diurnal period, and

132 A TREATISE ON ASTRONOMY. [cHAP. HI.

if not, can ascertain and allow for its rate. Thus, aiihough
his clock may not, and indeed cannot, either be set cor-
rectly, or go truly, yet by applying its error and rate (as
they are technically termed), he can correct its indications,
and ascertain the exact sidereal times corresponding to
them, and proper to his, locality. This indispensable
operation is called getting his local time. For simplicity
of explanation, however, we shall suppose the clock a
perfect instrument ; or, which comes to the same thing,
its error and rate applied at every moment it is consulted,
and included in its indications.

(208.) Suppose, now, two observers, at distant sta-
tions, A and B, each independently of the other, to set
and regulate his clock to the true sidereal time of his
station. It is evident that if one of these clocks could
he taken up without deranging its going, and set down by
the side of the other, they would be found, on compari-
son, to differ by the exact difference of their local epochs ;
that is, by the time occupied by the equinox, or by any
star, in passing from the meridian of A to diat of B : in
other words, by their difference of longitude, expressed
in sidereal hours, minutes, and seconds.

(209.) A pendulum clock cannot be thus taken up and
transported from place to place without derangement, but
a chronometer may. Suppose, then, the observer at B
to use a chronometer instead of a clock, he may, by bodily
transfer of the instrument to the other station, procure a
direct comparison of sidereal times, and thus obtain his
longitude from A. And even if he employ a clock, yet
by comparing it first with a good chronometer, and then
transferring the latter instrument for comparison with the
other clock, the same end will be accomplished, provided
the going of the chronometer can be depended on.'

(210.) Were chronometers perfect, nothing more com-
plete and convenient than this mode of ascertaining dif-
ferences of longitude could be desired. An observer,
provided with such an instrument, and with a portable
transit, or some equivalent method of determining the
local time at any given station, might, by journeying
from place to place, and observing the meridian passages
of stars at each (taking care not to alter his chronome-

CHAP. III.J LONGITUDES FOUND BT CHRONOMETERS. 183

ter, or let it run down), ascertain their differences of lon-
gitude with any required precision. In this case, the
same time-keeper being used at every station, if, at one
of them, A, it mark true sidereal time, at any other, B,
it will be just so much sidereal time in error as the dif-
ference of longitudes of A and B is equivalent to : in
other words, the longitude of B from A will appear as the
error of the time-keeper on the local time of B. If he
travel westward, then his chronometer will appear con-
tinually to gain, although it really goes correctly. Sup-
pose, for instance, he set out from A, when the equinox
was on the meridian, or his chronometer at 0", and in
twenty-four hours (sid. time) had travelled 16° westward
to B. At the moment of arrival there, his chronometer
will again point to 0^ ; but the equinox will be, not on
his new meridian, but on that of A, and he must wait
one hour more for its arrival at that of B. When it
does arrive there, then his watch will point not to 0*, but
to 1**, and will therefore be 1** fwt on the local time of
B. If he travel eastward, the reverse wiU happen.

(211.) Suppose an observer now to set out from any
station as above described, and constantly travelling
westward to make the tour of the globe, and return
io the point he set out from. A singular consequence
will happen : he will have lost a day in his reckoning
of time. He will enter the day of his arrival in his
diary as Monday, for instance, when, in fact, it is Tues-
day. The reason is obvious. Days and nights are
caused by the alternate appearance of the sun and stars,
as the rotation of the eaiih carries the spectator round
to view them in succession. So many turns as he
makes round the centre, so many days and nights will
he experience. But if he travel once round the globe in
the direction of its motion, he will, on his arrival, have
Teally made one turn mott round its centre ; and if in
the opposite direction, one turn ItBB than if he had re>
mained stationary at one point of its surface : in the
former case, then, he will have witnessed one alteration
of day and night more, in the latter one less, than if he
liad trusted to the rotation of the earth alone to carry
him round. As the earth revolves from west to east, it
M

134 A TREATISE ON ASTRONOMY. [cHAP. IU»

follows that a westward direction of his journey, by
which he counteracts its rotation, will cause him to lose
a day, and an eastward direction, by which he conspires
with it, to gain one. In the former case, all his days
will be Longer ; in the latter, shorter than those of a
stationary observer. This contingency has actually hap-
pened to circumnavigators. Hence, also, it must neces-
sarily happen that distant settlements, on the same meri-
dian, will differ a day in their usual reckoning of time,
according as they have been colonized by settlers arriving
in an eastward or in a westward direction, — a circum-
stance which may produce strange confusion when they
come to communicate with each other. The only mode
of correcting the ambiguity, and settling the disputes
which such a difference may give rise to, consists in
having recourse to the equinoctial date, which can never
be ambiguous.

(212.) Unfortunately for geography and navigation,
the chronometer, though greatly and indeed wonderfully
improved by the skifl of modem artists, is yet far too
imperfect an instrument to be relied on implicitly. How-
ever such an instrument may preserve its uniformity of
rate for a few hours, or even days, yet in long absences
from home the chances of error and accident become so
multiplied as to destroy all security of reliance on even
the best. To a certain extent this may, indeed, be reme-
(ped by carrying out several, and using them as checks
on each other ; but, besides the expense and trouble, this
is oiily a palliation of the evil — the great and funda-
mental, — as it is the only one to which the determincUion
of longitudes by time-keepers is liable. It becomes ne-
cessary, therefore, to resort to other mea.ns of communi-
cating from one station to another a knowledge of its
local time, or of propagating from some principal station,
as a centre, its loc^ time as a universal standard with
which the local time at any other, however situated, may
be at once compared, and thus the longitudes of all places
be referred to- the meridian of such central point*

(213.) The simplest and most accurate method by
which this object can be accomplished, when circum-
stances admit of its adoption, is that by telegraphic signal.

CHAP, ni.] LONGITUDES DETERMINED BT SIGNALS. IBS

Let A and B be two observatories, or other stations, pro-
vided with accurate means of determining their respective
local times, and let us first suppose them visible from
each other. Their clocks being regulated, and their errors
and rates ascertained and applied, let a signal be made at
A, of some sudden and definite kind, such as the flash
of gunpowder, the explosion of a rocket, the sudden ex-
tinction of a bright light, or any other which admits of
no mistake, and can be seen at great distances. The
moment of the signal being made must be noted by each
observer at his respective clock or watch, as if it were
the transit of a star, or anj'"' astronomical phenomenon,
and the error and rate of the clock at each station being
applied, the local time of the signal at each is determined.
Consequently, when the observers communicate their
observations of the signal to each other, since (owing to
the almost instantaneous transmission of light) it must
have been seen at the same absolute instant by both, the
difference of their local times, and therefore of their
longitudes, becomes known. For example ; at A the
signal is observed to happen at 5** 0"* 0' sid. time at A,
as obtained by applying the error and rate to the time
shown by the clock at A, when the signal was seen there.
At B the same signal was seen at 5*" 4*" 0*, sid. time at B,
similarly deduced from the time noted by the clock at B,
by applying its error and ra,te. Consequently, the differ-
ence of their local epochs is 4" 0% which is also their differ-
ence of longitudes in time, or V 0' 0" in hour angle.

(214.) The accuracy of the final determination may
be increased by making and observing several signals at
stated intervals, each of which affords a comparison of
times, and the mean of all which is, of course, more to
be depended on than the result of any single comparison.
By this means, the error introduced by the comparison
of clocks may be regarded as altogether destroyed.

(215.) The distances at which signals can be rendered
visible must of course depend on the nature of the in-
terposed country. Over sea the explosion of rockets
niay easily be seen at fifty or sixty miles ; and in moun-
tainous countries the flash of gunpowder in an open
spoon may be seen, if 2. proper station be chosen for its

186 A TREATISE ON ASTRONOMY. [CHAP. III.

exhibition, at much greater distances. The interval be-
tween the stations of observation may also be increased
by causing the signals to be made not at one of them,
but at an intermediate point ; for, provided they are seen
by both partie;», it is a matter of indifference where they
are exhibited. Still the interval which could be thus
embraced would be very limited, and the method in con-
sequence of little use, but for the following ingenious
contrivance, by which it can be extended to any distance,
and carried over any tract of country however difficult.
(216.) This contrivance consists in establishing, be-
tweeen the extreme stations, whose difference of longi-
tude is to be ascertained, and at which the local times
are observed, a chain of intermediate stations, alternately
destined for signals and for observers. Thus, let A and
Z be the extreme stations. At B let a signal station be
established, at which rockets, &c. are fired at stated in-
tervals. At C let an observer be placed, provided with
a chronometer ; at D, another signal station ; at E, an-
other observer and chronometer ; and so on till the whole

J I L

A. T3 C 33 E j!' Z

line is occupied by stations so arranged, that the signals
at B can be seen from A and ; those at D, from C and
E ; and so on. Matters being thus arranged, and the
errors and rates of the clocks at A and Z ascertained by
astronomical observation, let a signal be made at B, and
observed at A and G, and the times noted. Thus the
difference between A's clock and C's. chronometer be-
comes known. After a short interval (five minutes for
instance) let a signal be made at D, and observed by C
and E. Then will the difference between their respec-
tive chronometers be determined; and the difference
between the former and the clock at A being already as-
certained, the difference between the clock A and chro-
nometer E is therefore known. This, however, supposes

CHAP. III.3 NATURAL SIGNALS. 137

that the intermediate chronometer C has kept true side-
real time, or at least a known rate, in the interval between
the signals. Now this interval is purposely made so
very short, that no instrument of any pretension to cha-
racter can possibly produce an appreciable amount of
error in its lapse. Thus the time propagated from A to
C may be considered as handed over, without gain or
loss (save from error of observation), to E. Similarly,
by the signal made at F, and observed at E and Z, the
time so transmitted to E is forwarded on to Z; and thus
at length the clocks at A and Z are compared. The
process may be repeated as often as is necessary to
destroy error by a mean of results ; and when the line
of stations is numerous, by keeping up a succession
of signals, so as to allow each observer to note al-
ternately those on either side, which is easily pr^
arranged, many comparisons may be kept running along
the line at once, by which time is saved, and other ad-
vantages obtained.* In important cases the process is
usually repeated on several nights in succession.

(217.) In place of artificial signals, natural ones, when
they occur sufficiently definite for observation, may be
equally employed. In a clear night the number of those
singular meteors, called shooting stars, which ma;y be
observed, is usuadly very great ; and as they are sudden
in their appearance and disappearance, and from the
great height at which they have been ascertained to take
place are visible over extensive regions of the earth's
surface, there is no doubt that they may be resorted to
with advantage, by previous concert and agreement be-
tween distant observers to watch and note them.t

(218.) Another species of natural signal, of still greater
extent and universality (being visible at once over a whole
terrestrial hemisphere), is afforded by the eclipses of
Jupiter's satellites, of which we shall speak more at large
when we come to treat of those bodies. Every such
eclipse is an event which possesses one great advantage

* For a complete account of this method, and the mode of deducing
the most advantageous result from a combination of all the observations,
see a paper on the difierence of longitudes of Green^ch and Paris, Phil*
Trans. 1836 ; by the author of this volume.

f This idea was first suggested by the late Dr. Maskelyne.
n2

188 A TREATISC: ON ASTRONOMY. [cUAP. III.

in its applicability to the purpose in question, viz. that
the time of its happening, at any fixed station, such as
Greenwich, can be predicted from a long course of pre-
vious recorded observation and calculation thereon found-
ed, and that this prediction is sufficiently precise and
certain, to stand in the place of a corresponding obser*
vation. So that an observer at any other station wher^
ever, "who shall have observed one or more of these
ellipses, and ascertained his local time, instead of waiting
for a communication with Greenwich, to inform him at
what moment the eclipse took place there, may use the
predicted Greenwich time instead, and thence, at once,
and on the spot, determine his longitude. This mode
of ascertaining longitui^ is, however, as will hereafter
appear, not susceptible of great exactness, and should
only be resorted to when others cannot be had. The
nature of the observation also is such that it cannot be
made at sea ; so that, however useful to the geographer,
it is of no advantage to navigation.

(219.) But such phenomena as these are of only occa-
sional occurrence ; and in their intervals, and when cut off
from all communication with any fixed station, it is indis-
pensable to possess some means of determining longi-
tudes, on which not only the geographer may rely for a
knowledge of the exact position of important stations onr
land in remote regions, but on which the navigator can
securely stake, at every instant of his adventurous course,
the lives of himself and comrades, the interests of his
country, and the fortunes of his employers. Such a me-
thod is afforded by Lunar Observations. Though we
have not yet introduced the reader to the phenomena of
the moon's motion, this will not prevent us from giving
here the exposition of the principle of the lunar method ;
on the contrary, it will be higldy advantageous to do so,
since by this course we shall have to deal with the
naked principle, apart from all the peculiar sources of
difficulty with which the lunar theory is encumbered,
but which are, in fact, completely extraneous to the
principle of its application to the problem of the longi-
tudes, which is quite elementary.

(220.) If there were in the heavens a clock furnished

CHAP, in.3 LUNAR METHOD OF LONOITUOBS. 189

with a dial-plate and hands, which always marked
Greenwich time, the longitude of any station would be
at once determined, so soon as the local time was
known, by comparing it with this clock. Now, the
offices of the dial-plate and hands of a clock are these:-—
the former carries a sat of marks upon it, whose position
is known ; the latter, by passing over and among these
markst informs us, by Uie place it holds with respect to
them, what it is o'clock, or what time has elapsea since
a certain moment when it stood at one particular spot.

(221.) In a clock the marks on the dial-plate are uni-
formly distributed all around the circumference of a cir-
cle, whose centre is that on which the hands revolve with
a uniform motion. But it is cletr that we should, with
equal certainty, though with much more trouble, teil
what o'clock it were, if the marks on the dial-plate
were tenequally distributed, — ^if the hands were eccentric,
and their motion not uniform, — provided we knew, Ist,
the exact intervals round the circle at which the hour
and minute marks were placed ; which would be the case
if we had them all registered in a table, from the results
of previous careful measurement : — ^2dly, if we knew
the exact amount and direction of eccentricity of the
centre of motion of the hands ; — eild, 3dly, if we were
folly acquainted with all the mechanism which put the
hands in motion, so as to be able to say at every instant
what were their velocity of movement, and so as to be
able to calculate, without fear of error, how much time
should correspond to so much angukw movement.

(2:^2.) The visible surface of the starry heavens is the
dial-plate of our clock, the stars are the fixed marks dis-
tributed around its circuit, the moon is the moveable
hand, which, with a motion that, superficially consider-
ed, seems uniform, but which, when carefully examined,
is found to be far otherwise, and regulated by mechanical
^ws of astonishing complexity and intricacy in result,
^ough beautifully simple in principle and design, per-
forms a monthly circuit among them, passing visibly
over and hiding, or, as it is called, occulting, some, and
g^ding beside and between others ; and whose position
them can, at any moment when it is visible, be

140 A TREATISE ON ASTRONOMY. [CHAF. HI.

exactly measuted by the help of a sextant, just as we
might measure the place of our clock-hand among the
marks on its did-plate with a pair of compasses, and
thence, from the known and calculated laws of its mo-
tion, deduce the time. That the moon does so move
among the stars, while the latter hold constantly, with
respect to each other, the same relative position, the no-
tice of a few nights, or even hours, will satisfy the com-
menxsing student, and this is all that at present we require.

(223.) There is only one circumstance wanting to
make our analogy complete. Suppose the hands of our
clock, instead of moving quite close to the dial-plate,
were considerably elevated above, or distant in front of
it. Unless, then, in viewing it, we kept our eye just in
the line of their centre, we should not^see them exactly
thrown or projected upon their proper places on the dial.
And if we were either unaware of this cause of optical
change of place, this paraUaX'-^r negligent in not
taking it into aceount-^we might make great mistakes
in reading the time, by referring the hand to the wrong
mark, or incorrectly appreciating its distance froip the
right. On the other handf if we took care to note, in
eveiy case, when we had occasion to observe the time,
the exact position oF the eye, there would be no difficulty
in ascertaining and allowing for the precise inHuenee
of this cause of apparent displacement. Now, this is
just what obtains with the apparent motion of the moon
among the stars. The former (as will appear) is com-
paratively hear to the earth — the latter immensely dis-
tant ; and in consequence of our not occupying the cen-
tre of the earth, but being carried about on its surface, and
constantly changing place, there arises aparaUax, which
displaces the moon apparently among the stars, and must
be allowed for before we can tell the true place she
would occupy if seen from the centre.

(224.) Such a clock as we have described might, no
doubt, be considered a very bad one ; but if it were our
only one, and if incalculable interests were at stake on
a perfect knowledge of time, we should justly regard it
as most precious, and think no pains ill bestowed in stu-
dying the laws of its movements, or in facilitating the

CHAP. ni.J LUNAR METHOD OF LONGITirOES. 141

means of reeding it correctly. Sach, in the {^ralkl we
are drawing, is the lunar theory, whose object is to
reduce to regularity the indications of this strangely
irregular-going clock, to enable us to predict, long before-
hand, and with absolutely certainty, wkereabotUa among
the stars, at every hour, minute, and second, in every
day of every year, in Greenwich local time, the moon
would be seen from the earth's centre, and will be seen
from every accessible point of its surface ; and such is
the lunar method of longitudes. The moon's apparent
angular distances from all those principal and conspicu-
ous stars which lie in its course, as seen from the earth's
centre, are computed and tabulated with the utmost care
and precision in almanacs published under national
control. No sooner does an observer, in any part of
the globe, at sea or on land, measure its actual distance
from any one of those standard stars (whose places in
the heavens have been ascertained for the purpose with
the most anxious solicitude), than he has, in fact, per-
formed that comparison of his local time with the local
times of every observatory in the world, which enables
him to ascertain his difference of longitude from one or
all of them.

(226.) The latitudes and longitudes of any number
of points on the earth's surface may be ascertained by
the methods above described; and by thus laying down
a sufficient number of principal points, and filling in the
intermediate spaces by local surveys, might maps of
counties be constructed, the outlines of continents and
islands ascertained, the courses of rivers and mountain
chains traced, and cities and towns referred to their pro-
per localities. In practice, however, it is found simpler
and easier to divide each particular nation into a series
of great triangles, the angles of which are stations con-
spicuously visible from each other. Of these triangles,
^^ angles only are measured by means of the theo-
dolite, with the exception of one side only of one trian-
gle, which is called a base, and which is measured with
every refinement which ingenuity can devise or expense
command. This base is of moderate extent, rarely sui^
passing six or seven miles, and purposely selected in a

142 A TREATISE ON ASTRONOMY. [ciIAP. III.

perfectly horizontal plane, otherwise conveniently adapt-
ed for purposes of measurement. Its length between its
two extreme points (which are dots on plates of gold or
platina let into massive blocks of stone, and which are,
or at least ought to he, in all cases preserved with almost
religious care, as monumental records of the highest im-
portance), is then measured, with every precaution to
insure precision,* and its position with respect to the
meridian, as well as the geographical positions of its ex-
tremities, carefully ascertained.

(226.) The annexed figure represents such a chain of
triangles. AB is the base, O, C, stations visible from

both its extremities (one of which, O, we will suppose
to be a national observatory, with which it is a principal
object that the base should be as closely and immedi-
ately connected as possible) ; and D, E, F, G, H, K,
other stations, remarkable points in the county, by whose
connexion its whole surface may be covered, as it were,
with a network of triangles. Now, it is ovident that the
angles of the triangle A, B, C being obseived, and one
of its sides, AB, measured, the other two sides, AC, BC,
may be calculated by the rules of trigonometry ; and thus
e^ch of the sides AC and BC becomes in its turn ^ bcLse
capable of being employed as known sides of other tri-
angles. For instance, the angles of the triangles ACG
and BCF being known by observation, and Aeir sides
AC andBC, we can thence calculate the lengths AG, CG,
and BF, CF. Again, CG and CF being known, and
the included angle GCF, GF may be calculated, and so

* The greatest possible error in the Irish bose of between seven and
eight miles, near Londonderry, is supposed not to exceed two inches.

CORRECTION FOR THE BARTu's SPHERICITY. 143'

on. Thus may all the stations be accurately determined
and laid down, and as this process may be carried on to
any extent, a map of the whole county may be thus coft-
structed, and filled in to any degree of detail we please.

(227.) Now, on this process there are two important
remarks to be made. The first is, that it is necessary
to be careful in the selection of stations, so as to form
triangles free froin any very great inequality in their an-
gles. For instance, the triangle KBF would be a very
improper one to determine the situation of F from obser-
vations at B and K, becauise the angle F being very acute,
a small error in the angle K would produce a great one
in the place of F upon the line BF, Such ill-conditioned
triangles, therefore, must be avoided. But if this be at-
tended to, the accuracy of the determination of the calcu-
lated sides will not be much short of that which would
be obtained by actual measurement (were it practicable) ;
and, therefore, as we recede from the base on all sides
as a centre, it will speedily become practicable to use oh
bases the sides of much larger triangles, such as GF,
GH, HK, &c.; by which means the next step of the
operation will come to be carried on on a much larger
scale, and embrace far greater intervals, than it would
have been safe to do (for the above reason) in the imme-
diate neighbourhood of the base. Thus it becomes easy
to divide the whole face of a country into great trian"
gles of from 30 to 100 miles in their sides (according to
the nature of the ground), which, being once well deter«
mined, may be afterwards^ by a second series of subordi-
nate operations, broken up into smaller ones, and these
again into others of a still minuter order, till the final fill-
ing in is brought within the limits of personal survey and
draftsmanship, and till a map is constructed, \^ith any
required degree of detail.

(228.) The next remark we have to make is, that all
the triangles in question are not, rigorously speaking,
plane, but spherical — existing on the surface of a sphere,
or rather, to speak correctly, of an ellipsoid. In very
small triangles, of six or seven miles in the side, this
may be neglected, as the difference is imperceptible ; but
in the larger ones it must be taken into consideration.

144 A TREATISE ON ASTRONOMY. [cHAP. lU.

It is evident that, as every object used for pointing the
telescope of a theodolite has some certain elevation, not
only above the «oi7, but above the level of the sea, and as,
moreover, these elevations differ in every instance, a re-
duction to the horizon of all the measured angles would
appear to be required. But, in fact, by the constructiolf
of the theodolite (act. 155), which is nothing more than
an altitude and azimuth instrument, this seduction t« made
in the very act of reading off the horizontal angles. L«t
^ £ be the centre of the earth ;
>A, B, C, the places on its sphe- ■
rical surface, to which three
stations, A, P, Q, in a country
are referred by radii E, A,
EBP, ECQ, If a theodolite
he stationed at A, the axis of its
horizontal circle will point to E
when truly adjusted, and its
plane will be a tangent to the
sphere at A, interseeliDg the ra-
dii EBP, ECQ, at M and N,
above the spherical surface.
The telescope of the theodolite,
it is true, its pointed in succes-
sion to P, and Q ; but the readings off of its azimuth
circle give— no/ the angle PAQ between the directions
of the telescope, or between the objects P, Q, as seen
fifom A ; but the azimuthal angle MAN, which is the
measure of the angle A of the spherical triangle BAC.
Hesee arises this remarkable circumstance, — that the sum
of Uie three observed angles of any of the great triangles
in geodesical operations is always found to he rather more
than 180'' : were the earth's surface a plane, it ought to
be exactly 180** ; and this excess, which is called the
spherical excess, is so far from being a proof of incorrect-
ness in the work, that it is essential to its accuracy, and
oSm at the same time another palpable proof of the
earth's sphericity.

(229.) The true way, then, of conceiving the subject
of a trigonometrical survey, when the spherical form of
the earQi is taken into consideration, is to regard the net-

CHAP, in.] PROJECTIONS OF THE IP9EIIS. « 145

work of triangles with which the country is covered, as
the bases of an assemblage of pyramids converging to the
centre of the earth. The theolodite gives ua the true
measures of the angles included by the planes of these
pyramids ; and the surface of an imaginary sphere on
&e level of the sea intersects them in an assemblage of
spherical triangles, above whose angles, in the radii pro-
longed, the real stations of observation are raised, by the
Alperficial inequalities of mountain and valley. The ope-
rose calculations of spherical trigonometry which this
consideration would seem to render necessary for the re-
ductions of a survey, are dispensed with in practice by a
very simple and easy rule, called the rule for the spheri-
cal excess^ which is to be found in most works on trigo-
nometry.* If we would take into account the ellipticity
of the eacth, it may also be done by appropriate processes
of calculation, which, however, are too abstruse to dwell
upon in a work like the present.

(230.) Whatever process of calculation we adopt, the
result will be a reduction, to the level of the sea, of all the
triangles, and the consequent determination of the geo-
graphical latitude and longitude of every station observed.
Thus we are at length enabled to construct maps of
countries ; to lay down the outlines of continents and
islands ; the courses of rivers ; the direction of mountain
ridges, and the places of their principal summits ; and
all those details which, as they belong to physical and
statistical, rather than to astronomical geography, we
need not here dilate on. A few words, however, will be
necessary respecting maps, which are used as wefi in
astronomy as in geography.

(231.) A map is nothing more than a representation,
upon a plane, of some portion of the surface of a sphere,
on which are traced the particulars intended tp be ex-
pressed, whether they be continuous outlines qft points.
Now, as a spherical surfacet can by no contrivance
be extended or projected into a plane, without undue

* Laidnef'g Trigonometry, prop. j^. WoodhooM's ditto, p^ 148. Itt

t We here nej^Iect the ellipticity of the earth, which, ibr such a pur^
poee SM map-making, is too trifling to have any material mftnenc^

N

•^6

A TREATISE ON ASTRONOMY. [cHAP. III.

enlargement or contraction of some parts in proportion
to others ; and as the system adopted in so extending or
projecting it will decide what part shall be enlarged or
relatively contracted, and in what proportions ; it follows,
that when large portions of the sphere are to be mapped
down, a great difference in their representations may
subsist, according to the system of projection adopted.

(232.) The projections chiefly used in maps are the
orthographic^ stereographies and Mercator*9, In the
orthographic projection, eveiy point of the hemisphere
is referred to its diametral plane or base, by a perpendicular
let fall on it, so that the representation of the hemisphere
thus mapped on its base, is such
as it would actually appear to
an eye placed at an infinite dis-
tance from it. It is obvious,
from the annexed figure^ that
in this projection only the
central portions are represented
of their true forms, while all the exterior is more and
more distorted and crowded together as we approach the
edges of the map. Owing to this cause, the orthogra-
phic projection, though very good for small portions of
the globe, is of little service for large ones.

(233.) The atereographic projection is in great mea*
sure free from this defect. To understand this projection.

CHAP. 111.3 MKRCATOr's PROJECTION. 147

we must conceive an eye to be placed at E, one extremity
of a diameter, ECB, of the sphere, and to view the
concave surface of the sphere, every point of which, as
P, is referred to the diametral plane ADF, perpendicular
to EB by the visual line PME. The stereographic pro-
jection of a sphere, then, is a true perspective represen-
tation of its concavity on a diametral plane ; and, as
fluch, it possesses some singularly elegant geometrical
properties, of which we shall state one or two of the
principal.

(234.) And first, then, all circles on the sphere are re-
presented by circles in the projection. Thus the circle
X is projected into x. Only great circles passing through
the vertex B are projected into straight lines traversing
the centre C : thus, BPA is projected into CA.

2dly. Every very small triangle, GHK, on the sphere,
is represented by a similar triangle, g A A, in the projec-
tion. This is a very valuable property, as it insures a
general similarity of appearance in the map to the reality
in all its parts, and enables us to project at least a hemi-
flphere in a single map, without any violent distortion
of the configurations on the surface from their rea||^rms.
As in the orthographic projection, the borders of the
hemisphere are unduly crowded together; in the stereo-
graphic, their projected dimensions are, on the contrary,
somewhat enlarged in receding from the centre.

(235.) Both Siese projections may be considered na-
tural ones, inasmuch as they are really perspective re-

60

40
20

20

40

^

«0

1

€5

m

60

40
20

presentations of ^e surface on a plane. Mercator's is
entirely an artificial one, representing the sphere as it

lis A TREATISE ON ASTRONOMY. [CHAP. III.

cannot be seen from any one point, but as it might be
seen by an eye carried successively over every part of it.
In it, the degrees of longitude^ and those of latitude^
bear always to each other their due proportion; the
equator is conceived to be extended out into a straight
line, and the meridians are straight lines at right angles
to it, as in the figure. Altogether, the general character
of maps on this projection is not very dissimilar to
what would be produced by referring every point in the
globe to a circumscribing cylinder, by lines drawn from
die centre, and then unrolling the cylinder into a plane.
Like the stereographic projection, it gives a true repre-
sentation, as to /orm, of every particular small part, but
varies greatly in point of scale in its different regions ;
the polar portions in particular being extravagancy en-
larged; and the whole map, even of a single hemisphere,
not being comprizable within any finite limits.

(336.^ We shall not, of course, enter here into any
geograpnical details ; but one result of maritime discovery
on die great scale is, so to speak, massive enough to call
for mention as an astronomical feature. When the con-
timui^iiand seas are laid down on a globe (and since the
discovery of Australia we are sure that no very extensive
tracts of land remain unknown, except perhaps at the
south pole), we find that it is possible so to divide the
globe into two hemispheres, that one shall contain nearly
aU the land; the other being almost entirely sea. It is
a fact, not a little interesting to Englishmen, and, com-
bined with our insular station in that great highway of
nations, the Atlantic, not a little explanatory of our com-
mercial eminence,*that London occupies nearly the centre
of the terrestrial hemisphere. Astronomically speaking,
the fact of this divisibility of the globe into an oceanic
and a terrestrial hemisphere is important, as demonstra-
tive of a want of absolute equality in the density of the
solid material of the two hemispheres. Considering the
whole mass of land and water as in a state of equili-
brium, it is evident that the half which protrudes must
of necessity be buoyant : not, of course, that we mean
to assert it to be lighter than water, but, as compared
with the whole globe, in a less degree heavier than

CHAP. in. J DETERMINATION OE HEIGHTS.

140

that fluid. We leave to geologists to draw from these
premises their own conclusions (and we think them ob-
vious enough) as to the internal constitution of the globe,
and the immediate nature of the forces which sustain its
continents at their actual elevation ; but in any future
investigations which may have for their object to explain
the local deviations of the intensity of gravity, from
what the hypothesis of an exact elliptic figure would
require, this, as a general fact, ought not to be lost sight of.

(237.) Our knowledge of the surface of our globe is
incomplete, unless it include the heights above the sea
level of every part of the land, and the depression of the
bed of the ocean below the surface over all its extent.
The latter object is attainable (with whatever difliculty
and however slowly) by direct sounding ; the former by
two distinct methods : the one consisting in trigonome-
trical measurement of the diflferences of level of all the
stations of a survey ; the other, by the use of the baro-
meter, the principle of which is, in fact, identical with
that of the sounding line. la both cases we measure the
distance of the point whose level we would know from
the surface of an equilibrated ocean: only in the one
case it is an ocean of water ; m the other, of air. In
the one case our sounding line is real and tangible ; in
the other, an imaginary one, measured by the length of
the column of quicksilver the superineumbent air is ca-
pable of counterbalancing.

(238.) Suppose that instead of air, the earth and
ocean were covered with oil; and that human life could
itibsist under such .cixeumstanees* Let ABODE be a

continent, of which the portion ABC projects above the

water, but is covered by the oil, which also floats at an

n2

150 A TREATISE ON ASTRONOMY. [cHAP. III.

uniform depth on the whole ocean. Then if we would
know the depth of any point D below the sea level, we
let down a plummet from F. But if we would know the
height of B above the same level, we have only to send
up a float from B to the surface of the oil ; and having
dane the same at C,a point at the sea level, the difference
of the two float lines gives the height in question,

(23d.) Now, though the atmosphere differs from oil
in not having a positive surface equally definite, and in
not being capable of carrying up any float adequate to
such a use, yet it possesses dl the properties of a fluid
really essential to the purpose in view, and this in par-
ticular ; that, over the whole surface of the globe, its
strata of equal density are parallel to the surface of equi-
librium, or to what would he the surface of the sea, if
prolonged under the continents, and therefore each or
any of them has^ all the characters of a definite surface to
measure from, provided it can be ascertained and identi-
fied. Now the height at which, at any station B, the
mercury in a 'barometer is supported, infonns us at
once how much of the atmosphere is incumbent on B,
or, in ojt^r words, in what stratum of the general at-
mosphere (indicated by its density) B is situated:
whence we are enabled finally to conclude, by mechani-
cal reasoning,* at what height above the sea level that
degree of density is to be found over the whole surface
of the globe. Such is the principle of the application of
the barometer to the measurement of heights. For de-
tails, the reader is referred-to other works.f

(240.) Possessed of a knowledge of the heights of
stations above the sea, we may connect all stations at the
same altitude by level lines, the lowest of which will be
the outline of the sea-coast ; and the rest will mark out
the successive coast-lines which would take place were
the sea to rise by regular and equal ascensions of level
over the whole world, till the highest mountains were
submerged. The bottoms of valleys and the ridge-lines

♦ See Cab. Cycl Pneumatics, art. 143.

t Biot, Astronomie Physique, vol. 3. For tables, see the work of BioC
cited. Also those of Oltmaim, annually published by the French board
of lonntndes in their Annuaire : and Mr. Baily*B Collection of Astrono-
uucal Tables and Formulie.

CHAP. IV.] URANOGRAPHT. 161

o£ hills are determined by their property of intersecting
all these level lines at right angles, and being, subject to
that condition, the shortest and longest courses respec-
tively which can be pilrsued from the summit to the sea.
The former constitute the water-courses of a country ;
the latter divide it into drainage basins : and thus origi-
nate natural districts of the most ineffaceable character,
on which the distribution, limits, and peculiarities of hu-
man communities are in great measure dependent

CHAPTER IV.

OF URANOGRAPHY.

Comtnictioa of celettial Mapi and Globes by Observadoni of right Ab-
cension and Declinatioa — Celegtial Objeots distinguished into fixed
and erratic — Of the Constellations — ^Natural Regions in the Heavens
—The Milky Way— The Zodiac— Of the Ecliptic— Celestial Latitudes
and Longitades— Precession of the Equinoxes — Nutation — ^Aberration
— Uranographical Problems.

(241.) The determination of the relative situations of
objects in the heavens, and the construction of maps and
globes which shall truly represent their mutual configu-
rations, as well as of catalogues which shall preserve a
more precise numerical record of the position of each, is
a task at once simpler and less laborious than that by
which the surface of the earth is mapped and measured*
Every star in the great constellation which appears to
revolve above us, constitutes, so to speak, a celestial sta-
tion; and among these stations we may, as upon the
earth, triangulate, by measuring with proper instruments
their angular distances from each other, which, cleared
of the effect of refraction, are then in a state for laying
down on charts, as we would the towns and villages of a
country ^ and this without moving from our place, at least
for all the stars which rise above our horizon.

(242.) Great exactness might, no do»ubt, be attained
by this means, and excellent celestial charts constructed ;
but there is a far simpler and easier, and, at the same
time, infinitely more accurate course laid open to us, if

152 A TREATISE ON ASTRONOMY. [cHAP. IV.

we take advantage of the earth's rotation on its axis, and
by observing each celestial object as it passes our meri-
diai^, refer it separately and independently to the celes-
tial equator, and thus ascertain its place on the surface
of an imaginary sphere, which may be conceived to re-
volve with it, and on which it may be considered as pro-
jected.

(243.) The right ascension and declination of a point
in the heavens correspond to the longitude and latitude
of a station on the earth ; and the place of a star on a
cielestial sphere is determined, when the former elements
are known, just as that of a town on a map, by knowing
the latter. The great advantages which the method of
meridian observation possesses over that of triangula-
tion from star to star, are, then, 1st, that in it every star
is observed in that point of its diurnal course, when it is
best seen and least displaced by refraction. 2dly, that
the instruments required (the transit and mural circle)
are the simplest and least liable to error or derangement
of any used by astronomers. 3dly, that all the observa-
tions can be made systematically, in regular succession,
and with equal advantages ; there being here no ques-
tion about advantageous or disadvantageous triangles,
&c. And, lasdy, that, by adopting this course, the very
quantities which we should otherwise have to calculate
by long and tedious operations of spherical trigonometry,
and which are essential to the formation of a catalogue,
are made the objects of immediate measurement. It is
almost needless to state, then, that this is the course
adopted by astronomers. ^

(244.) To determine the right ascension of a celestial
object, all that is necessary is to observe the moment of
its meridian passage with a transit instrument, by a clock
regulated to exact sidereal time, or reduced to such by ap-
plying its known error and rate. The rate may be ob-
tained by repeated observations of the . same star at its
successive meridian passages. The error, however, re-
quires a knowledge of the equinox, or initial point from
which all right ascensions in the heavens reckon, as lon->
gitudes do on the earth from a first meridian.

(245.) The nature of this point will be explained pre

CHAP. lY. j RIGHT ASCENSIONS AND DECLINATIONS. 158

sently ; but for the purposes of uranography, in so far as
they concefn only the actual Qonfigarations of the stars
inter 9e, a knowledge of the equinox is not necessary.
The choice of the equinox, as a zero point of right as-
censions, is purely artificial, and a matter of convenience :
but as on the earth, any station (as a national observa-
tory) may be chosen for an origin of longitudes ; so in
uranography, any conspicuous star may be selected as an
initial point from which hour angles may be reckoned,
and from which, by merely observing differences or in-
tervals of time, the situation of all others may be de-
duced. In practice, these intervals are affected by cer^
tain minute causes of inequality, which must be allowed
for, and which will be explained in their proper places.

(246.) The declinations of celestial objects are ob-
tained, 1st, By observation of their meridian altitudes^
with the mural circle or other proper instruments. This
requires a knowledge of the geographical latitude of the
station of observation, which itself is only to be obtained
by celestial observation. 2dly, And more directly by ob-
servation of their polar distances on the mural circle,
as explained in art. Id6, which is independent of any
previous determination of the latitude of the station ;
neither, however, in this case, does observation give
directly and immediately the exact declinations. The
observations require to be corrected, first for refraction,
and moreover for those minute causes of inequality which
have been just alluded to in the case of right ascensions.

(247.) In this manner, then, may the places, one
^unong the other, of all celestial objects be ascertained,
and maps and globes constructed. Now here arises a
very important question. How far are these places per-
manent? Do these stars and the greater luminaries of
heaven preserve for ever one invariable connexion and
nlatioQ of place inter s€, as if they formed part of a
solid though invisible firmament; and, like the great
natural landmarks on the earth, preserve immutably the
lame distances and bearings each from the other ? If so,
^ most rational idea we could form of the universe
would be that of an earth at absdiute rest in the centre,
and a hollow crystallins sfdiere circulating round if, and

154 A TREATISE ON ASTRONOMY. [CHAP.IV.

carrying sun, moon, and stars along in its diurnal mo-
tion. If not, we must dismiss all such notions, and
inquire individually into the distinct history of each ob-
ject, with a view to discovering the laws of its peculiar
motions, and whether any and what other connexion
subsists between them.

(248.) So far is this, however, from being the case,
that observation, even of the most cursory nature, are*
sufficient to show that some, at least, of the celestial
bodies, and those the most conspicuous, are in a state
of continual change of place among the rest. In the
case of the moon, indeed, the change is so rapid and re-
markable, that its alteration of situation with respect to
such bright stars as may happen to be near it, may be
noticed any fine night in a few hours ; and if noticed on
two successive nights, cannot fail to strike the most care-
less observer. With the sun, too, the change of place
among the stars is constant and rapid ; though, from tlie
invisibility of stars to the naked eye in the day-time, it
is not so readily recognised, and requires either tlie use
of telescopes and angular instruments to measure it, or
a longer continuance of observation to be struck with it
Nevertheless, it is only necessary to call to mind its
greater meridian altitude in summer than in winter, and
the fact that the stars which come into view at night
vary with the season of the year, to perceive that a great
change must have taken place in that interval in its re-
lative situation with respect to all the stars. Besides the
sun and moon, too, there are several other bodies, called
planets, which, for the most part, appear to the naked
eye only as the largest and most brilliant stars, and which
offer the same phenomenon of a constant change of place
among the stars ; now approaching, and now receding
from, such of them as we may refer them to as marks ;
and, some in longer, some in shorter periods, making,
like the sun and moon, the complete tour of the heavens.

(249.) These, however, are exceptions to the general
rule. The innumerable multitude of the stars wluch are
distributed over the vault of the heavens form a constel-
lation, which preserves, not only to the eye of the casual
observer, but to the nice examination of the astronomer.

CHAP. IV.3 FIXED AND ERRATIC STARS. 155

a uniformity of aspect which, when contrasted with
the perpetual change in the configurations of the sun,
moon, and planets, may well be termed invariable*
It is not, indeed, that, by the refinement of «xact mea*
surements prosecuted from age to age, some small
changes of apparent place, attributable to no illusion
and to no terrestrial cause, cannot be detected in some
of them; — such are called, in astronomy, the proper
motions of the stars ; — ^but these are so excessively slow,
that their accumulated amount (even in those stars for
which they are greatest) has been insufficient, in the
whole duration of astronomical history, to produce any
obvious or material alteration in the appearance of the
starry heavens.

(250.) This circumstance, then, establishes a broad
distinction of the heavenly bodies into two great classes ;
—the fixed, among which (unless in a course of obser-
vations continued for many years) no change of mutual
situation can be detected ; and the erratic, or wandering
—{which is implied in the word planet*)— including the
sun, moon, and planets, as well as the singular class of
bodies termed comets, in whose apparent places among
the stars, and among each other, the observation of a few
days, or even hours, is sufficient to exhibit an indisputa-
ble alteration.

(251.) Uranography, then, as it concerns the fixed
celestial bodies (or, as they are usually called, the fixed
9tar8)j is reduced to a simple marking down of their re-
stive places on a globe or on maps ; to the insertion on
^t globe, in its due place in the great constellation of
the stars, of the pole of the heavens, or the vanishing
point of parallels to the earth's axis ; and of the equa-
tor and place of the equinox : points and circles these,
which though artificial, and having reference entirely to
our earth, and therefore subject to all changes (if any) to
which the earth's axis may be liable, are yet so con-
venient in practice, that they have obtained an admission
(with some other circles and lines), sanctioned by usage,
in all globes and planispheres. The reader, however,
will take care to keep them separate in his mind, and to
* nx^vnrnt, a wanderer.

156 A TREATISE ON ASTRONOMY. QcHAP. IV.

familiarize himself with the idea rather of two or more
celestial globes, superposed and fitting on each other, on
one of which — a real one — are inscribed the stars ; on
the others those imaginary points, lines, and circles
which astronomers have devised for their own uses, and
to aid their calculations; and to accustom himself to
conceive in the latter, or artificial, spheres a capability
of being shifted in any manner upon the surface of the
other; so that, should experience demonstrate (as it
does) that these artificial points and lines are brought,
by a slow motion of the earth's axis, or by other secular
variations (as they are called), to coincide, at very dis-
tant inter vids of time, with different stars, he may not
be unprepared for the change, and have no confusion to
correct in his notions.

(252.) Of course we Jo not here speak of those un-
couth figures and outlines of men and monsters, which
are usually scribbled over celestial globes and maps, and
serve, in a rude and barbarous way, to enable us to talk
of groups of stars, or districts in the heavens, by names
which, though absurd or puerile in their origin, have
obtained a currency from which it would be difficult,
and perhaps wrong, to dislodge them. In so far as they
have really (as some have) any slight resemblance to (he
figures caUed up in imagination by a view of the more
sjplendid "constellations," they have a certain conve-
nience ; but as they are otherwise entirely arbitrary, and
correspond to no natural subdivisions or groupings of
the stars, astronomers treat them lightly, or altogether
disregard them,* except for briefly naming remarkable
stars, as a Leonis, /3 Scorpii, &c. &c., by letters of the
Greek alphabet attached to Ihem. The reader will find
them on any celestial charts or globes, and may compare
them with the heavens, and there learn for himself their
position.

* This duMoaid is neither supercilious nor causeless. The crastBlla'
tionsseem to have been almost purposely named and delineated to cause
as much confusion and inconvenience as possible. Innumerable snakes
twine through long and contorted areas of the heavens, where no me-
mory can follow them ; bean, lions and fishes, laive and small, northern
and southern, confuse all nomenclature, &c. A oetter system of oon-
stellations might have been a material help as an artiiiciai memory.

CHAP. IV.3 THE MILKY WAY. — ^THK ZODIAC. 167

(253.) There are not wanting, however, natural dis-
tricts in the heavens, which offer great peculiarities of
character, and strike every observer: such is the milky
wayy that great luminous band, which stretches, every
evening, all across the sky, from horizon to horizon,
and which, when traced with diligence, and mapped
down, is found to form a zone completely encircling tht
whole sphere, almost in a great circle, which is neitlier
an hour circle, nor coincident with any other of our
astronomical grammata. It is divided in one part of its
coarse, sending off a kind of branch, which unites again
with the main body, after remaining distinct for about
150 degrees. This remarkable belt has maintained,
6t>m the earliest ages, the same relative situation among
the stars ; and, when examined through powerful tele-'
scopes, is found (wonderful to relate !) to consist entirely
of stars scattered by milHons, like glittering dust, on
the black ground of the general heavens.

(254.) Another remarkable region in the heavens is
the zodiac, not from any thing peculiar in its own con-
stitution, but from its being the area within which the
apparent motions of the sun, moon, and all the greater
planets are confined. To trace the path of any one of these,
it is only necessary to ascertain, by continued observa-
tion, its places at successive epochs, and entering these
upon our map or sphere in sufficient number to form a
series, not too far disjoined, to connect them by lines
from point to point, as we mark out the course of a ves-
sel at sea by mapping down its place from day to day.
Now when this is done, it is found, first, that the appa-
rent path, or track, of ^e sun on the surface of the hea-
vens, is no other than an exact great circle of the sphere
which is called the ecliptic, and which is inclined to the
equinoctial at an angle of about 23° 28', intersecting it at
two opposite points, called the equinoctial points, or
equinoxes, and which are distinguished from each other
by the epithets vernal and autumnal ; the vernal being
that at which the sun crosses the equinoctial from south
to north ; the autumnal, when it quits the northern and
enters the southern hemisphere. Secondly, that the
moon and all the planets pursue paths which, in like
O

168 A TREATISE ON ASTRONOMY. [CHAP. IV.

manner, encircle the whole heavens, but are not, like
that of the sun, great circles exactly returning into them-
selves and bisecting the sphere, but rather spiral curves
of much complexity, and described with very unequal
velocities in their different parts. They have all, how-
ever, this in common, that die general direction of their
motions is the same with that of the sun, viz. from west
to east, that is to say, the contrary to that in which both
they and the stars appear to be carried by the diurnal
motion of the heavens ; and, moreover, that they never
deviate far from the ecliptic on either side, crossing and
recrossing it at regular and equal intervals of time, and
confining themselves within a zone, or belt (the zodiac
already spoken of), extending 9° on either side of the
ecliptic.

(255.) It would manifestly be useless to map down on
globes or charts the apparent paths of any of those bodies
which never retrace the same course, and which, there-
fore, demonstrably, must occupy at some one moment
or other of their history, every point in the area of' that
zone of the heavens within which they are circum-
scribed. The apparent complication of their movements
arises (that of the moon excepted) from our viewing
them from a station which is itself in motion, and would
disappear, could we shift our point of view and observe
them from the sun. On the other hand, the apparent
motion of the sun is presented to us under its least in-
volved form, and is studied, from the station we occupy,
to the greatest advantage. So that, independent of the
importance of that luminary to us in other respects, it is
by the investigation of the laws of its motions in the first
instance that we must rise to a knowledge of those of all
the other bodies of our system.

(256.) The ecliptic, which is its apparent path among
the stars, is traversed by it in the period called the side-'
reed year, which consists of 365^* 6*" 9" 9'-6, reckoned
in mean solar time, or 366* 6*" 9" 9''6, reckoned in si-
dereal time. The reason of this difference (and it is this
which constitutes the origin of the difference between
solar and sidereal time) is, that as the sun's apparent
annual motion among the stars is performed in a con*

CHAP. IV.] THE ECLIPTIC. — SIDEREAL YEAR. 159

trary direction to the apparent diurnal motion of both
sun and stars, it comes to the same thing as if the diur-
nal motion of the sun were so much slower than that of
the stars, or as if the sun lagged behind them in its
daily course. Where this has gone on for a whole year,
the sun will have fallen behind the stars by a whole
circumference of the heavens— or, in other words — ^in a
year, the sun will have made fewer diurnal revolutions,
by one, than the stars. So that the same interval of time
which is measured by 366* 6\ Ac. of sidereal time, if
reckoned in mean solar days, hours, Ac. will be called
365** 6*", &c. Thus, then, is the proportion between
the mean solar and sidereal day established, which,
reduced into a decimal fraction, is that of 1-00273791 to
1. The measurement of time by these different stand-
ards may be compared to that of space by the standard
feet, or ells of two different nations ; the proportion of
which, once settled, can never become a source of error.

(257.) The position of the ecliptic among the stars
may, for our present purpose, be regarded as invariable.
It is true that this is not strictly the case ; and on com-
paring together its position at present with that which
it held at the most distant epoch at which we possess
observations, we find evidences of a small change, which
theory accounts for, and whose nature will be hereafter
explained ; but that change is so excessively slow, that
for a great many successive years, or even for whole
centuries, this circle may be regarded as holding the
same position in the sidereal heavens.

(258.) The poles of the ecliptic, like those of any
other great circle of the sphere, are opposite points on
its sur&ce, equidistant from the ecliptic in every direc-
tion. They are of course not coincident with those of
the equinoctial, but removed from it by an angular in-
terval equal to the inclination of the ecliptic to the equi-
noctial (23° 28'), which is called the obliquity of the
ecliptic. In the annexed figure, if Fp represent the north
and south poles (by which, when used without qualifi-
cation we always mean the poles of the equinoctial),
and EQAV the equinoctial, VSAW the ecliptic, and K^,
its poles — the spherical angle QVS is the obliquity of the

160 A TREATISE ON ASTRONOMY. [CHAP* lY.

ecliptic, and is equal in angular measure to PK or SQ.
If we suppose the sun's apparent motion to be in the
direction VSAW, V will be the vernal and A the au-
tumnal equinox. S and W, the two points at which
the ecliptic is most distant from the equinoctial, are
termed solstices, because, when arrived there, the snn
ceases to recede from the equator, and (in that sense, so
far as its motion in declination is concerned) to stand
still in the heavens. S, the point where the sun has* the
greatest northern declination, is called the summer sol-
stice, and W, that where it is farthest south, the vnnter.
These epithets obviously have their origin in the depend-
ence of the seasons on tJie sun's declination, which will
be explained in the next chapter. The circle EKPQ^p,
which passes through the poles of the ecliptic and equinoc-
tial, is called the solstitial colure ; and a meridian drawn
through the equinoxes, PVpA, the equinoctial colure.

(259.) Since the ecliptic holds a determinate situation
in the starry heavens, it may be employed, like the equi-
noctial, to refer the positions of the stars to, by circles
drawn through them from its poles, and ^therefore per-
pendicular to it. Such circles are termed, in astronomy,
circles of latitude— the distance of a star from the eclip-
tic, reckoned on the circle of latitude passing through it,
is called the latitude of the stars — and the arc of the

ecliptic intercepted between the vernal equinox and this
circle, its longitude. In the figure X is a star, PXR a

CHAP. IV.3 CELESTIAL LONGITUDES AND LATITUDES. 161

circle of declination drawn through it, by which it is
referred to the equinoctial, and KXT a circle of latitude
referring it to the ecliptic— then, as VR is the right
ascension, and RX the declination, of X, so also is VT
its longitude, and TX its latitude. The use of the terms
longitude and latitude, in this sense, seems to have ori-
ginated in considering the ecliptic as forming a kind of
natural equator to the heavens, as the terrestrial equator
does to the earth*-the former holding an invariable po-
sition with respect to the stars, as the latter does with
respect to stations on the earth's surface. The force of
this observation will presently become apparent.

(260.) Knowing the right ascension and declination of
an object, we may find its longitude and latitude, and vice
versa. This is a problem of great use in physical astro-
nomy. The following is its solution : In our last figure,
EKPQ, the solstitial cohire is of course 90'' distant
iirom V, the vernal equinox, which is one of its poles-^
so that VR (the right ascension) being given, and also
Y£, the arc ER, and its measure, the spherical angle
£PR, or KPX, is known. In the spherical triangle
KPX, then we have given, 1st, The side P K, which,
being the distance of the poles of the ecliptic and equi-
noctial, is equal to the obliquity of the ecliptic ; 2d, The
side PX, the polar distance^ or the complement of the
declination RX ; and 3d, the included angle KPX ; and
therefore, by spherical trigonometry, it is easy to find the
other side KX, and the remaining angles. Now KX is
the complement of the required latitude XT, and the
angle PKX being known, and PKV being a right
angle (because SV is 90°), the angle XKV becomes
known. Now this is no other than the measure of the
longitude VT of the object. The inverse problem is
resolved by the same triangle, and by a process exactly
similar.

(261.) The same course of observations by which the
padi of the sun among the fixed stars is traced, and the
ecliptic marked out among them, determines, of course,
the place of the equinox V upon the starry sphere, ^t
that time — a point of great importance in practical astro-
Momy, as it is the origin or zero point of right ascension
o2

162 A TREATISE ON ASTRONOMY. [CHAP. lY.

Now, when this process is repeated at considerably dis-
tant intervals of time, a very remarkable phenomenon is
observed ; viz. that the equinox does not preserve a con-
stant place among the stars, but shifts its position, travel-
ling continually and regularly, although with extreme
slowness, backwards^ along the ecliptic, in the direction
VW from east to west, or the contrury to that in which the
sun appears to move in that circle. The equinoctial point
thus moving, as it were, to meet the sun in his apparent an-
nual round, the sun arrives at the equinoctial point sooner ;
that is, the time of the equinox happens sooner than
it would otherwise do : hence the recession of the equi-
noctial point causes a prece&ston in the time of the equinox.
The amount of this motion by which the equinox travels
backward, or retrogrades (as it is called), on the ecliptic,
is 0° 0' 50"* 10 per annum^ an extremely minute quan-
tity, but which, by its continual accumulation from year
to year, at last makes itself very palpable, and that in a
way highly inconvenient to practical astronomers, by
destroying, in the lapse of a moderate number of yeaft,
the arrangement of their catalogues of stars, and making
it necessary to reconstruct them. Since the formation
of the earliest catalogue on record, the place of the equi-
nox has retrograded already about 30°. The period in
which it performs a complete tour of the ecliptic, is
25,868 years.

(262.) The immediate uranographical effect of the
precession of the equinoxes is to produce a uniform m-
crease of longitude in all the heavenly bodies, whether
fixed or erratic. For the vernal equinox being the initial
point of longitudes, as well as of right ascension, a re-
treat of this point on the ecliptic teUs upon the longi-
tudes of all alike, whether at rest or in motion, and pro-
duces, 80 far as its amount extends, the appearance of a
motion in longitude common to all, a« (^ the whole hea-
vens had a slow rotation round the poles of the ecliptic
in the long period above mentioQed, similar to what they
have in twenty-four hours round those of the equinoctial.

(263.) To form a just idea of this curious astronomi-
cal phenomenon, however, we must abandon, for a time,
the consideration of the ecliptic, as tending to produce

CHAP. IV. J PRECESSION OF THE EQUINC^ES. 168

confasion in our ideas ; for this reason, that the stability
of the ecliptic itself among the stars is (as already hinted,
art. 257) only approximate, and that in consequence its
intersection with the equinoctial is liable to a certain
amount of change, arising from its fluctuation, which
mixes itself with what is due to the principal uranogra-
phical cause of the phenomenon. This cause will be-
come at once apparent, if, instead of regarding the equi-
nox, we fix our attention on the pole of the equinoc-
tial, or the vanishing point of the earth's axis.

(264.) The place of this point among the stars is easily
determined, at any epoch, by the most direct of all astro-
nomical observations, — those with the mural circle. By
this instrument we are enabled to ascertain at every mo-
ment the exact distance of the polar point from any
three or more stars, and therefore to lay it down, by
triangulating from these stars, with unerring precision,
on a chart or globe, without the least reference to the
position of the ecliptic, or to any other circle not natu-
rally connected with it. Now, when this is done with
proper diligence and exactness^ it results that, although
for short intervals of time, such as a few days, the place
of the pole may be regarded as not sensibly variable, yet
in reality it is in a state of constant, although extremely
slow motion ; and, what is still more remarkable, this
motion is not uniform, but compounded of one principal,
uniform, or nearly uniform, part, and o^er smaller and
subordinate periodical fluctuations : the former giving
rise to the phenomena o( precession ; the latter to another
distinct phenomenon called nutation. These two phe-
nomena, it is true, belong, theoretically speaking, to one
and the same general head, and are intimately connected
together, forming part of a great and complicated chain
of consequences flowing from the earth's rotation on its
axis : but it will be of advantage to present clearness to
consider them separately.

(265.) It is found, then, that in virtue of the uniform
part of the motion of the pole, it describes a circle in the
heavens around the pole of the ecliptic as a centre, keep-
ing constantly at the same distance of 23^ 28' from it,
in a direction from east to west, and with such a velocity,

164 .A TREATISE ON ASTRONOMY. [^CHAP. IV.

that the annual angle described by it, in this its imaginary
orbit, is 50"' 10; so that the whole circle would be de-
scribed by it in the above-mentioned period of 25,868
years. It is easy to perceive how such a motion of the
pole will give rise to the retrograde motion of the equi-
noxes ; for in the figure, art. 259, suppose the pole P in
the progress of its motion in the small circle POZ round
K to come to O, then, as the situation of the equinoctial
EVQ is determined by that of the pole, this, it is evi-
dent, must cause a displacement of the equinoctial, which
will take a new situation, EUQ, 90° distant in every
part from the new position O of the pole. The point U,
therefore, in which the displaced equinoctial will inter-
sect the ecliptic, i. c. the displaced equinox, will lie on
that side of V, its original position, towards which the
motion of th€ pole is directed, or to the westward.

(266.) The precession of the equinoxes thus conceived,
consists, then, in a real but very slow motion of the pole
of the heavens among the stars, in a small circle round
the pole of the ecliptic. Now this cannot happen with-
out producing corresponding changes in the apparent
diurnal motion of the sphere, and fiie aspect which the
heavens must present at very remote periods of history.
The pole is nothing more than the vanishing point of the
earth's axis. As this point, then, has such a motion as
described, it necessarily follows that the earth's axis must
have a conical motion, in virtue of which it points suc-
cessively to every part of the small circle in question.
We may form the best idea of such a motion by noticing
a child's peg-top, when it spins not upright, or that amus-
ijQg toy the te-to-tum, which, when delicately executed,
and nicely balanced, becomes an elegant philosophical
instrument, and exhibits, in the most beautiful manner,
the whole phenomenon, in a way calculated to give at
once a clear conception of it as a fact, and a considerable
insight into its physical cause as a dynamical effect. The
reader will take care not to confound the variation of the
position of the earth^s axis in space with a mere shifting
of the imaginary line about which it revolves, in its inte-
rior. The whole earth participates in the motion, and
goes along with the axis as if it were really a bar of iron

CHAP. IV.] NUTATION. 106

driven through it. That such is the case is proved by
the two great facts : Ist, that the latitudes of places on
the earth, or their geographical situation with respect to
the poles, have undergone po perceptible change from
the earliest ages. 2dly, that the sea maintains its level,
which could not be the case if the motion of the axis
were not accompanied with a motion of the whole mass
of the earth.

(267.) The visible effect of precession on the aspect
of the heavens consists in the apparent approach of
some stars and constellations to the pole and recess of
others. The bright star of the Lesser Bear, which we
call the pole star, has not always been, nor will always
continue to be, our cynosure : at the time of the con-
struction of the earliest catalogues it was 12^ from the
pole — it is now only l"* 24', and will approach yet nearer,
to within half a degree, after which it will again recede,
and slowly give place to others, which will succeed it in
its companionship to the pole. After a lapse of about
12,000 years, the star a, Lyrae, the brightest in the north-
ern hemisphere, will occupy the remarkable situation of
a pole star, approaching within about 6° of the pole.

(268.) The nutation of the earth's axis is a small and
slow subordinate gyratory movement, by which, if sub-
sisting alone, the pole would describe among the starsr
in a period of about nineteen years, a minute ellipsis,
having its longer axis equal to 18''*6, and its shorter to
13"*74 ; the longer being directed towards the pole of
the ecliptic, and the shorter, of course, at right angles to
it. The consequence of this real motion of the pole is
an apparent approach and recess of all the stars in the
heavens to the pole in the same period. Since, also, the
place of the equinox on the ecliptic is determined by the
place of the pole in the heavens, the same cause will
give rise to a small alternate advance and recess of the
equinoctial points, by which, in the same period, both
the 'longitudes and the right ascensions of the stars will
be also alternately increased and diminished.

(269.) Both these motions, however, although here
considered separately, subsist jointly ; and since, while
in virtue of the nutation, the pole is describing its little

166 A TREATISS ON ASTRONOMY. [CHAP. IV.

ellipse of 18"*5 in diameter, it is carried by the greater
and regularly progressive motion of precession over so
much of its circle round the pole of the ecliptic as cor-
responds to nineteen years, — that is to say, over an angle
of nineteen times 50"* 1 round the centre (which, in a
small circle of 23° 28' in diameter, corresponds to 6' 20",
as seen from the centre of the sphere) : the path which
it will pursue in virtue of the two motions, subsisting
jointly, will be neither an ellipse nor an exact circle, but
a gently undulated ring like that in the figure (where,
however, the undulations are much exaggerated). (See
fig. to art. 272.)

