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Full text of "A treatise on astronomy"

Presented to the 

LIBRARY of the 


Miss M. Robertson 




F.R.S.L. &E. M.R.I.A. F.R.A.S. F.G.S. M.C.U.P.S. 





^--V ^r^ 

" Et quoniam eadem natura cupiditatem ingenult hominibus veri 
inveniendi, quod facillimi apparet, cum vacui curis, etiam quid in 
coelo fiat, scire avemus : his initiis inducti omnia vera diligimus ; id 
est, fidelia, simplicia, constantia ; turn vana, falsa, fallendia odimus." 

Cicero, de Fin. Jion. ct jMcil. ii. 14. 

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



Introduction Page 7 


General Notions Form and Magnitude of the Earth Horizon and 
its Dip Tlie Atinos])here Refrartion Twilight Appearances 
resuhing from diurnal Motion Parallax First Step towards form- 
ing an Idea of the Distance of the Stars Definitions - - - 14 


Of the Nature of astronomical Instruments and Observations in gene- 
ral Of sidereal and solar Time Of the Measurement of Time 
Clocks, Chronometers, the Transit Instrument Of the Measure- 
ment of angular Intervals Application of the Telescope to Instru- 
ments destined to tliat Purpose Of the Mural Circle Determina- 
tion of polar and horizontal Points Tlie Level Plumb Line 
Artificial Horizon Collimator Of compound Instruments with 
co-ordinate Circles, the Equatorial Altitude and Azimuth Instru- 
ment Of the Sextant and Reflecting Circle Principle of Repeti- 
tion 66 



Of the Figure of the Earth Its exact Dimensions Its Form that of 
Equilibrium mothlied by Centrifugal Force Variation of Cravity 
on its Surface Statical and dynamical Measures of Gravity The 
Pendulum Gravity to a Spheroid Other Effects of Earth's Rota- 
tion Trade-winds Determination of geographical Positions Of 
Latitudes Of Longitudes Conduct of a trignometrical Survey 
Of Maps Projections of the Sphere Measurement of Heights by 
the Barometer -.-..-.... 105 



Construction of celestial Maps and Glolx^s by Observations of right 
Ascension and Declination Celestial Objects distinguished into 
fixed and erratic Of the Constellations Natural Regions in the 
Heavens The Milky Way The Zodiac Of the Ecliptic Celes- 
tial Latitudes and Longitudes Precession of the Equinoxes Nu- 
tation Aberration Uranographical Problems .... 151 




OF THE sun's motion. 


Apparent Motion of the Sun not unifonn Its apparent Diameter also 
variable Variation of its Distance concluded Its apparent Orbit 
an Ellipse about the Focus Law of tlie angular Velocity Equa- 
ble Description of Areas Parallax of the Sun Its Distance and 
Magnitude Coperniean Explanation of the Sun's apparent Motion 
Parallelism of the Earth's Axis The Seasons Heat received 
from the Sun in diflerent Parts of the Orbit 17 


Of the Moon Its sidereal Period Its apparent Diameter Its Paral- 
lax, Distance, and real Diameter First Approximation to its Orbit 
An Ellipse about the Earth in the Focus Its Eccentricity and 
Inclination Motion of the Nodes of its Orbit Occultations Solar 
Eclipses Pliases of the Moon Its synodical Period Lunar 
Eclipses Motion of the Apsides of its Orbit Physical Constitution 
of the Moon Its Mountains Atmosphere EJotation on Axis 
Libration Appearance of the Earth from it - - - 203 


Of terrestrial Gravity Of the Law of universal Gravitation Paths 
of Projectiles ; apparent, real The Moon retained in licr Orbit by 
Gravity Its Law of Diminution Laws of elliptic Motion 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 Gravity at its Surface Disturbing Eflect of the Sun on 
the Moon's Motion 221 



Apparent Motions of the Planets Their Stations and Retrograda- 
tinns ^The Sun their natural Centre of Motion Inferior Planets 
Their Phases, Periods, &c. Dimensions and Form of their Orbits 
Transits across tlie Sun Superior Planets, 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 ultra-zodiacal Pla- 
nets Physical Peculiarities observable in each of the Planets - 231 



Of the Moon, as a Satellite of the Earth General Proximity of Satel- 
lites to their Primaries, and consefjuent Subordination of their 
Motions Masses of the Primaries concluded from the Periods of 
their Satellites Maintenance of Kepler's Laws in the secondary 
Systems Of Jupiter's Satellites Their Eclipses, &c. Velocity 
of Light discovered by their Moans Satellites of Saturn Of 
Uranus 272 




Great Number of reoorded Comets The number of unrecorded 
probably much greater Description of a Comet Comets without 
Tails Increase and Decay ol' their Tails Their Motions Sub- 
ject to the general Laws of planetary Motion Elements of their 
Orbits Periodic Return of certain Comets Halley's Encke's 
Biela's Dmiensions of Comets Their Resistance by the Etlier, 
gradual Decay, and passible Dispersion in Space .... 284 



Subject propotmded Superposition of small Motions Problem of 
three Bodies Estimation of disturbing Forces Motion of Nodes 
Changes of Inclination Compensation operated in a whole 
Revolution of the Node Lagrange's Theorem of the Stability of 
the Inclinations Chaniie of Obliquity of the Ecliptic Precession 
of the Equinoxes Nutation Theorem resjiecting forced Vibra- 
tions Of the Tides Variation of Elements of the Planet's Orbits 
Periodic and secular Disturbing Forces considered as tangen- 
tial and radial Effects of tangential Force 1st, in circular Orbits ; 
2d, in elii|)tic Compensations effected Case of near Commen- 
Rurability of mean Motions The great Inequality of Jupiter and 
Saturn explained The long Inequality of Venus and the Earth 
Lunar Variation Effects of the radial Force Moan Eflcct of the 
Period and Dimensions of the disturbed Orbit Variable Part of 
its Elfect Lunar Evoction Secular Acceleration of the Moon's 
Motion Permanence of the Axes and Periods Theory of the secu- 
lar Variations of the Eccentricities and Perilielia Motion of the 
lunar Apsides Lagrange's Theorem of the Stability of the Ec- 
centricities Nutation of the lunar Orbit Perturbations of Jupi- 
ter's Satellites 294 



Of the Stars generally ^Tlieir distribution into Classes according to 
their apparent Magnitudes Their apparent Distribution over the 
Heavens Of the Alilky Way Annual Parallax Real Distances, 
probable Dimensions, and Nature of the Stars Variable Stars 
Temporary Stars Of double Stars ^Their Revolution about each 
other in elliptic Orbits Extension of the Law of Gravity to such 
Systems Of coloured Stars Proper Motion of the Sun and Stars 
Systematic Aberration and Parallax Of compound sidereal 
Systems Clusters of Stars Of Nebulas Nebulous Stars Annu- 
lar and planetary Nebulae Zodiacal Light - . - . . 349 





Of the Calendar 381 

Synoptic Table of the Elements of the Solar System - - -389 

Synoptic Table of the Elements of the Orbits of the Satellites, so far 

as they are known 390 

I. The Moon 390 

II. Satellites of Jupiter 390 

III. Satellites of Saturn 391 / 

IV. Satellites of Uranus 391 ''^ 

Index - 393 





(1.) In entering upon any scientific pursuit, 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 hastily adopted 
notions respecting the objects and relations he is about 
to examine as may tend to embarrass or mislead him ; 
and to strengthen liimself, by something of an effort and 
a resolve, for the unprejudiced admission of any con- 
clusion which shall appear to be supported by careful 
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, 
stands 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 




the points of view may be in whicli 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 everyone, 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 stractures, either of art 
or nature, is divested by the astronomer of its a'ttrilnite 
of fixity, and conceived by him as turning swiftly on its 
centre, and at the same time moving onwards through 
space with ^reat 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 approacliing 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 Avorlds ; several of them vastly greater and far more 
curiously furnished tlian 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 efiulgent 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 point; so lost 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 hardly anything 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 


our belief when once admitted. In the conduct, therefore, 
of this volume, we shall take it for granted that our 
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 comes 
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 detinite 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 didactic 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 time 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 matured and ascer- 
tained character of the science itself, render this course 
both practicable and eligible. Practicable, because there 
is now no danger of any revolution in astronomy, like 
those which are daily changing the features of the less 
advanced sciences, supervening, ta 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 measured steps from the known to the un- 
known, maybe more advantageously devoted to each 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. 


Writing only to l)e unilerstood, and to communicate as 
imicli information in as little space as possible, consist- 
ently witli its distinct and rjfeciual communication, we 
can atTord to make no sacrifice to system, to form, or to 

(5.) We shall take for granted, from the outset, the 
Copcrnican 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 tlie form- 
ality of demonstration or the superfluous tedium of 
eulogy, calling to mind that important remark of Bacon : 
" Theoriarum vires, arcta et quasi se mutuo sustiaente 
partium adaptatione, qua, quasi in orbem coherent, lir- 
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 should 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 geometry and 
trigonometry, botli 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 Cy- 
clopaedia which profess to treat of the several subjects in 
question. Of course, the more of such knowledge he 
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- 

* Tlie confirmation of theories relies on the compact adaptation of 
their parts, hy which, like those of an arch or dome, they mutually 
sustain each other, and form a coherent whole. 


cal studies with the present work (though of such, at 
least in tlie latter predicament, we trust the number will 
be few), that its utmost pretension is to place them on 
the threshold of this particular wing of the temple of sci- 
ence, or rather on an eminence exterior to it, whence 
they may obtain something like a general notion of its 
structure ; or, at most, to give those who may wish to 
enter, aground-plan of its accesses, and put them in pos- 
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, a sound and sufficient know- 
ledge of mathematics, the great instrument of all exact in- 
quiry, tvithout ivhich no man can ever make such ad- 
vances in this or any other of the higher departments of 
science, as can entitle him to form an independent opi- 
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 illustrations to 
the necessities of such an intercourse. Propositions 
which to the one are almost identical, are theorems of 
import and difficulty to tire other ; nor is their evidence 
presented in the same way to the mind of each. In 
teaching such propositions, under such circumstances, 
the appeal has to be made, not to the pure 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, not 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 similar 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 of 
science to meet its exigencies. The diflerenee 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 
the faith of the student (though in some complex cases 
indispensable, where illustrative explanation would defeat 


its 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 Imnd, althougli it is something- now 
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 chielly by the way 
of illustration and familiar parallels, it is yet not impossi- 
ble that those who are already well acquainted with 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 acquired 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 uncommon, 
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 Avhich 
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 him, 
and the numerical and geometrical relations of his pro- 
blems all clear to his conception, when his forces are 
estimated and his lines measured, nay, when even he has 
followed up the application of his technical processes, and 


fairly arrived at his conclusion, there is still something 
wanting in his mind not in the evidence, 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 of ac- 
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 as 
such to his mind : he has not seen, as it were, the pro- 
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, will 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 paraphrased (if we may use the expression) by 
any single instance adducible in the ordinary course of 
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 more concealed effects of planetary perturba- 
tion than we had acquired by their mathematical investi- 
gation in detail, we may reasonably hope that the endeavour 
will not always be unattended with a similar success in 

(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 methods 
of observation, or the formulaj he requires prepared to 
his hand, or their demonstrations drawn out in detail. In 
all these the present work will be found meagre, and quite 
inadequate to his wants. Its aim is entirely different ; be- 
ing to present in each case the mere ultimate rationale of 
facts, arguments, and processes ; and, in all cases of mathe- 



muticiil fvpi)liculion, avoidiiiir wliatcver would tend to en- 
cunihor its pages with aloebraic or geometrical symbols, 
to place under his inspection that central thread of com- 
mon sense on which the pearls of analytical research are 
invariably strung ; but which, by the attention the latter 
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 thejn. 
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 hoiv little can be accomplish- 
ed in carrying such pi'ocesses 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 the human rea- 
son, that Ave 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 comprehension. 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. 


General Notions Form and Magnitude of the Earth Horizon and its 
Dip The Atmosphere Refraction Twihght Appearances result, 
ing from diurnal Notion Parallax First Step towards forming an 
Idea of the 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 


traced by the effects produced, and established by legi- 
timate reasoning, form the assemblage of objects to 
which the attention of the astronomer is directed. The 
term astronomy* itself, which denotes the law or rule 
of the astra (by which the ancients understood not only 
the stars properly so called, but the sun, the moon, and 
all the visible constituents of the heavens), sufficiently 
indicates this ; and, although the term astrology, which 
denotes the reason, theory, or interpretation 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 plane- 
tary influences, tlie same meaning originally attached 
itself to that epithet. 

(12.) But, besides the stars and other celestial bodies, 
the earth itself, regarded as an individual body, is one 
principal object of the astronomer's consideration, and, 
indeed, the chief of all. It derives its importance, in a 
practical as well as theoretical sense, not only from its 
proximity, and its relation to us as animated beings, 
who draw from it the supply of all our wants, but as the 
station from which we see all the rest, and as the only 
one among them to which we can, in the first instance, 
refer 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 tlie 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, which 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 

* A<rT>if , a Star ; i'ii/ot, a law ; or i-i^wsn', to tend, as a shepherd his flock ; 
so that y-o-Tfovoiioj means "shepherd of tlie stare." The two etymologies 
are, however, coincident. 

t A o^c,-, reason, or a luorJ, die vehicle of reason; the interpreter of 


hours of the day or night, or at difTercnt seasons 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 efibrt 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 modifica- 
tions 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 all the 
information Ave 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 really 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 
change, 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 


apparent, the results of the shifting of our own point of 
view ; and that, if we woukl ever arrive 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, Avhat 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 inhaljitants : we must remember that it is of the earth 
us a ivhole, 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 motion, 
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 that 
we are in motion. We perceiv6 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 wefecl. When, 
for example, we are carried along in a carriage with the 
blinds down, or with our eyes closed (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 



with a multitude of objects wliich participate with our- 
selves and each other in the common progress of the 
whole mass, that we 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 avc 
drop it, it lights at our feet. Insects buzz around us as 
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 respects, difler- 
ent ; the air, not being carried along with us, drifts away 
smoke and other light bodies such as 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 massiye objects and our own move- 
ments are concerned, remains complete ; and 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 ive 
form a part. 

" Provehimur portu, terraeque urbesque recediuit." 

(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 Avill see, or think he sees, the whole 
landscape thrown into rotatioji, 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 all beyond it, forwards, or in the 
direction in which he moves : but let him withdraw his 
eye from that object, and fix it on another, a nearer 


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 what their mo- 
tions are, we must subduct, or allow for, any such pa- 
rallactic motion which may exist. 

(18.) In order, however, to conceive the earth as in 
motion, we must form to ourselves a conception of its 
shape and size. Now, an object cannot have shape and 
size, unless it is limited on all sides by some definite 
outline, so as to admit of our imagining it, at least, dis- 
connected from other bodies, and existing insulated in 
space. The first rud(? notion we form of the earth is 
that of a flat surface, of indefinite extent in all directio;is-" 
from the spot where we stand, above which are the air 
and sky ; below, to an indefinite profundity, solid mat- 
ter. This is a prejudice to be got rid of, like that of the 
earth's immobility ; 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 u)here 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 senses 
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 to 
all our notions of solid matter, to suppose it to have 
made its way through the substance of the earth. It 
must, therefore, have gone under it, and that not by a 
mere subterraneous channel ; for if we notice the points 
where it sets and rises for many successive days, or for 
a whole year, we shall find them constantly shifting, 
round a very large extent of the horizon ; and, besides, 
the moon and stars also set and rise again in all points 


of the visible horizon. The ronrhision 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 
SiV/f, 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 night to us, and vice versa ; where, in short, 

" redit a nobis Aurora, diemque reducit. 
Nosque uhi primus equis orieiis affiavit anhelis, 
lUic sera ruberis acceiidit lumina Vesper." Georg. 

(19.) As soon as we have familiarized ourselves with 
the conception of an earth wilhont foundaf ions or fixed 
supports existing insulated in space from contact of 
every thing external, it becomes easy to imagine it in 
motion or, rather, difiicult to imagine it otherwise ; for, 
since there is nothing to retain it in one place, should 
any causes of motion exist, or any 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 earth. 

(20.) Let us first examine Avhat we can actually see 
of its shape. Now, it is not on land (unless, indeed, on 
uncommonly level and extensive plains) that Ave can see 
any thing of the general 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 u'hole 
earth, are yet too considerable, with respect to ourselves 
and to that small 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, Avhich 
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 ; 


and, secondly, from the fact of all its parts appearing at 
the same distance from us, and that evidently a mode- 
rate one; and, thirdly, from this, that its apparent 
diameler, measured with an instrument called the dijJ 
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; jElna, Teneriff'e, 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 have receded from 
its surface, while yet the absolute quantity of 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, ap- 
pears always circular, can be no other than a sphere or 

(23.) A diagram will elucidate this. Suppose the 
earth to be represented by the sphere LHNQ, whose 
centre is C, and let A, G, M be stations at different 
elevations above various points of its surface, represent- 
ed by a, f, m, respectively. From each of them (as 
from M) let a line be drawn, as MNn., a tangent to the 
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 MOo,MV p,M Q,q, the 


point of contact N will murk out on the surface the circle 
N O P Q. The area of this circle is the portion of the 

eartli's surface visible to a spectator at M, and the angle 
NMQ included between the two extreme visual rays is 
the measure of its ap])arent 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, viz. the line MN, increases ; and, 3dly, that 
the angle MNQ becomes less obtuse, or, in other words, 
the apparent angular diameter of the earth diminishes, 

'Ogi^ji, to terminate 


being nowhere so great as 180, or two right angles, 
but falling short of it by some sensible quantity, and that 
more and more the hio'her we ascend. The figure ex- 
hibits three states or stages of elevation, with the 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 imagine 
?iNM, MQq 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 perfect straight- 
ness till M is In-ought 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. M?, which is perpendicular to the general 
surface of the sphere at m, is also the direction in which 
a plumb-line* would hang ; for it is an observed fact, 
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 perpendicular to a line or surface truly adjusted by 
a spirit-level. * Suppose, then, that at our station M we 
were to adjust a line (a wooden ruler for instance) by a 
spirit-level, with perfect exactness ; tlien, if we suppose 
the direction of this line indefinitely prolonged both 
ways, as XMY, the line so drawn will be at right 
angles to Mm, and therefore parallel to xmy, the tan- 
gent to the sphere at m. A spectator placed at M will 
therefore see not only all the vault of the sky above this 
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 is 
the angular breadth of this redundant zone the angle 
YMQ, by which the visible horizon appears depressed 
below the direction of a spirit-level that is called the 
dip of the horizon. It is a correction of constant use in 
nautical astronomy. 

* See this instrument described in chap. II. 


(25.) From the foregoing explanations it appears, 
1st, 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- 
viations from the general outline of the whole mass, as 
we consider an orange not tlie less round for tlie rough- 
nesses on its rind. 2dly, That the appearance of a visi- 
ble horizon, or sea ofling, 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 v.diile 
to pursue the general notion thus acquired into 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 to 
each other, and held together as a whole. 

(26.) 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 ivhole of them. We only see their upper 
parts. Their bases where they rest on, or rise out 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 of the ship is 
small, a spectator, S, situated at some certain height 
above the sea, sees the whole of the ship, even to the 
water line where it rests on the sea, as at A. As it re- 
cedes it diminishes, it is true, in apparent size, but still 
the zvhole is seen down to the water line, till it reaches 
the visible horizon at B. But as soon as it has passed 
this distance, not only does the visible portion stiil con- 
tinue to diminish in apparent size, but tlie hull begins to 
disappear bodily, as if sunk below the surface. When 
it has reached a certain distance, as at C, its hull 
has entirely vanished, but the masts and sails remain, 
presenting the appearance c. But if, in this state of 





things, the spectator quickly ascends to a higher station, 
T, whose visible horizon is at D, the hull 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 d, and at length the whole disappears : 
Avhile yet the distinctness with which the last portion of 
the sail d is seen is such as to satisfy us that were it not 
for the interposed segment of the sea, ABODE, the dis- 
tance TE 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 fas 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 Art and B h (which, 
for simplicity, we will suppose to be exactly equal) are 
knov/n, as well as their exact horizontal interval aD6, 



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 aDb to be the sphere of 
the earth, and C its centre in the figure CUbB, we 
know Db. the length of the arch of the circle between 
D and b, viz. half the measured interval, and bB, tho 
excess of its secant above its radius which is the height 
of B, data which, by tlie solution of an easy geometrical 
problem, enable us to find the length of the radius DC. 
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 : 

The eurtlt's 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 eartli, yet it will suf- 
fice to aflbrd 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 other 
over still water, and in average atmospheric circum- 
stances, at a distance of about 8 miles. But 10 feet is 
Jhe 528th part of a mile, so tliat half their distance, or 
4 miles, is to the height of each as 4 x 528 or 2112 : 1, 
and 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 been accustomed to consider as of vast size, so as 
to interpose a few steps between it and our ordinary ideas 


of dimension. We have before likened the inequalities 
on tlie earth's surface, arising from mountains, valleys, 
buildings, &c. to the roughnesses on the rind of an 
orange .''compared with its general mass. The compa- 
rison is quite free from exaggeration. The highest moun- 
tain known does not excectf five miles in perpendicular 
elevation: this is only one 1600th part of the earth's 
diameter ; consequently, on a globe of sixteen inches in 
' diameter, such a mountain would be represented by a 
protuberance 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 thin 
writing paper; and the highest hills would be represented 
by tlie 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 impercepti])le without a magnifier. 
(.31.) The greatest depth of sea, probably, does not 
much exceed the greatest elevation of the continents ; 
and would, of course, 1)e represented by an excavation, 
in about the same proportion, into 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 leu 
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 film, and its adhesion to the 
surface, are altogether difl'erent from those which govern 
me 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- 


posed to the view of MM. Biot and Gay-Lussac, in tlieir 
celebrated aeronautic expedition to the enoi'mous heiglit 
of 25,000 feet, or rather less than five miles. To esti- 
mate the proportion of tlie 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 alwve the surface. 
The proportion, therefore, of the visilile area, in this 
case, to the whole earth's surface, is that of five miles to 
8000, or 1 to 1600. Tlie portion visible from ^tna, 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 insufliicient supply of ah\ 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 a 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 thirtietli of the 
whole mass of the atmosphere : that at 10,600 feet of 
perpendicular elevation (which 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. 
Fi'om the progression of these numbers, as well as, a pri- 
ori, from the nature of the air itself, which is compressi- 
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 

* Tlie height of /Etna above the Mediterriinean (as it results from a 
barometrical measurement of my own, made in July, 1824, under very 
favourable circumstances) is 10,872 English feet. Author. 


continually get above more and more of the air, and so re- 
lieve ourselves more and more from the pressure Avith 
wliich it weighs upon us, yet the amount of this additional 
relief, or \he ponderable quantity o{ 2\x 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 
whicli regulate its dilation and compression, is sufficient 
to sliow that, at an altitude above the surface of the earth 
not exceeding the hundreth part of its diameter, the tenui- 
ty, or rarefiction, 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 
allurd the slightest perceptible indications of its presence. 

(34.) Laying out of consideration, therefore, at pre- 
sent, all nice questions as to the probable existence of a 
definite limit to the atmosplicre, beyond which tliere 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's surface than the 
hundredth part of its diameter as void of air, and of course 
of clouds (which are nothing but visible vapours, diffused 
and floating in tlie air, sustained by it, and rendering 
it iurljid 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 tlie air is about an eighth part of 
what it is at the level of the sea. 

(.S5.) We are thus led to regard the atmosphere of air, 
with the clouds it supports, as constituting a coating of 
equable or nearly equal)le thickness, enveloping our globe 
on all sides ; or rat)u3r as an aerial ocean, of which the 
surface of the sea and laud constitutes the bed, and whose 
inferior portions or strata, within a few miles of the 
earth, contain by far tlie greater part of the Avhole mass, 
the density diminishing Avith extreme rapidity as we re- 
cede upwards, till, witliin a very moderate distance (such 
as would be represented I))^ the sixth of an inch on the mo- 
del we have before spoken of, and which is not more in pro- 

c 3 


portion to the globe on which it rests, than the downy 
skin of a peach in comparison Avitli 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 hint- 
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, as water poured into the sea, and distribute itself 
among the mass beneath. With the truth of this conclu- 
sion, however, astronomy has little concern ; all tlie 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 equally 
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 sfeneral surface of the sea and land, and each of 
which is rarer, or specifically lighter, than that immedi- 
ately beneath it ; and denser, or specifically heavier, than 
that immediately above it. This kind of distribution of 
its ponderal)le mass is necessitated by the laws of the 
equilibrium of fluids, whose results barometric observa- 
tions demonstrate to be in perfect accordance with expe- 

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





lion of its strata tlie inequalities of mountains and valleys 
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 
witli all transparent media, of refracting the rays of light, 
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 otherwise situated than 
they would to the same spectator, had the atmosphere no 
existence ; it thus produces a false impression respecting 
tlieir places, v/hich 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 situated from us 
at any assigned moment. 

(39.) Suppose a spectator placed at A, any point of the 
earth's surface KAA;, and let hi. Mm, N?*, represent 

the successive strata or layers, of decreasing density, into 
which we may conceive the atmosphere 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- 


rection of the straight line AS. But, in reality, when 
the ray of light SA reaches tlie atmosplicrc, suppose at d, 
it will, hy the laws of optics, begin to bend downwards, 
and take a more inclined (Urection, as d 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(/A, it will describe a curve Sdcb 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. Tlie 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 ^e/imrZ 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 I^ecn 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 As, n tangent to the curve 
SDCBA, at A. But because the curve described by the 
refracted ray is concave downwards, the tangent As, 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, dian 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 woidd therefore be invisible but for the effect of 
refraction, are, by that efl'ect, raised altove it and brought 


into sight. Thus, the sun, when situated at P below the 
true horizon, All, of the spectator, becomes visible to him, 
as if it stood at p, by the refracted ray FqrtA, to which 
Ap is a tangent. 

(41.) The exact estimation of the amount of atmo- 
spheric refraction, or the strict determination of the angle 
SAs, by which a celestial object at any assigned altitude, 
HAS, is raised in appearance above its true place, is, un- 
fortunately, a very difficult subject of physical inquiry, 
and one on which geometers (from Avhom 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 from 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 affected by 
its moisture ; and this, 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 

(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 an Arabic word of this signification. 


one of the most important and indispensable of all astro- 
nomical tables, since it is only by the nse 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 admit 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 

(43.) 1st. In the zenith there is no refraction; a ce- 
lestial object, situated vertically over head, is seen 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 zenitli. But this rule, which is not far 
from tlie tiiith, at moderate zenith distances, ceases to 
give correct results in the vicinity of the horizon, where 
the law becomes much more complicated in its ex- 

4thly. The average amount of refraction, for an ob- 
ject half-wa)^ between the zenith and liorizon, or at an 
apparent altitude of 45, is about 1' (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 tlie horizon, its whole disk is in 
reality below it, and would be entirely out of sight and 

*Vide " Requisite Tables to be used with the Nautical Almanac." 
See a]po Nautical Almanac for 1833, Dr. Pearson's Astronomical Tables, 
and Mr. Baily's Astronomical Tables and FoitouIbd, 


concealed by the convexity of the earth but for the bend- 
ing round it, which the rays of light have undergone in 
neir passage through the air, as alluded 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 suii and 
moon above the horizon. But even after they are set, 
the inlluence of the atmosphere still continues to send 
us a poflion 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. Tlie whole course of a sun- 
beam penetrating through the chink of a window-shutter 
into a dark room, is visible as a bright line in the air; 
and even if it be stifled, or let out through an opposite 
crevice, the liglit 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 broken 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 afford us that secondary 
illumination, which is twilight. The course of such rays 
will be immediately understood from the annexed figure, 
in which ABCD is the earth ; A a point on its surface, 
where the sun S is in the act of setting ; its last lower 


ray SAM just grazing the surface at A, while its superior 
rays SN, SO, traverse the atmosphere above A withoui 
striking the earth, leaving it finally at the points PQR, 
after being more or less 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, O, D, each 
more remote than the last from A, and each more deeply 
involved in the earth's shadoiv, 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 tvv^ilight from the whole sky. 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 PR.r, Avhich is traversed by the sun's rays, 
and which lies above the visible horizon BR of B, it re- 
ceives a twilight, which is strongest at R, the point im- 
mediately 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, PQ.? 
of the lenticular segment, thus illuminated, lies above 
the horizon, CQ, of that place : here, then, the twilight 
is feeble, and confined to a small space in and near the 


horizon, which the sun has quitted, wliile 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 pointy 
from every point of the sky. The generally diflused 
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 darkness ; the stars Avould 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, bv thus brinof- 
ing together masses of air of very unequal temperatures, 
produces partial reflections and refractions at their 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 layers of 
the atmosphere, it will be evident, that whenever 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 moun- 
tains, seen from the plains, or other terrestrial stations, 
at different levels, seen from each other, immersed in it. 
Every difference of level, accompanied, as it must be, 
with a difference of density in the aerial strata, must also 
have, corresponding to it, a certain amount of refraction i 



less, indeed, than what would he produced hy the whole 
atmosphere, but still often of very appreciable, and even 
considerable, amount. This refraction between terres- 
trial stations is termed ferresfrial refraction, to distin- 
guish it from that total effect which is only produced on 
celestial objects, or such as are beyond the atmosphere, 
and which is called celestial 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 approaches 
the horizon, a flattened or oval outline ; its horizontal 
diameter being visibly greater than that in a vertical di- 
rection. When very near tlie horizon, this flattening is 
evidently more considerable on the lower side than on 
the upper ; so that the apparent form is neither circular 
nor elliptic, but a species of oval, which deviates more 
from a circle below than above. This singular effect, 
which any one may notice in a fine sunset, arises from the 
rapid rate at which the refraction increases in approach- 
ing the horizon. Were every visible point in the sun's 
circumference equally raised by refraction, it would still 
appear cii'cular, though displaced : but the lower portions 
being more raised than the upper, the vertical diameter is 
thereby 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 same. The 
dilated size (generally) of the sun or moon, when seen 
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 illusion of the judgment arising from the 
terrestrial objects 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 their insulation in the 
expanse of sky leads us 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 


eyes almost exactly the same, and the moon a materially 
less angle, than when seen at a great altiuule in the sky, 
owing to the effect of what is called parallax, to be ex- 
plained presently, ^j _ 

(48.) After what has been said of the small extent of 
the atmosphere in 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 
they obviously form no part of the earth, 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 
them while speaking of their refractions as existing in 
the immensity of space beyond, and situated, perhaps, 
for any thing we can perceive to the contrary, at enor- 
mous distances from us and from each other. 

(49.) Could a spectator exist unsustained by the earth, 
or any solid support, he would see around him at one 
view the whole contents of space the visible consti- 
tuents of the universe : and, in the absence of any means 
of 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 ah 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 than for the idea that all were 
equidistant from him, 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 appareyit 
proportions, and the likeness is not taxed with incorrect- 


ness, though a man in the foreground should be repre- 
sented hirger than a mountain in the distance. So it is 
to a spectator of the heavenly bodies pictured, projected, 
or mapped down on that imaginary sphere we call tlie 
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 tlie great mass on which he stands, from seeing into 
all 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 the horizon will bring within the scope 
of vision a little more than a hemisphere, and refraction, 
wherever he may be situated, will enable him to look, 
as it were, a little round the corner; but the zone thus 
9.dded 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 the va- 
pours 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, un- 
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 different region 
pf space ; he would never obtain a sight of almost one 

*Such as the following, for instance: The late Mr. Sadler, the cele^ 
brated aeronaut, ascended in a balloon from Dublin at about 2 o'clock in 
the afternoon, and was wafted across the channel. About sunset he ap- 
proached the English coast, when the balloon descended near the surface 
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 sunrise. He subsequently descended in Wales, and 
witnessed a second sunset on the same evening. I have this anecdote 
from Dr. Lardner, who was present at his ascent, and read his owp acr 
cpuntofthe voyvigQ.^^ Author. 


half the objects external to our atmosphere. But if any 
of these cases be supposed, more, or all, may come into 
view accordincf 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 him to a person standing 
in a park close to a large tree. The massive obstacle 
presented by its trunk cuts off his view of all those parts 
of the landscape which it occupies as an object; but by 
walking round it a complete successive view of the 
whole panorama may be obtained. Just in the same 
way, if we set off from any station, as Loudon, 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, travelling with us south- 
wards round the sphere, sinks in succession beneath 
them. The novelty and splendour of fresh constella- 
tions thus gradually brought into view in the clear calm 
nights of tropical climates, in long voyages to the south, 
is dwelt upon by all who have enjoyed this spectacle, 



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



the accompLinying figure, exhibiting three successive 
stations of a traveller, A, B, C, with the horizon cor- 
responding to each, will place this pi'ocess in clearer 
evidence than any description. 

(52.) Again : suppose the earth itself to have a mo- 
tion of rotation on its centre. It is evident that a spec- 
tator at rest (as it appears to him) on any part of it will, 
imperceived 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 familiar objects in it, which serve 
him for landmarks and directions, retain, with respect 
to himself or to each other, the same invariable situa- 
tions. The perfect smoothness and equality of the 
motion of so vast a mass, in which every object he sees 
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. Re- 
curring to the figure 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, C ; or whether, the 
earth remaining at rest, he has transferred himself per- 
sonally 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 difterence will be in his view of the tree it- 
pelf, 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 under hia 

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


be constantly depressing itself below those objects which 
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 suc- 
cessively hiding the latter. As the horizon of every such 
spectator, however, appears to him motionless, all such 
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 set ; while those it has 
just disclosed, and from which it is receding, will seem 
to be rising above it. 

(54.) If we suppose this rotation of the earth to con- 
tinue in one and the 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 
from which it set out when the spectator began his obser- 
vations, it is manifest that every thing will 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, the 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 
that great law of periodicity,* which, as we shall see, 
pervades all astronomy ; by which expression we under- 

* rTjeioJof, a going round, a circulation or revolution 


Stand the continual reproduction of tlie same phenomena, 
in the same order, at equal intervals ol' 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 with 
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 
the influence of external agency are conceivable, in which 
the poles 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 any surrounding medium. The complete ab- 
sence of such obstructions draws with it, of necessity, 
the strict fulfilment of the two conditions above men- 

(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, the permanence of the 
axis of rotation, 'the 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 

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


tail how the liypothesis of the rotation of tlie earth about 
an axis accords with the plienoniena which the diurnal 
motion of the heavenly bodies ofl'ers to our notice, it will 
be proper to describe, with precision, in what that diur- 
nal 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 first 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 catch 
his attention in the twilight ; and more and more will 
appear as the darkness increases, till the Avhole sky is 
overspangled with them. When he has awhile admired 
the calm magnificence of this glorious spectacle, the 
theme of so much song, and of so much thought, a 
spectacle which no one can view without emotion, and 
without a longing desire to know something of its na- 
ture and purport, let him fix his attention more particu- 
larly on a few of the most brilliantstars, such as he can- 
not fail to recognise affuin 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 a^ain 
with their respective points of reference, after a moderate 
interval, as the night 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^- 


ing their motions, on tlie 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 Avhich 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 sovith and east, describe larger segments of their cir- 
cles above the horizon, remain proportionally longer in 
sight, and set precisely as far to the westward of south 
as they rose to the eastward ; while such as rise exactly 
in the east remain just twelve hours visible, describe a 
semicircle, 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 
arcs are more than semicircles. But the magnitudes of 
the circles themselves diminish, as we go from the east, 
northward ; the greatest of all the circles being described 
by those which rise exactly in the cast 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 all 
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 bright star, called 
the Pole Star, which is easily recognised by the very 

* We suppose our observer to be stationed in some northern latitude ; 
BOme where in Europe, for example. 


small circle it describes : so small, indeed, that, without 
paying particular attention, and referring its position very 
nicely to some fixed mark, it may easily be supposed at 
rest, and be, itself, mistaken for the common centre about 
which all the others in that region describe their circles ; 
or it may be known by its configuration with a very 
splendid and remarkable constellation or group of stars, 
called by astronomers the Great Bear. 

