NATURAL PHILOSOPHY ,

BE ING

HEADS OF LECTURES

DEL IVERED IN THE UN IV ERSITY OF ED INBURGH,

JOHN I’LAYFAIR,

PEOFESSOI OF NATURAL PE ILOSOPHY IN THE UN IVERSITY OF EDINBURGH,

FELLOWOF THE R OY AL SOC IETY OF LONDON ,

encnn'

ru w or THE nou n socm '

rv or znm nvn c n ,

u mr u n : or Tm:n ow“. uzm cu socm '

rv or h um mus .

VOL . II .

EDINBURGH

PR INTED B Y NE ILL 4 CO.

ARCHIBALD C ONSTABLE AND COMPANY, EDINBURGH,AND LONGMAN, nun s'

r, nuns, 0 1 mm BROWN,AND CADELL 8: Bu rns,

LONDON.

18 14 .

CONTENTS.

Ss cr . 8.

Eclipses of the Moon ,

Eclip s es of the Sun ,

Number of Eclipses,

9. Plan ets,

Orbits of the Plane ts,

Rotation of the Planets,

Secondary Planets,

Ring of Saturn ,

Comets,

A berration of L ight, and the Nuta

tion of the Earth‘

s A x is,

Dimen sions of the Solar Svstem,

The A n nual Parallax, and Distance of

the Fixed Stars,

A rrznnrx . On the M ethod of deter

min ing by observation the con

stan t coeffic ients in an assumed

or given Function of a variable

Quantity,

PA RT II. Pn r sxcu . ASTRONOMY .

Szcr . l . Forces w hich retain the Planets in

their Orbits,

2. Forces w hich disturb the Elliptical

Motions of the Planets, 245Disturbance of the Moon

’

s Motion, 247

3. Disturbance in the Motions of the

Primary

CONTENTS. vii

Primary Planets, from their actionon one another , Page 265

Disturbance in the Motion of the Se

tellites of Jupiter from their ac

ouone another,

General result from the theory of

the Planetary Disturbances, 288

5. A ttraction of Spheres and Spheroids, 291

6. Figure of the Earth, 296

7 . Procession of the Equinoxes, 308

Variation of the Diurnal Rotation , 318

Variation of the Obliquity of the

Ecliptic,

8. Explanation of the Phenomena of the

T ides, 326

Principle ofUniversal Gravitation, 339

OUTLINES

6 OUT L INES or NATURA L PHILOSOPHY .

a . The d irection of gravity passes nearly through thecentre of the earth.

5. The Zenith ahd the Nadir are the Poles of the

horizon .

c . If a plane parallel to the plane of the horizon of

any place , pass through the centre of the earth, it.

w ill cut the celestial sphere in a circle hardly di

stinguishable from the former . It is called the

Rational, and the other the Sensible Horizon .

d. The Horizon of any p lace divides the visible from

the invisible Hemisphere. This supposes the eye

to be situated on the surface of the earth : if it is

elevated above the surface , the apparent horizon is

a circle parallel to the former, but low er , being the

base of a cone w hich has the eye of the spectator

for its vertex , and ofWhich - the superficws t ouches

the superficies of the earth all] round.

10 . C ircles, the planes of wh ich pass through

the Zen ith and Nadir of any place , are c alled

Vertical Circ les, and are perpendicular to the ho

rizon of the place .

a . The Meridian’

of any place, is one of the vertical

circles of that place .

6. The vertical circle that is at right angles to the

plane of the meridian, is called the Pr ime Ver ti

cal.

8 our tms s or'

tram “. rn i t osorav.

altitude or e levation at the pole above the horizon

of that plac e.

0

141. The ele vation of the pole at any plac e is

found , by observing one of th e stars, which are so

n ear it as not to se t, w hen it passe s the meridianabove , and again w hen it passes the merid ian un

der the pole that is, by obse rving its greatest an d

its least altitude s above the horizon half the sum

of the two altitudes of the star, is the elevation of

the pole, or the Latitude of the place .

For the star is alw ays at the same distance from

the pole so that on the me ridian it w as as much

above the pole in the one obse rvation, as it w as

b elow in the other.

a . T hose stars never se t at any place of which the

polar distance is equal to the latitude of the

b. This observation requires- the use of sa instrument

whieh can be placed accurate ly ia the plane of themeridian , such as the A stronomical C irc le , the Qua.

dran t, &c . or one likeHa nu v’

s Sextant, wh ichcan

de termiDe the grea test or least altitude of a star

above the horizon , w ithout any previous inquiry

into the position of the meridian.

15. To

a sr nonouz. 9

T5§ Tmfin dfM eridian Line , or that in which

the p lanentm eridian in te rsectsme hotinon of

any place .

O bse rve the alt itude of a star w hen it is on the east

side of the meridian, md mafi dn thc hmizw ,

the point w hich is in the same vertical circle , or

w hich has the same azimuth w ith it. Observe the

star a it has the same altitude again on the

w est side of the meridian , and mark in like manner the point on the horizon w hich is in the same

vertical circle w ith it . The line that bisects the

angle made by lines drawn from the plaee of oh»

serration to the tw i) poin ts thus marked on the

horizon , is the Meridian L ine.

For w hen the altitudes of a star on opposite sides of

the meridian are equal, its azimuths, or the angles

which its verticals make with the meridian, are

1 6 . The meridian and the latitude be ing thus

found , if the merid ian altitude of any star be oh

se rved . its d istance from the pole, or from the

e quator, is de te rmined.

a. For the altitude of the equator, or of the point

where the equator cuts the meridian, is km meingthe same w ith the complement of the latitude gandthe ditl

’

ermce between the meridian altitude of the

star and of the equator is the declination of thestar.

10 our nms s or NATURA L rm hosoen r .

otherw ise on the same .

Thns the places of the stars, as north or south of a

giyea plane, that of the equator, are de termin ed ;hut those pupa , to be fully known, must be de te r

Now; of the circles at right anglea to the equator ,

or the cme les of declinat ion ,.

no one is fixed 1n its

position, but A ll y? them revolve uniformly w ith

the heavens ; so that it is only by the computation

of time that one of them can he d istinguished from

another .

The circle of declination which passes through the

point in the equator which the sun occup ie s ,at the

vernal equinox, has some advan tages ahnye the

rest, as a lin e to w hich the stars are to be referred .

called the Colurc, and its position ,

hisee , for a

instant, may be determin ed by means if a

regulated Accord ing to the follow ing me .

1 7 The iime from a star heing on the meri

d lan ,toits next comrng to the me ridian, is al

w ays of the seine length it is called a Side rial

Day.

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12 oun mns or NATURA L rurt osornr .

Observe the hour by the clock wheutbe sun’

s oen tre

tion of the sun at the same instnnt . Let tbe nnye

at which the sun’

s path cuts the equator, or w hat

is called the Obliquity of the Eclip tic,

find an arch e, such that sin — i

c has two values, let the least of them, turned in to

time , When the sun’

s declin ation is north,

the hour w hich the clock should have marked at

the time of the observation is d , if it be before ,and 1211 — 03 if it be after the solstice . Whenthe sun

’

s declination is south, 12 o’

is the hour,

if it be before , and 24 — e’

, if it he after tbe sol

stice . The error thus found may either be corree

ted or allow ed for . The clock is afte rw ards to beregularly compared w ith the soothing of the

stars.

20 . If a meridian c ircle pass through any star,

the arch of the equator in tercepted be tw een that

c ircle and the poin t of the ve rnal equinox,is call

ed the R iglzt Ascension of the star ; and if a clock

be regulated as above desc ribed, the time of a

star’s passage over the merid ian , w hen turn ed in

to degrees, w ill be equal to its right asc en sion .

11 . Suppose

a srs ouomr . 13

a . Suppose the hour accm'ding to sidet ial tithg wlten a

m0

24 111 0 1: 6°

30”

Right Ascen . =

T he pl'

eceditlg ohservatiom , for determining the pla

m of the summ'

e anmpposed mhe made on the

meridian ; and such, when they can be obtained , on

ac count of their simplicity, are prd crable to all

others. It often la ppens, how ever, that the stars

must be observed when they are not on the meri

dian , and their position s, w ith respect to the im

moveable circles of the sphere , must then be deri

T he angle-which the mer idian of a star makes w ith

the meridian of the place of observation , is called

the star’

s Horary Angle, as it is the angle w hich

measures the time betw een the instant of observe s

t ion , and the star’

s passage over the meridian.

2 1 . Of these five quan tities, the Declinat ion,

the A ltitude, the Azimuth , the Horary A ngle

of a star, and the Latitude of the place of obse r

va t ion , if any three be give n , the othe r tw o may

b e found from the re solut ion of the sphe rical tri

angle

1 4 0 0 1 1 1142 3 or NA TURA I. rnrt osormr.

angle PZS, fig. 1. contained by the arches join ingthe pole , the zen ith, and the star.

This general problem contains tw enty cases, of

w hich those that follow are the most useful in

astronomy.

In all of these, w e suppose d to be the declination ,

a the altitude , h the horary angle , 2 the azimuth

of a star ; and l the latitude of the place of obser

vation .

22 . Suppose the altitude and azimuth of a star

to be observed, and the latitude to be also know n ;

it is required to find the dec lination, and the ho

rary angle of the star.

Here a , z, I are given , to find at and h, or, in the sphe

rical triangle PZS, the sides ZP, ZS are given ,

(the complemen ts of l and a) and the angle PZS,

between them to find the side PS, and the angle

P.

By letting fall a perpendicular from S on the meri

dian , w e may obtain a solution by case l st and 2d,

of oblique- angled spherical triangles.

The solution may also be expressed analyticallythus

s sr nouostr . 15

Fin d an angle as, so that

tan 3: cos 2‘ x cot a, then

tan z x sin x

sin a x sin

coa xand sin d

T he latter formula isuseful, when the declination of

a star is to be determined by observationsmade out

of.the meridian .