(270.) These movements of precession and nutation
are common to all the celestial bodies both fixed and er-
ratic ; and this circumstance makes it impossible to attri-
bute them to any other cause than a real motion of the
earth's axis, such as we have described. Did they only
affect the stars, they might, with equal plausibility, be
urged to arise from a real rotation of the starry heavens,
as a solid shell round an axis passing through the poles
of the ecliptic in 25,868 years, and a real elliptic gyration
of that axis in nineteen years : but since they also affect
the sun, moon, and planets, which, having motions inde-
pendent of the general body of the stars, cannot without ex-
travagance be supposed attached to the celestial concave,*
this idea falls to the ground ; and there only remains,
then, a real motion in the earth by which they can be
accounted for. It will be shown in a subsequent chapter
that they are necessary consequences of the rotation of
the earth, combined with its elliptical figure, and the un-
equal attraction of the sun and moon on its polar and
equatorial regions.

(271.) Uranographically considered, as affecting the
apparent places of the stars, they, are of the utmost im-
portance in practical astronomy. When we speak of the
right ascension and declination of a celestial object, it
becomes necessary to state what epoch we intend, and

* This argument, cogent as it is, acquires additional and decisive force
from tiiie law of nutation, which is dependent on the position, for the time,
of die lunar orbit. If we attribute it to a real motion of the celestial
sphere, we must then maintain that sphere to be 'kept in a constant state
of tremor by the motion of tbe moon !

EQUATIONS FOR PRECESSION AND NUTATION. 167

whether we mean the mean right ascension ; cleared, that
is, of the periodical fluctuation in its amount, which
arises from nutation, or the apparent right ascension,
which being reckoned from the actual place of the vernal
equinox, is affected by the periodical aidvance and recess
of the equinoctial point thence produced — and so of the
other elements. It is the practice of astronomers to re-
duce, as it is termed, all their observations, both of right
ascension and declination, to some common and conve- •
nient epoch— such as the beginning of the year for tem-
porary purposes, or of the decade, or the century for
more permanent uses, by subtracting from them the
whole effect of precession in the interval ; and, moreover,
to divest them of the influence of nutation by investigar
ting and subducting the amount of change, both in right
ascension and declination, due to the displacement of the
pole from the centre to the circumference of the little el-
lipse above mentioned. This last process is technically
termed correcting or equating the observation for nuta-
tion ; by which latter word is always understood, in as-
tronomy, the getting rid of a periodical cause of fluctua-
tion, and presenting a result, not as it was observed, but
as it would have been observed, had that cause of fluc-
tuation had no existence.

(272.) For these purposes, in the present case, very
convenient formules have been derived, and tables con-
structed. They are, however, of too technical a charac-
ter for this work ; we shall, however, point out the man-
ner in which the investigation is conducted. It has been
shown in art. 260 by what means the right ascension and
declination of an object are derived from its longitude
and latitude. Referring to the figure of that article, and
supposing the triangle KPX orUiographically projected
on the plane of the ecliptic as in the annexed figure : in
the triangle KPX, KP is the obliquity of the ecliptic,
KX the co-latitude (or complement of latitude), and the
angle PKX the co-longitude of the object X. These
are the data of our question, of which the first is con-
stant, and the two latter are varied by the effect of pre-
cession and nutation ; and their variations (considering
the minuteness of the latter effect generally, and the

168 A TREATISE ON ASTRONOMY. ^CHAP. IV.

small number of years in comparison of the whole period
of 25,868, for which we ever require to estimate the
effect of the former) are of that order which may be
regarded as infinitesimal in geometry, and treated as such
without fear of error. The whole question, then, is re-
duced to this : — In a spherical triangle KPX, in which
one side KX is constant, and an angle K, and adjacent

side KP vary by given infinitesimal changes of the po-
sition of P : required the changes thence arising in the
other side PX, and the angle KPX ? This is a very
simple and easy problem of spherical geometry, and be-
ing resolved, it gives at once the reductions we are seek-
ing ; for PX being the polar distance of the oliject, sad
the angle KPX its right ascension plus 90°, their va-
riations are the very quantities we seek. It only re-
mains, then, to express in propeir form the amount of the
precession and nutation in longitude and latitude^ when
their amount in right ascension and declination will im-
mediately be obtained.

(273.) The precession in latitude is zero, since the
latitudes of objects are not changed by it : that in lon-
gitude is a quantity proportional to the time at the rate
of 50'''10 per annum. With regard to the nutation in
longitude and latitude, these are no other than the ab^
scissa and ordinate of the little ellipse in which the pole

CHAP. IV.J ABERRATION OF LIOHT. 109

moves. The law of its motion, however, flierein, cmmot
be understood till the reader has been made acquainted
with the principal features of the moon's motioii on
which it depends. See chap. XI.

(274.) Another consequence of what has been shown
respecting precession and nutation is, that Mefe^ time
as astronomers use it, t. e, as reckoned from the trunit
of the equinoctial point, is, not a mean or unifotmh
flowing quantity^ being affected by nutation ; ana,
moreover, that so reckoned, even when cleared of the
periodical fluctuation of nutation, it does not strictly
correspond to the earth's diurnal rotation. As tlie sun
loses one day in the year on the stars, by its direct mo^
tion in longitude; so the equinox gains one day in
25,868 years on them by its retro gradation. We ought,
therefore, as carefully to distinguish between mean and
apparent sidereal as between mean and apparent sdlar
time.

(275.) Neither precession nor nutation chance the
apparent places of celestial objects inter se. We see
tiiem, so far as these causes ffo, as they are, though firom
a station more or less unstable^ as we see distant land
objects correctly formed, though appearing td rise and
M when viewed from the heaving deck of a ship in the
act of pitching and rolling. But there is an optic«d canse»
independent of refraction or of perspective, which dis-
places them one among the other, and causes us to view
the heavens under an aspect always to a certain slight
extent false ; aiid whose influence must be estimated and
allowed for before we can obtain a precise knowted^ of
the place of any object. This cause is what is calM
the aberration of light ; a singular and surprising eflM
arising from this, 3iat we occupy a station not at r*st
^t in rapid motion ; and that the apparent directions of
the rays of light are not the s&me to a spectator in mo-
tion as to one at rest. As the eistimation of its effect be-
longs to uranography, we must explain it here, though,
^ 80 doing, we must anticipate some of the results to be
delaHed in subsequent chapters.

. (276.) Suppose a shower of rain to fall perpendicubriy

wi a dead calm ; a person exposed to the shoWer, who

P

170

A TREATISE ON ASTRONOMY.

[chap.

IV.

should stand quite still and upright, would receive the
drops on his hat, which would thus shelter him, but if
he fan forward in any direction they would strike him in
the face. The effect Would be the same as if he remained
still, and a wind should arise of the same velocity, and
drift them against him. Suppose a ball let fall from a
point A^ above a horizontal line £F, and that at B were
placed to receive it the open mouth of an inclined hollow

QA.

tube PQ ; if the tube ^ere held Immoveable, the ball
would strike on its lower side, tut if the tube were car-
ried forward in the direction EF, with a velocity properly
adjusted at every instant to that of the ball, while pre-
serving its inclination, to the horizon, so that when the
ball in its natural descent reached C, the tube should
have been carried into the position RS, it is evident that
the ball would, throughout its whole descent, be found
in the axis of the tube ; and a spectator, referring to the
tube the motion of the ball, and carried along with the
former,, unconscious of its motion, would fancy that the
ball had been moving in the inclined direction RS of the
tube's axis.

(277.) Our eyes and telescopes are such tubes. In
whatever manner we consider light, whether as an ad-
vancing wave in a motionless ether, or a shower of
atoms traversing space, if in the interval between the
rays traversing the object-glass. of the one or the cornea
of the other {dt which moment they acquire that' con-
vergence which directs them to a certain point in fixed

CHAP. nr.J CORRECTION FOR ABERRATION. 171

spaet)^ the cross wires of the one or the retina of the
other be slipped aside, the point of convergence (which
remains unchanged) will no longer correspond to the in-
tersection of the wires or the central point of our visual
area. The object then will appear displaced ; and the
amount of this displacement is abermtion.

(278.) The earth is moving through space with a ve-
locity of about 19 miles per second, in an elliptic path
round the sun, and is therefore changing the direction
of its motion at every instant. Light travels with a ve-
locity of 192,000 miles per second, which, although
much greater than that of the earth, is yet not infinitely
80. Time is occupied by it in traversing any space, and
in that time the earth describes a space which is to the
former as 19 to 192,000, or as the tangent of 20"-5 to
radius'. Suppose now APS to represent a ray of light
from a star at A, and let the tube PQ be that of a tele-
scope so inclined forward that the focus formed by its
object-glass shall be received upon its cross wire, it is
evident from what has been said, that the* inclination of
the tube must be such as to make PS : SQ : : velocity of
light : velocity of the earth, : : tan. 20"*5 : 1 ; and,
therefore, the angle SPQ, or PSR, by which the axis of
the telescope must deviate from the true direction of the
star, must be 20"'5.

(279.) A similar reasoning will hold good when the
direction of the earth^s motion is not perpendicular to
the visual ray. If SB be the
true direction of the visual
ray, and AC the position in
which the telescope requires
to be held in the apparent di-
rection, we must still have the
proportion BC : BA : : velo-
locity of light ; velocity of the earth : : rad. : sine of 20"*5
(for in such small angles it matters not whether we use
the sines or tangents).. But we have, also, by trigono-
metry, BC : BA : : sine of BAC : sine of ACB or CBD,
which last is the apparent displacement caused by aber^,
ration. Thus it appears that the sign of the aberration, or
(since the angle is Extremely small) the aberration itself,

173 A TREATISE ON ASTRONOKY. [OHAP. IV.

'^ propprtioQal to the sine of the angle made by tlie earth's
motion in space with the visual ray, and is therefore a
maximum when the line of sight is perpendicular to the
direction of the earth's motion.

(280.) The uranographical effect of aberration, then,
is to distort the aspect of the heavens, causing all the
stars to crowd, as it were, directly towards that point in
the heavens which is the vanishing point of all lines
parallel to that in which the earth is for the 'moment
moving. As the earth moves round the sun in the plane
of the ecliptic, this point must lie in that plane, 90^ in
advance of the earth-s longitude, or 00^ behind the sun*«,
and shifts of course continually, describing the circum-
ference of the ecliptic in a year. It is easy to demon-
strate that the effect on each particular star will be to
mike it apparendy describe a small ellipse in the heaVens,
having for its centre the point in which the star would
be seen if the earth were at rest.,

(2^1.) Aberration then affects the apparent right as-
censions and declinations of all the stars, and Uiat by .
quantities easily calculable. The forinulse most conve-
nient for that purpose, and whiph, systematically embrac-
ing s(t the same time the corrections for precession and
nutation, enable the observer, with the utmost readiness,
to disencumber his observations of right ascension and
d^lination of their influence, have been constructed by
Prof. BesseU and tabulated in the appendix to the first
volume of the Transactions of the Astronomical Society,
where they will be found accompanied with an extensive
catalogue of the places, for 1830, of the principal fixed
stars, one of the most useful and best arranged works
of the kind which has ever appeared.

(282.) When the body from which the visual ray
emanates is, itself, in motion, the best way of conceiving
the effect of aberration (independendy of theoretical
views respecting the nature of light)* is as follows. The

* Tlie HMuIti of the undulatory and corpuscular theories of light, in
die matter of aberratiaiD. are, in the main, the same. We BwintMe enm.
There is, however, a minute diSTerenee even of numerical results. In
the undulatory doctrine, the propagation of light takes place with eaqal
velocity in all directions whether the lumma^ he at nst or in motwn.
In the corpuscular, with an ezcen of velocitjr ia the directioii of Uie

OBAF. IV.3 URANOGRAPHICAL PROBLEMS. 178

ray by which we «ee any object is not that which it emits
at the moment we look at it, but that which it did emit
some dme before, viz, the time occupied by light in trar
versing^e interval which separates it from us. The.
aberration of such a body-then arising from the earth's
velocity must be applied as a correction, not to the line
joiniiig the earth's place at the moment of observation
-with that occupied by the foody at the same moment^
hut at that antecedent instant When the ray quitted it.
Hence it is easy to derive the rule given by astronomical
writers for the case of a moving object. From the known
laws of its motion and the earth^e, calculate its apparent
or relative angular motion in the time taken by light to
traverse its distance from the earth, -This is its cierrar
tiony and its effect is to displace it in a direction contrary
to its apparent relative motion among, the stars.

We shall conclude this chapter with a few uranog^-
phieal problems of frequent practical occurrence, which
may be resoled by the rules of spherical trigonometry.

(283.) Of the following five quantiUes, given any three,
to find one inrboth the others.

Ist, The latitude of the place ; 2d, the declination of an
^3jeet ; 3d, its hour angle east or west from the meridian $
4th, its altiliude ; 5th, its azimuth.

In the figure of art. 94, P is the pole,^ the zenidi, and
S the star; and the five quantities above mentioned, or
their complements, constitute the sides and angles of the
spherical triangle PZS ; PZ being the co-latitude, PS
the co-declination or polar distance ; SPJ^ the hour an-
gle ; PS the co-altitude or zenith distance ; and PZS the
azimuth. By the solution of this spherical triangle, then,
a]l problems involving the relations between these quanti-
ties may be resolved.

(284.) For example, suppose the time of risihg or set-
ting of the sun or of a star were required, having given
its right ascension and polar distance. The star rises

motion over that in the contrary equal to twice ijie velocity of the body's
motion. In the caws, then, of a body moving wi^ equal velocity directly
to and directly from the earth, the aberrations will be aUkeda the midu-
latory, but di&rent on the corpuscular hypothesis. The utmost diflep-
ence which can arise from this cause in cur nyttem (wmot amount tp
• above six tiieiisandths of a second.

174 A TBXATU^ ON ASTROMOMY. [cHAP. IT.

when apparently on the horizon, or really about 34' be-
low it (owing to refraction), so that, af the moment of its
apparent rising, its zenith distance is 90° d4'3sZS. Its
polar distance PS being also given, and the cp-latitude ZP
of the place, we )iave given the three sides of the trian-
gle, to fine} th^ hour angle ZPS, which, being known, is
to be added to or subtracted from the star's right ascen-
sion, to give the sidereal time of setting or rising, which,
if we please, may be converted into solar time by the
proper rules and tables.

(285.) As another example of the same triangle, we
may propose to find the I0C9I sidereal time, and &e lati-
tude of the place of observation, by observing equal
altitudes of the same 9tar east and west of the meri-
dian, and noting the interval of the observations in side-
real time. . .

The hour apgles corresponding to equal altitudes of a
fijoed st^ being equal, the hour angle east or west will be
measured by half the observed interval of the -observa-
tions. Ja our triangle, then, we have given this hour an-
gle ZPS, the polar distance PS of the star^ and ZS, its
QOr^titpde at the moment c^ observation. Hence we miay
(ud PZ> the co-latitude of the place* Moreover, thue
hour angle of die star being kuowu* and also its right as-
cep^idn, the poiut of the equinoctial is known, whidi is
on the meridian ajt the moment of observation; and,
Aerefore, the local sidereid jlime at that moment. This
is a very useful x^servation for determining the latitude
and time at anjinkaown station.

(28.Q.) It is often of use to know the situation of tbe
ecliptic in the vi^ibl^ hisavjens at any instant ; that is to
s^y, the pioint? ^her^ it cut9 the horizon^ and the altitude
of its highest point, or, as it is sometimes called, die
noruigeeifnal point of the ecliptic, as well as the longitude
of this point on the ecliptic iteelf from the equinox.
Tb^^e, j»n4 all questions refemble to the same data and qu«.-
sita, are resolved by the spherical triangle ZPE, formed
by the zenith Z (considered as the pole of the horizon),
the pole of the equinoctial P, and the pole of the ecliptie
£. The sidereal time being given, and also the right
ascension of the pole of the ecliptic (which is always ihf^ .

C^AP. tV.J VRANOGRAPHICAL PROBLEVS* 175

z

same, viz. 18^ 0" 0*), (he hour angle ZPE- of that point
is known. Then, in this triangle we have given PZ, the
co-latitude; P£, the polar distance of the pole of the
ecliptic, 33'' 28', and the angle ZPE ; from which we
may find, 1st, the side ZE, which is easily seen to be
equal to the alliiude of the nonagesimal point sought ;
and, 2dly, the ^angler PZE, which is the azimuth of the
pole of ihe «dipi;dc, and which, therefore, being added to
and .subtracted from 00°, gives the azimuths of £e eastern
and western intersections of the ecliptic with the horizon.
Lastly, the longitude of the nonagesimal point may be
had, by calculating in the samie triangle the angle PE2«9
which is its complement.

(287.) The angle of situation of a star is the angle in-^
eluded at the star betifireen circles of latitude and of decli-
nation passing through it. To determin|^it in any pro»
posed case, we must resolve the triangle rSE, in which
are given PS, P£, and the angle SPE, which is the dif-
ference between the star's right ascension and 18 hours ;
from which it is ejasy to find ^e angle PiSE required.
This an^e is c^use in many inquiries in physical astro^
' nomy. It is called in most hodkB on astronomy the aLn-
^^o£ poutioH; but the latter expression has become
otherwise, and more conveniently, aj^ropriated.

(288.) From these instances, ihe manner of treating
such questions in uranography as depend on spherlcsd
trigonometry will be evident, and will, for the most part,
ofifer little difficulty, if the student will bear in min^, as a

176 A TREATISE ON ASTRONOMY. []CHAP. V.

practical maxim, rather to consider the poles of the great
circles which his question refers to, than the circles
themselves.

CHAPTER V.

Apparent Motion of the Sun not uniform— Its apparent Diameter nlfiova-
nable — ^Variation of its Distance concluded— Its apparent Orbit an El-
lipse about the Focus— Law of the angular Velocity — Equable Descrip-
tion of Areas — Parallax of the Sun — Its Distance and Magnitude —
Copemican Explanation of the Sun's apparent Motion — Parallelism of
the ibiarth's Axis— The Seasons — ^Heat received from the Sun in di^r^

' ent Parts of the Orbit

(289.) In the foregoing chapters, it has been shown
that the apparent path of the sun is a great circle of the
sphere, which it performs in a pericwi of one sidereal
year. From this it follows, that the line joining the
earth and sun lies constantly in one planer and that,
therefore, whatever be the real motion from which this
apparent motion arises, it must be confined to one plane,
which is called the plane of the ecliptic,

(290.) We have already seen (art. 118) that the sun's
motion in right ascension among the stars is not uniform.
This is partly accounted for by the obliquity of the eclip-
tic, in consequence of which equal variations in longitude
do not correspond to equal changes of right ascension.
But if we observe the place of the sun daily throughout
the year, by the transit and circle, and from these calcu-
late the longitude for each day, it will still be found that,
even in its own proper path, its apparent angular motion
is far from uniform. The change of longitude in twenty-
four mean solar hours averages 0° 59' 8"*33 ; but about
the 31st of December it amounts to 1° T 9"*9, and about
the 1st of July is only 0* 57' 11 "-5. Such are the ex-
treme limits, and such the mean value of the sun's appa-
rent angular velocity in its annual orbit.

(291.) This variation of its angular velocity is accom-
panied with a corresponding change of its (Ustance from
US. The change of distance is recognised by a variatioii

CHAF. V.J FORM OF THE 80LAR ORBIT. 1T7

observed to take place in its apparent diameter, when
measured at different seasons of the year, with an instru-
ment adapted for that purpose, called a heliometer,* or,
by calculating from the time which its disk takes to tra*
verse the meridian in the transit instrument. The great-
est apparent diameter corresponds to the 31st of Decem
ber, or to the greatest angular velocity, and measures 32'
35"*6 ; the least is 31' 31'''0, and corresponds to the 1st
of July ; at which epochs, as we have seen, the angular
motion is also at its extreme limit either way. Now, as
we cannot suppose the sun to alter its real size periodi-
cally, the observed change of its apparent size can only
arise from an actual change of distance. And the sines
or tangents of sucli small arcs being proportional to the
arcs themselves, its distances from us, at the above-named
epoch, must be in the inverse proportion of the apparent
^ameters. • It appears, therefore, that the greatest, the
mean, and the least distances of the sun from us are in
&e respective proportions of the numbers 1*01679,
1*00000, and 0*98321 ; and that its apparent angular ve-
locity diminishes as the distance increases, and vict versa*
(292.) It follows from this, that the real orbit of the
sun, as referred to the earth supposed at rest, is not a
circle with the earth in the centre. The situation of the
earth within it is eccentric^ the eccentricity amounting to

0*01679 of the mean distance, which may be regarded as
our unit of measure in this inquiry. But besides this,
the form of the orbit is not circular, but elliptic. If from
any point O, taken to represent the earth, we draw a line,
OA, in some fixed direction, from which we then set
off a series of angles, AOB, A06, &c. equal to the ob-
* iHxiOk, thesun;. and Mir($iv^ to measure.

178 A TREATISE ON ASTRONOMY. [cHAP. V,

served longitudes of the sun throughout the year, and in
these respective directions measure off from O the dis-
trances OA, OB, OC, &;c. representing the distances
deduced from the observed diameter, and then connect
all the extremities A, B, C, &c. of these lines by a con-
tinuous curve, it is evident this will be a correct represen-
tation of the relative orbit of the > sun about the earth.
Now, when this is done, a deviation from the circular
figure in the resulting curve becomes apparent; it is
found to be evidently longer than it is broad — ^that is to
say, elliptic, and the point O to occupy not the centre,
but one of the foci of the ellipse. The graphical process
here described is sufficient to point out, the general figure
of the curve in question ; but for the purposes of exact
verification, it is necessary to recur to the properties of
the ellipse,* and to express the distance of any one of its
points in terms of the angular situation of that point with
respect to the longer axis, or diameter of the ellipse.
This, however, is readily done ; and when numerically
calculated, on the supposition of the eccentricity being
such as above stated, a perfect coincidenpe is found to
subsist between the distances thus computed, and those
derived from the measurement of the apparent diameter.
(293.) The mean distance of the earth and sun being
taken for unity, the extremes are 1»01679 and 0*98321.
But if we compare, in like manner, the mean or average
angular velocity with the extremes, greatest and least,
we shall find these to be in the proportions of 1*03386,
1*00000, and 0*96614. The variation of the sun's an*
giUar velocity, then, is much greater in proportion than
that of its distance— fully twice as great ; and if we ex-
amine its numerical expressions at different periods, com-
paring them with the mean value, and also with the cor-
responding distances, it will be found, that, by whatever
fraction of its mean value the distance exceeds the mean,
the angular velocity will fall short of its mean or average
quantity by very nearly twice as great a fraction of the
latter, and vice versa. Hence we are led to conclude
that the angular velocity is in the inverse proportion, not
of the distance i^imply, but of its sqiXare ; so that, to com-
* See Conic Sections, by the Rev. H. P. Hamilton.

CHAP, v.] LAWS OF ELLIPTIC MOTION. 179

pare the daily motion in longitude of the 8un» at one
point, A, of its path, with that at B, we must state the
proportion thus : —

OB* : OA' : : daily motion at A : daily motion atB. And
this is found to be exactly verified in every part of the orbit.

(294.) Hence we deduce anotlier remarkable conclu*
sion — viz. that if the sun be supposed really to move
round the circumference of this ellipse, its actual speed
cannot be uniform^ but must be greatest at its least dis-
tance, and less at its greatest. For, were it uniform, the
apparent angular velocity would be, of course, inversely
proportional to the distance ; simply because the same
linear change of place, being produced in the same time
at different distances from the eye, must, by. the laws of
perspective, correspond to apparent angular displacements
inverisely as those distances. Since, then, observation
indicates a more rapid law of variation in the angular
velocities, it is evident that mere change of distance, un-
accompanied with a change of actual speed, is insuffi-r
cient to account for it ; and that the increased proximity
of the sun ta the earth must be accompanied with an
actual increase of its real velocity of motion along its path,

(295.) This elliptic form of Uie sun's path, the eccen-
tric position of the earth within it, and the unequal speed
with which it is actually traversed by the sun itself, all
tend to render the calculation of its longitude from theory
(i. e. from a knowledge of the causes and nature of its
motion) difficult, and indeed impossible, so long as the
law of its actual velocity continues unknown. This laiOf
however, is not immediately apparent. It does not come
forward, as it were, and present itself at once, like the
elliptic form of the orbit, by a direct comparison of an-
gles and distances, but requires an attentive consideration
of the whole series of observations registered during an
entire period. It was not, therefore, without much pain-
ful and laborious calculation, that it was discovered by
Kepler (who was also the first to ascertain the elliptic
form of the orbit), and announced in the following terms :
Let a line be always supposed to connect the sun, sup-
posed in motion, with the earth, supposed at rest ; then,
as the sun moves along its ellipse^ this line (which in

180 A TREATISE ON ASTRONOMY. j]cHAP. ▼.

called in astronomy the radius vector) will deserve or
sweep over that portion of the whole area or surface of
the ellipse which is included between its consecutive
positions : and the motion of the San will be such that .
equal areas are thus swept over by the revolving radius
vector in equal times, in whatever part of4he circom-
fbrence of the ellipse the sun may be moving.

(296.) From this it necessarily follows, that in un-
equal times, the areas described must be proportional to
the times. Thus, in the figure of art. 292, the time in
which the sun moves from A to B, is the time in which
it moves from C to 1), as the area of the elliptic sector
AOB is to the area of the feector DOC.

(297.) The circumstances of the sun's apparent annual
motion may, therefore, be summed up as follows :— It is
performed in an orbit lying in one plane passing through
the earth's centre, called the plane of the ecliptic, and
whose projection on the heavens is the great circle so
called. In this plane, however, the actual path iM not
t^ircnlaf, but elliptical f having the earth, not in its ceBtre,
but in one focus. The eccentricity of this ellipie is O^Ol 679,
in parts of a unit eqnal to the mean distance, or half the
longer diameter of the ellipse; and the motion of the sun
in its circumference is so regulated, that equal areas of the
ellipse are passed over by the radius vector in equal times.

(298.) What we have here stated supposes no know-
ledge of the sun's actual distance from the earth, nor,
consequently, of the actual dimensions of its orbit, nor
of the body of the SUn itself. To come to any conclu-
sions on these points, we must first consider by what
means ive can arrive at any knowledge of the distance of
an object to which we have no access. Now, it is ob-
vious, that its parallax alone can afford us any informa-
tion bn this subject. Parallax may be generally defined
to be the change of apparent situation of an object
arising from a change of real situation of the observer.
Suj)pose, then, PABQ to represent the earth, C its centre,
and S the sun, and A, B two situations of a spectator, or,
which comes to the same thing, the stations of tM'o spec-
tators, both observing the sun S at the same instant. The
spectator A will see it in the direction ASa, and will re-

CHAP. y. J DIURNAL OR GEOCENTiaC PARALLAX. 181

fer it to a point a in the infinitely distant sphere of the
fixed stars, while the spectator B will 6ee it in the direc-

tion BS6, and refer it to b. The angle included between
these directions, or the measure of the celestial arc ad, by
which it is displaced, is equal to the angle ASB ; and if
this angle be known, and the local situations of A and B,
with the part of the earth's surface AB included between
them, it is evident that the distance CS may be calculated.

(399.) Parallax, however, in the astronomical accepta-
tion of the word, has a more technical meaning. . It is
restricted to the difference of apparent posi&ons of any
celestial object when viewed from a station on the »iar-
face of the earth, and from its centre. The centre of
the earth is the general station to which all astronomical
observations are referred : but, as we observe from Ae
surface, a reduction to the centre is needed; and the
amount of this reduction is called parallax. Thus, the
snn being seen from the earth's centre, in the direction
CS, and from A on the surface in the direction AS, the
angle ASC, included between these two directions, is the
ps^allax at A, and similarly BSG is that at B.

Parallax, in this sense, may be distinguished by the
epithet diurnal, ox geocentric, to discriminate it from
the annual, or heliocerUricf of which more hereaflter.

(300.) The reduction for parallax, then, in any pro-
posed case, is obtained from the consideration of the
triangle ACS, formed by the spectator, the centre of the
earth, and the object observed ; and since the side CA
prolonged passes through the observer's zenith, it is
evident that the effect of parallax, in this its technical
acceptation, is always to depress the object observed in
a vertical circle. To estimate the amount of this de-
pression, we have, by -plane trigonometry,
CS : CA : : sine of CASa^sine of ZAS : sine of ASC.
Q

182 A TREATISE ON ABTRONOMY. [cHAP. V.

(301.) The parallax, then, for objects equidistant from
the earth, is proportional to the sines of their zenith dis-
tances. It is, therefore, at its maximum when the* body-
observed is in the horizon. In this situation it is called
the korizontal paraUax ; and when this is known, since
small arcs are proportional to their sines, the parallax at
any given altitude is easily had by the following rule :—

Parallax = (horizontal parallax) x sine of zenith dis-
tance.

The horizontal parallax'is given by this proportion : —

Distance of object : eartli's radius : : rad. : sine of ho-
rizon^ parallax.

It is, therefore, known, when the proportion of the
object's distance to the radius of the earth is known;
and vice verm — if by any method of observation we can
come at a knowledge of the horizontal parallax of an
object, its distance, expressed in units equid to the earth's
radius, .becomes known.

(302.) To* apply this general reasoning to the case of
the SUA. Suppose two observers— one in the nbrthero,
the other in the southern hemisphere — at stations on the
same meridian, to observe on the same day the meridian
altitudes of the sun's centre. Having thence derived
the apparent zenith distances, and cleared them of the
effects of refraction, if the distance of the sun were equal
to that of the fixed stars, the sum of the zenith distances
thus found would be precisely equal to the sum of the
latitudes north and south of the places of observation.
For the sum in qfuestion would then be equal to the
angle ZCX, which is the meridional' distance of the
stations across the equator. But the effect of parallax
being in both cases to inerease the apparent zenith dis-
tances, their observed sum will be greater than the sum
of the latitudes, by the whole amount of the two paral-
laxes, or by the an^e ASB. This angle, then, is
obtained by subducting the sum of the latitudes from
that of the zenith distances ; and this once determined,
the horizontal paraUax is easily found, by dividing the
angle so determined by the sum of the sines of the two
latitudes.

(303.) If the two stations be not exactly on the same

GHAP. V.J PARALLAX OF THE SUN. 183

meridian (a condition very difficult to fulfil), the same
process will apply, if we take care to allow for the
change of the sun's actual zenith distance in the interval
of time elapsing between its arrival on the meridians of
the stations. This change is readily ascertained, either
from tables of the sun's motion, grounded on the ex-
perience of a long course of observations, or by actual
observation of its meridional altitude on several days
before and after that on which the observations for paral-
lax are taken. Of course, the nearer the stations are to
each other in longitude, the less is this interval of time ;
and, consequently, the smaller the amount of ^is correc-
tion ; and, therefore, the less injurious to the accuracy
of the final result is any uncertainty in the daily ehange
of zenith distance which may arise from imperfection
in ^e solar tables, or in the observations made to deter-
mine it.

(304.) The horizontal parallax of the sun has been
concluded from observations of the nature above de-
scribed, performed in stations the most remote from each
other in latitude, at which observatories have been in-
stituted. It has also been deduced from other methods
of a more refined nature, and susceptible of much greater
exactness, to be hereafter described. Its amount, so
obtained, is about 8"*6. Minute as this quantity is,
there can be no doubt that it is a tolerably correct ap-
proximation to the truth ; and in conformity with it, we
must admit the eun to be situated at a mean distance from
as, of no less than 23,984 times the length of the earth's
radius, or about 95,000,000 miles.

(305.) That at so vast a distance the sun should ap-
pear to us of the size it does, and should so powerfully
influence our condition by its heat and light, requires us
to form a very grand conception of its actual magnitude,
and of the scale on which those important processes ^re
carried on within it, by which it is enabled to keep up its
liberal and unceasing supply of these elements. As to
its actual magnitude we can be at no loss, knowing its
distance, and the angles under which its diameter appears
to us. An object, placed at the distance of 95^000,000
miles, and subtending an angle of 32' 3", must have «

184 A TREATISE ON A8TR0M0MT. [CHAP.V.

real diameter of 882,000 miles. Such, then, is the dia-
BMter of this stupendous globe. If we compare it with
. whai we have already ascertained of the dimensions of
our own, we shall find that in linear magnitude it exceeds
Ihe earth in the proportion of 111^ to 1, and in bu^ in
thatof 1,384,472 to 1.

(306.^ It is hardly possible to avoid associating our
eoncepiion of an object of definite globular figure, and of
such enormous dimensions, with some corresponding
attribute of massiv^ess and material solidity. That the
sttn is not a mere phantom, but a body having its own
peculiar structure and economy, our telescopes distinctly
inform us. They show us dark spots on its surface,
which slowly change their places and forms, and by
attending to whose situation, at different times, astrono-f
mers have ascertained that the sun revolves about an
axis inclined at a constant angle of 82*^ 40' to ike plane
of the eclip^, performing one rotation in a period of 25
days and in the same direction with the diurnal rotation
of 4)ie earth, $.. e, from west to east. ' Here, then, we
havi an analogy with our own globe ; the slower and
more majestic movement only corresponding with the
greater dimensions of the machinery, and impressing us
with the prevalence of similar mechanical lawsi and of,
at least, such a community of nature as the existence of
inertia and obedience to force may argue. Now, in the
exact proportion in which we invest our idea of this im-
mense bulk with the attribute of inertia, or weight, it be-
comes di^cult to conceive its eirculaUon round so comr
paradvely small a body as the earth, without, on the on«
hand, dragging it along, and displacing it, if bound to it
by some invisible tie ; or, on the ot£^r hand, if not so
held to it, pursuing its course alone in space, and leaving
the earth behind. If we tie two Stones together by a
sti^ng, and fling them aloft, we see them circulate about
a point between them, which is their common centre of
gravity ; but if one of them be greatly more ponderous
than the other, this common centre will be proportionally
nearer to that one, and even within its surface, so that the
smaller one will circulate, in fact, about the larger, which
will be comparatively but little disturbed from its place.

dhheksions and rotation op the earth. 185

(307.) Whether the earth move round the son, the sun
round the earth, or both round their common centre of
gravity, will maJie no difference, so far as appearances are
concerned provided the stars be supposed sufficiently dis-
tant to undergo no sensible apparent parallactic displace-
ment by the motion so attributed to the earth. Whether
they are so or not must still be a matter of inquiry ; and
from the absence of any measureable amount of such dis-
placement, we can conclude noting but this, that the
scale of the sidereal universe is so great, that the mutual
orbit of the earth and sun may be regarded as an imper>-
ceptible point in its comparison. Admitting, then, iu
eonformity with the laws of dynamics, that two bodies
connected with and revolving about each other in free
space do« in fact, revolve about their common centre of
gravity, which remains immoveable by their mutual ac-
tion, it becomes a matter of further inquiry, whereaboutB
between them the centre is situated. Mechanics teai^eis
us that its place will divide their mutual distance Ui t}ie
Inverse ratio of their weights or masses ;*^d calculations
grounded on phenomena, of which an account wA be
given further on, inform us that this ratio, in liie case M
Sie sun and earth, is actually that of 354,936 ito l,*^^-ithe
sun being, in that proportion, mere ponderous than the
earth. From this it will follow that the commoji point
about which they both circulate is only 267 miles firoxa the
«un's centre, or about yjVir^ P^^ pf ^ ^^^ diameter.

(308.) Henceforwaiti, then, in conformity ivith the
above statements, and with the Copemican view of our
system, we must learn to look upon the sun as the com-
paratively motionless centre about which ^e earth per-
forms an annual elliptic orbit of the dimensions and ecr
isentricity, and with a veloeity regulated according to the
law above assigned ; the sun occupying one of the foci
4>f the ellipse, and from that station quieUy disseminating
on all sides its light and heat; while the earth, travelling
round it, and presenting itself differently to it at different
times of the year and day, passes through the varieties of
4ay and liight, summer and winter, which we enjoy.

{800.) In this annual motion of the earth, its axis pre-
f Sm C9ib, Cyc. MBCHAiacs, Centre of Gravity.

a2

166 A TREATISE ON ASTRONOMY. [cHAP. V.

serves, at all times, the same direction as if the orbitual
movement had no existence ; and is carried round paral-
lel to itself, and pointing always to the same vanishing
point in the sphere of the fixed stars. This it is which
gives rise to the variety of seasons, as we shall now ex«
plain. In so doing, we shall neglect (for a reason which
will be presently explained) the ellipticity of the orbit,
and suppose it a circle, with the sun in the centre.
. (310.) Let, then, S represent the sun,- and A, B, C, D,
four positions of the earth in its orbit, fiO^ apart, viz* A

thai which it has on iie 2 1st of March, or at the time of
the vernal equinox ; B that of Ihe 21st of June, or the
summer solstice ; C that of the 21st* of September, or die
autumnal equinox ; and D that of the 21st of December,
orthe winter solstice. In each of these positions let PQ
represent the s^is of the earth about which its diumal
rotation is performed without interfering with its amraal
motion in its orbit. Then, since the sun can only en-
lighten orle half of the surface at once, viz. that turned
towards it, the shaded portions of the globe in its several
positions will represent the dark, and the bright, the en<-
lightened halves of the earth's surface in these posittons.
Now, 1st, in .the position A, the sun is vertically over the
intersection of the equinoctial F£ and the ecliptic HG*
It is, therefore, in the equinox ; and in this position liie
poles P, Q, both fall on the extreme confines of the en-
lightened side. In this position, therefore, it is day oyer
hdf the northern and half the southern hemisphere al
ahee ; and as the earth revolves on its axis, every point
of its surface describes half its diumal course in light, and

CHAP, v.] TEMPERATURE OF THE EARTH. 187

half in darkness ; in other words, the duration of day and
night is here equal over the whole globe : hence the terra
equinox. The . same holds good at the avtumnal equinox
on the position C.

(311.) B is the position of the earth at the time of the
northern, summer solstice. Here the north pole P, and
ft considerable portion of the earth's surface in its neigh-
bourhood, as far as B, are situated within the enlighten-
ed half. As the earth turns on its axis* 4n this position',
therefore, the whole of that part remains constantly en-
lightened ; therefore, at this point of its orbit, or at. this
season of the year, it is continual day at the north pole,
and in all that region of the earth which encirdes this
pole as far as B— -that, is, to the distance of 23^ 28' from
the pole, or within what is called, in geography, the arctic
eirde. On the other hand, the opposite or south pole Q,
with all the region comprised wi&in the antarctic circle^
as far as 23° 28' from the south pole, are immersed at
this season in darkness, during the entire diurnal rotation,
BO that it is here continual night.

(312.) With regard to that portion of the surface com-
prehended between the arctic and antarctic circles, it is
no less evident that the nearer any point is to the north
pole, the larger will be the portion of its diurnal course
comprised within the bright, and the smaller within the
dark hemisphere ;• that is to say, the longer will be its
day, and the shorter its night. Every station north of ^e
equator will have a day of more and a night of less than
twelve hour'i^ duration, and vice versa. All these phe-
nomena are exactly inverted when the earth comes to the
opposite point D of its orbit.

(313.) Now, the temperature of any part of the earth's
surface depends mainly, if not entirely, on its exposure to
the sun's rays. Whenever the sun )s above the horizon
of any place, that place is receiving heat ; when below,
parting with it, by the process called radiation ; and the
whole quantities received and parted with in the year
must balance each other M every station, or the equilibri-
um of temperature would not be supported. Whenever,
then, the sun remains more than twelve hours above the
horizon of any place^ and less beneath, the general tempe*

189 A TREATISE ON ASTRONOMY. [CHAP. V.

rature of that place will be above the average ; when the
reverse, below. As the earth, then, moves from A to B,
the days growing longer, and the nights shorter in the
northern hemisphere, the temperature of every part of that
hemisphere increases, and we pass from spring to sum-
mer, while at the same time the reverse obtains in the
southern hemisphere. As the earth passes from B to C»-
the days and nights again approach to equality^the ex-
cess of temperature in the northern hemisphere above the
mean state grows less, as well as its defect in the south-
ern ; and at the autumnal equinox, C, the xnean state is
once more attained. From thence to D, and, finally,
round again to A, all the same phenomena, it is obvious,
must again occur, but reversed, it being now winter in
the northern, and summer in the southern hemisphere.

(314.) All this is exactly consonant to observed fact.
The continual day within the polar circles in summer,
and night in winter, the general increase of temperature
and length of day as the sun approaches the elevated
pole, and the reversal of the seasbns in the northern and
isouthern hemispheres, are all facts too well known to
require further comment. The positiqns A, C of the
earth correspond, as we have said, to the equinoxes ;
those at B, D to the solstices. This term must be ex-
plained. If, at any point, X, of the orbit, we draw XP
the earth's axis, and XS to the sun, it is evident that the
angle PXS will be the sun's polar distance. Now, this
angle is at its maximum in the position D, and at ifi
minimum at B; being in the former case =90^+23^
28'== 103'' 28', and in the latter 90**— 23^ 28'=66'' 32%
At these points the sun ceases to approach to or to recede
from the pole, and hence the name solstice.

(315.) The elliptic form of the earth's orbit has but
a very trifling share in producing the variation of tem-»
perature corresponding to the difference of seasons. This
assertion may at first sight seem incompatible with what
we know of the laws of the communication of heat from
a luminary placed at a variable distance. Heat, like
light, being equally dispersed from the sun in all direc-
tions, and being spread over the surface of a sphere T'On^*
tinually enlarging as we recede from the centre, must of

CHAP, y.3 EQUAL PISTRIBUTIOir OF BBAT.

1S9

coarse diminish in intensity accoi^ing to the inverse pro-
portion of the surface of the sphere over which it is
spread ; that is, in the inverse proportipn of the square
of the distance. But we have seen (art. 293) that this
is also the proportion in which the angtdar velocity of
the earth about the sun varies. Hence it appears, that
ihe momentary supply of heat received by the earth from
the sun varies in the exact proportion of the angular ve-
locity, i, e. of the momentary increase of longitude; and
from this it follows, that equal amounts of heat are re-
ceived from the sun in passing over equal angles round
it, in whatever part of the ellipse those angles may be
situated. Let, tiien, S represent the sun ; AQMP the

eardi^s orbit ; A its nearest point to the sun, or, as it is
called, iJtie perihelion of its orbit ; M the farthest, or the
aphelion; and therefore ASM the axis of the ellipse.
Now, suppose the orbit divided into two segments by a
straight line PSQ drawn through the sun, and any how
situated as to direction ; &en, if we suppose the earth
lo circulate in the direction PAQMP, it will have passed
over 180^ of longitude in moving from P to Q, and as
many in moving from Q to P. It appears, therefore,
from what has been shown, that the supplies of heat re-
ceived from the sun will be equal in the two segments,
in whatever direction the line PSQ be drawn. They
will, indeed, be described in unequal times ; that in
which the perihelion A lies in a shorter, and the other
in a longer, in proportion to their unequal area ; but the
greater proximity of the sun in the smaller segment com-

190 A TREATISE ON ASTRONOMY. [cHAP. V.

pensates exactly for its more rapid description, and thus
an equilibrium of heat is, at it were, maintained. Were
it not for this, the eccentricity of the orbit would mate-
rially influence the transition of seasons. The fluctua-
tion of distance amounts to nearly -j^^^th of its mean quan-
tity, and consequently, the fluctuation in the sun's direct
heating power to double this, or ^ih of the whole.
Now, the perihelion of the orbit is situated nearly at the
place of the northern winter solstice ; so that, were it
not for the compensation we have just described, the
effect would be to exaggerate the diflerence of summer
and winter in the southern hemisphere, and to moderate
it in the northern ; thus producing a more violent alter-
nation of climate in the one hemisphere, and an approach
to perpetual spring in the other. As it is, however, no
such inequality subsists, but an equal and impartial dis-
tribution of heat and light is accorded to.both.^

(316.) The great key to simplicity of conception in
astronomy, and, indeed, in all sciences where motion is
concerned, consists in contemplating every movement as
referred to points which are either permanently fixed,
or so nearly so, as that their motions shall be too small
to interfere materially with and confuse our notions. In
the choice of these primary points of reference, too, we
must endeavour, as far as possible, to select such as have
simple and symmetrical geometrical relations of situa-
tion with respect to the curves described by the moving
parts of the system, and which are thereby fitted to per-
form the office of natural centres— advantageous sta-
tions for the eye of reason and theory. . Having learned
to attribute an orbitual motion to the earth, it loses this
advantage, which is transferred to the sun, as the fixed
centre a^ut which its orbit is performed. Precisely as,
when embarrassed by the earth's diurnal motion, we
have learned to transfer, in imagination, our station of
observation from its surface to its centre, by the appli-
cation of the diurnal parallax ; so, when we come to in-
quire into the movements of the planets, we shall find

*See Gdological Transaction!, 1832, **Qn the Astronomical Canaea
which may Inflaenoe Geological Phenomena," by the author of thia

MEAN AND TRUE LONOITtTDB OF TBIB mm* 191

onrselves continnallf embarrassed by Hie oibitaat afo-
tion of our point of view, unless, by the considexatioii of
the annual or heliocentric parallaXf as it may be tenned,
we consent to refer all our obserratioBs on then to the
centre of the sun, or rather to the common centre of gra-
vity of the sun, and the other bodies which are connect
ed .with it in our system. Hence arises the distinetion
between the geocentric and heliocentric place of an ob-
ject. The former refers its situation in space to an
imaginary sphere of infinite radius, having the centre of
the earth for its centre — the latter to one concentric, with
the sun. Thus, when we speak of the heliocentfie lon-
gitudes and latitudes of objects, we suppose the specta^
tor situated in the sun, and referring them, by circles
perpendicular to the plane of the ecliptic, to the greal
circle marked out in the heavens by the infinite prolonga*
tion of that plane.

(317.) The point in the imaginary concave of an in-
finite heaven, to which a spectator in the sun refers the
earth, must, of course, be diametrically opposite to that to.
which a spectator on the earth refers the sun's centre ;
consequently, the heliocentric latitude of the earth is
always nothing, and its heliocentric longitude always
equal to the sun's geocentric longitude +180®, The
heliocentric equinoxes and solstices are, Uierefore, the
same as the geocentric ; and to conceive them, we have
only to imagine a plane passing through the sun's centre,
parallel to the earth's equator, and prolonged infinitely
on all sides. The line of intersection of tiiis plane and
the plane of the ecliptic is the line of equinoxes, and the
solstices are 90** distant from it.

(318.) The position of the longer axis of the earth's
orbit is a point of great importance. In the figure (art*
315) let EGLI be the ecliptic, E the vernal equinox, L
the autumnal (t. e, the points to which the earth is re-
ferred from the sun when its heliocentric longitudes are
0** and 180** respectively). Supposing the earth's mo-
tion to be performed in the direction ECLI, the angle
ES\, or the longitude of the perihelion, in the year 1800
was 09** 30' 5" : we say in the year 1800, because, in
point of fact, by the operation of causes hereafter to be

198 A TREATISE ON ASTRONOMF. [CHAF. V.

explained* its position is subject to an extremely slow va-
riation of about 12" per annum to the eastward, and
which, in the progress of an immensely long period — of no
less than 20,984 years — carries the axis ASM of the
orbit completely round the whole circumference of the
ecliptic. But this motion must be disregarded for the
present, as well as many other minute deviations, to be
brought into view when they can be better understood.
(319.) Were the earth's orbit a circle, described with
a uniform velocity about the sun placed in its centre, no-
thing could be e^ier than to calculate its position at any
time, with respect to the line of equinoxes, or its longi-
tude, for we should only have to reduce to numbers the
}»'Oportion following ; viz. One year : the time elapsed ::
300^ : the arc of longitude passed over. The longitude
so calculated is called in astronomy the mean longitude of
the earth. But since the earth's orbit is neither circular,
nor uniformly described, this rule will not give us the true
place in the orbit at any proposed moment. Neverthe-
less, as the eeeenlaricity and deviation from the circle are
small, the trtte place will never deviate very far from that
so determined (which, for distinction's sake, is called the
mean place), and the former may at all times be calculated
from the latter, by applying to it a correction or equation
(as it is termed), whose amount is never very great, and
whose computation is a question of pure geometry, de-
pending on the equable description of areas by the earth
about the sun. For since, in the elliptic motion, accord-
ing to Kepler^s law above stated, ar^os not angles 9xe
described uniformly, the proportion must now be stated
thus; One year : tfie time elapsed : : the whole area of
the ellipse : the area of the sector swept over by the ra*
dius vector in that tim^. This area, therefore, becomes
known, and it is then, as above observed, a problem of
pure geometry to aiscertain the ong/e about the sun (ASP,
/ig. art. 315), which corresponds to any proposed frac-
tional area of the whole ellipse supposed to be contained
in the sector APS. Suppose we set out from A the pe«»
rihelion, then will the angle ASP at first increase more
rapidly than the mean Umgitudcy and will, therefore, do-
nag the whole semi-revolution from A to M, exceed it in

MEAN AND TRUE LONGITUDE OF THE SUN. 199

amount ; or, in other words, ihe, true place will be in ad-
yanfie of the mean : at M, one half of the year will have
elapsed, and one half the orbit have been described,
whether it be circular or elliptic. Here, then, the mean
and true places coincide ; but in all the other half of the
orbit, from M to A, the true place will fall short of the
mean, since at M the angular motion is slowest, and the
true place from this point begins to lag behind the mean
— ^to make up witli it, however, as it approaches A, where
it once more overtakes it.

(320.) The quantity by which the true longitude of the
earth differs from the mean longitude is called the equa-
tion of the centre, and is additive during all the half-year
in which the earth passes from A to M, beginning at 0°
0' 0", increasing to a maximum, and .again diminish**
ing to 2ero at M ; after which it becomes subtractive,
attains a maximum of subtractive magnitude between M
and A, and again diminishes to at A. Its maximuray
both additive and subtractive, is I'' 55' 33"'3,

(321.) By applying, then, to the eartb's mean longi-
tude, the equation of the centre corresponding to any
given time at which we would ascertain its place, the true
longitude becomes known ; and since the sun is always
seen from the eartl^ in 180° more longitude than the earth
from the sun, in this way also the sun's true place in i^e
ecliptic becomes known. The calculation of the equa*
tion of the centre is performed by a table constructed for
that purpose, to be found in all *♦ Solar Tables."

(322.) The maximum value of thcequation of the cen-
tre depends only on the ellipticity of the orbit, and may
be expressed in terms of the eccentricity. Vice veraat
therefore, if the former quantity can be ascertained by
observation, the latter may be derived from it ; because,
whenever the law, or numerical connexion, between two
quantitiea is known, the one can always be determined
from the other. Now, by assiduous observation of the
sun's transits over the mmdian, we can ascertain, for
eve^y day, its exact right ascension, and thence conclude
its longitude (art. 260). After this, it is easy to assign the
angle by which this observed longitude exceeds or &1U
short oiihe mean ; and the greatest aoiount of this excess

194 A TREA-nsE ON ASTRONOMT. [CHAP. ▼•

or defect which occurs in the whole year, is the maxi-*
mum equation of the centre. This, as a means of ascer-
taining the eccentricity of the orbit, is a far more easy and
accurate method than that of concluding its distance by
measuriifg its apparent diameter. The results of the two
methods coincide, however, perfectly.

(323.) If the ecliptic coincided with the equinoctial,
the effect of the equation of the centre, by disturbing the
uniformity of the sun' s apparent motion in longitude,
would cause an inequality in its time of coming on the
meridian on successive days. When the sun's centre
com^s to the meridian, it is apparent noon, and if its mo-
tion in longitude were uniform, and the ecliptic coincident
with the equinoctial, this wonld always coincide with
mean noon, or the stroke of 12 on a well-regulated solar
clock. But, independent of the want of uniformity in
its motion, the obliquity of the ecliptic gives rise to an-
other inequality in this respect ; in consequence of which
the sun, even supposing its motion in the ecliptic uniform,
would yet alternately, in its time of attaining the meri-
dian, -Anticipate and fall short of the mean noon as shown
by the clock. For the right ascension of a celestial ob-
ject, forming a side of a right-angled spherical trian-
gle, of which its longitude is the hypothenuse, it is
clear that the uniform increase of the latter must necessi-
tate a deviation from uniformity in the increase of the
foi^mcr.

(324.) These two causes, then, acting conjointly, pro-
duce, in fact, a very considerable fluctuation in the time
as shown per clock, when the sun really attains the
meridian. It amounts, in fact, to upwards of half an
hour ; apparent noon sometimes taking place as much as
16|^ min. brfore mean noon, and at others as much as 14^
min. afler. This difference between apparent and mean
noon is called the equation of time, and is calculated and
inserted in ephemerides for every day of the year, under
that title;- or else, which comes to the same thing,
the moment, in mean time, of the sun's culmination,
for each day, is set down as an astronomical phenome-
non to be observed.

(325.) As the sun, in its appsurent annual course, is

CHAP, y.]] TROPICAL AND ANOMALISTIC TEARS. 195

carried along the ecliptic, its declination is continually
yarying between the extreme limits of 23° 28' 40" north,
and as much south, which it attains at the solstices. It
is cx)nsequently always' vertical over some part or other
of that zone or belt of the earth's surface which lies be-
tween the north and south parallels of 23° 28' 40".
These parallels are called in geography the tropics ; the
northern one that of Cancer ^ and the southern of Capri'
com ; because the sun, at the respective solstices, is situ-
ated in the division or signs of the ecliptic so denomi-
nated. Of these signs there are twelve, each occupying
30° of its circumference. They commence at the vemcd
equinox, and are named in order — Aries, Taurus, Gemi-
ni, Cancer, Leo^ Virgo, Libra, Scorpio, Sagittarius, Ca-
pricomus, Aquarius, Pisces. They are denoted also by
the following symbols:— V, 8, n, 25, ^, W^ ^ ,Wl, ^,
V^9 ^9 X* The ecliptic itself is also divided into
signs, de^ees, and minutes, &e. thus, 5* 27^ 0' corres-
ponds to 177° 0' ; but this is beginning to be disused.

(326.) When the sun is in either tropic, it enlightens,
as we have seen, the pble on that side the equator, and
shines over or beyond it to the extent of 23° 28' 40".
The parallels of latitude, at this distance from either
pole, are called the polar circles, and are distinguished
from each other by tlie names arctic and antarctic. The
regions within these circles are sometimes termed frigid
zones, while the belt between the tropics is called the
torrid zone, and the immediate belts temperate zones.
These last, however, are merely names given for the
sake/)f naming; as, in fact, owing to the different dis-
tribution of land and sea in the two hemispheres,
zones of climate are not co-terminal with zones of lati-
tude.

(327.) Our seasons are determined by the apparent
passages of the sun across the equinoctial, and its alter-
nate arrival in the northern and southern hemisphere,
^ere the equinox invariable, this would happen at in-
tervals precisely equal to the duration of the sidereal
year ; but, in fact, owing to the slow conical motion of
the earth's axis described in art. 264, the equinox re-
^ats on the ecliptic, and meets the advancing sun some-

106 A TREATISE OK ASTRONOBHT. [CKAP. V.

vthsX before the whole sidereal circuit is completed. The
annual retreat of the equinox is 50"* 1, and this arc is
described by the sun in the ecliptic in 20' 19"*9. By
so much shorter^ then, is the periodical return of our
seasons than the -true sidereal revolution of the earth
round the sun. As the latter period, or sidereal year, is
equal to 365* 6^ 9" 9* '6, it follows, then, that the former
must be^nly 365* 6** 48" 49* '7 ; and this is what is meant
by the tropical year.

(328.) We have already mentioned that the longer
axis of the ellipse described by the earth has a slow mo-
tion of ir'*8 per annum in advance. From this it re-
sults, that whea the earth, setting out from the perihelion,
has completed one sidereal period, the perihelion will
have moved fotward by 11 "'S, which arc must be de-
scribed before it can again reach the perihelion. In so
doipg, it occupies 4' SO'-'*?, and this must therefore be
added to the sidereal period, to give the interval between
two consecutive returns to the perihelion. This in-
terval, then, is 365* tf* 13" 49* -3,* and is what is called
the anomalistic year* All these periods have their uses
in astronomy ; but that in which mankind in general are
most interested is the tropical year, on which the return
of the seasons depends, and which we thus perceive to
be a compound phenomenon, depending chiefly and di-
rectly on the annual revolution of the earth round the
sun, but subordinately also, and indirectly, on its rota-
tion round its own axis, which is what occasions the
precession of the equinoxes ; thus affording an instruc-
tive example of the way in which a motion, onc^. ad-
mitted in any part of our system, may be traced in its
influence on others with which at first sight it could not
possibly be supposed to have any thing to do.

(329.) As a rough consideration of the appearance of
the earth points out the general roundness of its form>
and more exact inquiry has led us first to the discovery
of its elliptic figure, and, in the further progress of re*
finement, to the perception of minuter local deviations

* These numbers, as well as all the other numerical data of our sys-
tem, are taken from Mr. Baily's Astronomical Tables and FormulsB un-
less the oontraiy is expressed.

CHAP, v.] PHYSICAL CONSTITUTION OF THE SUN. 197

from that figure ; so, in investigating the solar motions,
the first notion we obtain is that of an orbit, generally
speaking, round, and not far from a circle, which, on
more careful and exact examination, proves to be an
ellipse of small eccentricity, and described in conformity
with certain laws, as above stated. Still minuter in-
quiry, however, detects yet smaller deviations again
from this form and from these laws, of which .ye have
a specimen in the slow motion of the axis of the orbit
spoken of in art. 318 ; and which are generally compre-
hended under the name of perturbations and secular in-
equalities. Of these deviations, and their causes, we
shall speak hereafter at lengths It is the triumph of
physical astronomy to have rendered a complete account
of them all, and to have left nothing unexplained, either
in the motions of the sun or in those of any other of the
bodies of our system. But the nature of this explana-
tion cannot be understood till we have' developed the
law of gravitation, and carried it into its more direct
consequences. This will be the object of our three fol-
lowing chapters ; in which we shall take advantage of
the proximity of the moon, and its immediate connexion
with and dependence on the earth, to render it, as it
were, a stepping-stone to the general explanation of the
planetar3r movements.

(330.) We shall conclude this by describing what is
known of the physical constitution of the sun.

When viewed through powerful telescopes, provided
with coloured glasses, to take off the heat, which would
otherwise injure our eyes, it is observed to have fre-
quently large and perfectly black spots upon it, sur-
rounded with a kind of border, less completely dark,
called a penumbra. Some of these are represented at
(ij 6, e, plate iii. fig. 1, in the plate at the end of this
volume. They are, however, not permanent. When
watched from day to day, or even from hour to hour,
they appear to enlarge or contract, to change their forms,
and at length to disappear altogether, or to break out
«i^«w in parts of the surface where none were before.
In such cases of disappearance, the central dark spot
always contracts into a point, and vanishes before the
r2

198 A TREATISE OX ABTRONOMT. [CHAP. V.

border. Occasionally they break up, or divide into two
or more, and in those offer every evidence of that ex-
treme mobility which belongs only to the fluid state, and
of that excessively violent agitation which seems only
compatible with the atmospheric or gaseous state of mat-
ter. The scale on which their movements take place is
immense. A single second of angular measure, as seen
from the earth, corresponds on the sun's disc to 466
miles ; and a circle of this diameter (containing there-
fore nearly 220,000 square miles) is the least space which
can be distincfly discerned on the sun as a visile area.
Spots have been observed, however, whose linear dia-
meter has been upwards of 45,000 miles ;* and even, if
some records are to be trusted, of very much greater ex-
tent That such a spot should close up in six weeks'
time (for they hardly ever last longer), its borders must
approach at the rate of more than 1000 miles a day.

Many other circumstances tend to corroborate this
view of the subject. The part of the sun's disc not oc-
cupied by spots is far from uniformly bright. Its ground
is finely mottled with an appearance of minute, dark
dots, or pores, which, when attentively watched, are
found to be in a constant state of change. There is
nothing which represents so faithfully this appearance
as the slow subsidence of some flocculent chymical pre-
cipitates in a transparent fluid, when viewed perpen-
dicularly f^om above : so faithfully, indeed^ that it is
hardly possible not to be impressed with the idea of a
luminous medium intermixed, but not confounded, with
a transparent and non-luminous atmosphere, either float-
ing as clouds tn our air, or pervading it in vast sheets
and columns like flame, or the streamers of our northern
lights.

(331.) Lastly, in the neighbourhood of great spots, or
extensive groups of them, large spaces of &e surface are
often observed to be covered with strongly marked
curved, or branching streaks, more luminous than the
rest, called factdse, and among these, if not already
existing, spots frequently break out. They may,

* Majrer, Obt. Mar. 15, 1758. « Ingeni macala in sole oonspidebatnr
«i^ut diameter s J^ diam. ■olis."

CHAP, v.] PROBABLE NATURE OF THE SOLAR SPOTS. Ij90

perhaps, be regarded with most probability, as the
ridges of immense waves in the luminous regions of the
sun's atmosphere, indicative of violent agitation in, their
neighbourhood.

(332;) But what are the spots? Many fanciful notions
have been broached on this subject, but only one seems
to have any degree of physical probability, viz. that they
are the dark, or at least comparatively dark, solid body
of the sun itself, laid bare to our view by those Immense
fluctuations in the luminous regions of its atmosphere, to
which it appears to be subject. Respecting the manner
in which Uiis disclosure takes place, different ideas again
have been advocated. Lalande (art. 3240) suggests,
that eminences in the nature of mountains are actually
laid bare, and project above the luminous ocean, appear-
ing black above it, while their shoaling declivities pro-
duce the penumbrse, where the luminous fluid is less
deep. A fatal objection to this theory is the perfectly
uniform shade of the penumbra and its sharp termination,
both inwards, where it joins the spot, and outwards,
where it borders on the bright surface. A more proba«
ble view has been taken by Sir William Herschel,* who
considers the luminous strata of the atmosphere to be
sustained far above the level bf the solid body by a
tiansparent elastic medium, carrying on its upper sur-
face (or rather, to avoid the former objection, at some
considerably lower level witUn its depth,) a cloudy
stratum which, being strongly illuminated n*om above,
reflects a considerable portion of the light to our eyes,
and forms a penumbra, while the solid body, shaded by
the clouds, reflects none. The temporary removal of
both the strata, but more of the upper than the lower, he
supposes eflfected by powerful npward currents of the
atmosphere, arising, perhaps, from spiracles in the body,
or from local agitations. See flg. I. d, plate III.