(60.) He will further observe that the apparent rela- 
tive situations of all the stars among one another is not 
changed by their diurnal motion. In Avliatever parts of 
their circles they are observed, or at whatever hour of the 
night, they form with each other the same identical groups 
or configurations, to which the name of constellations 
has been given. It is true, that, in different parts of their 
course, these groups stand differently with respect to the 
horizon ; and those towards the north, when in the course 
of their diurnal movement they pass alternately 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 same 
points towards the pole. In short, he will perceive that 
the whole assemblage of stars visible at once, or in suc- 
cession, in the heavens, may be regarded as one great 
constellation, which seems to revolve with a uniform mo- 
tion, as if it formed one coherent mass ; or as if it were at- 
tached to the internal surface of a vast hollow 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 mentioned. 

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


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 co-extensive with the whole surface of the 
sphere, being in reality nothing less than a universe of 
luminaries surrounding the earth on all sides, 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 nobis semper sublimis ; at ilium 
Sub pedibus noxatra videt, manesque profundi." Virgil. 

One pole rides high, one, plimged beneath -the main, 
Seeks the deep night, and Pluto's dusky reign. 

(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 to 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 longerpart of the twenty-four hours. 
They will continue, however, to circulate about the same 


point that is, holding the same invariable position icith 
respect to them in the concave of the heavens among the 
stars ; but this point itself will become gradually 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 constellations 
surrounding it will come into view ; at first momentarily, 
but by degrees for longer and longer times in each diur- 
nal revoluUon reahzing, 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, called the equa- 
tor, 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 successive 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 over 
him the whole hemisphere with which he began his 
night's observation will have been carried down beneath 
him, and the entire opposite one brought up from below. 
(65.) If he pass the equator, and travel still farther 
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 original station de- 
scribed their whole diurnal circles above his horizon, and 
never set, now describe them entirely below it, and never 



rise, 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 vertically under the northern pole of the heavens, 
at which none of the stars would either rise or set, but 
each Avould 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 at 
the pole itself, have proved to be in exact correspondence 
with its near proximity. A similar remark applies to the 
south pole of the earth, 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. 

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




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 right 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 him to the east point of his horizon, 
with hardly any perceptible deviation. The reason of 
this is, that the position of every object is refeiTed by lis 
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 axis 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, APB=cPp, at the near object P than at the remote 
one Q, 


(69.) This apparent angular motion of an object on 
our sphere of vision,* arising from a cliange 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 diflerence 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, DBP=DAP + APB, whence 

(70.) It follows from this, that the amount of paral- 
lactic motion arising from any given change of our point 
of view is, cseteris jxiribus, 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 view, 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 Avalk, 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 
diameter, 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 vicAV. 

* The ideal sphere without us, to which we refer the places of objects, 
and which we carry along with us wherever we go, is no doubt inti- 
mately connected by association, if not entirely dependent on that ob- 
scure perception of sensation in the retinae 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, 
&c. are really mapped down, as we have supposed them in the text to 
be, on the imaginary concave of the heavens. Wlien the whole surface 
of the retinae is excited by light, habit leads us to assoc^iate it with the 
idea of a real surface existing without us. Thus we become impressed 
with the notion of a sky and a heaven, but the concave surface of the 
retinas itself is the true seat of all visible angular dimension and angular 
motion. The substitution of the retina for the Aeawens would be awkward 
and inconvenient in language, but it may always be mentally made, 
(See Schiller's jiretty enigma on the eye in his Turoiidot.) 


(71.) Whatever notion, in other respects, we may 
form of the stars, it is quite clear they must be im- 
mensely distant. Were it not so, the apparent angular 
interval between any two of them seen over head would 
be much greater than wlien seen near the horizon, and 
the constellations, instead of preserving the same ap- 
pearances and dimensions during their whole diurnal 
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 ab, AB, a b, are three difierent positions of 

the same stars, as they would, if near the earth, be seen 
from a spectator S, under the visual angles aSb, ASB. 
No such change of apparent dimension, however, is ob- 
served. The nicest measurements of the apparent an- 
gular distance of any two stars inter se, taken in any 
parts of their diurnal course, (after allowing for the un- 
equal effects of refraction, or when taken at such times 
that this cause of distortion shall act equally on both,) 
manifest i2ot the slightest perceptible variation. Not 
only this, but at whatever point of the earth's surface the 
measurement is performed, the results are absolutely 
identical. No instruments ever yet invented by man 
are delicate enough to indicate, by an increase or dimi- 
nution of the angle subtended, that one point of the 
earth is nearer to or further from the stars than another. 
(72.) The necessary conclusion from this is, that the 
dimensions of the earth, large as it is, are comparatively 
nothing, absolutely imperceptible, when compared with 



the interval which separates the stars from the earth. If 
an observer walk round a circle not more than a few 
yards in diameter, and from diflerent 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, by the difference of the results, of his 
change of distance from them arising from liis change 
of place, although that difference may be so small as to 
produce no change in tlieir general aspect to his unas- 
sisted sight. This is one of the innumerable instances 
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 cases 
on the mere judgment of the eye. To so great a nicety 
have such observations been carried by the aid of an 
instrument called a theodolite, that a circle of the dia- 
meter above mentioned may thus be rendered sensible, 
may thus be detected to have a size, and an ascertainable 
place, by reference to objects distant by fully 100,000 
times its own dimensions. Observations, differing, it is 
true, somewhat in method, but identical in principle, 
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 
the distance of the stars from the earth cannot be so 


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 such a distance, to a spectator with our 
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 undiscernible. 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 will be of evanescent 
breadth to his eye, and will only mark out a great circle in 
the heavens, Avhich, like the vanishing point in perspec- 
tive to which all parallel lines in a picture appear to 
converge, is, in fact, the vanishing line to which all 
planes parallel to the horizon offer a similar appearance 
of ultimate convergence in the great panojxmia 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 the heavens 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 view bounded by 
the rational 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 
horizon, coiTesponding to their common zenith. 


(75.) Now, so far as appearances g-o, it is clearly the 
same thing whether the heavens, that is, all space, with 
its contents, revolve round a spectator at rest 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 horizon (which must be supposed to turn 
with him), will be the same in both supposition-3. And 
since, as has been shown, appearances are also, so far as 
the stars are concerned, the 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 Copernican astronomy adopts the latter as 
the true explanation of these phenomena, avoiding there- 
by the necessity of otherwise resorting to 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 the earth without derangement of their 
relative situations inter se. Such a contrivance 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 such a supposition, as Avill appear when 
we come to treat of these bodies. On the other hand, that 
a spherical mass of moderate dimensions (or, rather, 
when compared with 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 those 
general laws which, so far as we know, regulate the mo- 
tions of all material bodies, is so 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 offer 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' 


ameter about which it revolves, with a uniform motion, 
from 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 


(78.) Def. 2. The jioles 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.) Def. 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 7iadir* are the two points 
of the sphere of the heavens, vertically over tlie specta- 
tor's head, and vertically under his feet ; they are, there- 
fore, the vanishing points of all lines mathematically pa- 
rallel 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-r-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 
from 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 Arabic worjs. Nadir corresponds evidently to the German 
nieder (down) 


(81.) Def. 5. Vertical circles of the sphere are ^eat 
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 equinocticd. 

(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/es^za/wienV/iVm of a spectator stationed at that place. 
When we speak of the meridian of a spectator, we intend 
the celestial meridian, which is a vertical 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 his horizon. 

(85.) Def. 9. Jlzimuth 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.] latitude and loxgitude. 69 

gined, called an altitude and azimuth instrument, which 
will 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 degrees, 
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 
51 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 of 51 28' 40". 
(88.) Def. 12. The /og"i7w(Ze 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. Eng- 
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. Hereafter, when we 
speak of longitude, we reckon from Greenwich. The 
longitude of a place is, therefore, measured by the arc 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 

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 ivestu'ard 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 


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 longi- 
tudes of places which occur in them, without referring 
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 may be conceived to be seen at once 
by one at the centre of the globe. It 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 it 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 i 
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 Geo- 
graphy and Uranography,* that astronomers have 
odopted different terms (viz. declination, and right 
* Tt), the earth; yea^Mv, to describe or represent: ovfuve;, the heavens. 


ascension) to represent those arcs in the heavens which 
correspond to latitudes and longitudes on the earth. It 
is for this reason that they term 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. All 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 
circles 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 preference, a point in the 
equinoctial called the equinox, 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 to 360, which com- 
pletes the circle ; or, in time, in hours, minutes, and 
seconds, from h. to 24 h. The apparent diurnal motion 
of the heavens being contrary to the real motion of the 



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 whicli 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 its poles; EQ its eqiia-' 
tor; AB the parallel of latitude of the station A on its 
surface ; AP parallel to SON, the direction in which an 
observer at A will see the elevated pole of the heavens ; 
and AZ, the prolongation of the terrestrial radius CA, 


that of his zenith. NAES will be his meridian ; NGS 
that of some tixed station, as Greenwich; and GE, or 
the spherical angle GNE, his longitude, and EA his la- 
titude. Moreover, if ns be a plane touching the surface 
in A, this will be his sensible horizon ; wAs marlied on 
that plane by its intersection with his meridian will be 
his meridian line, and n and s the north and south points 
of his horizon. 

(94.) Again, neglecting the size of the earth, or con- 
ceiving him stationed at its centre, and referring every 
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 celes- 
tial horizon ; Pp the elevated and depressed poles of 

the heavens ; HP the altitude of the pole, and HPZEO 
his meridian; ETQ, a great circle perpendicular to Pp, 
will be the equinoctial ; and ifT represent the equinox, 
'V T will be the 7'ight ascension, TS the declination, and 
PS the polar distance of any star or object S, referred to 
the equinoctial by the hour circle PST/j; and BSD will 
be the diurnal circle it will appear to describe about the 
pole. Again, if we reier it to the horizon by the vertical 
circle ZSA, HA will be its azimuth, AS its altitude, and 
ZS its zenith distance. H and O are the north and 
pouth, and ew the east and west points of his horizon, 


or of the heavens. Moreover, if HA, Oo, be small cir- 
cles, or parallels of declination, touching the horizon in 
its north and south points, HA will be the circle of per- 
petual apjiurition, between which and the elevated pole 
the stars never 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 foi 
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 difl^'erent 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 the 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. 

(98.) 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 


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

(99.) All the heavenly bodies culminate (i. e. come to 
their greatest altitudes) on the meridian ; which is, there- 
fore, the best situation to observe them, being least con- 
fused by the inequalities and vapours of the atmosphere, 
as well as least displaced by refraction. 

(100.) All celestial objects within the circle of perpe-. 
tual apparition come twice on the meridian, above the hori- 
zon, in every diurnal revolution ; once above and once 
beloiv the pole. These are called their iqyper and lower 

(101.) We shall conclude this chapter by calling the 
reader'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 well 
as the night ; and that, in proportion to the 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 noonday, unless in that part of the 
sky which is very near the sun, by those who possess the 
means of pointing a telescope accurately to the proper 
places. Indeed, from the bottoms 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 ourselves heard it stated by a celebrated optician, 
that the earliest circumstance 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. 




Of the Nature of astronomical Instruments and Observations in general 
Of sidereal and solar Time Of the Measurement of Time Clocks, 
Chronometers, the Transit [nstrnment Of the Measurement of angular 
Intervals Application of the Telescope to Instruments destined 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 reflecting Circle Princi- 
ple of Repetition. 

(102.) Our first chapter has been devoted to the 
acquisition chiefly of preliminary notions respecting the 
globe Ave inhabit, its relation to the celestial objects which 
surround it, and the physical circumstances under which 
all astronomical observations must be made, as well as to 
provide ourselves with a stock of technical words of 7nost 
frequent and familiar use in the sequel. We might now 
proceed to a more exact and detailed statement of the 
facts 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 tliey ground their 
conclusions ; in other words, of ascertaining by measure- 
ment the apparent and real magnitudes with which they 
are 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 error can 
certainly be perceived and distinctly measured, that he 
can satisfy himself whether any theory ofiers so close an 
approximation, in its numerical results, to actual phe- 
nomena, as will justify him in receiving it as a true repre- 
sentation of nature. 

(103.) Astronomical instrument-making may be justly 
regarded 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 


an easy thing, by one un^icquaintcd 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, and 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 the application of 
telescopes 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 workmanship, 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 owai weight, be- 
come perceptible and measurable. An angle of one mi- 
nute occupies, on the circumference of a circle of 10 
inches in radius, only about 3^0^^^ P^^"^ ^^ ^'^ inch, a quan- 
tity too small to be certainly dealt wdth wdthout the use 
of magnifying glasses ; yet one minute is a gross quan- 
tity in the astronomical measurement of an angle. With 
the instruments now employed in observatories, a single 
second, or the 60th part of a minute, is rendered a dis- 
tinctly visible and appreciable quantity. Now, the arc 
of a circle, subtended by one second, is less 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 
j-yL-th part of an inch ; a quantity requiring a poM'erful 
microscope to be discerned at all. Let any one figure to 
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 
subdivision of the degrees so marked off into minutes, and 
of these again into seconds. Such a work has probably 
baffled, and will probably for ever continue to baffle, the 
utmost stretch of human skill and industry ; nor, if exe- 
cuted, could it endure. The ever varying fluctuations of 
heat and cold have a tendency to produce not merely tern- 


porary and transient, but permanent, uncompensated 
changes of form in all considerable masses of ihose metals 
which 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 thousand 
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'ceuvre, 
but to instruments of moderate prices and dimensions, and 
in ordinary use, which, on due consideration, must ap- 
pear very surprising. But tliough 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 always 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 
hands. He must, therefore, endeavour so to combine his 
observations, so to choose his opportunities, and so to 
familiarize himself with all the causes which may pro- 
duce instrumental derangement, and with all the pecu- 
liarities of structure and material of each instrument he 
possesses, as not to allow himself to be misled by their 
errors, but to extract from their indications, as far as possi- 
ble, all that is true, and reject all that is 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 
rneasurement, his constant care and vigilance must be 
directed to the detection and compensation of errors, 
either by annihilating, or by taking account of, and aU 
'pwing for them. Now, if we examine the sources from 


which errors mar arise in any iiistriimontal 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 tlie air, 
vary also the form and position of the instruments used, 
by altering relative magnitude and the tension of their 
parts ; and others of the like nature. 

(106.) 2dly, Errors of observation : such as arise, for 
example, from inexpertness, defective vision, slowness 
in seizing the exact instant of occurrence of a pheno- 
menon, or precipitancy in anticipating it, &c. ; 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, &c. 

(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 ^rs^ 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 cylindrical, be slightly 
flattened, or elliptical, if it be not exactly (as it is in- 
tended it should) concentric with the circle it carries ; 
if this circle (so called) be in reality 7iot 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, Avhich every observer has to 
contend with. 

C108.) The other subdivision of instrumental errors 
comprehends .such as arise from an instrument not being 
placed in the posifio7i it ouglit to have ; and from those 


of its parts, whicli are made purposely moveable, 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 other way, become appreciable in 
delicate astronomical observations : others, again, are 
consequences of imperfect Avorkmanship, as where an 
instrument once well adjusted will not remain so, but 
keeps deviating and shifting. But the most important 
of this class of errors arise from the non-existence of 
natural indications, other than those afforded by astrono- 
mical observations themselves, whether an instrument 
has or has not the exact position, with respect to the 
horizon and its cardinal points, the axis of the earth,, 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, they actually vitiate, to 
their full extent, the results of any observations in which 
they subsist. Being, however, in their nature casual 
and accidental, their effects necessarily lie sometimes 
one way, sometimes the other ; sometimes diminishing, 
sometimes tending to increase the results. Hence, 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 subdiied, by setting 
them to destroy one another, as no longer sensibly to 
vitiate any theoretical or practical conclusion. This is 
the great and indeed only resource against such errors not 
merely to the astronomer, but to the investigator of nu' 
pierical results in every department of physical i^esearch, 

{110.) With regard to errors of adjustment and 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 m.ar 
chines never formed a circle, drew a straight line, or 
erected a perpendicular, nor ever placed an instrument 
in perfect adjustment, unless accidentally ; and then only 


during an instant of time. This does not prevent, how- 
ever, that a great approximation to all these desiderata 
should be 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 course of astrono- 
mical observations will make distinctly evident. The 
imperfect products of man's 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 theoretical conclusions and 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 laAvs 
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 them, make their appearance, and become subjects of 
further inquiry. Now, it invariably happens (and the 
reason is evident) that the first glimpse we catch of such 
subordinate laws the first form in which they are 
dimly shadowed out to our minds is that of errors. 
We perceive a discordance between 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 


then inquire, to what amount of error their determina- 
tions can, by possibility, be liable. If their limit of pos- 
sible error exceed the observed deviation, we at once 
condemn 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 defined than before. We are now sure that 
we are on the traces of a law of nature, and we pursue 
it till Ave have reduced it to a definite statement, and 
verified it by repeated observation, under every variety 
of circumstances. 

(112.) Now, in the course of this inquiry, it will 
not fail to happen that other discordances will strike us. 
Taught by experience, we suspect the existence of some 
natural law, befoi'e unknown ; we tabulate (i. e. draw out 
in order) the results of our observations ; and we per- 
ceive, in this synoptic statement of them, distinct indi- 
cations of a regular progression. Again Ave 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 theory 
of our instrument ; Ave suppose defects in its struc- 
ture, and, by the aid of geometry, we trace their in- 
fluence in introducing actwd errors into its indications. 
These errors have their laivs, Avhich, so long as we 
have no knowledge of causes to guide us, may be con- 
founded with laAvs of nature, and are mixed up Avith 
them in their effects. They are not fortuitous, like 
errors of observation, but, as they arise from sources 
inherent in the instrument, and unchangeable Avhile it 
and its adjustments remain unchanged, they are reduci- 
ble to fixed and ascertainable forms ; each particular 
defect, Avhether of structure or adjustment, producing its 
own appropi'iate form of error. When these are tho- 
roughly 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: Ave have detected, by direct observation, an in- 
strumental defect. 


(113.) 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 instru- 
ment, by any assigned deviation in this respect ; that is 
to say, what would be the disagi'cement 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 devia- 
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 exactly 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 from 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 utmost import- 
ance to practice, and one of which a complete knowledge 
will enable an observer, with very moderate instrumental 
means, to attain a degree of precision which might seem 
to belong only to the most refined and costly. In the 
present work, however, we have no further concern with 
it. The few astronomical instruments we propose to de- 
scribe in this chapter will be considered as perfect both in 
construction and adjustment. 

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



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 tlie 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 forced 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 other, the apparent diurnal orbit of every 
star is distorted from a circular into an oval form, its 
lower segment being flatter than 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 squeezed 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 difficulty. 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 it a salvo for the effect of refraction, 
or by making a distinction between the apparent diurnal 
orbits, as affected by refraction, and the fri<e 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 instrument ally, i. e. by 
no ling, by timekeepers, their successive arrivals on the 
ijiitji'tdian, than we find differences which cannot be ac- 


counted for 1)y any error of obsorvution. All the stars, 
it is true, occupy the same interval of time between their 
successive appulses to the meridian, or to any vertical 
circle ; but tliis is a very different one from that occupied 
by the sun. It is palpably shorter : being, in fact, only 
SG*" 53' 4-09", instead of 24 hours, such hours as our 
common clocks mark. Here, then, we have already two 
dijft^rent days, a sidereal and a solar ; and if, instead of 
the sun, we ol)serve the moon, we find a third, much 
longer tlian either, a lunar day, whose average dura- 
tion is 24*' 54 of our ordinary time, which last is solar 
time, being of necessity conformable to the sun's succes- 
sive reappearances, on which all the business of life de- 

(116.) Now, all the stars are found to be unanimous 
in giving the same exact duration of 23'' 56' 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 their own, independent of the rotation of tlie 
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 small hole in a plate of metal nail- 
ed to some immoveable support), notice the sviccessive 
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 asrcrtaining the rate of a 
clock or watch, being exceedingly accurate if a few precautions are at- 
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 planet) disappears shall be- 
quite smooth; as otherwise variable rcdectioii may transfer the point of 
disappearance 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. 


come up to the wall, from which, by taking half the in- 
terval he will 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 the times of 
the daily arrival of the sun on the meridian, irregulari- 
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 the 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 pressed upon our notice 
between the actual 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 upon 
us. The sun's apparent motion is not only not the same 
Avith that of the stars, but it is not (as the latter is) uni- 
form. It is subject to fluctuations, 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 insti-u- 
ment called the transit instrument, especially destined 
for such observations, and to attend minutely to ad 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 continu- 
ally smaller and smaller, Avhich were before obscured, or 
mixed up with errors of observation and instrumental im- 
perfections. In short, we may not 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 
pea unagitated by tide or waves. The great annual fluc- 
tuation above noticed may be compared to the daily varia- 


ions 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 rinplings, to tlie series of whose subordi- 
nation we can perceive no end. 

(119.) With the causes of these irregulainties 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 tlie 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 
this country), even when they use mean solar time, de- 
part from the civil reckoning, commencing their day at 
noon, and reckoning the 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 the civil reckoning substi- 

(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 
they must, if we consider that, 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 consid^erable inconvenience, especially as respects 
places differing very widely in situation, 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- 



cumstance, 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 every part of the globe. Its origin will be ex- 
plained more fully at a moi'e 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 Imv of any observed 
motion in a celestial 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 compare such laws, therefore, with 
observation, we must possess a register of the observed 
situations of the object in question, and of the times ivhen 
they were observed. 

(122.) The measurement of time is performed by 
clocks, chronometers, clepsydras, and hour-glasses : the 
two former are alone used in modern astronomy. The 
hour-glass is a coarse and rude contrivance for measuring, 
or rather 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 
determinate orifice, is susceptible 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 the bottom of 
a vessel, kept constantly full to a fixed height. The 


Stream is intercepted at the moment of noting any event, 
and directed aside into a receiver, into which it continues 
to run, till the moment of noting any other event, when 
the intercepting cause is suddenly removed, the stream 
flows in its original course, and ceases to run into the 
receiver. The weight of mercury received, compared 
with the weight received in an interval of time observed 
by the clock, 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 chronometer,* 
are the instruments on which the asti'onomer depends 
for his knowledge of the lapse of time. These instru- 
ments are now brought to such perfection, that an irregu- 
larity in the rate 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 
ascertained within a ievf tenths of a second, by their use. 
In proportion as intervals are longer, the risk of error, as 
well as the amount of error risked, becomes greater, be- 
cause the ai'cidental errors of many days may accumu- 
late ; and causes producing a sIoav progressive change in 
the rate of going may subsist unperceived. It is not safe, 
therefore, to trust the determination of time to clocks, 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. But if this be done, the longest intervals 
maybe fixed with the same 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 ; the fractional parts only, at either end, are mea- 
* x>">f, time ; i"Tti,to measure. 


sured 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- 

(124.) In the culmination, or transit (i. 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 instrum.ent with wliich 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 wes-t points of the 
horizon, or at right angles to the plane of the meridian of 
the place of observation. The extremities of the axis 
are formed into cylindrical pivots of exactly equal diame- 
ters, which rest in notches formed in metallic 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-l'"' 
zontal, by levelling it with a level made to rest on the 


pivots. By the latter adjustment the axis is brought pre- 
cisely into the east and west direction, the criterion of 
which is furnished by the observations themselves made 
with the instrument, or by a well-defined object called a 
meridian mark, originally determined by such observa- 
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 dcpcription, 
that, if the central line of the telescope (iliat wliich joins 
the centres of its object-glass and eye-glass, and which 
is called in astronomy its line of coUimalion) be once well 
adjusted at right angles to the axis of the ti-ans.ii, it will 
never quit the plane of the meridian, when the instrument 
is turned round on its axis. 

(126.) In the focus of the eye-piece, and at right an- 
gles to the length of the telescope, is placed a system of 
one horizontal and five equidistant vertical threads or 
wires, as represented in the annexed figure, which always 
appear in the field of view, when properly illuminated, 

by day 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 a lateral (horizontal) motion ; and 
it is by this means brought to sitch a position, that the 
middle one of the vertical wires shall intersect the line of 
collimation of the telescope, where it is arrested and 
permanently fastened. In this situation it is evident 
that the middle thread will be a visible representation of 
that portion of the celestial meridian to wliich 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 


noted by the clock or chroaomeler, which forms an in- 
dispensable accompaniment of the transit instrument. 
For greater precision, tlie moments of its crossing all tlie 
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 will, 
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, Ave 
may be sure that the line of collimation 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, an 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 witli 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 
tlie 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 the help of the earth's diurnal motion, 
which carries equal arcs of the equinoctial across the 
meridian, in equal times, at 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. Pearson's Treatise on Practical Astronomy. Also Bianchi 
Sopra lo Stromenio de' Passagi. Ephein- di Milano, 1824, 


firm and tlurable material, and mechanically subdivided 
into equal parts, such as degrees, minutes, &c. Let 
A BOD be such a circle, divided into 360 degrees (niun- 


bered in order from any point 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, ab, is exactly parallel to 
the plane of the circle, 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 avis 6 of the tube be directed to 
one of them, S, and clamped. Then will a mark on the 
arm 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 division and frac- 
tional part thus noted, and reduced into degrees, minutes, 
and seconds, is to be set down us the reading of the limb 


corresponding to that position of the tube ab, where it 
points to the object S. Tlie same must then be done for 
the object T ; the tube pointed to it, and the limb " read 
off.''"' It is manifest, then, that, if the lesser of these 
readings be subtracted from the greater, tlidr 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 
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 Avith 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, at pp, on the circle AB, 


whose axis, D, works in the solid metallic centring E, 
from Avhich originates an arm, F, carrying at its ex- 
tremity an index, or other proper mark, to point out and 
read ofl' the exact division of the circle at B, the point 
close to it. 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- ct) ; or a vernier (fig. b) ; or, 



lastly, a compound microscope (fig. c), represented in 
section (in fig. d), and furnished with a cross in the 
common focus, of its ohject 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 
mod-e of accomplishing the latter object with any re- 
quired exactness has been explained in the last article. 
With regard to the graduation of the limb, being merely 
of a mechanical nature, we shall pass it Avithout remark, 
further than this, that, in the present state of instrument 
making, the amount of error from this source of inaccu- 
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 vision with the naked eye 
can command. But if, in place of these simple but 
coarse contrivances, the tube itself be converted into a 
telescope, 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 that a greater 
degree of exactness may be attained in the pointing of 



the tube tlian 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 afford 
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 the 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 Avhich 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 modern astronomy, without which all other re- 
finements in instrumental Avorkmanship 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, 

* The honour of this capital improvement has been successfully vin- 
dicated by Derham (Phil. Trans, xxx. 603) to our young, talented, and 
unfortunate countryman Gascoigne, from his correspondence with Crab- 
tree and Horrockes, in his (Derham's) possession. The pa.ssages cited 
by Derham from these letters leave no doubt that, so early as 1640, 
Gascoigne had applied telescopes to his quadrants and sextants, with 
threads in the common focus of 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 appeareth not, or it is not otherwise 
light enough." These inventions were freely communicated by him to 
Crabtree, and through him to his friend 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 battle 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 was revived, or re-invented, in 
1667, by Picard and Auzout (Lalande, Astron. 2310), after which its use 
became universal. Morin, even earlier than Gascoigne (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 Morin. 
(3ee Lalande, ttbi supra.) 


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, Avith 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 tlie 
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 diurnal 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 v/ith the intersection of the wires, 
would constantly remain pointed to some portion or other 
of this line, which would thercibre continue to appear 
in its field as a luminous 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's 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 continusd to see it 
ever so long. The reading off of the limb may then be 


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

(134.) This is the principle of the mvu-al 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 powerfvd 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 altitudes 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 coui'se (when connected 
for refraction), equal to twice the polar distance of the 



CHAP. 11.] 

star ; 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- 
urnal 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 is the pole ; and if we suppose hp o to be the mural 
circle, having S for its centre, b a cp d will be the points 
on its circumference corresponding to BACPD in the 
heavens. Now, the arcs b a, b c,b d, and c d are given 
immediately by observation ; and since CP=PD, we 
have also cp=p d, and each of them =i'C d, consequently 
the place of the polar po'mt, as it is called, upon the limb 
of the circle becomes known, and the arcs pb,p a,p c, 
which 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 degi-ee and a half from the pole ; it 
is, therefore, the star usually and almost solely diosen 
for this important purpose ; the 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 
each other, so that their correction is easily and safely 
applied. The brightness of the pole star, too, allows 
it to be easily observed in the daytime. In consequence 



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 13 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 pass 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 point on the limb of 
the mural circle once determined, becomes an origin, 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 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- 
ference of the circle shall be employed, and different 
graduations brought into use, by which inequalities of 
division may be detected and neutralized. This is ac- 
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, is the horizontal 
point, which, being once known, becomes in like man- 
ner an origin, or zero point, from which altitudes are 
reckoned, The principle of its determination is ulti' 


mately nearly the same with tliat of the polar point. 
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 below the horizontal 
point as the otlier 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 reflected in the still, 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 diff'erence of refrac- 
tion at the moments of observation). The arc inter- 
cepted 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 the 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 verified, as a transit may, by the reversal of the 
two ends of its axis, east for west. Nothing, however, 
prevents a divided circle being permanently fastened on 
the axis of a transit instrument, near to one of its extre- 
mities, so as to revolve with it, the reading off" being per- 
formed by a microscope fixed on one of its piers. Such 
an instrument is called a transit circle, or a meridian 
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 off" by the lateral microscope. 

(141.) The determination of the horizontal point on 
the limb of an instrument is of such essential importance 
in astronomy, that the student should be ma4e acquaint- 


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 ultiirately assumed by such a line, 
admitting its perfect fiexibility , is that of gravity, or per- 
pendicular to the surfac3 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 v.S3, that it is at present almost uni- 
versally abandoned, for the more convenient and equally 
exact instrument the level. 

(142.) The level '"I 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 v/i!l cccupy 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 b, two points distant by the 
length of the bubble ; then, if the instrument be so placed 
that 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 place, and run towards 
the elevated side. Suppose, now, that we would ascer- 
tain whether any given line PQ be horizontal ; let the 
base of the level CD be set upon it, and note the points 


a b, between Avhich the bubble is exactly contained ; then 
turn the level end for end, so that C shall rest on Q, and 
D on P. If then the bubble continue to occupy the same 
place between a and b, 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- 
nomical levels are furnished with a divided scale, by 
which the places of the ends of the bubble can be nicely 
marked ; and it is said that they can be executed with 
such 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 
firmly 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- 
sect it by its horizontal Avire, and in this position clamp 
it fast. Let L be a level fastened at right angles to an 
arm, LEF, furnished with a microscope, or vernier at F, 
and, if we please, another at E. Let this arm be fitt2d by 
grinding on the axis C, but capable of moving smoothly 
on it without carrying it round, and also of being clamped 
fast on it, so as to prevent it from moving until required. 
While the telescope is kept fixed on the obiect S, let the 


level be set so as to bring its bubble to the marks a b, 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 oil' at F, and 
then let the Avhole apparatus be reversed by turning its 
horizontal axis end for end, ivithout imclamping 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 b. 
Then we are sure that the telescope has now the same 
inclination to the horizon the other ivay, 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 ouce 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 hoi'izontal 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 


certainly is, in innumerable cases, the most convenient 
means of ascertaining the Iwrizontul 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 its 
focus, and fastened horizontally, or as nearly so as may 
be, on a flat iron fioat, 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 
icross 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 difference of the readings on its limb 
will be twice the zenith distance of either ; 
whence, as in the last article, the horizontal 
or zenith point is immediately determined.* 

* Another, and, in many respects, preferable form of the floating colli- 
mator, in whicli the telescope is vertical, and whereby the zenith point is 
directly ascertained, is described in the Phil. Trans. 1828, p. 257 bv the 
same author. 

**'., '\. 

\// : 







(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 parallel to the horizontal thread of the system 
of wires therein, they may, by giving a fine motion 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, Avhich 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, Avhich 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. Noav, 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 tlirough 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, &c. 

(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 




amount of its an^ilar motion in each must be measured 
on two circles co-ordinate to each other, \vhose planes 
must be parallel to those in which the telescope movies. 
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 hy 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 
hemispliere, one of which 
has for its poles the pro- 
longations of the principal 
axis or the vanishing points 
of a system of lines parallel 
to it, and the other passes 
always through these poles ; 
for the 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 instalment, at right angles, the great circles, 


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

(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 Gil 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, or, as it was formerly called, a parallactic 
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 the 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 GH, the point to which 
it is directed will always lie in the small 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 


the entire diurnal circle, and by observing, at proper in- 
tervals, other considerable stars whose places are well 

(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 lieavens. 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 bvibble ought 
not to sliift 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 tlie azimuthal 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 Littrowontlie Adjustment of the Equatorial. Mem. Astron. So^, 
vol. ii. p. 4&. 


altitude about the western horizon, that it had at the mo- 
ment of the first observation above the eastern. At this 
point let the motion he arrested, and the liorizontal circle 
be again read off. The difiereace 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, Avhen 
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 the 
moment of the star's meridian passage, which may thus 
be determined without a transit ; and, vice versa, the 
error of a clock or chronometer may by this process be 
discovei'ed. For this last purpose, it is not necessary 
that our instrument should be provided witli a horizontal 
circle at all. Any means by which altitudes can be mea- 
sured will enable us to determine the moments when the 
same star arrives at equal altitudes in the eastern and 
western halves of its diurnal course ; and, these once 
known, the iiistaat 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 wliich grazes the horizon, through their whole 
diurnal course, the exact apparent form of their diurnal 
orbits, or the ovals into whicli 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 inhich the refraction 
itself takes place (i. e. in altitude), is made a matter of 
direct observation. 

CHAP. 11. J hadley's sextant. 101 

(155.) The zenith sector and the theGiloUte 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 
Avhich the telescope never requires to be elevated more 
than a few degrees, and in which, therefore, the vertical 
circle is either dispensed with, or executed on a smaller 
scale, and with less delicacy ; while, on the other hand, 
great care is bestowed on securing the exact perpendicu- 
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 horizontal 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 one 
determined, either by measuring its distance from the 
visil:)le horizon (such as the sea-ofling, allowing for its 
dip), or from its own reflection on the surface of mercury. 
It is the sextant, or quadrant, commonly called Hadley' s^ 
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 communicated it to Dr. Halley, who suppressed it. The 
description of the instrument was found, after the death of Halley, 
among his papers, in Newton's own handwriting, by his executor, who 
communicated the papers to the Royal Society, twenty-five years after 
Newton's death, and eleven after the publication of Hadley's invention, 
which might be, and probably was, independent of any knowledge of 
Newton's, though Hutton insitiuates the contrary. 