T he former is useful, when the time is to be found bythe mere observation of a star. It is more usual,

w hen this is inquired afi er, to have the declination

of the star given as in the following problem.

2 3. The declination and the altitude of a star

b e in g given , as also the latitude, to find the hora

ry angle a nd the azimuth .

Here the three sides of the triangle ZPS given, to

find the angles at P and Z.

If d’

complement of Dec.

a’

complement of A lt .

l’

complement of Lat .

16 OUTL INES or NATURAL ra t t osorar.

d' + a’ — l'

By the first fortunla, the hour can always be deter

mined from astronomical observations, if the tim0

of the star’s passing the meridian is known .

The second formula serves to find the meridian, than

the observation of the altitude of a know n star.

The meridian , how ever, is better found by ehser

ving‘

a star when it has the same altitude on the

e ast and w est sides of the meridian , and bisectingthe difference of the azimuths, as in 15.

24 . Let a, h and I be given , to find (I that is,

the altitude, the horary angle , and the latitude , to

find the declination .

0 . Here two sides of the triangle ZPSare given, and

the angle opposite to one of them, to find the third

side . This falls under Case 7. of oblique- angled

spherical triangles.

6. Find

ASTRONOMY. 17

5. Find an are s , such that tan x: cos b x cos I.

cos x x sin p .Next find an arc y, so that cosysin

-

I

then the complement ofDec . y x.

This solution is from its nature ambiguous ; the sum

ofy and somust be taken , when the perpendicular

from the,zenith, on the circle of declination, falls

within the triangle ; their difference, when it falls

w ithout.

25. Le t a, d, 2 be given to find I,or to find the

latitude from obse rving the altitude and azimuthof a star, and know ing also its declination.

Here, again, in the triangle ZPS, two sides are gi.

ven , and the angle opposite to one of them, to find

the third side , the complement of latitude . The

perpendicular must he let fall from the star on

the meridian ; and the distance of this perpendicu

lar, first from the zenith, and then from the pole ,

is found as in the last case ; the sum or difference

is the complemen t of the latitude .

26 . Le ti

a, d and 11, be given to find I that is,

the altitude and herary angle be ing obse rved of a

star, of w hnch the declinat ion is known , to find

the latitude.

VOL . 11 .

[ 8'

OUTL INES OF NA TURAL PHILOSOPHY.

Theperpendicular is to be let fall, as‘ in ,the last case,

from the star on the meridian ; and there beingtwo sides of the spam triangle given, m l the

g‘

le Opposite to one of them, the calculation is

This M M “useful for finding the latitude, when

two equal altitudes of a star are observed , and the

inta val of time between the observations. The

half of the i n terval gives the horary angle, and so

the latitude may be found as above .

27 . In the above formulas it may happen , that

a 2 0 , or that the star is in the horizon, or 90°

from the zenith . The horary angle ts then found,

if the lat itude and dec lination are given , from a

right angled triangle , of,

w hich one of the sides,

ght angle , is the e levat ion of the

the arch be tw een the star, whenrising crea ting, and the meridian ; ans the by

pothenuse is the distance of the star from.

the

I n this case, the hursry angle , (converted into time),is the time of ha lf the stay of the star above the

horizon , (or under it) , and if it be ealled H,

obs H tan ! x ten d .

The other side of the tr iangle , or the azimuth of the

rising or setting star, is also called the Amplitude,

and

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a sr aonomr . 21

30 . Suppose that the right ascension ~and dc

c linat ion of a star are given , to find its longitude

a nd latitude . Find an are r , such that

cot x 1: sin 2 x cot Dec .

The dehlination , if north, is reckoned positive , ifsouth, negative, and s has the same sign w ith it .

L et y z x— v, being reckoned positive for thathalf of the ecliptic which is north of the equa

tor, and negative for the opposite , or w hich

chines to the same, pos itive when the right ascen

sion is lgss than a semicircle, negative when it is

greater .

Then tan Long.

cosz-x tan R ight Asa ,

‘

cos x

and tan Lat z sin Long. x tw y .

T his rule , which is Dr Mu s eu m’s, is quite free

tion AB the right ascension , SD the latitude , and

AD the longitude of the star. See V inc e’

s Act .

101. 1 . p . 39. L A Lann a computes the angde BAS,

B 3 and

‘

ss OUTLINES OF NATURAL PHI LOSOPHY.

and the arc AS, in the triangle ABS. Hence the

angle SAD being known , and the side AS, in the

triangle ADS, the sides AD and DS are found .

This is somew hat more prolix . Cannon , 144 8,

(2d edit .) considers the subject differently, butbrings out the same solution w ith Dr Muns

nvnz.

D1:Lsunan has computed tables, for facilitating this

computation, as it occurs very often in astrono

‘

If the star is in the ecliptic , or Lat. = 0 , x

and y 0 ,‘

t herefore cosy l , and

tan R ight A sc .

cosptan Long

The changes are obvious, that would make the same

formulas apply to’

the converse of the problem,

viz. to find the right ascension and declination from

the longitude and latitude being given .

3 1 . By the methods now explained , and chief

ly'by the observations made on the meridian

,os

talogues of the stars have been formed , in w hich

the ir place s are all set dow n in respec t of the c in

cles of the sphe re , viz. the Ec liptic or the Equator,

but most comm’

only the forme r.

a . Htrrs ncn ns began the first catalogue of the fixed

stars, 120 yearsA . C . J esus,”

says Punv, mm

d iam

a sr aonomr . 23

T his catalogue w as iti'

terw ards published by Proseur , w ith some additions, and contained 1022

b. The use of the telescope increased the nmnber ot'

the visible stars, as w ell as the precision w ith which

the ir places were asc ertained. The Britannic Ca

talogue , published by h um an ) in 1669, contain.

e d nearly 3000 fixed stars. Great additions have

been made to this catalogue , particuhrly by LA

C ast e s , who determined the places of nearly10 ,000 in tbe southern bemispbere .

MrWannasron, in 1784 , published a catalogue , in

which the stars are arranged accord ing to their

right ascension and declination .

The difl'

erent volumes of the Connoi'

umwe dc: Tam,

contain a catalogue of more than fixed

La L anna and his nephew observed , in the space of

twelve years, fixed stars, and determined

the ir place s exac tly . Mém. dc (”

Academic des Satan

ccs , Paris 1789. 1790 . See Mos ruc ns, vol. xv.

p. &c .

a Some catalogues, though onot very numerous, are

highly valued for the ir accuracv, such as that of Ls

C amus, contain ing 515 stars : of Hu nt e r , con .

taining 389, Naut . A lmanack, 1773 ; of Mar e n ,

B 4. containing

21 oun m s s or NA TURA L en rrosorn r .

containing 998 ; of M a s c ara s , containing only

36 ; but the places determined w ith sueh exactness,

as to render it extremely valuable . See his Astro

nomical Observations, 1770 .

Bon s’

s Atlas, Berlin , 1797, is the largest catalogue ,

reduced into one body, that bas appeared ; it eon

tains

J. The number of the stars, how ever, visible w ith

the telescope , far exceeds whatever can be reckon

ed . The bright tract in the heavens, called the

M ilkyWay , seems to ow e its w hite appearance to

an incredible multitude of stars, which the eye can

not distinguish. Dr Han scum. has seen

stars pass through the field of his telescope m a

quarter of an hour ; though the field w as not more

than 15’ in diameter .

39 . The most obvious d istinc tion among the

stars, is founded on the ir d ifl'

erent magnitudes.

T hose of the first magn itude , are distinguiihcd by

part icular names ; the re are only ten visible in

Europe , w hich all astronomers have agreed to

belong to that class.

a . The stars visible to the naked eye, are divided, inall, in to six claw es, not very accurately separated

from one another . The an cients counted 15of the

ASTRONOMY. 86

Dr HALLEY, Phil. Tram . N° 364. Also Vine s ,

vol..r. p. 505:

b. If c be the class reckoned from the first, 13 x c“is

nearly the number of stars in that class .

33. The fixed stars are not scattered over the

h eavens indisc riminately , but are disposed in

t'

o'w hich , from

'

the most remote an t iqui

given fromce rtain figure s of

an imals, conce ived to be connec ted w ith them,

w hich are called Constella tions .

a . n the beginning of astronomical sc ience , it was

o

I

nly by such a device as this that men could speak

pf the stars, or describe them to one another . It

is a remain of the ancient picture w riting, that pre

ceded alphabetical language .

b. The number ot’

the ancien t constellations was fi ;

24 M in the

southern hemisphere , and 10 composed out of

groups, not included in the ancient m gemen t.

The stari hf each constellation are distinguished

by the Greek letters , disposed inreferenee to their

magnRude and positionfl

For the method’

of distinguishing the constellations,see L s Lan na , tom. 1 . 8m.

34 . Many of the stars, w hich, to the

'

naked eye , or

fibroug h telescopes of small pow e r, appear single ,

are

26 our t ms s or warm s rnxtosornr .

are found , w ith highe r magnifiers, to consist of

tw o, some time s three or more Stars, ex treme ly

near-

to one anothe r.

a . Dr m asca ra has observe'

d no less t han

thm multiple stam of which on ly 42

before . In some of thsm, the small

very unequal ; the largest a reddish

smallest a sky- blue, inclining to green .

669 .

85. The fixed stars are not entire ly exemptf ram

change several stars w hich are men tioned by the

ancien t astronome rs having now ceased td be vi

sible , and some be ing now visible to the naked

eye , w hich are not in the anc ient catalogues.

M andin lm Vi nca, §704 .

b. A star of the same kind w as seen by K e n na in

1604 ; and several similar facts are recorded .

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28 ob i -

t i ne s or NATURA L en i t osornr .

and of:an irregular shape , are discovered ; and on

are resolved into a multitude of small stars, dis

tinct ly separate , but extreme ly near to one ano

a . The fl i'

lkyWay is a space of this kind,visible to

the naked eye , and encompassing the w hole he a .

vens . Dr HEs sc n e n’

s telescope discovers it to con

sist of a vast multitude of stars, 81 .