(333.) The region of the spots is confined within
about 30° of the sun's equator, and, from their motion on
the surface, carefully measured with micrometers, is ^»
certained the position of the equator, which is a plane
inclined 7° 20' to the ecliptic, and intersecting it in a line
♦Phfl. Trans. 1801.

200 A TREATISE ON ASTRONOMY. [cHAP. V.

whose direction makes an angle of 80° 21' with that ot
the equinoxes. It has been also noticed (not, we think,
without great need of further confirmation), that extinct
spots have again broken out, after long intervals of time,
on the same identical points. of the sun's globe. Our
knowledge of the period of its rotation (which, according
to Delambre's calculations, is 25''*01154, but, .according
to others, materially different,) can hardly be regarded as
sufficiently precise to establish a point of so much nicety.
(334.) That the temperature at the visible surface of
the sun cannot be otherwise than very elevated, much
more so than any artificial heat produced in our furnaces ,
or by chemical or galvanic processes, we have indications
of several distinct kinds : 1st, From the law of decrease
of radiant heat and light, which, being inversely as the
squares of the distances, it follows, that the heat received
on a given area exposed at the distance of the earth, and
on an equal -area at the visible surface of the sun, must
be in the proportion of the area of the sky occupied .by
the sun's apparent disc to the whole hemisphere, or as 1
to about 300000. A far less intensity of solar radiation,
collected in the focus of a burning glass, suffices to dis-
sipate gold and platina in vapour. 2dly, From the fa-
cility with which the c^orific rays of the sun traverse
glass, a property which is found to belong to the heat of
artificial fires in the direct proportion of their intensity.*
ddly, From the fact, that 4io most vivid fiames disappear,
and the most intensely ignited solids appear only as black
spots on the disk of the sun when held between it and
the eye.t From this last remark it follows, that the body
of the sun, however dark it may appear when seen through
its spots, may, nevertheless, be in a state of most intense

* By direct measurement with the ocftnometer, an instrument I have
long employed in such inquiries, and whose indications are liable to none
of those sources of fallacy which beset the usual modes of estimation, I
find that out of 1000 calorific solar rays, 816 penetrate a sheet of plate
glass 0-12 inch thick; and that of 1000 rays wnich have passed through
one such plate, 859 are capable of passing through another. — Author.

tThe ball of ignited guick-lime, m Lieutenant Drummond's oxy-hvdro-
gen lamp, gives the nearest imitation of the solar splendour wnicn has
yet been produced. Jlle appearance of this asainst the sun was, how*
eyer, as described ih.tti imperfect trial at whidi I was present The
experiment ought to he repeated under favourable circumstances.—
Author

/

CSAP. v.] ACTION OF THE SUN's RAXS ON THE EARTH. 201

ignition. It idoes not, howei^r, follow of necessity thst
if fiiust be so. The contrary is at least physically possi-
ble, A perfectly reflective canopy would effectually de-
fend it from the radiation of the luminous regions above
its atmosphere, and no heat would be conducted down-
wards through a gaseous' medium increasing rapidly in
density. That the penumbral clouds are highly reflect-
ive, the fact of their visibility in such a situation can
leave no doubt.

(336.) This immense escape of heat by radiation, we
may also remark, will fully explain the constant state of
tumultuous agitation in which the fluids composing the
visible surface are maintained, and the continual genera-
tion and filling in of the pored, without having recjourse
to internal causes. The mode of action here alluded to
is perfectly represented to the eye in the disturbed sub-
sidence of a precipitate, as described in art. 330, when
the fluid from which it subsides is warm, and losing heat
from its surface.

(336.) The sun's rays are the ultimate source of al-
most every motion which takes place on the surface o(,
the earth. By its heat are produced all winds, and those
disturbances in the electric equilibrium of the atmosphere
which give rise to the phenomena of terrestrial magnet-
ism. By their vivifying action vegetables are elaborated
from inorganic matter, and become, in their turn, the sujH
port of animals and of man*, and the sources of those* ^
great deposites of dynamical efficiency whiph are laid up
^r human use in our coal strata. By them the waters,
of the sea are made to circulate in vapour through the
air, and irrigate the land, producing springs and rivers.
By them are produced all disturbances of the chymical
equilibrium of the elements of nature, which, by #i series
of compositions and decompositions, give rise to new
products, and originate a transfer of materials. Even
the slow degradation of the solid constituents of the sur-
face, in which its chief geological changes consist, and
their diffusion among the waters of the ocean, are entirely
due to the abrasion of the wind and rain, and the alter-
nate action of the seasons ; and when y^ consider the
immense transfer ot matter so produced, the increase of

\

202 A TREATISE ON ASTRONOMY. [CHAP. Y^

pressure ovsr large spaces in the bed of the ocean, and.
diminution over corresponding portions of the land, we
are not at a loss to perceive how the elastic power of
subterranean fires, thus repressed on the one hand and
relieved on the other, may break forth in points when
the resistance is barely adequate to their retention, and
thus bring' the phenomena of even volcanic acti<rit|r under
the general law of solar influence.

(337.) The great mystery, however, is to conceive
how so enormous a conflagration (if such it be) can be
kept up. Every discovery in chymical science here
leaves us completely at a loss, or rather, seems to remove
farther the prospect of probable explanation. If conjec-
ture might be hazarded, we should look rathec to the
known possibility of an indefinite generation of heat by
friction, or to its excitement by the electric discharge,
Aan to^ny actual combustion of ponderable fuel, whe-
ther solid or gaseous, for the origin of the solar radiation.*

* EUectricity traversing excessively rarefied air or vapours, nves out
light, and, doubtless, also heat May not a continual current of electric
matter be constantly circulating iii tne sun's immediate neighbouriiood,
or traversing the planetary spaces, and exciting, in the upper regions of
its atmosphere, those phenomena of which, on however diminutive a
scale, we have yet an unequivocal manifestation in our aurora borealis ?
The possible analogy of the solar hght to that of the aurora hira been
distiactly insisted on by Iny father, in his paper already cited. It would
be a highly curious subject of experimental inquiry, how far a mere re-
du])licatJon of sheets of flame, at a distance one behind the other (by
which their light might be brought to any required intensity), would com-
t&unicate to the heat of the resulting compound ray the penetrating cha-
racter which distinguishes the solar calorific rays. We may also owerve,
that the tranquillity of the sun's polar, as compared with its equatorial
pegions (if its spots be really atmoipheric), cannot be accounted for by its
rotation on its axis only, but must arise from some cause external to the
sun, as we see the belts of Jupiter and Saturn, and bur trade- winds, arise
fiom a cause, external to these planets, combining itself with their rota-
tion, which alone can produce no motions when once the form of equili-
* brium is attained.

The prumatic analysis of the solar beam exhibits in the spectrum a
series of ** fixed lines,'"^ totally unlike those which belong to the light of
anyr known terrestrial flame. This may hereafter lead. us to a clearer
insight into its origin. But, before we can draw any conclusions from
such an indication, we must recollect, that previous to reaching us it has
undergone the whole absorptive action of our atmosphere, as well as of
the sun's. Of the latter we know nothing, and may conjecture every
thing ; but of the blue colour of the former we are sure \ and if this l^
an inherent (i. e. an absorptive) colour, the air must be expected to a^
«m the spectrum after the analogy of other j^ured media, which often
(and etpeciaUy light Uue media) leave unabSorbed portions sepamted by
dark intervalis. It deserves inquiry, therefore, whether some or all the

CHAP. Tl.] C» THB MOON. 208

CHAPTER VI.

Of the MoaiL— Its sidereal Period — Its apparent Diameter — Its Parallai;
Distance, and real Diameter — First Approximation to its Orbit — An
Ellipse about Ae Earth in the Focos^ItB Eccentricity and Indina*-
tion — ^Motion ef the Nodes of its Orbit— Occultations— Solar Eclipse*
— Phases of the Moon — ^Its synodical Period — Lunar Eclipses-
Motion of t|ie Apsides of its Orbit— Physical Constitution of the Moon
— ^Its Mountmtis — Atmosphere — ^Rotation on Axis — ^libration — Ap-
pearance of i^e Earth from it

(388.) The moon, like the sun, appears to advance
among the stars with a morement contrary to the general
diurnal motion of the heavens, but much more rapid, so
as to be very readily perceived (as we have before ob-
served) by a few hours' cursory attention on any moon-
light night. By this continual advance, which, though
sometimes quicker, sometimes slower, is never intermit-
ted or reversed, it makes the tour of the heavens in a
mean or average period of 27'*?^ 43°* 11**5, returning,
in that time, to a position among the stars nearly coin-
cident with that it had before, and which would be ex-
actly so, but for causes presently to be stated.

(339.) The moon, then, like the sun, apparently de-
scribes an orbit round the earth, and this orbiU cannot be
very different from a circle, because the apparent angular
diameter of the fuU moon is not liable to any great extent
of variation.

(340.) The distance of the moon from the e^rth is
concluded from its horizontal parallax, which igray be found
either directly, by observations at r^ote geographical
stations, exactly similar t^ those described in art. 302,
in tiie case of the sun, or by means of the phenonieBa
called oceultations (art. 346), irom which also its appar
rent diameter is most readily and correctly found. From
such observations it •results that the mean or average dis-

fized lines observed by Woliaston and Fraunhofer may not have their
origin in our own atmosphere. Experiments made on lofljr mountains^
orUie cars c€ balloons, on the one hand, and on the other with reflected
beams which have been made to traverse several miles of additional air
near the surface, wQuld decide, thie point The absorptive effect of the
nm*8 atmoei^ere, and possibly also of the medium surronndinf it (what-
ever it be), Which resists the motions of comets, cannot be thos eliminated.

u-

9M A TREATISE ON ABTROXOHY. [cHAP. ▼!•

tance of the centre of the moon from that of the earth
is 59-9643 of the earth's equatorial radii, or about
237,000 miles. This distance, great as it is, is little
more than one fourth of the diameter of the sun's body,
so that the globe of the sun would nearly twice include
the whole orbit of the moon ; a consideration wonderfully
calculated to raise our ideas of that stupendous lumi-
nary!

(341.) The distance of the moon's centre from an ob-
server at any station on the earth's surface, compared
with its apparent angular diameter as measured from that
station, will give its real or linear diameter. Now, the
former distance is easily calculated when the distance
from the earth's centre is known, and the apparent Zenith
distance of the moon also determined by observation;
for if we turn to the figure of art. 298, and suppose S the
moon, A the station, and C the earth's centre, the dis-
tance SO, and the earth's radius CA, two sides of the
triahgle AGS are given, and the angle CAS, which is the
supplement of ZAS, the observed zenith distance, whence
it is easy to find AS, the moon's distance from A« From
such observations and calculations it results, that the
real diameter of the moon is 2160 miles, or about 0*2729
of that of«the earth, whence it follows that the bulk of
the latter being considered as 1, that of the ^former will
be 0*0204, or about ^V-

(342.) By a series of observations, such as deseribed
in art. 340, if continued during one or more revolutions
of the moon, its real distance may be ascertained at every
point of its orbit ; and if at the. same time its apparent
places in the heavens be obserned, and reduced by means
of its parallax to the earth's centre, their angukr intei^
vals will become known, so that the path of the moon
may then be laid down on a chart supposed to represeit
the plane in which its orbit lies, just as was explained in
the case of the solar ellipse (art. 292). Now, when this
is done, it is found that, neglecting certain small (though
very perceptible) deviations (of which a satisfactory ac-
count will hereafter be rendered), the form of the appa^
rent orbit, like that of the sun, is elliptic, but consider-
ably more eccentric, the eccentricity amounting to 0*05484

CHAP. VI.3 REVOLUTION OF THE MOON's K0DE8. M0^

of the mean distance, or the major semi-axis of the ellipse,
and the earth's centre being' situated in its focus.

(343.) The plane in which this orbit lies is not the
ecliptic, however, but is inclined to it at an angle of 5^
8' 48'% which is called the inclination of the lunar orbit,
and intersects it in two opposite points, which are called
its node — the ascending node being that in which the
moon passes from the southern side of the ecliptic to the
northern, and the descending the reverse. The points
of the orbit at which the moon is nearest to, and farthest
from, &e earth, are called respectively its perigee and
apogee, and the line joining them and the earth the line
of apsides,

(344.) There are, however, several remarkable cir^
cumstances which interrupt the closenees of the analogy,
which cannot fail to strike the reader, between the mo-
tion of the moon around the earth, and of the earth round
the sun. In the latter case, the ellipse described remains,
during a great many revolutions, unaltered in its position
and dimensions ; or, at least, ^e changes which it under-
goes are not perceptible but in a course of very nice ob»
servations, which have disclosed, it is true, the existenee
of " perturbations," but of so minute an order, that, in
ordinary parlance, and for common purposes, we may
leave them unconsidered. But this cannot be done in
the case of the moon. Even in a single revolution, ite
deviation from a perfect ellipse is rery sensible. It does
not return to the same exact position among the stars
from which it set out, thereby indicating a continual
change in the plane of its orbit. And, in effect, if we
trace by observation, from month to month, the point
where it traverses tihe ecliptic, we shall find diat ^ nodes
of its orbit are in a continual state of retreat upon the
ecliptic. Suppose O to be the earth, and A bad that
portion of die plane of the ecliptic whteh is intersected
by the moon, in its alternate passages through it, from
south to north, and vice versa ; and let ABCDEF be a
portion of the moon's orbit, embracing a complete side-
real revolution. Suppose it to set out from the ascending
node, A ;> then, if the orbit lay all in one plane, passing
through 0, it would have a, the opposite point in the
S

206 A TREATISE ON ASTRONOMY. [CHAP. TI.

ecliptic, for its descending node ; after passing which, it
would again aseead at A. But, in fact, its real path car-

J)

ries it not to a, but along a certain curve, ABC, to C a,
point in the ecliptic less than 180° distant from A ; so
that the angle AOC, or the arc of longitude described
between the ascending and the descending node, is some-
what less th^n 180°. It then pursues its course below
the ecliptic, along the curve CDE, and rises again above
it, not at the point c, diametrically opposite to C, but at
a point £, less advanced in longitude. On the whole,
then, the arc described in longitude between two conse-
cutive passages from south to north, through the plane
of the ecliptic, falls short of 360° by the angle AOE ;
or, in other words, the ascending node appears to have
retreated in one lunation, on ^e plane of the ecliptic by
that amount. To complete a sidereal revolution, then, it
must still go on to describe an arc, AF, on its orbit,
which will no longer, however, bring it exactly back to
A, but to a point somewhat above it, or having north lati-
tude,

(345.) The actual amount of this retreat of the moon's
node is about 3' W'^ per-diem^ on an average^ and in
a period of 6793'39 mean solar days, or about 18*6 years,
the ascending node is carried round in a direction con-
trary to the moon's motion in its orbit (or from east to
west) over a whole circumference of the ecliptic. Of
course, in the middle of this period the position of the
orbit must have been precisely reversed from what it was
at the beginning. Its apparent path, then, will lie among
totally different stars and constellations at difierent parts
of this period ; and, this kind of spiral revolution being
continually kept up, it will, at one time or other, cover
with its disc every point of the heavens within that

CHAP. VI.] ECLII»S£S AND OCCtJLTATIONg. 207

limit of latitude or distance from the ecliptic which its
inclination permits ; that is to say, a belt or zone of the
heavens, of 10* 18' in breadth, having the ecliptic for its
middle line. Nevertheless, it still remains true that the
actual place of the moon, in consequence of this motion,
deviates in a single revolution very little from what it
would be were the nodes at rest. Supposing ^e moon
to set out from its node A, its latitude, when it comes to
F, having completed a revolution in longitude, will not
exceed 8' ; and it must be borne in mind that it is to ac-
count for, and represent geometrically, a deviation of this
small order, that the motion of the nodes is devised.

(346.) Now, as the moon is at a very moderate dis-
tance from us (astronomically speaking), and is in fact
our nearest neighbour, while the sun and stars are in
comparison immensely beyond it, it must of necessity
happen, that at one time or other it must pas8 over and
occult or eclipse every star and planet within the 2one
above described (and, as seen from the surface of earth,
even somewhat beyond it, by reason of parallax, which
may throw it apparently nearly a degree either way
from its place as seen from the centre, according to the
observer's station). Nor is the sun itself exempt from
being thus hidden, whenever any part of the moon's
disc, in this her tortuous course, comes to overlap any
part of the space occupied in the heavens by that lumi-
nary. On these occasions is exhibited the most striking
and impressive of all the occasional phenomena of astro-
nomy, an eclipse of the suny in which a greater or less
portion, or even in some rare conjunctures the whole, of
its disc is obscured, and, as it were, obliterated, by the
superposition of that of the moon, which appears upon
it as a circularly-terminated black spot, producing a
temporary diminution of daylight, or even nocturnal
darkness, so that the stars appear as if at midnight. In
other cases, when, at the moment that the moon is cen-
trally superposed on the sun, it so happens that her dis-
tance from the earth is such as to render her angular
diameter less than the sun's, the very singular pheno-
menon of an annular solar eclipse takes place, when
the edge of the sun appears for a few minutes as a nar-

A TRSATISS ON ASTRONOMY. [CHAF. Vt.

row ring of light, projecting on all sides beyond the dark
circle occupied by the moon in its centre.

(347.) A solar eclipse can only happen when the sua
and moon are in conjunction, Uiat is to say, have the
sanie, or nearly the same, position in the heavens, or the
same longitude. It will presently be seen that this con-
dition can only be fulfilled at the time of a new moon^
through it by no means follows, that at every conjunction
there niU8t be an eclipse of the sun. If the lunar orbit
coincided with the ecliptic, this would be the case, but
as it is inclined to it at an angle of upwards of 5^, it is evi-
dent that the conjunction, or equality of longitudes, may
take place When the moon is in the part of her orbit too
rexdote from the ecliptic to permit the discs to meet and
overlap. It is easy, however, to assign the limits within
which an eclipse is possible. To this end we must con-
sider, that, by the effect of parallax, the moon's appa*
rent edge may be thrown in ani/ direction, according to
a spectator'^ geographical station, by any amount not
exceeding the horizontal parallax. Now, this comes to
the same (so far as the possibility of an eclipse is con-
cerned) as if the apparent diameter of the moon, seen
from ^e earth's centre, were dilated by twice its hori-
zontal parallax ; for, if, when so dilated, it can touch or
overlap the sun, there must be an eclipse at some part or
other of the earth's surface. If, then, at the moment of
the nearest conjunction, the geocentric distance of the
centres of the two luminaries do not exceed the sum of
their semidiameters and of the moon's horizontal paral-
lax, there will be an eclipse. This sum is, at its maxi-
mum, about r H' ^T*. In the spherical triangle SNM»

liien, in which S is the sun's centre, M the moon's, SN
the ecliptic, MN the moon's orbit, and N the node, wa

CHAP. VI.] LIMITS OF A SOLAR ECLIPfiE. 209

may suppose the angle NSM a right angle, SMa=l® 34'
27", and the angle MNS=5° 8' 48", the inclination of
the orbit. Hence we calculate SN, which comes out
16** 68'. If, then, at the moment of the new moon, the
moon's node is farther from the sjun in longitude than
this limit, there can be no eclipse ; if within, there may,
and probably will, at some part or other of the earth.
To ascertain precisely whether there will or not, and,
if there be, how great will be the part eclipsed, the solar
and lunar tables must be consulted, the place of the node
and the semidiameters exactly ascertained, and the local
parallax^ and apparent augmentation of the moon's dia-
meter due to the difference of her distance from the
observer and from the centre of tfie earth (which may
amount to a sixtieth part of her horizontal diameter),
determined ; after which it is easy, from the above coi^
siderations, to calculate the amount overls^pped of the
two discs, and their moment of contact.

(348.) The calculation of the occultation of a star
depends on similar considerations. An occultation is
possible, when the moon's course, xu seen from the
earth's centre, carries her within a distance from the
star equal to tiie sum of her semidiameter and horizontal
parallax; and it wiU happen at any partictdar spotf
when her apparent path, as seen from that spot, carries
her centre within a distance equal to the sum of her
augmented semidiameter and actttal parallax. The de-
tails of these calculations, which are somewhat trouble-
some, must be sought elsewhere.*

(349.^ The phenomenon of a solar eclipse and of an
occultation are highly interesting and instructive in a
physical point of view. They teach us that the moon
is an opaque body, terminated by a real and sharply de-
fined surface intercepting light like a solid. They prove
to us, also, that at those times when we cannot see the
maoQ, she really exists, and pursues her course, and
that wkea we see her only as a crescent, however nar-
row, the whole globular body is there^ filling up the de-
ficieiKt o:utline« though unseen. For occultations take
place indifierendy at the dark and bright, the visible and

* Woodhouse's Astronomy, vol. i. See also Trans. Ast Soc. vol. i. p. 32Si
s2

dlO A TRSATI8B ON ASTRONOMY. £cHAP. VI.

invisible ontline, whichever happens to be towards the
direction in which the moon is moving ; with this only
difference, that a star occulted by the bright limb, if the
phenomenon be watched with a telescope, gives notice*
by its gradual approach to the visible edge, when to ex-
pect its disappearance, while, if occulted at the dark
limb, if the moon, at least, be more than a few days
old, it is, as it were, extinguished in mid-air, without
nodce or visible cause for its disappearance, which, as
it happens inatantaneouMlr/t and without the slightest
prf vious diminution of its light, is always surprising ;
and, if the star be a large and bright one, even startling
from its suddenness. The reappearance of the star, too,
when the moon has passed over it, takes place in those
cases when the bright side of the moon is foremost, not
at the concave outline of the crescent, but at the invisible
outline of the complete circle, and is scarcely less sur-
prising, from its suddenness, than its disappearance in
the o£er case.*

(350.) The existence of the complete circle of the disc,
even when the moon is not full, does not, however, rest
onLy on the evidence of occultations and eclipses. It
may be seen, when the moon is crescent or waning, a few
days before and after the new tnoonj with the naked eye,
as a pade round body to which the crescent seems attach-
ed, and somewhat projecting beyond its outline (whieh
is an opdcal illusion arising (torn the greater intensity of
its light). The cause of Siis appearance will presently
be explained. ' Meanwhile the fact is sufficient to show

* There is an opttcad illmion of • very ttnnge and unaccountable na-
ture which haa o jten been remarked in occultations. The star appears
to advance actually ttpon and wUhin the edge of the disc before it disap-
pears, and that sometimes to a considerable depth. I have never nysBif
wimeawd this sinfular efiect, but it rests oa. most unequivocal testimony.
I have called it an optical illusion ; but it is hardy posnUe that a star
may shine on such occasions tiuough deep fissures in the substance of
the moon. The occuUalioos of dose double stars ought to be nartowly
watched, to see whether both individuals are thus prvjecUd^ as well as
ibr other purposes connected with their theoiy. I will only hint at one,
viz. that a double star, too dose to be seen divided with any telesoopa^
may yet be detected to be double by the mode of its disappearance.
Shottfd a considemUe star, Ibr instance, instead of undergoing instantar
neous and complete extinction, go out by two distinct steps, following
dose upon each other; ^rst losing a portion, then the whwe remaiiidffr
of its light, we may be sure it is a douole star, though wO cannot see the
indivichials separately. — Author.

CHAP.YI.] PHilSES OF TUK MOOH. 811

that the moon is not inherently laminons like the nav^
but that her light is of an adventitaoiu nature. And itf
crescent form, increasing regularly from a narrow
fiemicircular line to a complete circular disc, corres*
ponds to the appearance a {^obe would present, one he-
misphere of which was bladL, the other white, when dif*
ferentiy turned towards the eye, so as to present a great*
er or less portion of each. The obvious conclusion from
this is, that the moon is such a globe, one half of which
is brightened by the rays of some luminary sufficiently
distant to enlighten the complete hemisphere, and suffi-
ciently intense to give it the degree of splendour we see.
Now, the sun alone is competent to such an effect Its
distance and light suffice ; and, moreover, it is invariably
observed that, when a crescent, the bright edge is towarae
the «tm, and that in propoition as the moon in her monthly
•course becomes more and more distant from the sun, the
breath of the crescent increases, and vice versa.

(351.) The sun's distance being 23084 radii of Htu
€arth, and the moon's only 60, the former is nearly 400
times the latter. Lines, therefore, drawn from the sun
to every part of the moon's orbit may be regarded as paip-
Allel. Suppose, now, O to be the earth, AB€D« ^kc»

various positions of the moon in its orbit, and S tibe sun,
at the vast distance above stated ; as is shown, then, in
the figure, the hemisphere of the lunar globe turned to-
wards it (on the right) will be bright, the opposite dark,
wherever it may stand in its orbit Now, in the position
A, when in conjunction with the sun, the dark part is

212. A TREATISE ON ASTRONOMY. [cHAP. Tip

entirely turned towards O, and the bright from it. In
this case, then, the moon is not seen, it is new moon*
When the moon has come to C, half the bright and half
the dark hemisphere are presented to O, and the same in
the dpposite situation 6 : these sure the first and third
quarters of the moon. Lastly, when at E^ the whole
bright face is towards the earth, the whole dark side from
it, and it is then seen wholly bright orfullmoonw In the
intermediate positions BDFH, the portions of the
bright face presented to O will be at first less than half
the visible surface, then greater, and finally less again,
till it vanishes altogether, as it comes round again to A.

(352.) These monthly changes of appearance, or
phase9, as they are called^ arise^ then, from the moon, an
opaque body, being illuminated on one side by the sun,
and reflecting from it, in all directions, a portion of the
light BO received. Nor let it be thought surprising that
a solid substance thus illuminated should appear to shine
and again illuminate the earth. It is no more than a
white cloud does standing off upon the clear blue sky.
By day, the moon can hardly be distinguished in bright-
ness from such a cloud ; and, in the dusk of evening,
clouds catching the Last rays of the sun appear with 3
dazzling splendour, not inferior to the seeming brightness
of the moon at night. That the earth sends also such a
light to the moon, only probably more powerful by rea-
son of its greater apparent size,* is agreeable to optical
principles^ and explains the appearance of the dark por-
tion, of the young moon completing its crescent (art. 850).
For, when die moon is nearly new to the earth, the lat-
ter (so to speak) is nearly full to the former; it then illu*
minates its dark half by strong eartMight ; and it is 9
portion of this, reflected fo^ck again, which ma]&es it visi-
ble to us in the twilight sky. As the moon gains age,
the earth oflfers it a less portion of its bright side, and tho
phenomenon in question dies away,

(353.) The lunar month is determined by the recur*
ren^ee of its phases ; it reckons from new moon to new

, ^thit ftpparent diameter of' the moon is 2Sf from the earth ; that of th»
ear^ 9^e]^ from the moon is twice her horizontal parallax, or JL** Ml* Th^
apparent surfaces, therefore, are as (114)3 . (32)3, or as 13 : 1 nearly.

CHAP. VI.] SYNODICAL RSVOLVTION OF THB VOON. 213

moon ; that ist from leaving its conjunction with the mm
to its retarn to conjunction. If the sun stood still, like a
fixed star, the interval between two conjunctions would
be the same as the period of the moon's sidereal revolu-
tion (art. 338^ ; but, as the sun apparently advances in
the heavens m the same direction with the moon, only
slower, the latter has more than a complete sidereal pe-
riod to perform to come up with the sun again, and will
require for it a longer time, which is the lunar month,
or, as it is generally termed in astronomy, a synodieal
period. The difTerenee is easily calculated by consider-
ing that the superfluous arc (whalever it be) is described
by the sun with his velocity of 0^*08565 per diem, in
the same time that the moon describes that arc pku a
complete revolution, with her velocity of 18***17640;>er
diem$ and, the times of description being identical, the
spaces are to each other in the proportion of the veloci-
ties.* From these data a slight knowledfe of arithmetic
wiU asfllce to derive the are in question, and the time
of te^description by the moon ; which, being die ejccess
of ife synodic over the sidereal period, the former will
be liad, and will appear to be 29** 12** 44" 2*'87.

(354.) Supposing the position of the nodes of the
moon's orbit to permit it, when the moon stands at A
(or at the new moon), it will intercept a part or the
whole of the sun's rays, and cause a solar eclipse. On
the other hand, when at £ (or at the full moon), the
earth O will intercept the rays of the sun, and east a
$hadow on the moon, thereby causing a lunar eclipse.
And this is perfectly consonant to fact, such eclipses
never happening but at the exact time of the full moon.
But, what is still more remarkable, as confirmatory of
the position of the earth's sphericity, this shadow, which
we plainly see to enter upon, and, as it were, eat away
the disc of the moon, is always terminated by a circultur
outline, though, from the greater size of the circle, it is

*Let V and V be the mean angular Telocitie%« die flupeifluont arc;

ttienV:«::l-f.«:«;andV-«:v:;l:«,wfaenoe«kifiM]nd,a]id-adie

time of dewDbing «^ or the diflerenee of the tidereal and lynodioal peii*
oda. We thaU have occaakm for tbia again.

214 A TREATISE ON ASTRONOMY [cHAP. VI.

only partially seen at any one time. Now, a body
which always casts a circular shadow must itself be
spherical.

(355.) Eclipses of the sun are best understood by re-
garding the sun and moon as two independent luminaries,
each moving according to known laws, and viewed from
the earth ; but it is also instructive to consider eclipses
generally as arising from the shadow of one body thrown
on another by a luminary much larger than either. Sup-
pose, then, A6 to represent the sun, and CD a spherical
body, whether earth or moon, illuminated by it. If we
join and prolong AC, BD ; since AB is greater than CD,
these lines will meet in a point E, more or less distant
from the body CD, according to its size, and within the
space CED (which represents a cone, since CD and AB

are spheres), there will be a total shadow. This shadow
is called the umbra, and a spectator situated within it
can see no part of the sun's disc. Beyond the umbra
are two diverging spaces (or rather, a portion of a single
conical space, having K for its vertex), where if a
spectator be situated, as at M, he will see a portion only
(AONP) of the sun's surface, the rest(BONP) being ob-
scured by the earth. He will, therefore, receive only
partial sunshine ; and the more, the nearer he is to the
exterior borders of that cone which is called the penum-
bra. Beyond this he will see the whole sun, and be in
full illumination. All these circumstances may be per-
fectly well shown by holding a small globe up in the

CHAP. VI. J SITNODICAL REVOLUTION OF THB MOON. 21 5

sun, and receiving its shadow at different distances on a
sheet of paper.

(356.) In a hinar eclipse (represented in the upper
figure), the moon is seen to enter the penumbra first, and
by degrees, get involved in the umbra, the former sur-
rounding ^e latter like a haze. Owing to the great size
of the earth, the cone of its umbra' always projects
far beyond the moon ; so that, if, at the time of the
eclipse, &e moon's path be properly directed, it is sure
ta pass through the umbra. This is -not, however, the
ease in solar eclipses. It so happens, from ibe adjust-
ment of the size and distance of the moon, that the ex-
tremity of her umbra always falls near the earth, but
sometimes attains and sometimes falls short of its surface.
In the former case (represented in the lower figure), a
black spot, surrounded by a fainter shadow, is formed,
beyond which &ere is no eclipse on any part of the
earth, but within which there may be either a total or
partial one, as the spectator is within the umbra or
penumbra. When the apex of the umbra falls on the
surface, the moon at that point will appear, for an in-
stant, to jtist cover the sun ; but, when it falls short,
there will be no total eclipse on any part of the earth ;
but a spectator, situated in or near the prolongation of
the axis of the eon^, will see the whole of th^ moon on
the sun, although not large enough to cover it, t. e. he
will witness an annular eclipse.

(357.) Owing to a remarkable enough adjustment of
the periods in which the moon's synodical revolution,
and that of her nodes, are performed, eclipses return after
a certain period, very nearly in the same order and of the
same magnitude. For 223 of the moon's mean synodi-
eal revolutions, or lunations^ as they are catled, wiU be
found to occupy 6585*32 days, and nineteen complete
synodical revolutions of the node to occupy 6585*78.
The difference in the mean position of the node, then, at
the beginning and end of 223 lunations, is nearly insen-
sible ; so that a recurrence of all eclipses within that in-
terval must take place. Accordingly this period of 223
lunations, or eighteen years and ten days, is a very im-
portant one in the calculation of eclipses. It is supposed

316 A TREATISE ON ASTRONOMY. [CHAF. H*

to have been known to the Chaldeans, under the name of
the aaroa ; the regular return of eclipses having been
known as a physical fact for ages before their exact the-
ory was understood.

(358.) The commencement, duration, and magnitude
of a lunar eclipse are much more easily calculated than
those of a solar, being independent of the position of the
spectator on the earth's surface, and the same as if view-
^ from its centre. The common centre of the umbra
and penumbra lies always in the ecliptic, at a point oppo-
site to the sun, and the path described by the moon in pass-
ing through it is its true orbit, as it stands at the moment
of the full moon. In this orbit, its position, at every in-
stant, is known from the lunar tables and ephemeris ; and
all we have, therefore, to ascertain is, the moment when
the distance between the moon's centre and the centre of
the shadow is exactly equal to the sum of the semidiame-
teis of the moon and penumbra<t or of the moon and
ttm6ni, to know when it enters upon and leaves ihem re-
spectively.

(359.) The dimenswns of the shadow, ai the place
where it crosses the moon's path, require U9 ^ know
the distances of the sun and moan ai the time. These
are variable ; but are calculated and set down, as well as
their semidiameteis, for every day, in the ephemeris, so
that none of the data are wanting. The sun's distance is
easily calculated from its elliptic orbit ; but the moon's
is a matter of more difficulty, for a reason we will now
explain.

(300.) The moon's orbit, as we have before hinted, i»
not, strictly speaking, an ellipse returning into itself, by
reason of the variation of the plane in which it lies, and
the motion of its ^odes. But even laying aside this con-
sideration, the axis of the ellipse is itself constantly
changing its direction in space, as has been already stated
of the solar ellipse, but much more rapidly ; making a
complete revolution, inthef isame direction with the moon's
own motion, in 3332*5753 mean solar days, or about
nine years, being about 3° of angular motion in a whole
revolution of the moon. This is the phenomenon known

CSAP, VI.] PHYSICAL CONDITION OF THB HOON. 217

by the name of the revolution of the moon's apsides. Its
cause will be hereafter explained.' Its immediate effect
is to produce a variation in the moon's distance from the
earth, which is not included in &e laws of exact elliptic
motion. In a single revolution of the moon, this varia-
tion of distance is trifling ; but in the course of many it
becomes considerable, as is easily seen, if we consider
that in four years and a half the position of the axis will
be completely reversed, and the apogee of the moon will
occur where the perigee occurred before.

(361.) The best way to form a distinct conception of
the moon's motion is to regard it as describing an ellipse
about the earth in the focus, and, at the same time, to re-
gard this ellipse itself to be in a' twofold state of revolu-
tion ; 1st, in its own plane, by a continual a;ivance of its
axis in that plane ; and 2dly, by a continual tilting mo-
tion of the plane itself, exactly similar to, but much more
rapid than, that- of the earth's equator produced by the
conical motion of its axis described in art. 266.

(362.) The physical constitution of the moon is better
known to us than that of any other heavenly body. By
the aid of telescopes, we discern inequalities in its sur-
face which can be no other than mountains and valleys-—
for this plain reason, that we see the shadows cast by the
former in the exact proportion as to length which they
ought to have, when we take into account the inclination
of the sun's rays to that part of the moon's surface on
which they stand. The convex outline of the limb turned
towards tiie sun is always circular, and very nearly
smooth ; but the opposite border of the enlightened part,
which (were the moon a perfect sphere) ought to be an
exact and shai^ply defined ellipse, is always observed to
be extremely ragged, and indented with deep recesses
and prominent points. The mountains near this edge
cast long black shadows, as they should evidently do,
when we consider that the sun is in the act of rising or
setting to the parts of the moon so circumstanced. But
as the enlightened edge advances beyond them, i, e. as
the sun to them gains altitude, their shadows shorten ;
and at the full moon, when all the light falls in our line

218 A TREATISE ON ASTRONOMY. [cHAP. TT*

of sight, no shadows ajre seen on any part of her surface.
From micrometrical measures of the lengths of the sha-
dows of many of the more conspicuous mountains, taken
under the most favourable eircumstanees, Uie heights of
many of them have been calculated ; the highest being
about 1| English miles in perpendicular altitude. The
existence of such mountains is corroborated by their ap-
pearance as small points or islands of light beyond the
ejttreme edge of the enlightened part, which are their
tops catching the sunbeams before the intermediate
plain, and which, as the light advances, at length connect
themselves with it, and appear as prominences from the
general edge.

(363.) The generality of the lunar mountains present a
striking uniformity and singularity of aspect. They are
wonderfully numerous, occupying by far the larger por-
tion of the surface, and almost universally of an exactly
circular or cup-shaped form, foreshortened, however, into
ellipses towards the limb ; but the larger have for the
most part flat bottoms within, from which rises centrally
a small, steep, conical hill. They offer, in short, in its
highest perfection, the true volcanic character, as it may
be seen in the crater of Vesuvius, and in a map of the
volcanic districts of the Campi Phlegraei* or the Puy de
Ddme. And in some of the principal ones, decisive
marks of volcanic stratification, arising from successive
deposites of ejected matter, may be clearly traced with
powerful telescopes.! What is, n^oreover, extremely
singular in the geology of the moon is, that although no-
thing having the character of seas can be traced (for the
dusky spots which are commonly called seas, when
closely examined, present appearances incompatible with
the supposition of deep water), yet there are large re-
gions perfectly level, and apparently of a decided alluvial
character.

(364.) The moon has no clouds, nor any other indi-
cations of an atmosphere. Were there any, it could not
fail to be perceived in the occultations of stars and the
phenomena of solar eclipses. Hence its climate must

* See Breislak's map of the environs of Napl«8, Imd Desmareflt's of
Auvergne.

t Front my own observations. — Author,

GHAP. YI.] PHYSICAL CONDITION OF THE MOON. 219

be very extraordinary; the alternation being that of un-
mitigated and burning sunshine, fiercer than an equatorial
noon, continued for a whole fortnight, and the keenest
severity of frost, far exceeding that of our polar winters,
for an equal time. Such a disposition of things must
produce a constant transfer of whatever moisture may
exist on its surface, from the point beneath the sun to
that opposite, by distillation in vacuo after the manner
of the little instrument called a cryophoros. The con-
sequence must be absolute aridity below the vertical sun,
constant accretion of hoar frost in the opposite region,
and, perhaps, a nartow zone of running water at the
borders of the enlightened hemisphere. It is possible,
then, that evaporation on the one hand, and condensation
on the other, may to a certain extent preserve an equili-
brium of temperature, and mitigate the extreme severity
of both climates.

(365.) A circle of one sedbnd in diameter, as seen
from the earth, on the surface of the moon, contains
about a square mile. Telescopes, therefore, must yetbe
greatly improved, before we could expect to see signs of
inhabitants, as manifested by edifices or by changes on
the surface of the soil. It should, however, be observed,
that, owing to the small density of the materials of th^
• moon, and the comparatively feeble gravitation of bodies
on her surface, muscular force would there go six times
as far in overcoming the weight of materials as on the
earth. Owing to the want of air, however, it seems im-
possible that any form of life analogous to those on earth
can subsist there. No appearance indicating vegetation,
or the slightest variation of surface which can fairly be
ascribed to- change of season, oan any where be discerned.
(366.) The lunar summer and winter arise, in fact,
from the rotation of the moon on its own axis, the peried
of which rotation is exactly equal to its sidereal revolu-
tion about the earth, and is performed in a plane 1° 30'
11" inclined to the ecliptic, and therefore nearly coinci-
dent with her own orbit. This is the cause why we al-
ways see the same face of the moon, and have no know-
ledge of the other side. This remarkable coincidence
of two periods, which at first sight would seem perfectly

220 A TREAllSE ON ASTRONOMY. [CHAP. VI.

distinet, is said to be a consequence of the general laws
to be explained hereafter.

(367.) The moon's rotation on her ^xis is uniform;
but since her motion in her orbit (like that of the sun) is
not so, we are enabled to look a few degrees round the
equatorial parts of her visible border, on the eastern or
western side, according to circumstances ; or, in other
words, the line joining the centres of the earth and moon
fluctuates a little in its position, from its mean or average
intersection with her surface, to the east or westward.
And, moreover, since the axis about which she revolves
is not exactly perpendicular to her orbit, her poles come
alternately into view for a small space at the edges of her
disc. These phenomena are known by the name of /t-
bratians. In consequence of these two distinct kinds of
libration, the same identical point of the moon's surface
is not always the centre of her disc, and we therefore get
sight of a zone of a few degrees in breadth on all sides
of the border, beyond an exact hemisphere.

(308.) If there be inhabitants in the moon, the earth
must present to them the extraordinary appearance of a
moon of nearly 2° in diameter, exhibiting the same phases
as we 'see the moon to do, but immoveMy fixed in their
sky (or, at least, changing its apparent place only by the
small amount of the libration), while the stars must seem .
to pass slowly beside and behind it. It will appear
clouded with variable spots, and belted with equatorial
and tropical zones corresponding to our trade-winds ; and
it may be doubted whether, in their perpetual change, the
ou^nes of our continents and seas can ever be clearly
discerned.

CHAP. vn.J ON TERRESTRIAL GRAVITY. 221

CHAPTER VII.

Of terrestrial Gravity— Of the Law of universal Gravitation— Paths of
Projectiles ; am)arent, real— The Moon retained in her Orbit by Gravi^
— ^Its Lew of Diminution — ^Laws of elliptic Motion— Orbit of the £artn
round the Sun in accordance with these Laws— Masses of the Earth
' and Sun compared— Density of the Sun-~Force of Gravity at its Sur-
face — ^Disturbing Effect of the Sun on the Moon's Motiqn*

(369.) The reader has now been made acquainted with
the chief phenomena of the motions of the earth in its
orbit round the sun, and of the moon about the earth.
We come next to speak of the physical cause which
maintains and perpetuates thes^jnotions, and causes the
massive bodies so revolving toV^viate continually from
the directions they would naturally seek to follow, in
pursuance of the first law of motion,* and bend their
courses into curves concave to their centres. ■ ^ ,, — ,

(370-) Whatever attempts may have been made by
metaphysical writers to reason away the connexion of
cause and effect, and fritter it down into the unsatisfacto-
ry relation of habitual sequence yf it is certain that the
conception of some more real and intimate connexion is
quite as strongly impressed upon the human mind as that
of the existence of an external world, — the vindication
of whose reality has (strange to say) been regarded aa
an achievement of no common merit in the annals of this
branch of philosophy. It is our own immediate «on-
sciousness of effort^ when we exert force to put matter
in motion, or to oppose and neutralize force, which gives
us this internal conviction of power and causation so far
B£ it refers^ to the material world, and compels us to be-
lieve that whenever we see material objects put in motion

♦ See Cab. Cyc. Mechanics, chap. iii.

t See Brown " On Cause and Efiect,"— a work of great acuteness and
aabtlety <j£ reasoning on some points, but in which the whole train of ar-
^ment is vitiated Inr one enormous evenight; the omission, namely, of
a distinct and immeaiate personal consciousness of causation in his enu-
meration of that sequence of events^ hy which the volition of the rniiid is
made to terminate in the motion of material objects. I mean the con*
eciousness oi effort, as a thing entirely distinct from mere de«ire or volition.
on the one hand, and from mere spasmodic contraction of musclen on the
Other. Brown, 3d edit Edin. 1818, p. M.-^Author,
T 2

1

BS2 ▲ TREATISE OK ASTRONOMY. [cHAP. Til,

from a state of rest, or deflected from their rectilinear
paths, and changed in their velocities if already in motion
it is in consequence of such an i^tfokt somehow exerted,^
though not accompanied with our consciousness. That |
such an effort should be exerted with success through an
interposed space, is no more difficult to conceive than
that our hand should communicate motion to a stone,
with which it is demonstrably not in contact.

(371.) All bodies with which we are acquainted, when
raised into the air abd quietly abandoned, descend to the
earth's surface in lines perpendicular to it. They are t
therefore urged thereto by a- force or effort, the direct or 1
indirect result of a consciousness and a wiU existing ^
somewhere, though beyond our power to trace, which ^
force we term gravity And whose tendency or direction,
as universal experience teaches, is towards the earth's
centre ; or rather, to speak stricdy, with reference to its
spheroi dal figure, perpendicular t o the surface of still
water. / but if we cast a body obliquely into the air,
^ this tendency, though not extinguished or diminished, is
materially modified in its ultimate effect. The upward
impetus we give the stone is, it is true, after a time de-
stroyed, and a downward one communicated to it, which
ultimately brings it to the surface, where it is opposed in
its further progress, and brought to rest. But all the
while it has been continually deflected or bent aside from
its rectilinear progress, and made to describe a curved
line concave to the earth's centre ; and having a highest
point, vertex, or apogee, just as the moon has in its orbit,
where the direction of its motion is perpendicular to the
radius.

(372.) When the stone wWch we fling obliquely up»
wards meets and is stopped in its descent by the earth's
surface, its motion is not towards the centre, but inclined
to the earth's radius at the same angle aa when it quitted
our hand. As we are sure that, if not stopped by the
resistance of the earth, it would continue to descend, and
that obliquely, what presumption, we may ask, is there
that it would ever reach the centre, to which its motion,
in no part of its visible course, was ever directed ? What
reason have we to believe that it might not rather circw-

CHAP. VII.J MOTION OF A PROJECTILE. 323

I^te round it, as the moon does round, the earth, returning
again to the point it set out from, after completing an
elliptic orhit of which the centre occupies the lower
focus ? And if so, is it not reasonable to imagine that the
same force of gravity may (since we know that it is ex-
cited at all accessible heights above the surface, and even
in the highest regions of the atmosphere) extend as far
as 60 radii of the earth, or to the moon ? and may not
tills be the power — ^for some power there mtist be—
which deflects her at every instant from the tangent of
her orbit, and keeps her in the elliptic path which expe-
rience teaches us she actually pursues ?

(373.) If a stone be whirled round at the end of a
string, it will stretch the string by a centrifugal force,*
which, if the speed of rotation be sufficiently increased,
will at length break the string, and let the stone escape*
However strong the string, it may, by a sufficient rotatory
velocity of the stone, be brought to the utmost tension it
will bear without breaking ; and if we know what weight
it is capable of carrying, the velocity necessary for iJiis
purpose is easily calculated. Suppose, now, a string to
connect the earth's centre, with a weight at its surface,
whose strength should be just sufficient to sustain that
weight suspended from it. Let us, however, for a mo-
ment imagine gravity to have no existence, and that the
weight is made to revolve with the limiting vthcUy
which that string can barely counteract : then will its
tension be just equal to the weight of the revolving body ;
and any power which should continually urge the body
towards the centre with a force equal to its weight would
perform the office, and ftiight supply the place of the
Btring, if divided. Divide it, then, and in its place let
gravity act, and the body will circulate as before ; its ten-
dency to the centre, or its weigMj being just balanced by
its centrifugal force. Knowing the radius of the earth,
we can calculate the periodical time in which a body so
balanced must circulate to keep it up ; and this appears
to be l*^ 23- 22*.

(374.) If we make the same calculation for a body at
the distance of the moon, supposing its weight or gra*
* See Cab. Cyc Mbchanicb, chap. viiL

224 A TREATISE ON ASTRONOMY. [cHAP. VH*

vity the same aa at the earth* s surface, we shall find the
period required to be 10** 45™ 30*. The actual period of
the moon's revolution, however, is 27* 7** 43" ; and hence
it is clear that the moon's velocity is not nearly sufficient
to sustain it against such a power, supposing it to revolve
in a circle, or neglecting (for the present) the slight ellip-
ticity of its orbit. In order that a body at the distance
of the moon (or the moon itself) should be capable of
keeping its distance from the earth by the outward effort
of its centrifugal force, while yet its time of revolution
should be what the moon's actually is, it will appear (on
executing the calculation from the principles laid down
in Cab. Cyc. Mechanics) that gravity, instead of being
as intense as at the surface, would require to be very
nearly 3600 times less energetic; or, in other words,
that its intensity is so enfeebled by the remoteness
of the body on which it acts, as to be capable of
producing in it, in the same time, only -37^7^^ P^^ ^^
the motion which it would impart to the same mass
of matter at the earth's surface.

(375.) The distance of the moon from the earth's
centre is somewhat less than sixty times the distance
from the centre to the surface, and 3600 : 1 : : 60^ : 1' ;
so that the proportion in which we must admit the earth's
gravity to be enfeebled at the moon's distance, if it be
really the force which retains the moon in her orbit, must
be (at least in this particular instance) that of the squares
of the distances at which it is compared. Now, in such
a diminution of energy with increase of distance, there
is nothing prima facie inadmissible. Emanations from
a centre, such as light and heat, do really diminish in in-
tensity by increase of distance, and in this identical pro-
portion; and though we cannot certainly argue much
from this analogy, yet we do see that the power of mag-
netic and electric attractions and repulsions is actually
enfeebled by distance, and much more rapidly than in
the simple proportion of the increased distances. The
argument, therefore, stands thus: — On the one hand,
gravity is a real power, of whose agency we have daily
experience. We know that it extends to the greatest ac^
cessible heights, and far beyond ; and we see no reason

CHAP. Vn.] ATTRACTION OF 8FHEKE6. 235

for drawing a line at any particular height, and there as-
serting that it must cease entirely ; though we have ana-
logies to lead us to suppose its energy may diminish
rapidly as we ascend to great heights from the surface,
such as that of the moon. On the other hand, we are
sure the moon is urged towards the earth by some power
which retains her in her orbit, and that the intensity of
this power is such as would correspond to a diminished
^avity, in the proportion— otherwise not improbable —
of the squares of the distances. If gravity be not that
power, diere must exist some other ; and, besides this,
gravity must cease at some inferior level, or the nature
of the moon must be different from that of ponderable
matter ; — for if not, it would be urged by both powers,
and therefore too much urged, and forced inwards from
her path.

(376.) It is on such an argument that Newton is un-
derstdod to have rested,- in the first instance, and provi-
sionally, his law of universal gravitation, which may be
thus abstractly stated: — ** Every particle of matter in
the universe attracts every other particle, with a force
directiy proportioned to the mass of the attracting par-
ticle, and inversely .to the square of the distance between
them." In^his abstract and general form, however, the
proposition is not applicable to the case before us. The
earth and moon are not mere particles, but great spherical
bodies, and to such the general law does not immediately
apply ; and, before we can make it applicable, it becomes
necessary to inquire what will be the force with which a
congeries of particles, constituting a solid mass of any as-
signed figure, will attract another suc]i collection of mate-
rial atoms. This problem is one purely dynamical, and, in
its general form, is of extreme difficulty. Fortunately,
however, for human knowledge, when the attracting and
attracted bodies are spheres, it admits of an easy and di-
rect solution. Newton himself has shown {Princip.
b. i. prop. 75) that, in that case, the attraction is pre-
cisely the same as if the whole matter of each sphere
were collected into its centre, and the spheres were
single particles there placed ; so that, in this case, the
general law applies in its strict wording. * The effect of

286 A TREATISE ON ASTRONOMY. [cHAP. VH,

the trifling; deviation of the earth from a spherical form
is of too minute an order to* need attention at present.
It is, however, perceptible, and may be hereafter noticed.
(377.) The next step in the Newtonian argument is
one which divests the law of gravitation of its provisional
character, as derived from a loose and superficial consi-
deration of the lunar orbit as a circle described with an
average or mean velocity, and elevates it to the rank of
a general and primordial relation, by proving its applica-
bility to the state of existing nature in all its detail of
circumstances. This step consists in demonstrating, as
. he has done* (Princip, i. 17, i. 75), that, under the in-
fluence of such an attractive force mutually urging two
spherical gravitating bodies towards each other, they
will each, when moving in each other's neighbourhood,
be deflected into an orbit concave towards the other, and
describe, one about the other regarded as fixed, or both
round their common centre of gravity, curves whose
forms are limited to those figures known in geometry by
ihe general namie of conic sections. It will depend, he
■flhows, in any assigned case, upon the particular circum-
stances of velocity, distance, and direction, which of
these curves shall be described, — wljether an ellipse, a
circle, a parabola, or an hyperbola; but one or other it
jmust be ; and any one of any degree of eccentricity it
:niay be, according to the circumstances of the case ; and,
dn ail cases, the point to which the motion is referred,
whether it be the centre of one of the spheres, or their
common centre of gravity, will of necessity be the Jhcus
of the conic section described. He shows, furthermore
(^Princip. i. 1), that in every case, the angular velocity
with which the line joining their centres moves, must be
inversely proportional to the square of their mutual dis-
tance, and that equal areas of the curves described will
be swept over by their line of junction in equal times.

(378.) AJl this is in conformity with what we have
stated of the solar and lunar movements. Their orbits

* We refer ibr these fundamental propositions, as a point of duty, to
the immortal work in which they were fiipt propoimded. It is impdssi-
hie for us in this volume to go mto these illveitigations : even did our
limits permit, it would be utterly inconsistent wiA our plan ; a general
idea, however, of their conduct will be given in the next chapter.

CHAP. VII.] MASS OF THE SUN. 227

are ellipses, but of different degrees of eccentricity ; and
this circumstance already indicates the general applica-
bility of the principles in question.

(379.) But here we have already,, by a natural and
ready implication (such is always the progress of gene-
ralization)j taken a further and most important step, al-
most unperceived. We have extended the action of
gravity to the case of the earth and sun, to a distance
immensely greater than that of the moon, and to a body
apparently quite of a different nature from either. Are
we justified in this ? or, at all events, are there no modi-
fications introduced by the change of data, if not into
the general expression, at least into the particular inter-
pretation, of the law of gravitation ? Now, the moment
we come to numbers, an obvious incongruity strikes us.
When we calculate, as above, from the known distance
of the sun (art. 304), and from the period in which the
earth circulates about it (art. 327), what must be the cen-
trifugal force of the latter by which the sun's attraction
is balanced (and which, therefore, becomes an exact
measure of the sun's attractive energy as exerted on the
earth), we find it to be immensely greater than would
suffice to counteract the earth's attraction on an equal
body at that distance — greater in the high proportion of
354936 to 1. It is clear, then, that if 5ie earth be re-
tained in its orbit about the sun by solar attraction^ con-
formable in its rate of diminution with the general law,
this force must be no less than 354036 times more in-
tense than what the earth would be capable of exerting,
ceteris paribus, at an equal distance.

(380.) What, then, are we to understand from this
result ? Simply this, — that the sun attracts as a collec-
tion of 354036 earths occupying its place would do, or,
in other words, that the sun contains 354036 times the
mass or quantity of ponderable matter that the earth con-
sists of. Nor let this conclusion startle us. We have
only to recall what has been already shown in art. 305,
of the gigantic dimensions of this magnificent body, to
perceive that, in assigning to it so vast a mass, we are
not outstepping a reasonable proportion. In fact, when
we come to compare its mass with its hulk, we find its

328 A TREATISE ON ASTRONOMY. [CHAP. VZI«

density* to be less than that of the earth, being no more
than 0*2543. So that it must consist, in reality, of far
lighter materials, especially when we consider the force
under which its central parts must be condensed. This
consideration renders it highly probable that an intense
heat prevail? in its interior, by which its elasticity is re-
inforced, and rendered capable of resisting this almost
inconceivable pressure without collapsing into smaller
dimensions.

(381.) This will be more distinctly appreciated, if we
estimate, as we are now prepared to do, tiie intensity of
gravity at the sun's surface.

The attraction of a sphere being the same (art. 376)
as if its whole mass were collected in its centre, will, of
course, be proportional to the mass directly, and the
square of the distance inversely ; and, in this case, the
distance is the radius of the sphere. Hence we con-
clude, t that the intensities of solar and terrestrial gravity
at the surfaces of the two globes are in the proportions
of 27*9 to 1. A pound of terrestrial matter at the sun's
surface, then, would exert a pressure equal to what 27*9
such pounds would do at the earth's. An ordinary man,
for example, would not only be unable to sustain his own
weight on the sun, but would literally be crushed to
atoms under the load.|

(382.) Henceforward, then, we must consent to dis-
miss all idea of the earth's immobility, and transfer that
attribute to the sun, whose ponderous mass is calculated
to exhaust the feeble attractions of such comparative
atoms as the earth and moon, without being perceptibly
dragged from its place. Their centre of gravity lies, as
we have already hinted, almost close to the centre of
the solar globe, at an interval quite imperceptible from
our distance ; and whether we regard the earth's orbit as
being performed about the one or the other centre makes
* The density of a material body is as the man directly, and the
volume inversely : hence density of Q • density of : : ^^- : 1 i
0-2643:1.

t Solar gravity : terrestrial : • 1^, : -^^ii 279 : 1 ; the respec
tive radii of die sun and earth being 440000, and 4000 mUes.'

X A mass weighing 12 stone or 170 lbs. on the earth, would produce •
pressure of 4600 lbs. on the sun.

<?IIAP. Vn.] DISTURBANCE OF THE HOON's ORBIT. 999

no appreciable difference in any one phenomenon of
astronomy.

(383.) It is in consequence of the mutual gravitation
of all the several parts of matter, which the Newtonian
law supposes, that the earth and moon, while in the act
of revolving, monthly, in their mutual orbits about their
common centre of gravity, yet continue to circulate,
without parting company, in a greater annual orbit round
tlie sun. We may conceive this motion by connecting
two unequal balls by a stick, which, at their centre of
gravity, is tied by a long string, and whirled round.
Their joint systems will circulate as one body about the
common centre to which the string is attached, while yet
they may go on circulating round each other in subor-
dinate gyrations, as if the stick were quite free from any
such tie, and merely hurled through the air. If the earth
alone, and not the moon, gravitated to the sun, it would
' be dragged away, and leave the moon behind---and vice
versa ; but, acting on both, they continue" together under
its attraction, just as the loose parts of the earth's sur-
face continue to rest upon it. It is, then, in slrictness,
not the earth or the moon which describes an ellipse •
around the sun, but their common centre of gravity. The
effect is to produce a small, but very perceptible, monthly
equation in the sun's apparent motion as seen from the
earth, which is always t^dten into account in calculating
the sun's place.

(384.) And here, i, e, in the attraction of the sun, we
have the key to all those differences from an exact
elliptic movement of the moon in her monthly orbit,
which we have already noticed (arts. 344, 360), viz,
to the retrograde revolution of her nodes ; to the direct cir-
culation of the axis of her ellipse ; and to all the other
deviations from the laws of elliptic motion at which we
have further hinted. If the moon simply revolved about
the earth under the influence of its gravity, none of these
phenomena would take place. Its orbit would be a per-
fect ellipse, returning into itself, and always lying in one
and the same plane : that it is not sOj is a proof that
some cause disturbs it, and interferes with the earth's
attraetiion ; and this cause is no other than the sun's at-
U

230. A TREATISE ON ASTRONOMY. [cHAP. VII.

traction— or rather, that part of it which is not equally
exerted on the earth.

(385.) Suppose two stones, side by side, or otherwise
situated with respect to each other, to be let fall together ;
then, as gravity accelerates them equally, they will re-
tain their relative positions, and fall together as if they
formed one mass. But suppose gravity to be rather
more intensely exerted on one than the other; then
would that one be rather more accelerated in its fall, and
would gradually leave the other; and thus a relative
motion between them would arise from the difference of
action, however slight.

(386.) The sun is about 400 times.more remote than
the moon; and, in consequence, while the moon de-
scribes her monthly orbit round the earth, her distance
from the sun is alternately ^j^th part greater and as
much less than the earth's. Small as this is, it is yet
sufficient to produce a perceptible excess of attractive .
tendency of the moon towards the sun, above that of the

N qM ^

■fc- s

earth when in the nearer point of her orbit, M, and a
corresponding defect on tJie opposite part, N ; and, in
the intermediate positions, not only will a difference of
forces subsist, but a difference of directions also ; since,
however small the lunar orbit MN, it is not a point, and,
therefore, the lines drawn from the sun S to its several
parts cannot be regarded as strictly parallel. If, as we
have already seen, the force of the sun were equally ex-
erted, and in parallel directions on both, no disturbance
of their relative situations would take place ; but from
, the non-verification of these conditions arises a disturb-'
ing force, oblique to the line joining the moon and earth,
which in some situations acts to accelerate, in others to
retard, her elliptic orbitual motion ; in some to draw the
earth from the moon, in others the moon from the earth.
Again, the lunar orbit, though very nearly, is yet not
quite coincident with the plane of the ecliptic ; and hence
the action of the sun, which is very nearly parallel to the
lajBt mentioned plane, tends to draw her somewhat out

CHAP. Tin.] SOLAR SYSTEV. 231

of the plane of her orbit, and does actually do so— pro-
ducing the revolution of her nodes, and other phenomena
less striking. We are not yet prepared to go into the
subject of these perturbations, as they are called ; but
they are introduced to the reader's notice as early as
possible, for the purpose of reassuring his mind, should
doubts have arisen as to the logical correctness of our
argument, in consequence of our temporary neglect of
them while working our way upward to the law of
gravity from a general consideration of the moon's orbit.

CHAPTER VIII.

OF THE SOLAR SYSTEM.

Apparent Motions of the Flaneto— Their Stations and Retrogradations —
The Sun their natural Centre of Motion— Inferior Planets-— Their
Phases, Periods, &o. — Dimensions and Form of their Orbits — ^Transits
across the Sun — Superior Planets — ^Their Distances, Periods, &c. —
Kepler's Laws and their Interpretation — Elliptic Elements of a Planet's
Orbit— Its heliocentric and geocentric Place — Bode's Law of planetary
Distances— The four ultra-zodaical Planets — ^Physical Peculiarities ob-
servable in each of the Planets.