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

the inclination of the reflecting' surfaces to each other." 
Let AB be tlie liinb, or graduated arc, of a portion of a 
circle 60 in extent, but divided into 120 equal parts. 
On the radius CB let a silvered plane glass D be fixed, 
at riglit angles to the plane of the circle, and on the 
moveable radius CE let another such silvered glass, C, 
be fixed. The glass D is permaneatl^' fixed parallel to 
AC, and only one half of it is s'dveied, the other half 
allowing objects to be seen through it. The glass C is 
wholly silvered, and its plane i.s parallel to the length 
of the moveable radius CE, at the extremity E, of which 
a vernier is placed to read ofl' the divisions of the limb. 
On the radius AC is set a telescope F, through which 
any object, Q, may be seen by direcl rays Avhich pass 
through the unsilvered portion of the glass D, while 
another object, P, is seen through the same telescope 
by rays, which, after reflection at C, have been thrown 
upon the silvered 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, 
and by moving the radius CE will (if the reflectors be 
truly perpendicular to the plane of the circle) meet and 
pass over, without obliterating each other. The motion, 
however, is arrested when they meet, and at this point 
the angle included between the 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 


only half as distant as would correspond to degrees, the 
arc BE, when read off, as if the graduations were whole 
degrees, will, in fact, read double its real amount, and 
therefore tlie numbers to read off will express not the 
angle ECB, but its double, the angle subtended by the 

(158.) To determine the exact distances between the 
stars by direct observation is comparatively of little ser- 
vice ; but in nautical astronomy the measurement of 
their distances from the moon, and of their altitudes, is 
of essential importance ; 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 ; and the image of the star observed, 
seen by reflection, is brought to coincide with the boun- 
dary of the sea seen by direct rays. Thus the altitude 
above the sea-line is found ; and this corrected for the 
dip of the horizon (art. 24) gives the true altitude of the 
star. On land, an artificial horizon may be used (art. 139), 
and the consideration of dip is rendered unnecessary. 

(159.) The reflecting circle is an instrument destined 
for the same uses as the sextant, but more complete, the 
circle being entire, 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 of which 
the error of graduation and of reading is reduced. This 
is altogether a very refined and elegant instrument. 

(160.) We must not conclude this chapter without 
mention of the " principle of repetition ;" an invention 
of Borda, by which the error of graduation may be di- 
minished to any degree, and, practically speaking, anni- 
hilated. LetPQ be two objects which we may suppose 
fixed, for purposes of mere explanation, and let KL be a 
telescope moveable on O, the common axis of two cir- 
cles, AML and a 6 c, of which the former, AML, is ab- 
solutely fixed in the plane of the objects, and carries the 
graduations, and the latter is freely moveable on the axis. 
The telescope is attached permanently to the latter circle, 
and moves with it. An arm OaA carries the index, or 


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


detached at pleasure. Suppose, now, the telescope di- 
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 I'ound, over an arc AB, 
on the graduated liml? 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 graduation 
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 outer circle; then, 
having made the bisection of P, clamp the arm to b, and 
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, BC, equal to the angle POQ. Now again 
yead 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 with 
the same error of graduation as before. Let this pro- 
cess be repeated as ofteii as we please (suppose ten 
times) ; then will the final arc ABCD read off on the 
eircl? be ten times the required angle, affected by the. 


joint errors of all the ten observations, but only by the 
same constant error of graduation, which depends on the 
initial and final readings off alone. Now the errors of 
observation, when numerous, tend to balance and destroy 
one another ; so that, if sufficiently multiplied, their in- 
fluence will disappear from the result. There remains, 
then, only the constant error of graduation, which comes 
to be divided in the final 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 unknown 
cause, which, probably, must be sought for in imperfect 



Of the FigTirp of the Earth Tts exact Dimensinns Its Form that of Equi- 
librium modiiicd by centrifugal Force Variation of Gravity on its 
Surface Statical and Dynamical Measures of Gravity The Pendu- 
lum Gravity to a Spheroid Other Effects of Earth's Rotation Trade 
Winds Determination of geographical Positions Of Latitudes Of 
Longitudes Conduct of a trigonometrical Survey Of Maps Pro- 
jections of the Sphere Measurement of Heights' by the Barometer. 

(161.) Geography is not only the most important of 
the practical l)ranehes of knowledge to Avhieh astronomy 
is applied, but is also, theoretically speaking, an essen- 
tial part of the latter science. The earth being the ge- 
neral 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 us, 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- 
graphical determinations, and to give at the same time 


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

(1G2.) 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 com.munities of men. With physical 
^nd 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 continua- 
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 acquiesced 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 surface 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 
flattened, in the manner of an orange, having the diame- 
ter which coincides with the axis about gi^th part shorter 
than the 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 eye or hand would not 

Chap. hi. J figure of the earth. 107 

detect the flattening, since the diflerence of diameters, ill 
a globe of sixteen inches would amount only to jV^^ ^^ 
an inch. In all common parlance, and for all ordinary 
purposes, then, it would still be called a globe ; while, 
nevertheless, by careful measurement, the difljerence 
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 geometers 
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 ia 
easy to demonstrate, however, that its deviation from the 
circular form, arising from so very slight an " ellipticity" 
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 led 
to notice so small a deviation from perfect sphericity* 
How we are led to this conclusion, as a practical result^ 
will appear, when we have explained the means of de- 
termining with accuracy the dimensions of the whole, oi" 
any part of the earth. 

(165.) As Vie cannot grasp the earth, nor recede frottl 
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 upoil 
it, and apply our diminutive measures to comparatively 
small parts of its vast surface in succession, it become^ 
necessary to supply, by geometrical reasoning, the defect 
of our physical powers, and from a delicate and 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* 141 5926 to 
! 0000000), we have only to ascertain the length of the 
entire circumference of any great circle, such as a meri- 
dian, in miles, feet, or any other standard units, to knOW 
the diameter in units of the same kind. Now the cir-* 
cumference of the whole circle is known as soon as wj 


know the exact lengtli of any aliquot part of it, such as 
1 or 3^6 o*^li P^i"t ; and, this being not more than about 
seventy miles in lengtli, is not l)eyoncl the limits of very 
exact measurement, and could in fact, be measured (if 
we knew its exact termination at each extremity) Avithin 
a veiy few feet, or, indeed, inches, by methods presently 
to be particularized. 

(J 66.) Supposing, then, we were to begin measuring 
with all due nicety from any station, in the exact direc- 
tion 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 
Avould then be at once resolved. It only remains, there- 
fore, to inquire by what indications we can be sure, 1st, 
that we have advanced an exact degree ; and, 2dly , that Ave 
have been measuring in the exact direction of a great circle. 

(167.) Now, the earth has no landmarks on it to in- 
dicate degrees, nor traces inscribed on its surface to guide 
lis in such 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 little 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 Avhich are of equal permanence and stability with the 
earth itself. Such marks are afforded by the stars. By 
observations of their m'eridian altitudes, performed at any 
station, and from their known polar distances, Ave 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 Avhere 
we may establish the requisite instnmients. When our 
latitude, then, is found to have diminished a degree, Ave 
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 obserA'ations described in art, 137, 
and although local difficulties may oblige us to deviate in 
our measurement from this exact direction, yet if Ave 
Keep a strict account of the amount of this deviation, a 


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

(169.) Such is the principle of that most important 
geographical operation, the measurement of an arc of 
the meridian. In its detail, however, a somewhat modi- 
fied course must 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 more convenient method 
of measuring from one good observing station to another, 
about a degree, or two or three degrees, as the case may 
be, apart, 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 affect 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 
effect this, we take care to select for observation, at the 
extreme stations, some star which passes through or near 
the zeniths of both. The amount of refraction, within a 
few degrees of the zenith, is very small, and its fluctua- 
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, and at the other to pass one degree south or north 
of the zenith, we are sure that the geographical latitudes, 
or the altitudes of the pole at the two stations, must dif- 
fer by the same amount. 

(171.) Granting that the terminal points of one degree 



can be ascertained, its length may be measured by tlie 
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 error, 
with proper care, can hardly exceed a single second. 
Supposing we grant the possibility of ten feet of error in 
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 spherical figure is not very remote from 
the truth, 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 stancferd feet, as resulting from actual measure- 
ment, made with all 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 extracted from among the 
data given m Professor Airy's excellent paper " On the Figure of the 
Earth," in the Encyclopaedia Metropolitana. 




of Middle 
of the Arc 



of the 





C6 20 10 

. 1''.3719' 



Russia - 

58 17 37 

3 35 5 




52 35 45 

3 57 13 


Roy, Kater. 


4() 52 2 

8 20 


Lacaille, Cassini. 

France - 

44 51 2 

12 22 13 




42 59 

2 9 47 



America, U. S. - 

Sd 12 

1 28 45 


Mason, Dixon. 

Cape of Good Hope 

33 18 30 

1 13 ]7i 




16 8 22 

15 57 40 


Lambton, Everest. 


12 32 21 

1 34 5t) 




1 31 

3 7 3 


Condaniine, &c. 

It is evident from a mere inspection of the second and 
fourth coUimns of this table tliat the measured length of 
a degree increases with the latitude, being greatest near 
the poles, and least near the equator. Let 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 
earth smoothly turned in wood, it would be, as already- 
observed, so nearly spherical, that neither by the eye nor 
the touch, unassisted by instruments, could we detect any 
deviation from that form. Suppose, 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 


& 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 severarother 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, B, in 
which the light can enter in the middle between the globe 
and plate, or any other, G, where the latter tilts by pres- 
sure, or admits the light under its ed^es, we are sure that 
the curvature of the surface at B is less, and at C greater 
than at A. 

(174.) What we here do by the application 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 
Pleasured distance of advance, we 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 
eentre find some given star), mo7'e 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, Avhen, 
in order to produce the same change of horizon with 
respect to the pole (suppose 1), 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 
&f ihe earth is sensibly greater at the equator than to- 
wards the poles ; or, in other words, that the earth is 
not spherical, but flattened at the poles, or, which comea 
io the same, protubpi-ant at the equator. 

(175.) Let NABDEF represent a meridional section 
of the eartli, 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, ak, bB, rfD, ^'G, 
E, be the respective directions of ihe plumb-line at" the 
stations N, A, B, D, G, E, of which we will suppose N 



to be at the pole and E at the equator ; then will tlie tan- 
gents to the surface at these pouits respectively bo per- 
pendicular to these directions ; and, consequently, if each 
pair, viz. nN and oA, 6B and (ID, gG and eE, be pro- 
longed till they intersect each other (at the points x, y, z), 
the angles N.rA, B^D, OzE, will each be one degree, 
and, therefore, all equal ; so that the small curvilinear 
arcs NA, BD, GE, may be regarded as arcs of circles 
of one degree each, described about x, y, z, as centres. 
These are what in geometry are called centres of curva- 
ture, and the radii a:N or xA, yB or yl), zG or zE,, re- 
present 7'adii of curvature, by which the curvatures 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 radii, and as the arc NA is greater than BD, and 
that again than GE, it follows that the radius Nx must 
be greater than By, and By than Ez. Thus it appear* 
that the mutual intersections of the plumb-lines will not, 
as in the sphere, all coincide in one point C, the centre, 
but will be arranged along a certain curve, xyz (which 
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. 



(176.) In the flattening of a round figure at two op- 
posite points, and its protuberance at points rectangularly- 
situated to the former, we recognise the distinguishing 
feature of the elliptic form. Accordingly, the next and 
simplest supposition that we can make respecting the 
nature of the meridian, since it is proved not to be a 
circle, 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 revolve about its shorter axis NS. This 
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 E, the extremity of the 
longer axis is the least, and at that of the shorter axis, 
the greatest it admits, and the form of its evolute agrees 
with that here represented.* Assuming, then, that it is 
an ellipse, the geometrical properties of that curve ena- 
ble us to assign the proportion between the lengths of its 
axes which shall correspond to any proposed rate of va-. 
nation in its curvature, as well as to fix; upon their ab- 
solute lengths, con-esponding to any assigned length of 
the degree in a given latitude. Without troubling the 
reader with the investigation (which may be found in 
any work on the conic sections), it will be sufficient to 
state that the lengths which agree on the whole best with 
the entire series of meridional arcs which have been 
satisfactorily measured, are as follow :t 

Feet. Miles. 

Greater ox erjiialorial d iameter = 41 ,847,426 =7925-648 

Le.sser or polar diameter" = 41,707,620= 7899170 

Diftererif-e of diameters, or polar com- > _ 139306= 26478 
pression S 

Tlie proportion of the diameters is very nearly that of 
298 : 299, and their difference ^^y of the greater, or a 
very little greater than g-l^. 

(177.) Thus we see that the rough diameter of 8000 
miles we have hilhcrlo used is rather too great, the ex- 
cess being about 100 miles, or ^\\h part. We consider 
it extremely improbable that an error to the extent of 

*The dotted lines are the portions of the evolute belonging to the othef 
t See Profess. Ally's Essay before cited. 


five miles can subsist in the diameters, or an 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 (24,899). 

(178.) The supposition of an elliptic form of the 
earth's section through the axis is recommended by its 
simplicity, and confirmed by comparing the numerical 
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 
small, 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 
observed deviations to either local or, if general, to com- 
paratively small causes. 

(179.) Now, it is highly satisfactory to find that the 
general elliptical figure thus practically 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 spliere, 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 ; and the water on 
its 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 26 miles which we 
know to exist. It is not less evident that a mountain 
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 on 
the side of a hill. The consequence, therefore, would 


be the formation of two great polar seas, liemmed in all 
round by equatorial land. Now, this is by no means 
the case in nature. The ocean occupies, indifferently, 
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 poiver. 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 Jigure 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 facie 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 potcer 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 spin 
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 
all 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 


centrifugal force, and a state of eqiii/ibrmin is attained. 
The experiment is a very easy and instrnctive one, and 
is admirably calculated to show how \he foi'tn of equili- 
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 diminisli ; the elevated outward portion will de- 
scend, and the depressed central rise, while all the time a 
perfectly smooth surface is maintained, till the rotation is 
exliausted, when the water resumes its horizontal state. 

(181,) Suppose, then, a globe, of the size of tlie 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 reeede from the axis, 
A rotation might, indeed, be conceived so swift as io flirt 
the whole ocean from the surface, like Avater from a mop. 
But this would require a far greater velocity than what 
we now speak of. In the case supposed, the iveight 
of the Avater would still keep it on the earth : and the 
tendency to recede from the axis could only be satisfied, 
therefore, by the Avater leaving the poles, and floAving 
towards the equator; there heaping itself up in a ridge, 
just as the water in our pail accumulates against the side ? 
and being retained in opposition to its weiglit, or natural 
tendency towards the centre, by the pressure thus caused. 
This, hoAvever, could not take place Avithout 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 gi-eat equato- 
rial continent and polar seas Avould be formed 5 in the 
latter, protuberant land Avould appear at the poles, and a 
zone of ocean be disposed around the equator. This 
would be the first or immediate effect. Let us noAV see 
what Avould afterAvards happen, in the two cases, if things 
were alloAved to take their natural course. 

(182.) The sea is constantly beating on the land, 
grinding it doAvn, and scattering its worn off particles and 
fragments, in the state of mud and pebbles, over its bed, 


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 /ori of eqidUbrium 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 polar 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 and (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 -^-^ th part 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 than at the 
poles, where no such counteracting force exists ; and 
* See Cab. Cyc. Mechanics, c. viii. 


where, in consequence, the water may be considered as 
specifically heavier. Taking this principle as a guide, 
and combining it with the laws of gravity (as developed 
by Newton, and as hereafter to be more fully explained), 
mathematicians have been enabled to investigate, a pri' 
ori, 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 found 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 than an oblate ellipsoid, of a degree of 
ellipticity 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 assumed 
solely on account of the easy explanation it ofl'ers of the 
apparent diurnal motion of the heavens. 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 was 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- 


tion must generate a centrifugal force (see art. 184), the 
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 gi-eat- 
est care, and in every accessible part of the globe, have 
fully demonstrated the fact of a regular and progressive 
increase in the weiglits of bodies corresponding to the 
increase of latitude, and fixed its amount and the law of 
its progression. From these it appears, tliat 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 194 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-' 

(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 that they 
will still 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 
M'e assert that a body weighing 194 pounds at the equa- 
tor will weigh 195 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 Avould 
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, x4, and 
conducted (supposing such a thing possible) over other 



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 -weight^/ would preponde- 
rate ; and to restore the equipoise 
the weight x must be increased byy-ijth 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 Avith 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 C let a spiral spring 
G 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 
gropes the agate, a contact which can _ 
be made with the utmost imaginable fen ii i li lii ^ ll il'igiiiiHi iiiliiiit 
delicacy. Let these weights be noted ; the weight F de- 
tached ; the spring G carefully lifted off its hook, and 


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 half 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 Avill produce a further 
extension of ^Jogth of an inch, a quantity which cannot 
possibly be mistaken in such a contact as that in question. 
Thus we should be provided with the means of mea- 
suring the power of gravity at any station to within 
Toi 00"^^ 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 difierent 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 for the use of the pendulj- j, must 
depend on the degree of permanence and uniformity of action 6 dynngs, 
on the constancy or variabihty of the effect of temperature, on their elas- 
tic force, on the possibility of transporting them, absolutely unaltered, 
from place to place, &c. The great advantages, however, which such 
an apparatus and mode of observation would nosse^'" f'. p ' jf conve- 
nience, cheapness, portability, and expedition, over th pre>'- ' 1-vborious, 
tedious, and expensive process, render the atteippiw6jiA"5tn making. 


case be coimtcd, the intensities of the forces will be to 
each other inversely as the squares of the numbers of 
oscillations made, and thus their 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, Avhen 
transported to London, the same pendulum makes 86,535 
vibrations in the same time. Hence we conclude, that 
the intensity of the force virging the pendulum down- 
wards at the equator is to that at London as 86400 to 
865^5, or as 1 to 1-00315; or, in other Avords, 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 1 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-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 2-g-9 expressing tlie centrifugal force at the equator. 
The difference by which the former fraction exceeds the 
latter is ji-^, 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 cenljpfugal 
forc^ *^ centrifugal force produces an ellipticity in the 
form O' lie earth itself; and this very ellipticity of form 
modifier its power of attraction on bodies placed at its 
surface, and thus gives rise to the difference in question. 
Here, tlien, we liave tlie same cause exercising at once a 
direct ati'l an indirect influence. The amount of the former 


is easily calculated, that of the latter Avith 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 l)ody (considered as undimi- 
nished by a centrifugal force) is the 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 in 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 tliat, 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 Avith 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 treated Avith 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 Aveight of a body in going from the equator 
to th%pole by almost exactly j-g-pth part ; which, toge- 
ther with 2^-f7rth due to the centrifugal force, make up the 
whole quantity, y-g-jth, observed. 

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


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 bidk, 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 

(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 23 1 north, and as much south of the equator; 
and that the whole of that zone or belt of the earth's sur- 
face 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, flows 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 



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 same time lag, or hang back, 
and drag tfpon it in the direction opposite to the earth's 
rotation, i. 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 mass 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 tlie 
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 rotation, 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 incumlient on it, in point of motion. 

(198.) It follows from this, then, that as the winds on 
both 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, wiiere this is very distinctly, and, so far as I am aware, for 
the first lime, reasoned out, Author. 


crease very slowly in the immediate vicinity of the equa- 
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 easterly 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 another. 

(199.) The result, then, 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-ivinds.* 

(200.) 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 consequence 
generally impossible ; and 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 surface 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. 


westerly relative direction ; and when at length it returns 
to the surface, in its circulation, which it must do more 
or 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 
rotation affects the atmosphere on the great scale, we omit all considera- 
tion of local periodical winds, such as monsoons, &c. 

It seems worth inquiry, wliether hurricanes in tropical climates may 
not arise from portions of the upper currents prematurely diverted dowTi- 
wards before their relative velocity has been sufficiently reduced by fric- 
tion on, 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 accoimt 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 general, a rapid transfer, either way, in latitude, of any 
mass of air which local or temporary causes might carry above the im- 
mediate reach of the friction of the earth's surface, would give a fearful 
exaggeration to its velocity. Wherever such a mass should strike the 
earth, a hurricane might arise ; and should two such masses encounter 
in mid-air, a tornado of any degree of intensity on record might easily 
result from their combination. Author. 


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

(202.) Tlie latitude of a station on a sphere woukl be 
merely the length of an arc of the meridian, intercepted 
between tlie 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 difterences 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 longitude, whose exact de- 
termination is a matter of more difficulty. The reason 
is this : as 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, i. 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 meridian (i. e. differing in latitude) 
the heavens present difierent aspects at all 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 diflerently inclined 
to their horizons, and differently divided by them, and 
attain different altitudes. On the other hand, to ob- 
servers situated on the same parallel (i. e. differing only 
in longitude) the heavens present the same aspects. 
Their visible portions are the same ; and the same stars 
describe circles equally inclined, and similarly divided 


by their lionzons, and attain the same altitudes. In tho 
former case there is, in the latter there is not, any thing 
in the appearance of the lieavens, 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 ivesterhj 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, i. e. before the earth's rotation from 
west to east carries him back again to the line joining 
the star and the earth's centre from which he set out. 

(205.) 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 altittides. In 
that of stations differing only in longitude, it comes to 
the meridian at the same altitude, 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 difierence 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 


must leani to distinguish between time, in the abstract, 
as common to the whole universe, and therefore reckoned 
from an epoch independent of local situation, and local 
time, which reckons, at each particular place, from an 
epoch, or initial instant, determined by local convenience. 
Of time reckoned in the former, or abstract manner, 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 local 
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"' 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 ; Avhich is, therefore, entirely a Zoc/ 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 neces- 
sary to name the place, or particularize ivhat mean time 
we intend. On the other hand, a date by equinoctial 
time is absolute, and requires no such explanatory ad- 

(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 
his clock he knows when the equinox passed. At that 
moment his clock ought to have marked 0" 0' ; and if 
it did not, he knows and can correct its error, and by the 
agreement or disagreement of the en-ors assigned by each 
star he can ascertain whether his clock is correctly regu- 
lated to go twenty-four hours in one diurnal period, and 


if not, can ascertain and allow for its rate. Thus, although 
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 
be 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 that 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- 


ter, or let it run down), ascertain their difierences 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 15 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 O**, but 
to l**, and will therefore be i** fast on the local time of 
B. If he travel eastward, the reverse will 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 
to the point he set out from. A singular consequence 
Avill 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 earth 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 
really made one turn more round its centre ; and if in 
the opposite direction, one turn less 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 
had trusted to the rotation of the earth aloiie to carry 
him round. As the earth revolves from west to east, it 


134 A tUEAtlBE ON ASTRONOMY. [cHAP. lit. 

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' 
earily 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 pi'oduce 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 skill of modern 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- 
died by carrying out several, and using them as checks 
on each other ; but, besides the expense and trouble, this 
is only a palliation of the evil the great and funda- 
mental, as it is the only one to which the determination 
of longitudes by time-keepers is liable. It becomes ne- 
cessary, therefore, to resort to other means 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 local 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. 


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 any 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 
difterence 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 atB, 
by applying its error and rate, Consequently, the differ- 
ence of their local epochs is 4"" 0% which is also their differ- 
ence of longitudes in time, or 1 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 afl'ords a comparison of 
times, and the mean of all which is, of course, more to 
he 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 
may easily be seen at fifty or sixty miles ; and in moun- 
tainous countries the flash of gunpowder in an open 
spqon may be seeq, if a proper station be chosen for its 


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 parties, it is a matter of indifference where they 
are exhibited. Still the interval which could be thus 
embraced would be very limited, and tlie 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 

\* :'* :* 

A B c D :e ji" z 

line is occupied by stations so arranged, that the signals 
at B can be seen from A and C ; 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 C, 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 


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 pre- 
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 may be 
observed, is usually 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 Avhole 
terrestrial hemisphere), is afibrded by the eclipses of 
Jupitei-'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 combinaticai of all the observations, 
see a paper on the difference of longitudes of Greenwich and Paris, Phik 
Trans. 1826 ; by the author of this volume. 

t This idea was first suggested by the late Dr. Maskelyne, 



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 
eclipses, 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 longitudes 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 on 
land in remote regrions, 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 highly 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 


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 set of marks upon it, whose position 
is known ; the latter, by passing over and among these 
marks, informs us, by the place it holds with respect to 
them, what it is o'clock, or what time has elapsed since 
a certain moment wiien 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 wliich the hands revolve with 
a uniform motion. But it is clear that we should, with 
equal certainty, though with much more trouble, tell 
what o'clock it were, if the marks on the dial-plate 
were i<nequally distributed, if the hands were eccentric, 
and their motion not uniform, provided we knew, 1st, 
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 ; and, 3dly, if we were 
fully 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 angular movement. 

(222.) 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, tlie 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 
laws of astonishing complexity and intricacy in result, 
though 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 
gliding beside and between others ; and whose position 
amono" them can, at any moment when it is visible, be 


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

(223,) There is only one circumstance wanting ta 
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 
ihe 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 parcdlax or negligent in not 
taking it into account we might make great mistakes 
in reading the time, by referring the hand to the wrong 
mark, or incorrectly appreciating its distance from the 
Tight. On the other hand, if we took care to note, in 
every 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 influence 
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 near 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 a parallax, 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 contre. 

(3.24.) Such a clock as we have described might, no 
doubt, be considered a very bad one ;- but if it were our 
anil/ 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- 
dyhig the laws of its movements, or in facilitating th 


means of reading it correctly. Such, in the parallel 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 Avith absolutely certainty, whereabouts amo;ig 
the stars, at every hour, minute, and second, in every 
day of every year, in Greenwich local time, the moon 
ivoidd 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. 

(225.) 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, 
the 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 sur- 
passing six or seven miles, and purposely selected in a 



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 op 
platina let into massive blocks of stone, and which are, 
or at least ought to be, in all cases preserved with almost 
religious car, 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, carefidly 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 ol)servatory, with which it is a principal 

pbject that the base should be as closely and immedi- 

9.tely 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, 

\vith a netvy^ork of triangles. Now, it is evident that tlie 

angles of the triangle A, B, C being obseived, and one 

of its sides, AB, measured, the other two sides, AC, BC, 

^piay be calculated by the rules of trigonometry ; and thusi 

each of the sides AC and BC becomes in its turn a base 

capable of being employed as known sides of cHher tri^ 

angles. For instance, the angles of the triangles ACQ 

^nd BCF being known by observation, and theii sides 

AC andBC, we can thence calculate th|.lengths AG, CG, 

pnd 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 base of between seven and 
f^ght miles, ne^- Lonilpnderry, is sin)jTosed not to exceed two i\icUea. 

doRilEctioN FOR The earth's sphericity. 143 

mi. 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 con* 
structedj and filled in to atiy 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 from 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, because 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 as 
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, maybe 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, with 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, thi^ 
may be neglected, as the difference is imperceptible 5 but 
in the larger ones it must be taken into eonsidei'atiotif 


It is evident that, as every object used for pointing the 
telescope of a theodolite has some certain elevation, not 
only above the soil, 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 construction 
of the theodolite (art. 155), which is nothing more than 
an altitude and azimuth instrument, this reduction is made 
in the very act of reading off' the horizontal angles. Let 

E be the centre of the earth ; 
Lk, 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 
be 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, intersecting the ra- 
dii EBP, ECQ, at M and N, 
above the spherical surface. 
The telescope of the theodolite, 
it is true, is pointed in succes- 
sion to P, and Q ; but the readings off of its azimuth 
circle give not the angle PAQ between the directions 
of the telescope, or between the objects P, Q, as seen 
from A ; hut the azimuthcd angle MAN, which is the 
measure of the angle A of tbe spherical triangle BAG. 
Hence arises this remarkable circumstance, that the sum 
of the three observed angles of any of the great triangles 
in geodesical operations is always found to be rather more 
than 180 : were the earth's surface a plane, it ought to 
be exactly 180 ; and this excess, Avhich 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 
ofTers 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 earth is taken into consideration, is to regard the net- 


work of triangles with Avhich the country is coveredj as 
the bases of an assemblage of pyramids converging to the 
Centre of the earth. The theolodite gives us the true 
measures of the angles included by the planes of these 
pyrainids ; and the surface of an imaginary sphere on 
the 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 
superficial 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 earth, it may also be done by appropriate processes 
of calculation, which, howevef) 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 
tidges, and the places of their principal summits ; and 
all those details which, as they belong to physical and 
statistical, rather than to asti'onomical geography, wd 
need not here dilate on. A (ew words, however, will be 
necessary respecting maps, which are used as well ill 
astronomy as in geography. 

(231.) A map is nothing more than a fepresentation,- 
tlpon a plane, of some portion of the surface of a spherCj 
on which are traced the particulars intended to be ex-= 
pressed, whether they be continuous outlines or points* 
Now, as a spherical surface! can by no contrivance 
be extended or projected into a plane, without undud 

* Lardner's Trigonometry, prop. 94. Woodhouse's ditto, p. I48. 1st 

t We here neglect the ellipticity of the earth, which, for such a pur- 
pose as map-making, is too trifling to have any material influence. 



enlargement or contraction of some parts in proportion 
to others ; and as the system adopted in so extending or 
projecting it will decide xvhat 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, stereographic, and Mercator''s. In the 
orthographic projection, every 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 stereographic projection is in great mea- 
sure free from this defect. To understand this projection, 


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 
such, it possesses some singularly elegant geometrical 
properties, of which we shall state one or two of the 

(234.) And first, then, all circles on the sphere are re- 
presented by circles in the projection. Thus the circle 
Xis projected into x. Only great circles passing through 
the vertex B are projected into straiglit 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, ghk, 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- 
sphere in a single map, without any violent distortion 
of the configurations on the surface from their real forms. 
As in the orthographic projection, the borders of the 
hemisphere are imduly crowded together ; in the stereo- 
graphic, their projected dimensions are, on the contrary, 
somewliat enlarged in receding from the centre. 

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
















. 1 (\ 




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


cannot be seen from any one point, but as it might be 
geen 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 
Jine, 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 i-eferring eveiy point in the 
globe to a circumscribing cylinder, by lines drawn frorn 
the centre, and then unrolling the cylinder into a plane, 
Jjike the stereographic projection, it gives a true repre-- 
gentation, as io form, of every particular small part, but 
yaries greatly in point of scale in its different regions j 
the polar portions in particular being extravagantly en-; 
Jarged ; and the whole map, even of a single hemisphere, 
ftot being con^prizable within any finite limits. 

(236.) We shall not, of course, enter here into any 
geographical details ; but one result of maritipie discovery 
pn the great scale is, so to speak, massive enough to call 
for mention as an astronomical feature. When the con-i 
tinents and seas are laid down on a globe (and since the 
discovery of Australia we are sui-e that no very extensive 
tracts of land remain unknown, except perhaps at the 
south pole), we find that it is possiljle so to divide the 
globe into two hemispheres, that one shall contain nearly 
<jtll the land ; the other being- almost entirely sea. It i^ 
|i fact, not a little interesting to Englishmen, and, com-, 
tined with our insular station in that great highway of 
^^ations, the Atlantic, not a little explanatory of our com- 
]fnercial eminence, that London occupies nearly the centre 
pf the terrestrial hemisphere. Astronomically speaking, 
the fact of this divisibility of the globe into an oceanic 
^nd a terrestrial hemisphere is important, as demonstra^ 
tive of a want of absolute equality in the density of the 
^olid material of the two hemispheres. Considering the 
^hole mass of land and water as in a state of equili- 
brium, it is evident that the half which protrudes must 
pf necessity be buoyant : not, of course, that we mean 
to assert it to be lighter than water, but, as compared 
witU the whole globe, in a less degree heavier thaQ 



that fluid. We leave to g-cologists to draw from these 
premises their own conckisions (and Ave 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 hypotliesis of an exact elliptic figure would 
require, this, as a general fact, oughtnot to be lostsight 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 difficulty 
and however slowly) by direct sounding ; tlie former by 
two distinct methods : the one consisting in trigonome- 
trical measurement of tlie diff'ercnces 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. In 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 ; in the other, of air. In 
the one case our sounding line is real and tan^il^le ; in 
the other, an imaginary one, measured by the length of 
the column of quicksilver the superincumbent 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 
subsist under such circumstances. Let ABODE be a 

continent, of which the portion ABO projects above the 
water, but is covered by the oil, which also floats at an 



uniform depth on the whole ocean. Then if we woukl 
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 
done the same at C, a point at the sea level, the difference 
of the two float lines gives the height in question. 

(239.) Now, though the atmosphere diff'ers from oil 
jn 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 all 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 ivoidd be the surface of the sea, if 
frolo)iged under the continents, and therefore each or 
sny of them has all the characters of a definite surface to 
measure from, pix)vided it can be ascertained and identic 
fied. Now the height at which, at any station B, the 
mercury in a barometer is supported, informs us at 
mcc how much of the sitmosphere is incumbent on B, 
or, in other words, in U'hat stratum of the general at- 
mosphere (indicated by its density) B is situated ; 
whence we aj-e 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- 
iails, the reader is refeiTed 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 Avere 
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-linea 

* See Cab. Cycl. Pneumatics, art. 143. 

t Biot, Astronomie Physique, vol. 3. For tables, see the work of Biot 
cited. Also those of Oltmauri, annually published by the French board 
of longitudes in their Annuaire: and Mr. Baily's Collection of Astrono^ 
nicl Tables and Forinute. 


of 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 pursued 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. 



Construction of celestial Maps and Globes by Observations of right As- 
cension and Deolination Celestial Objects distinguished into fixed 
and erratic Of the Constellations Natural Regions in the Heavens 
The Milky Way The Zodiac Of the Echptic Celestial Latitudes 
and Longitudes Precession of the Equinoxes Nutation Aberration 
Uranographical Prablema. 

(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 doubt, 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 


we take advantage of the earth's rotation on its axis, and 
by observing each celestial object as it passes our meri- 
dian, 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- 

(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 
celestial 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, lastly, 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 


sently ; but for the purposes of uranography, in so far as 
they concern only the actual configurations of the stars 
inter se, 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 natioTial 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 aff'ected 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 meridiem 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. 136, 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 tliosc 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 
among 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 
' relation of place inter se, as if they formed part of a 
solid though invisible firmament; and, like the great 
natural landmarks on the earth, preserve immutably the 
same distances and bearings each from the other ? If so, 
the most rational idea we could form of the universe 
would be that of an earth at absolute rest in the centre, 
an4 a hollow crystalline sphere circulating round it, aucl 


carrying sun, moon, and stars along in its tliurnal 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 the 
invisibility of stars to the naked eye in the day-time, it 
is not so readily recognised, and requires either the use 
of telescopes and angular instruments to measure it, or 
a lonsfer 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 witii 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 mvdtitude of the stars which 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, 


a uniformity of aspect which, when contrasted Avith 
the perpetual change in the configurations of the sun, 
moon, and planets, may well be termed invariable. 
It is not, intleed, that, by the refinement of exact 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 singidar 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 
stars), is reduced to a simple marking down of their re- 
lative places on a globe or on maps ; to the insertion on 
that 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, arc 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 
* HAavnTHf , a wanderer. 