Other two nab-Le n ear the South Pole , distinguish

able by the naked eye, are called by sailors the

to be eommsed of ytam

b. The other nebuhz are not visible but with telescopes . Hovonns discovered one in Orion

’

s Sword ,

that appeared a bright spot on a dark ground, andseemed like an opening into some brighter

gion .

9. Several more usbs lc had been observed before Dr

He nson“ , by Hu an g , CAssi i i i , LA Cum , &e .

and a catalogue of103 w as published by the French

astronomers, in .

the Connoissance des Ten s for

1783. Dr Hznsc nxi. has given a catalogue of

Trans . 1786, 1789, &c .

asc ns i. has also d iscovered nebulous

single stars, surrounded by a faintluminoes

a sr nortomr . £9

may arise eigfigr

the body in that di

o s»w iry so the appearance of

of the heaven ly bodies, round

the earth, may e ither be produced by the real t e

the rotat ion at the earth, on its axis, from w e st to

40 . It is no objec tion to the supposition , that

of his motion . The motion which any

w ith respec t to those bodies, and all“

fig; one take place just as if the mo

tion common to them all had no existence .

80 our t ms s or . NATURA L r n i i osor n r .

4 1 . The reality of the diurnal revolut

heavens, is liable toa great objection , as

that a circular motion, in the same di

common to an immense number

mm, and entirely de tached from one,

that this motion is so regulated , that

tim e are all pe rformed in the same tim

plane s paralle l to one another .

The revolution of a detached body about a

about an axis, cann ot take place w ithoconstan tly, acting, to draw it out of

line in which it has at er ery instan t a

continue its motipa, The revolution

body, like them on MM mas ses? fromone original impulse ; ‘

its con tinuance remain s no

new action , but is a consequence of theWmatter .

It w as to obviate the difficulty arising

tached and distant situation of the’

bod

one anothe r , that the hypothesis of

orbs w as inven ted . To those w ho do

in the se orbs, the diurnal revolution hf the

can have no probability.

asr aasiom r. 3 1

42. The physical and meehfliical obje etioas to

the rotation of the earth being en tirely obviated,while they press so hard upon the

o

opposite hypo

thesis ; the simplicity of the uplanations afl'

orded

hy the forme ry jm ifin us in admitting it as the

cause of the apparen t diurnal revolution of the

heavens, at least till some fac t , or some princ iple

inconsistent w ith it is discovered.

Sizer . II.

O F“M E ATM OSPUERKZA L REPRACTKOR .

43; A’

new of light, t in passing through t he at

motphere, inh e ri t intoml curve , in t he same ve rt i

eai planeWith the original ray, and concave to

ward the‘

Surface“

of the earth. But the objec t

[mmw hich the ray come s, is m in the direc t ion

which the ray has w hen it ente rs the eye , and

appears elevated above its t rue p lace .

and some t imes

the AstronomicalM a caw .

a . The d irect of the atmospheri c reflection alters the

place of an object only i n a vertical plane ; it ih

“98 888

32 ou'

r tms s or star dus t eni t osor nr .

creases the altitude, but does not affect the azi

muth .

6. Hence all the altitudes, measured as in the prece

ding sections; require corrections to be applied to

them before the true altitudes are obtained .

The method ofmaking these‘

correctiont isnow to be

4 4 . From the princ iples of opt ic s, it is know n ,

that the rays which pass through the strata of the

atmosphe re at right angles, or wh ich come from

stars in the zenith, su'

fi'

e r no refrac tion , and that,

at all other elevat ions, the quan t ity of the refrac

tion is nearly as the tangen t of the zen ith dis

tance .

a . If x be the true distance of a star from the zenith,

and y the M on , so that the apparent distance

is x -

y, then, by the nature of refraction, sin 3:is

to sin (3: y) in a constant ratio, suppose that of

m to n ; and since sin (x— y) sin x . . cosy

sin x . cosy- cos x . siny,

cos t” 0 e

“

sinm ‘v' Novv if y be very

small, cosy I nearly ; and therefore siny

22: tan s . As sin y z y

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36 ourmn s s or NA TURA L PHILOSOPHY.

a . Dr BRADLEY has given a formula,

Let 6 : height of the mercury in the

inches, 1: height of Faua znn zrr’

a

s : the zenith distance , r : the m

computed by the rule , the

6 7°

expression for the density or specifi

and In that case, there

49 . Not only are the stars elevated h

tion , but all te rrestrial objects

the same cause , by an angle

the straight line draw n from the eye to

makes w ith the tange nt to the pa th of the ray at

the point w he re it en te rs the eye .

The refraction of the heavenly and t

is difl'

eren tly estimated . That of th

is the angle con tained be tw een the

en

by the atmosphere ,

curve w hen it enters t

ASTRONOMY. 37

tangent at the eye and the chord of the arch inter

cepted betw een the Object and the eye .

Near the earth’s surface; the curvilineal path of the

ray of light may be supposed nearly to comcxde

w ith the circle of equal curvature .

50 . If the e levat ion of the top of a moun tain

from a poin t in the plane below ,and the depre s

s ionof that point from the top of the mountain, be

b oth obse rved at the same time , the angle sub tend

ed at the earth’s cen tre , by the d istanée be tw een

them,added to the obse rved e levation , and the

sum d imin ished by the depression ,is double of

the refrac t ion .

This supposes the path of a ray of light, for a small

part, to coincide w ith a circle . If, in fig. the

arch from B to A be the path of a ray, and if AHand BFbe perpendicular to A C , BC , in A and B

the tangent EA , EB being draw n to the path of the

ray, HAE is the apparen t elevation of B from A ,

and FBE the apparent depression of A from B ;

the true elevation being HA B, and the true de

pression FBA . It is evident, that FBA

BAH AGB, that is, true Elev. true Dep .

Hor . ang. But true Elev. z app . Elev. R efi ;

and trueDep. app . Dep. R ef. therefore 2 Ref.

Hor. ang. app . Elev. app . Dep.

51 . The

88 oue s s or NATURA L PHILOSOPHY.

51 . The o terrestrial refrac tion found by means

of the preced ing theorem, w hen the elevation is

not very great, varies from i of the angle

subtended by the horizontal distanc e of the Oh

jec ts ; and the radius of curvature of the ray,

the refore , varies from tw ice to tw e lve times the

radius of the earth .

In the mean state (if the atmosphere , the refraction is

about of the horizontal angle , and the radius of

curvature of the ray seven times the radius of the

earth.

The terrestrial refraction must vary w ith the densityof the air , that i s, w ith the barometer and thermo

meter. The great differences, how ever, remarkedin this proposition , must be ow ing to some other

cause .

In the measurement of heights, the angle of elevation

should b e diminished by one - fourteen th of the angle

corresponding to the horizontal distance , Sappo

sing the refraction to be of the mean quanti ty.

52. The effect of refrac tion may also be allow ed

for, by comput ing the corre c tion of curvature , as

in 24 8 . vol. I and taking one - seven th of it,for

the number of feet, by w hich the objec t is ren

de red

as t aonomr . 39

dared by the refrac tion higher than it ought to

be.

“L is the length of the horizontal line in English

2

miles, the correction fpr curvature in feet is

and for refraction

In the measurement Of heights i t must be observed,

that the one of these corrections is Opposed to the

other.

In‘

the next section , the method of estimating the

angle subtended by the horizontal distance w ill be

explained . The subject of terrestrial refraction

require s to be farther investigated by observations

of elevations and depressions made at the same

tith e, and w ith a reference to the states of the

barometer and thermometer.

Secr . III.

FIGURE OF THE EARTH.

53. .Tn e figure of the earth is understood to be

determined by a surface at eve ry point perpend i

C 4 cular

40 OUTLINES or NA TURA L PHILOSOPHY.

cular to the direction Of gravity, or to the direc

t ion of the plumb- line

, 24 . b).

This. surface is the same that the on would havé if it

w ere continued all round the earth ; or, if w e w ere

to trace curve lines, by levelling from a given poin t

round the earth, in every direction , till they

turned in to themselves, the superficies in which all

these lines w ould lie, is that w hich w e consider as

the superficies of the earth. The given point may

be supposed any one, on the level of the sea.

Q .

The figure'

bounded by this superficies, is that w hich

is really measured by‘the combined me thods of

astronomy and practical geometry, and is to be

carefully distinguished from the actual figureof the

earth, including all its inequalities ; or from an ave.

rage figure that should leave out as much of Solidma tter above it, as is included of empty space un .

der it .

541. The length of an arch Of the'

me ridian, tra

c ed on the supe rfic ies above defined, may be mea

sured by Obse rving the latitude of the tw o ex tre

m it ies Of the arc , and then measuring the distance

be tw een the se poin ts in fathoms, toises, or any

othe r know n measure .

The distance, as measured on the surface, divided bythe degrees, and parts of a degree contained in the

difference

AS’

ERONOMN. 4 1

diflgmnce ofwe hmm w giveme kngmof a

Eas t-Osm an“ was the who

A

si a

-

l, this me thod

to the estimation of the s circumference . By

uded the ch cumference Of

. H 1 1 stadia . MON‘

I’

UCLA ,Hislm'

rc

tom. 1 . p. i t? Qde ed it .

the distance Of the arch requhe d, and nmrly north

and south of one another. A series of triangles is

then to he carried poin t to the other

by means of stations taken on the taps of hills or

other elevated

and the bearings of the sidhsrm b of the ma idian d the

given .

or the sides or the .. gles in know nmeasures, or fathoms, is next found by mea .

suring a hue on a level ground, and connecting it

M anglesw ith 4 he sides Of one of the triangla .

In all t we proceed as if the triangles were plane ,whereas they are in fact spherical, and the three

“M

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4-4 OUT L INES or NA TURA L PHILOSOPHY.