(387.) The sun and moon are not the only celestial
objects which appear to have a motion independent of
that by which the great constellation of the heavens is daily
carried round the earth. Among the stars there are seve-
ral, — and those among the brightest and most conspi-
cuous, — which, when attentively watched from night to
night, are found to change their relative situations among
the rest; 8om*e rapidly, others much more slowly. These
are called planets. Four of them — Venus, Mars, Ju-
piter, and Saturn — are remarkably large and brilliant ;
another. Mercury, is also visible to the naked eye as a
large star, but, for a reason which will presently appear,
is seldom conspicuous ; a fifth, Uranus, is barely dis-
cernible without a telescope ; and four others — Ceres,
Pallas, Vesta, and Juno — are never visible to the naked
eye. Besides these ten, others yet undiscovered may
exist ; and it is extremely probable that such is the case,
--*ihe multitude of telescopic stars being so great that

333 A TREATI8B ON ASTRONOMY. [cHAP. yiU*

only a. small fraction of their number has been siifficiently
notioed to ascertain whether they retain the same places
or not, and the five last-mentioned planets having all been
discovered within half a century from the present time.

(388.) The apparent motions of the planets are much
more irregular than those of the sun or moon. Generally
speaking, and comparing their places at distant times,
they all advance, though with very different etverage or
taean velocities, in the same direction as those lumina-
ries, u 6. in opposition to the apparent diurnal motion, or
from west to east : all of them make the entire tour of
the heavens, though under very different circumstances ;
and all of them, with the exception of the four telescopic
planets, — Ceres, Pallas, Juno, and Vesta (which may
therefore be termed ultra-zodiacal), — are confined in
their visible paths within very narrow limits on either
side the ecliptic, and perform their movements within
that zone of the heavens we have called above the Zo*
diac (art. 254).

(389.) The obvious conclusion from this is, that
whatever be, otherwise, the nature and law of their mo-
tions, they are all performed nearly in the plane of the
ecliptic, — that plane, namely, in which our own motion
about Uie sun is performed. Hence it follows, that we
see their evolutions, not in plan, but in section ; their
real angular movements and linear distances being all
fofethortened and confounded uhdistinguishably, while-
only their deviations from the ecliptic appear of their
natttM magnitude, undiminished by the effect of per-
spective.

(390.) The apparent motions of the suit and moon»
though not uniform, do not deviate very greatly from
uniformity ; a moderate acceleration and retardation,
accountable for by the ellipticity of their orbits, being all
that is remarked. But the case is widely different with
the planets : sometimes they advance rapidly ; then re-
lax in their apparent speed—come to a momentary stop ;
and then actually reverse their motion, and run back upon
their former course, with a rapidity at first increasing,
then diminishing, tUl the reversed or retrograde motion
ceases altogether. Another station j or moment of ap*

C«AF. VIII.3 NODES OF A PLANET's OBBIT. 233

parent rest or indecision, now takes place ; after which
the movement is again reversed, and redumes its original
direct character. On the whole, however, the amount
of direct motion more than compensates the retrograde ;
and by the excess of the former over the latter, the gra-
dual advance of the planet from west to east is main-
tained. Thus, supposing the zodiac to be unfolded into
a plane surface (or represented as in Mercator's projec-
tion, art. 234, taking tJhie ecliptic EC for its ground line),
the track of a planet, when mapped down by observation

from day to day, will offer the appearance PQRS, &c. ;
the motion from P to Q being direct, at Q stationary,
from Q to R retrograde, at R again stationary, from R
to S direct, and so on. .

(391.) In the midst of the irregularity and fluctuation
of this motion, one remarkable feature of uniformity is
observed. Whenever tJie planet crosses the ecliptic, as
at N in the figure, it is said (like the moon) to be in its
node ; and as the earth necessarily lies in the plane of
the ecliptic, the planet cannot be apparently or urano-
graphically situated in the celestial circle so called, with-
out being really and locaUy situated in that plane. The
visible passage of a planet through its node, then, is a
phenomenon indicative of a circumstance in its real mo-
lion quite independent of the station from which we view
it. Now, it is easy to ascertain, by observation, when a
planet passes from the north to the south side of the
ecliptic : we have only to convert its right ascensions
and declinations into longitudes and latitudes, and the
change from north to south latitude on two successive
days will advertise us on what day the transition took
place ; while a simple proportion, grounded on the ob-
served state of its motion in latitude in the interval,
will suffice to fix the precise hour and minute of its ar-
rival on the ecliptic. Now, -this being done for several
transitions from side to side of the ecliptic, and their
u2

^4 A TREATISE ON ASTRONOMY. [CHAP. Till.

dates thereby fixed, we find, universally, that the interval
of time elapsing between the successive passages of each
planet through the same node (whether it be the ascend-
ing or the descending) is always alike, whether the planet
at the moment of such passage be direct or retrograde,
swift or slow, in its apparent movement.

(392.) Here, then, we have a circumstance which,
while it shows that the motions of the planets are in fact
subject to certain laws and fixed periods, may lead us
very naturally to suspect that the apparent irregularities
and complexities of their movements may be owing to
our not seeing them from their natural centre (art. 316),
and from our mixing up with their own proper motions
movements of a parallactic kind, due to our own change
of place, in virtue of the orbitual motion of the earth
about the sun.

(393.) If we abandon the earth as a centie of the pla-
netary motions, it cannot admit of a moment's hesitation
where we should place that centre with the greatest pro-
bability of truth. It must surely be the sun which is
entitled to the first trial, as a station to which to refer
them. If it be not connected wkh them, by any physical
relation^ it at least possesses the advantage, which the
«arth does not, of comparative immobility. But after
what has been shown in art. 380, of the immense mass
of tliat luminary, and of the office it performs to us as a
quiescent centre of our orbitual motion, nothing can be
more natural than to suppose it may perform die same
to other globes which, like the earth, may be revolving
round it ; and these globes may be visible to us by its
light reflected from them, as the moon is. Now there
are many facts which give a strong support to the idea
ihat the planets are in this predicament.

(394.) In the first place, the planets really are great
globes, of a size commensurate, with the earth, and seve-
ral of them much greater. When examined through
powerful telescopes, they are seen to be round bodies, of
sensible and even c^ considerable apparent diameter, and
ofibring distinct and characteristic peculiarities, which
show them to be solid masses, each possessing its indi-
vidual structure and mechanism ; and that, in one in-

CHAP. Vin.3 APPARENT DIAMETEIUI OF THE PLANETS. 236

stance at least, an exceedingly artificial ai^d complex one.
(See the representations of Jupiter* Saturn, and Mars,
in plate I.) That their distances from us are great,
much greater than that of the moon, and some of them
•even greater than that of the sun, we infer from the
■«mallness of their diurnal parallax, which, even for the
nearest of them, when most favourably situated, does
not exceed a few seconds, and for the more remote ones
is almost imperceptible. From the comparison of the
diurnal parallax of a celestial body, with its apparent
4iemidiametet, we can at once estimate its real size. For
the parallax is, in fact, nothing else than the apparent se-
midiameter of the earth as seen from the^body in ques-
tion (art. 298, et seq.); and, the intervening distance
being the same, the real diameters must be to each other
in the proportion of the apparent ones. Without going
into particulars, it will sudice to state it as a general re-
sult of that comparison, that the planets are all of them
incomparably smaller than the sun, but some of them as
large as the earth, and others much greater.

(395.) The next faot respecting them is, that their
distances from us, as estimated from the measurement
of their angular diameters, are in a continual state of
change, periodically increasing and decreasing within
certain limits, but by no means corresponding with the
supposition of regular circular or elliptic orbits described
by them about the earth as a centre or focus, but main-
taining a constant and obvious relation to their apparent
angular distances, or elongations from the sun. For ex-
ample ; the apparent diameter of Mars is greater when
in opposition (as it is called) to the sun, i. f . when in
the opposite part of the ecliptic, or when it comes on
the meridian at midnight,-*«'being then about 18",— but
diminishes rapidly from the amount to about 4", which
is its apparent diameter when in conjunction, or when
seen in nearly the same direction as that luminary. This,
and facts of a similar character, observed with respect to
the apparent diameters of the other planets, clearly point
out the sun as having more than an accidental relation
to their movements. .

(396.) Lastly, certain of the planets, when viewed

236 A TREATISE ON ASTRONOMY, [cHAP. VlII.

through telescopes, exhibit the appearance of phases
like those of the moon. This proves that they are
opaque bodies, shining only by reflected light, which
can be no other than the sun's ; not only because there
is no other source of light external to them sufficiently
powerful, but because the appearance and succession of
the phases themselves are (like their visible diameters)
intimately connected with their elongations from the sun,
as will presently be shown.

(397.) Accordingly, it is found, that, when we refer
the planetary movements to the sun •as a centre, all that
apparent irregularity which they offer when viewed from
the earth disappears at once, and resolves itself into one
simple and general law, of which the earth's motion, as
explained in a former chapter, is only a particular case.
In order to show how this happens, let us take the case
of a single planet, which we will suppose to revoke
round the sun, in a plane nearly, but not quite, coinci-
dent with the ecliptic, but passing through the sun, and
of course intersecting the ecliptic in a fixed line, which
is the line of the planet's nodes. This line must of
course divide its orbit into two segments ; and it is evi-
dent that, so long as the circumstances of the planet's
motion remain otherwise unchanged, the times of de-
scribing these segments must remain the same. The
interval, then, between the planet's quitting either node,
and returning to the same node again, must be that in
which it describes one complete revolution round the
sun, on its periodic time; and thus we are furnished
with a direct method of ascertaining the periodic time
of each planet. .

(398.) We have said (art. 388) that the planets make
the entire tour of the heavens under very different cir-
cumstances. This must be explained. Two of them —
Mercury and Venus — ^perform this circuit evidently as
attendants upon the sun, from whose vicinity they never
depart beyond a certain limit. They are seen sometimes
to the east, sometimes to the west of it. In the former
case they appear conspicuous over the western horizon,
just after sunset, and are called evening stars : Venus,
especially, appears occasionally in this situation with a

CH>U>. TUI.]] MOTIONS OF THB INFERIOR PLANETS. 237

darzling lustre ; and in favourable circumstances *may
be observed to cast a pretty strong shadow.* When
they happen to be to the vi'est of the sun, they rise be-
fore^ that luminary in the morning, and appear over the .
eastern horizon as morning stars : they do not, how-
ever, attain the same elongation from the sun. Mer-
<niry never attains a greater angular distance from it
than about 29°, while Venus extends her excursions on
either side to about 47°. When they have receded from
thu sun, eastward, to their respective distances, they
recnain for a time, as it were, immoveable with respect to
ity and are carried along withr it in the ecliptic with a
motion equal to its own; but presently they begin to
approach it, or, which comes to the same, their motion
in longitude diminishes, and the sun gains upon them.
As this approach goes on, their continuance above the
h(wizon after sunset becomes daily shorter, till at length
they set before the darkness has become sufficient to
allow of their being seen. For a time, then, they are
not seen at all, unless on very rare occasions, when they
are to be observed passing across the sun's disc as
smalli round, well-defined black spots, totally different
in appearance from the solar spots (art. 330). These
phenomena are emphatically called transits of the re-
spective planets across the sun, and take plax^e when
the earth happens to be passing the line of their nodes
wliile they are in that part of their orbits, just as in the.
account we have given (art. 355) of a solar eclipse.
After having thus continued invisibly for a time, however,
they begin to appear on the other aide of the sun, at first
showing themselves only for a few minutes before sun-
rise, and gradually longer and longer as they recede from
him. At this time their motion in longitude is rapidly
retrograde. Before they attain their greatest elongation,
liowever, they become stationary in the heavens ; but
. their recess from the sun is still maintained by the ad-
vance of that luminary along the ecliptic, which continues
to leave them behind, until, having reversed their motion,
• It must be thrown upon a white ground. An open window in a
whitewaahed room is the best exposure. In this situation, I have ob-
«enr(3d not only the shadow, but the diffiacted iringes edging its outlinev—
Author.

288 A TREATISE ON ASTRONOMY. [cHAP. VIII.

and become again direct, they acquire sufficient speed to
commence overtaking him— at which moment they have
their greatest western elongation ; and thus is a kind of
oscillatory movement kept up, while the general advance
along the ecliptic goes on.

(399.) Suppose PQ to be the ecliptic, and ABD tic
orbit of one of these planets (for instance. Mercury) ^
seen almost edgewise by an eye situated very nearly in
its plane ; S, the sun, its centre; and A, 6, D, S suc-
cessive positions of the planet, of which B and S are in
the nodes. If, then, the sun S stood apparently still in
the ecliptic, the planets would simply appear to oscillate
backwards and forwards from A to D, alternately passing
before and behind the sun ; iatnd, if the eye happened to
• lie exactly in the plane of the orbit, transiting his dis&
in the former case, and being covered by it in the lattei.
But as the sun is not so stationary, but apparently cat*
ried along the ecliptic PQ, let it be supposed to movd
over the spaces ST, TU, UV, while the planet in eack
case executes one quarter of its period. Then will it»
orbit be apparently carried along with the sun, into the
successive positions represented in the figure ; ani
while its read motion round the sun brings it into the irc5-
spective points B, D, S, A, its apparent movement in the
heavens will seem to have been along the wavy or zig-.
zag line ANHK. In this, its motion in longitude wii
have been direct in the parts AN, NH, and retrograde ia
the parts HnK ; while at the turns of the zigzag, at H^
K, it will have been stationary.

(400.) The only two planets — Mercury and Venus—
whose evolutions are such as above described, are called
inferior planets ; their points of farthest recess from the
sun are called (as above) their greatest eastern and west-
ern elongations s and their points of nearest approach to.
it, their inferior and superior conjunctions ; the former

I

csulP. yui.] elongations of inferior planets. 939

utrhen the planet passes between the earth and the suoy
•the latter when behind the sun.

(401.) In art. 398 we have traced the apparent path
of an inferior planet, by considering its orbit in section »
or as viewed from a point in the plane of the ecliptic.
Let us now contemplate it in plan, or as viewed from a
station above that plane, and projected on it. Suppose,
then, S to represent the sun, abed the orbit of Mer-
iury, and ABCD a part of that of the earth — the direc-
tion of the circulation being the
same in both^ viz. that of the
arrow. When the planet stands
at a, let the earth be situated at
^B A, in the direction of a tangent,
oA, to its orbit ; then it is evi-
dent that it will appear at its
greatest elongation from the
sun; the angle aAB, which
measures their apparent interval
as seen from A, being then great-
er than in any other situation of a upon its own circle.

(402.) Now, this angle being known by observation,
we are hereby furnished with a ready means of ascer-
taining, at least approximately, the distance of the planet
from the sun, or ^e radius of its orbit, supposed a cir-
cle. For the triangle SAa is right-angled at a, and con-
sequently we have Sa : SA : : sin. SAa : radius, by which
proportion the radii So, SA of the two orbits are directly
compared. If the orbits were both exact circles, this
would of course be a perfectly rigorous mode of pro-
ceeding : but (as is proved by the inequality of the re-
sulting values of Sa obtained at different times) this is
not the case ; and it becomes necessary to admit an ec-
centricity of position, and a deviation from the exact cir-
cular form in both orbits, to account for this difference.
Neglecting, however, at present this inequality, a mean
or average value of Sa may, at least, be obtained from
the frequent repetition of this process in all varieties of
situation of the two bodies. The calculations being per-
formed, it is concluded that the mean distance of Mer-
cury from the sun is about 36000000 miles ; and that of

240 A TREATISE ON ASTRONOMY. [cHAP. n\U

Venus, similarly derived, about 68000000 : the radiw
of the earth's orbit being 95000000.

(403.) The sidereal periods oif the 'planets maybeot)-
tained (as before observed), with a considerable approach
to accuracy, by observing their passages through the
nodes of their orbits ; and, indeed, when a. certain very
minute motion of these nodes (similar to that of th^
moon's nodes, but incomparably slower) is allowed for>
with a precision only limited by the imperfection of the
appropriate observations. By such observation, so cor*
rected, it appears that the sidereal period of Mercury jb
8r» 23'* 15"* 43-9'; and that of Venus, 224* W 49»
8*0*. These periods, however, are widely different from •
the intervals at which the successwe appearances of the
two planets at their eastern and western elongations from
the sun are dbserved to happen. Mercury is seen at its
greatest splendour as an evening star, at average intervals
of about 116, and Venus at intervals of about 584 days*
The difference between the sidereal and tynodical re-
volutions (art. 353) accounts for this. Referring agaiR
to the figure of art. 401, if the earth stood stiU at A,
while the planet advanced in its orbit, the lapse of a si-
dereal period, which should bring it round again to a,
would also reproduce a similar elongation from the sun.
But, meanwhile, the earth has advanced in its orbit ui
the same direction towards E, and therefore the next
greatest elongation on the same side of the sun will hap-
pen— -ndt in the position ak. of the two bodies, but in
some more advanced position, eE. The determinalion
of this posi^n depends on a calculation exactly siaffiar
to what has been explained in the article referred to ;
and we need, therefore, only here state the resulting
synodical revolutions of the two planets, which Come
out respectively 115-877*, and 583-920*.

(404.) In this interval, the planet will have described
a whole revolution plus the arc a c, and the earth only
the arc ACE of its orbit. During its lapse, the inferior
conjunction will happen when the eardi has a certain
intermediate situation, B and the planet has reached 6» a
point between the sun and earth. The greatest elongar
tion on the opposite side of tiie wan will happen wlm

CHAP. Vni.J STNODICAL KEYoujrwvk. 241

the earth has come to C, and the planet to c, where the
line of junction Cc is a tangent to the interior circle on
the opposite side from M. Lastly, the superior con-
junction will happen when the eardi arrives at D, and
the planet at d in the same line prolonged on the other
side of the sun. The intervals at which these phenome--
na happen may easily be computed from a knowledge of
the synodical periods and the radii of the orbits.

(405.) The circumferences of circles arc in Ae propor-
tion of their radii. If, then, we calculate the circumfe-
rences of the orbits of Mercury and Venus, and the earth,
arid compare them with the times in which their revblu-
tions are performed, we shall find that the actual veloci-
ties with which they move in their orbits differ greatly ;
that of Mercury being about 109400 miles per hour, of
Venus 80060, and of the earth 6808O. Froto thi's it fol-
lows, that at the inferior conjunction, or at 6, either
planet is moving in the same direction as the earth, but
with a greater velocity ; it will, therefore, leave the earth
behind it : arid the apparent motion of the planet viewed
from the earth, will be as if the planet stood still, and
the earth moved in a contrary direction from what it
really doe^. In this situation, then, the apparent motioft
of the planet must be. contrary to the apparent motion of
the sun ; and, therefore, retrograde. On the other hand,
at the superior conjunction, the real motion of the planet
being in the opposite .direction to that of the earth, the
relative motion will be the same as if the planet stood
still- and the earth advanced with their united velocities
in its own proper direction. In this situation, then, the
apparent motion will be direct. Both these results are in
accordance with observed fact.

(406.) The stationary points may be determined by
the following consideration. At a or c, the points of
greatest elongation, the motion of the plariet is directly
to or from the earth, or along their line of junction, while
that of the earth is nearly perpendicular to it. Here,
then, the apparent motion must be direct. At 6, tlie in-
ferior conjunction, we have seen that it must be retro-
grade, owing to the planet's motion (which is there, as
well as the earth's, perpendicular to the line of junction)

X

242 A TREATISE ON ASTRONOMY. [CHAP. Vllt.

suipassing the earth's. Hence, the stationary points
ought to lie, as it is found by observation they do, be-
tween a and b, or c and b, viz. in such a position that
the obliquity of the planet's motion with respect to the
line of junction shall just compensate for the excess of
its velocity, and cause an equal advance of each extre-
mity of that line, by the motion of the planet atone end»
and of the earth at the other : so that, for an instant of
time, the whole line shall move parallel to itself. The
question thus proposed is purely geometrical, and its
solution on the supposition of circular orbits is easy ;
but when we regard them as otherwise than circles
(which they really are), it becomes somewhat complex
— ^too much so to be here entered upon. It will sufficu
to state the results which experience verifies, and which
assigns the stationary points of Mercury at from 15° to
20° of elongation from the sun, according to circum-
stances ; and of Venus, at an elongation never varying
much from 29°. The former continues to retrograde
during about 22 days ; the latter about 42.

(407.) We have said that some of the planets exhibit
phases Uke the moon. This is the case with both Mer-
cury and Venus ; and is readily explained by a consi-
deration of their orbits, such as we have above supposed

them. In fact, it requires little more than mere inspec-
tion of the figure annexed, to show, that to a spectator
situated on the earth E, an inferior planet, illuminated
by the sun, and therefore bright on the side next to him,
and dark on that turned from him, will appear /«/// at the
superior conjunction A; gibbous (i. t, more than half

CHAP. VII1,J TRANSITS OF VENtJS. 243

fiill, like the moon between the first and second quarter)
between that point and the points BC of its greatest
elongation ; half-mooned at these points ; and crescent*
shaped, or hornecT, between these and the inferior con-
junction D. As it approaches this point, the crescent
ought to thin off till it vanishes altogether, rendering the
planet invisible, unless in those cases where it transits
the sun's disc, and appears on it as a black spot. All
these phenomena are exactly conformable to observation ;
and, what is not a little satisfactory, they were predicted
as necessary consequences of the Copernican theory be*
fore the invention of the telescope.*

(408.) JThe variation in bri^tnesS of Venus in differ-
ent parts of its apparent orbit is very remarkable. This
arisen from two Causes : 1st, the varying proportion of
its visible illuminated area to its whole disc ; and, 2dly,
the varying angular diameter, dr whole apparent magni-
tude of the "disc itself. As it approaches its inferior con-
junction from its greater elongation, the half-moon be-
cpmea a crescent, which thins off^ but this is more than
compensated, for some time,, by the increasing apparent
magnitude, in consequence of its diminishing distance.
Thus the total light received from it goes on increasing,
till at length it attains a maximum, which takes place
when the planet's elongation is about 40^.'

(409.) The trarjsits of Venus are of very rare occur-
rence, ts^ing place alternately at intervals of 8 and 113
years, or thereabouts.* As astronomical phenomena, they
are, however, extremely important ; since they afford the
best and most exact means we possess of ascertaining
the sun's distance, or its parallax. Without going into
the niceties of calculation of this problem, which, owing
to the gireat multitude of circumstances to be attended to,
are extremely intricate, we shall here explain its prin-
ciple, which, in the abstract, is very simple and obvious.
Let E be the earth, V Venus, and S the sun, -and CD the
portion of Venus's relative orbit which she describes
while in the act of transiting the sun's disc. Suppose
AB two spectators at opposite extremities of that dia-

* See Essay on Tint Study of Natural Philosophy, Cab. Cyclo.-
Vol. XIV. p.^9.

24* ' TREATISE ON ASTRONOMY. [CHAt.Vin.

meter of the earth which is perpendicular to the ecliptic,
and, to avoid complicating the case, let us lay out of

consideration the earth's ^rotation, aofd suppose A, -B, to
retain that situation during the. whole tiiho pf the transit.
Then, at any moment when the spectator ^i ^ sees the
centre of Venus projected at a on the sun's disc, he at B
will see it projected at 6. If then one or othef" spectator
could suddenly transport himself from 4 to B, he would
see Venus suddenly displaced on the disc from a to 6 ;
and if he had any means of noting accurately the place
' of the points on the disc, either by micron^trical mea-
sures from its edge, or by other means, he might ascer-
tain the angular measure of a 6 as seen from the earth.
Now, since AVa, B V6, are straight lines., and therefore
make equal angles on each side V, ab will be to AB as
the distance of Venus from the sun is to its distance from
the earth, or as 68 to 27, or nea,rly as 2s to 1 : a ^, therefore, .
occupies on the sun's disc a space 2^ times as great as the
earth's diameter ; and its angular measure is therefore
equal to about 2s times the earth's apparent diameter at
the distance of the sun, or (which* is the same thing) to
five tim68 the sun's horizontal parallax (art. 298). Any
«rror, therefore, which may be conmiitted in measuring
« by will entail only one Mth of that error on the hori-
. zontal parallax concluded from it.

(410.) The thing to be ascertained, therefore, is, in
fact, neither more nor less than the breadth of the zone
PQRS, pq r e, included between . the extreme apparent
paths of the centre of Venus across the sun's disc, from
its entry on one side to its quitting it on the other. The
whole business of. the observers ^t A, B, therefore, re-
solves itself into this ; — to ascertain, with all possible
care and precision, each at his own station, this path —
where it enters, where it quits, and what segment of the

CHAP. Vlll.J TRANSIT OF VENDS. 246

flun's disc it cuts off. Now, one of the most exact ways
in which (conjoined with careful micrometric measures)
this can be done, is by noting the time occupied in the
whole transit: for the rdative angular motion of Venus
being, in fact, very precisely known from the tables of her
motion, and the apparent path being very nearly a straight
line, these times give us a measure {on a very enlarged
scale) of the lengths of the chords of the segments cut
off; and the sun's diameter being known also with great
precision, their versed sines, and therefore their differ-
ence, or the breadth of the zone required, becomes
knoi^n. To obtain these times correctly, each observer
must ascertain the instants of ingress and egress of the
centre, * To do this, he must note, 1st, the instant when
the'ferst visible Itapression or notcli on the edge of the
disc at P is produced, or the first external contact $ 2dly,
when the planet is just wholly immersed, and the
broken edge of $h6 disc just closes a^in at Q, or the
first internal contaet ; and lastly, he must make the same
observations at the egress at B, S. The mean of the in-
ternal and external contacts gives, the entry and egress
of the planet*s centre.

(411.) The modifications introduced into this process
by the earth's rotation on its axis, and by other -geogra-
phical stations of the observers thereon than here sup-
posed, are similar in their principles to those which enter
inV) the calculation of a solar eclipse, or the occultaiioii of
a star hy the dioon, only more refined. Any considera-
tion of them, however, here, would lead us too far ^ but
in the view we have taken of the subject, it affords an
admirable example of the way in which minute elements
in astronomy may become magnified in their effects, and,
by being made subject to ipeasurement on a greatly en-
larged scale, or by substituting the measure of time for
space, may be aiscertained with a degree of precision
adequate to every purpose, by only watching favourable
opportunities, and taking advantage of nicely adjusted
combinations of circumstance. . So important has this
observation appeared to astronomers, that at the last
transit of Venus, in 1769, expeditions were fitted out, on
the most efiicient scale, by the British, French, Russian,
x2

946 A TKEATISE ON ASTRONOMT. [cHAF.Vm.

and oUier governments, to the remotest comers of the
globe,- for tKe express purpose of performing it. The
' QBlebratex^ eiLpedition of Captain Cook to Otaheite was
one of them. The general result of all the observations
^ade on tlii« most memorable occasion gives 8"'5776
for the sun's horizontal parallax.

("412.) The orbit of Mercury is very elliptical, the ec-
. centricily being nearly one fourth of the mean distance.
This appears from the inequality of the greatest elonga-
tions from the sun, as observed at different times,, and
iwiiich var5r between the limits 16° 12' and 28° 48','and,
from exact* measures of such elongations, it is not diffi-
cult to show that the orbit of Venus also is slightly ec-
centric, and that both these planets, in fact, describe
ellipses, having the sun in their commocL focus.'

(413.) Let us now consider the superior planets, or
those whose orbits enclose on all sides that of the earth.
That ttfey -do so is proved by several circumstances :-^
Ifit, They are not,' like the inferior |)lanets, confined to
certain limits of elongation from the sun, but appear at
all distances from it, even in the opposite quarter of the
heavens j or, as it is called, in opposition ; which could
not happen, did not the earth at such times place itself
betweim them and the sun: 2dly, They never .appear
horned^ like Venus or Mercury, nor even semilunar.
Those, on the contrary, which, from the minuteness of
Iheit parallax, we cooifilude to be the most distant from
tts, viz. Jupiter, Saturn, and Uranus, qcvj^ appear other-
wise than round ; a sufficient proof, of itself, that we see
them always in a direction not very remote from that in
which the sun's rays illuminate them ; and that, there-
fore, we occupy a station which is never very widely re-
moved from the centre of their orbits, or, in other words,
that the earth's orbit is entirely enclosed within theirs,
and of comparatively small diameter. One only of them,
Mars, exhibits any perceptible p^o^e, and in its defi-
eienf(^ from a circular outline, never surpasses a mode-
rately gibbous appearance— the enlightened portion of
the disc being never less than seven-eighths of the whole.
To uBderstand this, we need only cast our eyes on the
ani^xed jggure, in which E is the earth, at its apjparent

CHAP. VIII.] BISTANCBS OF SUPERIOR PLANETS.

247

greatest elongation from the sun S, as
«een from Mars, M. In this position,,
the angle SM£, included between the
lines SM and £M, is at its maximum;
and* therefore, in this state of things, a
spectator on theearth is enabled to see a
greater portion of the dark hemisphere
of Mars than in any other situation. The
extent of the phase, then, or greatest ob-
servable degree of gibbosity affords a
measure — a sare, al4:hough a coarse and
rode one— of the angle SME,and tl^ere-
fore of the proportion of the distance
SM, of MaAi to SE, that of the earth
from the sua, by which it appears that
the diameter of the orbit of Mars can- i£

not be less than H that of the earth's. The phases of
Jupiter^ Satuhi, and'Uranus being imperceptiblfe, it fol-
lows that their orbits must indude not only that of the
earth, but of Mars also.

(414.) AH the superior planets are retrograde in ^eir
apparent motions when in opposition^ and for som^e time
before and after; but they differ greatly. from each other,
both in the extent of their arc c^ retrogradationy in the
duration of their retrograde movement, and in its rapidity
when swiftest. It is more extensive and rapid in the
case of Mars than of Jupiter, (tf Jupiter than of Saturn*
and of that plimet than Uranus. The angular veloci^
with which a planet appears to retrograde is easily ascer-
tained by observing its apparent- place in the heavens
from day today; and from such observations, made about
the time of opposition, it is eafiy to conclude the relative
magnitudes of their orl»ts as co^ipared witli the earth's,
supposing their periodic times known. For, from these,
their mean angular veloeities are known also, being in-

versely as the times. Suppose, then, Ec to be a very
ismall portion of the earth's orbit, and Mm a correspond-

348 A TREATISE ON ASTRONOMY. [cttAP. VHI.

ing portion of that of a superior planet, described on the
day of opposition, about the sun S, on which day Uie
three bociies'lie in one straight line SEMX. Then the
angles ESc and MSm are given. Now, if c m be joined
and prolonged to meet SM continued in X, the angle eXB,
which is equal to the alternate angle Xet/, is evidently the
retrogradation of Mars on that day, and is, therefore, also
given.. Ec, therefore, and the angle EXc, being given in
the right-angled triangle EcX, the side EX is easily cal-
culated, and thus SX' becomes known. Consequently,
in the triangle SmX, we have given the side SX and the
two angles mSX andmXS, whence the other sides, Sm^
mXf are easily determined. - Now, Sm is no other than
the radius of the orbit of the superior jdanet required,
which in this calculation is supposed circular as well as
that of the earth ; a supposition not exact, but sufficiency
so to afford a satisfactory approximiation to the dimen-
sions of its orbit, and which, if the process be often re-
peated, in every variety of situation at which the oppo-
sition can occur, will ultimately aibrd an average or
mean valtie of its diameter fully to be depended upon^.

(415.) To apply this principle, however, to practice,
it is necessary to know the periodic times of the several
planets. These may be obtained directly, as has been
already stated, by observing the intervals of their pas-
sages through the ecliptic ; but owing to the very small
inclination of the orbits *of some of them to its plame,
they cross it so obliquely that the precise moment of
their arrival on it is not ascertainable, unless by very nice
observations. A better method consists in determining,
from the observations of. several successive days, the
exact moments of their arriving inappoaitionvrxih the sun,
the criterion of which is a difference of longitudes be-
tween the sun and planet of exactly 180°. The interval
between successive oppositions thus obtained is nearly
one synodical period ; and would be exactly ^o, were the
planet's orbit and that of the earth both circles, and uni-
formly described ; but as that is found not to be the case
(and the criterion is, the ineqiudity of successive synod-
ical revolutions so observed), the average of a great num-
ber, taken in all varieties of situation in which the oppo-

ctSAF. yiii.j Kepler's law of periodic times. 240

sUions occur, will be freed from the elliptic inequality, .
and may be taken as a mean sy nodical period. From
this, by the considerations employed in art. 353, and by
the process of calculation indicated in the note to that
article, the. sidereal, periods are readily obtained. The
accuracy of this determination will, of course, be greatly
increased by embracing a long interval between the ex-
treme observations employed. In point of fact, that in-
terval extends to nearly 2000 years in the cases of tfe
planets known to the ancients, who have riecorded tJieir
observations of them in a manner sufficiently careful to
be made use of. Their periods may, therefore, be r€gaid-
ed as ascertained with the utmost, exactness. Their nu-
merical values will be found stated, as well as the meaoi
distances, and all the other elements of the planetary
orbits, in the synoptic. table at the end of the volun»t^,t0
which (to avoid repetition) the reader is once for aUre^
ferred.

(41 6.) In casting our eyes down the list of the plaiivt^.-
ary distances, and comparing them with the periodle-
times, we' cannot but be struck with a certain correspo*i—
enee. The greater the distance, or the larger the orbi;.
evidently the longer the period. The order of the pla*--
xiets» beginning from the sun, is the same, whether w^
arrange them according to their distances, or to the time
they occupy in completing their revolutions ; and is as
follows :— Mercury, Venus, Earth, Mars — the four ultra-
zodiacal planets — ^Jupiter, Saturn, and Uranus. Never- •
theless, when we come to examine the numbers express-
ing them, we find that the relation between the two series
IS not that of simple proportional increase. The periods
increase more than in proportion to the distances. Thus,
the period of Mercury is about 88 days, and that of the
Earth 365"^being in proportion as 1 to 4*15, while their
distances are in the less proportion of 1 to 2*56 ; and a
similar remark holds good in every instance. Still, tlie
ratio of increase Of the times is not so rapid as that of
the squares of the distances. The square of 2*56 is
6*5536, which is considerably greater than 4* 15, An in-
termediate rate of increase, between the simple proportion
of the distances and that of their squares, is therefore

250 A TREATISE ON ASTRONOMY. [CHAP. VIII.

clearly pointed out by the sequence of the numbers; but
it required no ordinary penetration in the illustrious Kep-
ler, backed by uncommon perseverance and industry, at
a period when the data themselves were involved in ob-
scurity, and when the processes of trigonometry and of
numerical calculation were encumbered with difficulties,
of which the more recent invention of logarithmic tables
has happily left us no conception, to perceive and demon-
strate the real law of their connexion. This connexion
is expressed in the following proposition : — ** The squares
of the periodic times of any two planets are to each
other, in the same proportion as the cubes of their mean
distances from the sun." Take, for example, the earth
and Mars,* whose periods are in the proportion of
3652564 to 6869796, and whose distances from the sun
is that, of 100000 to 152369; and it will be found, by-
anyone who will take the trouble to go through the calcu-
lation, that —

(3652564)»: (6869796)*:: (lOOOOO)^: (152369)3.

(417.) Of all the laws to which induction from pure
observation has ever conducted man, this third law (as
it is called) of Kepler may justly be regarded as the most
remarkable, and die most pregnant with important conse-
quences. When we contemplate the constituents of the
planetary system from the point of view which this rela-
tion affords us, it is no longer mere analogy which strikes
us— -no longer a general resemblance among them, as
individuals independent of each other, and circulating
about the sun, each according to its own peculiar nature,
and connected, with it by its own peculiar tie. The re-
senblance is now perceived to be a true fcmiily likeness ;
they are bound up in one chain — ^interwoven in one web of
mutual relation and harmonious agreement — ^subjected to
one pervading influence, which extends from the centre
to th(B farthest limits of that great system, of which all of
them, the earth included, must henceforth be regarded as
members.

(418.) The laws of elliptic motion about the sun as a

* The expression of this law of Kepler requires a slight modificatiQa

when we come to the pxtreme nicety of numerical ealoulation, for the

greater planets, due to the influence of their iQqsses, This correction is

imperceptible ibr the earth and Mars,

CHAP. VIII.2 INTERPRETATION OF KSFLEr's LAWS. 251

focus, and of the equable description of areas by lineb
joining the sun and planets, were originally established
by Kepler, from a consideration of the observed motions
of Mars ; and were by him extended, analogically, to all
the other planets. However precarious such an extension
might then have appeared, modem astronomy has com-
pletely verified it as a matter of fact, by the general coinci-
dence of its results with entire series of observations of
the apparent places of the planets. These are found to
accord satisfactorily with the assumption of a particular
ellipse for each planet, whose magnitude, degree of eccen-
tricity, and situation in space, are numerically assigned
in the synoptic table before referred to. It is true, that
when observations are carried to a high degree of preci-
sion, and when each planet is traced Uirough many suc-
cessive revolutions, and its history carried back, by the
aid of calculations founded on these data, for many centu-
ries, we learn to regard the laws of Kepler as onlyjirst
appTOOcimations to the much more complicated ones
which actually prevail ; and that to bring remote observa-
tions into rigorous and mathematical accordance with
each other, and at the same time to retain the extremely
convenient nomenclature and relations of the elliptic
svsTEM , it becomes necessary to modify, to a certain ex-
tent, our verbal expression of the laws, and to regard the
numerical data or elliptic elements of the planetary orbits
as not absolutely permanent, but subject to a series of
extremely slow and almost imperceptible changes. These
changes may be neglected when we consider only a few
revolutions ; but going on from century to centu^, and
continually accumulating, they at length produce consider-
able departures in the orbits from their original state*
Their explanation will form the subject of a subsequent
chapter ; but for the present we must lay them out of
considieration, as of an order too minute to affect the gene-
ral conclusions with which we are now concerned. By
what means astronomers are enabled to compare the re-
sults of the elliptic theory with observation, and thus
satisfy themselves of its accordance with nature, will be
explained presently.

(419.) It will first, however, be proper to point but

2fii A TtttfAtlErs Off AflVftoNdirf . fcHAp. Ttrr*

what particular theoi'etical conclusion is involved in each
of the three laws of Itepler, Considered aM satisfactorily
established, — what indication each of them separately
affords of the mechanical forces prevalent in our system ,
and the mode iii which its parts are connected^ — and how,
when thus considered, they constitute the basis on whicjx
the Newtonian explanation of the mechanism cf( the hea-
vens is mainly supported. To begin with the first law,
that of the equable description of areas .-*— Since the pla-
nets move m curvilinear paths, they mw*^ (if they be bo-
dies obeying the laws of dynamics) be deflected from
their otherwise natural rectilinear progress by force. And
from this law, taken as a matter of observed fact, it fol-
lows, that the direction of such force, at every point of
the orbit of each planet, always passes through the sun.
No matter from what ultimate cause the power which is
called gravitation originates — ^be it a virtue lodged in
the sun as its receptacle, or be it pressure from without,
or the resultant of many pressures or solicitations of un-
kno\i^n fluids, magnetic or electric ethers, or impulses —
dtill, when finally brought under our contemplation, and
summed up into a single resultant energy, its direction
it, from every point on all sides, towards the sun's cen-
tre. As an abstract dynamical proposition, the reader
win fin^ it demonstrated by Newton, in the 1st proposi-
tion of the Principitt, with an elementary simplicity to
which we really could add nothing but obscurity by ampli-
fication, that any body, urged towards a certain central
point by a force continually directed thereto, and thereby
deflected into a curvilinear path, will describe about that
centre equal areas in equal times ; afnd vice versa, that
snch equable description of area^ is itself the essential
criterion of a continual direction of the acting force to-
wards the centre to which this character belongs. The
first law of Kepter, then, gives us no information as to the
nature or intensity of the force urging the planets to the
sun ; the only conclusion it involves, is that it does so
urge them. It is a property of orbitual rotation under
the infiuence of qentral forces generalty, and as such, we
daily see it exemplified in a thousand femiliar instances.
A sinlple experimental illustration of it is to tie a bullet

CflA^. Vlli.] INTERPRETATION OF KEPLER's LAWS. 253

to a thin string, and, having whirled it round with a mo-
derate velocity iii a vertical plane, to draw the end of the
string through a smal^ ring, or allow it to coil itself round
the finger, or a cylindrical rod held very firmly in a hori-
zontal position. The bullet will then approach the centre
of motion iii a spiral line ; and the increase not only of its
angular but of its linear velocity, and the rapid diminution
of its periodic time when near the centre, will express,
more clearly than any words, the compensation by which
its uniform description of areas is maintained under a
constantly diminishing distance. If the motion be re-
versed, and the thread allowed to uncoil, beginning with
a rapid impulse, the velocity will diminish by the same
degrees as it before increased. The increasing rapidity
of a dancer's pirouette^ as he draws in his Umbs and
straightens his whole person, so as to bring every part of
his frame as liear as possible to the axis of his motion, is
another instance where the connexion of the observed
effect with the central force exerted, though equally real,
is much less obvious.

(420.) The second law of Kepler, or that which as-
serts that the planets describe ellipses about the sun as
their focus, involves, as a consequence, the law of solar
gravitation (so be it allowed to call the force, \<rhatever it
be, which urges them towards the sun) as exerted on each
individual planet, apart from all connexion with the rest.
A straight line, dynamically speaking, is the only path
which can be pursued by a body absolutely freef and un-
der the action of no external force. All deflection into a
curve is evidence of the exertion of a force ; and the
greater the deflection in equal times, the more intense the
force. Deflection from a straight line is only another
word for curvature of path ; and as a circle is character-
ized by the uniformity of its curvature in all its parts — so
is every other curve (as an ellipse) characterized by the
particular law which regulates the increase and diminu-
tion of its curvature as we advance along its circumfe-
rence. The deflecting force, then, which continually
bends a moving body into a curve, may be ascertained,
provided its direction, in the first place, and, secondly,
the law of curvature of the curve itself, be known. Both
these enter as elements into the expression of the force. A
Y

254 A TREATISE ON ASTKONOMT. [cHAP. TUI.

body may describe, for instance, an ellipse, under a great
variety of dispositions of the acting forces : it may glide
along it, for example, as a bead upon a polished wire,
bent into an elliptic form ; in which case the acting force
is always perpendicular to the wire, and the velocity is
uniform. In this case the force is directed to no fixtd
centre, and there is no equable description of areas at all.
Or it may describe it as we may see it done, if we sus-
pend a ball by a vtry long string, and, drawing it a little
aside from the perpendicular, throw it round with a gen-
tle impulse. In this case the acting force is directed to
the centre of the ellipse, about which areas are described
equably, and to which a force proportional to the distance
(the decomposed result of terrestrial gravity) perpetually
urges it. This is at once a very easy experiment, and a very
instructive one, and we shall again refer to it. In the
case before us, of an ellipse described by the action of a
force directed to the focu8t the steps of the investigation
of the law of force are these : 1st, Tlie law of the areas
determines the actual velocity of the revolving body at
every point, or the space really run over by it in a given
minute portion of time ; 2dly, The law of curvature of the
ellipse determines the linear amount of deflection from the
tangent in the direction of the focus, which corresponds
to that space so run over ; 3dly, and lastly, The laws of
accelerated motion declare that the intensity of the acting
force causing such deflection in its ovm direction^ is mea-
sured by or proportional to the amount of thatdeflection»
and may therefore be calculated in any particular position,
or generally expressed by geometrical or algebraic sym-
bols, as a law independent of particular positions, when
that deflection is so calculated or expressed. We have
here the spirit of the process by which Newton has resolved
this interesting problem. For its geometrical detail, we
must refer to the 3d section of his Principia. We know
of no artificial mode of imitating this species of elliptic
motion ; though a rude approximation to it— enough,
however, to give a conception of the alternate approach
and recess of the revolving body to and from the focus,
and the variation of its velocity — may be had by suspend-
ing a small steel bead to a fine and very long silk fibre,
and setting it to revolve in a small orbit round the pole of

CHAP. Vtll.] INTERPRETATION OF KEPLER^S LAWS. 255

a powerful cylindrical magnet, held upright, and verti-
cally under the point of suspension.

(421.) The third law of Kepler, which connects the
distances and periods of the planets by a general rule,
bears with it, as its theoretical interpretation, this im-
portant consequence, viz. that it is one and the same
force, modified only by distance from the sun, which
retains all the planets in their orbits about it. That the
attraction of the sun (if such it be) is exerted upon all
the bodies of our system indifferently, without regard to
the peculiar materials of which they may consist, in the
exact proportion of their inertiae, or quantities of matter ;
that it is not, therefore, of the nature of the elective at-
tractions of chymistry, or of magnetic action, which is
powerless on other substances than iron and some one
or two more, but is of a niore universal character, and
extends equally to all the material constituents of our
system, and (as we shall hereafter see abundant reason to
admit) to those of other systems than our own. This
law, important and general as it is, results, as the sim-
plest of corollaries, from the relations established by
Newton in the section of the Frincipia referred to
(prop. XV.), from which proposition it results, that if
the earth were taken from its actual orbit, and launched
anew in space at the place, in the direction, and with
the velocity of any of the other planets, it would describe
the very same orbit, and in the same period, which that
planet actually does, a very minute correction of the pe-
riod only excepted, arising from the difference between
the mass of the earth and that of the planet. Small as the
planets are compared to the sun, some of them are not,
as the earth is, mere atoms in the comparison. The
strict wording of Kepler's law, as Newton has proved in
his fifly-ninth proposition, is applicable only to the case
of planets whose proportion to the central body is abso-
lutely inappreciable. When this is not the case, the
periodic time is shortened in the proportion of the
square root of the number expressing the sun's mass
or inertia, to that of^ the sum of the numbers expressing
the masses of the sun and planet ; and in general, what-
ever be the masses of two bodies revolving round each
other under the influence of the Newtonian law of gra-

2U A TflBATISE ON ASTRONOUY. [cHAP. VHI.

vity, the square of their periodic time will be expressed
by a fraction whose numerator is the cube of their mean
distance, i, e. the greater semi-axis of their elliptic orbit,
and whose denominator is the sum of their masses.
When one of the masses is incomparably greater than
the other, this resolves itself into Kepler's law; but
when this is not the ease, the proposition thus general-
ized stands in lieu of that law. In the system of the sun
and planets, however, the numerical correction thus in-
troduced into the results of Kepler's law is too small to
be of any importance, the mass of the largest of the
planets (Jupiter) being much less than a thousandth
part of that of the sun. We shall presently, however,
perceive all the importance of this generalization, when
ve come to speak of the satellites.

(423.) It will first, however, be proper to explain by
what process of calculation the expression of a planet's
elliptic orbit by its elements can be compared with ob-
servation, and how we can satisfy ourselves that the
numerical data contained in a table of such elements for
the whole system does really exhibit a true picture of
it, and affora the means of determining its state at every
iB3tant of time, by the mere application of Kepler's laws.
Now, for each planet, it is necessary for this purpose to
know, 1st, the magnitude and form of its ellipse ; 2dly,
the situation of this ellipse in space, with respect to the
ecliptic, and to a fixed line drawn therein ; 3dly, the
local situation of the planet in its ellipse at some known
epoch, and its periodic time or mean angular velocity,
or, as it is called, its mean motion.

(423.) The magnitude and form of an ellipse are de-
l^rmined by its greatest length and least breadth, or its
(two principal axes ; but for astronomical uses it is pre-
ferable to use the semi-axis major (or half the greatest
length^, and the eccentricity or distance of the focus
from the centre, which last is usuayy estimated in parts
of the former. Thus, an ellipse, whose length is 10
and breadth 8 parts of any scale, has for its major semi-
axis 5, and for its eccentricity 3 «uch parts ; but when
estimated in parts of the semi-axisj regarded as a unit,
the eccentricity is expressed by the fraction |.

(424). The ecliptie is the plane to which an inhabit-

CRAP.VIII.^ BLEMENTS OF A PLANET^S ORBIt. 257

ant of the earth most naturally refers the rest'of the solar
system, as a sort of ground-plane ; and the sD^is of its
orbit might be taken for a line of departure in that plane
or origin of Angular reckoning. Were the zxis'Jitedf
this would be the best possible origin of longitudes ; but
as it has a motion (though an excessively slow one),
there is, in fact, no advantage in reckoning from the axis
more than from the line of the equinoxes, snd astrono-
mers therefore prefer the latter, taking account of its. va-
riation by the effect of precession, and restoring; it, by
calculation at every instant, te a fi^ced position. Now,
to d^termino, the situation of the ellipse described by a
planet with respect to this plane, three elements inquire
to be knoyvn: — 1st, the inclination of the plane of the
planet^s orbit to the plane of the ecliptic ; SMly, the line
in which these two planes intersect each other, which of
necessity passes through the sun, and whose position
with respect to the line of the equinoxes is therefore
given by stating its longitude. This line is called the
line of the nodes. When the planet is in this line, in
tiie act of passing from the south to the north side of
the ecliptic, it is in its ascending node, and its longitude
at that moment is the element called the longitude of the
node. These two data determine the situation of the
plane of the orbit ; and there only remains, for the com-
plete determination of the situation of the planet's ellipse,
to know how it is placed in that plane, which (since its
focus is necessarily in the sun) i« ascertained by stating
the longitude of its perihdion, or die place which (he
extremity of the axis nearest the sun occupies, when
orthographically projected on the ecliptic,

(425.) The dimensions and situation of the planet's
orbit thus determined, it only remains, for a complete
acquaintance with its history, to determine the circum-
stances of its motion in the orbit so precisely fixed.
Now, for this purpose, all that is needed is to know the
moment of time when it is either at the perihelion, or
at any other precisely determined point of its orbit, and
its whole period ; for these being known, the law of the
areas determines the place at every ^other instant. This
moment is called (when the perihelion is the point
chosen) the perihelion passage^ or, when some point of
y2

258 A TREATISE ON ASTRONOMY. [CHAP. VIII,

the orbit is fixed upon, without special reference to the
perihelion, the epoch,

(426.) Thus, then, we have seven particulars or ele-
ments, which must be numerically stated, before we can
reduce to calculation the state of the system at any
given moment. But, these known, it is easy to ascertain
Uie apparent positions of each planet, as it would be seen
from ^ the sun, or is seen from the earth at any time.
The former is called the heliocentric, the latter the geo-
centric, place of the planet.

. (427.) To commence with the

^ra^^—;— — —^heliocentric places. Let S re-

/\ ""^\;^^=^2|ilH/ the sun; APN the orhit

rp / V _^g^^^t^ of the planet, being an ellipse,
"~~;;;:::^p^^^*^'^ having the sun S in its focus»

^ and A for its perihelion ; and let

/)aN T represent the projection of the orbit on the plane
of the ecliptic, intersecting the line of equinoxes S ^ in
T» which, therefore, is the origin of longitudes. Then
will SN be the line of nodes ; and if we suppose B to
lie on the south, and A on the north side of the ecliptic,
and the dhrection of the planet's motion to be from o to
A, N will be the ascending node, and the angle T SN the
longittcde of the node. In like manner, if P be the place
of the planet at any time, and if it and the perihelion A
be projected on the ecliptic, upon the points ;? a, the angles
T Sp, T Sa, will be the respective heliocentric longitudes
of the planet, and of the perihelion, the former of which
is to be determined, and the latter is one of the given
elements. Lastly, the angle pSF is the heliocentric lati-
tude of the planet, which is also required to be known.

(428.) Now, the time being given, and also the mo-
ment of the planet's passing the perihelion, the interval,
or the time of describing the portion AP of the orbit, is
given, and the periodical time, and the whole area of the
ellipse being known, the law of proportionality of areas
to the times of their description gives the magnitode of
the area ASP. From this it is a problem of pure geo*
metry to determine the corresponding angle ASP, which
is cafied the planet's tnte anomaly. This problem is of
the kind called transcendental, and has been resolved by •
a great variety of processes, some more, some less in*

CHAP. VIII.] HELIOCENTRIC PLACE OF A PLANET. 259

tricate. It offers, however, no peculiar difficulty, and is
practically resolved with great facility by the help of
^bles constructed for the purpose, adapted to the case of
each particular planet.*

(429.) The true anomaly thus obtained, the planet's
angular distance from the node, or the angle NSP, is to
be found. Now, tlie longitudes of the perihelion and
node being respectively V a and T N, which are given,
their difference aN is also given, and the angle N of the
spherical right-angled triangle ANa, being the inclina-
tion of the plane of the orbit to the ecliptic, is known.
Hence we calculate the arc NA, or the angle NSA,
which, added to ASP, gives the angle NSP required.
And from this, regarded as the measure of the arc NP,
forming the hypothenuse of the right-angled spherical
triangle PN;?, whose angle N, as before, is known, it
is easy to obtain the other two sides, Np and Pp. The
latter, being the measure of the angle j^SP, expresses
the planet's heliocentric latitude ; the former measures
the angle NSp, or the planet's distance in longitude
from its node, which, added to the known angle T SN,
the longitude of the node, gives the heliocentric longitude.
This process, however circuitous it may appear, when
once well understood, may be gone thitough numerically,
by the aid of the usual logarithmic and trigonometrical
tables, in little more time than it wHl have taken the
reader to peruse its description.

(430.) The geocentric differs from the heliocentric

place of a planet by reason of that parallactic change of

apparent situation which arises from the earth's motion

in its orbit. Were the planets' distance as vast as those

* It will readily be understood, that, except in the case of miifbrm cir-
cular motion, an equable description of areas about any centre is incom-
patible with an equable description of an^2e«. The object of the problem
«n the text is to pass from the area, supposed known, to the an^, sup-
posed unknown : in other words, to denve the true amount of luigular
motion from the perihelion, or the true anomaly fiom what is technically
icalled the menu anomaly, that is, the mean angular motion which would
tiave been performed had the motion in an^ 2e oeen uniform instead of
the motion in area. It happens, fertunately, that this is the simplest of
«ll problems of the transcendental kind, and can be resolved, in Ifae
most difficult case, by the rule of" false position," or tnal and ennr» in a
very few minutes. Nay* it may even be resolved inirtantly en inspec-
tion by a simple and easily constructed piece of mechanism, of D^ch the
leader may see a description in the Cambridflre Philosophical Tfoam^
iJffPM, vol. IV. p. 4S5f by the author of this work.

260 A TREATISE ON ASTRONOMY. [cHAP. Till.

of the stars, the earth's orbitaal motion would be insen-
sible when viewed from them, and they would always
appear to us hold the same relative situations among the
fixed stars, as if viewed from the sun, t. e, they would
then be seen in their heliocentric places. The differ-
ence, then, between the heliocentric and geocentric
places of a planet is, in fact, the same thing with its par
rallax arising from the earth's removal from the centre
of the system and its annual motion. It follows from
this, that the first step towards a knowledge of its
amount, and the consequent determination of the ap-
parent place of each planet, as referred from the earth to
the sphere of die fixed stars, must be to ascertain the
proportion of its linear distances from the earth and
from the sun, as compared with the earth's distance from
the sun, and the angular positions of all three with re-
spect to each other.

(431.') Suppose, Aerefore, S to represent the sun, E
the earth, and P the planet ; S V the line of equinoxes,
V E Ae earth's orbit, and Pp a perpendicular let fall
from the planet on the ecliptic. Then will the angle
SPE (according to the general notion of parallax con-
veyed in art. 69) represent the parallax of the planet
arising from the change of sta-
tion from S to E, EP will be
the apparent direction of the
i^ planet seen from E ; and if SQ
be drawn parallel to Ep, the
^ ^^ angle V SQ will be the geo-

"^ *■ ^^ centric longitude of the planet,

while Y SE represents the heliocentric longitude of the
earth, and V Sp that of the planet. The former of
these, T SE, is given by the solar tables ; the latter,
T Sp is found by the process above described (art. 429).
Moreover, SP is the radius vector of the planet's orbit,
and SE that of the earth's, both of which are determined
from the known dimensions of their respective ellipses,
and the places of the bodies in them at the assigned time.
Lastly, the angle PSp is the planet's heliocentric lati-
tude.

(432.) Our object, then, is, from all these data, to de-
termine the anirle T SQ and PEp, which is the geocen-

CHAP. VIII.] DISCOVERY OF THE PLANETS. 201

trie latitude. The process, then, will stand as follows :
1st, In the triangle SPp, right-angled at P, given SP,
and the angle PSp (the planet's ra£ns vector and helio-
centric latitude,) find Sje>, and Pp ; 2dly, In the triangle
S£p, given Sp (just found), S£ (the earth's radius
vector), and the angle ESp (the diflference of heliocen-
tric longitudes of the earth and planet), find the angle
S/)E, and theside Ep. The former being equal to 3ie
alternate angle pSQ, is the parallactic removal of the
planet in longitude, which, added to T Sp, gives its helio-
centric longitude. The latter, Ep (which is called the
curtate distance of the planet from the earth), gives at
once the geocentric latitude, by means of the right-angled
triangle PEp, of which Ep and Pp are known sides,
and the angle PEp is the longitude sought.

(433.) The calculations required for these purposes
are nothing but the most ordinary processes of plane
trigonometry ; and, though somewhat tedious, are nei-
ther intricate nor difficult. When executed, however,
they afford us the means of comparing the places of
the planets actually observed with the elliptic theory,
with the utmost exactness, and thus putting it to the se-
verest trial ; and it is upon the testimony of such compu-
tations, so brought into comparison with observed facts,
that we declare that theory to be a true representation of
nature.

^434.) The planets Mercury, Venus, Mars, Jupiter,
ana Saturn, have been known from the earliest ages in
which astronomy has been cultivated. Uranus was dis-
covered by Sir W. Herschel in 1781, March 13, in the
course of a review of the heavens, in which every star
visible in a telescope of a certain power was brought
under close examination, when the new planet was im-
mediately detected by its disc, under a high magnifying
power. It has since been ascertained tor have been ol^
served on many previous occasions, with telescopes of
insufficient power to show its disc, and even entered in
catalogues as a star; and some of the observations which
have been so recorded have been used to improve and
extend our knowledge of its orbit The discovery of the
ultra-zodiacal planets dates from the first day of 1801,
Vhen Geres was discovered by Piazzi, at Palermo ; a

262 A TREATISE ON ASTRONOMY. [cHAP. VIII.

discovery speedily followed by those of Juno by Pro-
fessor Harding, of Gottingen ; and of Pallas and Vesta,
fay Dr. Olbers, of Bremen. It is extremely remarkable
that this important addition to our system had been in
some sort surmised as a thing not unlikely, on the ground
that the intervals between the planetary orbits go on
doubling as we recede from the sun, or nearly so. Thus,
the interval between the orbits of the earth and Venus is
nearly twice that between those of Venus and Mercury ;
that between the orbits of Mars and the earth nearly
twice that between the earth and Venus ; and so on.
The interval between the orbits of Jupiter and Mars,
however, is too great, and would form an exception to
this law, which is, however, again resumed in the case
of the three remoter planets. It was, therefore, thrown
out, by the late Professor Bode of Berlin, as a possible
surmiise, that a planet might exist between Mars and
Jupiter ; and it may easily be imagined what was the as-
tonishtnent of astronomers to find four, revolving in orbits
tolerably well corresponding with the law in question.
No account, a priori^ or from theory, can be given of 4his
singular progression, which is not, like Kepler's laws,
strictly exact in its numerical verification ; but the cir-
cumstances we have just mentioned lead to a strong be-
lief that it is something beyond a mere accidental coinci-
dence, and belongs to the essential structure of the
system. It has been conjectured that the ultra-zodiacal
planets are fragments of some greater planet, which
formerly circulated in that interval, but has been blown
to atoms by an explosion ; and that more such fragments
exist, and may be hereafter discovered. This may
serve as a specimen of the dreams in which astronomers,
like other speculators, ccasionaUy and harmlessly indulge.
(435.) We shall devote the rest of this chapter ta an
account of the physical peculiarities and probable condi-
tion of the several planets, so far as the former are known
by observation, or the latter rest on probable grounds of
conjecture. In this, three features principally strike us,
as necessarily productive of extraordinary diversity in the
provisions by which, if they be, like our earth, inhabited,
animal life must be supported. There are, first, the dif«
ference in their respective supplies of light and heat from

GHAP. Vin.3 APPEARANCES OF THE PLANETS. 268

the sun ; secondly, the difference in the intensities of the
gravitating forces which must subsist at their surfaces, or
, the different ratios which, on their several globes, the
inertise of bodies must bear to their weights ; and, third-
ly, the difference in the nature of the materials of which,
from what we know of their mean density, we have
every reason to believe they consist. The intensity of
solar radiation is nearly seven times greater on Mercury
than on the earth, and on Uranus 330 times less ; the
proportion between the two extremes beings that of
upwards of 2000 to one. Let any one figure to himself
the condition of our globe, were the sun to be septupled,
to say nothing of the greater ratio ! or were it diminished
to a seventh, or to a 300th of its actual power ! Again,
the intensity of gravity, or its efficacy in counteracting
muscular power and repressing animal activity on Jupiter
is nearly tiiree times that on the Earth, on Mars not more
than one third, on the Moon one sixth, -and on the four
smaller planets probably not more than one twentieth ;
giving a scale of which the extremes are in the proportion
of sixty to one. Lastly, the density of Saturn hardly
exceeds one eighth of the mean density of the earth, so
that it must consist of materials not much heavier than
cork. Now, under the various combinations of elements
so important to life as these, what immense diversity
must we not admit in the conditions of that great problem,
the maintenance of animal and intellectual existence and
happiness, which seems, so far as we can judge by what
we see around us in our own planet, and by the way in
which every corner of it is crowded with living beings, to
form an unceasing and worthy object for the exercise of
the Benevolence and Wisdom which presides over all I

(436.) Quitting, however, the region of mere specula-
tion, we will now show what information the telescope
affords us of the actual condition of the several planets
within its reach. Of Mercury we can see littie more than
that it is round, and exhibits phases. It is too small,
and too much lost in the constant neighbourhood of the
Sun, to allow us to make out more of its nature. The
real diameter of Mercury is about 3200 miles : its appa-
rent diameter varies from 5" to 12". Nor does Venus
offer any remarkable peculiarities : although its real dia-

264 A TREATISE ON i^TRONOMT. [cHAP. YIH.

meter is 7800 miles, and although it occasionally attains
the considerable apparent diameter of 61", which is
larger than that of any other planet, it is yet the most dif-
ficult of them all to define with telescopes. The intense
lustre of its illuminated part dazzles the sight, and exag-
gerates every imperfection of the telescope ; yet we see
clearly that its surface is not mottled over with permanent
spots like the moon ; we perceive in it neither mountains
nor shadows, but a uniform brightness, in which some-
times we may, indeed, fancy obscurer portions, but can
seldom or never rest fully satisfied of the fact. It is from
some observations of this kind that both Venus and Mer-
cury have been concluded to revolve on their axes in
about the same time as the Earth. The most natural
conclusion, from the very rare appearance and want of
permanence in the spots, is, that we do not see, as in the
Moon, the real surface of these planets, but only their
atmospheres, much loaded with clouds, and which may
serve to mitigate the otherwise intense glare of their sun-
shine.