156 A tREAtlSE ON A^tRONOMIr. [cHAP. iVi 

familiarize himself with the idea rather of tivo 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 seculat 
variations (as they are called), to coincide, at very dis- 
tant intervals 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 do 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 the 
figures called up in imagination by a view of the more 
splendid " 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, of altogether 
disregard them,* except for briefly naming remarkable 
stars, as a. Leonis, ^ Scorpii, &c. &;c., by letters of the 
Greek alphabet attached to them. The reader will find 
them on any celestial charts or globes, and may compare 
them with the heavens, and there learn for himself their 

*This disregard is neither supercilious nor causeless. The constella' 
tions seem 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; bears, lions and fishes, large and small, northern 
and southern, confuse all nomenclature, &c. A better system of con-- 
stellations might have been a material help t*s an artificial memory. 


(253.) There are not wanting, however, natural dis- 
tricts in the heavens, which ofi'er great peculiarities of 
character, and strike every observer: such is the milky 
way, that great luminous band, which stretches, every 
evening, all across the sky, from horizon to horizon, 
and wlaich, when traced with diligence, and mapped 
down, is found to form a zone completely encircling the 
whole sphere, almost in a great circle, which is neither 
an hour circle, nor coincident with any other of our 
astronomical grammata. It is divided in one part of its 
course, 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, 
from 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 millions, like glittering dust, on 
the black ground of the general heavens. 

(254.) Another remarkable region in the heavens ia 
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 gi-eater 
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 the 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 



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 mucli complexity, and described with very unequal 
velocities in their different parts. They have all, how- 
ever, this in common, tliat the general direction of their 
motions is the same with that of the sun, viz. from tvest 
to east, that is to say, the contrary to that in whicli 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 f.iemselves 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 Avhich, 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 wliich 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 tlie 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- 
real year, which consists of 365"* 6" 9" 9'- 6, reckoned 
in mean sqlar 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- 


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 siui 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*", &c. of sidereal time, if 
reckoned in mean solar days, hours, &c. will be called 
SeS'' &^, Sic. Tims, 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 surface, 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 obliqiuty of the 
ecliptic. In the annexed figure, if Pja 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 Kk, 
its poles the spherical angle QVS is the obliquity of the 


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 uu- 
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 sun 
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 Avhere the sun has the 
greatest northern declination, is called the summer sol- 
stice, and W, that where it is farthest south, the tvinter. 
These epithets obviously have their origin in the depend- 
ence of the seasons on the sun's declination, which will 
be explained in the next chapter. The circle EKPQA;j3, 
which passes through the poles of the ecliptic and equinoc- 
tial, is called the solstitial colure ; and a meridian drawn 
through the equinoxes, PV/jA, 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 


circle of declination drawn throiigli it, by which it is 
referred to the equinoctial, and KXT a circle of latitude 
referring it to the ecliptic then, an 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 surtace. The force of 
this observation will presently become apparent. 

(260.) Knowing the right ascension and declination of 
an object, we niay 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 colure is of course 90 distant 
from V, the vernal equinox, which is one of its poles 
so that VR (the right ascension) being given, and also 
VE, the arc ER, and its measure, the spherical angle 
EPR, 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 eas)' 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 

(261.) The same course of observations by which the 
path 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, at 
that time a point of great importance in practical astro- 
nomy, as it is the origin or zero point of right ascension 



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 contrary 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 /jrecesston 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' 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 years, 
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 in- 
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 tells upon the longi- 
tudes of all alike, whether at rest or in motion, and pro- 
duces, so far as its amount extends, the appearance of a 
motion in longitude common to all, as if the whole hea- 
vens had a slow rotation round the poles of the ecliptic 
in the long period above mentioned, 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 


confusion 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 other smaller and 
subordinate periodical fluctuations : the former giving 
rise to the phenomena of jurecession; 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, 


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 sitviation of the equinoctial 
EVQ is determined by tliat of the pole, this, it is evi- 
dent, must cause a displacement of the equinoctial, which 
Avill 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. e. the displaced equinox, will lie on 
that side of V, its original position, towards which the 
motion of tlje 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 the 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- 
ing 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 


driven through it. That such is the case is proved by 
the two great facts: 1st, that tlie latitudes of places on 
the earth, or their geographical situation with respect to 
the poles, have undergone no pei'ceptible 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 1 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. Lyrre, the brightest in the north- 
ern hemisphere, will occupy the remarkable situation of 
a pole star, approaching within about 5 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 stars, 
in a period of about nineteen years, a minute ellipsis, 
having its longer axis equal to 18"'5, 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 tlie nutation, the pole is describing its little 


ellipse of 18"*5 in diameter, it is carried by the gi-eater 
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 tAvo motions, subsisting 
jointly, will be neither an ellipse nor an exact circle, but 
a gently nndulated ring like that in tlie iigure (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 tlie 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 wliich 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 the law of nutation, which is dependent on the position, for the time, 
of the lunar orbit. If we attribute it to a real motion of the celestial 
(Sphere, we must then maintain that splicre to bo kept in a constant state 
oi tremor by the motion of the moon I 


whether we ineau t!ie 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 advance 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 subti-acting from them the 
whole effect of precession in the interval ; and, moreover, 
to divest them of the influence of nutation by investiga- 
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 formulae 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 oljject are derived from its longitude 
and latitude. Referring to the figure of that article, and 
supposing the triangle KPX orthographically 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-longilude 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 efiect generally, and the 


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 object, and 
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 proper 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 


moves. The law of its motion, however, therein, cannot 
be understood till the reader has been made acquainted 
with the principal features of the moon's motion on 
Avhich it depends. See chap. XI. 

(274.) Another consequence of what has been shown 
respecting precession and nutation is, that sidereal time 
as astronomers use it, i. e. as reckoned from the transit 
of the equinoctial point, is, not a mean or uniformly 
Jlotving quantity, being affected by nutation ; and, 
moreover, tliat so reckoned, even when cleared of the 
periodical lluctuation of nutation, it does not strictly 
correspond to the earth's diurnal rotation. As the 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 solar 

(275.) Neither precession nor nutation change the 
apparent places of celestial objects inter se. We see 
them, so far as these causes go, as they are, though from 
a station more or less unstable, as we see distant land 
objects correctly formed, though appearing to rise and 
fall when viewed from the heaving deck of a ship in the 
act of pitching and rolling. But there is an optical cause, 
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 ; and whose influence must be estimated and 
allowed for before we can obtain a precise knowledge of 
the place of any object. This cause is what is called 
the aberration of light ; a singular and surprising effect 
arising from this, that we occupy a station not at rest 
but in rapid motion ; and that the apparent directions of 
the rays of light are not the same to a spectator in mo- 
tion as to one at rest. As the estimation of its effect be- 
longs to uranography, we must explain it here, though, 
in so doing, we must anticipate some of the results to be 
detailed in subsequent chapters. 

(276.) Suppose a shower of rain to fall perpendicularly 
in a dead calm ; c person exposed to the shower, who 



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


tube PQ ; if the tube were held immoveable, the ball 
would strike on its lower side, but 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 sliould 
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 tlie 
tube the motion of the ball, and carried along with the 
former, unconscious of its motion, wovdd 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 coraea 
of the other {at ivhich moment they acquire that con- 
vergence which directs them to a certain point in fixed 


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

(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 tliat of the earth, is yet not infinitely 
so. 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 tliat 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 tlie 
proportion BC : BA : : velo- Ji iV> 

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) tlie aberration itself, 


is proportional to the sine of the angle made by the 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 90 behind the sun's, 
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 
make it apparently 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. 

(281.) Aberration then affects the apparent right as- 
censions and declinations of all the stars, and that by 
quantities easily calculable. The formulae most conve- 
nient for that purpose, and which, systematically embrac- 
ing at the same time the corrections for precession and 
nutation, enable the observer, with the utmost readiness, 
to disencumber his observations of right ascension and 
declination of their influence, have been constructed by 
Prof. Bessel, 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 (independently of theoretical 
views respecting the nature of light)* is as follows. The 

* The results of the undulatory and corpuscular theories of light, in 
the matter of aberration, are, in the main, the same. We say in the main. 
There is, however, a minute difii^rence even of numerical results. In 
the undulatory doctrine, the propagation of light takes place with equal 
velocity in all directions whether the luminary be at rest or in motion. 
bi the corpuscular, with an excess of velocity in the direction of the 


ray by which we see any object is not that which it emits 
at the moment we look at it, but that which it did emit 
some time before, viz. the time occupied by light in tra- 
versing the 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 
joining the earth's place at the moment of observation 
with that occupied by the body at the same moment, 
but 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'' s, calcidate its apparent 
or relative angular motion in the time taken by light to 
traverse its distance from the earth. This is its aberra- 
tion, 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 uranogra- 
phical problems of frequent practical occurrence, which 
may be resolved by the rules of spherical trigonometry. 

(283.) Of the following five quantities, given any three, 
to find one or both the others. 

1st, The latitude of the place ; 2d, the declination of an 
object ; 3d, its hour angle east or west from the meridian ; 
4th, its altitude ; 5th, its azimuth. 

In tlie figure of art. 94, P is the pole, Z the zenith, 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 ; SPZ the hour an- 
gle Ti-^S the co-altitude or zenith distance ; and PZS the 
azimuth. By the solution of this spherical triangle, then, 
all problems involving the relations between these quanti- 
ties may be resolved. 

(284.) For example, suppose the time of rising 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 the velocity of the body's 
motion. In the cases, then, of a body moving with equal velocity directly 
to and directly from the earth, the aberrations will be alike on the undu- 
latory, but different on the corpuscular hypothesis. The utmost difier- 
ence which can arise from this cause in our s^yslem cannot amount to 
above six thousandths of a second. 



when apparently on the horizon, or really about 34' be- 
low it (owing to refraction), so that, at the moment of its 
apparent rising, its zenith distance is 90 34'=ZS. Its 
polar distance PS being also given, and the co-latitude ZP 
of the place, we have given the three sides of the trian- 
gle, to find the 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 local sidereal time, and the lati- 
tude of the place of observation, by observing equal 
altitudes of the same star east and west of the meri- 
dian, and noting the interval of the observations in side- 
real time. 

The hour angles corresponding to equal altitudes of a 
fixed star being equal, the hour angle east or west will be 
measured by half the observed interval of the observa- 
tions. In our triangle, then, we have given this hour an- 
gle ZPS, the polar distance PS of the star, and ZS, its 
co-altitude at the moment of observation. Hence we may 
find PZ, the co-latitude of the place. Moreover, the 
hour angle of the star being known, and also its right as- 
cension, the point of the equinoctial is known, which is 
on the meridian at the moment of observation ; and, 
therefore, the local sidereal time at that moment. This 
is a Vtry useful observation for determining the latitude 
and time at an unknown station. 

(286.) It is often of use to know the situation of the 
ecliptic in the visible heavens at any instant ; that is to 
say, the points where it cuts the horizon, and the altitude 
of its highest point, or, as it is sometimes called, the 
nonagesimal point of the ecliptic, as well as the longitude 
of this point on the ecliptic itself from the equinox. 
These, and all questions referable to the same data andquae- 
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 ecliptic 
E. The sidereal time being given, and also the right 
ascension of the pole of the ecliptic (which is always the 



same, viz. 18'' C" 0'), the hour angle ZPE of that point 
is known. Then, in this triangle we have given PZ, the 
co-latitude ; PE, the polar distance of the pole of the 
ecliptic, 23 28', and the angle ZPE ; from which we 
may find, 1st, the side ZE, which is easily seen to be 
equal to the altitude of the nonagesimal point sought ; 
and, 2dly, the angle PZE, which is the azimuth of the 
pole of the ecliptic, and which, therefore, being added to 
and subtracted from 90, gives the azimuths of the eastern 
and western intersections of the ecliptic with the horizon. 
Lastly, the longitude of the nonagesimal point may be 
had, by calculating in the same triangle the angle PEZ, 
which is its complement, 

(287.) The angle of situation oi ^ star is the angle in- 
cluded at the star between circles of latitude and of decli- 
nation passing through it. To determine it in any pro- 
posed case, we must resolve the triangle PSE, in which 
are given PS, PE, and the angle SPE, which is the dif- 
ference between the star's right ascension and 18 hours ; 
from which it is easy to find the angle PSE required. 
This angle is of use in many inquiries in physical astro- 
nomy. It is called in most books on astronomy the an- 
gle of position ; but the latter expression has become 
otherwise, and more conveniently, appropriated. 

(288.) From these instances, the manner of treating 
sucli questions in uranography as depend on spherical 
trigonometry will be evident, and will, for the most part, 
offer little difficulty, if the student will bear in mind, as a 


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


OF THE sun's motion. 

Apparent Motion of the Sun not uniform Its apparent Diameter also va- 
riable Variation of its Distance concluded lis apparent Orbit an El- 
lipse about the Focus Law of the angular Velocity Equable Descrip- 
tion of Areas Parallax of the Sun lis Distance and Magnitude 
Copernican Explanation of the Sun's apparent Motion Parallelism of 
the Earth's Axis The Seasons Heat received from the Sun in differ- 
ent Parts of the Orbit. 

(289.) In the foreg-oing chapters, it has been shown 
that the apparent path of the sun is a great circle of the 
sphere, which it performs in a period of one sidereal 
year. From this it follows, that the line joining the 
earth and sun lies constantlj^ in one plane; 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 59' 8"-33 ; but about 
the 31st of December it amounts to 1 1' 9"'9, and about 
the 1st of July is only 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 Avith a corresponding change of its distance from 
us. Tlie change of distance is recognised by a variation 


observed to take place in its apparent diameter, Avhen 
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 Decern 
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 such 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 
diameters. It appears, therefore, that the greatest, the 
mean, and the least distances of the sun from us are in 
the respective proportions of the numbers 1*0 1679, 
1-00000, and 0-98321 ; and that its apparent angular ve- 
locity diminishes as the distance increases, and vice 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/on?i 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, AOG, &c. equal to the ob- 
* iHkioi, the sun; and /"i-^i'v, to measure. 


served lono;-itiules of the sun tlirouorhont 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 readUy done ; and when numerically 
calculated, on the supposition of the eccentricity being 
such as above stated, a perfect coincidence 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 eartli 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 
angidar 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 aw- 
gidar 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 tioice as great a fraction of the 
latter, and vice versa. Hence Ave are led to conclude 
that the angidar velocity is in the inverse proportion, not 
of the distance simply, but of its square : so that, to com^ 
* Seo Conic Seciions, hv the Rev. H. P. Ilavnilton, 


pare the daily motion in longitude of the sun, 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 at B. And 
this is found to be exactly verified in every part of the orbit. 

(294.) Hence we deduce another 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 Avould 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 
inversely 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- 
cient to account for it ; and that the increased })roximity 
of the sun to the earth must be accompanied with an 
actual increase of its real velocity of motion along its path. 

(295.) This elliptic form of the 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 oi its actual velocity continues unknown. This laiv, 
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 mucli 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 is 


called in astronomy the radius vector) will describe 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 sun will be such that 
equal areas are thus swept over by the revolving radius 
vector in equal times, in whatever part of the circum- 
ference of the ellipse the sim may be moving. 

(296.) From this it necessarily follows, that in iin- 
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 D, as the area of the elliptic sector 
AOB is to the area of the sector 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 
Avhose projection on the heavens is the great circle so 
called. In this plane, however, the actual path is not 
circidar, but elliptical ; having the earth, not in its centre, 
but in one focus. The eccentricity of this ellipse is 0-01 679, 
in parts of a unit equal 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 we 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 on 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. 
Suppose, 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 two spec- 
tators, both observing the sun S at the same instant. The 
spectator A will see it in the direction ASa, and will re- 


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


tion BSi, and refer it to h. The angle included between 
these directions, or the measure of the celestial arc a b, 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. 

(299.) Parallax, however, in the astronomical accepta- 
tion of tlie word, has a more technical meaning. It is 
restricted to the difference of apparent positions of any 
celestial object when viewed from a station on the stir- 
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 the 
surface, a reduction to the centre is needed ; and the 
amount of this reduction is called parallax. Thus, thd 
sun 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 
parallax at A, and similarly BSC is that at B. 

Parallax, in this sense, may be distinguished by the 
epithet diurnal, or geocentric, to discriminate it frolii 
the annical, or heliocentric ; of which more hereafteri 

(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 ia 
evident that the effect of parallax, in this its technical 
acceptation, is always to depress the object observed ill 
a vertical circle. To estimate the amount of this de 
pression, we have, by plane trigonometry, 
CS : CA : : sine of CAS=sine of ZAS : sine of ASC. 

Q w 


(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 horizonful parallax ; 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- 

The horizontal parallax is given by this proportion : 

Distance of object : earth's radius : : rad. : sine of ho- 
rizontal parallax. 

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

(302.) To apply this general reasoning to the case of 
the Sim. Suppose two observers one in the northern, 
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 w^ere equal 
to that of the fixed stars, the sum of the zenith distances 
thus found would be precisely equal to tlie sum of the 
latitudes north and south of the places of observation. 
For the sum in question would then be equal to tlie 
angle %C]ji, which is the meridional distance of the 
stations across the equator. Bat the effect of parallax 
being in both cases to increase 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 angle 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 parallax is easily found, by dividing the 
angle so determined by the sum of the sines of the two 
..lafitu^es. :c '"^ - f / ,. fj i j ^ ..^ , ^ ,- ; 

(3Q3.) If the two stations be not exactly on the same 


meridian (a condition A'ery 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 Avhich 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 this correc- 
tion ; and, therefore, the less injurious to the accuracy 
of the final result is any uncertainty in the daily change 
of zenith distance which may arise from imperfection 
in the 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 
otlier 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 sun to be situated at a mean distance from 
us, 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 are 
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 a 


real diameter of 882,000 miles. Such, then, is the dia-. 
meter of this stupendous glolie. If we compare it with 
what we have already ascertained of the dimensions of 
our own, Ave shall find that in linear magnitude it exceeds 
the earth in the proportion of III2 to 1, and in bulk in 
that of 1,-384,472 to 1. 

(306.) It is hardly possible to avoid associating our 
ooncepticn of an object of definite globular figure, and of 
such enormous dimensions, with some corresponding 
?tttribute of massiveness and material solidity. That the 
gun is not a mere phantom, but a body having its own 
peculiar structure ad economy, our telescopes distinctly 
tnform us. They show us dark spots on its surface, 
vyhich slowly change their places and forms, and by 
attending to whose situation, at different times, astrono^ 
triers have ascertained that the sun revolves about an 
axis inclined at a constant angle of 82 40' to tlie plane 
of the ecliptic, performing one rotation in a period of 25 
days and in the same direction with the diurnal rotation 
af tlie earth, i. e. from west to east. Here, then, we 
liave 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 laws, and of, 
at least, such a community of nature as the existence of 
inei-tia and obedience to force may argue. Now, in the 
exact proportion in which we invest our idea of this im-i 
mense bulk with the attribute of inei-tia, or Aveight, it be- 
comes difficult to conceive its circulation round so com^ 
paratively small a body as the earth, without, on the one 
hand, dragging it along, and displacing it, if bound to it 
by some invisible tie ; or, on the other hand, if not so 
held to it, pursuing its course alone in space, and leaving- 
^he earth behind. If v/e tie two stones together by a 
string, 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 gi'eatly 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 fort, about the larger, which 
will be comparatively but little disturbed from its place. 


(307.) Whether the earth move round the sun, the sun 
round the earth, or both round their common centre of 
gravity, will make no difference, so far as appearances are 
concerned provided the stars be supposed sufficiently dis- 
tant to undergo no sensilde 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 nothing 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, in 
conformity with the laws of dynamics, that two bodies 
connected with and revolving al^out each other in free 
space do, in fact, revolve about their common centre of 
gravity, Avhich remains immoveable by their mutual ac- 
tion, it becomes a matter of further inquiry, U'hereahouls 
between them the centre is situated. Mechanics teaches 
us that its place will divide their mutual distance in the 
inverse ratio of their iveights or masses ;* and calculations 
grounded on phenomena, of which an account will be 
given further on, inform us that this ratio, in the case of 
the sun and earth, is actually that of 354,936 to 1, the 
sun being, in that proportion, more ponderous than the 
earth. From this it will follow tliat the common point 
about which they both circulate is only 267 miles from the 
sun's centre, or about ^^^^ g^th part of its own diameter. 

(308.) Henceforward, then, in conformity with the 
above statements, and with the Copernican view of our 
system, we must learn to look upon the sun as the com- 
paratively motionless centre about which the earth per- 
forms an annual elliptic orbit of the dimensions and ec- 
centricity, and with a velocity regulated according to the 
law above assigned ; the sun occupying one of the foci 
of the ellipse, and from that station quietly 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 
day and night, summer and winter, which we enjoy. 
(309.) In this annual motion of the earth, its axis pre- 
* See Cab. Cyc. Mechanics, Centre of Gravity. 



serves, fit all times, the same direction as if the orbitiial 
movement liad no existence ; and is carrieel 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) tlie 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, 90 apart, viz. A 

that which it has on the 21st of March, or at the time of 
the vernal equinox ; B that of the 21st of June, or the 
summer solstice ; C that of the 21st of September, or the 
autumnal equinox ; and D that of the 21st of December, 
or the winter solstice. In each of these positions let PQ 
represent the axis of the earth about Avhich its diurnal 
Yotation is performed without interfering with its annual 
motion in its orbit. Then, since the sun can only en^ 
lighten one 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 positions. 
Now, 1st, in the position A, the sun is vertically over tlie 
intersection of the equinoctial FE and the ecliptic IIG, 
It is, therefore, in the equinox ; and in this position the 
poles P, Q, both fall on the extreme confines of the en-, 
lightened side. In this position, therefore, it is day over 
half the northern and half the southern hemisphere at 
once ; and as the earth revolves on its axis, every point 
of its surface describes half its diunial course in light, ajicl 


half in darkness ; in other words, the duration of day and 
night is here equal over the whole globe : hence the term 
equinox. The same holds good at the autumnal 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 
a 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 in 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 encircles 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 
circle. On the other hand, the opposite or south pole Q, 
with all the region comprised within the antarctic circle, 
as far as 23 28' from the south pole, are immersed at 
this season in darkness, during the entire diurnal rotation, 
so 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 the 
equator will have a day of more and a night of less than 
twelve hour's 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 is 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 othfer at 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- 


rature of that place will be above the average ; when the 
reverse, below. As the eartli, tlien, 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 
southei'n 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 mean 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 Avinter in 
the nortliern, 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 seasons in the northern and 
southern hemispheres, are all facts too well known to 
require further comment. The positions 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 ^joZj- distance. Now, this 
angle is at its maximum in the position D, and at its 
minimum at B; being in the former case =90 + 23'' 
28' = 103 28', and in the latter 90 23^ 28' = 66 33'. 
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 con- 
tinually enlarging as we recede from the centre, must of 


course diminish in intensity according to the inverse pro- 
portion of the surface of the sphere over which it is 
spread ; that is, in the inverse proportion of the square 
of the distance. But Ave have seen (art. 293) that this 
is also the proportion in which the angular velocity of 
the earth about the sun varies. Hence it appears, that 
tlie momentary supply of heat received by the earth from 
the sun varies in the exact proportion of the angular ve- 
locity, i. e. oi the momentary increase of lotigitude ; 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 
gituated. Let, then, S represent the sun ; AQMP the 

earth's orbit ; A its nearest point to the sun, or, as it is 
called, the perihelion of its orbit ; M the farthest, or tho 
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 ; then, if we suppose the earth 
to circulate in the direction PAQiMP, 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 cow^ 


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 -joth of its mean quan- 
tity, and consequently, the fluctuation in the sun's direct 
heating power to double this, or ^jih 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 difl'erence 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 oflfice of natural centr3S advantageous sta- 
tions for the eye of reason and theory. Having learned 
to attribute an orbilual motion to the earth, it loses this 
advantage, which is transferred to the sun, as the fixed 
centre about 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, Ave shall find 

* See Geological Transactions, 1832, " On the Astronomical Causes 
which may iiilluence Geological Phenomena," by the author of thia 


ourselves continually embarrassed by the orbitual mo- 
tion of our point of view, unless, by the consideration of 
the annual or heliocentric para/lax, as it may be termed, 
we consent to refer all our observations on them to the 
centre of the sun, or rather to the common centime of gi'a- 
vity of the sun, and the other bodies which are connect- 
ed with it in our system. Hence arises the distinction 
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 heliocentric 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 great 
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 tlie 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, therefore, 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 this 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 ECLI be the ecliptic, E the vernal equinox, L 
the autumnal {i. e. the points to which the earth is re- 
ferred from the sun ivhen its heliocentric longitudes are 
and 180 respectively). Supposing the earth's mo- 
tion to be performed in the direction ECLI, the angle 
ESA, or the longitude of the perihelion, in the year 1800 
was 99 30' 5" : Ave say in the year 1800, because, in 
point of fact, by the operation of causes hereafter to be 


explained, its position is snbject 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 easier 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 
proportion following; viz. One year :the time elapsed :: 
360 : the arc of longitude passed over. The longitude 
so calculated is called in astronomy the vnerm 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 eccentricity and deviation from the circle are 
small, the true 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, areas not angles are 
described uniformly, the proportion must now be stated 
thus ; One year : the time elapsed : : the whole area of 
the ellipse : the area of the sector swept over by the ra- 
dius vector in that time. This area, therefore, becomes 
known, and it is then; as above observed, a problem of 
pure geometry to ascertain the angle about the sun (ASP, 
fig. 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 longitude, and will, therefore, du" 
ring the whole semi-revolution from A to M, exceed it ia 

Mean and True longitude of the sun. 193 

amount ; or, in other words, the trite place will be in ad- 
vance 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 with 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", increasing to a maximum, and again diminish- 
ing to zero at M ; after which it becomes subtractive, 
attains a maximum of subtractive magnitude between M 
and A, and again diminishes to at A. Its maximum, 
both additive and subtractive, is 1 55' 33"-3. 

(321.) By applying, then, to the earth'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 earth in 180 more longitude than the earth 
from the sun, in this way also the sun's true place in the 
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 the equation of the cen- 
tre depends only on the ellipticity of the orbit, and may 
be expressed in terms of the eccentricity. J^ice versa, 
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 
quantities is known, the one can always be determined 
from the other. Now, by assiduous observation of the 
sun's transits over the meridian, we can ascertain, for 
every 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 falls 
short of the mean ; and the greatest amount of this excess 



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 conch;ding its distance by 
measuring its apparent diameter. Tlie results of the two 
methods coincide, however, perfectly. 

(323.) If the ecliptic coincided Avith 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 
comes to the meridian, it is apparent yjoon, and if its mo- 
tion in longitude were uniform, and the ecliptic coincident 
with the equinoctial, this would 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 tliis 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 

(324.) These two causes, then, acting conjointly, pro- 
duce, in fact, a very considerable fluctuation in the time 
as shown per clock, v/hen 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. before mean noon, and at others as much as 14i 
min. after. 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 apparent annual course, is 


carried along the ecliptic, its declination is continually 
varying- between the extreme limits of 23 2^' 40" north, 
and as much south, which it attains at the solstices. It 
is consequently 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- 
corn; 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 vernal 
equinox, and are named in order Aries, Taurus, Gemi- 
ni, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Ca- 
pricornus, Aquarius, Pisces. They are denoted also by 
the following symbols: T, 8, H, 25, ^, n, ^ ,"1, /, 
\3, ^, >. The ecliptic itself is also divided into 
signs, degrees, and minutes, &c. 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 pole on that side the equator, and 
shines over or beyond it to the extent of 23 2!^' 40". 
The parallels of latitude, at this distance from either 
pole, are called the polar circles, and are distinguished 
from each other by the 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 of 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- 

(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. 
Were the equinox invariable, this Avould 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. 2G4, the equinox re- 
treats on the ecliptic, and meets the advancing sun somC' 


what before the whole sidereal circuit is completed. The 
annual retreat of the equinox is 50"-l, 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 SeS"* 6*' 9 9' -6, it follows, then, that the former 
must be only 365' 5'' 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 lias a slow mo- 
tion of 11 "'8 per annum in advance. From this it re- 
sults, that when the earth, setting out from the perihelion, 
has completed one sidereal period, the perihelion will 
have moved forward by 11""8, which arc must be de- 
scribed before it can again reach the perihelion. In so 
doing, it occupies 4' 39"*7, 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'' 6*^ IS"" 49' -3,* and is what is called 
the ano7nalistic 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 tlie 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 aflfording an instruc- 
tive example of the way in which a motion, once 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 FormuLse un 
less the contrary is expressed- 


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- 
(5[uiry, however, detects yet smaller deviations again 
from this form and from these laws, of which we 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 length. 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 v>diich 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 
planetary 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 Avith a kind of border, less completely dark, 
called a penumbra. Some of these are represented at 
a, b, c, 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 
anew 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 



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 465 
miles ; and a circle of this diameter (containing there- 
fore nearly 220,000 square miles) is the least space which 
can be distinctly discerned on the sun as a visible 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, Avhen viewed perpen- 
dicularly from above : so faithfully, indeed, that it is 
hardly possible not to be impressed with the idea of a 
luminous medium intermixed, bvit not confounded, with 
a transparent and non-luminous atmosphere, either float- 
ing as clouds in our air, or pervading it in vast sheets 
and columns like flame, or the streamers of our northern 

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

* Mayer, Obs. Mar. 15, 1758. " Ingens macula in sole conspiciebatup 
ciyus diameter = ^V ^^^^' soUs." 


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

(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 tlie manner 
in which this 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 penumbrae, 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 of the solid body by a 
transparent elastic medium, carrying on its upper sur- 
face {or rather, to avoid the former objection, at some 
considerably loiver level within its depth,) a cloudy 
stratum which, being strongly illuminated from 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 effected by powerful upward currents of the 
atmosphere, arising, perhaps, from spiracles in the body, 
or from local agitations. See fig. 1. 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 as- 
certained the position of the equator, which is a plane 
inclined 7 30' to the ecliptic, and intersecting it in a line 
*Phil. Trans. 1801. 


whose direction makes an angle of 80^ 21' with that ol 
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 whicli the calorific 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.* 
3dly, From the fact, that the most vivid flames 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.f From this last remark it follows, that the body 
of the sun, hov/ever dark it may appear when seen through 
its spots, 7nay, nevertheless, be in a state of most intense 

* By direct measurement with the ac/Zrtome/er, an instrument I have 
long employed in such inquiries, and whose indications are hable to none 
of those sources of fallacy whicli beset the usual modes of estimation, I 
find that out of 1000 calorific solar rays, 816 penetrate a sheet of plate 
glass 012 inch thick; and that of 1000 rays which have passed through 
one such plate, 859 are capable of passing through another.- A ;/ior. 

t The ball of ignited quick-lirae, in Lieutenant Drummond's oxy-hydro- 
gen lamp, gives the nearest imitation of the solar splendour which has 
yet been produced. The appearance of this against the sun was, how- 
ever, as described in an imperfect trial at which I was present. The 
experiment ought to be repeated under favourable circumstances. 


ignition. It does not, however, follow of necessity that 
it must 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 doul)t. 

(33.5.) This immense escape of heat by radiation, we 
may also remark, will fully explain the constant state of 
tumultuous agitation in Avhich tlie fluids composing the 
visible surface are maintained, and the continual genera- 
tion and filling in of the pores, without having recourse 
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 of 
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 sup- 
port of animals and of man, and the sources of those 
great deposites of dynamical efficiency which are laid up 
for human use in our coal strata. By them the waters 
of the sea are made to circulate in vapour through the 
ait, 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 a 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 we consider the 
immense transfer of matter so produced, the increase of 


pressure over 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 activity 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 rather to the 
known possibility of an indefinite generation of heat by 
friction, or to its excitement by the electric discharge, 
than to any actual combustion of ponderable fuel, whe- 
ther solid or gaseous, for the origin of the solar radiation.* 

* Electricity traversing excessively rarefied air or vapours, gives out 
light, and, doubtless, also heat. May not a continual current of electric 
matter be constantly circulating in the sun's immediate neiglibourhood, 
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 light to that of the aurora has been 
distinctly insisted on by my faiher, in his paper already cited. It would 
be a highly curious subject of experimental inquiry, how far a mere re- 
duplication of sheets of tlame, at a distance one behind the other (by 
which their light might be brought to any required intensity), would com- 
municate to the heat of the resulting compound ray the penetrating cha- 
racter which distinguishes the solar calorific rays. We may also observe, 
that the tranquillity of the sun's polar, as compared with its equatorial 
regions (if its spots be really atmospheric), cannot be accounted for by its 
rotation on its axis only, but 7nn.1t arise from some cause external t5 the 
sun, as we see the belts of Jupiter and Saturn, and our trade-winds, aiise 
from a cause, external to these planets, combining itself with iheir rota- 
tion, which alone can produce no motions when once the form of equili. 
brium is attained. 

The prismatic analysis of the solar beam exhibits in the spectrum a 
series of " fixed lines," totally vmlike those which Ijelong to the liglit of 
any 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 be 
an inherent (;'. e. an absorptive) colour, tlie air must be expected to act 
on the spectrum after the analogy of other coloured media, which often 
(and especially light blue media) leave unabsorbed portions separateti by 
dark intervals. It deserves inquiry, therefore, whether some or all the 



Of the Moon Its sidereal Period Its apparent Diameter Its ParallaS, 
Distance, and real Diameter First Approximation to its Orbit Art 
Ellipse about the Earth in the Focus Its Eccentricity and Inclina- 
tionMotion of the Nodes of its Orbit Occultations Solar Eclipses 
Phases of the Moon Its synodical Period Lunar Eclipses 
Motion of the Apsides of its Orbit Physical Constitution of the Moon 
Its Mountains Atmosphere Rotation on Axis Libration Ap- 
pearance of the Earth from it. 

(388.) The moon, like the sun, appears to advance 
among the stars with a movement contrary to the general 
diurnal motion of the heavens, but much more rapid, so 
as to be very readily perceived (as Ave 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*^7^ 43"" ll'-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 orbit cannot be 
very different from a circle, because the apparent angular 
diameter of the full moon is not liable to any gi-eat extent 
of variation. 

(340.) The distance of the moon from the earth is 
concluded from its horizontal parallax, which may be found 
either directly, by observations at remote geographical 
stations, exactly similar to those described in art. 302, 
in the case of the sun, or by means of the phenomena 
called occultations (art. 346), from which also its appa- 
rent diameter is most readily and correctly found. From 
such observations it results that the mean or average dis- 

fixed lines observed by WoUaston and Fraunhofer may not have their 
origin in our own atmosphere. Experiments made on lofty mountains, 
or the cars of 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, would decide this point. The absorptive effect of the 
sun's atmosphere, and possibly also of the medium surrounding it (what- 
ever it be), which resists the motions of comets, cannot be thus eliminated. 


tance of the conlre 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 tlie 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 SC, and the earth's radius CA, two sides of the 
triangle ACS 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 ^\. 