Let Ann be one half of the meridian , A and B

points in the Equator, C the centre of the earth, D

the Pole , EA a perpendicular to the meridian at

E, a poin t of which the latitude is A : EGA , Hthe centre of curvature , F a perpendicular on the

axis ; then EG is the normal, or H, GF the sub

normal S ; let CF x, and FE z y ; then

x2

) But, by the property of the

subtangents of the ellipses,

s n cos A, and y : n sin A ; therefore , by anbatiq

tution,

a“(b

2sin ’

A2

a2cos A

2

) 64 ,

62

(a2cos A

2sin A

and n

2

Now , by §56. r n ’ therefore,

a"b2

(a2cos A

”sin A

Z

);0

If D be the length of a degree in lat . A, andm the number Of degrees in an

are equal to the radius, then r mD,

58. In,

ASTRONOMY: 45

58. In an ellipse w he re the eccentric ity is

small, or w here a and b d iffe r but by a small

quantity 0, this general formula may be reduced

tomore simplic ity, by extracting the root of the

denominator, and rejec ting the pow e rs of 6 greater than the first we have then

' mD = a (l - i —c

—c

i

a a

t . This value Ofm.

Dmay be changed into another, moreconvenient in calculation , by substituting for sin a?2

l—cos 2 7.

2from which is obtained

6. At the Equator, A 0, and cos 2 A : I ; so that

mD= a (l a — 2 c .

13. At the Pole, x= 90°, 2 A= 180

°§ and since

cos — l , mD = a + c.

The degree of the meridianat the equator, is there

fore to the degree at the pole as a 2 c to

a + c.

d. In the parallel of 2 A=90°, and c0 32 A=O

46 OUT L INES or NATURA L PHIL OSOPHY.

The radius of curvature at the parallel of or

m D, is nearly an arithme tical mean between the

radius of the equator and half the earth’

s axis.

The degree in the parallel of 4-5is also an arithme ti

cal mean betw een any two degrees equally distan t

from it on the north and on the south.

e. The degree in any latitude is understood to be that

ofw hich the middle point is in that latitude .

If, therefore , D be found by actual measurement in

any two known latitudes, w e shall have tw o equa

tions, in w hich a and c are the only unknow n

quantities, and from which, therefore , theymay be

determined .

59. The lengths of tw o degrees, of w hich the

middle poin ts are in given latitude s, be ing know n

it,is required from thence

O

td de termine the diame

ter of the equator, and the axis of the earth, that.

is,the longe r and the shorter axis Of the ellipt ic

me ridian .

a. L et D and D’

be the given degrees, (the least , or thatnearest the equator, being D), A and A

’ ‘

the lati

tude s of their middle points, a the semitransverse

ax is of the meridian , c the difference of the semi .

axis ; w e have the equations,

4 8

c.

OUT L INES OF NA TURA L PHIL OSOPHYo

degree , in any latitude , above the degree at the

equator, w hen divided by the square of the sine of

the latitude , should always give the same quotien t:

or the excess of the degrees of the ‘

meridian above

the degree at t he equator, should be as the squares

of the sine'

s of the latitudes.

Since 0

m—r— v

(D, D)

3 sin (A’

A) x sin (A A)’

D’ D sin (A’

A) x sin (A’

If, then, D'

and D are two contiguous degrees, so

that — D sm (2 A + 10

) x sin

and since sin 1 ° 01746, D’

- D

3 c x . 01745sin (2 A

The contiguous degrees, therefore , differ by a quan

tity proportional to the sine of tw ice the middle

latitude The difference is a maximum, w hen

2 A 1 or when the middle latitude is

450

Asrson omr . A9

The quantity5; is called the C ompressicn , and deter

.

mines the spee ies of the'

ellipsh .

d.We shall now take for the determination of the fi

gure and magnitud e of the earth, the five arches

subjoined , as those that have been measured w ith

the greatest care, and the best in strumen ts ; as

being the largest also, and the most distant from

one another .

fathoms .

I . 0° 56749 60480 2 Peru.

II . 1 1 56755 Ind ia.

III . 46 570 1 1 607594 France .

IV. 52 57074 608256 England .

V . 66 57 192 60952 4 Lapland .

As five quan tities may be combined, two and two, in

ten differen t w ays, ten results may be deduced from

the comparison of these degrees ; and if the merid ian

were truly elliptical, and if there w ere no error in

the observations, all these results w ould coinc ide .

As the latter supposition cann ot be expected to hold,

we must look for some d ifference in the results,

and must choose only those combinations, in which

the degrees are considerably distan t from one ano

ther, because in tha t w ay the errors Of observat ion

will least affect the conclusions . Such are the lat

and 311, l st and 4 th, l st an d 5th ; 2d and 3d , 2d

and 4 th, 2d and 5th ; 3d and 5th.

V00

1“ II. c. The

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52 our us s s or NATURA L rm sosornr .

Sw a nse a. Stockholm, 1805.

k. Of all these degrees, it may be said,

of 30 tomes m the length, or 2” in the

the arch, is more than

If the hypothesis of an

them nearer than this

dered as having the su

if it is found that these arches cannot he

w ith the elliptic hypothesis, w ithout

greater errors than those just mentioned

pothesis must be either rejeete d, or

60 . The five degrees in the last artic

in giving ve ry n early the same compress

earth at the poles, and may all be repre

the same equat ion, to an exac tness

the limits that have been assigned (559

a. By combining the degrees in the seven

tioned , it w ill be found, that 2is betw een s ome

and 9 0325.

5. The men be tween these, or £ 032, is that which, on‘

the whole, seems the nearest to the truth . Itmakesthe sum of all the errors in the five degree s amoun t

only to 21 toises, taking them w ith the same sign .

Taking them with their proper signs, they nearly

destroy

ASTRONOMY. 53

destroyone another . The compressionmay therefore

he stated at .0032 and the equation

which doe s most nearly represent the degrees of

the meridian, w ill from'

thence come out

57ol 1t cos2 A

In fathoms,

D 607594 72 cos2 A

In mile s, D 3 299 cos 2 A.

2 .Hence, by the formulas, 59. b.)

Toises. Fathom Miles.

c 1046958 1 1 158 8

q 3271743 00 assesse s 3962 349

1. 3861273 42 34 75700 .

Radius of curvature for the parallel of

0

45° a 3266508 2 1

miles . The miles meant here are English miles.

d . The circumference of the Elliptic Meridian may be

found nearly by multiplying the mean degree, or

that in the parallel of by360 . The result is

miles.

54 ‘ oun ms s os -NATURAL PHILOSOPHY.

The circumference:of the Equator is 24806 16“

miles,

a little more than 4-0 miles greater than the prece

din’

gJi

The ‘

ch'

cu'

nfa enoe :of the Meridian may be found

more t ecumtely hy‘the thdomm for the M tifica

tion of the ellipsis . See LA Camus, Lego” E le

mentaz'

rcs dc Math. 5954 . A lso Base Mctn'

t ,

tom. 11 . p. 676.

The Frendhffromtheir late measurement, comparedw ith that in Perl , make the compression 0 0324 ,and the quadrant of the meridian 51311 1 1 toises

w hich gives for the entire c ircumference546848 154fathoms, or 248561 2 miles ; about 1 mile greater

than the result ob tained above ,

e . TheGeographicalMile, or that of w hich there are

60 in the length of the mean degree, is fa

tliliihs,’

53560756 feet.

61 . The scmidiama er belonging to any lat i.

tude A, is nearly equal to a (l n sin2

A), w he re

n denotes or the compression ,’

as before, (g58.

a. This is found, by expressing the“

semidim ter of

the elliptic meridian, in terms of the latitude , in a

manner

56 c ou rs e s or NATURA L PHILOSOPHY.

that poin t, the centre of curvature of this sec tion ,

at the point where it cuts the meridian , is the

point in which the direc tion of gravity, or of the

plumb- line , intersec ts the axis of the earth.

a . The direction of gravity, if the earth he a solid of

revolution , passes alw ays through the axis of the

earth. If, therefore , w e conceive the plumb- line

to be carried over an indefin itely small arch of the

perpendicular:

to the meridian , either to the east or

w est, its direct ion w ill intersect the axis at the

same point w here it intersected it before , w hich

point, therefore, ‘is the centre of curvature of the

arch, or the same,

w ith K (fig. BK is greater

than EH, nnd~the dpgree ofthe perpendicular arch

is greater,than the degree of the meridian in the

same ratio

The radius of cury ature‘

of the a rch perpendicular

to the meridian, is'therefore the normal of the me .

ridian , relatively to its shorter axis, or it is

a‘l

63. If D be the degree of the meridian, at a

point of which the lat itude is A, and A the degre e

of the curve perpendicular to the meridian at'

the

same poin t,

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66 oun m s s or -na‘

rm t PHILOSOPHY.

sw im ,w henare '

given in meson w ithre‘%peé€M isfisnperficfies.

t ] L ll "Mla v lh

'

rhati thefletevhlinations thns aflbrdeflm r be the

l n m lest pouiblw the two planes ought toh e at

m m am a and if t he tmpum..aiais @ne hn ing a m m the planes ahonu pm

case hf the earth, the plane of the Equatortion fixed by the diurnal motion , is

ut as one of t

ton fi determinfl by finding its M-tnde m above

it"cs The other circle to which the ipositidnof places on

the earth s surface is to be referred, must necessa

rily be a Meridian (a) ; but as none of the meri

dians i s d ist inguished from another , by any circum

stance ia the diurnal ,motion , of w hich they all

partake alike , the particular meridian that is to befixed on for the de termination of geographical poo

sitie na, is a matter of arbitrary arrangement.

d,When a meridian is chosen for a first meridian , or

that to w hich all positions are to be referred , it is

net by directly measuring the distance from it that

such a reference is made, bat by measuring the

angle which the plane of the meridian passing

asr aonom‘

r . 6 1

through anygiven place, makes with theaof

the first‘

m n . This angle'

is ‘called a..

gitudc of the place , and the diurnal motion fi r

nhhes us w it h the means of de te rlnining i t. It '

is measnred by the arch p f the equator, inter .