(437.) The case is very diflerent with Mars. In this'
planet we discern, with perfect distinctness, the outlines
of what may be continents and seas. (See plate I. Jig,
1, which represents Mars in its gibbous state, as seen on
the 16th of August,1830,in the 20-feet reflector at Slough.)
Of these, the former are distinguised by that ruddy colour
which characterizes the light of this planet (which always
appears red and fiery), and indicates, no doubt, an ochrey
tihge in the general soil, like what the red sandstone dis-
tricts on the Earth may possibly offer to the inhabitants
of Mars, only more decided. Contrasted with this (by
a general law in optics), the seas, as we may caU them,
appear greenish.* These spots, however, are not always
to be seen equally distinct, though, when seen, ihcy offer
always the same appearance. This may arise from the pla-
net not being entirely destitute of atmosphere and clouds ;t
and what adds greatly to the probability of this is the ap-
pearance of brilliant white spots at its poles,-— one of which

* i have noticed the phenomena described in the text on many occa-
sions, but never more distinct than on the occasion when the (uawing
was made from which the fiffure in plate I. is engraved. — Author,

t It has been surmised to have a very extensive atmoephere, but od no
sufficient or even (dausible grounds.

OHAP. Vin.3 APPEARANCES OF THE PLANETS. 205

appears in our figure, — ^whichhave been conjectured with
a great deal of probability to be snow ; as they disappear
when they have been long exposed to the sun, and are great-
est when just emerging from the long night of their polar
winter. By watching the spots during a whole night,
and on successive nights, it is found that Mars has a ro-
tation on an axis inclined about 30° 18' to the ecliptic,
and in a period of 24** 39" 21' in the same direction ais the
earth's, or from west to east. The greatest and least appa-
rent diameters of Mars are 4'' and 18'', and its real dia-
meter about 4100 miles.

(438.) We come now to a much more magnificent pla-
net, Jupiter, the largest of them all, being in diameter no
less than 87,000 miles, and in bulk exceeding that of the
Earth nearly 1300 times. It is, moreover, dignified by
the attendance of four moons, satellites, or secondary
planets, as they are called, which constantly accompany
and revolve about it, as the moon does round the earth,
and in the same direction, forming with their principal,
or primary, a beautiful miniature system, entirely analo-
gous to that greater one of which their central body is
itself a member, obeying the same laws, and exemplifying,
in the most striking and instruqtive manner, the preva-
lence of the gravitating power as the ruling principle of
their motions : of these, however, we shall speak more
at large in the next chapter.

(439.) The disc of Jupiter is always observed to be
crossed in one certain direction by dark bands or belts,
presenting the appearance in plate hjig. 2, which repre-
sents this planet as seen on the 23d of September, 1832,
in the 20-feet reflector at Slough. These belts are, how-
ever, by no means alike at all times ; they vary in breadth
and in situation on the disc (though never in their general
direction). They have even been seen broken up, and
distributed over the whole face of the planet : but this
phenomenon is extremely rare. Branches running out
from them, and subdivisions, as represented in the figure,
as well as evident dark spots, like strings of clouds, are
by no means uncommon; and from these, attentively
watched, it is concluded that this" planet revolves in the
surprisingly short period of 9** 55" 50* (sid. time), on an
axis perpendicular to the direction of the belts. Now, it
Z

266 A TREATISE ON ASTRONOMY. [CHAP. VUI.

is very remarkable, and forms a most satisfactory com-
ment on the reasoning by which the spheroidal figure of
the earth has been deduced from its diurnal rotation, that
the outline of Jupiter's disc is evidently not circular, but
elliptic, being considerably flattened in the direction of its
axis of rotation. This appearance is no optical illusion,,
but is authenticated by micrometrical measures, which,
assign 107 to 100 for the proportion of the equatorial and
polar diameters. And to confirm, in the strongest man-
ner, the truth of those principles on which our former
conclusions have been founded, and fully to authorize
their extension to this remote system, it appears, on calcu-
lation, that this is really the degree of oblateness which
corresponds, on those principles, to the dimensions of Ju-
piter, and to the time of his rotation.

(440.) The parallelism of the belts to the equator of
Jupiter, their occasional variations, and the appearances
of spots seen upon them, render it extremely probable
that they subsist in the atmosphere of the planet, forming
tracts of comparatively clear sky, determined by currents
analogous to our trade- winds, but of a much more steady
and decided character, as might indeed be expected from
the immense velocity of its rotation. That it is the
comparatively darker body of the planet which appears
in the bells is evident from this, — Aat they do not come
up in all their strength to the edge of the disc, but fade
away gradually before they reach it. (See plate I.
Jig, 2.) The apparent diameter of Jupiter varies from
30" to 46".,

(441.) A still more wonderful, and, as it may be
termed, elaborately artificial mechanism, is displayed in
Saturn, the next in order of remoteness to Jupiter, to which
it is not much inferior in magnitude, being about 70,000
miles in diameter, nearly 1000 times exceeding the earth
in bulk, and subtending an apparent angular diameter at
the earth, of about 16". This stupendous globe, be-
sides being attended by no less than seven satellites or
moons, is surrounded with two broad, fiat, extremely
thin rings, concentric with the planet and with each
other ; both lying in one plane, and separated by a very
narrow interval from each other throughout their whole
circumference, as they are from the planet by a much

CHAP. viii.J OF Saturn's rings. 207

wider. The dimensions of this extraordinary appendage
are as follows :* —

Miles.

Exterior diameter of exterior ring = 176418.

Interior ditto....... = 156272.

Exterior diameter of interior ring = 151690.

Interior ditto = 11733a

Equatorial diameter of the body = 79160.

Interval between the planet and interior ring =s 19090.

Interval of the rings , = 1791.

Thickness of the rings not exceeding = 100.

The figure (Jig* 3, plate L) represents Saturn surrounded
by its rings, and having its body striped with dark belts,
somewhat similar, but broader and less strongly marked
than those of Jupiter, and owing, doubtless, to a similar
cause. That the ring is a solid opake substance is shown
by its throwing its shadow on the body of the planet,
on the side nearest the sun, and on the other side re-
ceiving that of the body, as shown in the figure. From
the parallelism of the belts with the plane of the ring,
it may be conjectured that the axis of rotation of the
planet is perpendicular to that plane ; and this conjec-
ture is confirmed by the occasional appearance of ex-
tensive dusky spots on its surface, which when watched,
like the spots on Mars or Jupiter, indicate a rotation in
10** 29" 17* about an axis so situated.

(442.) The axis of rotation, like that of the* earth,
preserves its parallelism to itself during the motion of
the planet in its orbit ; and the same is also the case
with the ring, whose plane is constantly inclined at the
same, or very nearly the same, angle to that of the orbit,
and, therefore to the ecliptic, viz. 28° 40' ; and intersects
the latter plane in a line, which makes an angle with the
line of equinoxes of 170°. So that the nodes of the
ring lie in 170° and 350° of Ipngitude. Whenever, then,
the planet happens to be situated in one or other of these
longitudes, as at AB, the plane of the ring passes through
the sun, which then illuminates only the edge of it ;
:and as, at the same moment, owing to the smsdlness of
the earth's orbit, £, compared with that of Saturn, the

* These dimensions are calculated from Prof. Struve's micrometric
measures, Mem. Art Soc. iii. 301, with the exception of the thickness of
the ring, which is concluded from my own observations, during its gra-
dual extinction now in progress. The interval of the rings here stated
is possibly somewhat too smalL

269 A TREATISE ON AI^TRONOHY. [cHAP. Tin*

earth is necessarily not far out of that plane, and must,
at aU events, pass through it a little before or after that
moment, it only then appears to us a very fine straight
line, drawn across the disc, and projecting out on each
side — indeed, so very thin is the ring, as to be quite in-
visible, in this situation, to any but telescopes of extra-
ordinary power. This remarkable phenoJmenon takes
place at intervals of 15 years, but the disappearance of
the ring is generally double, the earth passing twice
through its plane before it is carried past our orbit by
the slow motion of Saturn. This second disappearance
is now in progress.* As the planet, however, recedes
from these points of its orbit, the line of sight becomes
gradually more and more inclined to the plane of the
ring, which, according to the laws of perspective, ap-
pears to open out into an ellipse which attains its greatest
breadth when the planet is 90^ from either node, as at
CD. Supposing the upper part of the figure to be north,
and the lower south of the ecliptic, the north side only
of the ring will* be seen when the planet lies in the
semicircle ACB, and the southern only when in ADB.
At the time of the greatest opening, the longer diameter
is almost exactly double the shorter.

(443.) It will naturally be asked how so stupendous
an arch, if composed of solid and ponderous materials,
can be sustained without collapsing and falling in upon
the planet? The answer to this is to be found in a swift
rotation of the ring in its own plane, which observation
has detected, owing to some portions of the ring being
a little less bright Qian others, and assigned its period at
10** 29"^ 17% which, from what we know of its dimen-
sions, and of the force of gravity in the Saturnian sys-
tem, is very nearly the periodic time of a satellite re-
volving at the same distance as the middle of its breadth.
It is the centrifugal force, then, arising from this rotation,
which sustains it; and, although no observation nice
enough to exhibit a difference of periods between the
outer and inner rings have hitherto been made, it is more
than probable that such a difference does subsist as to

* The disappearance of tbe ringB ii complete, when obieryed with a
lefleetor eighteen inches in aperture, and twenty feet in focal lengdk
April 29. 1&3.— iliiMor.

CHAP. vin.J OF ^Saturn's rinos. 269

place each independently of the other in a similar state
of equilibrium.

(444.) Although the rings are, as we have said, very
nearly concentric with the body of Saturn, yet recent
micrometrical measurements of extreme delicacy have
demonstrated that the coincidence is not mathematically
exact, but that the centre of gravity of the rings oscillates
round that of the body describing a very minute orbit,
probably under laws of much complexity. Trifling as
this remark may appear, it is of the utmost importance
to the stability of die system of the rings. Supposing
them mathematically perfect in their circular form, and
exactly concentric with the planet, it is demonstrable
that they would form (in spite of their centrifugal force)
a system in a state of unstable equilibriumf which the
slightest external power would subvert— not by causing a
rupture in the substance of the rings — ^but by precipita-
ting them, unbroken, on the surface of the planet. For
the attraction of such a ring or rings on a point or sphere
eccentrically situate within them, is not the same in all
directions, but tends to draw the point or sphere towards
the nearest part of the ring, or away from the centre.
Hence, supposing the body to become, from any cause,
ever so little eccentric to the ring, the tendency of their
mutu^ gravity is, not to correct but to increase this eo*>
centricity, and to brinff the nearest parts of theiia toge-
ther. (See chap. XI.} Now, external powers, capable
of producing such eccentricity, exist in the attractions
of the satellites, as will be shown in chap. XI. ; and in
order that the system may be stable, and possess within
itself a power of resisting the first inroads of such a ten-
dency, while yet nascent and feeble, and opposing them
by an opposite or maintaining power, it has been shown
that it is sufficient to admit the rings to be loaded in some
part of their circumference, either by some minute in-
equality of thickness, or by some portions being denser
than others. Such a load would give to the whole ring
to which it was attached somewhat of the character of a
heavy and sluggish satellite, maintaining itself in an
orbit with a certain energy sufficient to overcome minute
causes of disturbance, and establish an average bearing
an its centre. But even without supposing the existence
z2

!t7& A TREATI8B ON ASTR01iOM7. [cHAP. TIII«

of any such ioady-^-^f which, after all, we have no
proof, — and granting, therefore, in its full extent, the
general instability of the equilibrium, we think we per-
ceive, in the periodicity of all the causes of disturbance,
a sufficient guarantee of its preservation. However
homely be the illustration, we can conceive nothing more
apt in every way to give a general conception of this
maintenance of equilibrium under a constant tendency
to subversion, than the mode in which a practised hand
will sustain a long pole in a perpendicular position rest-
ing on the finger by a continual and almost imperceptible
variation of the point of support. Be that, however, as
it may, the observed oscillation of the centres of the rings
about that of the planet is in itself the evidence of a
perpetual contest between conservative and destructive
powers — ^both extremely feeble, but so antagonizing one
another as to prevent the latter ffom ever acquiring an
uncontrollable ascendancy, and rushing to a catastrophe^

(445.) This is also the place to observe, that, as the
smallest difference of velocity between the body and rings
must infallibly precipitate the latter on the former, never
more to separate (for they would, once in contact, have
attained a position of stable equilibrium, and be held to-
gether ever after by an immense force) : it follows, either
that their motions in their conamon orbit round the sun
must have been adjusted to each other by an external
power, with the minutest precision, or that the rings must
have been formed about the planet while subject to their
common orbitual motion, and under the full and free in-
fluence of all the acting forces. '

(446.) The rings of Saturii must present a magnificent
spectacle from those regions of the plsmet which lie above
^eir enlightened sides, as vast arches spanning the sky
from horizon to horizon, and holding an invariable situa-
tion among the stars. On the other handi^ in the regions
beneath tl^ dark side, a solar eclipse of fifteen years in
duration, under their shadow, must afford (to our ideas)
an inhospitable asylum to animated beings, iS compeur
sated by the faint light of the satellites. But we shall do
wrong to judge of the fitness or unfitness of their eon-
dition from what we see around us, when, perhaps, the
jrery combinations which convey to our minds only im-

OHAP. Till.] GENERAL VIEW OF THE BOLAR 8TSTEM. 271

ages of horror, may be in reality theatres of the most
striking and glorious displays of beneficent contrivance.

(447.) Of Uranus we see nothing but a small, round,
uniformly illuminated disc, without rings, belts, or dis*
cemible spots. Its apparent diameter is about 4'\ from
which it never varies much, owing to the smallness of
our orbit in comparison of its own. Its r^ diameter is
about 35,000 miles, and its bulk 80 times that of the
earth. It is attended by satellites — ^two at least, probably
five or six — whose orbits (as will be seen in the next
chapter) offer remarkable peculiarities.

(448.) If the immense distance of Uranus precludes
all hope of coming at much knowledge of its physical
state, the minuteness of the four ultra-zodiacal planets
is no less a bar to any inquiry into theirs. One of them,
Pallas, is said to have somewhat of a nebulous or hazy
appearance, indicative of an extensive and vaporous at-
mosphere, little repressed and condensed by die inade-
quate gravity of so small a mass. No doubt the most
remarkable of their peculiarities must lie in this condi-
tion of their state. A man placed on one of them would
spring with ease 60 feet high, and sustain no greater
shock in his descent that he does on the earth from leap-
ing a yard. On such planets giahts might exist ; and
those enormous animals, which on earth require the buoy-
ant power of watet to counteract their weight, might
there be denizens of the land. But of such speculation
there is no end.

(449.) We shall close this chapter with an illustration
calculated to convey to the minds of our readers a gene-
ral impression of Uie relative magnitudes and distances
of the parts of our system. Choose any well levelled
field or bowling green. On it place a globe, two feet m
diameter ; this will represent the Sun ; Mercury will be
represented by a grain of mustard seed, on the circum-
ference of a circle 164 feet in diameter for its orbit ;
Venus a pea, on a circle 284 feet in diameter; the Earth
also a pea,' on a circle of 430 feet ; Mars a rather large
pin's head, on a circle of 664 feet ; Juno, Ceres, Vesta,
and Pallas, grains of sand, in orbits of from 1000 to 1200
feet ; Jupiter a moderate-sized orange, in a circle nearly
half a mile across ; Saturn a small orange, on a circle of

272 A TREATISE ON ASTRONOMY. [CHAP.IX.

four-fifths of a mile ; and Uranus a full-sized cherry, or
small plum, upon the circumference of a circle more than
a mile a&d a half in diameter. As to getting correct no-
tions on this suhject by drawing circles on paper, or,
flliil worse, from those very chilcUsh toys called orreries,
it is out of the question* To imitate ihe motions of the
pianets, m th» above-mentioned orbits. Mercury must
describe its own diameter in 41 seconds ; Venus, in 4"»
14* ; the earth, in 7 minutes ; Mars, in 4"*^ 48»; Jupiter,
ta 2* 56" ; Saturn, in 3* 13" ; and Uraaus, in 2*' 16".

CHAPTER IX.

OF THE SATELLITES.

Of Ike Mooiit a» a Satellite of the Earth— General Proximit]^ of SatelUfea
to dieir Primaries, an^ consequent Subordination of their Motions —
Masses of the Primaries concluded from the Periods of their SateUites
'oMaintenanoe of Kepler's Laws in the secondary Systems— Of Jupi-
ter's SateUites— Their Eclipses, &c. — Velocity of light discovered py
their Means— Satellites of Saturn— Of Uranus.

(450.) In the annual circuit of the earth about the sun,
it is constantly attended by its satellite the moon, which
revolves round it, or rather both round their common
centre of gravity'; while this centre, strictly speaking,
and not either of the two bodies thus connected, moves
in an elliptic orbit, undisturbed by their mutual action,
just as the centre of gravity of a large and small stone
tied together and flung into the air describes a parabola
as if it were a real material substance under the earth's
attraction, while the stones circulate round it or round
each other, as we choose to conceive the matter.

(451.) If we trace, therefore, the real curve actually
described by either the moon's or the earth's centres, in
virtue of this compound motion, it will appear to be, not
an exact ellipse, but an undulated curve, like that repre-
aented in the figure to article 272, only that the number
of undulations in a whole revolution is but 13, and their
actual deviation from the general ellipse, which serves
them as a central line, is comparatively very much smaller;
ao much so, indeed, that every part of the curve described

CVAF. IZ. j OF THE SATEI^LlTEfl. 278

by either the earth or moon is concave towards the sun^
The excursions of the earth on either side of the ellipse,
indeed, are so very small as to be hardly appreciable. In
fact, the centre of gravity of the earth and moon lies al-
ways within the surface of the earth, so that the monthly
orbit described by the earth^s centre about the common
centre of gravity ^is comprehended within a spac^ less
than the size of the earth itself. The effect ts, neverthe-
less, sensible, in producing an apparent monthly dis-
placement of the sun in longitude^ of a parallactic kind^
ivhich is called the meiutrual equation; whose greatest
amount is^ however, less than the sun's horizontal parM-
lax, orthan'8'6".

(452.) The moon, as we have seen, is about 60 radii
of the earth distant from the centra of the latter. Its
proximity, therefore, to its centre of attraction, thus esti-
mated, is much greater than that of the planets to the
sun ; of wiiich. Mercury, the nearest, is 84, and Uranus
2026 solar radii from its centre. It is owing to this prox-
.imity that the moon remains attached to the earth as a
sateUite. Were it much farther, the feebleness of its
gravity towards the earth would be inadequate to produce
that alternate acceleration and retardation in its motion
about the sun, which divests it of the character of an in-
dependent planet, and keeps its movements subordinate
to those of the earth. The one would outrun, or be left
behind the other, in their revolutions round the sun (by
reason of Kepler's third law), according to the relative
dimensions of their heliocentric orbits, after which the
whole influence of the earth would be confined to pro-
ducing some considerable periodical disturbance in the
moon's motion, as it passed or was passed by it in each
synodical revolution.

(453.) At the distance at which the moon really is
from us, its gravity towards the earth is actually less than
towards the sun* That this is the case, appears suffi-
ciently from what we have already stated, that the moon's
reai path, even when between the earth and sun, is con*
cave towards the latter. But it will appear still more
clearly if, from the known periodic times* in which the

* A and r ndii of two orbito (supposed ciicular), F and p the periodic
timea; then the arci in qnegtion(A and a) are to each other aa p- to -;

274 A TREATISE ON ASTRONOMY. [cHAP. IX.

earth completes its annual and the moon its monthly orbit,
and from die dimensions of those orbits, we calculate the
-amount of deflection, in either, from their tangents, in
equal very minute portions of time, as one second.
These are the versed sines of the arcs described in that
time in the two orbits, -and these are the measures of the
acting forces which produce these deflections. If we
execute the numericsd calculation in the case before us,
we shall find 2*209 : 1 for the proportion in which the
intensity of the force which retains the earth in its orbit
round the sun actually exceeds that by which the moon
is retained in its orbit about the earth.

(454.) Now the sun is 400 times more remote from
the earth than the moon is. And, as gravity increases as
the squares of the distances decrease, it must follow that,
at eqiud distances, the intensity of solar would exceed
that of terrestrial gravity in the above proportion, aug-
mented in the further ratio of the square of 400 to 1 ;
that is, in the proportion of 354936 to 1 ; and therefore,
if we grant that the intensity of the gravitating energy is
commensurate with the mass or inertia of the attracting
body, we are compelled to admit the mass of the earth
to be no more than sst^-sv ^^ ^^^ ^^ ^^® ^^°*

(455.) The argument is, in fact, nothing more than a
recapitulation of what has been adduced in chap. Vn.
(art. 380.) But it is here re*-introduced, in order to show
how the mass of a planet which is attended by one or
more satellites can be as it were weighed against the sun,
provided we have learned from observation the dimen-
sions of the orbits described by the planet about the sun,
and by the satellites about the planet, and also the periods
in wWch these orbits are respectively described. It is
by this method that the masses of Jupiter, Saturn, and
Uranus have been ascertained. (See Synoptic Table.)

(456.) Jupiter, as already stated, is attended by four
satellites, Saturn by seven ; and Uranus certainly by two,
and perhaps by six. These, with their respective pri-
maries (as the central planets are called), form in each

and since the vened sines are as the squares of the arcs directly and the

R r
ladii inversely, these are to each other as =^ to ^^; and in this ratio ara

the forces acting on the revolving bodies in either case.

CHAP. IX.] OF THE SATELLITES. 275

case miniature systems, entirely analogous, in the ge-
neral laws by which their motions are governed, to 3ie
great system in which the sun acts the part of the pri-
mary, and the planets of its satellites. In each of these
systems the laws of Kepler are obeyed, in the sense,
that is to say, in which they are obeyed in the planetary
system — approximately, and without prejudice to the
effects of mutual perturbation, of extraneous interference,
if any, and of that small but not imperceptible correction
which arises from the elliptic form of the central body.
Their orbits are circles or ellipses of very moderate ec-
centricity, the primary occupying one focus. About this
they describe areas very nearly proportional to the times ;
and the squares of the periodical times of all the satellites
belonging to each planet are in proportion to each other
as the cubes of their distances. The tables at the end
of the volume exhibit a synoptic view of the distances
and periods in these several systems, so far as they are
at present known; and to all of them it will be observed
that the same remark respecting their proximity to their
primaries holds good, as in the case of the moon, with a
similar reason for such close connexion.

(457.) Of these systems, however, the only one
which has been studied with great attention is that of
Jupiter ; partly on account of the conspicuous brilliancy
of its four attendants, which are large enough to offer
visible and measurable discs in telescopes of great pow-
er ; but more for the sake of their eclipses, which, as
they happen very frequently, and are easily observed,
afford signals of considerable use for the determination
of terrestrial longitudes (art. 218). This method, in-
deed, until thrown into the back ground by the greater
facility and exactness now attainable by lunar observa-
tions (art. 219), was the best, or rather the only one
which could be relied on for great distances and long in-"
tervals.

(458.) The satellites of Jupiter revolve from west to
east (following the analogy of the planets and moon), in
planes very nearly, although not exactly, coincident with
that of the equator of the planet, or parallel to its belts.
This latter plane is inclined 3** 5' 30" to the orbit of the
planet, and is therefore but little different from the plane

276 A TREATISE ON ASTRONOMY. [CHAP. IX,

of the ecliptic. Accordingly, we see their orbits pro-
jected very nearly into straight lines, in which they ap-
pear to oscillate to and fro, sometimes passing before
Jupiter, and casting shadows on his disc (which are
very visible in good telescopes, like small round ink
spots), and sometimes disappearing behind the body, or
being eclipsed in its shadow at a distance from it. It is
by these eclipses that we are furnished with accurate
data for the construction of tables of the satellites* mo-
tions, as well as with signals for determining differences
of longitude. .

(459.) The eclipses of the satellites, in their general
conception, are perfectly analogous to those of the moon,
but in their detail they differ in several particulars.
Owing to the much greater distance of Jupiter from the
sun, and its greater magnitude, the cone of its shadow or
umbra (art. 355) is greatly more elongated, and of far
greater dimension, than that of the earth. The satel-
lites are, moreover, much less in proportion to their
primary, their orbits less inclined to its ecliptic, and of
(comparatively) smaller dimensions, than is the case with
the moon. Owing to these causes, the three interior
satellites of Jupiter pass through the shadow, and are
totally eclipsed, every revolution ; and the fourth, though,
from the greater inclination of its orbit, it sometimes
escapes eclipse, and may occasionally graze as it were
the border of the shadow, and suffer partial eclipse, ye^
this is comparatively rare, and, ordinarily speaking, its
eclipses happen, like those of the rest, each revolution.

(460.) These eclipses, moreover, are not seen, as is
the case with those of the moon* from the centre of their
motion, but from a remote station, and one whose situa-
tion with respect to the line of i^adow is variable.
This, of course, makes no difference in the times of the
eclipses, but a very great one in their visibility, and in
their apparent situations with respect to the planet at the
moment of their entering and quitting the shadow.

(461.) Suppose S to be the- sun, £ the earth in its
orbit EF6K, J Jupiter, and at the orbit of one of its
satellites. The cone of the shadow, then, will have its
vertex at X, a point far beyond the orbits of all the sar
tcllites ; and the penumbra, owing to the great distance

CHAP. IX.3 ECLIPSES OF JTJPITEr's SATELLITES. 277

of the snn, and the consequent smallness of the angle its
disc subtends at Jupiter, will hardly extend, within the

liiBiis of the satellites' orbits, to any perceptible distance
beyond the shadow, — for which reason it is not repre«
sented in the figure. A satellite revolving from west to
east (in the direction of the arrows) will be eclipsed
when it enters the shadow at a, but not suddenly, be-
cause, like the moon, it has a considerable diameter seen
from the planet ; so that the time elapsing from the first
perceptible loss of light to its total extinction will be that
which it occupies in describing about Jupiter an angle
tqual to its apparent diameter as seen from the centre
of the planet, or rather somewhat more, by reason of the
penumbra ; and the same remark applies to its emer-
gence at 6. Now, owing to the difference of tdescopes
and of eyes, it is not possible to assign the /^reet^e mo-
ment of incipient obscuration, or of total extinction at a,
nor that of the first glimpse of light falling on the satel-
lite at b, or the complete recovery of its light. The ob-
servation of an eclipse, then, in which only the immer-
sion, or only the emersion, is seen, is incomplete, and
inadequate to afford any precise information, dieoretical
or practical. But, if both the immersion and emersion
can be observed vnih the same telescope, and by the
same person, the interval of the times will give the du-
ratieoy and their mean the exact middle of the eclipse,
when the satellite is in the line SJX, i, e, the true mo-
ta&ai of its opposition to the sun. Such observations,
and such only, are of use ior determining the periods and
other particulars of the motions of the satellites, and for
afTording data of any material use for the calculation of
terrestrial longitudes. The intervals of the eclipses, it
2A

278 A TREATISE ON ASTRONOMY. QcHAP. IX.

will be observed, give the synodic periods of the satel-
lites' revolutions ; from which their sidereal periods must
be concluded by the method in art. 353 (note).

(462.) It is evident, from a mere inspection of our
figure, that the eclipses take place to the west of the
planet, when the earth is situated to the west of the line
SJ, t. e. before the opposition of Jupiter ; and to the
east, when in the other half of its orbit, or after the op-
position. When the earth approaches the opposition, the
visual line becomes more and more nearly coincident
with the direction of the shadow, and the apparent
place where the eclipses happen will be continually-
nearer and nearer to the body of the planet. When the
earth comes to F, a point determined by drawing bF to
touch the body of the planet, the emersions will cease
to be visible, and will thenceforth, to an equal distance
on the other side of the opposition, happen behind ttie
disc of the planet. W'^hen the earth arrives at G (or H)
the immersion (or emersion) will happen at the very
edge of the visible disc, and when between G and H (a
very small space) the satellites will pass tmeclipsed be-
hind the limb of the planet.

(463.) When the satellite comes to m, its shadow will
be thrown on Jupiter, and will aippear to move across it
as a black spot till the satellite comes to n. But the satel-
lite itself will not appear to enter on the disc till it comes
up to the line drawn from £ to the eastern edge of the
disc, and will not leave it till it attains a similar line
drawn to the western edge. It appears 'then that the
shadow will precede the satellite in its progress over the
disc before the opposition^ and vice versa. In these
transits of the satellites, which, with very powerful
telescopes, may be observed with great precision, it fre-
quently happens that the satellite itself is discernible on
the disc as a bright spot if projected on a dark belt ; Imt
occasionally also as a dark spot of smaller dimensions
than the shadow. This curious fact (observed by Schroe-
ter and Harding) has led to a conclusion that certain
of the satellites have occasionally on their own bodies,
or in their atmospheres, obscure spots of great extent.
We say of great extent; for the satellites of Jupiter,
small as they appear to us, are really bodies of con-

CHAP. IX.] OP Jupiter's satellites. 279

siderable size, as the following comparatiye table will
show.*

Mean apparent
diameter.

Diameter in
miles.

Mass.t

Jupiter
Ist satellite

M

M

4th

1105
0-911

1-488
1-273

87000
2508
9068
3377
2890

1-0000090
00000173
0-0000232
00000685
00000427

(464.) An extremely singular relation subsists be-
tween the mean angular velocities or mean motions (as
they are termed) of the three first satellites of Jupiter.
If the mean angular velocity of the first satellite be added
to twice that of the third, the sum will equal three times
that of the second. From this relation it follows, that if
from the mean longitude of the first added to twice that
of the third, be subducted three times that of the second,
the remainder will always be the same, or constant, and
observation informs us that this constant is 180**, or two
right angles ; so that, the situations of any two of them
being given, that of the third may be found. It has been
attempted to account for this remarkd,ble fact, on the
theory of gravity by their mutual action. One curious
consequence is, that these three satellites cannot be all
eclipsed at once ; for, in consequence of the last-men-
tioned relation, when the second and third lie in the
same direction from the centre, the first must lie on the
opposite s and therefore, when the first is eclipsed, the
other two must lie between the sun and planet, throwing
its shadow on the disc, and vice versa. One instance only
(so far as we are aware) is on record when Jupiter has
been seen ivithoiU satdlites ; viz. by Molyneux, Nov.
2 (old style), 1681.*

(465.) The discovery of Jupiter's satellites by Galileo,
one of the first-fruits of the invention of the telescope,
forms' one of the most memorable epochs in the history
of astronomy. The first astronomical solution of the
great problem of " the longitude^* — ^the most important
for the interests of mankind which has ever been brought
under the dominion of strict scientific principles, dates

* Strove, Mem. Ast Soc. iu. 301. t Laplace, Mec. Col. liv. viiL $ 37.
t Molyneux^ Optics, p. 371 .

!)80 A TREATISE ON ASTRONOMY. [CHAP. IX.

immediately from their discovery. The final and ccmi-
clusive establishment of the Copernican system of as-
tronomy may also be considered as referable to the dis-
covery and study of this exquisite miniature system, in
which the laws of the planetary motions, as ascertained
by Kepler, and especially that which connects their
periods and distances, were speedily traced, and found
to be satisfactorily maintained. And (as if to accumulate
historical interest on this point) it is to the observation of
their eclipses that we owe the grand discovery of the
aberration of light, and the consequent determination of
the enormous velocity of that wonderful element. This
we must explain now at large.

(466.) The earth's orbit being concentric with that of
Jupiter and interior to it (see Jig, art. 460), their mutual
distance is continuali^y varpng, the variation extending
from the wm to the difference of the radii of the two
orbits, and the difference of the greater and least dis-
tances being equal to a diameter of the earth's orbit.
Now, it was observed by Roemer (a Danish astronomer,
in 1675), on comparing together observations pf eclipses
of the satellites during many successive yea^, that the
eclipses at and about the opposition of Jupiter (or its
nearest point to the earth) took place too soon — sooner,
that is, tiian* by calculation from an average, he expected
them ; whereas those which happened when the earth
was in the part of its orbit most remote from Jupiter
were always too late. Connecting the observed error ia
their computed times with the variation of distance, he
concluded, that, to make the calculation on an average
period correspond with fact, an allowance in respect of
time behooved to be made proportional to the excess or
defect of Jupiter's distance from the earth above or below
its average amount, and such that a difierenee of distance
of one diameter of the earth's orbit should correspond to
16" 26**6 of time allowed. Speculating on the probable
physical cause, he was naturally led to think of the
gradual instead of an instantaneous propagation of light.
This explained every particular of the observed phe*
nomenon, but the velocity required (102000 miles per
second) was so great as to startle many,. and, at all events,
to require confirmation. This has been afforded since.

CHAP. IX.] SUCCESSIVE TRANSMISSION OF LIGHT* 281

and of the most unequivocal kind, by Bradley's discovery
of the aberration of light (art. 275). The velocity of light
deduced from this last phenomenon differs by less than one
eightieth of its amount from that calculated from the
eclipses, and even this difference will no doubt be de-
stroyed by nicer and more rigorously reduced observations.

(467.) The orbits of Jupiter's satellites are but little
eccentric ; those of the two interior, indeed, have no per-
ceptible eccentricity; their mutual action produces in
them perturbations analogous to those of the planets
about the sun, and vhich have been diligently investi-
gated by Laplace and oljiers. By assiduous observation
it has been ascertained that they are subject to marked
fluctuations in respect of brightness, and that these fluc-
tuations happen periodically, according to their position
with rqspect to the sun. From this it has been con-
cl^ded, apparently with reason, that they turn on their
axes, like our moon, in periods equal to their respective
sidereal revolutions about their primary.

(468.) The satellites of Satui:n have been much less
studied than those of Jupiter. The most distant is by
far the largest, and is probably not much inferior to Mars
iu size. Its orbit is also materially inclined to the plane
of the ring, with which those of sdl the rest nearly coin-
cide. It is the only one of the number whose theory
has been at all inquired into, further Itian suffices to
verify Kepler's law of the periodic times, which holds
good, fnutatis mutandis^ and under the requisite reser-
vations, in this as in the system of Jupiter. It exhibits,
like those of Jupiter, periodic defalcations of light,
which prove ij;s revolution on its axis in tfie time of a
sidereal revolution about Saturn. The nex,t in order (pro-
ceeding inwards) is tolerably conspicuous ; the three next
very minute, and requiring pretty powerful telescopes to
see them ; while the two interior satellites, which just
skirt the edge of the ring, and move exactly in its plane,
have never been discerned but with the most powerful
telescopes which human art has yet constructed, and
then only under peculiar circumstances. At the time of
the disappearance of the ring (to ordinary telescopes)
they have been seen* threading like beads the almost .
*^ymy &ther, in 1769, with a reflecting telescope four feet in aperture
9a9

298 A TREATISE ON ASTRONOMY. [CHAP. IX.

infinitely thin fibre of light to which it is then reduced, and
for a short time advancing off it at either end, speedily to
return, and hastening to their habitual concealment.
Owing to the obliquity of the ring, and of the orbits of
the satellites to Saturn's ecliptic, there are no- eclipses of
the satellites (the interior ones excepted) until near the
time when the ring us seen edgewise.

(469.) With the exception of the two interior satel-
lites of Saturn, the attendants of Uranus are the most dif-
ficult objects to obtaia a sight of, of any in our system.
Two undoubtedly exist, and four more have been sus-
pec^d. These two, however, offer remarkable and, in-
deed, quite unexpected and unexampled peculiarities.
Contrary to the unbroken analogy of the whole planet-
ary systeBBh— whether of primaries or secondari6s — ^the
planes of their orbits are nearly perpendicular to the
ecUptiCf being inclined no less than 78^ 58' to that plane,
and in these orbits their motions are retrograde; that is
to say, their positions, when projected on the ecliptic,
instead of advancing from west to east round the centre i
of their primary, as is the case with every other planet
and satellite, move in the opposite direction. Their
OffUta are nearly or quite circular, and they do not appear
to have any sensible, or, at least, any rapid motion of
nodes, or to have undergone any material change of indi-
BAti^n, in the course, at least, of half a revolution of their
primary round the dun.*

* These anomalous peculiarities, which seem to occur at the extreme
limits of our system, as if to prepare us for further departure from ott its
analogies, in other svstemt which may yet be disckMed to us, have hith-
orto vested on the sole testimony of their discoverer, who alone had ever
obtained a view of them. I am happy to be able, from my own obsenra-
tions from 1838 to the present time, to confirm, in the amplest waaxxan, mgr
fiuhar's reniltB.— AulAor. i

GHAF. X.J NUMBER OF COMETS. 98SI

CHAPTER X.

OF COMETS.

Creat Number of recorded Cometi— The number of unrecorded proba-
bly much greater— Description of a Comet-Comets without Tails-
Increase and Decay of their Tai]»^Their Motion»-~Subjeot to the
ceneral Laws of planetary Motion— Elements of their Orbits— Periodic
Return of certain Comets— Hane^*s—Encke*8—Biela*B— Dimensions of
Comets- Their Resistance by tha Ether, giadual Decay, and possibla
Dispersion in Spaca.

(470.) The cxtraordmary aspect of comets, their rapid
and seemingly irregular motions, the unexpected manner
in which they often burst upon us, and the imposing
magnitudes which they occasionally assume, have in aM
ages rendered them objects of astonishment, not unmixed
with superstitious dread to the uninstructed, and an enig-
ma to those most conversant with the wonders of crea-
tion and the operations of natural causes. Even now,
that we have ceased to regard their movements as irregu-
lar, or as governed by other kws than those which retain
the planets in their orbits, their intimate nature, and the
offices they perform in the economy of our system, are
as much unknown as ever. No rational or even plausible
account has yet been rendered of those immensely volu-
minous appendages which they bear about with them,
and which are known by the name of their tails, (though
improperly, since they often precede them in their mo-
tions), any more than of several other singularities which
they present.

(471.) The number of comets which have been astro-
aomically observed, or of which notices have been re-
corded in history, is very great, amounting to several
hundreds ;* and when we consider that in the earlier ages
of astronomy, and indeed in more recent times, before the
invention of the telescope, only large and conspicuous

* See eatalogaes in the Almagest^ Riocioli ; PiBgre*s Cometogiaphia ;
Delambro's Astiron. vol. iii.; Astronomische Abhandlungen, No. 1.
(which contains the elements of all the orbits of comets which have been
4»mpated to the time of its publication, 1823) ; also, a catalogne now in
pn^gresi, by the Rev. T. J. Hussey. Lon. & £d. PhiL Mag. vol. iL Na d.
€t sea. In a list cited by Lalande firom the lat vol of the Tables de BM-
Ihif 700 comets are enumemted.

284 A TREATISE ON ASTRONOHY. [CHAP. Z.

ones were noticed ; and that, since due attention has been
paid to the subject, scarcely a year has passed without
the observation of one or two of these bodies, and that
sometimes two and even three have appeared at once ; it
will be easily supposed that their actual number must be
at le^st many thousands. Multitudes, indeed, must es*
cape all observation, by reason of their paths traversing
only that part- of the heavens which is above the horizon
in .the daytime. Comets so circumstanced can only be-
come visible by the rare coincidence of a total eclipse of
the sun,-— a coincidence which happened, as related by
Seneca, 60 years before Christ, when a large comet was
actually observed very near the sun. Several, however,
stand on record as haying been bright enough to be seen
in the daytime, even at noon and in bright sunshine.
Such were the comets of 140,2 and 1532, and that which
appeared a little bfsfore the assassination of C^saur, and
was {afterwards) supposed to have predicted his death.
(472.) That feelings of awe and astonishment should
be excited by the sudden and unexpected appearanpe of
a great comet, is no way surprising ; being, in fact, ac-
cording to the accounts we nave of such events, one of
the most brilliant and imposing of all natural phenomena.
Comets consist for the most part of a large and splendid
but ill defined nebulous mass of light, called the head,
which i;3 usually much brighter towards the centre, and
blFers the appearance of a vivid nucleus^ like a star or pla-
net. From the head, and in a directioQ opposite to that
in which the sun is situated from the comet, appear tp
diverge two streams of light, which grow broader an<j[
naore diffused at a distance from the head, ^nd which
sometimes close in and unite at a little distance behind
it, sometimes continue distinct for a great part of their
course ; produ<cing an effect like that of the trains left by
some bright meteors, or like the diverging fire of a sky-
rocket (only without sparks or perceptible motion). This
is the tail. This magnificent appendage attains occasion-
ally an immense apparent length. Aristotle relates of the
tail of the comet of 371 a. c, that it occupied a third of
the hemisphere, or 60"" ; that of a. d. 1618 is stated to
have been attended by a train no less than 104° in length.
The comet of 1680, the most celebrated of njodern times.

CHAP. X.] SMALL DENSITY OF COMETS. 285

and on many accounts the most remarkable of all, with a
head not exceeding in brightness a star of the second
magnitude, covered with its tail an extent of more than
70° of the heavens, or, as some accounts state, 90®. The
figure {Jig* 2, plate II.) is a very correct representation
of the comet of 1819 — ^byno means one of the most con-
siderable, but the latest which has been conspicuous to.
the naked eye.

(473.) The tail is, however, by no means an invariable
appendage of comets. Many of the brightest have been
observed to have short and feeble taib, and not a few have
been entirely without them. Those of 1585 and 1763
offered no vestige of a tail ; and Cassini describes the
comet of 1682 as being as round and as bright as Jupiter.
On the other hand, instances are not wanting of comets .
furnished with many tails or streams of diverging light.
That of 1744 had no less than six, spread out like an im-
mense fan, extending to a distance of nearly 30® in length.
The tails of comets, too, are often curved, bending, in
general, towards the region which the comet has leU, as
if moving somewhat more slowly, or as if resisted in their
course.

(474.) The smaller comets, such as are visible only in
telescopes, or with difficulty by the naked eye, and which
are by far the most numerous, offer very frequently no*
appearance of a tail, and appear only as round or some-
what oval vaporous masses, more dense towards the cen-
tre^ where, however, they appear to have no distinct nu-
cleus, or any thing which seems entitled to be considered
as a solid body. Stars of the smallest magnitude remain
• distinctly visible, though covered by what appears to be
the densest portion of their substance ; although the same
stars would be completely obliterated by a moderate fog,
extending only a few yards from the surface of the earth.
And since it is an observed fact, that even those larger
comets which have presented the appearance of a nu-
cleus have yet exhibited no pha^e^, though we cannot
doubt that they shine by the reflected solar light, it fol-
lows that even these can only be regarded as great masses
of thin vapour, susceptible of being penetrated through
their whole substance by the sunbeams, and reflecting
them alike from their interior parts and from their sur-

286 A TREATISE ON ASTRONOMY. [CHAP. X.

faces. Nor will any one regard this explanation as
forced, or feel disposed to resort to a phosphorescent qua
lity in the comet itself, to account for the phenomena in
question, when we consider (what will be hereafter
shown) the enormous magnitude of the space thus illumi-
nated, and the extremely small mass which there is
ground to attribute to these bodies. It will then be evi-
dent that the most unsubstantial clouds which float in the
highest regions of our atmosphere, and seem at sunset to
be drenched in light, and to glow throughout their whole
depth as if in actual ignition, without any shadow or dark
side, must be looked upon as dense and massive bodies
compared with the filmy and all but spiritual texture of
a cornet^ Accordingly, whenever powerful telescopes
have been turned on these bodies, they have not failed to
dispel the illusion which attributes solidity to that more
condensed part of the head, which appears to the naked
eye as a nucleus ; though it is true that in some, a very
minute stellar point has been seen, indicating the exist-
ence of a solid body.

(475.) It is in all proba))ility to the feeble coercion of
the elastic power of their gaseous parts, by the gravitation
of so small a central mass, that we must attribute this ex-
traordinary developement of the atmospheres of comets.
If the earth, retaining* its present size, were reduced, by
any internal change (as by hollowing out its central
parts) to one thousandth part of its actual mass, its
coercive power over the atmosphere would be dimi-
nished in the same proportion, and in consequence the
latter would expand to a thousand times its actual bulk ;
and indeed much more, owing to the still farther dimi-
nution of gravity, by the recess of the upper parts from
the centre.

(476.) That the luminous part of a comet is something
in the nature of a smoke, fog, or cloud, suspended in a
transparent atmosphere, is evident from a fact which has
been often noticed, viz.— that the portion of the tail
where it comes up to, and surrounds the head, is yet
separated from it by an interval less luminous, as if sus-
tained and kept off from contact by a transparent stratum,
as we often see one layer of clouds laid over another
with a considerable clear space between. These, and

CHAP. X.j MOTIONS OF COMETS. 287

most of the other facts observed in the history of comets,
appear to indicate that the structure of a comet, as seen
in section in the direction of its length, must be that of
a hollow envelope, of a parabolic form, enclosing near its
vertex the nucleus and head, something as represented
in the annexed figure. This would account for the aj;^

parent division of the tail into two prineipal lateral
branches, the envelope being oblique to the line of sight
at its borders, and dierefore a greater depth of illumi-
nated matter being there exposed to the eye. In all proba-
bility, however, they admit great varieties of structure,
and among them may very possibly be bodies of widely
different physical constitutioqi.

(477.) We come now to speak of the motions of co-
mets* These are apparently most irregular and capri-
cious. Sometimes they remain in sight for only a few
days, at others for many months ; some move with ex-
treme slowness, others with extraordinary velocity ;
while not unfrequently, the two extremes of apparent
speed are exhibited by the same comet in different parts
of its course. The comet of 1472 described an arc of
the heavens of 120° in extent in a single day. Some
pursue a direct, some a retrograde, and others a tortuous
and very irregular course : nor do they confine them-
selves, like the planets, within any certain region of the
heavens, but traverse indifferently every part. Their
variations in apparent size, during the time they continue
visible, are no less remarkable than those of their velo-
city ; sometimes they make their first appearance as faint
and slow moving objects, with little or no tail ; but by
degrees accelerate, enlarge, and throw out from them this
appendage, which increases in length and brightness till
(as always happens in such casesj they approach the
sun, and are lost in his beams. After a time they again
emerge, on the other side, receding from the sun with a

$88 A TREATISE ON ASTRONOMY. [cRAP. X*

velocity at first rapid, but gradually decaying. ' It is after
^us passing &e san, and not till then, that they shine
fojrth in all their splendour, and that their tails acquire
their greatest length and developement ; thus indicating
plainly the action of the sun's rays as the exciting cause
of that extraordinary emanation. As they continue to
recede from the sun, their motion diminishes and the
tail dies away, or is absorbed into the head, which itself
grows continually feebler, and is at length altogether lost
sight of, in by far the greater number of cases never to
be seen more.

(478.) Without the clue furnished by the theory of
gravitation, the enigma of these seemingly irregular and
capricious movements might have remained for ever un-
resolved. But Newton, having demonstrated the pos*
sibility of any conic section whatever being described
about the sun, by a body revolving under the dominion
of that law, immediately perceived the applicability of
the general proposition to the ease of cometaiy orbits,
and the great comet of 1680, one of the most remark-
able on record, both for the immense leng^ of its tail
and' for Ae excessive doseness of its approach te Ae
sun (within one sixth of the diameter of that luminary),
afforded him an excellent opportunity for the trial of his
theory. The success of the attempt was complete. He
Moertained that this comet described about the sun as its
focus an elliptic orbit of so great an eccentricity as to be
undistinguishable from a parabola (which is the extreme,
or limiting form of the ellipse when the axis becomes
infinite), and that in this orbit the areas described a^ut
the sun were^ as ii^ the planetary ellipses, proportional
to the times. The representation of the apparent mo-
tions of this cmnet by such an orbit, throughout its wiiole
observed course, was found to be as complete as those
of the motions of the planets in their nearly eiTeuiar
paths. Fr^m that time it became a received troth, that
the motions of comets are regulated by the same general
laws as tlioee of the planets — the difference of the cases
consisting only in the extravagant elongation of their el-
bpses, and in the absence of any limit to the inclinations
of their planes to that of the ecliptic— or any general co-
incidenoe in the direction of the motions &om west to

CHAP. X.] PERIODICAL COMETS. 28$

east, rather than from east to west, like what is observed
among the planets.

(479.) It is a problem of pure geometry, from the
general laws of elliptic or. parabolic motion, to find the
situation and dimensions of the ellipse or parabola which
shall represent the motion of any given comet. In ge-
neral, three complete observations of its right ascension
and declination, with the times at which they were
made, suffice for the solution of this problem (which is,
however,ji very difficult one), and for the determination
of th^ elements of the orbit. These consist, mutatis
mutandis, of the same data as are required for the com-
putation of the motion of a planet : and, once deter-
mined^ it becomes very easy to compare them with the
whole observed course of the comet, by a process ex-
actly similar to that of art. 426, and thus at once to as-
certain their correctness, and to put to the severest trial
the truth of those general laws on which all such calcu-
lations are founded.

(480.) For the most part, it is found that the motions
of comets may be sufficiently well represented by para-
bolic orbits, — that is to say, ellipses whose axes are of
infinite length, or, at least, so very long that no appre-
ciable error in the calculation of their motions, during all
the time they continue visible, would be incurred by
supposing them actually infinite. The parabola is that
conic section which is the limit between the ellipse on
the one hand, which returns into itself, and the hyper-
bola on the other, which runs out to infinity. A comet,
therefore, which should ilescribe an elliptic patii, how-
ever long its axis, must have visited the sun before, and
must again return (unless disturbed) in some determinate
period, — ^but should its orbit be of the hyperbolic cha-
racter, when once it has passed its perihelion, it could
never more return within the sphere of our observation,
but must run ofif to visit other systems, or be lost in the
immensity of space. A very few comets have been as-
certained to move in hyperbolas, but many more in
ellipses. These then, in so far as their orbits can remain
unsdtered by the attractions of the planets, must be re-
garded as permanent members of our system.

(481.) The most remarkable of these is the comet of
2B

290 A TREATISE ON ASTRONOMY. [cHAP. X«

Halley, so called from the celebrated Edmund Halley,
who, on calculating its elements from its perihelion pas-
sage in 1682, when it appeared in great splendour, with
a tail 30^ in length, was led to conclude its identity with
the great comets of 1531 and 1607, whose elements he
had also ascertained. The intervals of these successive
apparitions being 75 and 76 years, Halley was encou-
raged to predict its re-appearance about the year 1759.
So remarkable a prediction could not fail to attract the
attention of all astronomers, and, as the time approached,
it became extremely interesting to know whether fhe at-
tractions of the larger planets might not materially inter-
fere with its orbitual motion. The computation of their
influence from the Newtonian law of gravity, a most
difficult and intricate piece of calculation, was undertaken
and accomplished by Clairaut, who found that the action
of Saturn would retard its return by 100 days, and that
of Jupiter by no less than 518, making in all 618 days,
by which the expected return would happen later than
on the supposition of its retaining an unaltered period —
and that, in short, the time of the expected perihelion
passage would take place within a month, one way or
other, of the middle of April, 1759. — ^I't actually hap-
pened on the 12th of March in 'that year. Its next re-
turn to the perihelion has been calculated by Messrs.
Damoiseau and Pontecoulant, and fixed by the former
on the fourth, and by the latter on the seventh of Novem-
ber, 1835, about a month or six weeks before which time
it may be expected to become visible in our hemisphere ;
and, as it will approach pretty near the earth, will very
probaWy exhibit a brilliant appearance, though, to judge
from the successive degradations of its apparent size and
the length of its tail in its several returns since its first
appearances on record (in 1305, 1456, &c.), we are not
now to expect any of uiose vast and awful phenomena
which threw our remote ancestors of the middle ages into
agonies of superstitious terror, and caused public prayers
to be put up in the churches against the comet and its
malignant agencies.

(482.) More recently, two comets have been especially
identified as having performed several revolutions about
the sun, and as having been not only observed and re-

CHAP. X.3 RESISTANCE EXPERIENCED BT COMETS. 291

corded in preceding revolutions, without knowledge of
this remarkable peculiarity, but have had already seve-
ral times their returns predicted, and have scrupulously
kept to their appointments. The first of these is the
comet of Encke, so called from Professor Encke, of Ber-
lin, who first ascertained its periodical return. It re-
volves in an ellipse of great eccentricity, inclined at an
angle of about 13® 22' to the plane of the ecliptic, and
in the short period of 1207 days, or about Bi years.
This remarkable discovery was made on the occasion of
its fourth recorded appearance, in 1819. From the el-
lipse then calculated by Encke, its return in 1822 was
predicted by him, and observed at Paramatta, in New
South Wales, by M. Rumker, being invisible in Europe :
since which it has been re-predicted, and re-observed in
all the principal observatories, both in the northern and
southern hemispheres, in 1825, 1828, and 1832. Its
next return will be in 1835.

(483.) On comparing the intervals between the suc-
cessive perihelion passages of this comet, after allowing
in the most careful and exact manner for all the disturb-
ances due to the actions of the planets, a very singular
fact has come to light, viz. that Ihe periods are continu-
ally diminishing, or, in otlier words, the mean distance
from the sun, or the major axis of the ellipse, dwindling
by slow but regular degrees. This is evidently the effect
which would be produced by a resistance experienced
by the comet from a very rare ethereal medium pervading
the regions in which it moves ; for such resistance, by
diminishing its actual velocity, would diminish also its
centrifugal force, and thus give the sun more power over
it to draw it nearer. Accordingly (no other mode of
accounting for the phenomenon in question appearing),
this is the solution proposed by Encke, and generally
received. It will, therefore, probably fall ultimately into
the sun, should it not first be dissipated altogether — a
thing no way improbable, when the lightness of its ma-
teriads is considered, and which seems authorized by the
observed fact of its having been less and less conspicuous
at each reappearance.

(484.) The other comet of short period which has
lately been discovered is that of Biela, so called from

292 A TREATISR ON ASTRONOMY. [CHAP. X.

M. Biela, of Josephstadt, who first arrived at this mte-
resting conclusion. It is identical with comets which
appeared in 1789, 1795, &c., and describes its mode-
rately eccentric ellipse about the sun in 6| years ; and
the last apparition having taken place according to the
prediction in 1832, the next will be in 1838. It is a
small insignificant comet, without a tail, or any appear-
ance of a solid nucleus whatever. . Its orbit, by a re-
markable coincidence, very nearly intersects that of the
earth ; and had the latter, at the time of its passage in
1832, been a month in advance of its actual place, it
would have passed through the comet— a singular ren-
contre, perhaps not unattended with danger.*

(485.) Comets in passing among and near the planets
are materially drawn aside from their courses, and in
some cases have their orbits entirely changed. This is
remarkably the case with Jupiter, which seems by some
strange fatality to be constantly in their way, and to
serve as a perpetual stumbling block to them. In the
case of the remarkable comet of 1770, which was found
by Lexell to revolve in a moderate ellipse in the period
of about 5 years, and whose return was predicted by him
accordingly, the prediction was disappointed by the comet
actually getting entangled among the satellites of Jupiter,
and being completely thrown out of its orbit by the at-
traction of that planet, and forced into a much larger el-
lipse. By this extraordinary rencontre, the motions of
the satellites suffered not the least perceptible derange-
ment — a sufficient proof of the smallness of the comet's
mass.

(486.) It remains to say a few words on the actual di-

* Should calculation establish the fact of a resistance experienced also
by this comet, the subject of periodical comets will assume an extreor*
dmary degree of interest. It cannot be doubted that many more will
be discovered, and by their resistance questions will come to be decided,
such as the following : — What is the law of density of the resisting medium
which surrounds the sun ? Is it at rest or in motion ? If the latter, in
what direction does it move ? Circularly round the sun, or traversing
space ? If circularly, in what plane ? It is obvious that a circular or
vorticose motion of the ether would acceUrafe some cfmett and retard
others according as their revolution was, relative to such motion, direct
or retrograde. Supposing* the neighbourhood of the sim to be filled with
a material fluid, it is not conceivable that die circulation of the planets
in it for ages should not have impressed upon it some degree of rotation
in tlieir own direction. And this may preserve them from the extreme
effects of accumulated rosistance. — AtUhor.

CHAP. X.] DIMENSIONS OF C01£ETS. 293

mensions of comets. The calculation of the diameters
of their heads, and the lengths and breadths of their
tails, offers not the slightest difficulty when once the
elements of their orbits are known, for by these we know
their real distances from the earth at any time, and the
true direction of the tail, which we see only foreshort-
ened. Now calculations instituted on these principles
lead to the surprising fact, that the comets are by far the
most voluminous bo£es in our system. The following
are the dimensions of some of those which have been
made the subjects of such inquiry.

(487.) The tail of the great comet of 1680, imme-
diately after its perihelion passage, was found by New-
ton to have been no less than 20000000 of leagues in
length, and to have occupied only, two days in its emis-
sion from the comet's body ! a decisive proof this of its
being darted forth by some active force, the origin of
which, to judge from the direqtion of the tail, must be
sought in the sun itself. Its greatest length amounted
to 41000000 leagues, a length much exceeding the
whole interval between the sun and earth. The tail of
the comet of 1769 extended 16000000 leagues, and that
of the great comet of 1811, 36000000. The portion of
the head of this last comprised within the transparent
atmospheric envelope which separated it from the tail
was 180000 leagues in diameter. It is hardly conceiv-
able that matter once projected to such enormous dis-
tances should ever be collected again by the feeble at-
traction of such a body as a comet— a consideration
which accounts for the rapid progressive diminution of
the tails of such as have been frequently observed.

(488.) A singular circumstance has been remarked
respecting the change of dimensions of the comet of
Encke in its progress to and retreat from the sun : viz.
that the real diameter of the visible nebulosity under-
goes a rapid contraction as it approaches, and an equally
rapid dilatation as it recedes from the sun. M. Valz»
who, among others, had noticed this fact, has accounted
for.it by supposing a real compression or condensation
of volume, owing to the pressure of an ethereal medium
growing more dense in the sun's neighbourhood. It is
very possible, however, that the change may consist in
2b2

294 A TREATISE ON ASTRONOMY. [cHAP. XI«

no real expansion or condensation of volume (further
than is due to the convergence or divergence of the dif-
ferent parabolas described by each of its molecules to or
from a common vertex), but may rather indicate the al-
ternate conversion of evaporable materials in the upper
regions of a transparent atmosphere, into the states of
visible cloud and invisible gas, by the mere ejffects of
heat and cold. But it is time to quit a subject so myste-
rious, and open to such endless speculation.

CHAPTER XI.

OF ■ PERTURBATIONS.

propounded — Saperpoeition of gmali Motions — ^Problem of three
Bodies — Estimation of disturbing Forces — Motion of Nodes — Changes
of Inclination— Compensation ooerated in a whole Revolutioa of me
Node — ^Lagranffe's Theorem or the Stability of the Inclinations —
Change of Obliquity of the Ecliptic — Precession of the Equinokes —
Nutation — ^Theorem respectinff forced Vifomtions— Of the Tides — ^Va-
riation of Elements of the Puinefs Orbite — Periodic and secular-
Disturbing Forces considered as tangential and radial — ESecta of tan-
gential Force — 1st, in circular Orbits ; 2dly, in elliptic^— CompepsatioDB
efiected— Case of near Commensurability of mean Motions— ^The great
Inequality of Jupiter and Saturn explamed— The long Ineauality of
Venus and the Earth — Lunar Variation — Eflfects of the radial Force —
Mean Effect on the Period and Dimensions of the disturbed Orbit—
Variable Part of its Effect — ^Lunar Evection — Secular Acceleration
of the Moon's Motion— Invariability of the Axes and Period»— Theory
of the secular Variations of the Eccentricities and Perihelia — ^Motkm
of the lunar Apside^^Lagrange's Theorem of the Stability of the
Eccentricities— Nutation of the lunar Orbit— Perturbations of Jupi-
ter's Satellites.

(489.) In the progress of this work, we have more
than once called the reader's attention to the existence
of inequalities in the lunar and planetary, motions not
included in the expression of Kepler's laws, but in some
sort supplementary to them, and of an order so far sub-
ordinate to those leading features of the celestial move-
ments, as to require, for their detection, nicer obrerva-
tions, and longer continued comparison between facts
and theories, than suffice for the establishment and veri-
fication of the elliptic theory. These inequalities are
known, in physical astronomy, by the name of pertur*

OHAP. XI.3 OF PERTURBATIONS. 295

bations. They arise, in the case of the primary planets,
from the mutual gravitations of these planets towards
each other, which derange their elliptic motions round
the sun ; and in that of the secondaries, partly from
the mutual gravitation of the secondaries of the same
system similarly deranging their elliptic motions round
their common primary, and partly from the unequal
attraction of the sun on them and on their primary.
These perturbations, although small, and, in most in-
stances, insensible in short intervals of time, yet, when
accumulated, as some of them may become, in the lapse
of ages, alter very greatly the original elliptic relations,
so as to render the same elements of the planetary
orbits, which at one epoch represented perfectly well
their movements, inadequate and unsatisfactory afler
long intervals of time.