(342.) By a series of observations, such as described 
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 observed, and reduced by means 
of its parallax to the earth's centre, their angular inter- 
vals will become known, so that the path of the moon 
may then be laid down on a chart supposed to represent 
the plane in Avhich 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 


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, the 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 closeness 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, the changes which it under- 
goes are not perceptible but in a course of veiy nice ob- 
servations, which have disclosed, it is true, the existence 
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, its 
deviation from a perfect ellipse is very 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 the ecliptic, we ^lall find that the nodes 
of its orbit are in a continual state of retreat upon the 
ecliptic. Suppose O to be the earth, and Ab ad that 
portion of the plane of the ecliptic which 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 O, it would have a, the opposite point in the 



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

ries it not to a, but along a certain curve, ABC, to C aj 
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 than 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 E, 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 the 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' 

(345.) The actual amount of this retreat of the moon's 
node is about 3' 10"'64 joer 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 Avas 
at the beginning. Its apparent path, then, will lie among 
totally different stars and constellations at different 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 



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 the 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.) NoWs 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 pass over and 
occult or eclipse every star and planet within the zone 
above described (and, as seen from the surface of earth, 
even somewhat beyond it, by reason of parallax, M'hich 
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 sim, 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 angnilar 
diameter less than the sun's, the very singular pheno- 
menon of an annular solar eclipse takes place, when 
the edge of tlie sun appears for a few minutes as a nar-. 


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 sun 
and moon are in conjunction, that is to say, have the 
same, 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 must 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 
remote 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 any direction, according to 
a spectator's 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 the earth's centre, were dilated by twice its hori- 
zontal parallax ; for, if, when so dilated, it can touch or 
overlap the sun, there inust 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 1 34' 27". In the spherical triangle SNM, 

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


may suppose the angle NSM a right angle, SM = 1 34' 
27", and the angle MNS = 5 8' 48", the mclmation of 
the orbit. Hence we calculate SN, which comes out 
16 58'. If, then, at the moment of the new moon, the 
moon's node is farther from the sun 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 the earth (which may 
amount to a sixtieth part of her horizontal diameter), 
determined ; after which it is easy, from the above con- 
siderations, to calculate the amount overlapped 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, as seen from the 
earth's centre, carries her within a distance from the 
star equal to the sum of her semidiameter and horizontal 
parallax; and it tvill happen at any particular spot, 
when her apparent path, as seen from that spot, carries 
her centre within a distance equal to the sum of her 
augmented semidiameter and actual 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 
moon, she really exists, and pursues her course, and 
that when we see her only as a crescent, however nar- 
row, the whole globular body is there, filling up the de- 
ficient outline, though unseen. For occultations take 
place indifferently at the dark and bright, the visible and 

* Woodhoase's Astronomy, vol. i. See also Trans. Ast. Soc. vol. 1. p. 325, 



invisible oulline, 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 tlie 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 
notice or visible cause for its disappearance, which, as 
it happens instantaneously, and without the slightest 
previous 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 other 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 moon, with the naked eye, 
as a pale round body to which the crescent seems attach- 
ed, and somewhat projecting beyond its outline (which 
is an optical illusion arising from the greater intensity of 
its light). The cause of this appearance will presently 
be explained. Meanwhile the fact is sufficient to show 

* There is an optical illusion of a very strange and unaccountable na- 
ture which has often been remarked in occultations. The star appears 
to advance actually upon and within the edge of the disc before it disap- 
pears, and that sometimes to a considerable depth. I have never myself 
witnessed this singular effect, but it rests on most unequivocal testimony. 
I have called it an optical illusion ; but it is barely possible that a star 
may shine on such occasions through deep fissures in the substance of 
the moon. The occultations of close double stars ought to be narrowly 
watched, to see whether both individuals are thus projected, as well as 
for other purposes connected with their theory. I will only hint at one, 
viz. that a double star, too close to be seen divided with any telescope, 
may yet be detected to be double by the mode of its disappearance. 
Should a considerable star, for instance, instead of undergoing instanta- 
neous and complete extinction, go out by two distinct steps, following 
close upon each other; first losing a portion, then the whole remainder 
of its liglit, we may be sure it is a double star, though we camiot see the 
individuals separately. Author. 


that the moon is not inherently luminous like the sun, 
but that her liglit is of an adventitious nature. And its 
crescent form, increasing regularly from a narrow 
semicircular line to a complete circular disc, corres- 
ponds to the appearance a globe would present, one he- 
misphere of which was black, the other wliite, when dif- 
ferently turned towards the eye, so as to present a great- 
er or less portion of eacli. The obvious conclusion from 
this is, that the moon is such a globe, one half of Avhich 
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 towards 
the sun, and that in proportion as the moon in her monthly 
course becomes more and more distant from the sun, the 
breadth of the crescent increases, and vice versa. 

(351.) The sun's distance being 2.3984 radii of the 
earth, 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 par- 
allel. Suppose, now, O to be the earth, ABCD, &c. 


my ^^m 


various positions of the moon in its orbit, and S the 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 


entirely turned towards O, and the bright from it, Iir 
this case, then, the moon is not seen, it is neiv 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 opposite situation G : these are the first and third 
quarters of the moon. Lastly, when at E, the whole 
brigfht face is towax-ds the earth, the whole dark side from 
it, and it is then seen wholly bright o\ full moon. 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 
phases, 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 so i-eceived. 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 ofl' 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 a 
dazzling splendour, not inferior to the seeming brightness 
of the moon at night. That tlie earth sends also such a 
light to the moon, orJy probably more powerful by rea- 
son of its greater apparent size,* is agi'eeable to optical 
principles, and explains the appearance of the dark por- 
tion of the young moon completing its crescent (art. 350). 
For, when the 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 earth-light ^ and it is a 
portion of this, reflected back again, which makes it visi- 
ble to us in the twilight sky. As the moon gains age, 
the earth ofl^ers it a less portion of its bright side, and the 
phenomenon in question dies away. 

(353.) The lunar month is determined by the recur- 
rence of its phases ; it reckons from new moon to new 

*The apparent diameter of the moon is 32' from the earth; that of the 
earth seen from the moon is twice her horizontal parallax, or 1" 54'. The 
apparent surfaces, therefore, are as (114)2 ; (32)2, or as 13 ; 1 nearly. 


moon ; that is, from leaving its conjunction with the sun 
to its return 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 in 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 synodical 
period. The difference is easily calculated by consider- 
ing that the superfluous arc (whatever it be) is described 
by the sun with his velocity of -98565 per diem, in 
the same time that the moon describes that arc yj/ws a 
complete revolution, with her velocity of 13"17640 joer 
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 knowledge of arithmetic 
will suffice to derive the arc in question, and the time 
of its description by the moon ; which, being the excess 
of the synodic over the sidereal period, the former will 
be had, 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, Avhen at E (or at the full moon), the 
earth O will intercept the rays of the sun, and cast a 
shadow 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 circular 
outline, though, from the greater size of the circle, it is 

* Let V and w be the mean angular velocities, x the superfluous arc ; 
thenV: w;; I+2:a;;and V v.v.-.l .- a;, whence a; is found, and- =the 


time of describing x, or the difference of the sidereal aad syiiodical peri- 
ods. We shall have occasion for tliis again. 


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

(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, AB 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 
gpace CED (which represents a cone, since CD and AB 

pre 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. 


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

(356.) In a lunar 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 the 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, the moon's path be properly directed, it is sure 
to pass through the umbra. This is not, however, the 
case in solar eclipses. It so happens, from the 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 there 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 
pemwibra. When the apex of the umbra falls on the 
surface, the moon at that point will appear, for an in- 
stant, to jzist 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 cone, will see the whole of the moon on 
the sun, although not large enough to cover it, i. e. he 
will witness an annular eclipse. 

(357.^ Owing to a remarkable enough adjustment of 
the periods in which the moon's sy nodical revolution, 
and that of her nodes, are performed, eclipses return aftef 
a certain period, very nearly in the same order and of the 
same magnitude. For 223 of the moon's mean synodi- 
cal revolutions, or lunations, as they are called, will 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 


to have been known to the Chaldeans, under the name of 
the saros ; 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- 
ed 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 ivhen 
the distance between the moon's centre and the centre of 
the shadow is exactly equal to the sum of the seinidiame- 
ters of the moon and penumbra, or of the moon and 
umbra, to know when it enters upon and leaves them re- 

(359.) The dimensions of the shadow, at the place 
where it crosses the moon's path, require us to know 
the distances of the sun and moon at the time. These 
are variable ; but are calculated and set down, as well as 
their semidiameters, for every day, in the epliemeris, so 
that none of the data are wanting. The sun's distance is 
easily calcidated from its elliptic orbit ; but the moon's 
is a matter of more difficulty, for a reason we will now 

(360.) The moon's orbit, as we have befoi'e hinted, is 
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 nodes. 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, in the same direction with the moon's 
own motion, in 3232*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 


by the name of the revohition of the moon's apsides. Its 
cause will be hereafter explamed. Its immediate effect 
is to produce a variation in the moon's distance from the 
earth, which is not included in the 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 advance of its 
axis in that plane ; and 2dly, by a continual tilting mo- 
tion of the piano 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, v/e 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 thd 
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 drl 
which they stand. The convex outline of the limb turned 
towards the sun is always circular, and very nearly 
smooth ; but the opposite border of the enlightened partj 
which (were the moon a perfect sphere) ought to be ari 
exact and sharply defined ellipse, is always observed to 
be extremely ragged, and indented with deep recessea 
and prominent points. The mountains near this edgd 
cast long black shadows, as they should evidently doj 
when we consider that the sun is in the act of rising or 
setting to the parts of the moon so circ^nnstanced. Bui 
as the enlightened edge advances beyond them, i. e. as 
the sun to them gains altitude, their shadows shorten 5 
and at the full moon, when all the light falls in our lind 



of sight, no shadows are seen on any part of her surface. 
From micrometrical measures of the lengths of the sha- 
dows of many of tlie more conspicuous mountains, taken 
under the most favourable circumstances, the 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 
extreme 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 bottom.s 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 Phlegr^ei* or the Puy de 
Dome. 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, moreover, 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, Avhcn 
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 

(304.) 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 Naples, and Desmarest's of 
t From ray (nvn o?J3ervation& Author. 


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 Avinters, 
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 narrow 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 second in diameter, as seen 
from the earth, on the surface of the moon, contains 
about a square mile. Telescopes, therefore, must yet be 
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 the 
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 tlie 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, can any where be discerned. 

(366.) The lunar summer and winter arise, in fact, 
from the rotation of the moon on its own axis, the period 
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 perio<ls, which at first sight would seem perfectly 


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

(367.) The moon's rotation on her axis is uniform ; 
but since her motion in her orbit (like that of the sun) is 
not so, Ave 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, tlie line joining the centres of the earth and moon 
fluctuates a little in its position, from its mean or average 
intersection Avith 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 knov/n by the name of li- 
prutions. 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. 

(368.) if there be inhabitants in the moon, the earth 
^aist present to them the extraordinary appearance of a 
piioon cf nearly 2 in diameter, exhibiting the same phases 
fts we see the moon to do, but hnmoveably fixed in their 
sky (or, at least, changing its apparent place only by the 
small amount of the libration), while the stars must seem 
^o pass slowly beside and behind it. It v/ill appear 
clouded with variable spots, and belted with equatorial 
^nd tropical zones corresponding to our trade-winds ; and 
it may be doubted whether, in their perpetual change, the 
outlines of our contineiits and seas can ever be clearly 



Of terrestrial Gravity Of the Law of universal Gravitation Paths of 
Projectiles ; apparent, real ^The Moon retained in her Orbit by Gravity 
Its Law of Diminution Laws of elliptic Motion Orbit of the Earth 
round the Sun in accordance with these Laws Masses of the Earth 
anil Sun compared Density of the Sun Force of Gravity at its Sur- 
face Disturbing Effect of tlie Sun on the Moon's Motion. 

(369.) The reader has now been made acquainted with 
the chief phenomena of the motions of tlie 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 these motions, and causes the 
massive bodies so revolving to deviate 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,! 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 wliose reality has (strange to say) been regarded as 
an achievement of no common merit in the annals of this 
branch of philosophy. It is our own immediate con- 
sciousness of effort, when Ave exert force to put matter 
in motion, or to oppose and neutralize force, which gives 
us this internal conviction of poiver and causation so far 
as 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 Effect," a work of great acuteness and 
subtlety of reasoning on some points, but in which the wliole train of ar- 
ginnent is vitiated by one enormous oversight ; the omission, namely, of 
a. dislinct and immediate personal consciovsriess of causation in his enu- 
meration of that sequence of evendi, by wliicli the volition of the mind is 
made to terminate in the motion of material objects. I mean the con- 
sciousness of effort, as a thing entirely distinct from mere desire or volition 
on the one hand, and from mere spasmodic contraction of muscles on the 
other. Brown, 3d edit Ediii. 1818, p. il. Author. 

T 2 


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 effort someAo?/? exerted, 
i^hough not accompanied with our consciousness. That 
such an effort should be exerted with success tlirough an 
interposed space, is no more difficult to conceive than 
that our hand should communicate motion to a stone, 
\yith Avhich it is demonstrubly not in contact. 

(.371.) All bodies with which we are acquainted, when 
raised into the air and quietly abandoned, descend to the 
(^ai'th's surface in lines perpendicular to it. They are 
therefore urged thereto by a force or eflbrt, the direct or 
indirect result of a consciousness and a loill existing 
somewhere, though beyond our pov/er to trace, which 
force Ave term gravity ; and whose tendency or direction, 
^s imiversal experience teaches, is towards the earth's 
centre ; or rather, to speak strictly, with reference to its 
spheroidal figure, perpendicular to the surface of still 
water. But if we cast a body obliquely into tlie air, 
^his tendency, though not extinguished or diminished, ia 
materially modified in its ultimate effect. The upward 
ijmpetus 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 
pointy vertex, or apogee, just as the moon has in its orbit, 
where the direction of its motion is perpendicular to the 

(372.) When the stone which 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 as when it quitted 
our hand. As we are sure that, if not stopped by the 
resistance of the earth, it woidd continue to descend, and 
that obliquehj, 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 circii-* 


late round it, as the moon does round the earth, returning 
again to the point it set out from, after completing an 
elliptic orbit of which the centre occupies the lower 
focus ? And if so, is it not reasonable to imagine that the 
same force of gi-avity may (since we know that it is ex- 
erted 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 
this be the power for some power there must 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 Ave know Avhat weight 
it is capable of carrying, the velocity necessary for this 
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 velociti/ 
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 might supply the place of the 
string, 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 iveight, 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 I'' 23 22^ 

(374.) If we make the same calculation for a body at 
the distance of the moon, supposing its iveight or gra'>^ 
* See Cab. Cyc. Mechanics, chap. viii. 


vity the same as at the earth'' s surface, we sliall find the 
period required to be 10'' 45 30". The actual period of 
the moon's revohition, 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 -^ oVo'^^ P^^'*- ^^ 
the motion which it Avould 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^ : P; 
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 v/hieh- 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 v/e 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 


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 
gravity, in the proportion otherwise not improbable 
of the squares of the distances. If gravity be not that 
power, there 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- 
derstood 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 
directly proportioned to the mass of the attracting par- 
ticle, and inversely to the square of the distance between 
them." In this 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 fioairc, will attract another sucli collection of mate-. 
rial atoms. This problem is one purely dynamical, and, m 
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 v/hole 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 ita strict wording. The effect of 


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 ai-gument 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 neighbourheod, 
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 
the general name of conic sections. It will depend, he 
shows, in any assigned case, upon the particular circum- 
stances of velocity, distance, and direction, which of 
these curves shall be described, whether an ellipse, a 
circle, a parabola, or an hyperbola ; but one or other it 
must be ; and any one of any degree of eccentricity it 
77iay be, according to the circumstances of the case ; and, 
in all 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 focus 
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 squai'e 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.) All this is in conformity with what we have 
stated of the solar and lunar movements. Their orbits 

* We refer for these fundamental propositions, as a point of duty, to 
the immortal work in which they were first propounded. It is impossi- 
ble for us in this volume to go into these investigations : even did our 
limits permit, it would be utterly inconsistent with our plan; a general 
idea, however, of their conduct will be given in the liext chapter, 


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), 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 tliat 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 eartli's attraction on an equal 
body at that distance greater in the high proportion of 
35493G to 1. It is clear, then, that if the 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 354936 times more in- 
tense than what the earth would be capable of exerting, 
caeferis 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 354936 earths occupying its place would do, or, 
in other words, that the sun contains 354936 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 mas3 with its bulk, we find its 


density* to be less than that of the earth, bein^ no more 
than 0-2513. 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 prevails in its interior, by which its elasticity is re- 
inforced, and rendered capable of resisting this almost 
inconceivable pressure without collapsing into smaller 

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

The attraction of a sphere being the same (art. 370) 
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 Ave con- 
clude,! 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 Ave regard the earth's orbit as 
being performed about the one or the other centre makes 

* The density of a material body is aa the mass directly, and the 
volume inversely : hence density of Q : density of @ : : ^~t 1 .' 

t Solar gravity : terrestrial : i^-^j : -^^-2 : : 279 : 1 ; the respec 
tive radii of the sun and earth beijig 410000, and 4000 miles. 

\ A mass weighing 12 stone or 170 lbs. on the eorth, would produce a 
preesujfe of 40O0 lbs. on tJie aun. 


no appreciable difference in any one phenomenon of 

(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, Avhile 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 
the 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 strictness, 
not the earth or the moon which describes an ellipse 
around the sun, but their common centre of gi-avity. 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 taken 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 so, is a proof that 
some cause disturbs it, and interferes with the earth's 
attraction ; and this cause is no other than the sun's at- 



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 eacli 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 l)e 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^^oth part greater and as 
much less than the earth's. Small as this is, it is yet 
sufhcient to produce a perceptible excess of attractive 
tendency of the moon towards the sun, above that of the 



earth when in the nearer point of her orbit, M, and a 
corresponding defect on the opposite part, N ; and, in 
the intermediate positions, not only will a difference of 
forces subsist, but a difference of directions also ; since, 
hoAvever 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 witli the plane of the ecliptic ; and hence 
the action of the sun, which is very nearly parallel to the 
last mentioned plane, tends to draw her somewhat out 

CHAP, vni.] SOLAR SYSTEM. 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 
suhjeci o( 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. 



Apparent Motions of the Planets ^Their Stations and Retrogradations 
The Sua their natural Centre of Motion Inferior Planets Tlieir 
Phases, Periods, &c. 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 iiltra-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 brightes.t and most conspi- 
cuous, which, when attentively watched from night to 
night, are found to change their relative situations among 
the rest; some rapidly, others inuch more slowly. These 
are called planets. Foitr 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, 
--e-tilie multitude of telescopic stars being so great that 

233 A TREATISE ON A6TR0N0MY. [cHAP. Vlll. 

only a small fraction of their number has been sufficiently 
noticed 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 average or 
7nean velocities, in the same direction as those lumina- 
ries, i. e. 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 the 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 
foreshortened and confounded undistinguishably, while 
only their deviations from the ecliptic appear of their 
natural magnitude, undiminished by the effect of per- 

(390.) The apparent motions of the sun 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, till the reversed or retrograde motion 
ceases altogether. Another station, or moment of ftp- 


parent rest or indecision, now takes place ; after which 
the movement is ajjain reversed, and resumes its orijjinal 
direct character. On the whble, however, the amount 
of direct motion more than compensates the retrograde ; 
and by the excess of the former over tlie 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 repi'esented as in Mercator's projec- 
tion, art, 234, taking the 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 the 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 iirano- 
graphkally situated in the celestial circle so called, with- 
out being really and locally situated in that plane. The 
visible passage of a planet through its notle, then, is a 
phenomenon indicative of a circumstance in its real mo- 
tion 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 



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 wliich, 
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 centre 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 Avhich is 
entitled to the first trial, as a station to which to refer 
them. If it be not connected with them by any physical 
relation, it at least possesses the advantage, which the 
earth does not, of comparative immobility. But after 
what has been shown in art. 380, of the immense mass 
of that 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 the 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 
that 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 of considerable apparent diameter, and 
ofiering distinct and characteristic peculiarities,, which 
show them to be solid masses, each possessing its indi- 
vidual structure and mechanism ; and that, in one in- 


stance at least, an exceedingly artificial and complex one. 
(See the representations of Jupiter, Saturn, and Mars, 
in plate I.) Tluit 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 
smallness of their diurnal parallax, which, even for the 
nearest of them, when most favourably situated, docs 
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 
semidiameter, 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 suffice 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 fact 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. e. 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 


through telescopes, exhibit the appearance of phases 
like those of the moon. This proves that they are 
opaque bodies, shining only by rellected light, which 
can be no other than the sun's ; not only because there 
is no other source of light external to them sufilciently 
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 revolve 
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, tlie 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 Avestern horizon, 
just after sunset, and are called evening stars : Venus, 
especially, appears occasionally in this situation witli a 


dazzling lustre ; and in favourable circumstances may 
be observed to cast a pretty strong shadow.* When 
they happen to be to the west 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- 
cury never attains a greater angular distance from it 
than aboiTt 29, while Venus extends her excursions on 
either side to about 47. When they have receded from 
the sun, eastward, to their respective distances, they 
remain for a time, as it were, immoveable icith resjiect to 
it, and are carried along Avith 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 
horizon 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 smi's disc as 
small, round, loell-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 place when 
tlie eartli 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 invisible for a time, however, 
they begin to appear on the other side 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, 
however, they become stationary in the heavens ; but 
their recess from the sun is still maintained by the ad- 
vance of that luminary along the ecliptiCi which continues 
to leave them behind, until, having reversed their motion, 

* It must be thrown upon a white ground. An open window if a 
whitewashed room is the best exposure. In this situation, I have ob- 
served not only the sliadow, but tlte dif&actcd fringes edging its outline, 


and become again direct, they acquire sufficient speed to 
commence overtaking him at which moment they have 
their greatest ivestern 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 the 
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, B, 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 ; and, if the eye happened to 
lie exactly in the plane of the orbit, transiting his disc 
in the former case, and being covered by it in the latter. 
But as the sun is not so stationary, but apparently car- 
ried along the ecliptic PQ, let it be supposed to move 
over the spaces ST, TU, UV, while the planet in each 
case executes one quarter of its period. Then will its 
orbit be apparently carried along with the sun, into the 
successive positions represented in the figure ; and 
while its real motion round the sun brings it into the re- 
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 will 
have been direct in the parts AN, NH, and retrograde in 
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; and their points of nearest approach to 
it, their inferior and superior conjunctions ; the former 


when the planet passes between the earth and the sun, 
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- 
cury, 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 o, let the earth be situated at 
A, in the direction of a tangent, 
a A, to its orbit ; then it is evi- 
dent that it will appear at its 
greatest elongation from the 
sun ; the angle oAS, M'hich 
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 the radius of its orbit, supposed a cir- 
cle. For the triansfle SArt is right-angled at a, and con- 
sequently we have Sa : SA : : sin. SAa : radius, by which 
proportion the radii Sa, 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 


Venus, similarly derived, about G8000000 : the radius 
of the earth's orbit being 95000000, 

(403.) The sidereal periods of the planets may be ob- 
tained (as before observed), with a considerable approach 
to accuracy, by observing their passages through the 
nodes of their orbits ; and, indeed, Avhen a certain very 
minute motion of these nodes (similar to that of the 
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 is 
87'* 23'^ 15'" 43-9; and that of Venus, 224'^ 16'^ 49" 
8'0'. These periods, however, are widely different from 
the intervals at which the successive appearances of the 
two planets at their eastern and western elongations from 
the sun are observed 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 betAveen the sidereal and synodical re- 
volutions (art. 353) accounts for this. Referring again 
to the figure of art. 401, if the earth stood still at A, 
while the planet advanced in its orbit, the lapse of a si- 
dei-eal 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 in 
the same direction towards E, and therefore the next 
greatest elongation on the same side of the sun will hap- 
pen not in the position A of the tv\'o bodies, but in 
some more advanced oosition, eE. The determination 
of this position depends on a calculation exactly similar 
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 115877^ and 583-920'^. 

(404.) In this interval, the planet will have described 
a whole revolution plus the arc a e, and the earth only 
the arc ACE of its orbit. During its lapse, the inferior 
conjunction Avill happen when the earth has a certain 
intermediate situation, B and the planet has reached b, a 
point between the sun and earth. The greatest elonga- 
tion on the opposite side of the sun will happen when 


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 
tlie opposite side from M. Lastly, the superior con- 
junction will happen when the earth arrives at D, and 
the planet at d in the same line prolonged on the other 
side of the sun. The intervals at which tliese 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 are in the propor- 
tion of their radii. If, then, we calculate the circumfe- 
rences of the orbits of INTercury and Venus, and the earth, 
and compare them with the times in which their revolu- 
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 68080. From this it fol- 
lows, that at the inferior conjunction, or at b, either 
planet is moving in the same direction as the earth, but 
with a greater velocity ; it will, therefore, leave the earth 
behind it : and the apparent motion of the planet viewed 
from the earth, will be as if tlie planet stood still, and 
the earth moved in a contrary direction from what it 
really does. In this situation, then, the apparent motion 
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. Botli these results are in 
accordance witli 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 planet 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 b, the 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, perpendicidar to the line of junction) 


surpassing 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 
tlie 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 at one 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 circle*^ 
(which they really are), it becomes somewhat complex 
too much so to be here entered upon. It will suffice 
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 like 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 x^jg 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 tliat turned from him, will appear /<</Z at the 
superior conjunction A ; gibbous {i. e. more than half 


full, like the m8on 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 horned, 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 Avhere 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.) The variation in brightness of Venus in differ- 
ent parts of its apparent orbit is very remarkable. This 
arises from two causes : 1st, the varying proportion of 
its visible illuminated area to its whole disc ; and, 2dly, 
the varying angular diameter, or whole apparent magni- 
tude of the disc itself. As it approaches its inferior con- 
junction from its greater elongation, the half-moon be- 
comes 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, wdiich takes place 
when the planet's elongation is about 40. 

(409.) The fiansits of Venus are of very rare occur- 
rence, taking 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 great 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 the Study of Natural Puilosophy, Cab. Cyclo; 
Vol. XIV. p. 269. 


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, and suppose A, B, to 
retain that situation during the whole time of the transit. 
Then, at any moment Avhen the spectator at A sees the 
centre of Venus projected at a on the sun's disc, he at B 
will see it projected at b. If then one or other spectator 
could suddenly transport himself from A 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 micrometrical 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, BV6, are straight lines, and therefore 
make equal angles on each side V, a b will be to AB as 
the distance of Venus from the sun is to its distance from 
the earth, or as G8 to 27, or nearly as 2^ to 1 : 6, 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 2h times the earth's apparent diameter at 
the distance of the sun, or (which is the same thing) to 
five times the sun's horizontal parallax (art. 298). Any 
error, therefore, which may be committed in measuring 
a b, will entail only one ^fifth 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, p q r s, 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 at 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 


sun's disc it cuts off. Now, one of the most exact Avays 
in which (conjoined with careful niicronieti-ic measures) 
this can be done, is by noting the time occupied in the 
whole transit : for the relative angular motion of Venus 
being, in fact, very precisely known from the tables of her 
motion, and the apparent path bei)ig 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 
of!'; and the sun's diameter being known also Avith great 
precision, their versed sines, and therefore their differ- 
ence, or the breadth of the zone required, becomes 
known. 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 first visible impression or notch on the edge of the 
disc at P is produced, or ihe first external contact ; 2dly, 
when the planet is just wholly immersed, and the 
broken edge of the disc just closes again at Q, or the 
first internal contact ; and lastly, he must make the same 
observations at the egress at R, 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 tliereon than here sup- 
posed, are similar in their principles to those Avhich enter 
into the calculation of a solar eclipse, or the occultation of 
a star by the moon, 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 efl^ects, and, 
by being made subject to measurement on a greatly en- 
larged scale, or by substituting the measure of time for 
space, may be ascertained 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 efficient scale, by the British, French, Russian, 



and other governments, to the rcmotost corners of the 
globe, for the express purpose of performing it. The 
celebrated expedition of Captain Cook to Otaheite was 
one of them. The general result of all the observations 
made on this most memorable occasion gives. 8"-5776 
for the sun's horizontal parallax. 

(412.) The orbit of Mercury is very elliptical, the ec- 
centricity being nearly one fourdi of the mean distance. 
This appears from the inequality of the greatest elonga- 
tions from the sun, as observed at difterent times, and 
which vary 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 common focus. 

(413.) Let us now consider the superior planets, or 
those whose orbits enclose on all sides that of the earth. 
That they do so is proved by several circumstances : 
1st, They are not, like the inferior planets, confined to 
certain limits of elongation from the sun, but appear at 
all distances from it, even in the opposite quarter of tlie 
heavens, or, as it is called, in opposition ; which could 
not happen, did not the earth at such times place itself 
between them and the sun : 2dly, They never appear 
horned, like Venus or Mercury, nor even semUunar. 
Those, on the contrary, which, from the minuteness of 
their parallax, we conclude to be the most distant from 
us, viz. Jupiter, Saturn, and Uranus, never appear other- 
wise than round ; a sufficient proof, of itself, that we see 
them always in a direction not very remote from that in 
Avhich 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 phase, and in its defi- 
ciency 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 understand this, we need only cast our eyes on the 
annexed figure, in which E is the earth, at its apparent 


greatest elongation from the sun S, as 
seen from Mars, M. In this position, 
the angle SME, included between tlie 
lines SM and EM, is at its maximum; 
and, therefore, in this state of things, a 
spectator on the earth 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 sure, although a coarse and 
rude one of the angle SME, and there- 
fore of the proportion of the distance 
SM, of Mars to SE, that of the earth 
from the sun, by which it appears that 
the diameter of the orbit of Mars can- M 

not be less than I2 that of the earth's. The phases of 
Jupiter, Saturn, and Uranus being imperceptible, it fol- 
lows that their orbits must include not only that of the 
earth, but of Mars also. 

(414.) All the superior planets are retrograde in their 
apparent motions when in opposition, and for some time 
before and after ; but they differ greatly from each other, 
both in the extent of their arc of retrogradation, 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, of Jupiter than of Saturn, 
and of that planet than Uranus. The angtdar velocity 
with which a planet appears to retrograde is easily ascer- 
tained by observing its apparent place in the heavens 
from day to day ; and from such observations, made about 
the time of opposition, it is easy to conclude the relative 
magnitudes of their orbits as compared with the earth's, 
supposing their periodic times known. For, from these, 
their mean angular velocities are known also, being in- 

versely as the times. Suppose, then, Ee to be a very 
small portion of the earth's orbit, and Mm a correspond- 


ing portion of that of a superior planet, described on the 
day of opposition, about t!ie sini S, on which day the 
three bodies lie in one straight line SEMX. Then the 
angles ESe and MSm ai-e given. Now, if e m be joined 
and prolonged to meet SM continued in X, the angle eXE, 
Avhich is equal to the alternate angle Xe?/, is evidently the 
retrogradation of Mars on that day, and is, therefore, also 
given. Ee, therefore, and the angle EXe, being given in 
the right-angled triangle EeX, 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 jSX and wXS, whence the other sides, S?w, 
mK, are easily determined. Now, S?7i is no other than 
the radius of the orbit of the superior planet required, 
which in this calculation is supposed circular as well as 
that of the earth ; a supposition not exact, but sufficiently 
so to afford a satisfactory approximation 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 afford an average or 
mean value 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 plane, 
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 in opj)Osition with 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 so, 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 ineqiicdity of successive synod- 
ical revolutions so observed), tlie average of a great num- 
ber, taken in all varieties of situation in which the oppo 

CHAP. viii.J Kepler's law of periodic times. 349 

sitions occur, will be freed from tlie elliptic inequality, 
and may be taken as a mean synocUcal 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 the 
planets known to the ancients, who have recorded their 
observations of them in a manner sufficienUy careful to 
be made use of. Their periods may, therefore, be regard- 
ed as ascertained w^ith the utmost exactness. Their nu- 
merical values will be found stated, as well as the mean 
distances, and all the other elements of the planetary 
orbits, in the synoptic table at the end of the volume, to 
which (to avoid repetition) the reader is once for all re- 

(416.) In casting our eyes down the list of the planet- 
ary distances, and comparing them with the periodic 
times, we cannot but be struck with a certain correspond- 
ence. The greater the distance, or the larger the orbit, 
evidently the longer the period. The order of the pla- 
nets, beginning from the sun, is the same, whether we 
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 3-56 ; and a 
similar remark holds good in every instance. Still, the 
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 


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 wlien 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,* v/hose 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 
any one who will take the trouble to go through the calcu- 
lation, that 

(3652564)2: (6869796)-:: (100000) 3; (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 the most pregnant with important conse- 
quences. When we contemplate the constituents of the 
planetary system from the point of view which tliis rela- 
tion affords us, it is no longer mere analogy Avhich 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. Tire re- 
semblance is iioAV perceived to be a true family 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 the farthest limits of that great system, of which all of 
them, the earth included, must henceforth be regarded as 


(418.) The laws of elliptic motion about the sun as a 

* The expression of this law of Kepler requires a slight modificalion 

when we come to the extreme nicety of numerical calculation, for the 

greater iilanets, due to the inOuence of their masses. This correction i^ 

<j!Qperceplible for the earth oxid Mars. 


focus, and of the equable description of areas by lines 
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, modern 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 
elUpse 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 through many suc- 
cessive revolutions, and its history carried back, hy the 
aid of calculations founded on these data, for many centu- 
ries, we learn to regard the laws of Kepler as only Jirst 
approximations 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 
SYSTEM, 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 century, 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 
consideration, 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 out 


what particular theoretical conclusion is involved in each 
of the three laws of Kepler, considered as satisfactorily 
established, what indication each of them separately 
affords of the mechanical forces prevalent in our system, 
and the mode in Avhich its parts are connected and how, 
when thus considered, they constitute the basis on which 
the Newtonian explanation of the mechanism of the hea- 
vens is mainly supported. To begin with the first law, 
that of the equable description of areas, Since the pla- 
nets move in curvilinear paths, thej must (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 poAver 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- 
known fluids, magnetic or electric ethers, or impulses 
still, when finally brought under our contemplation, and 
summed up into a single resultant energy, its direction 
is, from every point on all sides, towards the sicn''s cen- 
tre. As an abstract dynamical proposition, the reader 
will find it demonstrated by Newton, in the 1st proposi- 
tion of the Principia, 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 ; midi vice versa, that 
such equable description of areas is itself the essential 
criterion of a continual direction of tlie acting force to- 
wards the centre to wliich this character belongs. The 
first law of Kepler, 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 influence of central forces generally, and as such, we 
daily see it exemplified in a thousand familiar instances. 
A simple experimental illustration of it is to tie a bullet 


to a thin string, and, having whirled it round with a mo- 
derate velocity in a vertical plane, to draw the end of the 
string through a small 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 in 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 jnroucttc>, as he draws in his limbs and 
straightens his whole person, so as to bring every part of 
his frame as near 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, whatever 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 free, 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 



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 7io fixed 
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 very long string, and, drawing it a little 
aside from the perpendicular, tlirow 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 jjroportional 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 focus, the steps of the investigation 
of the law of force are these : 1st, The 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 own direction, is mea- 
sured by or proportional to the amount of that deflection, 
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 


a powerful cjlindrical 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 inertise, 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 more 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 Principia 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 tlie 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 diflerence 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 fifty-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- 


vity, the square of their periodic time will be expressed 
by a fraction whose numerator is tlie cube of their mean 
distance, i. c. the greater semi-axis of their elliptic orbit, 
and whose denominator is the sum of their masses. 
When one of the masses is incojnparably greater than 
the other, this resolves itself into Kepler's law ; but 
when this is not the case, the proposition thus general- 
ized stands in lieu of that law. In the system of the sun 
find 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 
\ye come to speak of the satellites. 