Wham the first meridian and the me

ridian of theuplace, and is reckoned eastm west,

according as the place is east or w est of the first

'The ancients took for the ir first meridian, or that

from which.their. longitude w as counted, th

than of the JFortunate Isles, a line passing, as they

conce ived, through the w estern extremity of the ha

bitable earth. Many of the moderns have employedthe same meridian , or rather that of the Island of

Farm, one of the most westerly of the Canariea. In

general, however, nations employ the meridian of

their own metropolis, or of the ir principal observa

tory ; as w e do that ofGreenw ich, the French that.

It has been proposed to tfi e the

that of' a point very remarkable in the natural history of the globe. It would be inconvenient to take

for a first mer idian any poin t where astronomicalobservations are not constantly made .

65. The hour, as reckoned unde r any two me

”

Tid im s,is d iffe ren t, and the d iffe rence is propor

fina l to the difiemnce of longitude , or the angle

w hich

64 our t m s s or NATURA L rn rt osor n r .

tioned . all the methods of finding the lopgitude may

be reduced .

l

68. If the latitude s of any two places are given ,

and also the ir difl'

e rence ot'

longitude , th’

e ir dis

tance may be found by sphe rical trigonomet ry.

a . It'

the earth is con sidered as a sphere ; then , in the

spherical triangle con tained by the arches joiningthe two places w ith one another, and with the pole ,

tw o sides are given , viz . the distances from the pole ,

or the complements of latitude, and the angle at the

pole , or the d ifferenc e of longitude ; and there

fore the 3d side may be found by the 2d case of

oblique- angled spherical triangles . This side is the

(fistance of the places expressed in degreea, &c. ;

and may be turned into miles, by multiplying bythe mean length of a degree , 60. b.)

If the angles at the base or the azimuths are also re

quired, it w ill be best to resolve the triangle by

Nar i an’

a Formula . See E lm . of Gen “. Edin .

1810, p. 378. See alsoWoonuoosa’

s Trigonome

try , p. 196.

6. But if the spheroidal figure is to be taken - into ac

coun t, the calculation becomes more complex . For

as , on this supposition , the direc tions of the plum

mets AD, BF, (fig at the tw o places, if the ir

latitudes are difi'

erent, do not meet the axis in the

same point, these three lines do not contain a solid

ASTaoNo . 66

angle , and therefore the rules of trigonometry can'

not be directly applied to them . If, how ever, C

be the.

cen tre of the spheroid , and if A C and BC

be joined, the angles PCA , PC B, are deduced from

the latitudes, 6 1 . 6. Then , in the solid angle at

C , are given the two plane angles PUA , PCB, and

the inclination of their planes, viz . the difference

of longitude , or the angle at P ; therefore the angle

ACB may be found by the same case of spherical

triangles as before . He nce the straight line

A B is also found, the radii CA , CB being given,

61 .

5. In this w ay also, are found the angles at the base

of the triangle PAB, or those w hich the plane

A CB makes w ith the planes AOP, BCP. These ;

how ever, are not the true azimuths, w hich are the‘

angles that the plane ADB makes w ith ADP, andthat ABE makes w ith PEB

To find these last if DB be draw n , then in the tri

angle BCD, BC , CD, and the angle BCD are gi

ven , w hence DB is found . Then in the triangle

ADB,all the three sides are given w herefore the

angle ADB may be found . Next, in the triangle

BED,the sides BE, ED, DB are given ; therefore

the angle EDB, and its supplement FDH are

found . Therefore the three plan'

e angles ADP,

ADB, FDH, w hich con tain the solid angle at D,

are given ; w hence the inclination of the planes

may be found , and therefore the angle w hich the

plane PAD makes w ith the plane ADB, that is,

Von. II. E the

66 OUTL INES or NATURA L p n rrosorur .

the angle PAB, such as it w ould be.

measured at

A . In the same w ay the azimuth at B may befound.

d .When these calculations are applied in small tri

angles, they naturally become much more simple .

The process now described , contains a general ao

lation of spheroidal triangles, w hich have one angle

at the pole , w hatever he the oblateness of the sphe

roid, and whatever he the magnitude of the tri

angles.

69. The A rtific ial Globe is a delineation of the

surface of the earth, and the c irc les belonging to

it, on the surface of a sphere , moveable about an

axis ; it se rves to give a correc t notion of the fi

gure and proportion of the parts into wh ich the

earth’s surface is e ither naturally or artific ially

d ivided , as w e ll as to resolve many of the pro

hlems of geography, w hen great accuracy is not

required .

A contrivance of the same kind is applied to the heaven s . The uses of the celestial and terrestrial globes

are“

fully explained in most of the treatises on

A stronomy and Geography.

70 . A Map is a representat ion of the w hole , or

of a part of the earth ’s surface on a plane ; and

though

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68 OUTLINES or NATURA L PHI LOSOPHY.

The eye is then supposed placed in the Opposite

pole, and the other hemisphere is in like mannerrepresented . It is in this w ay that the Maps of theWorld are usually constructed .

b. The stereographic projection has these two very

remarkable properties.

1 . All the circles of the sphere , both great and

small, are represented by circles in this pro

jection .

2. A ny twocircles cut oneanother in the projection , at the same angle in which they cut one

another on the surface of the sphere . A c

cordingly, the parallels of latitude 1n this pro

jection cut the meridians at right angles.

These properties contribute much to the simplicityand beauty of the construction , which, how ever,has this disadvantage , that the same drea on the

earth’s surface, is represented by a much larger

area near the equator, and especially tow ards the‘

edges of the projection , than at a greater distance .

Notw ithstanding of this, the stereographic projection is w ell adapted to Map s of theWorld, or oflarge portions of the globe .

72 . The construc tion called FLAMSTEnn ’s Pro

jection , (though it is rather a Developemen t than

a Projection), is very w ell contrived for the rep t e

scutation

A sr aouomr . 69

aentation of smalle r portions of the earth’s sur

face .

a . In this construction , a straight line is draw n for

the meridian of the middle of the map, on w hich

aremarked off equal distances, to denote degrees

of latitude . From a poin t in this lin e , as a centre ,

and w ith a radius that is to the length of the de

gree of latitude as the cotangen t of the middle lati

tude to an arch of 1 degree , an arch of a circle is

de scribed , to represen t the middle parallel of lati

tude . From the same cen tre are described other

arches, through the different points marked off on

the merid ian of the.

middle of the map, w hich re

present the different paralle ls of latitude . Oh any

one of these parallels, equal distances are se t off on

each side of the middle poin t, w hich are to the as

sumed degree of latitude , as the cosine of the lati

tude of that parallel to the radius. The degrees

of longitude are thus marked on each parallel, and

the curves w hich pass through the corresponding

points in the different parallels are meridians.

These are curved more and more on re tiring

from the middle of the map ; but unless the extent

is very great, they afford a very good replt sen ta.

firmof the convex surface .

b. This construction has a very remarkable property,viz. that the quadrilaterals in the map, included be

tw een the meridians and parallels of latitude , have

the same ratio to one another nearly, w ith the qua:

drilaterals which they represent on the surface of

E 3 the

70 OUT L INES or NA TURAL PHILOSOPHY.

the globe . See Mémoire sur la Projection dc:

Cortes Geog raphz'

ques , par M . HENRY, 4 to, Paris,1810, chap . 3m , p . Ste .

73. The construc t ion w h ich is called MERCA

TCR’S Proj ec t 1ou, is chiefly used for nautical

charts. In it the me ridian s are paralle l line s ; the

degree s of longitude are all equal ; the parallels of

lat itude are also paralle l line s and the d egre e s of

latitude increase on the chart in the same rat io that

the degrees of longitude diminish in the sphe re ,

or in the spheroid.

a . The consequence ofmaking the degrees of latitudeincrease in this chart, in the manner descr ibed, is,

that the degrees of latitude and longitude bear to

one another the same ratio that they actually do on .

the surface of the earth ; and as the meridians are

all parallel, the rhumb- lines, or the lines of azi

muth, are straight lines . Hence the great use ofthis construction in navigation . 1

b. This very ingenious contrivance is alluded to,

though obscurely, by P'rou mr . It w as first used

among the moderns by Mna cxroa , whose nam e it

bears ; but the principle of it w as first explained byED .Wa te n 'r in 1599, who shew ed that the parts

into w hich the meridian is divided , must be ia

versely as the cosines, or directly as the secants (if

the latitude ; and he taught how it might be con

structed

a sr aouoMr . 7 1

structed by the addition of the secants of a series

of arches taken in arithmetical progression .

It w as afterw ards observed, that the meridian line

thus divided , w as analogous to a scale of loga

rithmic tangents of the half- complements of the

latitudes ; this w as at first only know n as a fact,

but w as afterw ards demon strated by Ju l e s Gus:

c oRr , in his Exercitationes Geometrica , 1668. Dr

HAL LEv proved the same in a more concise man

ner, Phil. T rans . N 219 , and greatly improved

the construction of the chart.

For other me thods of constructing maps and charts,

see VARENw s, Sect . iv. chap . 32. Encyclopedic

Met/zodique, art . Cartes . L oR cNA , Princip ii dz'

Geog rqfia , 4 to, Verona, 1789. Traité dc Topo.

g raphic, St e . par L . PUt ssAN'

r, L iv. u. LA GRANGE,

Mém. dc Berlin, 1779.

72 oun mns or NATURAL PHILOSOPHY.

Sscr . V .

or PA RA LLAX ES.