(490.) When Newton first reasoned his way from
the broad features of the celestial motions, up to the
law of universal gravitation, as affecting all matter, and
rendering every particle in the universe subject to tho
influence of every other, he was not unaware of the
modifications which this generalization would induce
into the results of a more partial and limited application
of the same law to the revolutions of the planets about
the sun, and the satellites about their primaries, as their^
only centres of attraction. So far from it, that his ex-'
traordinary sagacity enabled him to perceive very dis-
tinctly how several of the most important of the lunar
inequalities take their origin, in this more general way
of conceiving the agency of the attractive power, espe-
cially the retrograde motion of the nodes, and the direct
revolution of the apsides of her orbit. And if he did
not extend his investigations to the mutual perturbations
of the planets, it was not for want of perceiving that such
perturbations must exist, and might go the length of
producing great derangements from the actual state of
the system, but owing to the then undeveloped state of
the practical part of astronomy, which had not yet at-
tained the precision requisite to make such an attempt
inviting, or indeed feasible. What Newton left undone,
however, his successors have accomplished; and, at
this day, there is not a single perturbation, great or small.

906 A TREATISE ON ASTRONOMY. [ciIAF. XI.

which observation has ever detected, which has not
been traced up to its origin in the mutual gravitation of
the parts of our system, and been minutely accounted
for, in its numerical amount and value, by strict calcula-
tion on Newton's principles.

(491.) Calculations of this nature require a very high
analysis for their successful performance, such as is far
beyond the scope and object of this work to attempt ex-
hibiting. The reader who would master them must
prepare himself for the undertaking by an extensive
course of preparatory study, and must ascend by steps
which we must not here even digress to point out. It will
be our object, in this chapter, however, to give some
general insight into the nature and manner of operation
of the acting forces, and to point out what are the cir-
cumstances which, in some cases, give them a high de-
gree of efficiency — a sort of purchase on the balance of
the system ; while, in others, with no less amount of
intensity, their effective agency in producing extensive
and lasting changes is compensated or rendered abortive ;
as well as to explain the nature of those admirable re-
sults respecting the stability of our system, to which the
researches of geometers have conducted them; and
which, under the form of mathematical theorems of great
beauty, simplicity, and elegance, involve the history of
the past and future state of the planetary orbits during
ages, of which, contemplating the subject in this point
of view, we neither perceive the beginning nor the
end.

(492.) Were there no other bodies in the universe but
the sun and one planet, the latter would describe an
exact ellipse about the former (or both round their com-
mon centres of gravity), and continue to perform its revo-
lutions in one and the same orbit for ever; but the
moment we add to our combination a third body, the at-
traction of this will draw both the former bodies out of
their mutual orbits, and, by acting on them unequally,
will disturb their relation to each other, and put an end
to the rigorous and mathematical exactness of their ellip-
tic motions, either about one another or about a fixed point
in space. From this way of propounding the subject,
we see that it is not the whole attraction of the newly in-

CHAP. XI.] OF PERTURBATIONS. 8W

troduced body which produces perturbation, but the dif-
Jtrence of its attractions on the two originally present.

(493.) Compared to the sun, all the planets are of ez«
ireme minuteness ; the m^ss of Jupiter, the greatest of
them all, being not more than one 1300th part that of the
8iin. Their attractions on each other, therefore, are all
very feeble, compared with the presiding central power,
and the effects of their disturbing forces are proportionally
minute, in the case of the secondaries, the chief agent
by which their motions are deranged is the sun itself,
whose mass is indeed great, but whose disturbing influ-
ence is immensely diminished by their near proximity to
their primaries, compared to their distances from the sun,
which renders the difference of attractions on both ex-
tremely smaU, compared to the whole amount. In this
case, the greatest part of the sun's attraction, viz. that
which is common to both, is exerted to retain both pri-
mary and secondary in their common orbit about itself,
and prevent their parting company. The small overplus
of force only acts as a disturbing power. The mean
value of this overplus, in the case of the moon disturbed
by the sun, is calculated by Newton to amount to no
higher a fraction than fSTZvu ^^ gravity at the earth's
surface, or y^^ of the principal force which retains the
moon in its orbit.

(494.) From this extreme minuteness of the intensities
of the disturbing, compared to the principal forces, and
the consequent smallness of their momentary effects, it
happens that we can estimate each of these effects sepa-
rately, as if the others did not take place, without fear
of inducing error in our conclusions beyond the limits
necessarily incident to a first approximation. It is a
principle in mechanics, immediately flowing from the
primary relations between forces and the motions they
produce, that when a number of very minute forces act
at once on a system, their joint effect is the sum or ag-
gregate of their separate effects, at least within such limits,
that the original relation of the parts of the system shall
not have been materially changed by their action. Such
effects supervening on the greater movements due to the
action of the primary forces may be compared to the
small ripplings caused by a thousand varying breezes on

298 A TREATISE ON ASTRONOMY. [CHAP. XI.

the broad and regular swell of a deep and rolling ocean,
which run on as if the surface were a plane, and cros6 in
all directions, withont interfering, each as if the other had
no existence. It is only when their effects become accu-
mulated in lapse of time, so as to alter the primary rela-
tions or data of the system that it becomes necessary to
have especial regard to 'the changes correspondingly in-
troduced into the estimation of their momentary efficiency,
by which the rate of the subsequent changes is affected,
and periods or cycles of immense length take their origin.
From this consideration arise some of the most curious
theories of physical astronomy.

(495.) Hence it is evident, that in estimating the dis-
turbing influence of several bodies forming a system, in
which one has a remarkable preponderance over all the
rest, we need not embarrass ourselves with combinations
of the disturbing powers one among another, unless where
immensely long periods are concerned ; such as consist
of many thousands of revolutions of the bodies in ques-
tion about their common centres. So that, in effect, the
problem of the investigation of the perturbations of a
system, however numerous, constituted as ours is, reduces
itself to that of a system of three bodies : a predominant
central body, a disturbing, and a disturbed ; the two lat-
ter of which may exchange denominations, according as
the motions of the one or the other are the subject of
inquiry.

(496.) The intensity of the disturbing force is conti-
nually varying, according to the relative situation of the
disturbing and disturbed body with respect to the sun. If
the attraction of the disturbing body M, on the central
body S, and the disturbed body P (by which designa-
tions, for brevity, we shall hereafter indicate them), were
equal, and acted in parallel lines, whatever might other-
wise be its law of variation, there would be no deviation
caused in the elliptic motion of P about S, or of each
about the other. The case would be strictly that of art.
385 ; the attraction of M, so circumstanced, being at
every moment exactly analogous in its effects to terres-
trial gravity, which acts in parallel lines, and is equally
intense on all bodies, great and small. But this is not
the case of nature. Whatever is stated in the subsequent

€TEAF. XI.]] PROBLEM OF THREE BODIES. 299

article to that last cited, of the disturbing effect of the
8iin and moon, is, mutatis mutandis, applicable to every
case of perturbation ; and it must be now our business to
enter, somewhat more in detail, into the general heads of
the subject there merely hinted at.

(497.) We shall begin with that part of the disturbing
force which tends to draw the disturbed body out of the
plane in which its orbit would be performed if undisturb-
ed, and, by so doing, causes it to describe a curve, of
Avhich no two adjacent portions lie in one plane, or, as it
is called in geometry, a curve of double curvature. Sup-
pose, then, APN to be the orbit which P would describe
about S, if undisturbed, and suppose it to arrive at P, at
any instant of time, and to be about to describe in the
next instant the undisturbed arc Pp, which, prolonged in
the direction of its tangent PjpR, will intersect the plane
of the orbit ML of the disturbing body, somewhere in the
line of nodes SL, suppose in R. This would be the case
if M exerted no disturbing power. But suppose it to do
SO5 then, since it draws both S and P towards it, in direc-
tions not coincident with the plane of P's orbit, it will
cause them bothj in the next instant of time^ to quit that
plane, but unequally : — first, because it does not draw
them both in parallel lines ; secondly, because they, being
unequally distant from M, are unequally attracted by it,
by reason of the general law of gravitation. Now, it is
by the difference of the motions thus generated that the
relative orbit of P about S is changed ; so that, if we
continue to refer its motion to S as a fixed centre, the dis-
turbing part of the impulse which it receives from M will
impel it to deviate from the plane PSN, and describe in
the next instant of time, not the arc Pp, but an arc P^,
lying either above or below Pp, according to the prepon-
derance of the forces exerted by M on P and S.

(498.) The disturbing force acts in the plane of the tri-
angle SPM, and may be considered as resolved into two ;
one of which urges P to or from S, or along the line SP,
and, therefore, increases or diminishes, in so far^is it is
effective, the direct attraction of S or P ; the other along
aline PK, parallel to SM, and which may be regarded as
either pulling P in the direction PK, or pushing it in a
contrary direction ; these terms being well understood to

300 A TREATISE ON ASTRONOMY. £cHAP. XI*

have only a relative meaning as referring to a supposed
fixity of S, and transfer of the whole effective power to P. .
The former of these forces,
acting always in the plane of
P's motion, cannot tend to
urge it out of that plane : the
latter only is so effective, and
/ that not wholly ; another reso^
lution of forces heing needed
to estimate its effective part.
But with this we shall not
concern ourselves, the object here proposed being only
to explain the manner in which the motion of the nodes
arises, and not to estimate its amount.

(499.) In the situation, or configuration, as it is termed^
represented in the figure, the force, in the direction PK,
is A^piUling force ; and as PK, being parallel to SM,lie»
below the plane of P's orbit (taking that of M's orbit for
a ground plane), it is clear that the disturbed arc P^, de-
scribed in the next moment by P, must lie below Pp*
When prolonged, therefore, to intersect the plane of M's
orbit, it will*meet it in a point r, behind R, and the line
Sr, which will be the line of intersection of the plane
SPg (now, for an instant, that of P's disturbed motion)^
or its new line of nodes, will fall behind SR, the undis-
turbed line of nodes ; that is to say, the line of nodes
will have retrograded by the angle RSr, the motions of
P and M being regarded as direct.

(500.) Suppose, now, M to lie to the left instead of the
right of the line of nodes, P retaining its situation, then
will the disturbing force, in the direction PK, tend to raise
P out of its orbit, to throw Fq above Pp, and r in advance
of R. In this configuration, tiien, tiie node will advance $
but so soon as P passes the node, and comes to the lower
side of M's orbit, although the same disposition of. the
forces will subsist, and Fq will, in consequence, continue
to lie above Pp, yet, in this case, the little arc P^ will
have to be prolonged backwards to meet our ground
plane, and, when so prolonged, will lie below the similar
prolongation of Pp, so that, in this case again, the node
will retrograde.
(501.) Thus we see that the effect of the disturbing

CHAP. XI.] MOTION GF THE N0DS8. 801

force, in the different states of configuration which the
^ bodies P and M may assume with respect to the node, is
to keep the line of nodes in a continual state of iluctna*
tion to and fro ; and it will depend on the excess of cases
favourable to its advance over those which favour its re-
cess, in an average of all the possible configurations,
whether, on the whole, an advance or recess of the node
shall take place.

(502.) If the orbit of M be very large compared with
that of P, so large that MP may, without material error,
be regarded as parallel to MS, which is the case with the
moon's orbit disturbed by the sun, it will be very readily
seen, on an examination of all the possible varieties of
configuration, and having due regard to the direction of
^e disturbing force, that during every single c>omplete
revolution of P, the cases favourable to a retrograde mo-
tion of the node preponderate over those of a contraury
tendency, the retrOgradation taking place over a largef
extent of the whole orbit, and being at the same time
more rapid, owing to a more intense and favourable action
of the force than the recess. Hence it follows that, oa
the whok^ during every revolution of the moon about the
earth, the nodes of her orbit recede on the ecliptic, con-
formable to experience, with a velocity varying from lu<r
nation to lunation. The amount of this retrogradation,
when calculated, as it may be, by an exact estimation of
all the acting forces, is found to coincide with perfect
precision wiSi that immediately derived from observation,
so that not a doubt can subsist as to this being the real
process by which so .remarkable an effect is produced.

(603.) Theoretically speaking, we cannot estimate
correctly the recess of the intersection of the moon's
orbit with the ecliptic, from a mere consideration of the
disturbance of one of these planes. It is a compound
phenomenon ; both planes are in motion with respect to
an imaginary fixed ecliptic, and, to obtain the compound
effect, we must also regard the earth as disturbed in its
relative orbit about the sun by the moon. But, on ac-
count of the excessive distance of the sun, the intensity
of the moon's attraction an U is quite evanescent, com-
pared witii its attraction on the earth : so that the per^
iurbative effect in this case, which is the difference of
2C

302 A TREATISE ON ASTRONOMY. [cHAP. XI,

the moon's attraction on the sun and earth, is equal to
the whole attraction of the moon on the earth. The ef-
. feet of this is to produce a monthly displacement of th©
centre on either side of the ecliptic, whose amount 19
easily calculated hy regarding their common centre of
gravity as lying strictly in the ecliptic. From this it ap-
pears, that the displacement in question cannot exceed a
small fraction of the earth's radius in its whole amount ;
and, therefore, that its momentary variation, on which the
motion of ^e node of the ecliptic on ^e moon's orbit
depends, must be utterly insensible.

(604.) It is otherwise with the mutual action of the
planets. In this case, both the orbits of the disturbed
and disturbing planet must be regarded as in motion.
Precisely on the above-stated principles it may be shown,
tiiat the effect of each planet's attraction on the orbit of
every other, is to cause a retrogradation of the node of
the one orbit on the other in certain configurations, and a
recess in others, terminating, like that of the moon, on
the average of many revolutions in a regular retrograda-
tion of the node of each orbit on every other. But since
this is the case with every pair into which the planets can
be combined, the motion ultimately arising from their
joint action on any one orbit, taking into the account the
different situations of all their planes, becomes a singu-
lar and complicated phenomenon, whose law cannot be
very easily expressed in words, though reducible to strict
numerical statement, and being in fact a mere geometri-
cal result of what is above stated.

(605.) The nodes of all the planetary orbits on ihetrtie
ecliptic then are retrograde, although (which is a most
material circumstance) they are' not all so on a fixed
plane, such as we may conceive to exist in the planetary
system, and to be a plane of reference unaffected by their
mutual disturbances. It is, however, to the ecliptic, that
we are under the necessity of referring their movements
from our station in the system ; and if we would transfer
our ideas to a fixed plane, it becomes necessary to take
account of the variation of the ecliptic itself, produced
by the joint action of all the planets.

(506.) Owing to the smallness of the masses of the
planets, and their great distances from each other, the re-

CHAP. XI.] CHANGE OF INCLINATIONS. 309

volutions of their nodes are excessively slow, being in
every case less than a single degree per century, and in
most cases not amounting to half that quantity. So far
as the physical condition of each planet is concerned, it
is evident that the position of their nodes can be of little
importance. It is otherwise with the mutual inclinationi^
of their orbits, with respect to each other, and to the
equator of each. A variation in the position of the eclip-
tic, for instance, by which its pole should shift its dis-
tance from the pole of the equator, would disturb our sea-
sons. Should the plane of the earth's orbit,'for instance,
ever be so changed as to bring the ecliptic^ to coincide
with the equator, we should have perpetual spring over
all the world ; and, on the other hand, should it coincide
with a meridian, the extremeis of summer and winter
would become intolerable. The inquiry, then, of the
variations of inclination of the planetary orbits inter «e,
is One of much higher practical interest than those of
their nodes.

(507.) Referring to the figure of art. 498^ ijt is evident
that the plane SPq, in which the disturbed body moves
' during an instant of time from its quitting P, is differently
inclined to the orbiC of M, or to a fixed plane, from the
original or undisturbed plane PSp. The difierence of
absolute position of these two planes in space is the an-
gle made between the planes PSR and PSr, and is there-
fore calculable by spherical trigonometry, when the angle
RSr or the momentary recess of the node is known, and
also the inclination of the planes of the orbits to each
other. We perceive, then, that between the momentary
change of inclination and the momentary recess of the
node there exists an intimate relation, and that the re-
search of the one is in fact bound up in that of the other.
This may be, perhaps, made clearer, by considering the
orbit of M to be not merely an imaginary line, but an
actual circular or elliptic hoop of some rigid material,
withput inertia, on which, as on a wire, the body P may
slide as a bead. It is evident that the position of this
hoop will be determined at any instant, by its inclinatioA
to the ground plane to which it is referred, and by the
place of its intersection therewith, or node. It will also
be determined by the momentary direction of P's motion.

M4 A tueatub on astronoht. [cbap.zi*

which (hating no inertia) it must ohey ; and any change
by which P should, in ^e next instant, alter its oibit,
would be equivalent to a shifting, bodily, of the whole
hoop, changing at once its inclination and nodes.

(508.) One immediate conclusion frpm what has been
pointed out above, is that where the orbits, as in the case
of the planetary system and the moon, are slightly in-
clined to one another, the momentary variations of the
inclination are of an order much inferior in magnitude to
those in the place of the node. This is evident on a
mere inspection of our figure, the angle RPr, being by
reason of the small inclination of the planes SPR and
RSr, necessarily much smaller than the angle RSr. In
proportion as the planes of the orbits are brought to coin-
cidence, a very trifling angular movement of Tp about PS
as an axis will make a great variation in the situation of the
point r, where its prolongation intersects the ground plane.
(509.) To pass from the momentary changes which
* take place in the relations of nature to the accumulated
effects produced in considerable lapses of time by the
continued action of the same causes, under circumstances
varied by these very effects, is the business of the integral
calculus. Without going into any calculations, however^
it will be easy for us to trace, by a few cases, the varying
influence of differences of position of the disturbing' and
disturbed body with respect to each other and to the node,
and from theise to demonstrate the two leading features
in this theory — the periodic nature of the change and
re-establishment of the original inclinations, and the
small limits within which these changes are confined.

(510.) Case 1.— When the disturbing body M is sito-
ated in a direction perpendieulac to the line of nodes, or

OBA£. XI.] MOTION OF THE NOBES.

doft

the nodes are in quadrature with it : M being the dis-
turbing body, and SN the line of nodes, the disturbing
force will act at P, in the direction PK ; being a pulling
force when P is in any part of the semicircle HAN, and
a pushing force in the whole of the opposite semicircle.
And it is easily seen that this force is greatest at A and
B, and evanescent at H and N. Hence, in the whole
semicircle HA, Fq will lie below Pp, and being pro-
duced backwards in the quadrant HA, and forwards in
AN, will meet the circle S6N« in the plane of M's
orbit, in points behind the nodes SN, the nodes being
retrograde in both cases. But the new inclination of
the disturbed orbit is, in the former case, ParA, which
is. less than PHa; and in the latter, Jya, which is
greater than PNa. In the other semicircle the direction
of the disturbing force is changed ; but that of the motion,
with respect to the plane of M's orbit, being also in
each quadrant reversed, the same variations of node and
inclination will be caused. In this situation of M, then,
the nodes recede during every part of the revolution of
P, biit the inclination diminishes throughout the quadrant
SA, increases again by the same identical degrees in the
quadrant AN, decreases throughout the quadrant N6,
and is finally restored to its pristine value at S. On the
average of a revolution of P, supposing M unmoved, the
nodes will have retrograded with their utmost speed, but
the inclination will remain unaltered.

(611.) Case 2.— Suppose the disturbing body now to
be fixed in the line of nodes, or the nodes to be in
syzygy, as in the annexed figure. In this situation the
direction of the disturbing force, which is always parallel
to SM, lies constandy in 3ie plane of P's orbit, and there-
2c2

t06 A TKSATI8S ON ASTRONOMY. [cHAP* Zl»

fore prodilees neither variation of inclination or motioii
of nodes.

(612.J Case 3. — Let us take now an intermediate
situation of M, and indicating by the arrows the directions
of the disturbing forces (which are pulling ones through-^
out all the semi-orbit Vhich lies towards M, and ptuhing
in the opposite), it will readily s^pear that the reasoning
of art. 510, will hold good in all that part of the orbit
which lies between T and N, and between V and H»
but that the effeet will be reversed by the reversal of the
direction of the motion with respect to the plane of M's
orbit, in the intervals HT and NV. In these portions,
however, the disturbing force is feebler than in the others,
being evanescent in the line fif quadraJturts TY, and in*
T

creasing to its maximum in the syzygies a b. The nodes
liien will recede rapidly in the former intervals, and ad-
vance feebly in the latter ; but since, as H approaches to
a, the disturbing force, by acting obliquely to the plane
of P's orbit, is again diminished in efficacy, still, pn the
average of a whole revolution, the nodes recede* On
the otiier hand, the inclination will now diminish during
the motion of P frtwn T to c, a poii^t 90° distant from
the node, while it increases not only during its whole
motion over the quadrant cN, but also in the rest of its
half revolution NV, and so for the other half. There
will, therefore, be an uncompensated increase of inclina-
tion in this position of M, on the average of a whole
revolution.

(513.) But this increase is converted into diminution
when the line of nodes stands on the other side of SM,
or in the quadrants Y^, Ta ; and still regarding M as
^xed« and supposing that the change of circumstances

ettAP. n.] OHAKQXS OF INCLINATIOH. .' Wf

arises not from the motion of M bat from that of the
node, it is evident that so soon as the line of nodes m
its retrograde motion has got past a, the circumstances
will be all exactly reversed, and the inclination will again
be augmented in each revolution by the very same steps ,
taken in reverse order by which it before diipinished.
On the average, therefore, of a- whole revolution of
THE NODE, the inclination- will be restored to its original
state. In fact, so far as the mean or average effect on
the inclination is concerned, instead of supposing M
&:ed in one position, we might conceive it at every in-
Btant divided into four equsd j)arts, and placed at equal
angles on either side of the line of nodes, in which case
it is evident that the effect of two of the |)arts would be
to precisely annihilate that of the others in each revo*
lution of P.

(514.) In what is said, we have supposed M at rest;
but the same conclusion, as to the mean and final results,
holds good if it be supposed in motion ; for in the
course of a revolution of ttie nodes, which, owing to the
extreme smallness of their motion, in the case of th0
planets, is of immense length, amounting, in most cases,
to several hundred centuries, and in that of the moon
is not less than 237 lunations, the disturbing body M
is presented by Ua own moHonj over and over again, in
every variety of situation to the line of nodes. Before
the node can have materially changed its position, M has
performed a complete revolution, and is restored to its
place ; so that, in fact (that small difference excepted
which arises from the recess of the node in one syno*
dical revolution of M)^ we may regard it as occupying at
every instant every point of its orbit, or rather as having
its mass distributed uniformly like a solid ring over its
whole circumference. Thus the compensation which
we have shown would take place in a whole revolution
of the node, does, in fact, take place in every synodic
period of M, that minute difference only excepted which
is due to the cause just mentioned. This difference,
then, and not the whole disturbing effect of M, is what
produces the effective variation of the inclinations, whe-
ther of the lunar or planetary oibits ; and this difference,
which remains uncompensated by the motion of M, is in

80^ A TREATI6E ON ASTRONOMY. [cHAP* XU

its turn coiqpensated by the motioB of the node during
its whole. revolution.

• (615.) It is clear, therefore, that the total variation of
the planetary inclinations must be comprised within very
narrow limits indeed. Geometers have accordingly de-
monstrated, by an accurate analysis of all the circum-
stances, and an exact estimation of the acting forces,
that such is the case ; and this is what Is meant by as-
serting the stability of the planetary system as to the
mutual inclinations of its orbits. By the rosearches of
Lagrange (of whose analytical conduct it* is impossible
here to give'any idea), thj following elegant theorem has
been demonstrated : —

*yjf the mdss of*evety planet be 'multiplied by the
square root of the major axis of its orbit, and thepro^
Met by the square iff the tangent of its inclination to a
fixed pUme, the sum of tfi^ihese products will be con-
stantly the same under the influence of their mutual at-
traction.^^ If the present situation of the plane of the
ecliptic be taken for that fixed plane (the ecliptic itself
being variable like the other orbits), it is found that this
sum is actually very small ; it must, therefore, always
remain so. This remarkable theorem alone, then, would
guarantee the stability of the orbits of the greater planets ;
but from what has above been shown, of the tendency of
each planet to work out a compensation on every other,
it is evident that the minor ones are not excluded from
this beneficial arrangement.

(516.) Meanwhile, there is no doubt that the plane
of the ecliptic does actually vary by the actions of the
planets. The amount of this variation is about 48" per
century, and has long been recognised by astronomers,
by an increase of the latitudes of all the stars in certain
situations, and their diminution in the opposite regions.
Its effect is to bring the ecliptic by so much per annum
nearer to coincidence with the equator; but from what
we have above seen, Chis diminution of the obliquity of
the ecliptic will not go on beyond certain very moderate
limits, after which (although in an immense period of
ages, being a compound cycle resulting from the joint
action of all the planets) it will again increase, and thus
^cillate backward and forward about a mean position*

CHAP. XI.] PRECESSION OF THE EQ.iriif70XE«f. 809 *

the extent of its deviation to one side and the other being
less than V 21'.

(517.) One effect of this variation of the plane of the
ecliptic, that which causes its nodes cm a fixed plaim
to change — is mixed up with the precession of the
equinoxes (art. 261), and undistinguishable from it, ex-
cept in theory.. This last-mentioned phenomenon is,
however, due |o another cause, analogoiis, it is true, in a
general point of view to those above considered, but
singularly modified by the ciicumstances under which it
IB produced. We shall endeavour to render these modi-
fications intelligible, as far as they can ba made so, with-
out the intervention of analytical formulae.

(518.) The. precession of the equihoxesj as we have
shown in art. 266, consists in a continual retrograda-
lion of the node of the earth's equator on the ecliptic,
and is,*therefore, obviously an effect so far analogous to
the general phenomenon of the retrogradation of the
nodes of the orbits on each other. The immense dis-
tance of the planets, however, compared with the size
of the earth, and the smallness of their masses com-
pared to that of the sun, puts their action out of the
question in the inquiry of its cause, and we must,
llierefore, look to the massive though distant sun, and
to our near though minute neighbour, the moon, for its
explanation. This will, accordingly, be found in their
disturbing action on the redundant matter accumulated
on the equator of the earth, by which its figure is ren-
dered spheroidal, combined with .the earth's rotation on
its axis. It is to the sagacity of Newton that we owe
the discovery of this singular mode of action.

(519.) Suppose in our figures (arts. 509, 510, 511)»
that instead of one body, P, revolving round S, there
were a succession of particles not coherent, but forming
a kind of fluid ring, free to change its form by any force
applied. Then, while this ring revolved round S in its
own plane, under the disturbing influence of the distant
body M (which now represents the moon or the sun,
as P does one of the particles of the earth's equator),
two things would happen : — 1st, Its figure would be
bent out of a plane into an undulated form, those parts
of it within the arcs Ve* and Td {fig. art. 511) being

310 A tREAtll^E ON ASTRONOMir. [cHAP. ZI.

rendered more inclined to the plane of M's orbit, and
those within the arcs cT, rfV, less so that they would
otherwise be. 2dly, the nodes of this ring, regarded as
a- whole, without respect to its change of figure, would
retreat upon that plane.

(520.) But suppose this ring, instead of consisting
of discrete molecules free to move independently, to be
rigid and incapable of such flexure, like the hoop we
. have supposed in art, 507, then it is evident that the
«ffort of those parts of it which tend to become more
inclined will act through the medium of the ring itself
(as a mechanical engine or lever) to counteract the
effort of those which have at the same instant a contrary
tendency. In so far only, then, as there exists an excess
on the one or the other side will .th^ inclination change,
an average being struck at every momelit of the ring's
motion ; just as was shown to happen in the view we
have taken of the inclinations, in every complete revolu-
tion of a single disturbed body, under the influence of a
fixed disturbing one.

(521.) Meanwhile, however, the nodes of the rigid
ring will retrograde., the general or average tendency of
the nodes of every molecule being to do so. Here, as
in the other case, a struggle will take place by the coun-
teracting efforts of the molecules contrarily disposed,
propagated through the solid substance of the ring ; and
thus, at every instant of time, an average Will be struck,
which average being identical in its nature with that ef-
fected in the complete revolution of a single disturbed
body, will', in every case, be in favour of a recess of the
node, save only when the disturbing body, be it sun or
moon, is situated in the plane of the earth's equator, or
in the case of the Jig, art. 510.

(522.) This reasoning is evidently independent of any
consideration of the cause which maintains the rotation
of the ring ; whether the particles be small satellites re-
tained in circular orbits under the equilibrated action of
attractive and centrifugal forces, or whether they be small
masses conceived as attached to a set of imaginary spokes
as of a wheel, centering in S, and free only to shift their
planes by a motion Of those spokes perpendicular to the
plane of* the wheel. This makes no diflference in the

OHAF* Xt,2 PRECESSION OF THB EQUINOXES. 91 1

general effect ; though the different velocities of rotation^
which may be impressed on such a system, may and
"will have a very gireat influence both on the absolute and
relative magnitudes of the two effects in question — the
motion of the nodes and change of inclination. This
will be easily understood, if we suppose the ring without
a rotatory motibn, in which extreme case it is obvious,
that so long as M remained fixed there would take place
no recess of nodes at all, but only a tendency of the ring
to tilt its plane round a diameter perpendicular to the
position of M, bringing it towards the line SM.

(523.) The motion of such a ring, then, as we have
been considering, would imitate, so far as the recess of
the nodes goes, the precession of the equinoxes, only that
its nodes would retrograde far more rapidly than the ob-
served precession, which is excessively slow. But now
conceive this ring to be loaded with a spherical mass
eclormously heavier than itself, placed concentrically
within it, and cohering firmly to* it, but indifferent, or very
nearly so, to any such cause of motion ; and suppose,
moreover, that instead of one such ring, there are a vast
multitude heaped together around the equatoir of such a
globe, so as to form an elliptical protuberance, enveloping
it like a shell on all sides, but whose mass, taken togethei,
should form but a very minute fraction of the whole
spheroid. We have now before us a tolera&le repre-
sentation of case of nature;* and it is evident that the
rings, having to drag round with them in their nodal re-
volution this great inert mass, will have their velocity of
retrogradation proportionally diminished. Thus, then, it
is easy to conceiye how a motion, similar to the preces-

* That a perfect sphere would be so inert and indifierent as to a revo-
lution of the nodes of its equator under the influence of a distant attract-
ing body appears from this — ^that the direction of. the resultant attraction
ofsuch a body, or of that single force which, opposed, would neutralize
and destroy its whole action, is necessarily in a Ime passing through the
centre of tne sphere, and, therefore, can nave no tendency to turn the
sphere one way or other. It may be objected bv the reader, that the
whole sphere may be conceived as consisting ot rings parallel to its
equator, of every possible diameter, and that, therefore, its nodes should
retrograde even without a protuberant equator. The inference is in-
correct, hot our limits will not allow us to go into an exposition of the
fiillacy. We should, however, caution him, generally, that no djmamical
subject is open to more mistakes of this kind, which nothing but the
ckmMt attention, in every varied point of view, will detect.

ai9^

A TREATISE ON ASTRONOMY. [CHAP. XI.

sion of the equinoxes, and, like it, characterized by ex*
treme slowness, will arise from the causes in action.

(^24.) Now a recess of the node of the earth's equar
tor, upon a given plane, corresponds to a conical motion
of its axis round a perpendicular to that pla^e. But in the
case before us, that plane is not the ecliptic, but the moon's
orbit for the time being ; and it may be asked how we
are to reconcile this with what is. stated in art. 266, re-
specting the nature of the motion in question. To this
we reply, that the nodes of the lunar orbit, being in a state
of continual and rapid retrogradation, while its inclination
is preserved nearly invariable^ the point in the sphere of
the heavens round which the pole of the earth's axis re-
volves (with that extreme slowness characteristic of the
precession) is itself in a state of continual circulation
round the pole of the ecliptic, with that much more rapid
motion which belongs to the lunar
node. A glance at the annexed
figure will explain this better than
words. P is the pole of the eclip-
tic, A the pole of Uie .moon's orbit,
moving round the small circle
ABGD in 19 years ; a the pole of
the earth's equator, which at each
moment of its progress has a cKree*
Hon perpendicular to the varying
position of the line Aa, and a vefo-
city depending on the varying in-
tensity of the acting causes during
the period of the nodes. This ve-
locity, however, being extremely small, when A comes
to B, G, D, E, the line Aa wiU have taken up the positions
B6, Cc, D/f, Ge, and the earth's pole a will thus, in one
tropical revolution of the node, have arrived at e, having
described not an exactly circular arc, but a single undu-
lation of a wave-shaped or epicycloidal curve, ab cdCf
with a velocity alternately greater and less than its mean
motion, and this will be repeated in every succeeding
revolution of the node.

(526.) Now this is precisely the kind of motion whidiy
as we have seen in art. 272, the pole of the earth's equa-
tor really has round the pole of the ecliptic, in coose

% cd

GHAP. XI.3 NTTTATION* 812^

qaence of the joint effects* of pTecession and nutation,
which are thus uranographically represented. If we
superadd to the effect of lunar precession that of the so-
lar, which alone would cause the pole to describe a circle
uniformly about P, this will only affect the nndnlatioiift
c^ our waved Curve, by extending thent in lengthf but
will produce no effect on the depth of ^ waves, or the
excursions of tlie earth's axis to and from the pole ef the
ecliptic. Thus we see that the two phenomena of nu-*
tation and precession are intimately conmected, or rather,
both of them essential constituent parts of one and the
same phenomenon. It is hardly necessary to state thai
a rigorous analysis of this great problemt by an exact es»
timation of all the acting forces and summation of their
dynamical effects,* leads to the precise value of the co-
efficients of precession and nutation, which observation
assigns, to them. The solar and lunar portions of the
precession of the equinoxes, that is to say, those portions
which are uniform, are to each other in the proportion
of about 2 to &. •

(526.) In the nutation of the earth's axis we have an
example (the first of its kind which 'has occurred to as)
of a periodical movement in one part of the system,
giving rise to a motion having the same precise period
in another. The motion of the moon's nodes is here,
we see, represented, though under a very different form,
yet in the same exact periodic time, by the movemeait
of a peculiar oscillatory kind impressed on the sdid
mass of the earth. We must not let the opportunity pass
of generalizing the principle involved in this result, as it
is one which we shadl find again and again exemplified in
every part of physical astronomy, nay, in every depart*
ment of natural science. It may.be stated as ** the prin*
ciple of forced oscillations, or of forced vibrations," and
thus' generally announced :—

Jf one part of any system connected either by mate"
rUu ties^ or by thp mutual attractions of Us members^
he continually maintained by any cause, whether inr
herent in the constitution of the system or external^ t^
itf in a state of regular periodic motion^ that motion
toill be propagated throughout the whole system^ and

* Vide F^. Airy'» MathMnatioal Tnu;H9d ad. p. 20Q» fte.
2D

814 A TREATISE ON ASTRONOMY. [cHAP. XI.

will give rite in every member of it, and in every part
of each member ^ to periodic movements executed in
equal periods toith that to which they owe their origiri,
though not necessarily synchronous with them in their
maxima and minima.*

The system may be farourably or unfavouiafely con-
stituted for such a transfer of periodic movements, or
favourably in some of its parts and unfavourably in
others ; and, accordingly as it is the one or the other,
the derivative oscillation (as it may be termed) will be
imperceptible in one case, of appreciable magnitude in
another, and even more perceptible in its visible effects
than the original cause, in a third; of this last kind we
have an instance in the moon's acceleration to be here-
after noticed.

(527.) It so happens that our situation on the ear&«
and the delicacy which our observations have attaioedy
enable us to make it, as it were, an instrument to feel these
forced vibrations*— these derivative motions, eonununi-
cated from various quarters, especially from our near
neighbour, the moon, much in the same way as we de-
tect, by the trembling of a board beneath us, th« secret
transfi^ of motion by which the sound of axT organ pipe
'is dispersed through the air, and carried do^n into the
earth. Accordingly, the monthly revolution of the moon,
and the annual motion of the sun, produce, each of them,
small nutations in the earth's axis, whose periods are
respectively half a month and half a year, each of which,
in this view of the subject, is to be regarded as otte por-
tion of a period consisting of two equal and similar parts.
But the most remarkable instance, by far, of diis propa-
gation of periods, and one of high importance to man-
kind, is that of the tides, which are iorced oscillations^
excited by the rotation of the earth in an ocean disturbed
from its figure by the varying attractions of the sun and
moon, each revolving in its own orbit, and propagating
its own period into the joint phenomenon.

(528.) The tides are a subject on which many persons

find a strange difficulty of conception. That the moon, by

* See a demonslmtioii of thi» theorem for the forced vibnUkni of tya^
terns connected by material ties of imperfect elasticity, in mv treatiae aa
Sound, Encyc. Metnm. art 323. The demonstration is easily extended
and generalised tp take in other wfut/trnt^Autkor.

CHAP. ^.] THS TIDES. 815

her attraction, should heap up the waters of the ^eaa
under her, seems to most persons very natural— that
the same cause should, at the same time, heap them up
on the opposite side, seems to many palpably absurd.
Yet nothing is more true, nor indeed more evident, when
we consider that it is not by her whole attraction, but by
the difierences of her attractions at the two surfaces and
at the centre that the waters are raised —that is to say,
by forces directed precisely as the arrows in our figure,
art. 510, in which we may suppose M the moon, and P
a particle of water on the earth's surface. A drop of
water existing alone would take a spherical form, by
reason of the attraction of its parts; and if .the same
drop were to fall freely in a vacuum under the influence
. of an uniform gravity, since every part would be equally
accelerated, the particles would retain their relative posi-
tions, and the spherical form be unchanged. But sup-
pose it to fall under the influence of an attraction acting
on each of its particles independently, and increasing
in intensity at every step of the descent, then the parts
nearer the centre of attraction would be attracted more
than the central, and the central than the more remote,
and the whole would be drawn out in the direction of the
motion into am oblong form ; the tendency to separation
being, however, counteracted by the attraction of the
particles on each other, and a form of equilibrium being
thus established. Now, in fact, the earth is constantly
falling to the moon, being continually drawn by it out
of its path, the nearer parts more and the remoter less
so than the central ; and tiius, at every instant, the moon's
attraction acts to force down the water at the sides, at
right angles to her direction, and raise it at the two ends
of the diameter pointing towards her. Geometry corro-
borates this view of the subject, and demonstrates that
the form of equilibrium assumed by a layer of water
covering a sphere, under the influence of the moon's at-
traction, would be an oblong ellipsoid, having the semi-
axis directed towards the moon longer by about 68 inches
than that transverse to it.

(529.) There is never time, however, for this spheroid
to be fully formed. Before the waters can take their
level, Uie moon has adiranced in her orbit, both diurnal

S16 A TREATISE ON AfllftONOMY. [CHAP« S.

^ttid mondily (for in this theory it will answer the pur-
pose of clearness better if we suppose the earth's diurnal
motion transferred to the sun and moon in the contliary
dii«ction), the vertex of the spheroid has shifted on the
earth's surface, and the ocean has to seek a new bearing.
The effect is to produce an immensely broad and exces-
sively flat wave (not a circulating current) f which follows,
or endeavours to follow, the apparent motions of the
moon, and must, in fact, if the principle of forced vibn^
tions be true, imitate by equal, though not by synchro-
rums, periods, all the periodicsd inequ^itiesof that motion.
When the higher or lower parts oif this wave strike our
eoasti, they experience what we caU high and low water.

(530.) The sun also produces precbely such a wave»
whose vertex tends to follow the apparent motion of the
san in the heavens, and also to imitate its periodic in*
equalities. This solar wave coexists with the lunar—
is sometimes superposed on it, sometimes transverse to it*
so as to partly neutralize it, according to the monthly
synodicfll configuration of the two luminaries. This al-
ternate matual reinforcement and destruction of the solar
and lunar tides cause what are called the spring and
neap tide»^the former bdng thmr sum, the latter their
difference. Although the real amount of either tide is,
at present, hardly within the reach of exact calculation,
yet their proportion at any one place is probably not
very remote from that of tiie ellipticities which would
belong to their respective spherdids, could an equilibrium
be attained. Now these ellipticities, for the solar and
lunar spheroids-, are respectively about two and five feet ;
so that the average spring tide will be to the ne^ as 7
to 3, or thereabouts.

r531.) Another effect of the combination of the solar
and lunar tides is what is called the priming and lagging
of the tides. If the moon alone existed, and moved in
th^ plane of the equator, the tideway (t. e. the interval
between two successive arrivals at the same |^ce of the
same vertex of the tide-wave) would be the lunar day
(art. 115) formed by the combination of tlie moon's si-
dereal period 'and that of the earth's diurnal motion.
Similarly, did the sun alone exist, and move always on
^ equator, the tide-day would be the mean soIeut day.

CHAR. XI.] tHE TIDES. 817

The actual tide-day, then, or the interval of the occur*
Tence of two successive . maxima of their superposed
waves, will vary as the separate waves approach to or
recede from coincidence ; hecause, when the vertices of
two waves, do not coincide, their joint height has its
maximum at a point intermediate between them. This
variation from uniformity in the lengths of successive
tide-days is particularly to be remarked about the time
of the new and full moon.

(532.) Quite different in its origin is that deviation of
the time of high and low water at any port or harbour,
from the' culmination of the luminaries, or of the theo-
retical maximum of their superposed spheroids, which
is called the *' establishment" of that port. If the water
were without inertia, and free from obstruction, either
owing to the friction of the bed of the sea — ^the narrow-
ness of channels along which the wave has to travel be-
fore reaching the port— their length, &c. &^, the times
above distinguished would be identical. But all these
causes tend to create a difference, tod to make that dif-
ference not alike at all ports. The observation of the
establishment of harbours is a point ef great maritime
importance ; nor is it of less consequence, theoretically
speaking, to a knowledge of the true distribution of the
tide waves over the globe.* In. making such observa-
tions, care must be taken not to confound tiie time of .
** slack water," when tiie current caused by the tide ceases
. to flow visibly one way or the other, and that of high or hw
water J when the level of the surface ceases to rise or fall.
These are totally distinct phenomena, and depend on ear
tirely different causes, though it is true they may some-
times coincide in point of time. Tkey are, it is feared,
too often mistaken one for the other by practical men ; a
circumstance which, whenever it occurs, must produce
the greatest confusion in any attempt to reduce the sys-
tem of the tides to distinct and intelligible laws.

(533.) The declination of the sun and moon materially
* The recent mvestigations of Mr. Lubbock, and thoM highly interart-
ing ones in which Mr. Whewell is updentood to be engaged, will, it is
to be hoped, apt only throw theoretical light on the very obscure sub*
jectof the tides, but (what is at present quite as much wanted) arouse
the attention of observers, and at the same time giTe it that right direcr
lion, bf pointing out wikat ought to be observed^ vyithmt which all <4pe|y
yation is lost -labour.

9d2

918 A TIIBATISE ON ASTRONOMT. [CHAP. XI.

sQeclB the tides at any particular spot. As the vertex of
the tide* ware tends to place itself vertically under the
luminary which produces it, when this vertical changes
its point ^ incidence on the surface, the tide-wave must
tend to shift accordingly, and thus, by monthly and an-
nual periods, must tend Uf increase and diminish altera ^
nately the principal tides. The period of the moon's
nodes is thus introduced into this subject ; her excursions
in declination in one part of that period being 29®, and
in another only 17", on either side the equator.

(534.) Geometry demonstrates that the efficacy of a
luminary in raising tides is inversely proportional to the
cube of its^distance. The sun and moon, however, by
reason of the ellipticity of their orbits, are alternately
nearer to and farther from the earth than their mean^ dis-
tonces. In consequence of thi», the efficacy of the sun
will fluctuate between the extremes 19 and 21, taking
20 for its mean value, and that of the moon between 43
and 59. Taking into account this cause of difference,
die highest spring tide will be to the lowest neap as 59
+21 to 43-^19, or as 80 to 24, or 10 to 3. Of all the
eauses of differences in the height of tides, however,
local situation is the most influential. In some places,
the tide- wave, rushing up a narrow channel, is suddenly
raised to an extraor£nary height. At Annapolis, for
instance, in the Bay t»f Fuhdy, it is said to raise 120
feet* Even at Bristol, the difference of high and low ^
water occasionally amounts to 50 feet.

(535.) The action (^ the sun and moon, in li|^e man
ner, produeeis tides in the atmosphere, which delieate
observations have been able to render sensible and mea*
surable. This effect, however, is extremely minute.

(536*) To^retum, now, to the planetary perturbations*
Let us next consider the changes induced by their mu-
tual action on the magnitudes and forms of Uieir orbits,
and in their positions therein in different situations with
respect to each other. In the £rst place, however, it
wiU be proper to explain the conventions under which
geometers and astronomers have alike agreed to use the
language and laws of the elliptic system, and to continue
to apply them to disturbed orbits, although those orbits
* BobMon*s Loetures on Mechanical Philosophy.

<IHAP. ZI.3 THB TIDES* 319

60 disturbed are no longer, in mathematicai starictnesSf
ellipses, or any known curves. This they do, partly on
account of the convenience of conception 'and calcula-
tion which attaches to this system, but mu^ more for
this reason — that it is found, and may be demonstrated
from the dynamical relations of the case, that the de-
parture of each planet from its ellipse, as determined at
any epoch, is capable of being truly represented^by sup-
posing the ellipse itself to be slowly variable, to change
its magnitude and eccentricity, and to shift its position
and the plane in which it lies according to certain laws,
while the planet all the time continues to move in this
ellipse, just as it would do if the ellipse remained in-
variable and the disturbing forces had no existence. By
thiB way of considering the subject, the whole petmanent
effect of the* disturbing forces is regarded as thrown upon
the orbit, while the relations of 3ie planet to that orbit
reihain unchanged, or only liable to brief and compara-
tively momentary fluctuation* This course of procedure,
indeed, is the most natural, and is in some sort forced upon
us by the extreme slowness with which the variations
of the elements develope themselves. For instance, the
fraction expressing the eccentricity of ihe earth's orbit
changes no more tfian 0*00004 in its amount in a cm^
iury / and the place of its perihelion, as referred to the
sphere of the heavens, by only 19' 39'' in the same
time* For several years, therefore, it would be next to
impossible to distinguish between an ellipse so varied
and one that had not varied at all ; and in a single revo*
Itttioh, the difference between the original elHpse and
the curve really represented by the varying one, is so
excessively minute, that if accurately drawn on a table*
«ix feet in diameter, the nicest examination with mi-
croscopes, continued along the whole oudinesof thetwo
curves, would hardly detect any perceptible interval be-*
tween them. Not to call a motion so minutely .conform-
ing itself to an elliptic curve, elliptic^ would be affectar
tion, even granting the existence of trivial departures
alternately on one side or on the other ; though, on the
other hand, to neglect a variation, which continues to
accumulate from age to age, till it forces itself on our
notice, would be wilful blindness.

320 ATIUEATISS ON ASTRONOMY. []CHAP. XI.

(587.) Geometers, then, have agreed in each single
revolution, or for any moderate interval of time, to re-
gard the motion of each planet as elliptic, and performed
according to Kepler's laws, with a reserve in favour of
certain very small and transient fluctuations, but at the
same time to regard all the elements of each ellipse as
in a continual, though extremely slow, state of change;
and, in tracing the effects of perturbation on the system,
they take account principally, or entirely, of this change
of the elements, as that upon which, after all, any mate-
rial change in the great features of the system will ulti-
mately depend.

(536.) And here we encounter the distinction between
what are termed secular variations, and such as are nr
pidly periodic, and are compensated in short intervals.
In our exposition of the variation of the inclination of a
disturbed orbit (art. 514), for in9tan^» we showed that,
in each single revolution of the disturbed body, the plane
of its motion underwent fluctuations to and fro in its
inclination to that of the disturbing body* which nearly
compensated each other; leaving, howevier, a portion
outstanding, which again is 9early compensated by the
revolution of the disturbing body, yet still leaving out-
standing and uncompensated a minute portion of the
change, which requires a whole revolution of the ifode
to compensate and bring it back to an average or jnean
value. Now, ^e two first compensations which jsra
operated by ike planets going through the succession of
conflguratioos with each other, and therefore in compar
ratively short periods, are called periodic variations;
and the deviations thus compensated are called inequa*
Utiea depending on configurations $ while the last,
which is operated by a period of the node (one of the
elements)^ has nothing to do with the configurations of
^e individual planets, requires an immense period ojf
time for its consummation, and is, therefore, distinguishr
ed from the former by the term secular variation.

(539.) It is true, that, to afford an exact representation
of the motions of a disturbed body, whether planet or
satellite, both periodical and secular variations, with
their corresponding inequalities, require to be express-
ed ; and, indeed, the former even more than the latter;

CAAP, XI.] VARIATIONS, PERIODIC AND SECULAR. d2t

Beeing that the secular inequalitiei^- are, in fact, nothing
but what remains after the mutual destructioil of a much
larger amount (as it very often is) of periodical. But
these are in thieir nature transient and temporary : they
disappear, and leaVe no trace. The planet is tempora-
rily drawn from its orbit (its slowly varying orbit), but
forthwith vetaims to it, to deviate j>resently as much the
other way, while the varied orbit accomodates and ad-
justs itself to the average of these excursions on either
side of it; and thus continues to present, for a succes-
sion of indefinite ages, a kind of medium picture of all
that the planet has been doing in their lapse, in which
ihe expression and character is preserved ; but the in-
dividual features are merged and lost. These periodic
inequalities, however, are, as we have observed, by no
means to be neglected, but they are taken account of by
a separate process, independent of the secular variations
of the elements.

(640.) In order to avoid complication, while endea-
vouring to give the reader an insight into both kinds of
variations, we shall henceforward conceive all the orbits
to lie in one plane, and confine our attention to the case
of two only, that of the disturbed and disturbing body,
a view of the subject which (as we have seen) compre-
hends the case of the moon disturbed by the sun, since'
any one of the bodies may be regarded a? fixed at plea.-
sure, provided we conceive all its motions transferred in

a contrary direction to each of the others. Suppose,
therefore, S to be the central, M the disturbing, and P
the disturbed body. Then the attraction of M acts on
P in the direction PM, and on S in the direction SM
And the disturbing part of M's attraction, being the dif-
ference only of these forces, will have no fixed direction.

322 , A TREATISE ON ASTRONOMY. [CHAP. XI.

but wiU acton P very differently, according to the configu-
rations of P and M. It will therefore be necessary, in
analyzing its effect, to resolve it, according to mechani-
cal principles, into forces acting according to some cer-
tain directions ; viz. along the radius vector SP, and per-
pendicular to it. The simplest way to do this, is to resolve
the attractions of M on both S and P in these directions,
and take, in both cases, their difference, which is the dis-
turbing part of M's effect. In this estimation, it will be
foimd then that two distinct disturbing powers originate ;
one, which we shall call the tangential force, acting in
the direction PQ, perpendicular to SP, and therefore in
that of a tangent to the orbit of P, supposed nearly a cir-
cle — ^the other, which may be called the radiid disturbing
force, whose direction is always either to or from S.

(541.) It is the former alone (art. 419). which disturbs
the equable description of areas of P about S, and is
therefore the chief cause of its angular deviations from
the elliptic place. For the equable description of areas
depends on no particular law of central force, but only
requires that the acting force, whatever it be, should be
directed to the centre ; whatever force does not conform
to this condition, must disturb the areas.

(542.) On the other hand, the radial portion of the dis^
turbing force, though, being always directed to or. from
the centre, it does not affect the equable description of
areas, yet, as it does not conform in its law of variation
to that simple law of gravity by which the elliptic figure
of the orbits is produced and maintained, has a tendency
to disturb this form ; and, causing the disturbed body P,
now to approach the centre nearer, now to recede farther
fron(i it, than the laws of elliptic motion would warrant, and
to have its points of nearest approach and farthest recess
otherwise situated than they would be in the undisturbed
orbit, tends to derange the magnitude, eccentricity, and
position of the axis of P's ellipse.

(543.) If we consider the variation of the tangential
force in the different relative positions of M and P, we
shall find that, generally speaking, it vanishes when P is
at A or C, see Jig. to art. 540, t. e, in conjunction with
M, and aliso at two points, B and D, where M is equi-
distant from S and P (or very nearly in the quadratures of

CHAP. X1.J EFFBCT8 OF THE TANGENTIAL FORCE. 323

P with M) ; and that, between A and B, or D, it tends to'
urge P towards A, while, in the rest of the orbit, its
tendency is to urge it towards C. Gonsequendy, the
general effect will be, that in P's progress through a com-
plete «^no(fiea/ revolution round its orbit from A, it will
first be accelerated from A up to B — ^thence retarded till
it arrives at G— thence again accelerated up to D, and
again retarded till its re-arrival at the conjunction A.^

(544.) If P's orbit were an exact circle, as well as M*b,
it is evident that the retardation which takes place during
the description of the arc AB would be exactly compen-
sated by the acceleration in the arc DA, these arcs being
just equal, and similarly disposed with respect to the
disturbing^ forces; and similarly, that the acceleration
through the arc BG would be exactly compensated by
the retardation along CD. Gonsequently, on the ave-
rage of each revolution of P, a compensation would take
place ; the period would remain unaltered, and all the
errors in longitude would destroy each other.

(545.) This exact compensation, however, depends
evidently on the exact symmetry of disposal of the parts
of the orbits on either side of the line GSM. If that
symmetry be broken, it will no longer take place, and in-
equalities in P*s motion will be produced, which extend
beyond the limit of a single revolution, and must await
their compensation, if it ever take place at all, in a re-
versal of the relations of configuration which jproduced
tiiem. Suppose, for example, that the orbit of P being
circular, that of M were elliptic, and that, at the moment
when P set out from A, M were at its greatest distsmee
from P; suppose, also, that M were so distant as to
make only a small part of iter whole revolution during a
revolution of P. Then it is clear that, during the whole
revolution of P, M's ^sturbing force would be on the
increase by the approach oi M, and that, in consequence,
the disturbance arising in each succeeding quadrant of
its motion, would over-compensate that produced in the
foregoing ; so that, when P had come round again to its
conjunction with M, there would be found on flie whole
to have taken place an over-compensation in favour of
an acceleration in the orbitual motion. This kind of ac-
tion would go on so long as M continued to approach S ;

324 A TRBATI8S ON ASTRONOMT. [cHAP. Xf«>

but when, in the progress of its elliptic moti<Mi, it began
again to recede, the reverse effect would take place, and a
retardation of P's orbttual motion would happen ; and so
on alternately, until at length, in the average of a great
many revolutions of M, in whidi the place of P in its
ellipse at the moment of conjunction should have been
situated in every variety of distance, and of approach
and recess, a compensation of a higher and remoter order,
among all those successive over and under-compensa-
tions, would have taken place, and a mean or average
angular motion would emerge, &e same as if no disturb-
ance had taken place*

(546.) The case is only a little more complicated, but
the reasoning very nearly similar, when the orbit of the
disturbed body is si^pposed elliptic. In an elliptic orbit,
tlie angular velocity is not uniform. The disturbed body
then remains in some parts of its revolution longer, in
others for a shorter time, under the influence of the ac-
celerating and retarding tangential forces, than is neces-
sary for an exact compensation ; independent, then, of
any approach or recess of M» there would, on this account
alone, take place an over oi under-compensation, and a
surviving, unextinguished perturbation, at the end of a
synodic period ; and, if the conjunctions otwoffB took
place on the same point of P'« e//tp«e, this cause would
constandy act one way, and an inequality would arise,
having no coihpensation, and which would at length, and
permanently, change the mean, angular motion of P«
But this can never be the case in the planetary system.
The mean motions (t. e. the mean angular velocities) of
the planets in their orbits, are incommen9urable to one
another. There are no two planets, for instance, which
perform their orbits in times exactly double, or triple,
the one of the other, or of which the one performs exact-
ly two revolutions while the other performs exactly three,
or five, and so on. If there were, the case in point would
arise. Suppose, for example, that the mean motions of
the distuibied and disturbing planet were exactly in the
proportion of two to five ; then would a cycle, consisting
of five of the shorter periods, or two of the longer, bring
them back exactly to the same configuration. It would
cause their conjunction, for instance, to happea once in

CHAP. XI.] THEORY OF JUPITER AND SATURN, 325'

every such cycle, in the same precise points of their orbits,
while in the intermediate periods of the cycle the other
configurations kept shifting round. Thus, then, would
arise the very case we have been contemplating, and a
permanent derangement would happen.

(547.) Now, although it is true that the mean motions
of no two planets are exactly commensurate, yet cases
are not Wanting in which there exists an approach to this
adjustment. And, in particular, in the case of Jupiter
and Saturn— 'that cycle we have taken for our example
in the above reasoning, viz. a cycle composed of five pe-
riods of Jupiter and two of Saturn — although it does nol
exactly bring about the same configuration, does so pretty'
nearly. Five periods of Jupiter are 21663 days, and two
periods of Saturn 21518 days. The difference is only
145 days, in which Jupiter describes, on an average, 12**,
and Saturn about 5°, so that after the lapse of the former
interval they will only be 5^ from a conjunction in the
same parts of their orbits as before. If we calculate the
time which will exactly bring about, on the average,
three conjunctions of the two planets, we shall find it to
be 21760 days, their synodical period being 7253*4 days.
In this interval Saturn will have described 8° 6' in excess
of two siderQsd revolutions, and Jupiter the same angle
in excess of fii^e. Every third conjunction, then, will
take place 8^ 6' in advance of the preceding, which is
near enough to establish, not, it is true, an identity with,
but still a great approach to the case in question. The
excess of action, for several such triple conjunctions (7
or 8) in successicm, will lie the same way, and at each
of them the motion of P will be similarly influenced, so
as to accumulate the effect upon its longitude ; thus giv-
ing rise to an irregularity of considerable magnitude and
very long period, which is wdl known to astronomers
by the name of the great inequality of Jupiter and Saturn.

(548.) The arc 8** 6' is contained 44$ times in the
whole circumference of 360° ; and accordingly, if we
trace round this particular conjunction, we shall find it
will return to the same point of the orbit in so many
times 21760 days, or in 2648 years. But the conjuno-
tion we are now considering, is only one out of three.
The other two will happen at points of the orbit about
2E

A TREATISE ON A8TH0N0MT. fcHAP. XI.

123° and 246'^ distant, and these points also mU advance
by the same arc of 8° 6' in' 21760 days. Consequently,
the period of 2648 years will bring them all i^ound, and
in that interval each of them will pass through that point
of the two orbits from which we commenced ; hence a
conjunction (one or other of the three) will happen at
that point once in one third of this period, or in 883
years; and this is, therefore, the cycle in which the
'* great inequality" would undergo its full compensation,
did .the elements of the orbits continue all that time in-
variable. Their variation, however, is considerable in so
long an interval ; and, owing to this cause, the period
itself is prolonged to about 918 years.

(649.) We have selected this inequality as a proper
instance of the action of a tangential disturbing force,
on account of its magnitude, the length of its period,
and its high historicsd* interest. It had long been re-
marked by astronomers, that on comparing together
modern with ancient observations of Jupiter and Saturn,
the mean motions of these planets did not appear to be
uniform. The period of Saturn, for instance, appeared
to have been lengthening throughout the whole of the
seventeenth century, and that of Jupiter shortening —
that is to say, the one planet was constantly lagging be-
hind, and the other getting in advance of its calculated
place. On the other hand, in the eighteenth century, a
process precisely the reverse seemed to be going on. It
is true, the whole retardations and accelerations observed
were not very great ; but, as their influence went on
accumulating, they produced, at length, material differ-
ences between the observed and calculated places of
both these planets, which, as they could not then be ac-
counted for by any theory, excited a high degre of atten-
tion, and were even, at one time, too hastily regarded as
almost subversive of the Newtonian doctrine of gravity.
For a long while this difference baffled every endeavour
to account for it, till at length Laplace pointed out its
cause in the near commensurability of the mean motions,
as above shown, and succeeded in calculating its period
and amount.

(550.) The inequality in question amounts, at itB
maximum^ to an alternate retardation and acceleration of

CHAP. XT.] THEORir OF JtJPITER AND SATURN. 827

about 0° '40' in the longitude of Saturn, and a Corres*
ponding acceleration or retardation of. about 0° 21' in
that of Jupiter. That an acceleration in the one planet
must necessarily be accompanied by a retardation in the
other, and vice versa ^ is evident, if we consider, that ac-^
tion and reaction being equal, and in contrary directions,
whatever momentum Jupiter communicates to Saturn in
the direction PM, the same momentum must Saturn com-
municate to Jupiter in the direction MP. The one, there-
. fore, will be dragged forward, whenever the other is
pulled back in its orbit. Geometry demonstrates, that,
on the average of each revolution, the proportion in
which this reaction will affect the longitudes of the two
planets is that of their masses multiplied by the square
roots of the major axes of their orbits, inversely, and this
result of a very intricate and curious calculation is fully
confirmed by observation.

(551.) The inequality in question would be much
greater, were it not for the partial compensation which
•is operated in it in every triple conjunction of the planets.
Suppose PQR to be Saturn's orbit, and pqr Jupiter's ;
and suppose a conjunction to take place at Pp, on the
line SA ;. a second at 123° distance, on the line SB ; a
third at 246° distance, on SC ; and the next at 368°, on
SD. This last-m6ntioned conjunction, taking place
nearly in the situation of the first, will produce nearly a
repetition of the first effect in retarding or accelerating
the planets ; but the other two, being in the most remote
situations possible from the first, will happen under en-
tirely different circumstances as to the position of the
perihelia of the orbits. Now, we have seen that a pre-

339 A TREATISE ON ASTRONOMY* [CHAP. XI.

sentation of the one planet to the other in conjunetion,
in a variety of situations, tends to produce compensation ;
and, in fact, the greatest possible amount of compensa-
tion which can be produced by only three configurations
is when they are thus equally distributed round the cen-
tre. Three positions of conjunction compensate more
than two, four than three, and so on. Hence we see
that it is not the whole amowit of perturbation, which is
thus accumulated in each triple conjunction, but only
that small part which is left uncompensated by the in- .
termediate ones. The reader, who possesses already
some acquaintance with the subject, will not be at a loss
to perceive how this consideration is, in fact, equivalent
to that part of the geometrical investigation of this in-
equality which leads us to seek its expression in terms
of the third order, or involving the cubes and products
of three dimensions of the eccentricities ; and how the
continual accumulation of smaH quantities, during long
periods, corresponds to what geometers intend when
they speak of small terms receiving great accessions of
magnitude by integration.