(422.) It will first, ]iov>^cver, 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 hoAV we can satisfy ourselves that the 
jiumerical data contained in a table of such elements for 
the whole system does really exhibit a tnie picture of 
Jt, and afford the means of determining its state at every 
Instant of time, by the mere applieation of Kepler's laws. 
Now, for each planet, it is necessary for this purpose to 
]{now, 1st, the magnitude and form of its ellipse ; 2dly, 
ihe situation of this ellipse in space, with respect to the 
ecliptic, and to a fixed line drawn therein ; 3dly, the 
Jocal situation of the planet in its ellipse at some known 
ppoch, and its periodic time or mean angular velocity, 
pr, as it is called, its mean motion. 

(423.) The magnitude and form of an ellipse are de-. 
termined 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 usually 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 such parts ; but when 
estimated in parts of the semi-axis, regarded as a unit, 
the eccentricity is expressed by the fraction |. 

^434)' The ecliptic is the plane tp which ?in ii^liabit-s 


ant of the earth most naturally refers the rest of the solar 
system, as a sort of ground-plane ; and the axis of its 
orbit might be taken for a line of departure in that plane 
or origin of angular reckoning. Were the axis Jixed, 
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, and 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, to a fixed position. Now, 
to determine the situation of the ellipse described by a 
planet with respect to this plane, three elements require 
to be known: 1st, the indbtation of the plane of the 
planet's orbit to the plane of the ecliptic ; 2dly, 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 l)y stating its longitude. This line is called the 
Une of the nodes. When the planet is in tliis line, in 
the 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 t/ie 
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) is ascertained by stating 
the longitude of its perihelion, or the place which the 
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 


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 
the apparent positions of each planet, as it wovild be seen 
from the sun, or is seen from the earth at any time. 
The former is called the heliofentric, the latter the geo- 
centric, place of the planet. 

(427.) To commence with the 
heliocentric places. Let S re- 
ly present the sun ; APN the orbit 
i-^/'of the planet, being an ellipse, 
having the sun S in its focus, 
and A for its perihelion ; and let 
]f)aN V represent the projection of the orbit on the plane 
of the ecliptic, intersecting the line of equinoxes S V 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 direction of the planet's motion to be from B to 
A, N will be the ascending node, and the angle T SN the 
iongitude 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 yj a, the angles 
*\p 8p, T Sa, will be the respective heliocentric longitudes 
of the planet, and of the perihelion, the former of whicli 
is to be determined, and the latter is one of the given 
elements. Lastly, the angle ;jSP 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 magnitude of 
the area ASP, From tliis it is a problem of pure geo- 
metry to determine the corresponding angle ASP, which 
is called the planet's true anomaly. This problem is of 
the kind called transcendental, and has been resolved by 
a great variety of processes, some more, some less in- 


tricate. It ofiers, however, no peculiar difRcnlty, and is 
practically resolved with great facility by the help of 
tables 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, the longitudes of the perihelion and 
node being respectively T a and T N, which are given, 
their difl'trence N is also given, and the angle N of the 
spherical right-angled triangle AN, 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, 
wiiich, 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/j, whose angle N, as before, is known, it 
is easy to obtain the other two sides, N/3 and Pp. The 
latter, being the measure of the angle ^^SP, expresses 
the planet's heliocentric latitude ; the former measures 
the angle NS;;, 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 through numerically, 
by the aid of the usual logarithmic and trigonometrical 
tables, in little more time than it will have taken the 
reader to peruse its description. 

(4.30.) 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 uniform cir- 
cular motion, an equable description of areas about any centre is incom- 
patible with an equable description of o^fes. The object of the problem 
in the text is to pass from the area, supix>sed knovvii, to the angle, sup- 
posed unknown : in other words, to derive the true amoimt of angular 
motion from the perihelion, or the true anomalii from what is teehnically 
called the mean anomaly, that is, the mean angular motion which would 
have been performed liad the motion in angle been unifonn instead of 
the motion in area. It happens, fortunately, that this is the simplest of 
all problems of the transcendental kind, and can be resolved, in the 
most diflicult case, by the rule of" fiilse position," or trial and error, in a 
very few minutes. Nay, it may even be resolved instantly on inspec- 
tion by a simple and easily constructed piece of mechanism, of which the 
reader may see a descrijition in tlie Cambridge Philosophical Transao 
tions, vol- IV. p. 425, by tbe author of this work. 


of the stars, the earth's orbitual 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, i. 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 pa- 
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 the 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, therefore, S to represent the sun, E 
the earth, and P the planet ; S T the line of equinoxes, 
T E the 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 
Jfi planet seen from E ; and if SQ 
be drawn parallel to E/>, the 
angle T SQ will be the geo- 
centric longitude of the planet, 
while T SE represents the heliocentric longitude of the 
earth, and T ^p 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 whicli 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 VSp is the planet's heliocentric lati- 

(432.) Our object, then, is, from all these data, to de- 
termine the anffle T SQ and PEp, which is the geocen- 


trie latitude. The process, then, will stand as follows : 
1st, In the triiinixle SP/j, right-ang-led at P, given SP, 
and the angle PS/; (the planet's radius vector and helio- 
centric latitude,) find Sp, and Pp ; 2dly, In the triangle 
SE;;, given S/j (just found), SE (the earth's radius 
vector), and the angle ES/J (the difference of heliocen- 
tric longitudes of the earth and planet), find the angle 
S/)E, and the side E7J. The former being eqival to the 
alternate angle />SQ, is the parallactic removal of the 
planet in longitude, which, added to T Sp, gives its helio" 
centric longitude. The latter, E^; (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 FEp, of which Ep and Fp are known sides, 
and the angle FEp 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 

(434.) The planets Mercury, Venus, Mars, Jupiter, 
and Saturn, have been known from the earliest ages in 
which astronomy has been cultivated. Uranus Avas 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 to have been ob- 
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, 
when Ceres was discovered by Piazzi, at Palermo ; g 


discovery speedily followed by those of Juno by Pro- 
fessor Harding, of Gottingen ; and of Pallas and Vesta, 
by 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 
surmise, that a planet might exist between Mars and 
Jupiter ; and it may easily be imagined what was the as- 
tonishment of astronomers to find four, revolving in orbits 
tolerably well corresponding with the law in question. 
No account, ci priori, or from theory, can be given of this 
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, ccasionally and harmlessly indulge. 
(435.) We shall devote the rest of this chapter to 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 


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 
inertiae 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 gi-eater on Mercury 
than on the eartli, and on Uranus 330 times less ; the 
proportion between the two extremes being 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 three 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 ! 

(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 little 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 TREAflSE ON ABTliOKOMY. [cHAt'. Vltl. 

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 ; Ave 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- 

(437.) The case is very different with Mars. In this 
planet we discern, Avith perfect distinctness, the outlines 
of Avhat may be continents and seas. (See plate I. fig, 
1, Avhich 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 ahvays 
appears red and fiery), and indicates, no doubt, an ochrey 
tinge in the general soil, like Avhat the red sandstone dis- 
tricts on the Earth may possibly offer to the inhabitants 
of Mars, only more decided. Contrasted Avitli this (by 
a general law in optics), the seas, as we may call them, 
appear greenish.* These spots, hoAvever, are not ahvays 
to be seen equally distinct, though, lohen seen, they offer 
ahvays 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 drawing 
was made from whicli the figure in plate I. is engraved. Author. 

t It has been surmised to have a very extensive atmosphere, but on no 
sufficient or even plausible grounds. 


appears in our figure, which have 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 daring 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*^ SO"" 2V in the same direction as 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 pri7nary, 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 instructive 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 I. Jig. 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 



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 dii'ection 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, that 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 diaixieter 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 79,000 
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, flat, 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. vin.J OF Saturn's rings. 267 

wider. The dimensions of this extraordinary appendage 

are as follows :* 


Exterior diameter of exterior ring = 176418. 

Interior ditto = 155272. 

Exterior diameter of interior ring = 151690. 

Interior ditto = 117339. 

Equatorial diameter of the body = 79160. 

Interval between the planet and interior ring = 19090. 

Interval of the rings = 1791. 

Thickness of the rings not exceeding = 100. 

The figure {fig- 3, plate I.) 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 tlie 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 wlien watched, 
like the spots on Mars or Jupiter, indicate a rotation in 
lO*" 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 longitude. 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 smallness of 
the earth's orbit, E, compared with that of Saturn, the 

* These dimensions are calculated from Prof. Struve's micrometric 
mea-sures, 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. 


oartli is necessarily not. far out of that plane, and must, 
at all events, puss 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 phenomenon 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 than 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 difierence does subsist as to 

* The disappearance of tVio rings is complete, when observed with ri 
reflector eighteen inches ia aperture, and twenty feet in focal length, 
4prU 2i), mSi. Author, 

CHAP. viii.J OF Saturn's rings. 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, 
probal^ly under laws of much complexity. Trifling as 
this remark may appear, it is of the \itmost importance 
to the stability of the system of the rings. Supposing 
them mathematically perfect in their circular form, and 
exactly concentric Avith the planet, it is demonstrable 
that they would form (in spite of their centrifugal force) 
a system in a state of unstable equilibrium, 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 
mutual gravity is, not to correct but to increase this ec- 
centricity, and to bring the nearest parts of them 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, Avhile 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 Avas 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 
on its centre. But even without supposing the existence 



of any such load, of 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 sufHcient 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 from 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 equiUbrium, and be held to- 
gether ever after by an immense force) : it follows, either 
that their motions in their common 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 Avhile subject to their 
common orbitual motion, and under the full and free in- 
fluence of all the acting forces. 

(446.) The rings of Saturn must present a magnificent 
spectacle from those regions of the planet which lie above 
their enlightened sides, as vast arches spanning the sky 
from horizon to horizon, and holding an invariable situa- 
tion among the stars. On the other hand, in the regions 
beneath the dark side, a solar eclipse of fifteen years in 
duration, under their shadow, must aflbrd (to our ideas) 
an inhospitable asylum to animated beings, ill compen- 
sated by the faint light of the satellites. But we shall do 
wrong to judge of the fitness or unfitness of their con- 
dition from what we see around us, when, perhaps, the 
very combinations which convey to our minds only im 


ages of horror, may bo in reality tlioatre.s of llie most 
striking and glorious displays of beneficent contrivance. 

(447.) Df Uranus we see nothing but a small, round, 
uniformly illuminated disc, without rings, belts, or dis- 
cernible 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 real 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 Avill 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 the 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 GO feet high, and sustain no greater 
shock in his descent that he does on the earth from leap- 
ing a yard. On such planets giants might exist ; and 
those enormous animals, which on earth require the buoy- 
ant power of water 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 the 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 Avill represent the Sun ; Mercury will be 
represented by a gi-ain 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 largo 
pin's head, on a circle of 654 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 


four-fifths of a mile ; and Uranus a full-sized cherry, or 
small plum, upon the circumference of a circle more than 
a mile and a half in diameter. As to getting correct no- 
tions on this subject by drawing circles on paper, or, 
still worse, from those very childish toys called orreries, 
it is out of the question. To imitate the motions of the 
planets, in the 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, 
in 2" 56" ; Saturn, in 3" 13'" ; and Uranus, in 2'' IG". 



Of the Moon, as a Satellite of the Earth General Proximity of Satellites 

to their Primaries, and consequent Subordination of their Motions 

Masses of the Primaries concluded from the Periods of their Satellites 
Maintenance of Kepler's Laws in Ihe secondary Systems Of Jupi- 
ter's Satellites Their Eclipses, &c. Velocity of Light discovered by 
their Means Satellites of Saturn Of Uranus. 

(450.) In the annual circuit of the eartli 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; Avhile 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- 
sented 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; 
so much so, indeed, that every part of the curve described 


by cither 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 space less 
than the size of the earth itself. The effect is, neverthe- 
less, sensible, in producing an apparent monthly dis- 
placement of the sun in longitude, of a parallactic kind, 
which is called the menstrual equation ; Avhose greatest 
amount is, however, less than the sun's horizontal paral- 
lax, or than 8*6". 

(452.) The moon, as we have seen, is about 60 radii 
of the earth distant from the centre 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 which, 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 tlie earth as a 
satellite. 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 
real 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 

* R and r radii of two orbits (supposed circular), P and p the periodic 

R r 

times ; then the arcs in question (A and a) are to each other as to - ; 


earth completes its annual and the moon its monthly orbit, 
and from the 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 numerical 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 squai'es of the distances decrease, it must follow that, 
at equal 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 3-4V3-6- of that of the sun. 

(455.) The argument is, in fact, nothing more than a 
recapitulation of what has been adduced in chap. VII. 
(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 weidied 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 which 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 versed sines are as the squares of the arcs directly and the 

R r 
radii iuverselv, these arc to each other as - to ; and in this ratio are 

the forces actinic on the revolving bodies in either case. 


case miniature systems, entirely analogous, in the ge- 
neral laws by which their motions are governed, to the 
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 Avhich 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 ths 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, 
aflbrd 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- 

(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 


of the ecliptic. Accordingly, we sec 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 difTerences 
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, yet 
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 Avhose situa- 
tion with respect to the line of shadow 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, E the earth in its 
orbit EFGK, 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 sa- 
tellites ; and the penumbra, owing to the great distance 


of the sun, and the consequent smaUness of the angle its 
disc subtends at Jupiter, will hardly extend, within the 

limits of the satellites' orbits, to any perceptible distance 
beyond the shadow, for which reason it is not repre- 
sented in the lijrure. A satellite revolvinff from west to 
east (in the direction of the arrows) will be eclipsed 
when it enters the shadow at o, but not suddenly, be- 
cause, like the moon, it has a considerable diameter seen 
from tlie 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 
equal to its apparent diameter as seen from the centre 
of the planet, or ratlier somew^hat more, by reason of the 
penumbra ; and tlie same remark applies to its emer- 
gence at b. Now, owing to the difference of telescopes 
and of eyes, it is not possible to assign the precise 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, theoretical 
or practical. But, if both the immersion and emersion 
can be observed ivith the same telescope, and by the 
same person, the interval of the times will give the du- 
ration, and their mean the exact middle of the eclipse, 
when the satellite is in the line SJX, i. e. the true mo- 
ment of its opposition to the sun. Such observations, 
and such only, are of use for determining the periods and 
other particulars of the motions of the satellites, and for 
affording data of any material use for the calculation of 
terrestrial longitudes. The intervals of the eclipses, it 

2 A 


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, i. 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, tlie 
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- bY 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 the 
disc of the planet. When 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 jmss unedipsed be- 
hind the limb of the planet. 

(463.) When the satellite comes to m, its shadow will 
be thrown on Jupiter, and will appear to move across it 
as a black spot till the satellite comes to n. But the satel- 
lite itself Avill not appear to enter on the disc till it comes 
up to the line drawn from E 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 Avill 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 ; but 
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.] OF Jupiter's satellites. 


siderable size, as the following comparative table will 

Mean apparent 

Diameter in 


1st satellite 






(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 tlie 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, 
tlie 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 remarkable 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 tliird lie in the 
same direction from the centre, the first must lie on the 
opposite; 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 vAthout satellites ; 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 

* Struve, Mem. Ast. Soc. iii. 301. t Laplace, Mec. Col. liv. viii. % 27. 
X Molyneux, Optics, p. 271. 


immediately from their discovery. The final and con- 
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 
liistorical 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 continually varying, the variation extending 
from the sum 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 of eclipses 
of the satellites during many successive years, that the 
eclipses at and about the opposition of Jupiter (or its 
nearest point to the earth) took place too soon sooner, 
that is, than, 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 
Avere always too late. Connecting the observed error in 
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 Jupitei-'s distance from the earth above or below 
its average amount, and such that a difference 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 (192000 miles per 
second) was so great as to startle many, and, at all events, 
to require confirmation. This has been afforded since, 


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 dilference 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 which have l)een diligently investi- 
gated by Laplace and others. 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 respect to the sun. From this it has been con- 
cluded, 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 Saturn 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 
in size. Its orbit is also materially inclined to the plane 
of the ring, witli which those of all the rest nearly coin- 
cide. It is the only one of the number whose theory 
has been at all inquired into, further than suflices to 
verify Kepler's law of the periodic times, which holds 
good, mutatis 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 its revolution on its axis in the time of a 
sidereal revolution about Saturn. The next 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* tlyeading like beads the almost 
* By my father, in 1789, with a reflecting telescope four feet in aperture. 

2 a2 


infinitoly thin fibro of light to which it is then reduced, and 
for a short time advancing olf 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 is 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 obtain a sight of, of any in our system. 
Two undoubtedly exist, and four more have been sus- 
pected. These two, however, offer remarkable and, in- 
deed, quite unexpected and unexampled peculiarities. 
Contrary to the unbroken analogy of tlie whole planet- 
ary system whether of primaries or secondaries the 
planes of their orbits arc nearly perpendicular to the 
ecliptic, 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 
of their primary, as is the case with every other planet 
and satellite, move in the opposite direction. Their 
orbits 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 incli- 
nation, in the course, at least, of half a revolution of their 
primary round the sun.* 

* These anomalous peculiarities, which seem to occur at the extreme 
limits of our system, as if to prepare us for further departure from all its 
analogies, in other systems which may yet be disclosed to us, have hith- 
erto rested 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 observa- 
tions li-om 1S28 to the present time, to confirm, in the amplest manner, my 
father's results. Author. 




Great Number of recorded Comets The number of unrecorded proba- 
bly much greater Description of a Comet Comets without Tails- 
Increase and Decay of their Tails Their Motions Subject to the 
general Laws of planetary Motion Elements of their Orbits Periodic 
Return of certain Comets Halley'sEncke'sBiela's Dimensions of 
Comets Tlicir Resistance by the Ether, gradual Decay, and iMssible 
Dispersion in Space. 

(470.) The extraordinary 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 all 
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 laws 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 wliich they bear about with them, 
and wliich 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- 
nomically 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 catalogues in the Almagest of Riecioli ; Pingre's Cometographia; 
Delambre's Astron. vol. iii. ; Astronomische Abhandlungen, No. 1. 
(which contains the elements of all the orbits of comets which have been 
computed to the time of its publication, 1823) ; also, a catalogue now in 
progress, by the Rev. T. J. liussey. Lon. & Ed. Phil. Mag. vol. ij. No. 9. 
et seq. In a list cited by Lalande from the 1st vol. of the Tables de Ber- 
lin, 700 comets are enumerated. 


ones were noticed ; and tliat, 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 least 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 having been bright enough to be seen 
in the daytime, even at noon and in bright sunshine. 
Such were the comets of 1402 and 1532, and that which 
appeared a little before the assassination of C^aisar, and 
was {afterwards') supposed to have predicted his death. 
(472.) That feelings of awe and astonishment should 
be excited b}; the sudden and unexpected appearance of 
a great comet, is no way surprising ; being, in fact, ac- 
cording to the accounts we have 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 is usually much brighter towards the centre, and 
offers the appearance of a vivid nucleus, like a star or pla- 
net. From the head, and in a direction opposite to that 
in which the sun is situated from the comet, appear to 
diverge two streams of light, which grow broader and 
more diffused at a distance from the head, and which 
sometimes close in and unite at a little distance behind 
it, sometimes continue distinct for a great part of their 
course ; producing 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 roost celebrated of modern times, 


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 {fig. 2, plate II.) is a very correct representation 
of the comet of 1819 by no 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 tails, 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 left, as 
if moving somewhat more slowly, or as if resisted in their 

(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 phases, 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- 


faces. Nor will any one regard this explanation as 
forced, or feel disposed to resort to a phosphorescent qua 
lity in tlie comet itself, to account for the phenomena in 
question, wlien we consider (what will be hereafter 
shown) the enormous magnitude of the space thus illumi- 
nated, and the exti'emely small mass which there is 
ground to attribute to these bodies. It will then be evi- 
dent that the most vmsubstantial 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 he looked upon as dense and massive bodies 
compared with the filmy and all but spiritual texture of 
a comet. 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 probability 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 tlie 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 


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 ap- 


parent division of the tail into two principal lateral 
branches, the envelope being oblique to the line of sight 
at its borders, and therefore 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 constitution. 

(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 mouths; some move with ex- 
treme slowness, others with extraordinary velocity ; 
while not unfrequeutly, the two extremes of apparent 
speed are exhibited by the same comet in diiferent 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 cases) they approach the 
sun, and are lost in his beams. After a time they again 
emerge, on the other side, receding from the sir^ ^vUh a 


velocity at first rapid, but gradually decaying. It is after 
thus passing the sun, and not till then, that they shine 
forth in all their splendour, and that their tails acquire 
tlieir 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 tlie 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 case of cometary orbits, 
and the great comet of 1680, one of tlie most remark- 
able on record, both for the immense length of its tail 
and for the excessive closeness of its approach to the 
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 
ascertained 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 about 
the sun were, as in the planetary ellipses, proportional 
to the times. The representation of the apparent mo- 
tions of this comet by sucli an orbit, throughout its whole 
observed course, was found to be as complete as those 
of the motions of the planets in their nearly circular 
paths. From that time it became a received truth, that 
the motions of comets are regidated by the same general 
laws as those of the planets tlie difference of the cases 
consisting only in the extravagant elongation of their el- 
lipses, and in the absence of any limit to the inclinations 
of their planes to that of the ecliptic or any general co- 
incidence in the direction of the motions from west to 


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 tliis problem (wliich is, 
however, a very difficult one), and for the determination 
of the elements of the orbit. These consist, mutatis 
mutandis, of the same data as are required for the com- 
putation of the motion of a jihuiet : 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 tlie limit between the ellipse on 
the one hand, whicli returns into itself, and the hyper- 
bola on the other, which runs out to infinity. A comet, 
therefore, which should describe an elliptic path, 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 off to visit other systems, or be lost in the 
immensity of space. A very few comets have been as- 
certained to move in hyperbolas, Init many more in 
ellipses. These then, in so far as their orbits can remain 
unaltered by the attractions of tlie planets, must be re- 
garded as permanent members of our system, 

(481.) The most remarkable of these is the comet of 



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, Avhose 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 the 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 Satuni would retard its retui-n 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. It 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 
probably 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 those 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- 


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 31 years. 
This remarkable discovery was made on the occasion of 
its iourth 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 the periods are continu- 
ally diminishing, or, in other 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- 
terials 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 


M. Biela, of Joscphstadt, who first arrived at this inte- 
resting conekision. It is identical with comets which 
appeared in 1789, 1795, Sic, 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 clianged. 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 i"eturn was predicted by him 
accordingly, the prediction was disappointed by the comet 
actually getting entangled among the satellites of Jupiter, 
and being completely throAvn out of its orbit by the at- 
traction of that planet, and forced into a much larger el- 
lipse. By this extraordinaiy rencontre, the motions of 
the satellites suffered not the least perceptible derunp^e- 
tnent a sufficient proof of the smallness of the comet's 

(486.) It I'emains to say a few words on the actual di- 

* Should calculation eslabtish the fact of a resistince experienced also 
by this comet, the subject of periodical c-omets will assume an extraor- 
dinary 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 
wliich surrounds the sun ? Is it at rest or in motion I If the latter, in 
what direction does it move ? Circularly round the sun, or ti-aversing 
space? If circularly, in what plane? It is obvious that a circular or 
vorticose motion of the ether would acceierate some comets and retard 
others according as their revolution was, relative to such motion, direct 
or retrtigrade. Su])[)osing the neighbourhood of the sini to bo filled with 
a material fluid, it is not conceivable that the circulation of the planets 
in it for ages should not have impressed upon it some degree of rotation 
in their own direction. And this may preserve them from the extreme 
effects of accumulated resistance. Autbor. 


niensions 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 orlnts 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 sur})rising fact, that the comets are by far the 
most voluminous bodies 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 direction 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 lias been remarked 
respecting the change of dimensions of the comet of 
Eneke 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 

2 B 2 


no real expansion or condensation of volume (further 
than is clue 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 atmospliere, into the states of 
visible cloud and invisible gas, by the mere effects of 
heat and cold. But it is time to quit a subject so myste- 
rious, and open to such endless speculation. 



Subject propounded Superposition of small Mutions IVoblem of thrre 
Bodies Eslimation of disturbing Forces Motion of Nudes Changes 
of Inclination Compensation operated in a whole Revolution of the 
Node Lagrange's Theorem of the Stability of the Inclinations 
Change of Obhquity of the Eclifitic Precession of the Equinoxes 
INutation Tlieorem respecting forced Vibrations Of ihe Tides Va- 
riation of Elements of the I'ianet's Orbits Periodic and secular 
Disturbing Forces considered as tangential and radial Eflects of tan- 
gential Force 1st, in circular Orbits; 2diy, in elliptic Compensations 
effected Case of near Commensurability of mean Motions The great 
Inecpiality of Jupiter and Saturn explained The long Ineipiality of 
Venus and the Earth Lunar Variation Etlijcts of the radial Force 
Mean Effect on the Period and Dimensions of the disturbed Orbit 
Variable Part of its Effect Lunar Eveclion Secular Acceleration 
of the Moon's Motion Invariabilily of the Axes and Periods Theoiy 
of the secular Variations of the Eccentricities and Perihelia Motion 
of the lunar Apsides^Lagrange's Theorem of the Stability of Ihe 
Eccentricities Nutation oi' llie liniar 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 observa- 
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- 


hations. Tliey arise, in the case of the primary planets, 
from the mutual gravitations of these planets towards 
each other, wliich derange their elliptic motions round 
the sun ; and in that of the secondaries, partly from 
tlie 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 perturhations, 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 perfecdy well 
their movements, inadequate and unsatisfactory after 
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 aflecting all matter, and 
rendering every particle in the universe subject to the 
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 satelHtes about their primaries, as their 
only centres of attraction. So far from it, that his ex- 
traordinary sagacity enaliled 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 Avant of perceiving that such 
perturbations inust 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, 


which observation has ever delected, which has not 
been traced up to its origin in the mutual gravitation of 
the parts of our s)'stein, 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 eonducted 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 

(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- 


trodiiced body which produces perturbation, but the dif- 
ference of its attractions on the two originally present. 

(493.) Compared to the sun, all the planets are of ex- 
treme minuteness ; the mass of Jupiter, the greatest of 
them all, being not more than one 1300th part that of the 
sun. 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 Avhose 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 small, 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 ga^-ro o ^^ gravity at the earth's 
surface, or ^^^ 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 rippiings caused by a thousand varying breezes on 


the broad and regular swell of a deep and rolling ocean, 
which run on as if the surface were a plane, and cross in 
all directions, without interfering, each as if the other had 
no existence. It is only when their eflecls 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 

(496.) The intensity of the disturbing force is conti- 
nually varying, according to the relative situation of the 
disturbing arhd 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 


article to that last cited, of the disturbing cft'ect of the 
sun 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 draAV 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 
which 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 wliich 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 Vp, which, prolonged in 
the direction of its tangent PpR, 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 
so, 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 both, in the next instant of time, to quit that 
plane, but iineqiiaUy : 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 Fq, 
lying either above or below P/9, 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 as 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 


have only a relative meaning as referring to a supposed 
fixity of S, and transfer of the whole efl'ective 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 being needed 
to estimate its efTective 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 ^pulling force ; and as PK, being parallel to SM,lies 
below the plane of P's orbit (taking that of M's orbit for 
a ground plane), it is clear that the disturbed arc Vq, de- 
scribed in the next moment by P, must lie beloiv Vp. 
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 
SP</ (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 P^' above Vp, and r in advance 
of R. In this configuration, then, the 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 Vq will, in consequence, continue 
to lie above Vp, yet, in this case, the little arc Vq will 
have to be prolonged backwards to meet our ground 
plane, and, when so prolonged, will lie belo2V 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 


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 fluctua- 
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 tliat 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 
the disturbing force, that during every single complete 
revolution of P, the cases favourable to a retrograde mo- 
tion of the node preponderate over those of a contrary 
tendency, the retrogradation taking place over a- larger 
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, on 
the whole, 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- 
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 with 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. 

(503.) 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 on it is quite evanescent, com- 
pared with its attraction on the earth : so that the per- 
turbative efi'ect in this case, which is the diflference of 



the moon's attraction on the sun and earth, is equal to 
the whole attraction of the moon on the earth. The ef- 
fect of this is to produce a monthly displacement of the 
centre on either side of the ecliptic, whose amount is 
easily calculated by 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, tlierefore, that its momentary variation, on which the 
motion of the node of the ecliptic on the moon's orbit 
depends, must be utterly insensible. 

(504.) 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 maybe shown, 
that the effect of each planet's attraction on the orbit of 
every other, is to cause a retrogradalion 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. 

(505.) The nodes of all the planetary orbits on the^rwe 
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- 


volutions of their nodes are excessively slow, being in 
every case less tlian 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 inclinations 
of their orbits, with respect to each other, and to the 
equator of each. A variation in the position of tlie 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 extremes of summer and winter 
would become intolerable. The inquiry, then, of the 
variations of inclination of the planetary orbits inter se, 
is one of much higher practical interest than those of 
their nodes. 

(507.) Referring to the figure of art. 498, it is evident 
that the plane SP^, in which the disturbed body moves 
during an instant of time from its quitting P, is diflerently 
inclined to the orbit of M, or to a fixed plane, from the 
original or undisturbed plane PSp. The difference 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 tlie 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, 
without 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 inclination 
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, 


which (having no inertia) it must obey ; and any change 
by which P should, in the next instant, alter its orbit, 
would be equivalent to a shifting, bodily, of the whole 
hoop, changing at once its inclination and nodes. 

(508,) One immediate conclusion from 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 HSr. In 
proportion as the planes of the orbits are brought to coin- 
cidence, a very trilling angular movement of Pp 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 
eftects 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 these 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 di.sturl)ing body M is situ- 
ated in a direction perpendicular to the line of nodes, or 





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 HAftfP(7 will lie below Vp, and being pro- 
duced backwards in the quadrant HA, and forvvards 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, PxA, Avhich 
"is less than PHa ; and in the latter, Vya, which is 
greater than PN. 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, but the inclination diminishes throughout the quadrant 
$A, 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 SL^i Oi^ the 
average of a revolution of P, supposing M unmoved, the 
nodes will have retrograded with their utmost speed, but 
the inclination Avill remain unaltered. 

(511.) 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 constantly in the plane of P's orbit, and there- 



fore produces neither variation of inclination or motion 
of nodes. 

(512.) Case 3. Let us take now an intermediate 
situation of M, and indicating by the arrows the directions 
of the disturbing forces (which are pulUng ones through- 
out all the semi-orbit which lies towards M, and pushing 
in the opposite), it will readily appear 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 effect 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, 
hoAvever, the disturbing force is feebler than in the others, 
being evanescent in the line of q^iadratures TV, and in- 


creasing to its maximum in the syzygics a h. The nodes 
then will recede rapidly in the former intervals, and ad- 
vance feebly in tlie latter ; but since, as II approaches to 
a, the disturbing force, by acting obliquely to the plane 
of P's orbit, is again diminished in efficacy, still, on the 
average of a whole revolution, the nodes recede. On 
the other hand, the inclination will now diminish during 
the motion of P from T to c, a point 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 

(513.) But this increase is converted into diminution 
when the line of nodes stands on the other side of SM, 
or in the quadrants Yb, Ta ; and still regarding M as 
fixed, and supposing that the change of circumstances 


arises not from the motion of M but iVoni that of the 
node, it is evident that so soon as tlic line of nodes in 
its retrograde motion has got past a, tlie 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 diminished. 
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 
fixed in one position, we might conceive it at every in- 
stant divided into four equal parts, 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 parts 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 the nodes, which, owing to the 
extreme smallness of their motion, in the case of the 
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 its own motion, 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, tliat minute difference only excepted which 
is due to the cause just mentioned. This difference, 
then, and not the whole disturbing eff'ect of M, is what 
produces the effective variation of the inclinations, whe- 
ther of the lunar or planetary orbits ; and this difference, 
which remains uncompensated by the motion of M, is in 


its turn rompensated by the motion of the node during 
its whole revolution. 

(51.^.) 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 tlie 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 researches of 
Lagrange (of whose analytical conduct it is impossible 
here to give any idea), the following elegant theorem has 
been demonstrated: 

" If the mass of every planet be multiplied by the 
square root of the major axis of its orbit, and the pro- 
duct by the Square of the tangent of its inclination to a 
fixed plane, the sum of all these products will be con- 
stantly the same under the influence of their mutual at- 
traction.^^ If the present situation of tlie 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 tlie 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 
Ave have above seen, this 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 
oscillate backward and forward about a mean position, 


the extent of its deviation to one side and the other being 
less than 1 21'. 

(517.) One effect of this variation of the plane of the 
ecliptic, that which causes its nodes on a fixed plane 
to change is mixed up with the precession of the 
equinoxes (art. 261), and undislinguishable from -it, ex- 
cept in theory. This last-mentioned phenomenon is, 
however, due to another cause, analogous, it is true, in a 
general point of view to those above considered, but 
singularly modified by the circumstances under which it 
is produced. We shall endeavour to render these modi- 
fications intelligible, as far as they can be made so, with- 
out the intervention of analytical formulae. 

(518.) The precession of the e([uinoxes, as we have 
shown in art. 2(50, consists in a continual retrograda- 
tion of the node of the earth's equator on the ecliptic, 
and is, therefore, oliviously 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 tlieir action out of the 
question in the inquiry of its cause, and we must, 
therefore, look to tlie massive though distant sun, and 
to our near though minute neighbour, the moon, for its 
explanation. This will, accordingly, be found in their 
disturl)inff action on the redundant matter accumulated 
on the equator of the earth, by which its figure is ren- 
dereil spheroidal, combined with the eartli'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 forni 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 Vc and T(/ {fig. art. 511) being 


rendered more inclined to the plane of M's orbit, and 
those within the arcs cT, f/V, less so that they would 
otherwise be. 2dly, the nodes of this ring, regarded as 
a Avhole, 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 
effort of those parts of it which tend to become more 
inclined will act through the medium of the rins: itself 
(as a mechanical engiiie 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 the inclination change, 
an average being struck at every moment 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, \\\e 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 difference in the 


general effect ; though the different velocities of rotation, 
which may be impressed on such a system, may and 
will have a very great iniluence both on the absolute and 
relative magnitudes of the two effects in question the 
motion of the nodes and change of inclination. This 
M'ill be easily understood, if we suppose the ring without 
a rotatory motion, 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 
enormously 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 equator of such a 
globe, so as to form an elliptical protuberance, enveloping 
it like a shell on all sides, but whose mass, taken together, 
should form but a very minute fraction of the whole 
spheroid. We have now before us a tolerable 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 conceive how a motion, similar to the preces- 

* That a perfect sphere would be so inert and indifferent 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 
of such a body, or of that single force which, opposed, would neutralize 
and destroy its whole action, is necessarily in a line passing through the 
centre of the sphere, and, therefore, can have no tendency to turn the 
sphere one way or other. It may be objected by the reader, that the 
whole sphere may be conceived as consisting of 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, but our limits will not allow us to go into an exposition of the 
fallacy. We should, however, caution him, generally, that no dynamical 
subject is open to more mistakes of this kind, which notliing but the 
closest attention, iii every varied point of view, will detect. 


sion of the equinoxes, and, like it, characterized by ex- 
treme slowness, will arise from the causes in action. 