HAvma obtained an accurate notion of the figure

of the earth, w e are enabled to measure the lines

either on its surface or in its interior, w hich must

serve as the bases from w hich, by the rules of tri

gonometry, w e are to deduce the distances of oh!

jects observed in the heavens. Though the fixed

stars are too far off, to have the ir distances thus

ascertained, there may be others, of which the dis

tances admit of be ing compared w ith the diame ter

of the earth. A ll distances that are not ascertainedby the direct application of a measuring line , are

determined on the same general principle ; that

is, from the change in the ir angular position , w hichis made by a know n change in the position of the

observer. This leads us to consider what is called

the Parallax of an Obj ect.

74 . The parallax of any objec t in the heavens,is the diffe rence of its angular position, as it w ould

be

11511 10 110 11 1 . 313

b e seen from the centre of the earth, and as it is

seen from a poin t on the surface .

The parallax of an object, is therefore the same

w ith the angle which the distance betw een the

centre and a given point on the surface subtends

at the object .

Though an object to have no parallax , ought, strict

ly speaking, to be at an infinite distance , yet it w ill

have no sen sible parallax, if its distance‘

19 very

great compared w ith the diameter of the earth . A n

angle of one - fourth of a second may be considered

as insensible ; so that if the radius of the earth

subtend an angle , at the distanceof any object, lessthan one - fourth of a second, that object w ill be seen,from all points of the earth

’

s surface, in the same

position .

Now , an archof 1 is 0 00004848 of the radius ; and

the fourth of a second is therefore

I00000 l212.

825082

and therefore, if a body is distant from the earth

by 825082 of its sem1diameters, i t can have no sen

sible parallax.

Though the centre of the earth i s a point from w hich

no observations can be made , yet as it is equally

related to all the points on the surface, the posi

tions of the heavenly=bodies may bemost conveni

t

OUT LINES OF NATURA L PHILOSOPHY .

ently referred to it . When a star is seen in the

zenith of any place , it is seen in the same position

as if it w ere view ed from the centre .

75. The parallax of a body at a given d istance

from the centre of the earth, is greatest when the

body is seen in the horizon . This is called the

Horizontal Parallax and the parallax at any gi

ven alt itude , or the quan tity by wh ich the true

altitude is dimin ished , is to the horizon tal paral .

lax as the cosine of the altitude to the radius.

If P be the horizontal parallax, p the parallax at the

altitude a, p P it cos a .

If r be the radius of the earth, supposing it spherical,

and a the distance of the body,5: sin P.

When P is very small, P 5, P being expressed,

not in degrees, but in parts of the radius to have it

in degrees, must be multiplied by 111, the num

ber of degrees in an arch equal to the radius .

If the horizontal parallax is known, the distance d

is known ; for d z fi .

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70 c ou rs e s or n am e s ] . rn rrosor n r.

he the horizontal parallax,

ang. ASC = s in ZAS, and

ang. BSC = s in Z’BS ; w hence

prmust be supposed negative, if the star is between

the zeniths .

b. If the star does not

.

change its declination, this

formula w ill give the horizontal parallax , though

the observer be not on the same meridian , because

the meridian altitude of such a star, is the same at

all places in the same latitude .

c . If there is a change of declination, and if the oh

servers are not under the same meri dian , then , byrepeated observation s of the zenith distance at any

of the places, the star‘

s change of declination for a

given in terval of time , may be found ; and so, from

the observed altitude at that place , the altitude at

any other place , under the same parallel, may he

found, if the difference of longitude is !know n .

Hence the altitude is known at the place where the

ASTRONGMY. 77

meridian of the second observer intersects the pa

rallel of the first, and thus the parallax is compu

ted as above .

This method of finding the parallax , was applied to

that of the 'moon , by LA CAILLE, who observed at

the Cape of Good Hope , and L A LANDE , who oh- s

served at Berlin, in 1751 . LA L ANDE, Astronomr'

e,

tom. 11 . 1650 .

77. Tw o obse rve rs being supposed, as in the

last article , the parallax of a star may be found,

by comparing it w ith a fixed star, (w hich has no

parallax ), at the time of its passage over the meri~

dian.

For if S be the object of which the parallax 1s to be

found, as before ; T a fixed star, that passes the

meridian at the same time , or nearly at the same

time w ith S ; the angles TAS, TBS, may be both

measured, and they are together equal to the angle

ASB because AT and BT are parallel.

The angle ASB being thus found, the rest of the cal

culation is as above .

La Cums andWARGI NTEN,‘ the former at the Cape,the latter at Stockholm, employed this method for

finding the parallax ofMars. LA LANDE, tom. 11.

6 1052; Vmcn, vol. 1. 158.

ASTRONOMY. 79

Thus changes are called the parallaxes in right

ascension and dec linat ion .

c ln fig . 8 . ifo he the me ridian , P the pole , Z the

zenith, S the true place of the star , depressed by its

parallax toin the direction of the vertical circle

ZSS’; let PS be a circle of declina tion passing

through the true place of the star, and PS'

anotber

such circle , passing through the apparent place .

If8M be draw n pe rpendiclar to PS’

,MS’ is the pa

rallax in declina tion, and the angle SFM, the pap

rallax in right ascension . The triangle SSM may

be con sidered as rect ilineal ; and if w e make theangle at S, or w hat is called the ang le of position,z y ,

the parallax in altitude , or SS'

, being p ,

we have MS’ : p x cosy : parallax in declina .

tion ; also SM Z‘

p x siny ; and hence the angle

COS

M I pmfi — flmwfllfl ifl n gbt ascen

6. 11ence itmay be shew n , that when the horizontal

As the parallax in right ascension is nothing, when

any body is on the merid ian , and increases on

either side , by compmi ng the difference be

tw een the t ime when the body, and a star near

to it , corne to the 1neridlan, wiflt the difi’erenee

hem the time s w hen they come to the same

hour circle, near the horiaony th parallax in rightascension

BU oun w as or NA TURA L ra rLosoq .

ascension may be found ; and thence the horizon)

tal parallax may be concluded. Vue n’a Astra” ;

my , vol . 1 . &c . See also LA Carat s , A ctroa

nomic, 65l . , &c . ; and LA Lanna, Liv. xx . tom. 34

&c . From the difference betw een the al

t itudes of the body and of the star, as they appear

on the meridian , and as they appear near the hori .

zon , the parallax in altitude may be still more di

rectly concluded .

Ss c '

r . V I.

MOT ION OF THE SUN

80 . THE great orb w hich is the source of light

and heat , and in w hose presence all other lum 1ua

rie s disappear, partakes, w ith the stars, in the d iur

nal morion ; but the time betw e en his passing the

meridian one day, and his passing it the n ext, is

greater than a syderial day, and , at a medium, ex

ceeds it by 8 minutes 56 seconds and a half'

nearly.

The sun , therefore, appears to go eastward amongthe fixed stars every day, by an arch correspond.

ing to or by 59’

8 .3. T his is the mean

rate ; the real motion of the sun is sometimes fas

ter, and sometimes slower, than this quantity.

Asraofioura

the meridian one day, and his passing it the next,

is called a mean sotar day.

whe mh r exceeds the fiderial day bym bfi' SSt ot‘

aiderial time ; and

the lengths are in the ratio of l .0027399 to l .

8 1 . The sun doe s not move eastw ard in the

plane of the equator, but in the plane of a circle

cutt ing it oblique ly in the two opposite poin ts, ai

ready re ferred to, of the V e rnal an d Autumn al

Equinoxes, and mak ing, w ith it, an angle of 23"27

’

30"n early.

mm the sun's motion is an in one pliné, may beshew n by observing his right ascension and decli

n ation, every day at noomand marking it od'

upon

a globe , on which a gre'

at circle , representing the

eqaator, has already bea i described ; or , whi ch “

better , by remarkmgflhat the fim of thfi tight wcension has alw ays the same rat io to the tangent

of the declination . This conld not be, tmless the

plane passing through the sun and the vernal equi

nox , made a constan t angle w ith the equator, such

that the radius had to its tant a'

‘

ratio the

Same with the preceditig.

The circle w hich the sun thus appears to describe in

the heayens , is called ' the Ecliptic, and thr engle

which it mahes with the m m , is ealled the abs:

Von. II. F

OUTL INES OF NA TURA L PHILOSOPHY.

My qf lk flcltplfc. A Year is the time which the

89 . During the apparent d iurnal revolution of‘

the ec lipt ic doe s not re

ma like the equator, so as to,

be alw ays vertical to the same poin t on the earth’s

surface . The plac es, how eve r, ove r w hich it

passes, and through the zeniths of w hich the sun

must pass in his annual course , are all contained

w ithin a zone , extending about 28° 28’on each

side of the equator.

a . This tract, which is called the Torr id Zone, is

88 . Each pole of the ecliptic is d istant from the

eor'

reSpond ing pole of the equator, by an arch

equal to the obliquity of the eclipt ic and in the

diurnal revolut ion , they de scribe the c ircles ’alled

Circla , and thc spaces within them m the Fr1’

g-id

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86 oun m as or NATURA L PHILOSOPHY.

tween the time and the declination, the letter.must

be supposed a maximum, or its fluxion must be

made equal to nothing. Thence the time of the

solstice, and also the declination itself, when a ma

xinmm, may both be found.

86 . The length of the year is de te rmined, by com

paring toge ther the t ime of the sun’s be ing in e ither

equinox , or in e ithe r tmpic , for one year, (found

as above ), w ith the t ime of his be ing in the same

point for anothe r ye ar, d istant from the former by

a conside rable number of years. The in terval

reckoned in days, and parts of a day, d ivided by

the number of years, gives the true length of the

year.

a . It w as soon found , that the year w as nearly equal

to 865days. But when tw o equinoxes, at the dis

tance of 60 years w ere observed , the inte rval w as

found to be nearly 21900 days ; this, d ivided by

60, gives which is nearly the length of the

year .

6. This, how ever, is too great ; and more accurate

observation has found it to be 365d .24-2264 or

365‘15h 48' Btor , Astron. vol. 11 . 40 .

2d edit.