(952.) Similar considerations apply to every case of
approximate commensurability which can take place
. among, the mean motions of any two planets. Such, for
instance, is that which obtains between the mean motion
of the earth and Venus — 13 times the period of Venus
being very nearly equal to 8 times that of the earth.
This gives rise to an extremely near coincidenqe of every
jSfth conjunction, in the same parts of each orbit (within
fi^ih part of a circumference), and therefore to a cor-
respondingly extensive accumulation of the resulting un-.
compensated perturbation. But, on the other hand, the
part of the perturbation thus accumulated is only that
which remains outstanding after passing the equalizing
ordeal of five conjunctions equally distributed round the
circle ; or, in the language of geometers, is dependent
on powers and products of the eccentricities and inclina^
tions of the fifth order. It . is, therefore, extremely mi-
nute, and the whole resulting inequality, according to
the recent elaborate calculations of professor Airy, to
whom it owes its detection, amounts to no more than a
few seconds at its maximum, while its period is no less

CHAP. XI.] THE moon's variatiok. 329

than 240 years. This example will serve to show to
what minuteness these inquiries have been carried in the
planetary theory.

(553.) In the theory of the moon, the tangiential force
gives rise to many inequalities, the chief of which is that
called the variation, which is the direct and principal
effect of that part of the disturbance arising from the al-
ternate acceleration and retardation of the areas from the
syzigies to the quadratures of the orbit, and vice versa ^
combined with the elliptic form of the orbit ; in conse-
quence of which, the same area described about the
focus will, in different parts of the ellipse, coirespond to
different amounts of angular motion. This inequality,
which at its maximum amounts to about 37', was fitst
distinctly remarked as a periodical correction of the moon's
place by Tycho Brahe, and is remarkable in the history
of the lunar theory, as the first to be explained by New-
ton from his theory of gravitation.

(554.) We come now to consider the effects of that
part of the disturbing force which acta in the direction of
the radius vector, and tends to alter the law of gravity,
and therefore to derange, in a more direct and sensible
manner than the tangential force, the form of the dis-
turbed orbit from that of an ellipse, or, according to the
view we have taken of the subject in art 536, to pirodnce
a change in its magnitude, eccentricity, and position in
its own plane, or in the place of its perihelion.

(555.) In estimating the disturbing force of M on P,
we have seen that the difference only of M's accelerative
attraction on S and P is to be regarded as effective as
such, and that the first resolved portion of M's attraction,
— ^that, namely, which acts at P in the direction PS —
not finding in the power which M exerts on P any cor-
responding part,' by which its effect may be nullified, is
wholly effective to urge P to\<rards S in addition to its
natural gravity. This force is called the addititious part
of the disturbing force. There is, besides this, another
power, acting also in the direction of the radius SPj
which is that arising; from the difference of actions of M
on S an^ P, estimated first in the direction PL, parallel
to SM, and then resolved into two forces ; one of which
is the tangential force, already considered, in the direction
2e2

330 A TREATISE ON ASTROKOMT. £cHAP. XI.

PK ; the other perpendicular to it, or in the direction PR.
This part of M's action is termed the ablatitious force, be-
cause it tends to diminish the gravity of P towards S ; and
it is the exeesB of the one of these resolved portions over
the other, which, in any assigned position of P and M,
constitutes the radial part of the disturbing force, and
respecting whose effects we are now about to reason.
(S66.) The estimation of these forces is a matter of no
difficulty when the dimensions of the orbits are given,
but they are too complicated in their expressions to find
any place here. It will suffice for our purpose to point
out their general tendency ; and, in the. first place, we
shall consider their mean or average effect. In order to
estimate what, in any one position of P, will be- the
mean action of M in all the situations it can hold with
respect to P, we have nothing to do but to suppose M
broken up, and distributed in the form of a thin ring
round the circumference of its orbit K we wopld take
aceount of the elliptic motion of M, we might conceive
the thickness of this ring, in its different parts, to be pro-
portional to the time which M occupies in every part of
its orbit, or in the inverse proportion of its angular
motion. But into this nicety we shall not go, but con-
teat ourselves, in the first instance, with supposing M's
orb^ circular and its motion uniform. Then it is clear
that the mean disturbing effect on P will be the difference
of attractions of that ring on the two points P and S, of
which the latter occupies its centre, the former is ec-
centric. Now the attraction of a ring on its centre is
manifestly equal in all directions, and therefore, estimated
in any one direction, is zero. On the other hand, on a
point P out of its centre, if within the ring, the resulting
attraction will always be outwards^ jlowards the nearest
point of the ring, or directly from the centre.* * But if P

* As this is a proposition which the equilibrium of Satam*s ring ran-
den not merely speculative or illustrative, it will be well to demonstrate
it; which may he done very simply, and without the aid of any ca^
cuius. Conceive a spherical shetl, and a point within it : every line
passing through the point, and terminating both ways in the shell, will,
of course, be equaUy inclined to its surfiice at either end, being a chord
of a spherical surfiace, and, therefore, sj^mmefricaUy related to all its
puts. Now, conceive a small double oona, or pyramid, having its apex
at the point, and formed by the conical motion of such a line round the
point Theu will the two portions of the spherioal shell, which farm dia

GSAP. XI.J EFFECTS OF THE RADIAL FORCE. 381

lie without the ring, the resulting force wiU act always
inwards, urging P towards its centre. Hence it appears
that the mean effect of the radial force will be different
in its direction, according as the orbit of the disturbing
body is exterior or interiolr to that of the disturbed. In
the former case it will diminish, in the latter will in-
crease, the central gravity.

,(567.) Regarding, still, only the mean effect, as pro-
duced in a great number of revolutions of both bodies, it
is evident that an increase of central force must be ac-
companied with a diminution of periodic time, and a
contraction of dimension of the orbit of a body revolving
with a stated velocity, and vice versd* This, then, is the
first and most obvious effect of the radial part of the dis-
turbing force. It alters permanently, and by a certain
mean and invariable amount, the dimensions of all the
orbits and the periodic times of all the bodies composing
the planetary system, from what they would be, did each
planet circulate about the sun uninfluenced by the at-
traction of the rest ; the angular motion of the interior
bodies of the system being thus rendered less, tod those
of the exterior greater, than on that supposition. . The
latter effect, indeed, might be at once concluded from this
obvious consideration — ^that all the planets revolving in-
teriorly to any orbit may be considered as adding to the
generd aggregate of the attracting matter within,, which
is not the less efficient for being distributed over spaee^
and maintained in a state of circulation.

bases of both the cones, or pyramids, be omilar and equally inclined to
their axes. Therefore their areas will be to each other as the squares of
their distances from the common apex. Therefore their attractions on it
will be equal, because the attraction is as the attracting matter directly,
and the square of its distance inversely. Now, these attractions act in

rwite directions, and, therefore, counteract each other. Therefore^
point is in eauilibrium between them ; and as the same is true of
every such pair or areas into which the spherical shell can be broken up,
therefore the poiut will be in equilibrium, however tUuated wiMn such a
spherical shell. Now take a ring, and treat it similarly, breaking its
Circumference up into pairs of elements, the bases o£lria7tgle$ formed by
lines passing through the attracted point Here the attracting elements,
bein|[ Une*t not surface$f are in the rimple ratio of the distances, not the
dimibcate, as they should be to maintain the equilibrium. . Therefore it
will not be maintained, but the nearest elements will have the supe>
rioiity, and the point will, on the whole, be urged towards the nearest •
part of the ring. ' The same is true of every linear ring, and is, therefon^
Irue of any assemblage of concentric ones forming a flat annulus, like the
ring •f Saturn.

332 A TREATISE ON ASTRONOMY. [cHAP. XI.

(658.) This effect, however, is one which we have no
means of measuring, or even of detecting, otherwise than
by calculation. For our knowledge of the periods of
the planets, and the dimensions of their orbits, is drawn,
from observations made on them in their actual state, and
therefore, under the influence of this constant part of
the perturbative action. Their observed m^an motions
are, therefore, affected by the whole amount of its in-
fluence ; and we have no means of distinguishing this
from the direct effect of the son's attraction, with which
it is blended. Our knowledge, however, of the masses
of the planets assures us that it is extremely small ; and
this, in fact, is all which it is at all important to us to
know, in the theory of their motions,

(559.) The action of the sun upon the moon, inlike
manner, tends, by its mean inflvence during many suc-
cessive revolutions of both bodies, to dilate permanently
the moon's orbit, and increase her periodic time.' But
this general average is not established, either in the case
of the moon or planets, without a series of subordinate
fluctuations due to the elliptic forms of their orbits, which
we have purposely neglected to take account of in the
above reasoning, and which obviously tend, in the average
of a great multitude of revolutions, to neutralize each
other. Iii the lunar theory, however, many of these
subordinate fluctuations are very sensible to observation,
and of great importance to a correct knowledge of her
motions. For example : — The sun*s orbit (referred to
the earth as flxed) is elliptic, and requires thirteen lunar
tions for its description, during which the distance of
the sun undergoes an alternate increase and jdiminution,
each extending over at least six complete lunations.
•Now, as the sun approaches the earth, its disturbing
forces of every kind are increased in a high ratio, and
vice versa. Therefore the dilatation it produces on the
lunar orbit, and the diminution of the moon's periodic
time, will be kept in a continual state of fluctuation, in-
creasing as the sun approaches its perigee, and dimi-
nishing as it recedes. And this is consonant to fact — the
observed difference between a lunation in January (when
the sun is nearest the earth) and in July (when it is
farthest) being no less than 35 minutes

CBAK XI.] THE UOON's SECULAR ACCEIERATIOK. 339

(560.) Another very remarkable and important effect
of this cause, in one of its subordinate fluctuations (ex-
tending, however, over an immense period of time), is
what is called the secular acctleration of the moofCM.
mean motion^ It had been observed by Dr. Halley, on
comparing together the records of the most ancient lu«
Bar eclipses of the Chaldean astronomers with those of
modem tim#s, that the period of the moon's revolution
at present is sensibly shorter than at that remote epoch ;
and this result was confirmed \f^ a further comparison
.of both sets of observations with those of the Arabian, as-
• trononi^rs of the eighth and ninth centuries. It appear-
ed from these comparisons, that tlie rate at which the
moan's mean motion increases is about 11 seconds per
oentury-:— a quantity small in itself, but becoming consi-
derable by its accumulatipn during a succession of ages.
This remarkable fact, like the great equation of Jupiter
and Sttujm, had been long the subject of toilsome inves-
tigation to geometers. Indeed, so difficult did it appear
to render any exact account of, that while some were on
the point' of again declaring the theory of gravity inade-
quate to its explanation, others were for rejecting altoge-
ther the evidence on which it rested, although quite a«
satisfactory as that on which most historical events ar6
eredited. It was in this dilemma that Laplace once more
stepped in to rescue physical astronomy from its re-
proach, by pointing out the real cause of the phenome-
non in question, which, when so explained, is one of the
most curious and instructive in the whole range of our
subject— one which leads our speculations further intQ
the past and future, and points to longer vistas in the dim
perspective of changes which our system has undergone
and is yet to undergo, than any other which observation
assisted by theory has developed.

(661.) If the solar ellipse were invariable, the alter-
nate dilatation and contraction of the moon's orbit, ex-
plained in art. 559, would in the course of a great many
revolutions of the sun, at length effect an exact com-
pensation in the distance and periodic time of the moon,
by bringing every possible step in the sun's change of
distance to correspond to every possible elongation of
the moon from the sun in her orbit. But this is not, ia

834 A TREATISE ON ASTROMOMT. [cHAT* Xn

fact, the case. The solar eclipse is kept (as we hare al-
ready hinted in art. 636, and as we shall very soon ex-
plain more fully) in a continual but excessively sloiw
state of change, by the action of the planets qn the earth.
Its axis, it is true, remains unaltered, but its eccentricity
is, and has been since the earliest ages, diminishing ;
and this diminution will continue (there is little reason
to doubt) till the eccentricity is annihilated altogether,
and the earth's orbit becomes a perfect circle ; after
which it will again opeh out into an ellipse, the eccen-
tricity will again increase, attain a certain moderate
amount, and then again decrease. The time reqiyred for
these revolutions, tiliough calculable, has not been calcu-
lated, further than to satisfy us that it is not to be repk-
oned by hundreds or by thousands of years. It is ^ pe-
riod, in short, in which the whple history of astronomy
and of the human race occupies but as it were a point,
during which all its changes are to be regarded as uni-
fonn. Now, it is by this variation in the eccentricity of
the earth's orbit that the secular acceleration of the moon
is caused. The compensation above spoken of (which,
if the solar ellipse remained unaltered, would be effect-
ed in a few years or a few centuries at furthest in the
mode already stated) will now, we see, be only imper-
fectly effected, owing to this slow shifting of one of the
essential data. The steps of restoration are no longer
identical with, nor equal to, those of change. The same
reasoning, in short, applies, with that by which we ex-
plained the long inequalities producied by the tangential
force. • The struggle up hill is not maintained on equal
terms with the downward tendency. The ground is all
the while slowly sliding beneath the feet of the antagonists.
During the whole time that the earth's eccentricity is
diminishing, a preponderance is given to the action over the
reaction ; and it is not till that diminution shall cease, that
the tables will be turned, and the process of ultimate re-
storation will commence. Meanwhile, a minute, outstand-
ing, and uncompensated effect is left at each recurrence,
or near recurrence, of the same configurations of the sun,
the moon, and the solar and lunar perigee. These ac-
cumulate, influence the moon's periodic time and mean
n^otion, and thus becoming repeated in every lunation.

CHAP. XI.3 THE moon's SEC17LAR ACCELESATION. 836

at length affect her longitude to an extent not to be over-
looked.

(562.) The phenomenon of which we have now
given an account is another and very striking example of
the propagation of a periodic change from one part of a
system to another. The planets have no direct^ appre^
eiable action on the lunar motions as referred to the earth.
Their masses are too small, and their distances too great,
for their dijQerence v( action on the moon and earth, ever
to become sensible. Yet ^eir effect on the earth's orbit
is thOs, we see, propagated through the sun to that of the
moon; and what is very remarkable, the transmitted
effect thus indirectly produced on the angle described by
the moon round the earth is more sensible to observa-
tion than that directly produced by them on the. angle
described by the earth round the sun.

(563.) The dilatation and contraction of the lunar and
planetary orbits, then, which arise from the action of the
radial force, and which tend to affect their mean mo-
tions, are distinguishable into two kinds ;— -the one per-
manent, depending on the distribution of the attracting
matter in the system, and on the order which each pla-
net holds in it ; the other periodic, and which operates
in length of time its own compensation. Geometers
have demonstrated (it is to Lagrange that we owe this
most important discovery) that, besides these, there ex-
ists no thivd class of effects, whether arising from the
radial or tangential disturbing forces, or from their com-
bination, such as can go on for ever increasing in one di-
rection without self-compensation ; and, in particular,
that the major axes of the planetary ellipses are not lia-
ble even to those slow secular changes by which the in-
clinations, nodes, and all the other elements of the sys-
tem, are affected, and which, it is true, are periodic, but
in a different sense from those long inequalities which
depend on the mutual configurations of the planets inter
se. Now, the periodic time of a planet in its orbit about
the sun depends only on the masses of the sun and pla-
net, and on the major axis of the orbit it describes, with*
out regard to its degree of eccentricity, or to any other
element. The mean sidereal periods of the planets,
therefore, such as result from an average of a sufficient

836 A TREATISX ON A8TR0KOUT. [cUAP. ZI.

number of revolutions to allow of the compensation of
' the last^mentioned inequalities, are unalterable by lapse
* of time. The length of the sidereal year, for example,
if concluded at this present time from observations em-
bracing a thousand revolutions of the earth round the
san (such, in short, as we now possess it), is the same
with that which (if we can stretch our imagination so
far) must result from a similar comparison of observa*
tions made a n^illion of years hence. •

(564.) This theorem is jo/stly regarded as the most
important, as a single result, of any which have hitherto
rewarded the researches of mathematicians. We shall,
therefore, endeavour to make clear to our readers, at
least the principle on which its demonstration rests ; and
although the complete application of that principle can*
not be satisfactorily made without entering into details
of calculation incompatible with, our objects, we shall
have no difficulty in leading them up to Uiat point where
those details must be entered on, and in giving such an
insight into their general nature as will render it evident
what must be their results when gone through.

(565.) It is a property of elliptic motion performed
under the influence of gravity, and in conformity with
Kepler's laws, that if the velocity with which a planet
moves at any point of its orbit be given, and also the
distance of that point from the sun, Uie major axis of the
orbit is thereby also given. It is no matter in what
direction the planet may be moving at that moment. This
will influence the eccentricity and the position of .its
ellipse, but not its length. This property of elliptic
motion has been demonstrated by Newton, and is one of
the most obvious and elementary conclusions from his
theory. Let us now consider a- planet describing an in«
definitely small arc of its orbit about the sun, under the
joint influence of its attraction, and the disturbing power
of another planet. This arc will have some certain cur-
vature and direction, and, therefore, may be considered
as an arc of a certain ellipse described about the sun as
a focus, for this plain reason — ^that whatever be the
curvature and direction of the arc in question, an ellipse
may always be assigned, whose focus shall be in the 8un»
and which shall coincide with it throughout the whole

CHAP. XI.] PERMANENCE OF THE MAJOR AXES. 33f

interval (supposed indefinitely small) between its extreme
points. This is a matter of pure geometry. It does not
follow, however, that the ellipse thus instantaneously
determined will have the same elements as that similarly
determined from the arc described in either the previous
or the subsequent instant. If the disturbing force did not
exist, this would be the case ; but, by its action, a vari-
ation of the elements from instant to instant is produced,
and the ellipse so determined is in a continual state of
change. Now, when the planet has reached the end of
the small arc under consideration, the question whether
it will in the next instant describe an arc of an ellipse
having the same or a varied axis will depend, not on the
new direction impressed upon it by the acting forces—
for the axis, as we have seen, is independent of that
direction — not on its change of distance from the sun,
while describing the former arc — for the elements of
that arc are accommodated to it, so that one and the same
axis must belong to its beginning and its end. The
question, in short, whether in the next arc it shall take
up a new major axis, or go on with the old one, will de-
pend solely on this — whether the velocity has undergone,
by the action of the disturbing force, a change incom-
patible with the continuance of the same axis. We say
by the action of the disturbing force, because the central
force residing in the focus can impress on it no such
change of velocity as to be incompatible with the per-
manence of any ellipse in which it may at any instant be
freely moving about that focus.

(566.) Thus we see that the momentary vacation of
the major axis depends on nothing but the momentary
deviation from the law of elliptic velocity produced by
the disturbing force, without the least regard to the
direction in which that extraneous velocity is impressed,
or the distance from the sun at which the planet may be
situated in consequence df the variation of the other
elements of its orbit. And as this is the case at every
instant of its motion, it will follow that, after the lapse
of any time, however great, the amount of change which
the axis may have undergone will be determined by the
total deviation from the original elliptic velocity produced
by the disturbing force ; without any regard to alteratioiis
2F

88§ A TREATISE ON ASTRONOMY. L.CHAP. XT«

which the action of that force may have produced in the
other elements, except in so far as the velocity may be
thereby modified. This is the point at which the exact
estimation of the effect must be intrusted to the calcu-
lations of the geometer. We shall be at no loss, hoW"
ever, to perceive that these calculations can only ter-
minate in demonstrating the periodic nature and ultimate
compensation of all the variations of the axis which cstn
thus arise, when we consider that the circulation of two
planets about the sun, in the same direction and in in-
commensurable periods, cannot fail to ensure their pre-
sentation to each other in every state of approach and
recess, and under every variety as to their mutual dis-
tance and the consequent intensity of their mutual action.
Whatever velocity, then, may be- generated in one by the
disturbing action of the other, in one situation, will in-
fallibly be destroyed by it in another, by the. mere efiTect
of change of configuration.

(567.) It appears, then, that the variations m the
major axes of the planetary orbits depend entirely on
cycles of configuration, like the great inequality of Ju-
piter and Saturn, or the long inequality of tiie Earth and
Venus above explained, which, indeed, may be regarded
as due to such periodic variations of their axes. In fact,
the mode in which we have seen those inequalities arise,
from the accumulation of imperfectiy compensated actions
of the tangential force, brings them directiy under the
above reasoning : since the efficacy of this force falls
almost wholly upon the velocity of the disturbed planet,
whose motion is always nearly coincident with or op-
posite to its direction.

(568.) Let us now consider the efifect of perturbation
in altering the eccentricity and the situation of the axis
of the disturbed orbit in its own plane. Such a change
of position (as we have observed in art. 318) actually
takes place, although very slowly, in the axis of the
earth's orbit, and much more rapidly in that of the
moon's (art. 360) ; and these movements we are now to
account for.

(569.) The motion of the apsides of the lunar and
planetary orbits may be illustrated by a very pretty me-
chanical experiment, which is otherwise instructive in

CHAP. XI.3 MOTION OF THE AP8IDE8. dl9&

giying an idea of the mode in which orbitual motion is
carried on under the action of central forces variable ac-
cording to the situation of the revolving body. Let a
leaden weight be suspended by a brass or iron wire to a
hook in the under side of a firm beam, so as to allow of
its free motion on all sides of the vertical, and so that
when in a state of rest it shall just clear the floor of the
room, or a table placed ten or twelve feet beneath the
hook. The point of support should be well secured
from wagging to and fro by the oscillation of the weight,
which should be sufficient to keep the wire as tightly
stretched as it will bear, with the certainty of not break-
ing. Now, let a very small motion be communicated to
the weight, not by merely withdrawing it from the ver-
tical and letting it fall, but by giving it a slight impulse
sideways. It will be seen to describe a regular ellipse
about the point of rest as its centre. If the weight be
heavy, and carry attached to it a pencil, whose point lies
ezacUy in the direction of the string, the ellipse may be
transferred to paper lighdy stretched and gendy pressed
against it. In these circumstances, the situation of the
major and minor axes of. the ellipse will remain for a
long time very nearly the same, though the resistance of
the air and the stiffness of the wire will gradually di-
minish its dimensions and eccentricity. But if the im-
pulse communicated to the weight be considerable, so as
to carry it out to a great angle (15® or 80° from the
vertical), this permanence of situation of the ellipse will
no longer subsist. Its axis will be seeh to shift its
position at every revolution of the weight, advancing in
the same direction with the weight's motion, by an uni-
form and regular progression, 'which at length will en-
tirely reverse its situation, bringing the direction of the
longest excursions to coincide with that in which the
shbrtest were previously made ; and so on, round the
whole circle ; and, in a word, imitating to the eye, very
completely, the motion of the apsides of the moon's orbit.
(570.) Now, if we inquire into the cause of this pro-
gression of the apsides, it will not be difficult of de-
tection. When a weight is suspended by a wire, and
drawn aside from the vertical, it is urged to the lowest
point (or rather in a direction at ifvery instant perpen-

940 A TRtATlBE ON ASTRONOMY. [cHAP. ZI«

dicular to the wire) by a force which varies as the sine
of the deiriation of. the wire from the perpendicular.
Now, the sines of very small arcs are nearly in the pro-
portion of th^ arcs themselves ; and the more nearly, as
the arcs are smaller. If, therefore, the deviations from
the vertical are so small 'thai we may neglect the curva-
ture of the spherical surface m which the weight moves,
and regard the curve described as coincident with its pro-
jection on a horizontal plane, it will be then moving
under the same circumstances, as if it were a revolving
body attracted to a centre by a force varying directly as
the distance ; and, in this case, the curve described would
be an ellipse, having its centre of attraction not in the
focus, but in the centre,* and Ae apsides of this ellipse
would remain fixed. But if the excursions of die weight
from the vertical be considerable, the force urging it
towards the centre will deviate in its law from the simple
ratio of the distances ; being as the nne^ while the cUs-
tattces are as the arc. Now the sine, though it continues
to increase as the arc increases, yet does not increase so
fast So soon as the arc has any sensible extent, the sine
begins to fail somewhat short of the magnitude which an
exact numerical proportionality would require ; and
therefore the force urging the weight towards Its centre
or point of rest, at great distances fadls, in like proportion,
somewhat short of that which would keep the body in its
precise elliptk orbit. It will no longer, therefore, have,
at those- greater distances, the same command over the
weightf in proportion to its speedy which would enable

it to deflect it from its rectilinear tangential course into an
ellipse. The true path which it describes will be le89

* Newton, Pnodp. 1 47. «

CHAl». tl.] MOTION OF THE APBWU. 341

curved in the remoter parts than is consisteni ^ith the
elliptic figure, as in the annexed cut; and, therefore, it •
vrill not so soon have its motion brought to be again at
right angles to the radius. It will require a longer con-
tinued action of 4he central force to do this ; and before
it is accomplished, more than a quadrant of its revolution
must be passed over in angular motion round the centre.
But this is only stating at length, and in a more circuitous
manner, that fact which is more brieiiy and summarily
expressed by saying that the apsides of its orbit are pro*
grissive.

(571.) Now, this is what takes place, mutatis mvh
tandis, with the lunar and planetary motions. The ac-
tion of the sun on the jxioon, for example, as we have
seen, besides the tangential force, whose effects we are
not now considering, produces a force in the direction
of the radius vector, whose law is not that of the earth's
direct gravity. When compounded, therefore, with the
earth's attraction, it will deflect the moon into an orbit
deviating from the elliptic figure, being either too much
curved, or too little, in its recess from the perigee, to
bring it to an apogee at exactly 180° from the perigee ;
— too much, if the compound force thus produced de-
crease al^ a slower rate than the inverse square of the
distance (t. e. be too strong in the remoter distances) ;
too little, if the joint force decrease faster than gravity,
or more rapidly than the inverse square, and be therefore
too weak at the greater distance. In the former case,
the curvature, being excessive, will bring the moon to
its apogee sooner than would be the case in an elliptic
orbit ; in the latter, the curvature is insufficient, and will

Fig. 1. /i^. 2.

therefore bring it later to an apogee. In the former case*
then, the line of apsides will retrograde ; in the latter*
^vance. {See Jig, 1 andyig". 2.)
^ (572.) Both these cases obtain in different configura-
f2

84ft A TREATISE ON ASTRONOKY. [cHAP. XI.

tions oC the sun and moon. In the ayzigies, the effect
of the sun's attraction is to weaken the gravity of the
earth by a force, whose law of variation, instead of the
inverse square, follows the direct proportional relation
of the distance ; while,' in the quadratures, the reverse
takes place — the whole effect of the radial disturbing
force here conspiring with the earth's gravity, but the
portion added being still, as in the former case, in the
direct ratio of the distance. Therefore die motion of
the moon, in and near the first of these situations, will
be performed in an ellipse, whose apsides are ia i state
of advance ; and in and near the latter, in a state of re-
cess. But, as we have already seen (art. 556), the ave-
rage effect arising from the muti^al counteraction of these
temporary values of the disturbing force gives the pre-
ponderance to the ablatitious or enfeebling power. On
the average, then, of a whole revolution, 2ie lunar apo-
gee will advance.

(573.) The above reasoning renders a satisfactory
enough general account of the advance of the lunar apo-
gee ; but it is not without considerable difficulty that it
can be applied to determine numerically the rapidity of
such advance : nor, when so applied, does it account for
the whole amount of the movement in question, as as-
signed by observation — ^not more, indeed, than about one
half of it ; the remaining part is produced by the tan-
gential force. ^ It is evident, that an increase of velocity
in the moon will have the same effect in diminishing the
curvature of its orbit as the decrease of central force,
and vice versd. Now the direct effect of the tangential
force is to cause a fluctuation of the moon's velocity
above and below its elliptic value, and therefore an alter-
nate progress and recess of the apogee. This would
compensate itself in each synodic revolution, were the
apogee invariable* But this is not the case ; the apogee
is kept rapidly advancing by the action of the radial
force, as above explained. An uncompensated portion
of the action of (he tangential force, "therefore, remains
outstanding (according to the reasoning already so often
employed in this chapter), and this portion is so distri-
buted over the orbit as to conspire with the former cause,
andj in fact, nearly to double its effect. This is what is

fSCAP. XI.3 ECCENTRICITIES :AND FERIHELI^. 343

meant by geometers, when they say that this past of the
motion of the apogee is due to the square of the dislprb*
ing force. The effect. of the- tangential force in disturb-
ing the apogee .would compensate itself, were it not for
the moticm which tHe apogee has already had impressed
upon it by the radial force ; and we have here, therefore,
disturbance reacting on disturbance.

(57i.) The curious and complicated effect of pertur-
bation, described in the last article, has given more trou-
ble to geometers than any other part of the lunar theory.
Newtqa •himself had succeded in tracing that part of the
'motion of the apogee which is due to the direct action
of the radial force ; but finding the amount only half
what observation assigns, he appears to have abandoned
tlie subject in despair. Nor, when resumed by his suc-
cessors, did the inquiry, for a very long period, assume^
a more promising aspect. On the contrary, Newton's
result appeared to be even minul^jely verified, and the ela-
borate investigations which were lavished upon the sub-
ject without success began to excite strong doubts whe-
ther this feature of the lunar motions could'be explained
at all by the Newtonian law of gravitation. The doubt
was removed, however, almost in the instant of its ori-
gin, by the ^ame geometer, Clairaut, who first gave it
currency, and who gloriously repaired the error of his
momentary hesitation, by demonstrating the exact coin-
cidence between theory and observation, when the effect
of the tangential force is properly taken into the account.
The lunar apogee circulates, as already stated (art. SGO),
in about nine years.

(575.) The same cause which gives rise to the dis-
placement of the line of apsides of the disturbed orbit
produces a corresponding change in its eccentricity*
This is evident on a glance at our figures 1 and 2 of
art. 571. Thus, in fig. 1, since the disturbed body, pro-
ceeding from its lower to its upper apsis, is acted on by
a force greater than would retain it in an elliptic orbit,
and too much curved, its whole course (as far as it is so
affected) will lie within the ellipse, as shown by the
dotted line ; and when it arrives at the upper apsis, its
distance will be less than in the undisturbed ellipse ; that
is to say, the eccentricity of its orbit, as estimated by

844 A TREA116E ON ASTRONOMY. [cHAP. 3tl.

the comparative distances of the two apsides from the
focus, will be diminished, or the orbit rendered more
nearly circular. The contrary effect will take place in
the case of ^g, 2. There exists, therefore, between the
momentary shifting of the perihelion of the disturbed
orbit, and the momentary variation of its eccentricity,
a relation much of the same kind with that which con-
nects the change of inclination with the motion of the
nodes ; and, in fact, the strict geometrical theories of .
the two cases present a close analogy, and lead to final
results of the very same nature. What the variation of
eccentricity is to the motion of the perihelion, the change
of inclination is to the motion of the node. In either
case, the period of the one is also the period of the
other ; and while the perihelia describe considerable an-
gles by an oscillatory motion to and fro, or circulate in
immense periods of time round the entire circle, the ec-
centricities increase and decrease by comparatively small
changes, and are at length restored to their original mag-
nitudes. In the lunar orbit, as the rapid rotation of the
nodes prevents the change of inclination from accumu-
lating to any material amount, so the still more rapid re-
volution of its apogee effects a speedy compensation in
the ^uctoations of its eccentricity, and never suffers
them to go to any material extent ; while the same causes,
by presenting in quick succession the lunar orbit in every
possible situation to all the disturbing forces, whether of
the sun, the planets, or the protuberant matter at the
earth's equator, prevent any secular accumulation of
small changes, by which, in the lapse of ages, its ellip-
ticity might be materially increased or diminished. Ac-
cordingly, observation shows the mean eccentricity of
the moon's orbit to be the same now as in the earliest
ages of astronomy.

(576.) The movements of the perihelia, and variations
of eccentricity of the planetary orbits, are interlaced
and complicated together in the same manner and neioly
by the same laws as the variations of their nodes and
inclinations. Each acts upon every other, and every
such mutual action generates its own peculiar period of
compensation ; and every such period, in pursuance of
the oijnciple of art^ 526, is thence propagated thrpughout

CHAP, n.3 STABILITV OF THE ECCENTRICITIES. 945

the system. Thus arises cycles upon cycles^ of whose
compound duration some notion may be formed, when
we consider what is the length of one such period in the
case of the two principal planets — Jupiter and Saturn',
Neglecting the action of the rest, the effect of their mu-.
tual attraction would be to produce a variation in the ec-^
centricity of Saturn's orbit, from 0*08409, its maximum^
to 0-01345, its minimum value; while that of Jupiter
-would vary between the narrower limits, 0*06036 and
0*02606 : the greatest eccentricity of Jupiter correspond-
ing to the least of Saturn, and vice versa. The period
in which these changes are gone through, would be 70414
years. After this example, it will be easily conceived
that many millions of years will require to elapse before
a complete fulfilment of the joint cycle which shall re- .
store the whole system to its original state as far as the
eccentricities of its orbits' are concerned,

(577.) The place of the perihelion of a planet's orbit
is of little consequence to its well-being; but its eccen-
tricity is most important, as upon this (the axes of.
the orbits being permanent) depends the mean tempera-
ture of its surface, and the extreme variations to which
its seasons may be liable. For it may be easily shown
that the mean anntud amount of light and heat received
by a planet from the sun is, cseteris paribus, as the minor
axis of the ellipse described by it.* Any variation^
therefore, in the eccentricity by changing tlie minor axis,
will alter the mean temperature of the surface. How
such a change will also influence the extremes of tempe-
rature appears from art. 315. Now, it inay naturally be
inquired whether, in the vast cycle above spoken of, in
which, at some period or other, conspiring changes may
accumulate on the orbit of one planet from several
quarters, it may not happen that the eccentricity of any
one planet— as the earth — may become exorbitantly
great, so as to subvert those relations which render it
habitable to man, or to give rise to great changes, at least,
in the physical comfort of his state. To this the re-
searches of geometers have enabled us to answer in the
negative. A relation has been demonstrated by Lagrange

* *' On the Astronomical Causes which may influence Geological Fhe-
somenft.*'— Greo^ Trans, 183S.

945 A TREATISB ON AlSTRONOMY. [cHAP. XI.

between tbe massets, axes of the orbits, and eccentrici-
ties of each planet, similar to what we have already stated
with respect to their inclinations, viz. that if the mass
of each planet be multiplied by the square root of the
axis of its orbit, and the product by the square of its
eccentricity, the sum of all such products throughout
the system is invariable j and as, in point of fact, this
sum is extremely small, so it will always remain. Now,
since the axis of the orbits are liable to no secular changes,
diis 18 equivalent to saying that no one orbit shall in>-
erease its eccentricity, unless at the expense of a com-
mon fund, the whole amount of which is, and must for
ever remain, extremely minute.*

(678.) We have hinted, in our last art. but one, at
perturbations produced in the lunar orbit by the protu-
berant matter of the earth's equator. The attraction of
a sphere is the same as if all its matter were condensed
into a point in its centre ; but that is not the case with
a spheroid. The attraction of such a mass is neither
exactly directed to its <5entre, nor does it exactly follow
the law of the inverse squares of the distances. Hence
will arise a series of perturbations, extremely small in
amount, but still perceptible, in the lunar motions ; by
which the node and the apogee will be affected. A more
remarkable consequence of this cause, however, is a smaU
nutation of the lunar orbit, exactly analogous to that which
the moon causes in the plane of the eardi's equator, by its
action on the same elliptic protuberance. And, in gene-
ral, it may be observed, that in the systems of planets
which have satellites, the elliptic figure of the primary
has a tendency to bring the orbits of the satellites to co-
incide with its equator, — a tendency which, though small
in the case of the earth, yet in that of Jupiter, whose el-
lipticity is very considerable, and of Saturn especially,
where the ellipticity of the body is reinforced by the at-
traction of the rings, becomes predominant over every
external and internal cause of disturbance, and produces

* There is nothing in this relation, however, taken per M, to Mcure
the smaller. planetfr—Mercury, Mare, Juno, Ceres, &c.-^finm acatas-
trophe, could thoy accumulate on themselves, or any one of them, the
whole amount or this eccentricity fund. But that can never be : Jupiter
and Saturn will alwa^ retain the lion's share of it A similar remaik
applies to the indinatumfund of art 515. These /uiMit, .be it ofaservedf
can never get into debt Every term of them is essentially positive.

BIASSES DETERMINED BY PERTURBATIONS. »4T

and maintains an almost exact coincidence of the planes
in question. Such, at least, is the case with the nearer
satellites. The more distant are. comparatively less af^
fected hy this cause, the difference of attractions between
a sphere and spheroid diminishing with great rapidity as
the distance increases. Thus, while the orbits of all the
six interior satellites of Saturn lie almost exactly in the
plane of the ring and equator of the planet, that of the
external satellite, whose distance from Saturn is between
sixty and seventy diameters of the planet, is inclined tx>
that plane considerably. On the other hand, this con-
siderable distance, while it permits the satellite to retain
its actual inclination, prevents (by parity of reasoning)
the ring and equator of the planet from being perceptibly
disturbed by its attraction, or being subjected to any ap-*
preciable movements analogous to our nutation and pre-
cession. K such exist, they must be much slower than
those of the earth ; the mass of this satellite (though the
largest of its system) being, as far as can be judged by its
apparent size, a much smaller fraction of that of Saturn
than the moon is of the earth ; while the solar preces-
sion, by reason of the immense distance of the sun, must
be q nite inappreciable.

(579.) It is by means of the perturbations of the
planets, as ascertained- by observation, and compared
with theory, that we arrive at a knowledge of the masses
of those planets, which, having no satellites, offer no
other hold upon them for this purpose. Every planet
produces an amount of perturbation in the motions of
every other, proportioned to its mass, and to the degree
of advantage or purchase which its situation in the sys-
tem gives it over their movements. The latter is a sub-
ject of exact calculation ; the former is unknown, other-
wise than by observation of its effects. In the determina-
tion, howpver, of the masses of the planets by this means,
theory lends the greatest assistance to observation, by
pointing out the combinations most favourable for elicit-
ing this knowledge from the confused mass of superposed
inequalities which affect every observed place of a planet;
by pointing out the laws of each inequality in its period-
ical rise and decay; and by showing how every parti-

"348 A TREATISE ON ASTRONOMY. [CHAP. XI*

eolar inequality depends for its magnitude on the mass
producing it. It is thus that the mass of Jupiter itself
(employed by Laplace in his investigations, and inter-
woven with all the planetary tables) has of late been as-
certained, by observations of the derangements produced
by it in the motions of the ultra-zodiacal planets, to have
been insufficiently determined, or rather considerably
mistaken, by relying too much on observations of its sa-
tellites, made long ago by Pound and others, with in-
,adequate instrumental means. The same conclusion has
been arrived at, and. nearly the same mass obtained, by
means of the perturbations produced by Jupiter on
Encke's comet. The error was one of great importance ;
the mass of Jupiter being by far the most influential ele-
ment in the planetary system, after that of the sun. It
is satisfactory, then, to have ascertained — as by his ob-
servations Professor Airy is understood to have recently
done— the cause of the error; to have traced it up to its
source, in insufficient micrometric measurements of the
greatest elongations of the satellites ; and to have found
it disappear when- measures taken with more care, and
with infinitely superior instruments, are substituted for
those before employed.

(580.) In the same way that the pertuibations of the .
planets lead us to a knowledge of tl^ir masses, as eom-
pai^d with that of the sun, so the perturbations of the
satellites of Jupiter have led, and those of Saturn's at-
tendants will, no doubt, hereafter lead, to a knowledge
of the proportion their masses bear to their respective
primaries. The system of Jupiter's satellites has. been
elaborately treated by Laplace ; and it is from his theory,
compared with innumerable observations of their eclipses,
that the masses assigned to them in art 463 have been
fixed. Few results of theory are more surprising, than
to see these minute atoms weighed in the same balance
which we have applied to the ponderous mass of the
sun, which exceeds the least of them in the enormous
proportion of 65000000 to 1 •

CHAP. Xlt.] OF SIDEREAL ASTROIIOXT. ,M9

CHAPTER XII.

OF SIDEREAL ASTRONOMY.

Of the Stan lenemlly— Their DUtribiitioii into Clanes acqpiding to their
anparent Sfagnitudes — ^Their Distribation over the Heaveiuh--Of th«
Muky Way— Annual Parallax — -Heal DistanceSt probable Dimen-
sions, and mture of the Stann-Vanable Stan— l^nnporary Stan —
Of double Stan— Their Revohition about each Other in elliptic Orbtts
— ^Extension of the Law of Gravi^ to such Systems— Or coloured
Stars — Proper Motion of the Sun and Stars — Systematic Aberration
and Parallax— Of compound ndereal Systems — Clusten of Stars— Of
Nebul»— Nebulous Stan — ^Annular and planetary Nebuh c J iodiacal
Light

(581.) Besides the bodies we have described in the
foregoing chapters, the heavens pres^it us with an in-
numerable mul^tude of other objects, which are called
generally by the name of stars. Though comprehending
individuals differing from each other^ not merely in
brightness, but in many other essential points, they all
agree in one attribute^-a high degree of permanenee as
to apparent relative situation. This has procured theili
the title of *^ fixed stars ;" an expression which is to be
understood in a comparative and not in an absolute senses
it being certain that many, and probable that aft are in m
st&te of motion, although too slow to be perceptible vnr
less by means of very delicate observations, coittiniied
during a long series of years.

(582.) Astronomers are in the habit of distinguisiiiBg
the stars into classes, according to their apparent bright-
Bess* These are termed magnitudes. The brightest
stars ore said to be of the first magnitude; those which fall
so far short of the first (degree of brightness as to make a
marked distinction are classed into the second, and so on
down to the sixth or seventh, which comprise the small-
est stars visible to the naked eye, in the clearest and dark-
est night. Beyond these, however^ telescopes continue
the range of visibility, and magnitudes from the 8th down
to the 16th are famUiar to those who are in the practice
of using powerful instruments ; nor does there seem the
least reason to assign a limit to this progression ; every
increase in the dimensions and power d instmmeniSy
which aucoessive improvements in optical science hsv»

2G

850 A TREATISE ON ASTRONOMY, [CHAF. XD.

attained, having brought into view multitudes innumerable
of objects invisible before ; so that, for any thing expe-
rience has hitherto taught us, the number of the stars
may be really infinite, in the only sense in which we can
assign a meaning to the word.

(583.) This classification into magnitudes, however,
it must be observed, is entirely arbitrary. Of a multitude
of bright objects, differing probably, intrinsically, both in
size and in splendour, and arranged at unequal distances
from us, one must of necessity appear the brightest, one
next below it, and so on. An order of succession (rela-
tive, of course, to our local situation among them) must
exist, and it is a matter of absolute indifference, where,
in that infinite progression downwards, from the one
brightest to the invisible, we choose to draw our lines of
demarcation. ' All this is a matter of pure convention.
Usage, however, has established such a convention^ and
though it is impossible to determine exactly, or a priori^
where one magnitude ends and the next begins, and al-
though different observers have differed in their magni-
tudes, yet, on the whole, astronomers have restricted
their first magnitude to about 15 or 20 principal stars;
their second to 50 or 60 next inferior ; their third to
about 200 yet smaller, and so on ; the numbers ineieas-
ing very rapidly as we descend in the scale of brightness,
the whole number of stars already registered down to- the
seventh magnitude, inclusive, amounting to 15000 or
20000.

(584.) As we do not see the actual disc of a star, but
judge only of its brightness by the total impression naade
upon the eye, the apparent "magnitude" of any star
will, it is evident, depend, 1st, on the star's distance ficoBt
us ; 2d, on the absolute magnitude of its illuminated sur-
face ; 3d, on the intrinsic brightness of that surface. Now,
as we know nothing, or next to nothing, of any of these
data, and have every reason for believing that each of
them may differ in different individuals, in the proportion
of many millions to one, it is clear that we are not to
expect much satisfaction in any conclusions we may draw
from numerical statements of the number of individuals
arranged in our artificial classes. In fact, astronomers
have not yet agreed upon any principle by which the

CHAP. XII.] LIGHT OF THE STARS. 351

magnitudes may be photometrically arranged, though a
leaning towards a geometrical progression, of which each
term is the half of the preceding, may be discerned.*
Nevertheless, it were much to be wished, that, setting
aside all such arbitrary subdivisions, a numerical estimate
should be formed, grounded on'pwcise photometrical ex-
periments, of the apparent brightness of each star. This
would afford a definite character in natural hiistory, and
serve as a term of- comparison to ascertain the changes
which may take place in them ; changes which we know
to happen in several, and may therefore fairly presume
to be possible in all. Meanwhile, as a first approxima-
tion, the following proportions of lighi, concluded from
Sir William Herschel'st experimental comparisons of a
few selected stars, naay be borne in mind : — '
Light of a star of the average 1st magnitude ss 100

2d =25

3d = 12?

4th rr: 6

5th = 2

. 6th . =1

By my own experiments, I have found that the light of
Sirius (the brightest of all the fixed stars) is about 324
times that of an average star of the 6th magnitude.^:

(585.) If the comparison of the apparent magnitudes
of the stars with their numbers leads to no definite con^
elusion, it is otherwise when we view them in connexion
with their local distribution over the heavens. If indeed
wa confine ourselves to the three or four brightest classes,
we shall find them distributed with tolerable impartiality
over the sphere ; but if we take in the whole amount
visible to the naked eye, we shall perceive a great and
rapid increase of number as we approach the borders of
the milky way. And when we come to telescopic mag-
nitudes, we find them crowded beyond imagination, along
the extent of that circle, and of the branch which it
sends oflf from it; so (art. 253) that in fact its whole light
is composed of nothing but stars, whose average magni-
tude may be stated at a^out the tenth pr eleventh.

(586.) These phenomena agree with the supposition

* Strove, Dorpat CataL of Double Stars, p. xxxv.

tPiuLTr. 1817. tTiana. Astroa Soc. iil 183. .

dM A tkeaube on astjionomy. [chap.soi.

tbal ihfi stats of our finaaraent, instead of being scattered
in all diiections iadifierently through space, form a stra-
tnm, of which the thickness is smaU, in comparison with
its length and breadth ; and in which the earth occupies
a place somewhere about the middle of its thickness, and
near the point where ait Isiubdivides into two principal
laminffi, inclined at a small angle to each other. For
it is certain* that, to an eye so situated, the apparent den-
sity of the stars, supposing them pretty equally scat-
tered thropgh the space they occupy, would be least in
a direction of the visual ray (as SA) perpendicular to
the lamina, land greatest in that of its breadth, as SB, SO,
SD ; increasing rabidly in passing from olie to the other
direction, just as we see a slight haze in the atmosphere
thickening into a decided fog bank near the horizon, by
the rapid increase of the mere length of the visual ray.
Accordingly, such is the view of the construction of the
starry firmameiit taken by Sir William Herschel, whose
powerful telescopes have effected a complete analysis of

this wonderful zone, and demonstrated the fact of its entire-
ly Consisting of stars. So crowded are they in some parts
of it, that by counting the stars in a single field of his tele-
scope, he was led to conclude that 50000 had passed under
his review in a zone two degrees in breadth, during a sin-
gle hour's observation. The immense distances at which
the remoter regions must be situated will sufficiently ac-
count for the vast predominance of small magnitudes
which are observed in it.

(587.) When we speak of the comparative repiote-
ness of certain regions of the starry heavens beyond
others, and of our own situation in them, the question
immediately arises. What is the distance of the nearest
fixed star ? What is the scale on which our visible fir-
mament is constructed ? And what proportion do ite di-
mensions bear to those of our own immediate system f
To this* however, astronomy has hitherto proved unable

CHAP. XII*] DISTANCE OF THE tftABS. 953

to supply an answer. All we know on the subject is ne-
gative. We have attained, by delicate observaticrns and
refined combinations of theoretical reasoning, to a correct
estimate, first, of the dimensions of the earth ; then,
taking that as a base, to a knowledge of those of its orbit
about the sun ; and again, by taking our stand, as it were,
on the opposite borders of the circumference of this orbit,
we have extended our measurements to the extreme verge
of our own system, and by the aid of what we know of
the excursions of comets, have felt our way, as it were,
a step or two beyond the orbit of the remotest known
planet. But between that remotest orb and the nearest
star there is a gulf fixed, to whose extent no observa-
tions yet made have enabled us to assign any distinct
approximation, or to name any distance, however im-
mense, which it may not, for any thing we can tell, sur-

(588.) The diameter of the earth has served us as the
base of a triangle, in the trigonometrical survey of our
system (art. 226), by which to calculate the distance of
the sun : but the extreme minuteness of the sun's paral-
lax (art. 304) renders the calculation from this ** ill-
conditioned" triangle (art. 227) so delicate, that nothing
but the fortunate combination of favourable circumstances,
afforded by the transits of Venus (art. 409) could ren-
der its results even tolerably worthy of reliance. But
the eartii'fi diameter is too small a base for direct triangu-
lation to the verge even of our own system (art. 449),
and we are, therefore, obliged to substitute the annual pa-
rallax for the diurnal, or, which comes to the same thing,
to ground our ealeulation on the relative velocities of the
earth and planets in their orbits (art. 414), when we
would push our tnangulation to that extent. It might be
naturally enough expected, that by this enlargement of
our base to the vast diameter of the earth's orbit, the
next step in our survey (art. 227) would be made at a
great advantage ;—^that our change of station, from side
to side of it, would produce a perceptible and measurable
amount of annud parallax in the stars, and that by its
means we should come to a knowledge of their distance.
But, after exhausting every refinement of observation, as-
tronomers have been unabde to ^me to any positive and
2 g2

m A TRKAtiSj? ^N AdTRONOHT. f CHAP. XU.

coineident conclusion upon this b^ad ; and it seems,
thereforet. demonstrated, that the amount of such paral-
lax, even for the nearest fixed star which has hitherto
been examined with the requisite attention, remains still
mixed up with, and concealed among, the errors inci-
dental to all astronomical determinations. Now, such is
the nicety to which these have been carried, that did the
quantity in question amount to a single second (i, e. did
the radius of the earth's orbit subtend at the nearest fixed
star that minute «ngle), it could not possibly have escaped
detection and universal recognition.

(589.) Radius is to the sine of 1", in round numbers, as
300000 to 1. Ia this proportion, then, at least must the
distance of the fixed stars from ^e sun exceed that of
the sun from the earth. The latter distance, as we have
already seen, exceeds the earth's radius in the proportion
of 24000 to 1 ; and, lastly, to descend to ordinary stand-
ards, the earth's radius is 4000 of our miles. The dis-
tance of the stars, then, cannot be so smaU as 4800000000
radii of the earth, or 19^0(^0<]jjM)C|^00 mile« S How much
larger it may be, we know not.

(590.) In such numbers, the imagination is lost. Tlie
only mode we have of conceiving such intervals at all is
by the time which it would require for light to traverse
&em. Now light, as we know, travels at the rate of 1 92000
miles per second. It would, therefore, occupy 100000000
seconds, or upwards of three years, in such a journey,
at the very lowest estimate. What, Aen, are we to
allow for the distance of those innumerable stars of
the smaller magnitude which the telescope discloses to
us ! If Ve admit the' light of a star of each magnitude
to be half that of the magnitude next above it, it will
follow that a star of the first magnitude will require to be
removed to 362 times its distance to appear no larger
liian one of the sixteenth. It follows, therefore, &at
anumg the countless multitude of such stars, visible in
tdescopes, there must be many whose light has taken at
least a thousand years to reach us ; and that when we
observe Uieir places, and note their changes, we are, in
fact, reading only their history of a thousand years' datet
thus wonderfully recorded. We cannot escape this con-
clusion, but by adopting as an alteroative an intrinsio

QHAP. XII.] INTRINSIC LlOHT OF THE STAM. :355

iiiferiority of light in o/Z-the smaller stars of the milky way.
We shall be better able to estimate the probability of tlus
alternative, when we have made acquaintance with other
sidereal systems, whose existence the telescope discloses
to usy and whose analogy will satisfy us that the view of
the subject we have taken above is in perfect harmony
with the general tenor of astronomical facts.

(591.) Quitting, however, the region of speculation, and
confining ourselves within certain Umits which we are sure
are less than the trutk, let us Employ the negative know-
ledge we have obtained respecting the distances of the
stars to form some conformable estimate of their real
magnitudes. Of this, telescopes afford us no direct
information. The discs which good telescopes show us
of the stars are not real, but spuriom — a mere optical
illusion.* Their light, therefore, must be our oply
g^de. Now Dr. Wollaston, by direct photometrical
experiments, open, as it would seem, to no objections,!
has ascertained the light of Sirius, as received by us, tp
be to that of the sun as 1 to 20000000000. The sun,
therefore, in order that it should appear to us no brighter
than Sirius, would require to be removed to 141400 times
itsaetual distance. We have seen, however, that the dis-
tance of Sirius cannot be so small as 200000 times tiiat of
the sun. Hence it follows, that, upon the lowest possible
computation, the light really thrown out by Sirius cannot
be so littie as double that emitted by the sun ; or that
Sirius must, in point of intrinsic splendour, be at least
equal to two sims, and is in all probability vastly greater.^

(5^.) Now, for what purpose sxe we to suppose such
magnificent bodies scattered through the abyss of space ?
Surely not to illuminate our nights, which an additional
moon of the thousandth part of the size of our own
would do much better, nor to sparkle as a pageant void
of meaning and reality, and bewilder us among vain
coi^ectures. Useful, it is true, they are to man as points
of exact and permanent reference; but he must have
studied astronomy to littie purpose, who can suppose

*SeeCab.CycOptici. t PhiL Trun. 1839, p. 94.

X Dr. WoUasUHit aMumiing, w we think he ii jperfecdy juiUfied in do-
Utt, a much lower limit of postihle parallax in Sihuf than we have adopt*
edinthe text, has concluded ihe mtrinaic li|fht of JSiriua to he useify
that^ioivteen auna.

350 A TREATISE ON ASTRONOMY. [cHAP.Xtl.

roan to be the only object of his Creator's care, or who
does not see in the vast and wonderful apparatus around us
provision for other races of animated beings. The planets,
as we have seen, derive their light from fiie sun ; but that
cannot be the case with the stars. These, doubtless, then,
are themselves suns, and may, perhaps, each in its sphere,
be the presiding centre round which other planets, or bo-
dies of which we can form no conception from any ana-
logy offered by our own system, may be circulating.

(593.) Analogies, however, more than conjectural, are
not wanting to indicate a correspondence between the
dynamical laws which prevail in the remote regions of
the stars and those which govern the motions of our own
system. Wherever we can trace the law of periodicity —
the re^lar recurrence of the same phenomena in the
same tunes — we are strongly impressed with the idea of
rotatory or orbitual motion. Among the stars are se-
vered which, though no way distinguishable from others
by any apparent change of place, nor by any difference
of appearance in telescopes, yet undergo a regular period-
ical increase and diminution of lustre, involving, in one
or two cases, a complete extinction and revival. These
are called periodical stars. One of the most remarkable
is the «tar OmicroHy in the constellation Cetua^ first no-
ticed by Fabricius in 1596. It appears about twelve Umes
in eleven years— or, more exactly, in a period of 334
days ; remains at its greatest brightness about a fort-
night, being then, on some occasions, equal to a large
star of the second magnitude ; decreases during about
three months, till it iKBcomes completely invisible, in
which state it remains during about five months, when
it again becomes visible, and continues increasing during
the remaining three months of its period. Such is the
general course of its phases. It does not always, how-
ever, return to the same ^egfee of brightness, nor in-
crease and diminish by the same gradations. Hevelius,
indeed, relates (Lalande, art. 794) that during the four
years between October, 1672, and December, 1676, it
d^d not appear at all.

(594.) Another very remarkable periodical star is that
called Algol, or /i Persei. It is usually visible as a star
pf the second magnitude, and such it continues for the

CKikP. ra.J, PERIODICAI. gTARS. 357

"spAce of U^ H*", when it suddenly begins to dipiinish in
splendour, and in about Si hours is reduced to the fourlk
magnitude. It then begins again to increase, and in Si
hours more is restored to its usual brightness, going
through all iu changes in 2^ 20*" 48"", or thereabouts.
This remarkable law of variation certainly appears
strongly to suggest the revolution, round it of some opake
body, which, wh^i interposed between us and Algol,
cuts off a large portion of its light ; and this is accord-
ingly the view taken of the matter by Goodricke, to
whom we .owe the discovery of this remarkable fact,* in
the year 1782 ; since which time tlie same phenomena
have continued to be observed, though with much less
diligence than their high interest would appear to merit.
Taken smy how, it is an indication of a high degree of
aciivityy in regions where, but for such evidences, we
might conclude all lifeless^ Our own sun requires nine
times this period. to perform a revolution on its own axis.
On the other hand^ the periodic time of an opake re-
volving body, sufficiently large, which should produce a
similar temporary obscuration of the sun, seen from a
fixed star, would be less than fourteen hours.

(595.) The following list exhibits specimens of pe*
riodical stars of every variety of period, so far as they can
be considered to be at present ascertained :*— *

Starts Name.

Period.

Vajriationof

Magnitude.

IK B.

K.

fi Peraei

2 90

48

2 to 4

Goodricke. 1782.
Palitr^h, 1783.
Goodricke, J784.

fCephei

5 8

37

3.4—5

fi Lyre

6 9

•0

3 - 4.5

Goodricke, 1784.

r Antinoi

7 4

15

3.4 — 4.5

Pigott, 1784.

• Herculia

60 6

3-4

Herieliel, 1796.

* Serpentia )
RA. 15l> Aim
PD. 740 15'

180 —

.—

7T —

Harding, 1896.

eCeti

334 —

_^

8-0

Fabriciua, 1596.

96^1 Ejdm
34 Fl. Cygni
490 M. Leonia

396r 81

6 — n

Kirch, 1687.

494 -^

—m.

4-10

Maraldi, 1704.

18 yeara

6 —

Janson, 1600.

Many yean

7—0

Koch, 1782.

• Sagittam

Ditto

3—6

Halley, 1676.

4^ Laonia

Pitto

6-0

Montanari, 1667.

* See note on page 358.

t Theae letten K fl. and M. refer to the Catalqguea of Bode, Flaxo-
gleed. and Mayer

358 A TREATISE ON ASTRONOMY. [cRAP. XII.'

The variations of these stare, however, appear fo be
affected, perhaps in duration of period, but certainly in
extent of change, by physical causes at present unknown.
The non-appearance of o Ceti, during four years, has al-
ready been noticed ; and to this instance we may add
that of ;t Cygni, which is stated by Cassini to have been
scarcely visible throughout the years 1699, 1700, and
1701, at those times when it ought to have been most
conspicuous.

(596.) These irregularities prepare us for other phe-
nomena of stellar variation, which have hitherto been re-
duced to no law of periodicity, and must be looked upon,
in relation to our ignorance and inexperience, as alto-
gether casual ; or, if periodic, of periods too long to have
occtfrred more than once within the limits of recorded
• observation. The phenomena we allude to are those of
temporary stars, \vhich have appeared, from time to time,
in different parts of the heavens, blazing foi'th with ex-
traordinary lustre ; and after remaining a while appa*
rently immoveable, have died away, and left no trace.
Such is the star which, suddenly appearing in the year
125 B. C, is said to have attracted the attention of Hip-
parchus, and led him to draw up ti catalogue of stars,
the earliest on record. Such, too, was the star which
blazed forth, A. D. 389, near * Aquila;, remaining for
three weeks as bright as Venus, and disappearing entire-
ly. In the years 945, 1264, and 1572, brilliant stars
appeared in the region of the heavens between Cepheus
and Cassiopeia ; and, from the imperfect account we have
of the places of the two earlier, as compared with that of
the last, which was well determined, as well as from the
tolerably near coincidence of the intervals of their appear-
ance, we may suspect them to be one and the same 8tar»
with a period of about 300, or, as Goodricke supposes,

* The same discoverjr appears to have been made nearly about the
same time by Palitzch, a farmer of Prolitz, near Dresden — a peapant by
station, an astronomer by nature-^who, from his familiar acquaintance
with the aspect of the heavens, had been led to notice amon^ so many
thousand stars this one as distinguished from the rest bv its variation, and
had ascertamed its period. The same Pftlitzch was also the first to re-
discover the predicted comiet of Halley in 1759, which he saw nearly a
month before any of the astronomers, who, anned with their telescopety
were anxiouslv watching its return. These anecdotes carry ub bacK to
the era of the tlhaldean shepherds.

CHAP. XU.3 TEMPORARY STARS. 360

of 150 years. The 4>pearance of the star of 1572 was
so sadden, that Tycho Brahe, a celebrated Danish astro-
nomer, returning one eyening (the Uth of November)
from his laboratory to his dwelling-house, was surprised
to find a group of country people gazing at a star, which
he was sure did not exist half an hour before. This
was the star in question. It was then as bright as
Sirius, and continued to increase. till it surpassed Jupiter
when brightest, and was visible at mid-day. It began
to diminish in December of the same year, and in March,
1574, had entirely disappeared. So, also, on: the 10th
of October, 1604, s^star of this kind, and not less bril-
liant, burst forth in the constellation of Serpentarius,
which continued visible till October, 1605.