(524.) Now a recess of the node of the earth's equa- 
tor, upon a given plane, corresponds to a conical motion 
of its axis round a perpendicular to that plane. 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 wdiat is stated in art. 266, re- 
specting the nature of the motion in question. To this 
Ave 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 rouncl 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 the moon's orbit, 
moving round tlie small circle 
ABCD in 19 years ; a the pole of 
the earth's equator, which at each 
moment of its progress has a direc- 
tion perpendicular to the varying 
position of the line Ao, and a velo- 
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, C, D, E, the line A will have taken up the positions 
B6, Cc, \)d, Ee, 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 ode, 
with a velocity alternately greater and less than its mean 
motion, and this will be repeated in every succeeding 
revolution of the node. 

(525.) Now this is precisely the kind of motion which, 
as we have seen in art. 272, the pole of the earth's equa- 
tor really has round the pole of the ecliptic, in conse 



quence of the joint effects of precession and nutation, 
which are thus uranographically represented. If we 
superadd to the elfeet of hinar precession that of the so- 
lar, which alone would cause the pole to describe a circle 
uniformly about P, this wjll only affect the undulations 
of our waved curve, by extending tliem in length, but 
will pi'odiice no effect on the depth of the waves, or the 
excursions of tlie earth's axis to and from the pole of the 
ecliptic. Thus we see that the two phenomena of nu- 
tation and precession are intimately connected, or rather, 
both of them essential constituent parts of one and the 
same phenomenon. It is hardly necessary to state that 
a rigorous analysis of this great problem, 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 
Avhich are uniform, are to each other in the proportion 
of about 2 to 5. 

(526.) In the nutation of the earth's axis we have an 
example (the. first of its kind which has occurred to us) 
of a periodical movement in one part of the system, 
giving rise to a motion having the same pi'ecise 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 movement 
of a peculiar oscillatory kind impressed on the solid 
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 shall 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 
tlius generally announced : 

If one pari of any system connected either by 7nafe- 
rial ties, or by the mutual attractions of its members., 
he continually maintained by any cause, whether in- 
herent in the constitution of the system or external to 
it, in a state of regular periodic motion, that motion 
will be propagated throughout the ivhole system, and 

* Vide Prof. Airy's Mathematical Tracts, 2d cd. p 200, &c. 


will give rise in every member of it, and in every -part 
of each member, to periodic movements executed in 
equal periods with that to which they owe their origin, 
though not necessarily synchronous ivith them in their 
maxima and minima* 

The system may be favourably or unfavourably 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 tliird; 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 earth, 
and the delicacy which our observations have attained, 
enable us to make it, as it were, an instrument to feel these 
forced vibrations these derivative motions, communi- 
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, the secret 
transfer of motion by which the sound of an organ pipe 
is dispersed through the air, and carried down 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 one por- 
tion of a period consisting of two equal and similar parts. 
But the most remarkable instance, by far, of this propa- 
gation of periods, and one of high importance to man- 
kind, is that of the tides, which are forced 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 diificulty of conception. That the moon, by 

* See a demoastration of (his theorem for the forced vibrations of sys- 
tems comiected by material ties of imperfect elasticity, in my treatise on 
Sound, Eneyc. Metrop. art. 323. The demonstration is easily extended 
and generalized to take in other systems. Author. 


her attraction, should heap up the \vaters of the ocean 
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 ivhole attraction, but by 
the differences 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 an 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 thus, 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 58 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, the moon has advanced in her orbit, both diurnal 


and monthly (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 contrary 
direction), 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), which follows, 
or endeavours to follow, the apparent motions of the 
moon, and must, in fact, if the principle of forced vibra- 
tions be true, imitate by equal, though not by synchro- 
nous, periods, all the periodical inequalities of that motion. 
When the higher or lower parts of this wave strike our 
coasts, they experience what we call high and low water. 

(530.) The sun also produces precisely such a wave, 
whose vertex tends to follow the apparent motion of the 
sun 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 
synodical configuration of the two luminaries. This al- 
ternate mutual reinforcement and destruction of the solar 
and lunar tides cause what are called the spring and 
neap tides the former being their 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 the ellipticities which would 
belong to their i-espective spheroids, 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 neap as 7 
to 3, or thereabouts. 

(531.) Another effect of the combination of the solar 
and lunar tides is what is called ih.e priming and lagging 
of the tides. If the moon alone existed, and moved in 
the plane of the equator, the tide-day (i. e. the interval 
between two successive arrivals at the same place of the 
same vertex of the tide-wave) would be the lunar day 
(art. 115) formed by the combination of the moon's si- 
dereal period and that of the earth's diurnal motion. 
Similarly, did the sun alone exist, and move always on 
the equator, the tide-day would be the mean solar day. 


The actual tide-day, then, or the interval of the occur- 
rence of two successive maxima of their superposed 
waves, will vary as the separate waves approach to or 
recede from coincidence ; becEruse, when the vei'tices 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. 

(.'5.32.) Quite different in its origin is that deviation of 
tlie 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 " esta1)lishment" of that port. If the water 
were without inertia, and free from obstruction, either 
owinof 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. &c., the times 
above distinguished would be identical. But all these 
causes tend to create a difference, and to make that dif- 
ference not alike at all ports. The observation of the 
establishment of harbours is a point of 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 the time of 
" slack water," when the current caused by the tide ceases 
to flow visibly one way or the other, and that of high or loiv 
water, when the level of the surface ceases to rise or fall. 
These are totally distinct phenomena, and depend on en- 
tirely different causes, though it is true they may some- 
times coincide in point of time. They 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 investigations of Mr. Lubbock, and those highly interest- 
ing ones in whicli Mr. Whevvell is understood to be engaged, will, it is 
to be hoped, not only throw theoretical light on the very otecure sub- 
ject of the tides, but (what is at present quite as much wanted) arouse 
the attention of observers, and at the same time give it that right direc- 
tion, by pointing out wliai ought to be observed, without v^hich all obser- 
vation is lost labour. 



affects the tides at any particular spot. As the vertex of 
the tide-wave tends to place itself vertically under the 
luminary which produces it, when this vertical changes 
its point of incidence on the surface, the tide-wave must 
tend to shift accordingly, and thus, by monthly and an- 
nual periods, must tend to increase and diminish alter- 
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 eitlier 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 fartlier from the earth than their mean dis- 
tances. In consequence of this, 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, 
the 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 
causes of differences in the height of tides, however, 
local situation is the most influential. In some places, 
the tide-wave, rusliing up a narrow channel, is suddenly 
raised to an extraordinary height. At Annapolis, for 
instance, in the Bay of Fmidy, it is said to raise 120 
feet.* Even at Bristol, the difference of high and low 
water occasionally amoimts to 50 feet. 

(535.) The action of the sun and moon, in like man 
ner, produces tides in the atmosphere, which delicate 
observations have been able to render sensible and mea- 
surable. This effect, however, is extremely minute. 

(536.) To return, now, to tlie planetary perturbations. 
Let us next consider the changes induced by their mu- 
tual action on the magnitudes and forms of their orbits, 
and in their positions therein in different situations with 
respect to each other. In the first place, however, it 
will 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 
* Robison's Lectures on Mechanical Philosophy. 


SO disturbed arc no longer, in mathematical strictness, 
ellipses, or any known curves. This they do, partly on 
account of the convenience of conception and calcula- 
tion vvhicli attaches to this system, but much more for 
this reason that it is found, and may be demonstrated 
from tlie 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 matrnitude 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 distar])ing forces had no existence. By 
this way of considering the subject, the whole permanent 
effect of the disturbing forces is regarded as thrown upon 
the orbit, while the relations of the planet to that orbit 
remain unchanged, or only liable to brief and compara- 
tively momentary ilnctuation. 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 the earth's orbit 
changes no moi'e than 0-00004 in its amount in a cen- 
tury ; 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- 
lution, the difi'erence between the original ellipse and 
the curve really represented by the varying one, is so 
excessively minute, that if accurately drawn on a table, 
six feet in diameter, the nicest examination Avith mi- 
croscopes, continued along the whole outlines of the two 
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 affecta- 
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. 


(537.) Geometers, then, have agreed in each single 
revohition, or for any ninderate interval of time, to re- 
gard the motion of each planet as elliptic, and performed 
according to Kepler's laws, willi 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. 

(538.) And here we encounter the distinction between 
what are termed secular variations, and such as are ra- 
pidly periodic, and are compensated in short intervals. 
In our exposition of the variation of the inclination of a 
disturbed orbit (art. 514), for instance, 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, however, a portion 
outstanding, which again is nearly 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 node 
to compensate and bring it back to an average or mean 
value. Now, the two lirst compensations which are 
operated by the planets going through the succession of 
configurations with each other, and therefore in compa- 
ratively short periods, are called periodic variations ; 
and the deviations thus compensated are called inequa- 
lities depending on conjigu rations ; while the last, 
which is operated by a period of the node (one of the 
elements'), has nothing to do with the configurations of 
the individual planets, requires an immense period of 
time for its consummation, and is, therefore, distinguish- 
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, Avith 
their corresponding inequalities, require to be express- 
ed ; and, indeed, the former even more than tlie latter; 


seeing that the secular inequalities are, in fact, nothing 
but what remains after the mutual destruction of a much 
larger amount (as it very oi'tcn is) of periodical. But 
these are in their 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 returns to it, to deviate presently 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 
the 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. 

(540.) 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 as 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, 


but will act on P very di (Trrently, accordino- to the configu- 
rations of P and M. It will therefore be necessary, in 
analyzing its effect, to resolve it, according to niecliani- 
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 tlie dis- 
turbing part of M's effect. In this estimation, it will be 
found 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 radial disturl)ing 
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 tlie chief cause of its angular deviations from 
the elliptic place. For the equable description of areas 
depends on no particular law of centi'al 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 liy 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 iiirther 
from 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, i. e. in conjunction with 
M, and also at two points, B and D, where JVl is equi- 
distant from S and P (or very nearly in the quadratures of 


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. Consequently, the 
general elfect will be, that in P's progress through a com- 
plete synodical i-evolution round its orbit from A, it will 
first be ac"9^1erated from A up to Br thence retarded till 
it arrives at C 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's, 
it is evident tliat the retardation which takes place during 
the description of the arc AB vv^ould be exactly compen- 
sated by the acceleration in the arc DA, these arcs being 
just equal, and similarly disposetl with respect to the 
disturbing forces ; and similarly, that the acceleration 
through the arc BC would be exactly compensated by 
the retardation along CD. Consequently, 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 CSM. 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 produced 
them. 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 distance 
from P ; suppose, also, that M were so distant as to 
make only a small part of its whole revolution during a 
revolution of P. Then it is clear that, during the whole 
revolution of P, M's disturbing force would be on the 
increase by the approach of 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 the 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 approacli S ; 


but when, in the progress of its elliptic motion, it began 
again to recede, the reverse effect would take place, and a 
retardation of P's orbitual motion would happen ; and so 
on alternately, until at length, in the average of a great 
many revolutions of M, in which 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, the 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 supposed elliptic. In an elliptic orbit, 
the 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 inlluence of the ac- 
celerating and retarding tangential forces, tlian 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 or under-compensation, and a 
surviving, unextinguished perturbation at the end of a 
synodic period ; and, if the conjunctions ahvays took 
2)la.ce on the same point of P's ellipse, this cause would 
constantly act one way, and an inequality would arise, 
having no compensation, 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 (i. e. the mean angular velocities) of 
the planets in their orbits, are incommensurable 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 disturbed and disturbing planet were exactly in the 
proportion of two to five ; then would a cycle, consisting 
of live of the shorter periods, or two of the longer, bring 
them back exacdy to the same configuration. It would 
cause their conjunction, for instance, to happen once in 


every such cycle, in the same precise points of tlieir orbits, 
ichile in the, intermediate periods ot" the cycle the other 
configurations kept shifting I'ound. TIius, then, Avould 
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 not 
exactly bring about the same configuration, does so pretty 
nearly. Five periods of Jupiter are 21063 days, and two 
periods of Saturn 21518 days. The difierence 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 descriljed 8 6' in excess 
of two sidereal revolutions, and Jupiter the same angle 
in excess of five. 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 tlie case in question. The 
excess of action, for several such triple conjunctions (7 
or 8) in succession, will lie tlie 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 well 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 conjunc- 
tion we are now considering, is only one out of three 
The other two will happen at points of the orbit abnt 

2 E 


123 and 246 distant, and these points also will advance 
by the same arc of 8" 6' in 21760 days. Consequently, 
the period of 2648 years will bring them all round, 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. 

(549.) 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 hiorh historical 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 jjettin"- 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 
accumvdating, 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 its 
maximum, to an alternate retardation and acceleration of 


about 49' in the longitude of Saturn, and a corres- 
ponding acceleration or retardation of about 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 m 
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 
eonfirmed 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-mentioned 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 diflerent circumstances as to the position of the 
perihelia of the orbits. Now, we have seen that a pre- 


sentation of the one planet to the other in conjunction, 
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 amount 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 small quantities, during long 
periods, corresponds to what geometers intend when 
they speak of small terms receiving great accessions of 
magnitude by integration. 

(552.) Similar considerations apply to every case of 
approximate commensura])ility 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 coincidence of every 
fifth conjunction, in the same j)arts of each orbit (within 
^igth 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 pertur])ation 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 onler. It is, therefore, extremely mi- 
nute, and the whole resulting inequality, according to 
the recent elaliorate calculations of professor Airy, to 
whom it owes its detection, amounts to no more than a 
few seconds at its maximum, wliile its period is no less 

CHAP. XI.] THE moon's VARIATION. 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 tangential 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 wliich, the same area described about the 
focus will, in different parts of the ellipse, correspond to 
different amounts of angular motion. This inequality, 
wliich at its maximum amounts to about 37', was first 
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 acts 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 produce 
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 tlie 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 iM'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 towards S in addition to its 
natural gravity. This force is called the additillous part 
of the disturbing force. There is, besides this, another 
power, acting also in the direction of the radius SP, 
which is that arising from the difference of actions of M 
on S and 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 

2e 2 



PK ; the other perpendicular to it, or in the direction PR. 
This part of M's action is termed the ablatitioiis force, be- 
cause it tends to diminish the gravity of P towards S ; and 
it is the excess of tlie 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. 
(556.) The estimation of these forces is a matter of no 
difficulty when tlie 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 efiect. In order to 
estimate what, in any one position of P, will be tlie 
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. If we would take 
account of the elliptic motion of M, we might conceive 
the thickness of this ring, in its ditferent 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- 
tent ourselves, in the first instance, with supposing M's 
orbit 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 unthin the ring, the resulting 
attraction will always be outivards, towards the nearest 
point of the ring, or directly from the centre.* But if P 

As this is a proposition which the equihbrium of Saturn's ring ren- 
ders not merely speculative or illustrative, it will be well to demonstrate 
it ; which may be done very simply, and without the aid of any cal- 
culus. Conceive a spherical shell, and a point within it: every line 
passing through the point, and terminating both ways in the shell, will, 
of course, be e(iually inclined to its surface at either end, being a chord 
of a spherical surface, and, therefore, symmetrically related to all its 
parts. Now, conceive a small double cone, or pyramid, having its apex 
at the p<jint, and formed by the conical motion of such a line round the 
point. Then will the two portions of the spherical shell, which form the 


lie without the ring, the resulting force will act always 
inwards, urging P towards its centre. Hence it appears 
that the mean effect of the radial force will be dilferent 
in its direction, according as the orbit of the disturbing 
body is exterior or interior to that of the disturbed. In 
the former case it will diminish, in the latter will in- 
crease, the central gravity. 

(557.) Regarding, still, only t!ie 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 versa. 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, and 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 
general aggregate of the attracting matter within, which 
is not the less efficient for being distributed over space, 
and maintained in a state of circulation. 

bases of both the cones, or pyramids, be similar 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 
opposite directions, and, therefore, counteract each other. Therefore, 
the point is in equilibrium between them ; and as the same is true of 
every such pair of areas into which the spherical shell can be broken up, 
therefore the point w'ill be in equilibrium, however situated within such a 
spherical shell. Now take a ring, and treat it similarly, breaking its 
circumference up into pairs of elements, the bases of triangles formed by 
lines passing through the attracted point. Here the attracting elements, 
being lines, not surfaces, are in the simple ratio of the distances, not the 
duplicate, as they should be to maintain the equilibrium. Therefore it 
vml not he maintained, but the nearest elements will have the supe- 
riority, 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, therefore, 
true of any assemblage of concentric ones forming a flat annulus, like the 
ring of Saturn. 


(558.) 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 tliem in their actual state, and 
therefore, under the influence of this constant part of 
the perturbative action. Their observed mean motions 
are, therefore, affected by tlie whole amount of its in- 
fluence ; and we have no means of distinguishing this 
from the direct effect of the sun's attraction, Avith 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, in like 
manner, tends, by its mean influence 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 whicli obviously tend, in the average 
of a great multitude of revolutions, to neutralize each 
other. In 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 fixed) is elliptic, and requires thirteen luna- 
tions for its description, during which the distance of 
the sun undergoes an alternate increase and diminution, 
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 


(560.) Another very remarkable and important effect 
of this cause, in one of its subordinate fluctuations (ex- 
tendiufi-, however, over an immense period of time), is 
what is called the secular acceleration of the moon's 
mean motion. It had been observed by Dr. Halley, on 
comparing together the records of the most ancient lu- 
nar eclipses of the Chaldean astronomers with those of 
modern times, that the period of the moon's revolution 
at present is sensibly shorter than at that remote epoch ; 
and this result was conlirmed by a further comparison 
of both sets of observations with those of the Arabian as- 
tronomers of the eighth and ninth centuries. It appear- 
ed from these comparisons, that the rate at which the 
moon's mean motion increases is about 11 seconds per 
century a quantity small in itself, but becoming consi- 
derable by its accumulation during a succession of ages. 
This remarkable fact, like the great equation of Jupiter 
and Saturn, had been long the subject of toilsome inves- 
tigation to geometers. Indeed, so diflicult 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 as 
satisfactory as that on which most historical events are 
credited. 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 into 
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. 

(561.) 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, in 


fact, the case. The solar eclipse is kept (as we have al- 
ready hinted in art. 536, and as we shall very soon ex- 
plain more fully) in a continual but excessively slow 
state of change, by the action of the planets on 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 open out into an ellipse, the eccen- 
tricity will again increase, attain a certain moderate 
amount, and then again decrease. The time required for 
these revolutions, though calculable, has not been calcu- 
lated, further than to satisfy us that it is not to be reck- 
oned by hundreds or by thousands of years. It is a pe- 
riod, in short, in which the whole 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- 
form. 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 produced 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. Tliese ac- 
cumulate, influence the moon's periodic time and mean 
motion, and thus becoming repeated in every lunation, 


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- 
ciable action on the lunar motions as referred to the earth. 
Their masses are too small, and their distances too great, 
for their difference of action on the moon and earth, ever 
to become sensible. Yet their effect on the earth's orbit 
is thus, we see, propagated tlirough 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 eartli round the sun. 

(583.) 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 third 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 


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 
sun (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;iillion of years hence. 

(564,) This theorem is justly 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 that point where 
those details must be entered on, and in giving such an 
insio-ht 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, the 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 sun, 
and which shall coincide with it throughout the whole 


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 tlie 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 variation of 
the major axis depends on nothing but tlie 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 of 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 alterations 



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 can 
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 efl'ect 
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 the 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 imperfectly compensated actions 
of the tangential force, brings them directly 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 effect of perturbation 
in altering tlie 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 insU'Uctive in 


giving 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 tlae 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 
exactly in the direction of the string, the ellipse may be 
transferred to paper lightly stretched and gently pressed 
against it. In these circumstances, the situation of the 
major and minor axes of the ellipse will remani for a 
long time very nearly the same, tliough 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 20 from the 
vertical), this permanence of situation of the ellipse will 
no longer subsist. Its axis will be seen 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, Avhich at length will en- 
tirely reverse its situation, bringing the direction of the 
longest excursions to coincide with that in which the 
shortest 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- 
gi-ession 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 every instant perpen- 


dicular to the wire) by a force which varies as the sine 
of the deviation of the wire from the perpendicular. 
Now, the sines of very small arcs are nearly in the pro- 
portion of the arcs themselves ; and the more nearly, as 
the arcs are smaller. If, therefore, the deviations from 
the vertical are so small that we may neglect the curva- 
ture of the spherical surface in 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 the apsides of this ellipse 
would remain fixed. But if the excursions of the 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 sine, while the dis- 
tances 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 fall 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 falls, in like proportion, 
somewhat short of that which would keep the body in its 
precise elliptic orbit. It will no longer, therefore, have, 
at those greater distances, the same command over the 
weight, in proportion to its speed, which would enable 

it to deflect it from its rectilinear tangential course into an 
ellipse. The true path Mddch it describes will be less 

* Newton, Princip. i. 47. 


curved in the remoter parts than is consistent with the 
elliptic figure, as in the annexed cut ; and, therefore, it 
will 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 the 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 briefly and summarily 
expressed by saying that the apsides of its orbit are pro- 

(571.) Now, this is what takes place, mutatis mu- 
tandis, with the lunar and planetary motions. The ac- 
tion of the sun on the moon, for example, as we have 
seen, besides the tangential force, whose efl^ects 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 at a slower rate than the inverse square of the 
distance (, 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. 

II2;. 2. 

^^ "^-v ^^\ 


V ' / 

\ /- 

therefore bring it later to an apogee. In the former case, 
then, the line of apsides will retrograde ; in the latter, 
advance. (See^o-. i and^g*. 2.) 

(572.) Both these cases obtain in different configura- 



tions of the sun and moon. In the syzigies, 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 the motion of 
the moon, in and near the first of these situations, Avill 
be performed in an ellipse, whose apsides are in a 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 mutual 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, the 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 difiicvdty 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 Avill have the same effect in diminishing the 
curvature of its orbit as the decrease of centi-al force, 
and vice versa. 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 the 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, 
and, in fact, nearly to double its effect. This is what is 


meant by geometers, when they say that this part of the 
motion of the apogee is due to the square of the disturb- 
ing force. The effect of the tangential force in disturb- 
ing the apogee would compensate itself, were it not for 
the motion which the apogee has already had impressed 
upon it by the radial force ; and we have here, therefore, 
disturbance reacting on disturbance. 

(574.) 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. 
Newton 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 
the 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 minutely 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 same geometer, Clairaut, who first gave it 
currency, and who gloriously repaired the error of his 
momentary hesitation, by demonsti-atlng 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. 360), 
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 ivithin 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 


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 fig. 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 orlnt, 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 fluctuations 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. 

(57G.) The movements of the perihelia, and variations 
of eccentricity of the planetary orbits, are interlaced 
and complicated together in the same manner and nearly 
by the same laws as the A'ariations 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 orinciple of art. 526, is thence propagated throughout 


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 etfect of their mu- 
tual attraction would be to produce a variation in the ec- 
centricity of Saturn's orbit, from 0-08409, its maximinn, 
to 0*01.345, 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 vcrtiu. The period 
in which these clianges 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 anmicil amount of light and heat received 
by a planet from the sun is, cxteris paribus, as the minor 
axis of the ellipse described by it.* Any variation, 
therefore, in the eccentricity by changing the minor axi?;, 
will alter the 7nean temperature of the surface. How 
such a change will also influence the extremes of tempe- 
rature appears from art. 315. Now, it may 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 gre^t 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 Phe- 
nomena." Geol. Trans. 1832. 


between the masses, 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 sy stein is invariable / 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, 
this is equivalent to saying that no one orbit shall in- 
crease its eccentricity, unless at the expense of a com- 
mon fund, the whole amount of which is, and must for 
ever remain, extremely minute.* 

(578.) 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 centre, 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 small 
ntltation of the lunar orbit, exactly analogous to that which 
the moon causes in the plane of the earth'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 se, to secure 
the smaller planets Mercury, Mars, Juno, Ceres, &c. from a catas- 
trophe, could ihcy accumulate on themselves, or any one of them, the 
whole amount of this eccentricity fund. But that can never be : Jupiter 
find Saturn will alwa)'s retain the lion's share of it. A similar remark 
applies to the inclination fund of art. 515. Theae funds, be it observed, 
can never get into debt. Every terra of them is essentially positive. 

Masses determined by perturbations. 347 

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 by this cause, the difference of attractions between 
a sphere and spheroid diminishing with great rapidity as 
the distance increases. Tlius, while the orbits of all the 
six interior satellites of Satui-n 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 to 
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. If 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 quite 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, however, 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- 


cular 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 perturbations of the 
planets lead us to a knowledge of their masses, as com- 
pared 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 enorraous 
proportion of 65000000 to 1. 




Of the Stars generally Their Distribution into Classes according to their 
apparent Magnitudes Their Distribution over the Heavens Of the 
Milky Way Annual Parallax Real Distances, probable Dimen- 
sions, and Nature of the Stars Variable Stars Temporary Stars 
Of double Stars Their Revolution about each Other in elliptic Orbits 
Extension of the Law of Gravity to such Systems Of coloured 
Stare Proper Motion of the Sini and Stars Systematic Aberration 
and Parallax Of compound sidereal Systems Clusters of Stars Of 
Nebute Nebulous Stars Annular and planetary Nebulae Zodiacal 

(581.) Besides the bodies we have described in the 
foregoing chapters, the heavens present us with an in- 
numerable multitude 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 permanence as 
to apparent relative situation. This has procured them 
the title of " fixed stars ;" an expression which is to be 
understood in a comparative and not in an absolute sense, 
it being certain that many, and probable that all are in a 
state of motion, although too slow to be perceptible un- 
less by means of very delicate observations, continued 
during a long series of years. 

(582.) Astronomers are in the habit of distinguishing 
the stars into classes, according to their apparent bright- 
ness. These are termed magnitudes. The brightest 
stars are 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 familiar 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 of instruments, 
which successive improvements in optical science have 



attained, having brought into view multitutles 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 indiffereuce, 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 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 increas- 
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 

(584.) As we do not see the actual disc of a star, but 
judge only of its brightness by the total impression made 
upon the eye, the apparent " magnitude" of any star 
will, it is evident, depend, 1st, on the star's distance from 
us ; 2d, on the absolute magnitude of its illuminated sur- 
face ; 3d, on the intrinsic brightness of thatsurface. 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 *thal 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 
liave not yet agreed upon any principle by which the 


magnitudes may be pliotometrically 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 precise photometrical ex- 
periments, of the apparent brightness of each star. This 
Avould afford a definite character in natural history, and 
serve as a term of comparison to ascertain the changes 
Avhich may take place in them ; changes whicli we know 
to happen in several, and may therefore fairly presume 
to be possilile in all. Meanwhile, as a first approxima- 
tion, the following proportions of light, concluded from 
Sir William Herschel'st experimental comparisons of a 
few selected stars, may be borne in mind : 

Light of a star of the average 1st magnitude := 100 

2d = 25 

3d =12? 

4th = 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- 
clusion, it is otherwise when we view them in connexion 
with their local distribution over the heavens. If indeed 
we 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 
tlie 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 off from it; so (art. 253) that in fact its whole light 
is composed of nothing but stars, Avhose average magni- 
tude may be stated at about the tenth or eleventh. 

(586.) These phenomena agree witli the supposition 

* Struve, Dorpat Catal. of Double Stars, p. xxxv. 

t Phil. Tr. 1817. f Trans. Astron. Soc. iii. 183, 


that the stars of our firmament, instead of being scattered 
in all directions indifferently through space, form a stra- 
tum, of which the thickness is small, 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 it sulnlivides into two principal 
laminae, 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 through the space they occupy, would be least in 
a direction of the visual ray (as SA) perpendicular to 
the lamina, and greatest in that of its breadth, as SB, SC, 
SD ; increasing rapidly in passing from one 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 firmament 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 wdll sufficiently ac- 
count for the vast predominance of small magnitudes 
which are observed in it. 

(587.) When we speak of the comparative remote- 
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 its di- 
mensions bear to those of our own immediate system ? 
To this, however, astronomy has hitherto proved unable 


to supply an answer. All we know on the subject is ne- 
gative. We have attained, by delicate observations and 
refined coniljinations 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 earth's 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 cmmial pa- 
rallax for the diurnal, or, which com.es to the same thing, 
to ground our calculation on the relative velocities of the 
earth and planets in their orbits (art. 414), when we 
would push our triangulation 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 annual 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 unable to come to any positive and 

3 G 2 


coincident conclusion upon this head ; and it seems, 
therefore, 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 angle), it could not possibly have escaped 
detection and universal recognition. 

(589.) Radius is to the sine of 1", in round numbers, as 
200000 to 1. In this proportion, then, at least must the 
distance of the fixed stars from the 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 sma// as 4800000000 
radii of the earth, or 19200000000,000 miles ! How much 
larger it may be, we know not. 

(590.) In such numbers, the imagination is lost. The 
only mode we have of conceiving such intervals at all is 
by the time which it would require for light to traverse 
them. 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, then, are we to 
allow for the distance of those innumerable stars of 
the smaller magnitude which the telescope discloses to 
us ! If 'we admit the lio-ht 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 
than one of the sixteenth. It follows, therefore, that 
among the countless multitude of such stars, visible in 
telescopes, there must be many whose light has taken at 
least a thousand years to reach us ; and that when we 
observe their places, and note their changes, we are, in 
fact, reading only their history of a thousand years' date, 
thus wonderfully recorded. We cannot escape this con- 
clusion, but by adopting as an alternative an intrinsic 


inferiority of light in all the smaller stars of the milky way. 
We shall be better able to estimate the probability of this 
alternative, when we have made acquaintance with other 
sidereal systems, whose existence the telescope discloses 
to us, 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 limits which we are sure 
are less than the truth, let us employ the negative know- 
ledge we have obtained respecting the distances of the 
stars to form some conformable estimate of tlieir real 
magnitudes. Of this, telescopes afford us no direct 
information. The discs which good telescopes show us 
of the stars are not real, but spurious a mere optical 
illusion.* Their light, therefore, must be our only 
guide. 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, to 
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 
its actual distance. AVe have seen, however, that the dis- 
tance of Sirius cannot be so small as 200000 times that of 
the sun. Hence it follows, that, upon the lowest possible 
computation, the light really thrown out by Sirius cannot 
be so little as double that emitted by the sun ; or that 
Sirius must, in point of intrinsic splendour, be at least 
equal to two suns, and is in all probability vastly greater.:}: 

(592.) Now, for what purpose are 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 
conjectures. Useful, it is true, they are to man as points- 
of exact and permanent reference ; but he must have 
studied astronomy to little purpose, who can suppose 

* See Cab. Cyc. Optics. t Phil. Trans. 1829, p. 21. 

X Dr. Wollaston, assuming, as we think he is perfectly justified in do- 
ing, a much lower limit ofpoasible parallax in Sirius tlian we have adopts 
ed in the text, has concluded the intrinsic light of Sirius to be nearly 
that of fourteen suns. 


man to he 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 the sun ; but that 
cannot be ihe 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. 

(59.3.) Analogies, however, more than conjectural, are. 
Hot 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 regular recurrence of the same phenomena in the 
same times we are strongly impressed with the idea of 
rotatory or orbitual motion. Among the stars are se- 
veral which, though no way distinguisliable from others 
by any apparent change of place, nor by any difference 
of appearance in telescopes, yet inidergo a regular period- 
ical increase and diminution of lustre, involving, in one 
or two cases, a complete extinction and revival. These 
are called periodicMl stars. One of the most remarkable 
is the star Omicron, in the constellation Cctus, first no- 
ticed l)y Fabricius in 159G. It appears about twelve times 
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 becomes 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 degree of brightness, nor in- 
crease and diminish by the same gradations. Hevelius, 
indeedj relates (Lalande, art. 79.4) that during the four 
years between October, 1072, and December, 1676, it 
did not appear at all. 

(594.) Another very remarkable periodical star is that 
called Algol, or ,3 Persei. It is usually visible as a star 
of the second magnitude, and such it continues for the 




space of 2'' 14'', when it suddenly begins to diminish in 
splendour, and in about 3k hours is reduced to the fourth 
magnitude. It then begins again to increase, and in 31 
hours more is restored to its usual brightness, going 
through all its 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, when 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 Coodricke, to 
whom we owe the discovery of this remarkable fact,* in 
the year 1782 ; since which time the same phenomena 
have continued to be observed, though with much less 
diligence tlian their high interest would appear to merit. 
Taken any how, it is an indication of a high degree of 
activity, in regions Avhere, 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 : 

Star's Name. 

(3 Persei 

y Cephei 
/3 LyrEE 
<r Aiitinoi 
X Herculis 
* Serpentis 
RA. 1511 4im 
PD. 740 15' 
% Cygni 
367 B. t Hydrae 
34 Fl. Cvffni 
420 M. Leonis 
X Sagittarii 
i/ Leonis 


2 20 43 









396 21 


18 years 
Many years 


ation of 

























6 - 


( Goodricke, 1782. 

I Palitzch, 1783. 
Goodricke, 1784. 
Goodricke, 1784. 
Pigott, 1784. 
Herschel, 1796. 

Harding, 1826. 

Fabricius, 1596. 
Kirch, 1687. 
Maraldi, 1704. 
Janson, 1600. 
Koch, 1782. 
Halley, 1676. 
Montanari, 1667. 

* See note on page 358. 

+ These letters B. Fl. and M. refer to the Cataloguee of Bode, Flam- 
eleed, and Mayer 


The variations of these stars, hov/ever, appear to 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 ;:^^ Cygni, which is stated by Cassini to have been 
scarcely visible througliout the years 1G99, 1700, and 
1701, at those times when it ought to have been most 

(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 
occurred more than once within the limits of recorded 
observation. The phenomena we allude to are those of 
temporary stars, which have appeared, from time to time, 
in different parts of the heavens, blazing forth with ex- 
traordinary lustre ; and after remaining a while appa- 
rently immoveable, have died away, and left no trace. 
Such is the star which, sitddenly appearing in the year 
125 B, C, is said to have attracted the attention of Hip- 
parchus, and led him to (h'aw up a 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, 126*, 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 eurl^'r, as compared with that of 
the last, which was w'ell 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 star, 
with a period of about 300, or, as Goodricke supposes, 

* Th,e s^nio. dist'ovcry appears to have been mnje nearly al)oiit the 
saiTie lime by PaHtzch, a fanner of ProlitTr, near Dresden a peasant by 
station, an astronomer by nature wfip, from his familiar acquaintance 
with the aspect of the hea\ ens. bad been h'd to notice among so many 
thousand stars lliis one as distinguished from the rest by its variation, and 
had ascertained its period. The same Palitzch was also the first to re^ 
discover the predicted comet of Halley in 1759, vvliidi he saw nearly a 
month before any of the astronomers, who, armed with their telescopes, 
W'ere anxiously watching its return. These auecjotes carry us back ^o 
^he era of the Chaldean shepherds. 


of 150 years. The appearance of the star of 1512 was 
so sudden, that Tycho Brahe, a celebrated Danish astro- 
nomer, returning one evening (the 11th 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, a star of this kind, and not less bi'il- 
liant, burst forth in the constellation of Serpentarius 
which continued visible till October, 1G05. 