88 OUTLINES or NM RA L e n t t osorur.

G - m ’ “ 8m m” 0 '

sin Dec . C

the R ight A scemitm.

respect of the fixed stars, to rec ede tow ards the

w est, at the rate of nearlyper annals , or about

1° in 72 years.

This motion is called the Procession of the EM

mers, the middle off the constellation A ries was at

the vernal equinox, from which it is now distant

ASTRONOMY . 89

more than 58° towards the east. t ancuoe

discovered the precession of the equinoxes, by e

comparison of his ow n w ith more ancient M ex-

va

tions .

90 . On accoun t of the precession of the equi

noxe s, the tropical year, or the time in which the

sun move s from the ve rnal equinox to the vernal

equinox again ,is le ss than the t ime in which he

move s from one star to the same star again ; the

ve rnal equinox having gone westward so as to

me et the sun .

T he time inwhich the sun goes fi'

om one fixed star

time i hat the sun takes to move overWi of hilorbit. This amounts to 0 14-1 19 of a day, or

so so that the siderial year is sosd 65

9 1 . The Obliquity Of the Ec liptic is also subject

to c hange , and appe ars to have been constantly di

m in ishing from the remotest date of ast ronomical

ob se rvation :its present rate of d iminut ion i s near

ly 50”in a cen tury .

A Chinese Ohservation Of the sun’s altitude at the

solstieu, as encient as the year 1 160 before 0hrist,

has

90 OUTL INES OF NA TURAL PHILOSOPHY.

has been preserved ; and from it LA Pu cs deduc

the obliquity at that time , 23° 54’ Comm it

sauce tics Tests, p. 4-32.

A series of observations, from the age of Pi t -run s,

dow n to the present time , confirms the same gene

rel result. LA Pu c s , ibid . LA Lu ng , Ad m .

2738 Vi nc e , Astra . r. 151 .

The mean obliquity for 1750 , w as determined very

exactly by LA Carm a and BRADLEY, to be 23° 28

'

19”

The obliquity, beside this progressive diminution , is

subject to slight periodic irregularities, w hich are

not here considered . The d iminution itself,

though apparently progressive , w ill be found after

w ards to be really periodical, and a part of a slow

vibration , by w hich the obliquity of the ecliptic

alternately increases and diminishes w ithin very

narrow limits.

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a sun’

s diame ter, measured

w ith h is place Observed in the

*t i, that it

“

the sun’s mean appa

m, his least diameter 113— 12, and

“s tanc e at any t ime from th e poin t

n he re his diame ter is least, his apat that time is m— n cos

m 32° 06' 2 and m— n : 31° 32' so that

n z z and or as 5945

97 . Because the distance Of the sun and earth

must be inve rsely as the apparent diameter of the

sun , therefore , if the distance be called y , y

Bwhere B is a constant quantity, to bem n cos

d e termined by Observation . Hence it can be show n,that the orbit Of

'

the sun isan e llipsis, having the

e arth in its focus.

BFrom the equation y

_m - n cos s

it is evident,

that y is greatest, when z= 0, and least when

Iftherefore , FA be taken (fig. equal to

B Band PP

m n’ they w rll represent the

greate st

94‘

OU'

rt rns s or NATURA L rn rt osornr .

he the orbit ot'

the sun ; the eartb being mwomat F, the place of the sumat

'

any time

the angle AFG being z z ; draw GHto AP ; then my

—n).

Now , y z FG, and y cos s H ; themefore

an . FG -

z m . PH+ (m— n) FA . Produce ” to

K, so that n . FK : (m— n) FA ; then m . F0

is . FH- t—a . FK z n . HK . Draw K L perpendi

cular to AK , and GL parallel to A K ; then

or m zzn zzGL zGF. The

lin e GF, then drawn to the given poin t F, has a

given ratio (of a le ss to a greater) to GL , the per

p endiculef di'awn w a line given in positioh ;

m o n o m enim d whas r n a e a .

cun K L the directrix, andZ-ithe ratio of the semi

two positions of the sun, in w hich his d iurnal mo

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96 oor t m s s or s n ou t. m n osora r .

99T Thepontien anci ' the spec ies of the sun's or

hit, being thus de termined, the calculation of his

plaee thf any give n time, is teduee d to the geo

metric problem,of d raw ing a line through the fo

cus of an eHipsig soas to cut ofl'

a sector betw e en

it and the t ransverse axis,having a given rat io to

the w hole elliptic area.

IfAGPH (fig. be the m'hit of the sun ; F the fo

cus in w hich the earth is placed ; ant he gle ce of

the mn at a given time ; then the time of- the sun

’s

dm ihihg thfi ereh’

PG, or of the rad‘

ins ve etor de

scribing the seetor PFG, is given ; and the time of

radius vector de

to the w hole e llipsis

given . If, from this, the M ie n d

line FG can be de termined , it is evi

da lt , that the angle PFG, and theW M G ,

are found.

This is known by the name of K epm s Problem , it

can only be resolved by approfimation .

a . The angle PFG, which measures the angular dis

tance of the Wil l iam the Pcrr'

gcc, or low er epsis

is called the true mummy.

6. Ha civcle b e deacrihed fi'om fiie centre F,With a

OUT L INES OF NATURA L PHILOSOPHY.

not far from the truth, in elliptical orbits of small

eccentricity. This solution w as first proposed byBunnunnus, a French astronomer, and w as after

w ards adopted and improved by Dr Sar aWa rm ,

and is known in this coun try by‘

the name of

Wann’

sHypothesis . Another solution is distinguish

ed, for the simplicity of the principles, and the e le

mentary nature of the reasoning employed, vit . tha t

given by the late DrMa r n awSr nw s a r , in'

the E din

burgh Physical and L iterary E ssays, vol. 11 . (1755)

p . and again republished in his Physical and

Mathematical Tracts, p. 40 4 . Among the other

lutions, those of Naw'rax, Pris . Math. lib . r.

prop. 30. Schol. of SIMPSON, Essays, 4 to, (1740 )

p. ofEm ma , Commen t. Petrop . tom. v u. and of

Ivoa r , Edinburgh Transactions, vol. v., (the latterextending to the most d ifficult case of the pro

blem, when the eccen tricity is great), are particu

larly commendable.

Of all these , how ever, it may be said, that though

excellent when a numerical calculation only is re

quired, yet when the solution is to be a step in the

investigation of other properties of the elliptic mo

tion , they cease to be ofuse , so that recourse must

be had to such gen eral theorems, as e xpress the

true anomaly in terms of the eccentricity,‘

and of

the mean anomaly. The first solution of this kindw as . given by C LAIRAULT , Theoric de to Lane,

31. lemma 3d , 8m. It w as afterwards tinproved

and extended by other mathematicians, particular

lr

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102 ourmmas or NA TURA L PHILOSOPHY.

103. Astronomical tables, constructed from the

data, and on the princ iples new explained , serve

to de termine the sun ’3 place ln the ecliptic , for

any instan t of time , eithe r past or future .

a . From the time of the sun’s passing through the peg

rigee , w hen his true and mean place coincide , his

mean place for any other t ime may be computed ,

by allow ing for the in terval an increase of longi

tude, at the rate of59’ p er dim , and thus the

mean anomaly is computed . From the mean ano

maly, is found the equation of the centre , con

tained in a table , which gives the quan tity of that

equation for every degree ofmean anomaly ; thence

is given the true anomaly, and of course the turd

longitude . The tables are so constructed, as to give

the mean place of the sun for the beginning of every

year , hence the mean place for any time ofany ye ar

i s easily found . When equations are thus ranged In

tables, the quantity by w hich they are found.

out

in the table , and on w hich their magn itude depends,are called the Arguments of the Equations.

The tables of the sun’

s motion , are in reality nothingelse than the expansion of the general M ule

contained in 100 .

Tim, from the fact of the s atiation Of the men’

s ap

parent diameter, compared wi th the reaction of

his angular velocity, w e have demanm d, that ,whether

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104 OUTL INES or NATURA L PHILOSOPHY.

rotation . The decisive facts have not yet occur

red, w hich are to determine, whether a motion of

translation does not also belong to the earth.

In order that the theory of the elliptical motion mayhave its conclusions extended either to the future or

the past, a very accurate measure and reckoning of.time are necessary, and w e are now

of the da ta by which these may w ith certain ty be

determined.

Of the Equation of time, and of the K alendatr,

The arrival of the Sun in the meridian , being a morqconspicuous phenomenon than that of a Star, has

been taken to mark the beginning and end of the

day, used for the purposes of civil life . Solar time,consists of days measured in this manner, and is

used by astronomers, as

‘

well as by the people at'

large . As tronomers begin the day at noon , and

reckon 24 hours round to noon again : in the com

mon reckoning, the day begins at midnight, and la

divided into 24 hours, w hich are counted by 12 and

12. L A PLACE has proposed, that In this the astro

nomers should follow the people, as, by beginningthe day at midnight , the whole of the sun s stayabove the horizon falls m the same day.

1041. The

106 OUT L INES or NAT URA L PHILOSOPHY.

mean and the apparen t t ime , and is called the

Equation (f Time .

a . The conversion of degrees and minutes, into

mean sOlar time , is performed by a different

rule from their conversion into siderial time . For

an hour of mean solar time, there must be reckon

bd'

ls° 9’ 2711847 of the equator ;'

and so on, in the

same proportion .

6. There are four times in the year, when the mean lon

gitude of the sun, and his true right ascension , are

equal to one another , and, at these times, the ap

parent and the mean time coincide . These times

happen , at present , about the 15th of April, the

i5th of June, the l st of September, and the 24 th

of December.

‘

From the first of the above periods

to the second, the apparent time is before the

mean , and the equation of time is subtractive , or

must be taken from the apparent time , to give the

mean . It is greatest at the 15th of May, w hen it

amounts to 3‘ 58"to be subtract ed . From the se

cond to the third interval, the equation is additive,

the mean time beingbefore the apparent ; and it be

comes amaximum about the 25th of July, w hen itamounts to 6’ It becomes negative be tw een

the third and fourth interval, and reaches its ma

ximum on the 2d or 3d of November, when it

amounts to 16’ 15”

subtractive . Dr HALs has

given a geome trical construction, ibr determin ing

the time when the apparent days are longest or

shortest, and also when they are equal'

to the

mean .