(597.) Similar phenomena, though of a less splendid
chaiacter^ have taken place more recently, as injiie case
of the star of the third magnitude discovered in 1670, by
Anthelm, in the head of the Swan ; which, after becom-
ing completely invisible, reappeared, and afier under-
going one or two singular fluctuations of light, during
two years, at last died away entirely, and has not since
been seen. On a careful re-examination of the .heavens,
too, and a comparison of catalogues, many stars are now
found to be missing ; and although there is no doubt that
these losses have often arisen from mistaken entries, yet
in many instances it is equally certain that there is no
mistake in the observation or entry, and that the star has
really been observed, and as really has disappeared from
the heavens.* This is a branch of practical astronomy
which has been too little followed up, and it is precisely
that in which amateurs of the science, provided with
only good eyes,. or moderate instruments, might employ
their time to excellent advantage.! It holds out a sure
promise of rich discovery, and is one in which astrono-

* The star 42 Viiginu is inserted in the Catalogue of the Astronomical
Society from Zach's Zodiacal Catalogue. I missed it on the 9lh of May,
1828, and have since repeatedly had its place in the field of view of my
90 feet reflector, ¥nthout perceiving it, unless it be one of two equal stars
of the 9th magnitude, very nearly m the place it must liave occupied. —
Author.

t " Ces variations des ^toiles sent bien dignes de I'attention des observ-
ateurs curieux . . . Un jour viendra, peut^tre, ou les sciences auront as-
sez d'amateurs pour qu'on puisse suffire & ces details." — Lcdande, art,
824<— Sorely that day is now arrived.

860 A TREATISE ON ABTROKOMY. QcHAF. XHv

men in established obsenralories aie almost of necessity
precluded from taking a patt by the nature of the ob»
servations required. Catalogues of the comparatire
bnghtness of the stars in each constellation have been
constructed by Sir Wm. Herscfael, with the express ob-
ject of facilitating these researches, and the reader will
find them, and a full account of his method of compari*
son, in the Phil. Trans. 1796, and subsequent years.

(598.) We come now to a class of phenomena of quite
a different character, and which give us a real and posi-
tive insight into th6 nature of at least some among the
stars, and enable us unhesitatingly to fledare them subject
to the same dynamical laws, and obedient to the same
power of gravitation, which govenis our own system.
Many of the stars, when examined with telescopes, are
found to.be double, i. e, to consist of two (in some cases
three) individuals plaeed near.together. This might be
attributed to accidental proximity, did it occur only in a
few instances ; but the lireqne&cy of this companionship,
the extreme closeness, and, in many cases, Ike near equal-
ity of the atars so conjoined, would alone lead to a strong
suspicion of a more near and intimate relation than m^re
casual juxtaposition. The bright star Castor, tor exam-
ple, when much magnified, is found to eonsist of two
stars of between the Siird and fourth magmtudey widiin
5" of each other. Stars of this magnitude, however,
are not so common in the heavens as to render it at all
likely that, if scattered at random, any two would ML so
near. But this is only one out of numerous suck in*
stances. Sir Wm. Herschel has enumerated upwards of
500 double staars, in which the individuals are within half
a minute of each other ; and to this list Professor Strove
of Dorpat, prosecuting the inquiry by the aid of instru-
ments more conveniently mounted for the purpose, has
recently added nearly five times that number. Other ob-
servers have still further extended the catalogue, already
so lar|e, without exha,usting the fertility of Sie heavens.
Among tlicse are great numbers in which the interval be-
tween the centres of the individuals is less than a single
second, of which f Arietis, Atlas Pleiadum, y Coronae, »
Coron«, j» and ^Herculb, arid t and x Qphiuchi, may be
cited as instaifees. They are divided into classes ao-

KFFECT OF I*ARiU.LAX ON A DOUBIfC STAR.

861

cording to their distancetB — ^the closest forming Hie first
class.

(599.) When these combinations were first noticed,
it was. considered that advantage might be taken of them^
to ascertain whether or not the annual motion of the earth
in its orbit might not produce a relative apparent displace*
ment of the individuals constituting a double star. Sup-
posing them to lie at a great distance one behind the other,
and to appear only by casual juxtaposition neaiiy in the
same line, it is evident that any motion of the earth must
subtend different angles at the two stars so juxtaposed,
and mast therefore produce different parallactic displace**
ments of them on the surface of the heavens, regarded
as infinitely distant. Every «tar, in consequence of ^e
earth's annual motion, should appear to describe in the
heavens a small ellipse (distinct from that which it would
appear to descnbe m consequence of the aberration of
light, and not to be confounded with it), being a section,
by the concave surface of the heavens, of an oblique
elliptic cone, having its vertex in the star, and the earth's
orbit for its base ; and this section will be of less diraen-

ftions the more distant is the star. If, then, we regard
two stars, apparently situated dose beside each other, but
in reality at very different distances, their parallactic el-
lipses wm be similar, but of different dimensions. Sup-
pose, for instance, S and « to be the positions of two
sta«| of such an^pacently or onticcdhf doable irtar 4U

30S A TREATISE ON ASTROHOIIY. [CHAF. ZIS

seen from the sun, and let ABOD, abed, be their p^-
rallactic ellipses ; then, since they will be at all times
similarly situated in these ellipses^ when the one star
is seen at A, the other will be seen at a. When the
the earth has made a quarter of a revolution in its orbit,
their apparent places will be B6 ; when another quarter,
Cc ; and when another, Dd. If, then, we measure care-
fully, with micrometers adapted for the purpose, their
apparent situation with respect to each other, at different
times of the year, we should perceive a periodical change,
both in the direction of the line joining them, and in the
distance between their centres. For the lines Aa and Qc
cannot be parallel, nor the lines B6 and Dd equal, unless
the ellipses be of equal dimensions, i. e, unless the two
stars have the same parallax, or are equidistant from the
earth.

(600.) Now, micrometers, properly mounted, enable
us to meastire very exactly both the distance between two
objects which can be seen together in the same field of a
telescope, and the position of the line joining them with
respect to the horizon, or the meridian, or any other de-
terminate direction in the heavens. The meridian is
chosen as the most convenient; and the situation of the
line of' junction between the two stars of a double star is
referred to its direction, by placing in the focus of the
eye-piece of a telescope, equatorially mounted, two cross
wires making a right angle, and adjusting their position
so that one of the two stars shall just run along it by its
diurnal motion, while the telescope remains at rest; noting
their situation ; and then turning the whole system of
wires* round in its own plane by a proper mechanical
movement, till the other wire becomes exactly pai^el to
their line of junction, and reading off on a divided circle
the angle the wires have moved through. Such an appa-
ratus is called a position micrometer ; and by its aid we
determine the angle of position of a double star, or tke
angle which their line of junction mak^s with the meri-
dian ; which angle is usually reckoned round the whole
circle, from to 360, beginning at the north and proceed-
ing in the direction north, following (or east) soath, pre-
ceding (or wcst)r

(601,) The advantages which this qipde of operHion

CHAP^ Xn.] «YST£ttATIC PARALLAX. 369

offers for the estimation of parallax are many «ind great-
In the first place, the result to be obtained, being depend*
ent only on the relative apparent displacement of the two'
stars, is unaffected by almost every cause which would
induce error in the separate determination of the place
of either by right ascension and declination. Refraction,
that greatest of all obstacles to accuracy in astronomical
determinations, acts equally on both stars ; and is there-
fore eliminated from the result. We have no longer any
thing to fear from errors of graduation in circles from
levels or plumb-lines— from uncertainty attending the
uranographieal reductions of aberration, precession, &c.
*— all which bear alike on both objects. In a woird, if we
suppose the stars to have no proper motions of their own
by which a reed change of relative situation may arise,
no other cause but their difference of parallax can pos-
sibly affect the observation.

(602.) Such were the considerations which first in-
duced Sir William Herschel to collect a list of double
stars, and to subject them all to careful measurements of
their angles of position and mutual distances. He had*
hardly entered, however, on these measurements, before
he was diverted from the original object of the inquiry
(which, in fact, promising as it is, still remains open and
untouched, though the only method which seems to of-
fer a chance of success in the research of parallax) by
phenomena of a very unexpected character, which at
once engrossed bis whole attention. Instead of finding,
as he expected, that annual fluctuation to and fro of one
star of a double star with respect to the other — that al-
ternate annual increase and decrease of their distance and
angle of position, which the parallax of the earth's an-
nual motion would produce — he observed, in many in-
stances, a regular progressive chafige ; in some cases
bearing chiefly on their distance — in others on their po-
sition, and advancing steadily in one direction, so as
clearly to indicate either a real motion of the stars them-
selves, or a general rectilinear motion of the sun and
whole solar system, producing a parallax of a higher
order than would arise from the earth's orbitual motion,
and which might be called systematic parallax.

(§03.) Supposing the two stars in motion indepiend^

3M A TRBilTISK ON ASTRONOMY. [CHAF. XII.

eiitly of each other, aad also the 8un» it is clear that for
the interval of a few years, these motions must be re-
garded as rectilinear and uniform. Hence, a very slight
acquaintance with geometry will suffice to show that the
app&rent motion of one star of a double star, referred to
the other as a centre, and mapped down, as it were, on a
{Aane in which that other shall be taken for a fixed or
zero point, can be no other than a right line« This, at
least, must be the case if the stars be independent of
each other; but it will be otherwise if they have a phy-
sical connexion, such as» for instance, real proximity 2iid
mutual gravitation would establish. In that case, they
would describe orbits round each other, and round their
common centre of gravity ; and therefore the apparent
path of either, referred to the other as fixed) instead of
being a portion of a straight line, would be bent into a
curve concave towards Uiat other. The observed mo-
tions, however, were so slow, that many years* observa-
tion was required to ascertain this point ; and it was not,
therefore, until the year 1803, twenty-five years from
the commencement of the inquiry, that any thing like a
positive conclusion could be come to respecting the rec-
tilinear or orbitual character of the oqserved changes of
position.

(604.) In that, and the subsequent year, it was dis-
tinctly announced by Sir William Herschel, in two
papers, which will be found in the Transactions of the
Royal Society for those years, that there exist sidereal
systems, composed of two stars revolving about each
other in regular orbits, and constituting what may be
termed binary stars, to distinguish Uiem from double
stars generally so called, in which these physically con^
nected stars are confounded, perhaps, with others only
optically double, or casually juxtaposed in the heavens
at difierent distances from the eye ; whereas the indir
viduals of a binary star are, of course, equidistant from
the eye, or, at least;^ cannot differ more in distance than
the semidiameter of the orbit they describe about each
other, which is quite insignificant compared with the
immense distance between them and the earth. Between
fifty and sixty instances of changes, to a greater or less
amount, in the angles of position of double stars, are ad-

CHAP. XII.3 ELLIPTIC ORBITS OF BINARY STARS. 365

dnced in the memoirs above mentioned ; many of which
are too decided, and too regularly progressive, to allow
of their nature being misconceived. In particular, among,
the more conspicuous stars, — Castor, > Vii'ginis, f Ursss,
70 Ophiuchi, «■ and » Coronas, ^ Bootis, * Cassiopeiss,
> Leonis, f Herculis, «f Cygni, /«* Bootis, • 4 and • 5 Lyrae,
X Ophiuchi, i«* Draconis, and ^ Aquarii, are enumerated
as among the most remarkable instances of the observed
motion ; and to some of them even periodic times of re-
volution are assigned, approximative only, of course, and
rather to be regarded as rough guesses than as results of
any exact calculation, for which the data were at the time
quite inadequate. For instance, the revolution of Castor
is set down at 334 years, that of y Virginis at 708, and
that of y Leonis at 1200 years.

(605.) Subsequent observation has fully confirmed
these results, not only in their' general tenor, but for the
most part in individual detail. Of all the stars above
named, there is not one which is not found to be fully
entitled to be regarded as binary ; and, in fact, this list
comprises nearly all the most considerable objects of that
description which have yet been detected, though (as at-
tention has been closely drawn to the subject, and ob-
servations have multiplied) it has, of late, begun to Extend
itself rapidly. The number of double stars which are
certainly known to possess this peculiar character is be-
tween thirty and forty at the time we write, and more
are emerging into notice with every fresh mass of obser-
vations which come before the public. Tbey require
excellent telescopes for their observation, being for the
most part so close as to necessitate the use of very high
magnifiers (such as would be considered extremely
powerful microscopes if employed to examine objects
within our reach), to perceive ian interval between the
individuals which compose them.

(606.) It may easily be supposed, that phenomena of
this kind would not pass without attempts to connect
them with dynamical theories. From their first disco-
very, they wiere naturally referred to the agency of some
power, like that of gravitation, connecting the stars thus
demonstrated to be in a state of circulation about each
other ; and the extension of the Newtonian law of gravi-^
2h2 '

306

A TRBATISB ON ASTRONOMY. [CHAP. ZII.

Utidn to these remote systems was a step so obvious, and
so well warranted by our experience of its all-sufficient
agency in our own, as to have been expressly or tacitly
made by every one who has given the subject any share
of his -attention. We owe, however, the first distinct
system of calculation, by which the elliptic elements of
the orbit of a binary star could be deduced from observa-
tions of its angle of position and distance at different
epochs, to M. Savary, who showed,* that the motions
of one of the most remarkable among them (| Urss)
were explicable, within the limits allowable for error of
observation, on the supposition of an elliptic orbit de-
scribed in the short period of 6Bf years. A different
process of computation has conducted Professor Encket
to an elliptic orbit for 70 Ophiuehi, described in a period
of seventy-four years ; and the author of these pages has
himself attempted to contribute his mite to these interest-
ing investigations. The following may be stated as the
chief resuks which have been hitherto obtained in this
branch of astronomy :—

Names of Bears.

Period of
Revolution.

Major Semi-
axis of
EUipae.

Eccentricity.

#

T-Leonis -
r Virnnis -
61 Cy^i
r Corona
Castor

70 Ophiuehi -
i UrsB
C Caneri
n Coronal

Years.
1300
638-9000

453-

386-6000
352-0600

80-3400

58*3685

55?

43-40

13'090
15-430 '

3-679

8086

4303

3-857

(607.) Of these, perhaps, the most remarkable is
y Virginis, not only on account of the length of its pe-
riod, but by reason also of the great diminution of ap-
parent distance, and rapid increase of angular motion
about each other, of the individuals composing it. It is
a bright star of the fourth magnitude, and its component
stars are almost exactly equd. It has been known to
consist of two stars since the beginning of the eighteenth
century, their distance being then between six and seven
seconds ; so that any tolerably good telescope would re*

» Coimoia. dea Temps, 1890. t Berlin Ephem. IfiS?

CRAP. xn.J ELLIPTIC ORBITS OF BINARY STARS. BtSU

solve it. Since that time they have been constantly ap^
proaching, and are at present hardly more than a single
second asunder ; so that no telescope, that is not of very
superior quality, is competent to show them otherwise
than as a single star somewhat lengthened in one direc-
tion. It fortunately happens, that Bradley, in 1718, no-
ticed, and recorded in the margin of one of his observa-
tion books, the apparent direction of their line of junction,
as being parallel to that of two remarkable stars, a and d
of the same constellation^ as seen by the naked eye ; and
this note, which ha^ been recently rescued from oblivion
by the diligence of Professor Rigaud, has proved of sig*
nal service in the investigation of their orbit. They are
entered also as distinct stars in Mayer's catalogue ; and
this affords also another means of recovering Uieir rela-
tive situation at the date of his observations, which were
made about the year 1756. Without particularising
individual measurements, which will be found in their
proper repositories,* it will suffice to remark, that their
whole series (which since the beginning of. the present
century has been very numerous and carefully made, and
which embraces an angular motion of 100*^, and a dimi-
nution of distance to one sixth of its former amount) is
represented with a demp of exactness fulhf equal to
that of observation item, by an ellipse of Uie dimensions
and period stated in the foregoing little table, and of
which the fiirthelr requisite particulars are as follows :—

Perihelion passage. August 18, 1834*

Inclination of orbit to the visual ray 229 58

Angle of position of the perihelion projected on the heavens - 96^ 24'
Anile of position of the line of nodes, or intersection of the ^ ^^ av
plane of thei orbit with the surface of the heavens )

{608.)' If the great length of the periods of some of
these bodies be remarkable, the shortness of those of
others is hardly less so. » Corons has already made a
complete revolution since its first discovery by Sir Wil-
liam Herschel, and is far advanced in its second period ;
and f Ursse, i Cahcri, and 70 Ophiuchi, have all accom-
plished by far the greater parts of their respective ellipses
«ince the same epoch. If any doubt, therefore, could re-
main ^ to the reality of their orbitual motions, or any

* See them collected in Mem. R. A«t Soc. voL v. p. i35.

308 ▲ THEATI8B ON ASTRONOHT. [cHAP. Xn«

idea of explaining them by mere parallactic changes, these
facts must suffice for their complete dissipation. We
have the same evidence, indeed, of their rotations about
each other that we have of those of Uranus and Saturn
about the sun; and the correspondence between their
calculated and observed places in such very elongated
ellipses, must be admitted to carry with it a proof of the
prevalence of the Newtonian law of gravity in their sys-
tems, of the very same nature and cogency as that of the
calculated and observed places of comets round the cen-
tral body of our own.

(609.) But it is not with the revolutions of bodies of
a planetary or cometary nature round a solar centre that
life are now concerned ; it is witii that of sun around sun
— each, perhaps, accompanied with its train of planets
and their satellites, closely shrouded from our view by
the splendour of their respective suns, and crowded into a
space bearing hardly a greater proportion to the enor-
mous intervfld which separates them^ than the distances
of the satellites of our planets from their primaries bear
to their distances from the sun itself. A less distinctly
characterized subordination would be incompatible with
&e stability of their systems, and with the planetary na-
ture of their orbits. Unless closely nestled under the
protecting wing of their immediate superior, the sweep
of their other sun in its perihelion passage round their
own might carry them off, or whirl them into orbits ut-
terly incompatible with the conditions necessary for the
existence of their inhabitants. It must be confessed, that
we have here a strangely wide and novel field for specu-
lative -excursions, and one which it is not easy to avoid
luxuriating in.

f6I0.) Many of the double stars exhibit the curious
«ia beautiful phenomenon of contrasted or complemen-
tary colours.* In such instances, the larger star is usu*'
ally of a ruddy or orange hue, while the smaller one ap-
pears blue or green, probably in virtue of that general
law of optics, which provides that when the retina is

* " other sura, perhaps.

With their attendant moons thou wilt descry,
Communicating male and female light,
(Which two ffreat sexes animate Uie world,)
Stored in each orb, perhaps, with some that live/

PQTodiM Utttt viiL 148.

GHAP. Xn.J CX>X.OURED 8tAR8» 9W

under the influence of excitement by any bright, coloured
light ; feebler lights, which seen alone would produce
no sensation but of whiteness, shall for the time appear
coloured with the tint complementary to that of the
brighter. Thus, a yellow colour predominating in the
light of the brighter star, that of the less bright one in the
same field of view will appear blue ; while, if the tint of
the brighter star verge to crimson, that of the other will
exhibit a tendency to green— ^r even appear as a vivid
green, under favourable circumstances. The former con-
trast is beautifully exhibited by i Cancri — the latter by y
Andromeds ; both fine double starsl If, however, the
coloured star be much the less bright of the two, it wiU
not materially aflfect the other. Thus, for instance, «
CassiopeiaB exhibits the beautiful combination of a large
white star, suad a small one of a rich ruddy purple. It is
by no means, however, intended to say, that in all such
cases one of the colours is a mere effect of contrast, and
it may be easier suggested in words, than conceived in
imagination^ what variety of illumination two «un^-r-a
red and a green, or a yellow and a blue one — ^must afford
a planet circulating about either; and what charming
contrasts and •* grateful vicissitudes"—^ red and a green
. day, for instance, alternating with a white one and with
darkness — ^might arise from the presence or absence of
one or other, or both, above tiie horizon. Insulated stars
of a red colour, almost as deep as that of blood, occur in
many parts of the heavens, but no green or blue star (of
any decided hue) has, we believe, ever been noticed un-
associated with a companion brighter than itself.

(611.) Another very interesting subject of inquiry, in
the physical history of the stars, is their proper motion.
^•^ priorif it might be expected that apparent motions of
some kind or other should be detected among so great a
multitude of individuals scattered through space, and with
nothing to keep them fixed. Their mutual attractions
even, however inconceivably enfeebled by distance, and
counteracted by opposing attractions from opposite quar*
ters, must, in the lapse of countless ages, produce some
movements — some change of internal arrangement— re-
sulting from the difierence of the opposing actions. And
it is a fact, that such apparent motions do exist, not only

970 ▲ TREATUIS ON ASTRONOMY. |^CHAP. Xn.

among single, but in many of the double stars ; which,
besides leyolring round each other, or round their com-
mon centre of gravity, are transferred, without parting
company, by a progressive motion common to both,
towards some determinate region. For example, the
two stars of 61 Cygni, which are nearly equal, have re-
mained constantly at the same, or very nearly the same,
distance, of 15", for at least fifty years past. Mean-
while they have shifted their local situation in the hea-
vens, in this interval of time, through no lees than 4' 23",
the annual proper motion of each star being 5"*3 ; by
which quantity (exceeding a third of their interval) this
system is every year carried bodily along in some un-
known path, by a motion which, for many centuries,
must be regarded as uniform and rectilinear. Among
stars not double, and no way differing from the rest in
any other obvious particular, (a Cassiopeiae is to be re-
marked as having the greatest proper motion of any yet
ascertained, amounting to 3"*74 of annual displacement*
And a great many others have been observed to be thus
constantly carried away from their places by smaller, but
not less unequivocal motions.

(612.) Motions which require whole centuries to ac-
cumulate before they produce changes of arrangement,
such as the naked eye can detect, though quite sufficient
to destroy that idea of mathematical fixity which pre-
cludes speculation, are yet too trifling, as far as practical
applications go, to induce a change of language, and lead
us to speak of the stars in common parlance as otherwise
than fixed. Too little is yet known of their amount and
directions, to allow of any attempt at referring them to
definite laws. It may, however, be stated generally, that
their apparent directions are various, and seem to have
no marked common tendency to one point more than to
another of the heavens. It was, indeed, supposed by Sir
William Herschel, that such a common tendency could
be made out ; and that, allowing for individual deviations,
a general recess could be perceived in the principal stars,
from that point occupied by the star ^ HercuUs, towards
a point diametrically opposite. This general tendency
was referred by him to a motion of Uie sun and solar
system in the opposite direction* No one, who reflects

CBAP. Xn.] MOTIONS OF THE Sim AND STARS. 371

with due attention on the subject, will be inclined'to deny
the high probability, nay certainty, that the sun has a
proper motion in some direction ; and the inevitable con*
sequence of such a motion, unparticipated by the rest,
must be a slow average apparent tendency of all the stars
to the vanishing point of lines parallel to that direction,
and to the region which he is leaving. This is the ne-
cessary effect of perspective ; and it is certain that it must
be detected by such observations, if we knew accurately
the apparent proper motions of all the stars, and if we
were sure that they were independent, i. e. that the.
whole firmament, or at least all that part which we see
in our own neighbourhood, were not drifting along
together, by a general set, as it were, in one direction, the
result of unknown processes and slow internal changes
going on in the sidereal stratum to which our system be-
longs, as we see motes sailing in a current of air, and
keeping nearly the same relative situation with respect
to one another. But it seems to be the general opinion
of astronomers, at present, that their science is not yet
matured enough to adOford data for any secure concluskms
of this kind one way or other. Meanwhile, a very in^
genious idea has been suggested by the present astron-
omer royal (Mr. Pond), vit, that a solar motion, if it
exist, and have a velocity at all comparable to that of
light, must necessarily produce a solar aberration^ in
consequence of which we do not see the stars disposed
as they really are, but too much crowded in the region
the sun is leaving, too open in that he is approaching.
(See art. 280.) Now this, so long as the solar velocity
continues the same, must be a constant effect which ol>-
servation cannot detect ; but should it vary, in the coarse
of ages, by a quantity at all commensurate to the velocity
of the earth in its orbit, the fact would be detected by a
general apparent rush of all the stars to the one or odier
quarter of the heavens, according as the sun's motion
were accelerated or retarded ; which observation would
not fail to indicate, even if it should amount to no more
than a .very few seconds. This consideration, refined
and remote as it is, may serve to give some idea of the
delicacy and intricacy of any inquiry into the matter of
proper motion ; since the last mentioned effect would ae-

d73 ▲ TREATISE ON ASTRONOMY. [cHAP. XU.

cessarily be mixed up with the systematic parallax, and
could only be separated from it by considering that the
nearer stars would be affected more than the distant ones
by the one cause, but both near and distant alike by the
other.

(613.) When we cast our eyes over the concave of the
heavens in a clear night, we do not fail to observe that
there are here and there groups of stars which seem to
be compressed together in a more condensed manner than
in the neighbouring parts, forming bright patches and
•clusters, which attract attention, as if they were there
brought together by some general cause other than casual
distribution. There is a group, called the Pleiades, in
which six or seven stars may be noticed, if the eye be
directed full upon it ; and many more if the eye be turned
carelessly aside, while the cUtention is kept directed*
upon the group. Telescopes show fifty or sixty large
«tars thus crowded together in a very moderate space,
comparatively insulated from the rest of the heavens.
The constellation called Coma Berenices is another such
group, more diffused, and consisting of much larger
stars.

(614.) In the constellation Cancer, there is a some-
what similar but less definite, luminous spot, called
Praesepe, or the bee-hive, which a very moderate tele-
scope — an ordinary night-glass, for instance — ^resolves
entirely into stars. In the sword handle of Perseus, also,
is another such spot, crowded with stars, which requires
rather a better telescope to resolve into individuals sepa-
rated from each other. These are called clusters of stars ;
and, whatever be their nature, it is certain that other laws
of aggregation subsist in these spots, than those which
have determined the scattering of stars over the general
surface of the sky. 'Hiis ccmclusion is still nu>re stnHigly

* It is a veiy remarkable fact, that the centre of the yira^ area la ^
far leas sensible to feeble impressions of liriit, than the exterior ixntions
of the retina. Few persons are aware of the extent to which this com-
parative insensibility extends, previous to trial. To appreciate it, let die
reader look alternately full at a star of the fifth magnitiide, and beside it ;
or choose two equally bright, and about 3^ or 4^ apart, and look fuU at
one of them, the probability is, he will see onlv the other: such, at least,
is my own case. The fiict accounts for the multitude of stars with which
we are impressed by a general view of the heavens ; their pandtj
when we come to count them.— Aulftor.

CHAP. XII.] CLU8TBR8 OF 8TAH8. 37?

pressed upon us, when we come to bring very powerfu]
telescopes to bear on these and similar spots. There are
a great number of objects which have been mistaken for
comets, and, in fact, have very much the appeiairance of
comets without tails : small round, or oval nebulous
specks, which telescopes of moderate power only show
as such. Messier has given, in the Connoia. des Tempt
for 1784, a list of the places of 103 objects of this sort ;
which all those who search for comets ought to be fami-
liar with, to avoid being misled by tfieir similarity of
appearance. That they are not, however, comets, Iheir
fixity sufficiently proves ; and when we come to examine
them with instruments of great power — such as reflectors
of eighteen inches, two feet or more in aperture— any
such idea is completely destroyed. They are then, for
the most part, perceived to consist entirely of stars
crowded together so as to occupy almost a definite out-
line, and to run up to a blaze of light in the centre,
where their condensation is usually the greatest. (See
Jig. 1, pi. ii., which represents (somewhat rudely^ the
thirteenth nebula of Messier's list (described by him as
nebuUuse sans etoiles)^ as seen in the 20 feet reflector at
Slough.)* Many of them, indeed, are of an exactly
round figure, and convey the complete idea of a globular
space filled full of stars, insulated in the heavens, and con-
stituting in itself a family or society apart from the rest,
and sul^ect only to its own internal laws. It would be
a vain task to attempt to count the stars in one of these
globular clusters. They are not to be reckoned by hun-
dreds : and on a rough calculation, grounded on the
apparent intervals between them at the borders (where
they are seen not projected on each other), and the angu-
lar diameter of the whole group, it would appear tliat
many clusters of this description must contain, at least,
ten or twenty thousand stars, compacted and wedged
together in a round space, whose angular diameter does
not exceed eight or ten minutes ; that is to say, in an
area not more than a tenth part of that covered by the
moon.

*^Thi8 beaatiful object wm fimt notioed by Hallejr in 1714 It ii via-
ble to the naked eye, between the Ktxn m and K ^rculu. In s night-
glaa it sj^pean exactly like a imall round comet
21

374 A TKEATISE ON ASTRONOMY* [cHAP. XU.

(615.) Perhaps it may be thought to savour of the
gigantesque to look upon the individuals of such a group
as suns like our own, and their mutual distances as equsiL
to those which separate our sun from the nearest fixed
star : yet, when we consider that their united lustre af-
fects the eye with a less impression of light than a star
of the fifth or sixth magnitude (for the largest of these
clusters is barely visible to the naked eye), the idea we
are thus compelled to form of their distance from us may
render even such an estimate of their dimensions familiar
to our imagination; at all events, we can hardly. look
upon a. group thus insulated, thus in aeipao totus, teres ^
atque rotundus, as not forming a system of a peculiar
and definite character. Their round figure clearly indi-
cates the existence of some general bond of union in the
nature of an attractive force ; and, in many of them,
there is an evident acceleration in the rate of condensa-
tion as we approach the centre, which is not referable to
a merely uniform distribution of equidistant stars through
a globular space, but marks an intrinsic density in their
state of aggregation greater at the centre than at the sur-
face of the mass. It is difficult to form any conception
of the dynamical state of such a system. On the one
hand, without a rotatory motion and a centrifugal force,
it is hardly possible not to regard them as in a state of
progressive collapse. On the other, granting such a mo-
tion and such a force, we find it no less difficult to recon-
cile the apparent sphericity of their form with a rotation
of the whole system round any single axis, without which
internal collisions would appear to be inevitable.* The
following are the places, for 1830, of a few of the prin-
cipal of these remarkable objects, as specimens of their
class :—

R.

A.

N. P. D.

R. A.

N. P. D.

H.

M.

o '

H. K.

O '

13

5

70 55

17 29

93 8

13

34

60 45

31 22

78 34

15

10

87 16

21 25

01 34

16

36

53 13

(616.) It is to Sir William Herschel that we owe the
most complete analysis of the great variety of thofe ob'
* See a note on thi& subject at the end of the work, p. 388.

CttAP. XII.] OF CLITSTERS OF STARS. 375

jecte which are generally classed under the common head
of Nebulae, but which have been separated by him into
—1st, Clusters of stars, in which the stars are clearly
distinguishable ; and these, again, into globular and ir-
regular clusters ; 2d, Resolvable nebulae, or such as ex^
cite a suspicion that they consist of stars, and which
any increase of the optical power of the telescope may
he expected to resolve into distinct stars; 3d, Nebulae
properly so called, in which there is no appearance
whatever of stars ; which, again, have been subdivided
into subordinate classes, according to their brightness
and size ; 4th, Planetary nebulae ; 5th, Stellar nebulae ;
and, 6th, Nebulous stars. The great power of his tele-
scopes has disclosed to us the existence of an immense
number of these objects, and shown them to be distri*
buted over the heavens, not by any means uniformly,
but, generally speaking, with a marked preference to a
broad zone crossing the milky way nearly at right
angles, and whose general direction is not very remote
from that of the hour circle of 0** and 12**. In some
parts of this zone, indeed— especially where it crosses
the constellations Virgo, Coma Berenices, and the Great
Bear — they arc assembled in great numbers ; being,
however, for the most part telescopic, and beyond the
reach of any but the most powerful instruments.

(617.) Clusters of stars are either globjilar, such as
we have already described, or of irregular figure. These
latter are, generally speaking, less rich in stars, and es-
pecially less condensed towards the centre. They are
also less definite in point of outline ; so that it is often
not easy to say where they terminate, or whether they
are to be regarded otherwise than as merely richer parts
of the heavens than those around them. In some of them
the stars are nearly all of a size, in others extremely dif-
ferent ; and it is no uncommon thing to find a very red
star much brighter than the rest, occupying a conspi-
cuous situation in them. Sir William Herschel regards
these as globular clusters in a less advanced state of con-
densation, conceiving all such groups as approaching, by
their mutual attraction, to the globular figure, and assem-
bling themselves togeUier from all the surrounding re-
gion, under laws of which we have, it is true, no othe?

370 ▲ TttEATISS ON ASTRONOMY* [CBAP. ^U*

proof than the observance of a gradation by which their
characters shade into one another, so that it is impossible
to say where one species ends and the other begins.

(618.) Resolvable nebulae can, of coarse, only be con-
sidered as clusters either too remote, or consisting of
stars intrinsically too faint to affect us by their individual
light, unless where two or three happen to be close
enough to make a joint impression, and give the idea of
a point brighter tlian the rest. They are almost univer-
sally round or oval — their loose appendages, and irregu-
larities of form, being as it were extinguished by the dis*
tance, and only the general figure of the more condensed
parts being discernible. It is under the appearance of
objects of this character that all the greater globular clus-
ters exhibit themselves in telescopes of insufficient opti->
cal power to show them well ; and the conclusion is
obvious, that those which the most powerful can barely
render resolvable, would be completely resolved by a
further increase of instrumental force.

(619.) Of nebulae, properly so called, the variety is
again very great. By far the most remarkable are those
represented in Jigs, 2 and 3, plate II., the former of
which represents the nebulae surrounding the quadruple
(or rather sextuple) star d in the constellation Orion ; the
latter, that about », in the southern constellation Robur
Caroli: the one discovered by Huygens, in 1656, and
figured as seen in the twenty feet reflector at Slough ;
the other by Lacaille, from a figure by Mr. Dunlop, PhiL
Trans. 1827. The nebulous character of these objects,
at least of the former, is very diflferent from what might
be supposed to arise from Uie congregation of an im-
mense collection of small stars. It is formed of little
flocky masses, like wisps of cloud; and such wisps
seem to adhere to many small stars at its outskirts, and
especially to one considerable star (represented, in the
figure, below the nebula), which it envelopes with a ne-
bulous atmosphere of considerable extent and singular
figure. Several astronomers, on comparing this nebula
with the figures of it handed down to us by its discoverer,
Huygens, have concluded that its form has undergone a
perceptible change. But when it is considered how dif-
ficult it is to represent such an object duly, and how en-

CHAP. XIl.^ OF NEBUKJE. 377

tirely its appearance will differ, even in the same tele-
scope, according to the clearness of the air, or other tem-
porary causes, we shall readily admit that we have no
evidence of change that can be relied on.

(620.) Plate II. ,^5^. 3, represents a nebula of a quite
different character. The original of this figure is in the
constellation Andromeda near the star f. It is visible to
the naked eye, and is continually mistaken for a comet,
by those unacquainted with the heavens. Simon Marius,
.who noticed it in 1612, describes its appearance as that
of a candle shining through horn, and the resemblance
is not inapt. Its form is a pretty long oval, increasing
by insensible gradations of brightness, at first very gra-
dually, but at last more rapidly, up to a central point,
which though very much brighter than the rest, is yet
evidently not stellar, but only nebula in a high state of
condensation. It has in it a few small stars ; but they
are obviously casual, and the nebula itself offers not the
slightest appearance to give ground for a suspicion of
its consisting of stars. It is very large, being nearly
half a de^ee long, and 15 or 20 minutes broad.

(621.) This may be considered as a type, on a large
scsde, of a very numerous class of nebulse, of a round or
oval figure, increasing more or less in density towards
the central point : they differ extremely, however, in
this respect. In some, the condensation is slight and
gradual ; in others great and sudden : so sudden, indeed,
that they present the appearance of a dull and blotted
star, or of a star with a slight burr round it, in which
case they are called stellar nebulae ; while others, again,
offer the singularly beautiful and striking phenomenon
of a sharp and brilliant star surrounded by a perfectly
circular disc, or atmosphere, of faint light in some cases,
dying away on all sides by insensible gradations ; in
others, almost suddenly terminated. These are nebulous
stars, A. very fine example of such a star is ,55 Andro-
medae R. A. 1** 43", N. P. D. 50** 7'. • Orionis and i of
the same constellation are also nebulous ; but the nebula
is not to be seen without a very powerful telescope. In
the extent of deviation, too, from the spherical form,
which oval nebula affect, a great diversity is observed :
some are only slightly elliptic ; others much extended
2i2

37^ ▲ TREATI8S ON AfiTRONOHT. [CHAP. Zn«

in length ; and in some, the extension so great, as to
give the nebala the character of a long, narrow, spindle-
shaped ray, tapering away at both ends to points. One
of the most remarkable specimens of this kind is in
R. A. 12»» 28~ ; F. P. D. 63** 4'.

(622.) Annular nebulae also exist, but are among the
rarest objects in the heavens. The most conspicuous
of this class is to be found exactly half way between the
stars li and y Lyne, and may be seen with a telescope of
moderate power. It is small, and particularly weU de-
fined, so as in fact to have much more the appearance
of a flat oval solid ring than of a nebula. The axes of
the ellipse are to each other in the proportion of about
4 to 5, and the opening occupies about half its diameter :
its light is not quite uniform, but has something of a
curdled appearance, particularly at the exterior edge ;
the centrad opening is not entirely dark, but is filled up
with a faint hazy light, uniformly spread over it, like a
fine gauze stretched over a hoop.

(623.) Planetary nebulas are very extraordinary ob*
jects. They have, as their name imports, exactly the
appearance of planets ; round or slightly oval discs, in
some instances quite sharply terminated, in others a
little hazy at the borders, and of a light exactly equable
or only a very little mottled, which, in some of them, ap-
proaches in vividness to that of actual planets. What-
ever be their nature, they must be of enormous magnitade.
' One of them is to be found in the parallel of y Aqnarii,
and about 5*" preceding that star. Its apparent diameter
is about 20''. Another, in the constellation Andromeda,
presents a visible disc of 12", perfectly defined and
round. Granting these objects to be equally distant
from us with the stars, their real dimensions must be
such as would fill, on the lowest computation, the whole
orbit of Uranus. It is no less evident that, if they be
solid bodies of a solar nature, the intrinsic splendour of
their surfaces must be almost infinitely inferior to that
of the sun's. A circular portion of the sun's disc, sub-
tending an angle of 20", would give a light equal to
100 JuU moona; while the objects in question are
hardly, if at all, discernible with the naked eye* The
uniformity of their discs, and their want of apparent

CHAF. Xn.] PLANBTARY NEBULJB. 879

central condensation, would certainly augar their light
to be merely superficial, and in the nature of a hollow
spherical shell : but whether filled with solid or gaseous
matter, or altogether empty, it would be a waste of
time to conjecture.

(624.) Among the nebnlfe which possess an evident
symmetry of form, and seem clearly entitied to be re-
garded as systems of a deifinite nature, however myste*
rious their structure and destination, the most remark-
able are the 51st and 27th of Messier's catalogue* The
former consists of a large and bright globular nebula
surrounded by a double ring, at a considerable distance
from the globe or rather a single ring divided through
about two fifths of its circumference into two laminiBy
and having one portion, as it were, turned up out of the
plane of the rest. The latter consists of two bright and
highly condensed round or slightly oval nebuke, united by
a short neck of nearly the same density. A faint nebu*
lous atmosphere completes the figure, enveloping them
both, and filling up the outline of a circumscribed ellipsei
whose shorter axis is the axis of symmetry of the sys-
tem about which it may be supposed to revolve, or the
line passing through the centres of both the nebulous
masses. These objects have never been properly de-
scribed, the instruments with which they were originally
discovered having been quite inadequate to showing the
peculiarities above mentioned, which seem to place Siem
in a class apart firom all others. The one ofiers obvions
analogies either with the structure of Saturn or with
that of our own sidereal firmament and milky way« The
other has little or no resemblance to any other known
object.

(625.) The nebuls furnish, in every point of view,
an inexhaustible field of speculation and conjecture.
That by far the larger share of them consist of stars
there can be little doubt ; and in the interminable range
of system upon system, and firmament upon firmament,
which we thus catch a glimpse of, the imagination is be-
wildered and lost. On the other hand, if it be true, as,
to say the least, it seems extremely probable, that a phos-
phorescent or self-luminous matter also exists, dissemi-
nated through extensive regions of space, in the manner

880 ▲ TREATISE ON ASTROXOMT. [cHAP. Xn*

of z cloud or fog — now assuming capricious shapes, like
actual cloads drifted by the wind, and now concentrating
itself like a cometic atmosphere around particular stars ;
what, we naturally ask, is the nature and destination of
this nebulous matter ? Is it absorbed by the stars in
whose neighbourhood it is found, to furnish, by its con-
densation, their supply of light and heat ? or is it pro-
gressively concentrating itself by the effect of its own
gravity into masses, and so laying the foundation of new
sidereal systems or of insulated stars ? It is easier to
propound such questions than to offer any probable reply
to them. Meanwhile, appeal to fact, by the method of
constant and diligent observation, is open to us ; and, as
the double stars have yielded to this s^le of questioning,
and disclosed a series of relations of the most intelligible
and interesting description, we may reasonably hope
that the assiduous study of the nebulee will, ere long, lead
to some clearer understanding of their intimate nature.

(626.) We shall conclude this chapter by the men-
tion of a phenomenon which seems to indicate the ex-
istence of some slight degree of nebulosity about the sun
itself, and even to place it in the list of nebulous stars.
It is called the zodiacal light, and may be seen any very
clear evening soon afler sunset, about the months of
April and May, or at the opposite season before sunrise,
as a cone or lenticular-shaped light, extending from the
horizon obliquely upwards, and following, generally,
the course of the ecliptic, or rather that of the sun's
equator. The apparent angular distance of its vertex
from the sun varies, according to circumstances, from
40^ to 90^, and the breadth of its base perpendicular to
its axis from 8^ to 30°. It is extremely faint and ill de-
fined, at least in this climate, though better seen in tro-
pical regions, but cannot be mistaken for any atmo-
spheric meteor or aurora borealis. It is manifestly in the
nature of a thin lenticularly-formed atmosphere, sur-
rounding the sun, and extending at least beyond the
orbit of Mercury and even of Venus, and may be .con-
jectured to be no other than the denser part of that me-
dium, which, as we have reason to believe, resists the
motion of comets ; loaded^ perhaps, with the actual ma^
terials of the tails of millions of those bodies, of which

CRAP. Xni.] OF THE CALENDAR. 3§1

they have been stripped in their fluccessive perihelion
passages (art. 487), and which may be slowly subsiding
into the san.

CHAPTER XIII.

OF THE CALENDAR.

(627.) Time, like distance, may be measured by com-
parison with standards of any length, and all that is
requisite for ascertaining correctly the length of any in-
terval, is to be able to apply the standard to the interval
throughout its whole extent without overlapping on the
one hand, or leaving unmeasured vacancies on the other;
to determine, without the possible error of a unit, the
number of integer standards which the interval admits
of being interposed between its beginning and end ; and'
to estimate precisely the fraction over and above an
integer, which remains when all the possible integers are
subtracted.

(626.) But though all standard units of time are equally
possible, theoretically speaking, all are not, practically,
equally convenient. The tropical year and the solar day
are natural units, which the wants of man and the busi-
ness of society force upon us, and compel us to adopt
as our greater and lesser standards for the measurement
of time, for all the purposes of civil life ; and that, in
spite of inconveniences which, did any choice exist,
would speedily lead to the abandonment of one or other.
The principal of these are their incommen8ur ability, and
the want of perfect uniformity in one at least of them.

(629.) The mean lengths of the sidereal day and year,
when estimated on an average sufficiently large to com-
pensate the fluctuations arising from nutation in the one,
and from inequalities of configuration in the other, are
the two most invariable quantities which nature presents
us with ; the former, by reason of the uniform diurnal
rotation of the earth — the latter on account of the inva^
riability of the axes of the planetary orbits. Hence it
follows that the mean solar day is also invariable. It is

362 A TREATISE ON ASTRONOMY. [cHAP. XIHi?

Otherwise with the tropical year. The motion of the
equinoctial points varies not only from tho retrograda^
tion of the equator on the ecliptic, but also partly from
that of the ecliptic on the orbits of all the other planets.
It is therefore variable, and this produces a variation in
the tropical year, which is dependent on the place of the
equinox (arts. 517, 328). The tropical year is actually
above 4*21' shorter than it was in the time of Hippar-
chus. This absence of the most essential requisite for
a standard, viz. invariability, renders it necessary, since
we cannot help employing the tropical year in our reck-
oning of time, to adopt an arbitrary or artificial value for
it, so near the truth, as not to admit of the accumulation
of its error for several centuries producing any practical
mischief, and thus satisfying the ordinary wants of civil
life ; while, for scientific purposes, the tropical year, so
adopted, is considered only as the representative of a
certain numher of integer days and a fraction — the day
being, in effect, the only standard employed. The case
is nearly analogous to the reckoning of value by guineas
and shillings, an artificial relation of the two coins being
fixed by law, near to, but scarcely ever exactly coincident
with, the natural one, determined by the relative market
price of gold and silver, of which either the one or the
other — ^whichever is really the most invariable, or the
most in use with other nations — may be assumed as the
true theoretical standard of value.

(630.) The other inconvenience of the standards io
question is their incommensurability. In our measure
of space, all our subdivisions are into aliquot parts : a
yard is three feet, a mile eight furlongs, &c. But a year
is no exact number of days, nor an integer number with
any exact fraction, as one third or one fourth, over and
above ; but the surplus is an incommensurable fraction,
composed of hours, minutes, seconds, &c., which pro-
duces the same kind of inconvenience in the reckoning
of time that it would do, in that of money, if we had
gold coins of the value of twenty-one shillings, with odd
pence and farthings, and a fraction of a farthing over. For
this, however, there is no remedy but to keep a strict re-
gister of the surplus fractions ; and, when tiiey amount
to a whole day, cast them over into the integer account.

CHAP. XIII.J OF THE CALENBAR.

(631.) To do this in the simplest and most eofiEvenie&t
manner is the object of a well-adjusted calendar. In the
Gregorian calendar, which we follow, it is accomplished,
with remarkable simplicity and neatness, by carrying a
little farther than is done above the principle oi an as-'
sumed or artificial year, and adopting two such years,,
both consisting of an exact integer number of days^i
viz. one of 365 and the other of 366, and laying down a
simple and easily remembered rule for the order in which
these years ahall succeed each other in the civil reckoning
of time, so that during the lapse of at least some thou-^
sands of years the sum of the integer artificial, or Gre-
gorian, years elapsed shall not differ from ^e same
number of real tropical years by a whole day. By this
contrivance, the equinoxes and solstices will always fall on
days similarly situated, and bearing the same name, in each
Gregorian year ; and the seasons will for ever correspond
to the same months, instead of running the round of the
whole year, as they must do upon any other system of
reckoning, and used, in fact, to do before this was adopted^
(632.) The Gregorian rule is as follows : — The years-
are denominated from the birth of Christ, according to*
one chronological determination of that event. Every
year whose number is not divisible by 4 without re--
mainder, consists of 365 days ; every year which is so-
divisible, but is not divisible by 100, of 366 ; every year
divisible by 100, but not by 400, again of 365 ; and every
year divisible by 400, again of 366. For example, the^
year 1833, not being divisible by 4, consists of 365
days; 1836 of 366; 1800 and 1900 of 365 each; but
2000 of 366. In order to see how near this rule will
bring us to the truth, let us see what number of days
10000 Gregorian years will contain, beginning with the
year 1. Now, in 10000, the numbers not divisible by 4
will be i of 10000, or 7500 ; those divisible by 100, but
not by 400, will in like manner be | of 100, or 75 ; so
that, in the 10000 years in question, 7575 consists of
366, and the remaining 2425 of 365, producing in all
3652425 days, which would give for an average of each
year, one with another, 365^**2425. The actual value of
the tropical year (art. 327) reduced inta a decimal frac-
tion, is 365*24224v so the error of the Gregorian rule oa

IM ▲ TREATISE ON ASTRONOMY. [CHAF. XIII.

10000 of the present tropical years is 2*6, or 7^ 14^ 24" ;
that is to say, less than a day in 3000 years ; which is
more than sufficient for all haman purposes, those of the
astronomer excepted, who is in no danger of being led
into error from this cause. Even this error might be
avoided by extending the wording of the Gregorian rule
one step farther than it| contrivers probably thought it
worth while to go, and declaring that years divisible by
4000 should consist of 365 days. This would take off
two integer days from the above calculated number, and
2*5 from a larger average ; making the sum of days in
100000 Gregorian years, 36524225, which differs only
by a single day from 100000 real tropical years, such as
they exist at present.

(633.) As any distance along a high road might,
though in a rather inconvenient and roundabout way, be
expressed without introducing error by setting up a series
of milestones, at intervals of unequal lengSis, so that
every fourth mile, for instance, should be a yard longer
than the rest, or according to any other fixed rule ; taking
care only to mark the stones, so as to leave room for no
mistake, and to advertise all travellers of the difference
of lengths and their order of succession ; so may any in-
terval of time be expressed correctly by stating in what
Gregorian years it begins and ends, taid whereabouts in
each. For this statement, coupled with the declaratory
rule, enables us to say how many integer years are to be
reckoned at 365, and how many at 366 days. The latter
years are called bissextiles, or leap-years, and the sur-
plus days thus thrown into the reckoning are called inr
terealary or leap-daya,

(634.; If the Gregorian rule, as above stated, had al-
ways been adhered to, nothing would be easier than to
reckon the number of days elapsed between the present
time and any historical recorded event. But this is not
the case ; and the history of the calendar, with reference
to chronology, or to the calculation of ancient obserraF^
tions, may be compared to that of a clock, going regularly
when left to itself, but sometimes forgotten to be wound
up ; and when wound, sometimes set forward, sometimes
backward, and that often to serve particular purposes and
private interests. Such» at least, appears to have been

CHAP. XIII.] OF TBE CALENDAR. 385

the case with the Roman calendar, in which our own
originates, from the time of Numa to that of Julius
Csesar, when the lunar year of 13 months, or 355 days,
was augmented at pleasure, to correspond to the solar,
by which the seasons are determined, by the arbitrary
intercalations of the priests, and the usurpations of Ae
decemvirs and other magistrates, till the confusion be-
came inextricable. To Julius Caesar, assisted by Sosi-
genes, an eminent Alexandrian astronomer and mathe-
matician, we owe the neat contrivance of the two years
of 365 and 366 days, and the insertion of one bissextile
after three common years. This important change took
place in the 45th year before Christ, which was the first
regular year, commencing on the 1st of January, being
the day of the new mioon immediately following the
winter solstice of the year before. We may judge of
the state into which the reckoning of time had fallen,,
by the fact, that, to introduce the new system, it was
necessary to enact that the previous year (46 b. c.)
should consist of 455 days, a circumstance whidi ob*"
tained it the epithet of ." the year of confusion."

(635.) The Julianrrule made every fourth year, with-
out exception, a bissextile. This is, in fact, an over-
correction ; it supposes the length of the tropical year to
be 365 jdy which is too great, and thereby induces . an
error of 7 days in 900 years, as will easily appear on
trial. Accordingly, so early as the year 1414, it began
to be perceived &at the equinoxes were gradually creep-
ing away from the 2l8t of March and Septeariber, wh^re
they ought to have always fallen had the Julian year
been exact, and happening (as it appeared) too early»
^ The necessity of a firesh and effectual reform in the ealen*
dar was from that time continually urged, and at length
admitted. The change (which took place under the
popedom of Gregcnry XIII.) consisted in the omission of
ten nominal days a^r the 4th of .October^ 1582 (so that
the next day was called the l&th, and fiot the 5th)^ and
the promulgation of the rule already esplaSoed for future
regulation. The change was adopted immediately in all
catholic countries ; but more slowly in protestant. In
England, ^' the change of style," as it was called, took
place after the 2d of September, 1752, eleven nominal
2K

380 A TREATISE ON ASTRONOMY. fcHAF. XtU*

days being then struck oat ; so that, the last day of Old
Style being the 2d, the first of New Style (the next day)
was called the 14th, instead of the- 3d. The same legis-
lative enactment which established the Gregorian, year
in England in 1752, shortened the preceding year, 1751,
by a full quarter. Previous to that time, the year waa
held to begin with the 25th March, and the year a. ih,
1751 did so accordingly ; but that year was not suffered
to run out, but was supplanted on the 1st January by
the year 1752, which it was enacted should com-
mence on that day, as well as every subsequent year^
Russia is now the only country in Ekirope in which the
Old Style is still adhered to, and (another secular year
having elapsed) the difference between the European and
Russian dates, amounts, at present, to 12 days.

(636.) It is fortunate for astronomy that Uie confusion
of dates and the irreconcilable contradictions which his-
torical statements too often exhibit, when confronted
with the best knowledge we possess of the ancient reck*
onings of time, affect recorded obserrations but little. An
astronomical observation, of any striking and well marked
phenomenon, carries with it, in most cases, abundant
means of recovering its exact date, when any tolerable ap*
proximation is afforded to it by chronological records ;.
and, so far from being abjecdy dependent on the ob-
scure and often contradictory dates which the compari-
son of ancient authorities indicates, is often ilself the
surest and most convincing evidence on which achiono-
logical epoch can be brought to rest. Remarkable eclipses,
for instance, now that the lunar theory is thoroughly un-
derstood, can be calculated back for several thousands of
years, without the possibility of mistaking the day of
their occurrence. And whenever any such eclipse is so
interwoven with the account given by an ancient author
of some historical event, as to indicate precisely the
interval of time between the edipsc amd the event, and
at the same time completely to identify the eclipse, that
date is recovered and fixed for ever.*

(637.) The days thus parcelled out into years, the

* See the remarkable calculations of Mr. Baily relative to the cel^
bnited solar eclipse which put an end to the battle between the kinfl»
of Media and Lydia, b. c. 610, Sept 3a PMl. Trans- ci. 880.

CHAP. XIII.] OF TBS CALENDAR. 387

next step to a perfect knowledge of time is to secure the
identification of each day, by imposing on it a name uni-
versally known and employed. Since, however, the
days of a whole year are too numerous to admit of load-
ing the memory with distinct names for each, all nations
have felt the necessity of breaking them down into par-
eels of a more moderate extent ; giving names to each
of these parcels, and particularizing the days in each by
numbers, or by some especial indication. The lunar
month has been resorted to in many instances ; and some
nations have, in fact, preferred a lunar to a solar chro-
nology altogether, as the Turks and Jews continue to do
to this day, making the year consist of 13 lunar months,
or 355 days.* Our own division into twelve unequal
months is entirely arbitrary, and often productive of con-
fusion, owing to the equivoque between the lunar and
calendar month. The intercalary day. naturally attaches
itself to February as the shortest.

* The Metonic ^cle, or the &ct, discovered by Meton, a Greek ma-
thematician, that 19 solar yean contain just 235 lunations (which in fact
they do to a very great degree of approximation), was duly appreciated
by the Greeks, as ensuring the correspondence of the solar and lunar
years, and honours were decreed to its discoTerer.

988 A TRXATI8B ON ASTRONOmT.

NOTE
0» Ike Cowiri litfi o n cf a COaAuUtr ChaUr, rtfemd to tn page 374.

If we suppose a globular space filled with equal stan, unifiNrmly dis-
peised throuffh it, and yery numerous, each of them attracting every
odier wid^a force inverwly as the square of the distance, the resultant
lane by which an^ one of them (those at the surftce alone excepted)
will be urged, in virtue of their joint attractions, will be directed towards
the common centre of the sphere, and will be directly as the distance
therefrom. This follows from what Newton has proved of the internal
attraction of a homogeneous sphere. Now, under such a law of force,
each particular star would describe a perfect ellipse about die conunon
centre of ^vitv, as its centre, and that^ in whafiBver plane and whatever
direction it might revolve. The condition, therefore, of a rotation of
the cluster, as a mass, about a single axis would be unnecessary. Each
ellipse, whatever mig^t be the proportion of its axes, or the uiclination of
its plane to the others, would be invariaUa m every pariicMiar, wod all
would be described in one common period, so that at the end of every
such period or aimics magnus of the svstem, every star of the cluster
(except the superficial ones) would be exactly re-esfiabtished in its
original position, thence to set tmt afresh and run the same unvarying
found finr an iiidefinite succession of ages. Supposing theur motions,
therefore, to be so adjusted at aiw one moment as that the orbits
should not intersect each other, and so that the maniitude of each star,
and the sphere of its more intense attraction, should bear but a small pro-
portion to die distance separating the individuals, such a svsteai^it is
obvious, might subsist, ana realize, in great measure, that aostiact and
ideal harmony, which Newton, in the 89th Proposition of the First Book
of the Prtaqpia, has shown to characterize a law of force direcdy as
the distance. See also Quarteriy Review, Na 94, p. 540.— AtilAor*

A TRBATISB ON ASTSONOMT.

Stvoptic Tablx or thx Eunirrs ov thi Solas Ststbk.

N. B.— The data for Vesta, Juno, Ceres, and Pallat are for January 1, IflSOL
The rest for January 1, 1801.

- Planers
name.

. Mean distance
from Sun, or
Semi-axis.

Mean Sidereal

Period in Mean

Solar Days.

Eccentricity in
ParUofthe
SemiaTis.

Mercury

Venus

Earth

Mars

Vesta

Juno

Geres

Pallas

Jupiter

Saturn

Uranus.

0-3870981
0-7233316
1-0000000
1-5236923
3-3678700
2*6690090
2-7672450
2-7728860
6-2027760
9-5387861
19-1823900

87-9692580

224-7007869

365-2563612

686-9796458

1325-7431000

1692-6608000

1681-3931000

1686-:538a000

4339-5848212

10759-2198174

30686-8208296

0-2055149 *

0-0068607

0^)167836

0-0933070

0-0891300

0-2578480

0-0784390

0-2416480

0-0481621-

0-0561505

0-0466794

Planers
name.

InclinaUoB to the
Eeliptie.

Longitude pf
ascending Node.

PeribeUon.

venus

Earth

Mars

Vesta

Juno

Geres

Pallas

Jupitur

Saturn

Ujanus

70 O' 9"-l
3 23 28 -5

450 57' 30" '9
74 64 12 '9

740 21' 46" -9
128 43 53 -1

99 30 5-0
332 23 66 -O
249 33 24 -4

53 S3 46 -0
147 7 31 -5
121 7 4-3

11 8 34 -O

89 9 29 -8
167 31 16 -1

1 51 6 -2

7 8 9-0

13 4 9-7

10 37 26 •2

34 34 55 -0

1 18 51 -3

2 29 35 -7
f> 46 %S -4

48 b 3-5
103 13 }8 -2

171 7 40 -4
80 41 24 H)

172 39 26 -8
98 26 18 -9

111 56 37 -4
72 59 35 -3

Planers
name.

Mean Longitude
at tbe Bpocb.

Mass in BUlionths
oftheSun's.

Equatorial Dia-
meter, the Sun*s
being 111-454.

Mercury

Venus

Earth

Mars

Vesta

J«w

Gerss

Pallas
Jupiter
Saturn
Uranus

166« 0^48'' -6
11 33 3 -0
100 39 10 -2
64 22 55 -5
278 30 -4
200 16 19 -1
123 16 11 -9
108 24 57 -9
112 15 23 -0
135 20 6 -5
177 48 23 •O

493628
2463836
2817409

392735

0-908
9'976
1-000
0-517

10-860
9-987
4-332

953570222

284738000

55809812

2k9

aoo

A TBBATI8K OM A8TB0N0MT.

Synoptic Table ov thk Elements of the Orbits oi
THE Satellites, so far as they are known.

N. B.— The diitancM are ezpfraMed in equatorial radii of llie pri-
nariM. The epoch is Jan. 1, 1801. The period^ dec* are ex-
preaeed in mean solar days.

L The Moon.

Mean distaBoe from earth
Mean sidereal revolution
Mean synodical ditto
Eccentiidty of orbit . .
Mean reyolution of nodes
Mean rerolution of apogee . ■
Mean longitude of node at epoch
Mean longitude of perigee at do.
Mean ineUnaMon of oifatt
Mean longitude of moon at epoch
Mass, that of earth being 1, .
Diameter in miles

29'-98317600
37^391661418
89^690588716
0^)54844200
6793d-391080
3332^575343

13° 63^ 17^ -7

266 10 7 -5

6 8 47 -9

118 17 8 -3

0-013517S

2160

n. Satellites of JmnrER.

Set

Mean Distoaee.

Sidereal
ReYOlution.

Inclination of

Orbit to that of

Jopiter.

Iffaasj tbat^
of^jiter

1
2
3

4

6*04868

9-62347

15*85034

26*99835

Id igh 38»
8 18 14
7 8 43
16 16 32

S« ft' 80"
Yanable
Variable

2 56 48

17328
23336
88497
42669

Tlie eccentricities of the 1st and 2d satellite are inaenaibie, tfiat
of the 3d and 4th small^ but Tariable in conw^enoe of their mutual
pertortntions.

A TREATISE ON ASTROWOMT.

301

III. Satellites of Saturn.

Sat.

Mean
Distance

Sidereal
Revolution.

' Eccentricities and Inclinations.

3*361
4*300
6-284
6*819
9*624
22«081
64*359

0* 22»» 38-"
1 8 63

1 21 18

2 17 46
4 12 26

16 22 41
79 7 66

The orbitB of the six interier
sateUites are neariy drcular,
and veiy nearly in &e plane of
the ring. That of the seyenth
18 considerably inclined to the
rest and ai^roaches nearer to
coincidence with the ecliptic

IV. Satellites of Uranus.

Bat

Bfean
Distance.

aiilereal Period.

IneliBatioBtoEeUptie.

11

18*120

64 21h 25"» 0-

Their orbits are inclined

2

17*022

8 16 66 6

about 799 68' ta the

31

19*846

10 23 4

ecliptic, and their motion

4

22-752

13 U 8 69

Is retrograde. The pe-

61

46*607

38 1 48

riods of the 2d and 4th

61

91*008

107 16 40

t

tion. The orbits appear
to be nearly drdes.

FlaUl.

^■2Ji!ifir Si2^^

J}'fanfy Sr.

9

i»1

• i':4 .

■•-» ^^:.^•

^

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