(597.) Similar phenomena, though of a less splendid 
character, have taken place more recently, as in the 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 after 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 disappeare-tl 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 Virginis is inserted in llie Catalogue of the Astronomical 
Society from Zach's Zodiacal Catalogue. I missed it on the 9th of Maj', 
1828, and have since repeatedly had its place in the field of view of my 
20 feet reflector, without perceiving it, unless it be one of two equal stars 
of the 9th magnitude, very nearly in the place it must have occupied. - 

t " Ces variation.? des etoiles sont bien dignes de I'attention desobserv- 
ateurs curieux . . . Un jour viendra, peut-etre, oii les sciences auront na* 
sez d'amateurs pour qu'on puisse suffire a ccs details." Lalande, art. 
834. Surely that day is now arrived. 


mers in established observatories are almost of necessity- 
precluded from taking a part by the nature of the ob- 
servations required. Catalogues of the comparative 
brightness of the stars in each constellation have been 
constructed by Sir Wm. Herschel, 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 diflerent character, and which give us a real and posi- 
tive insight into the nature of at least some among the 
stars, and enable us unhesitatingly to declare them subject 
to the same dynamical laws, and obedient to the same 
power of gravitation, which governs 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 placed near together. This might be 
attributed to accidental proximity, did it occur only in a 
few instances ; but the frequency of this companionship, 
the extreme closeness, and, in many cases, the near equal- 
ity of the stars so conjoined, would alone lead to a strong 
suspicion of a more near and intimate relation than mere 
casual juxtaposition. The bright star Castor, for exam- 
ple, when much magnified, is found to consist of two 
stars of between the third and fourth magnitude, within 
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 fall so 
near. But this is only one out of numerous such in- 
stances. Sir Wm. Herschel has enumerated upwards of 
500 double stars, in which the individuals are within half 
a minute of each other ; and to this list Professor Struve 
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 large, without exhausting the fertility of the heavens. 
Among these are great numbers in which the interval be- 
tween the centres of the individuals is less than a single 
second, of which s Arietis, Atlas Pleiadum, -y Coronas, 
Coronae, and ^ Herculis, and t and x Ophiuchi, may be 
cited as instances. They are divided into classes ac- 



cording to their distances the closest forming the first 

(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 nearly in the 
same line, it is evident that any motion of the earth must 
subtend different angles at the two stars so juxtaposed, 
and must therefore produce different parallactic displace- 
ments of them on the surface of the heavens, regarded 
as infinitely distant. Every star, in consequence of the 
earth's annual motion, should appear to describe in the 
heavens a small ellipse (distinct from that which it would 
appear to describe m consequence of the aberration of 
ligiit, 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 dimen- 

sions the more distant is the star. If, then, we regard 
two stars, apparently situated close beside each other, but 
in reality at very diff'erent distances, their parallactic el- 
lipses will be similar, but of diff'erent dimensions. Sup- 
pose, for instance, S and s to be tlie positions of two 
stars of such an apparently or optically double star a8 

2 H 


seen from the sun, and let ABCD, ab c d, be their pa- 
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 B^ ; when another quarter, 
Cc ; and when another, Drf. 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 sliould perceive a periodical change, 
both in the direction of the line joining them, and in the 
distance between their centres. For the lines A and Cc 
cannot be parallel, nor the lines B6 and J)d equal, unless 
the ellipses be of equal dimensions, i. e. unless the two 
stars have the same parallax, or are equidistant from the 

(600.) Now, micrometers, properly mounted, enable 
us to measure 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 licavens. 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 tlie telescope remains at rest; noting 
their situation ; and then turning the whole system of 
wires round in its own plane by a proper mechaniccd 
movement, till the otlicr wire Ijecomes exactly parallel 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 the 
angle which their line of junction makes 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) south, pre- 
ceding (or west). 

(601.) The advantages which this mode of operation 


offers for the estimation of parallax are many and great. 
Ill 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 Avould 
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 
uranographical reductions of aberration, precession, &;c. 
' all which bear alike on both objects. In a word, if we 
suppose the stars to have no proper motions of their own 
by which a real 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 his 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 
aiigle of position, which the parallax of the earth's an- 
nual motion would produce he observed, in many in- 
stances, a regular progressive change ; 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. 

(603.) Supposing the two stars in motion independ- 


ently of each other, and also the sun, 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 
apparent motion of one star of a double star, referred to 
the other as a centre, and mapped down, as it were, on a 
plane in which that otlier 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 and 
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 that 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 tiling like a 
positive conclusion could be come to respecting the rec- 
tilinear or orbitual character of the oqservcd changes of 

(604.) In that, and the subsequent year, it was dis- 
tinctly announced by Sir Willinm Herschel, in two 
papers, which wi?l be found in tlie Transactions of the 
Royal Society for those years, that there exist sidereal 
systems, composed of two stars revolving about each 
other in regidar orbits, and constituting what may be 
termed binary stars, to distinguish them from double 
stars generally so called, in which these physically con- 
nected stars are confounded, perhaps, with othei-s only 
optically double, or casually juxtaposed in the heavens 
at different distances from the eye ; whereas the indi- 
viduals of a binary star are, of coui-se, equidistant from 
tlie eye, or, at least, cannot differ nrore in distance than 
the semidiameter of the orbit they describe about each 
other, which is quite insignificant compared witli the 
immense distance between tlieni 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- 


duced 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, y Virginis, ^ Ursae, 
70 Ophiuclii, <r and Coronae, ^ Bootis, Cassiopeiae, 
y Leonis, ^ Herculis, J Cygni, f^ Bootis, s 4 and s 5 Lyrae, 
^ Ophiuclii, ^ Draconis, and ^ Aquarii, are enumerated 
as among the most remarkable instances of the observed 
motion; aitd'to some of them even periodic times of re- 
volution are assigned, approximative only, of course, and 
rather to be regarded as roug^h sfuesses than as results of 
any exact calculation, for wiiich 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 Lgonis at 1200 years. 

(605.) Subsequent observation has fully confirmed 
these residts, 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. They 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 an 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 were 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- 




tation 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 (| Ursae) 
were explicable, within the limits allowable for error of 
observation, on the supposition of an elliptic orbit de- 
scribed in the short period of 58| years. A different 
process of computation has conducted Professor Encket 
to an elliptic orbit for 70 Ophiuchi, described in a period 
of sevent3'-four years ; and tlu; author of these pages has 
himself attempted to contribute his mite to these interest- 
ing investigations, 'i'he following may be stated as the 
chief results which have been hitherto obtained in this 
branch of astronomy : 

Names of Stars. 

Teriod of 

Major Suini- 
axis of 


y liBonis 

y Virginia - 

(il V.y^m 

0- CoronsE 


70 Ophiuchi - 

I Ur:3SC 

't, Cancri 
V, Coronce 









(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 tlie fourth magnitude, and its component 
stars are almost exactly equal. 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- 

* Connois. des Temps, 1830. 

t Berlin Ephem. 1838 


solve it. Since that time they have been constantly ap- 
proaching, and arc 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 S 
of the same constellation, as seen by the naked eye ; and 
this note, which has 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 their 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 angidar motion of 100, and a dimi- 
nution of distance to one sixth of its former amount) is 
represented with a degree of exactness fully equal to 
that of observation itself by an ellipse of the dimensions 
and period stated in the foregoing little table, and of 
which the further requisite particulars are as follows : 

Perihelion passage. August 18, 1834* 

Inclination of orbit to the visual ray ..... 22 58 
Angle of position of the perihelion projected on the heavens 36 24' 
Angle of jiosition of the line of nodes, or intersection of the ) nno 23" 
plane of the orbit with the surface of the heavens J 

(608.) If the great length of the periods of some of 
these bodies be remarkable, the shortness of those of 
^Z ers is hardly less so. Corona) has already made a 
complete revolution since its first discovery by Sir Wil- 
liam Herschel, and is far advanced in its second period ; 
and I Ursae, ^ Cancri, and 70 Ophiuchi, have all accom- 
plished by far the greater parts of their respective ellipses 
since the same epoch. If any doubt, therefore, could re- 
main as to the reality of their orbitual motions, or any 

* See them collected in Mem- R. Ast Soc. vol. v. p. 35. 


idea of explaining thera 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 
we are now concerned ; it is with 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 interval 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 
the stability of their systems, and with the planetary na- 
ture of their orbits. Unless closely nestled under the 
protecting Aving of their immediate superior, the sweep 
of their other sun in its perihelion passage round their 
own might carry them oft', 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. 

(610.) Many of the double stars exhibit the curious 
and 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 suns, perhaps, 

With their attendant moons thou wilt descry, 
Communicating male and female light, 
(Which two great sexes animate tlie world,) 
Stored in each orb, perhaps, with some that live.* 

Paradise Lost, viii. 148. 


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 
brigliter. 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 or even appear as a vivid 
green, under favourable circumstances. 'J'he former con- 
trast is beautifully exhibited by < Cancri the latter by y 
Andromedae ; both fine double stars. If, however, the 
coloured star be much the less bright of the two, it will 
not materially affect the other. Thus, for instance, 
Cassiopeiae exhibits the beautiful combination of a large 
white star, and 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 suns 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"' a 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 the 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. 
^Ji priori, 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 intei'nal arrangement I'e- 
sulting from the difference of the opposing actions. And 
it is a fact, tliat such apparent motions do exist, not only 


among single, but in many of the double stars ; which, 
besides revolving 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 less 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, [j. Cassiopeia? 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 liave 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 clianges 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 ^ Herculis, towards 
a point diametrically opposite. This general tendency 
was referred by him to a motion of the sun and solar 
system in the opposite direction. No one, who reflects 


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 afford data for any secure conclusions 
of this kind one way or other. Meanwhile, a very in- 
genious idea has been suggested by the present astron- 
omer royal (Mi*. Pond), viz. 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 ob- 
servation cannot detect ; but should it vary, in the course 
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 other 
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 ne- 


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 

(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 tlie 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 attention is kept directed* 
upon the group. Telescopes show fifty or sixty large 
stars 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 

(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. This conclusion is still more strongly 

It is a very remarkable fact, that the centre of the visual area is by 
far less sensible to feeble impressions of light, than the exterior portions 
of the retina. Few persons are aware of the extent to which this com- 
parative insensibility extends, previous to trial. To appreciate it, let the 
reader look alternately full at a star of the fifth magnitude, and beside it ; 
or choose two equally bright, and about 3 or 4 apart, and look full at 
one of them, the probability is, he will see only the oilier : such, at least, 
is my own case. The fact accounts for the multitude of stars with which 
we are impressed by a general view of the heavens ; their paucity 
when we come to count them. Author. 


pressed upon us, when we come to bring very powerful 
telescopes to bear on these and similar spots. There are 
a gi-eat number of objects which have been mistaken for 
comets, and, in fact, have very much the appearance of 
comets without tails : small round, or oval nebulous 
specks, which telescopes of moderate power only show 
as such. Messier has given, in the Connois. des Temps 
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 their similarity of 
appearance. That they are not, however, comets, their 
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 
ncbuleuse sans etoiles), as seen in the 20 feet reflector at 
Slough.)* Many of them, indeed, arc 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 re?it, 
and subject 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 that 
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 

* This beautiful object was first noticed by Halley in 1714. It is visi- 
ble to the naked eye, between the stars /^ and < llerculis. In a niglit- 
glaas it appears exactly like a small round comet, 



(615.) Perhaps it may be thought to savour of the 
gigantesque to look upon tlie imlividuals of such a group 
as suns like our own, and their mutual distances as equal 
to those wliioli sepai'ate 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 tlie 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 seipso 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 tliem, 
there is an evident acceleration in the rate of condensa- 
tion as we approach the centre, Avliieh 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 tlie 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 tlie one 
hand, without a rotatory motion and a centrifugal force, 
it is liardly 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 : 



N. P 

. D. 



N. P. D. 







O ' 







93 8 







73 34 







91 34 





(616.) It is to Sir William Herschel that we owe the 
most complete analysis of the great variety of tliose ob- 
* Soe a note on tliis subject at the end of tiie work, p- 386. 


jects which are generally classed under the common head 
of Nebula, but- which have been separated by him into 
1st, Clusters of stars, in which the stars are cleai'ly 
distinguishable ; and these, again, into globular and ir- 
regular clusters ; 2d, Resolvable ne])ula3, or such as ex- 
cite a suspicion that they consist of stars, and which 
any increase of the optical power of the telescope may 
be expected to resolve into distinct stars ; 3d, Nebula? 
properly so called, in which there is no appearance 
whatever of stars ; which, again, have been subdivided 
into subordinate classes, accordingf to their brightness 
and size; 4th, Planetary nebulaj ; 5th, Stellar nebula?; 
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 
Kear-^they are 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 globular, 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 tlian 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 glolnilar figxn'C, and assem- 
bling themselves together from all the surrounding re- 
gion, under laws of which we have, it is true, no other 


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 course, only be con- 
sidered as clusters either too remote, or consisting of 
stars intrinsically too faint to affect us by their individual 
Kght, unless Avhere two or three happen to be close 
enough to make a joint impression, and give the idea of 
a point brighter than 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, Avould be completely resolved by a 
further increase of instrumental force. 

(019.) Of nebula?, 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 nebulas surrounding the quadruple 
(or rather sextuple) star 6 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 diflerent from what might 
be supposed to arise from the 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. XII.'] OF NEniTL^. 377 

tirely its appearance will differ, even in the same tele- 
scope, according to the clearness of the air, or other tem- 
porary causes, Ave shall readily admit that we have no 
evidence of change that can be relied on. 

(620.) Plate II. ,^7^^. 3, represents a nebula of a quite 
different character. The original of this figiire is in the 
constellation Andromeda near the star v. 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 degree long, and 15 or 20 minutes broad. 

(621.) This may be considered as a type, on a large 
scale, of a very numerous class of nebulaj, 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 pi'csent 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- 
meda R. A. 1** 43"", N. P. D. 50 '/'. s Orionis and / 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 



in length ; and in some, the extension so great, as to 
give the nebula 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 nebulee 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 /2 and y Lyrae, and may be seen with a telescope of 
moderate power. It is small, and particularly well 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 central 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. 

(6*23.) Planetary nebulae 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 sliarply 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 magnitude. 
One of them is to be found in the parallel of v Aquarii, 
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", Avoidd give a light equal to 
100 full 71100118; 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 


central condensation, woiikl certainly augur their light 
to be merely superlicial, 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 nebula) which possess an evident 
symmetry of form, and seem clearly entitled to be re- 
garded as systems of a definite 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 laminae, 
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 nebulas, 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 ellipse, 
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 them 
in a class apart from all others. The one offers obvious 
analogies eitiier 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 

(625.) The nebulae 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 


of a cloud or fog now assuming capricious shapes, like 
actual clouds drifted by the wind, and now concentrating 
itself like a cometic atmosphere around particular stars ; 
what, we naturally ask, is tlie 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 liglit and heat ? or is it pro- 
gressively concentratmg itself by the eftect 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 style 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 nebula; 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 after sunset, about the months of 
April and May, or at the opposite season before sunrise, 
as a cone or lenticulai'-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 boreaiis. It is manifestly in the 
nature of a thin lenticulany-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 


they have been stripped in their successive perihelion 
passages (art. 487), and which may be slowly subsiding 
into the sun. 



(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 possilile 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 

(628.) But though all standard units of time are equally 
possible, theoretically speaking, all are not, practically, 
equally convenient. The trppical 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 incommensurability, and 
the want of perfect uniformity in on'e 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 iluctuations arising from nutation in the one, 
and from inequalities of configuration in the other, are 
the two most invariable quantities which nature presents 
us Avith ; the former, by reason of the uniform diurnal 
rotation of the earth the latte? on account of the inva- 
riability of the axes of the planetary orbits. Hence it 
follows that tlie mean solar day is also invariable. It is 


Otherwise with the tropical year. The motion of the 
equinoctial points varies not only from the 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 tlierefore variable, and this produces a variation in 
the tropical year, which is dependent on the place of the 
equinox (arts, 517, 328). TJie tropical year is actually 
above 4*2 P 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 number 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 arlificial 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 otlier nations may be assumed as the 
true tlieoretical standard of value. 

(630.) The other inconvenience of the standards in 
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 they amount 
to a whole da5^, cast them over into the integer account. 


(631.) To do this in the simplest and most convenient 
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 of an as- 
sumed or artificial year, and adopting two such years, 
both consisting of an exact integer number of days, 
viz. one of 305 and the other of 366, and laying down a 
simple and easily remenibered rule for the order in which 
these years shall 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 the 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 denominatetl 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 360. 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 into a decimal frac- 
tion, is 305-34224, so the error of the Gregorian rule on 


10000 of the present tropical years is 2*0, or 2'' M*" 24 ; 
that is to say, less than a day in 3000 years ; which is 
more than sufficient for all human 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 its 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 nundjer, 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 eri'or by setting up a series 
of milestones, at intervals of unequal lengths, 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, and 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 in- 
tercalary or leap-days. 

(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 observa- 
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 Lave been 


tlie case with the Roman calendar, in which our own 
originates, from the time of Numa to that of Julius 
Caesar, Avhen 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 the 
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 moon 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.) 
sliould consist of 455 days, a circumstance which ob- 
tained it the epithet of " the year of confusion." 

(635.) The Julian rule made every fourth year, witli- 
out exception, a bissextile. This is, in fact, an over- 
correction ; it supposes the length of the tropical year to 
be 3654*1, 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 that the equinoxes were gradually creep- 
ing away from the 21st of March and September, where 
they ought to have always fallen had the Julian year 
been exact, and happening (as it appeared} too early. 
The necessity of a fresh and effectual reform in the calen- 
dar was from that time continually urged, and at length 
admitted. The change (which took place under the 
popedom of Gregory XIII.) consisted in the omission of 
ten nominal days after the 4th of October, 1582 (so that 
the next day was called the 15th, and not the 5th), and 
the promulgation of the rule already explained 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 



(lays being then struck out ; 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 yeai 
in England in 1753, shortened the preceding year, 1751, 
by a full quarter. Previous to that time, the year was 
lield to begin with the 25th March, and the year a. d. 
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 Europe in which tlie 
Old Style is still adhered to, and (another secular year 
having elapsed) tlie difference between the European and 
Russian dates amounts, at present, to 12 days. 

(6.S6.) It is fortunate for astronomy that the confusion 
of dates and the irreconcilable contradictions which his- 
torical statements too often exhibit, when confronted 
with the best knowledge we possess of tlie ancient reck- 
onings of time, affect recorded observations 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 abjectly dependent on the ob- 
scure and often contradictory dates which the compari- 
son of ancient authorities indicates, is often itself the 
surest and most convincing evidence on which a chrono- 
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 eclipse and 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 cele- 
brated solar ecliiise which put an end to tlie battle between tlie kings 
of Metlia and Lydia, b. c GIO, Sept. '30. Phil. Trans, ci. 220. 


next step to a perfect knowledge of time is to secure the 
idcntitication 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 tlie memory with distinct names for each, all nations 
have felt the necessity of breaking them down into par- 
cels 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 Metonio cycle, or llie fact, discovered by Meton, a Greek ma- 
thematician, that 19 solar years contain just 235 lunations (which in fact 
they do to a very great degree of approximation), was duly appreciated 
by the Greeks, as ensuring tlie corresiwndence of the solar and lunar 
years, atid honours were decreed to its discoverer. 



On Uie Constitution of a Globular Cluster, referred to tn page 374. 

If we suppose a globular space filled with equal stars, uniformly dis- 
pei-sed through it, and very numerous, each of them attracting every 
other with a force inversely as the square of the distance, the resultant 
force by which any one of them (those at the surface 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 the common 
centre of gravity, as its centre, and that, in whatever 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 might be the proporlion of its axes, or the inclination of 
its plane to the others, would be invariable in every particular, and all 
would be described in one common period, so that at the end of every 
such period or annus magnus of the system, every star of the cluster 
(except the superficial ones) would be exactly re-established in its 
original position, thence to set out afresh and run the same unvarying 
round for an indefinite succession of ages. Supposing their motions, 
therefore, to be so adjusted at any one moment as that the orbits 
should not intersect each other, and so that the magnitude of each star, 
and the sphere of its more intense attraction, should bear but a small pro- 
portion to the distance separating the individuals, such a system, it is 
obvious, might subsist, and realize, in great measure, that abstract and 
ideal harmony, which Newton, in the 89th Proposition of the First Book 
of the Frincipia, has shown to characterize a law of force directly as 
the distance. See also Quarterly Review, No. 94, p. 540. Author- 



Stnoptic Table or the Elements of tue Solab, Sistem. 

N. B. The data for Vesta, Juno, Ceres, and Pallas are for January 1, 1820. 
The rest for January 1, 1801. 


Mean distance 

Mean Sidereal 

Eccentricity in 

from Sun, or 

Period in Mean 

Parts of the 


Solar Days. 















































Inclination to the 

Loncitude of 

Longitude of 



ascending Node. 



7 0' 9"-l 

45 57' 30" -9 

74 21' 46" -9 


3 23 28 -5 

74 54 12 -91128 43 53 -ll 


99 30 5 -0 
332 23 56 -6 

1 51 6 -2 

48 3-5 


7 8 9-0 

103 13 18 -2 249 33 24 -41 


13 4 9-7 

171 7 40 -4 

53 33 46 -0 


10 37 26 -2 

80 41 24 -0 147 7 31 -5 


34 34 55 -0 

172 39 26 -8121 7 4 -3 


1 18 51 -3 

98 26 18 -9 

11 8 34 -6 


2 29 35 -7 

111 50 37 -4 

89 9 29 -8 


46 28 -4 

72 59 35 -3 

167 31 16 -1 


Mean Longitude 
at the Epoch. 

Mass in Billionths 
of the Sun's. 

Equatorial Dia- 
meter, the Sun's 


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 -0 



2 k2 



Synoptic Table of the Elements of the Orbits of 
THE Satellites, so far as they are known. 

N. B. The distances are expressed in equatorial radii of the pri- 
maries. The epoch is Jan. 1, 1801. The periods, &c. are ex- 

pressed in mean solar days. 

I. The Moon. 

Mean distance from earth 
Mean sidereal revolution 
Mean synodical ditto 
Eccentricity of orbit 
Mean revolution of nodes 
Mean revolution of apogee 
Mean longitude of node at epoch 
Mean longitude of perigee at do. 
Mean inclination of orbit 
Mean longitude of moon at epoch 
Mass, that of earth being 1, . 
Diameter in miles 

29'-982 17500 

13 53' 17" 

266 10 7 

5 8 47 

118 17 8 




II. Satellites of Jupiter. 


Mean Distance. 


Inclination of 

Orbit to that of 


Mass; that 
of Jupiter 








l^ 18^ 28"" 

3 13 14 

7 3 43 

16 16 32 

3 5' 30" 



2 58 48 


The eccentricities of the 1st and 2d satellite are insensible, that 
of the 3d and 4th small, but variable in consequence of their mutual 



III. Satellites of Saturn. 




Eccentricities and Inclinations. 





0-1 22'^ 38 
1 8 53 

1 21 18 

2 17 45 
4 12 25 

15 22 41 
79 7 55 

The orbits of the six interior 
satelhtes are nearly circular, 
and very nearly in the plane of 
the ring. That of the seventh 
is considerably inclined to the 
rest and approaches nearer to 
coincidence with the ecUptic. 

IV. Satellites of Uranus. 



Sidereal Period. 

Inclination to Ecliptic. 




21h 25'" 0^ 

Their orbits are inclined 




16 56 5 

about 78 58' to the 




23 4 

ecliptic, and their motion 




11 8 59 

is retrograde. The pe- 




1 48 

riods of the 2d and 4th 




16 40 

require a trifling correc- 
tion. The orbits appear 
to be nearly circles. 


Fui. Z. 

Fig. 3. 

Vihu'tr Sattp. 

riate Z^ 


Fi.-. ^ 

J.ylaoer . 


ho i. 

Vk .'. 


Air, 28. Mechanical laws for regu- 
lating its dilation and compres- 
sion ; rarefraction of, 29. Density 
of, 29. Refractive power of, af- 
fected by its moisture, 33. 

Angle of reflection equal to that of 
incidence, 91. 

Angles, measurement of, 82. 

Anomalistic and tropical years, 196. 

Apparent diurnal motion of the hea- 
venly bodies explained, 45. 

Apsides, their motion illustrated, 

Astronomical instruments, 66. 
Practical difficulties in the con- 
struction of 67. Observations in 
general, 68. 

Astronomy, 7. General notions 
concerning the science, 14. 

Atmosphere, 29. Refractive power 
of the, 31. General notions of its 
amount, and law of variation, 34. 
Reflective power of 36. 

Attraction, magnetic and electric, 
224. Of spheres, 225. Solar at- 
traction, 227. 

Azimuth and altitude instruments, 


Barometrical determination of 
heighls, 149. 

Biot, M., his aeronautic expedition, 

Bode's law of planetary distances, 

Bodies, effect of the earth's attrac- 
tion on, 124. Motion of, 222. 
Rule for determining the velo- 
city of 223. Problem of three, 

Borda, his invention of the principle 
of repetition, 103. 

Calendar, 381. Gregorian, 383. 
Juhan, 385. 

Cause and effect, 221 . 

Celestial refraction, 38. Maps, 151. 
Construction of, by observationa 
on right ascension and declina- 
tion, 152. Objects divided into 
fixed and erratic, 155. Longitudes 
and latitudes, 160. 

Centrifugal force, 118. 

Chronometers, 78. 

Circles, co-ordinate, 96. 

Clairaut, 124. 

Clepsydras, 78. 

Clocks, 78. 

Comets, their number, 283. Their 
tails, 284. Their constitution, 285. 
Their orbits, 287. Their predicted 
returns; Encke's, 291. Biela's, 
291. Their dimensions, 293. 

Copernican explanation of the sun'a 
apparent motion, 185. 

Dates, astronomical meaiia of fixing, 

Day, solar, civil measure of time, 
381. Sidereal, 381. 

Definitions of various terms employ- 
ed in astronomy, 56. 

Diurnal or geocentric parallax, 181. 


Earth, the, one of the principal ob- 
jects of the astronomer's conside- 
ration; opinions of the ancients 
concerning, 16. Real and appa- 
rent . motion of explained, 17. 
Form and magnitude of 19. Its 
apparent d iameter, 21 . A diagram 
elucidating the circular form of, 
22. Effect of the curvature of, 
24, Diurnal rotation of, 42. Poles 




of, 50. Figure of, lOG. Means-- 
of determining with accuracy tlie 
dimensions of the whole or any 
part of, explained, 107. Meridio- 
nal section of, 112. Exact dimen- 
sions of, 114. Its form that of 
erjuilibrium, modilied by centri- 
fugal force, 117. Local variation 
of gravity on its surface, 120. Ef- 
fects of the earth's rotation, 123. 
Correction for the sphericity of 
144. Tlie point of the earth's 
axis, 103. Conical movements of, 
164. Mutation of, Ifiij. Parallel- 
ism of 186. Proportion of its mass 
to that of the sun, 274. 

Ecliptic, the, 157. Its position among 
the stars, 158. Poles of 159. 
Plane of its secular variation, 308. 

Elliptic motion, laws of 179. 

Equations for precession and nuta- 
tion, 167. 

Equatorial or parallactic instru 
ment, 98. 

Equinoxes, precession of the, 162 
Uranographical effect of 162. 

Eccentricity ol'the planetaiy orbits 
its variation, 343. 

Explanation of the seasons, 186. 

Floating collimator, invented by 

Captain Kater, 95. 
Force, centrifugal, 223 


Gay-Lussac, his aeronautic expedi- 
tion, 28. 

Galileo discovers Jupiter's satellites, 

Geographical latitudes determined, 

Geography, outline of so far as it is 
to bo considered a part of astro- 
nomy, 105. 

Gravitation, law of universal, 222. 

Gravity, local, variation of 119. 
Statical measure of 121. Dyna- 
mical measure of 122. Terres- 
trial, 222. Diminution of, at the 
moon, 224. Solar, 229. 


Hadley's sextant, 101. 

Halley discovers the secular accele- 

ration of the moon's mean motion, 

Harding, Professor, 262. 
Herschel, Sir William, his view of 

tlie physical constitution of the 

sun, 198. 
Horizon, dip of the, explained, 23. 
Hour-glass, 78. 


Rater's floating collimator, 95. 

Kepler, the first who ascertained the 
elliptic form of the earth's orbit, 
179. His laws, and their inter- 
pretation, 250, 


Lalande, his ideas of the spots on 
the sun, 199. 

Laplace accounts for the secular ac- 
celeration of the moon, 333. 

Latitude, 59. Length of a degree 
of, 109. 

Level, description and use of 92. 

Light, aberration of 169. Urano- 
graphical effect of 172. Its velo- 
city proved by eclipses of Jupiter's 
satellites, 280. 

Longitudes, determination of by 
astronomical observation, 131. 
Differences found by chronome- 
ters, 132. Determined by tele- 
graphic signals, 134. 

Lunar eclipses, 215. 


Maclaurin, 124. 

Maps, construction of 141. Projec- 
tions chiefly used in, 146. The 
orthographic, stereographic, and 
Mercator's, 146. 

Menstrual equation, 273. 

Mercator's projection of the sphere, 

Mercury, the most reflective fluid 
known, 91. 

Meridian, or transit circle, for ascer- 
taining the right ascensions and 
polar distances of objects, 91. 

Microscope, compound, 85. 

Milky Way, 157, 351. 

Moon, the, its sidereal period; its 
apparent diameter, 203. Its paral- 
lax, distance, and real diameter 



204. The I'orm of its orbit, liki' 
that of the sun, is ellipiiL-, l)ut loii- 
sidorably more ecceutnc ; the ilrsl 
approximation to its orbit, 205. 
Molioiiti of tlie nodes of, 20j. Oc- 
ciiltations of, 21)7. Piiases of, 21 1. 
It.s synodical periods, 212. Revo- 
lutions of the apsides of, 210. 
Physical constitution of, 217. lis 
niountains, 218. Its aanuspliere, 

219. Rotation of; libration of 

220. Diminution of gravity at tiie ; 
distance of it from the earth, 224. 
Its gravity towards tlie earih; to- 
wards the sun, 273. Its motion 
disturbed by the sun's atlraclion, 

332. Acceleration of its moan 
motion; accounted for by Laplace, 

333. Motion, parallactic, 18. Ap- 
pearances resulting from diurnal 
motion, 19. Real and apparent 
motion of the earth described, HJ5. 
Of bodies, 222. Laws of elliptic 
motion, 226. Orbit of the earth 
round the sun in accordance with 
these law s, 227. 

Mural circle, 89. 


Nebulte, Sir W. Herschel'.-; disco- 
veries oi; 375. Resolvable, 376. 
Aimular, 3'/ 8. Planetary, 378. 

JNewton, his law of universal gravi- 
tation, 225. 

JVodes, their motion, 302. 

JMutation, its jjhysical causes, 313. 


Olbers, Dr., 202. 

Orbits, variation of their inclinations, 


Parallax, 52. 

Pendulum, 122. 

Perturbations, 294. Of the planeta- 
ry orbits, 319. 

Planet, method of ascertaining its 
mass, compared with that of the 
sun, when it has a satellite, 274. 

Planets, the, 231. Apparent motion 
of, 2 J2. Their stations and retro- 
gradations, 233. The sun their 
natural centre of motion, 234. 
Their apparent diameters and dis- 

tances irom the sun, 235. Motions 
of llu; iidbrior planets ; transits of, 
236. Elongations of, 238. Their 
sidereal periods, 240. Synodical 
revolutions of, 241. Phases of 
Mercury and Mars, 242. Transiis 
oi" Venus explained, 243. Supe- 
rior planets, 246. Their distances 
and periods, 247. Method for de- 
termining tlieir sidereal periods 
and distances, 248. Elliptic ele- 
ments of the planetary orbits, 251. 
Their heliocentric and geocentric 
places, 258. The four ultra-zodi- 
acal planets, discovered in 1801, 
201. The physical peculiarities, 
and prol'.able condition of the 
several planets, 202. Their u(> 
parent and real diameters, 205. 
Their periods unalterable, 335. 
Their masses discovered inde- 
pendently ol' satellites, 347. 

Polar and horizontal points, 90. 

Pole star, 46. Situation of 89. 

Precession, its physical causes, 309. 

Projectiles, motion of, 222. Curvili- 
near path of, 222. 


Rays of light, refraction of, 3L 

Keilecling circle, 103. 

llellectlon, angle of, equal to the of 
incidence, 91. 

Refraction, 31. Of the atmosphere 
31. Effects of, to raise all the 
heavenly bodies higher above the 
horizon in apjiearance than they 
are in re:ilitv, 32. General notions 
of its amount, and law of variation, 
34. Terrestrial refraction, 38 
Celestial refraction, 38. 

Repetition, jiriaciple of, invented by 
Borda, 103. 


Satellites, 272. Their motions round 
their primary analogous to those 
of the latter round the sun, 274. 
Of Jupiter, 275. Their masses, 348. 

Saturn, his satellites, 281. 

Sea, action of the on the land, 117. 

Seasons, explanation of the, 186. 

Sextant and reflecting circle, lOL 
Its optical property, 102. 

.^iderenl clock, (i2. 

Sidejeal year, 158. 



Sidereal time, reckoned by the di- 
urnal motion of tlie sturs, 02. 

Sirius, its intrinsic brilliancy, 355. 

Solar eclipses, 208. System, 231. 

Sphere, celestial, 39. Projections 
of, 146. 

Stars, 52. Distance of, from the 
earth, 53. Sidereal time reckoned 
by the diurnal motion of the, 62. 
Visible by day, 65. Fixed and 
erratic, 155. Their relative mag- 
nitude ; infinite number, 349. 
Their distribution in the heavens, 
351. Their distances, 352. The 
centres of planetary systems, 356. 
Periodical, 356. Temporary, 358. 
Double, 360. Binary, 364. Their 
orbits elliptic, 365. Their colours, 
368. Their proper motions, 369. 
Clusters of, 373. Globular clus- 
ters of, 374. Irregular clusters of, 
375. Nebulous, 377. 

Sun, apparent motion of the, not 
uniform, 176. Its apparent diame- 
ter also variable, 177. Its orbit 
not circular, but elliptical, 177. 
Variation of ils distance, 179. Its 
apparent annual motion, 180. 
Parallax of, 180. Its distance and 
magnitude, 183. Dimensions and 
rotation of, 184. Mean and true 
longitude of, 192. Equation of its 
centre, 193. Phy.sical conslitulion 
of, 197. Density of; force of gra- 
vity on its surface, 227. The dis- 
turbing eflect of, on the moon's 
motion, 228. 

Table, exhibiting degrees in differ- 
ent latitudes, expressed in British 

standard feet, as resulting from 
actual measurement, 111. 

Telescope, 85. Application of, the 
grand source of all the precision 
of modern astronomy, 86. Differ- 
ences of dechnation measured by, 

Terrestrial refraction, 38. 

Theodolite, construction of the, 144. 

Tides, their physical cause, 314. 

Time, mea.su rement of, 78. Its 
measures, 381. 

Trade-winds, 124, Explanation of 
this phenomenon, 125. Compen- 
sation of, 127. 

Transit instrument, 76. 

Trignometriftal survey, 142. 

Tropical and anomalistic years, 195. 

Twilight caused by the reflection 
of the sun and the moon on the 
atmosphere, 35. 


IJranographical problems, 173. 
Uranography, 151. 
Uranus, his satellites, 282. 


Varialions, periodic aaid secular, 

Year, tro])ical, the civil measure of 
time, 381. Sidereal, 381. 

Zodiac, the, 157. 
Zodiacal light, 380. 


T^vT-' - ^\'%^ 

I 0. 


I c 



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