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a sr aonomi‘.

Suppose, for example, that e ars .

volution of the sun ; then

so th'

at if w e

count three years, each.

of 365days, and a fourth

of 866, w e shall have exac tly four revolutions of

the sun i and at the beginn ing of the fifi b yearnhe

sun w ill be in the mme poin t of the ecliptic which

he was in at the beginning of the first . This is the

arrangement of what is ralled the Juhhn Kalendar ; .

and if the revolution of the stmgwe re exactly 365d

6h, it would be altogether perfect . It is called the

Julian Kalendar, and the Y ear thus computed, the

Julian Year, from Jus tus Cue/m , by w hom, w ith

the advice of the astronomer Sostosuas, in vas in

Tbe addition of a day, or a numba '

of days, to any

fixed period , at stated in tervals, is called Intercala

tion . T he year on which the intercalation fell w as

calledBisscxtslis , because the 6th of the K almds of

March w as tw ice counted. With us, it is called

109. As the true length of the sun ’s revolution

is not w hat has now been supposed, but instead of

is only 36548 4 2264 , the Julian year is

longer than'

the revolution of the sun by

(nearly 1 1m

and , therefore , before a

new year begins, the sun has passed the point in

the

1 10 oun w r s or na ruu t ra 1Losoe n r .

the eclipt ic w here the last year began , by a small

naeuonfl iz. caressx594s'

.

l‘

he Julian reckoning, therefene, falls cuntinunllybehind the sun, and the course of the season s,

hy a quanti ty which, how ever , is so small, that it

was long term it was chsefi 'éd.

m A t the time of the Gound q Ntoa in the yw fi é of

the Christian era, the J uitm talendum in tro

a” dueed into the C lmrch and at mat time the vernal

equinox fell on the QIst om h.

"0 n aeeoun t of

the imperfeefion in the mode ef feel onihgjust no .

titted , the reckoning tiell m mntly hehind the true

time ; so that in the year lm the m year

had fallen nearly 10 days, behind the‘

snn t aud the eqfi nm instead “M on ti“:

2l st, fell on the 1 1th MM ; 80 M the differ

ence w as nearly a day in 182 years. The conti

nuance of this.

erroneous

made the

and it was théi éfoi'e

dar, whichwas dene

the first step w'

as tfi correct the loaf; of the ten

days, by coun t ing the day after the 4thibf

’

October

1582, not the 5th, but the 15th of the month.

1 10 . A s the loss in the Julian kalendar awanuied to one day in 132 yearS, itWould ani bunt

'

to

three 111 996 years, or in the space nearly bf tam

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1 12 OUTL INES OF NAT URA L PHILOSOPHY.

tions in the kalendar, and'

tlie real motions of the

sun, should always be the least possible ,°

n not

thereby completely resolved . The modes of in ter

ce lation best suited to that object, require all the

integer numbers to be found which most nearly ex

press the ratio of the fraction 2 4-2264 to 1 . See

EULER , Elemcns d’Alg

-

ebrc, tom. Additions byLa Gasman, 20.

Secular Variations in the apparent Motion of theSun .

The variations in the sun

’s motionWhich have now

been described, are confined wi thin short periods,

during which they alternately increase and dimi

nish. There are others, which go on from one age

to another, and are either continually progressive ,

or circumscribed by periods of very long duration .

These are so slow , that they are'

only perceived bycomparing together observations made at a great

distance of t ime . They are called Secular inequa

1 1 1 . By comparing very distant observations, it

is found that the line of the apsides, or the longe r

axis of the sun ’s orbit , has a progre ssive motion ,

or a motion eastward ; so that the apsis recedes

from

Ast aonour

fromthe vernal equ1nox or by Dr. LAMBRa’s

‘Tables

‘

fi annually.

a. This motion includes the precession of the equi

noctial points, which is in the opposite direction ,

and amounts to so that the real motion of

the apsides eastw ard, in respect of the fixed stars,

is l l".65a- year, or 19'

in a century.

11. Hence there is a difference betw een the Tropical

Y ear, or the time‘

of the sun 3 resolution from equi

nox to equi nox, and w hat 18 called the Anomalie

iic Year , or the time of the sun s revolution frome ither apsis to the same apsis again . A s the apsis

has gone in the same direction w ith the sun over

62” in a year, the sun must come to the placewhere the apsis w as at the beginn ing of the year,

and must move over 62 ”more before the anoma

listie year is completed . The time required to this

is .0 1748 of a day, w hich, added to the tropical

year, gives 36.5d 259744 , or 36.5d 6h 14m 2“for theanomalistic . B lo'

r. Astron . tom. 11 . 91 .

r. The line of the apsides thus con tinually movinground, must at one period have coincided with the

line of the equinoxes . The low er aps18 or perigee

in 1750 , w as 278° .621 1 from the vernal equinox,

according to LA CAILLE and the higher apsis w as

therefore at the distance of The time re

quired to move over this arch, at the rate of 62 '

annually, is about 5722 years, w hich goes back

nearly 4000 before our era,—a period remarkable

Von. II . H for

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OUTLINES or n s ruaar. pun osop a r .

b. The secular variation of the equ tion of timefi a

differen t for every different state of that equation

it is greatest about the time w hen the sun is in the

perigee ; and a is then w as

The Sun ’s Rotation on his Axis .

1 14 . The face of the sun ,when view ed w ith a

telescope , though of a bright and in tense light,

ihr above that of any other objed't , is often

marked w ith dark Spots, w hich, when examined

from day to day,“

are found to trave rse the w hole

surfac e fmm e ast to w e st, in the spac e nearly of

fourteea ' days.

11 . These spots, though only visible w ith the telescope ,are sometimes so large, as to subtend an angle

nearly of one minute. Their number, position,and magnitude , are extremely variable . Each of

them is usually surrounded with a Penumbra , be

yond w hich is a borde r of light, more brilliant than

the rest of the sun’

s disk . When a spot is first d is

covered ou the eastern limb, it appears like a fine

line ; its breadth augments, as it approaches the

middle of the disk , from which it diminishes as it

goes over to the w estern limb, w here , at last, it an

tirelydisappears . The same spot,atter fourteen days,

is sometimes discovered again on the east side . It

10

1 1 8 oe t s or NATURA L PHILOSOPHY.

The Paths of the spots thus trac ed, are

obse rve d to be rectilineal at two opposite season s

of the year, the beginn ing of June and the beginn ing of Decembe r, and to cut the ecliptic nearly

at an angle of 7° Be tw een the first and the

second of the se seasons, the paths of the spots are

convex upward, or to the north, and acquire the ir

greatest curvature about the middle of that pe riod .

In the follow ing six mon ths, the paths of the spots

are convex tow ard the south, and go through the

same series of changes. They appear to be e lliptic arches.

In the beginning ofMarch and September, w hen the

opening of the elliptic paths is at its maximum, the

smaller axis is to the greater as 18 to 100. LA

L anna , Astronomic,3233.

1 17. The preceding appearances may be ex

plained, by supposing the spots to be opaque t o

d ies,attached to the luminous surface of the sun5

the sun having a revolut ion on an ax is inclined at

an angle of 7° 20'

to the axis of the ecliptic .

The apparent revolution of a spot i s performed in 27

days ; but in this time the spot has done more than

complete an entire revolution , having, in addition

to it, gone over an arch equal to that which the sun

has described in the same time in his orbit. This"reduces

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120 OUTL INES or NA TURA L rmLosornr .

The length of the Zodiacal Light, taken fiom the

sun upw ards to its vertex , is various, from 45° to

and even 120 °

The season most favourable for observing this phe.

nomenon , is about the beginning of March, after

sunse t : the axis extends toward Aldebaran , and

makes, w ith the horizon, an angle nearly of 64

The aspect of the Zodiacal Light is by not means

uniform ; it is much brighter in some years than

others. A remarkably brilliant appearance of it

w as observed at Paris, l 6th February 1769.

The Zodiacal Light appears to be inclined to the

ecliptic at an angle of and to cut it in the 18th

of Gemini ; so that i t is in the plane of the sun’

s,

equator, or perpendicular to his axis of rotation .

L A LANDE , Astron . 847 .

A s w e do not think that the decisive facts have

yet occurred, which are to determine w hether the. ap

parent motion of the sun is not to he explained by the

real motion of the earth, w e have employed entirely the

language that is suggested by the appearances. The

low er apsis of the elliptic prbit, or that nearest to the

focus in which either the sun or the earth is placed, has

AWRONOMY . 12 1

the Pen‘

gec . supposing that it is the poin t

sun buinotion approaches nearest to the

e con trary supposition , it w ould be

The same applies to ~ the terms

distance of the Sun and Earth, it

that t he distances w e have treat

not compared with any

such as the diame ter of the

The obse rvat ions on w hich such a compa

as this must be founded , and by which the

parallax is ascertained, cannot be explained

understood . In the

that even by help of

it may be infer

less than 10"

of the sun’s solstitial altitudes,

w e saw , how the latitude may be deduced .

In order to make the deduction w ith accuracy, the

sun‘

s altitudes must be corrected for the parallax ;

and it may be easily shewn , that if the horizontal pa

rallax of the sun is so great as the lat. of Green

w ich, for example , determined w ithout makingthe due allow ance for it, w ould be erroneous, by

The latitude de termined by observations of

the stars, is free from this error, and therefore

ought not to agree w ith the former by Now ,

the correction necessary to be made , in order that

the determination deduced from the altitude of the

sun ,

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1521 OUTL INES or NATURA L rnxt osornr .

is nearly a straight line , and the disk a semicircle .

The diminution continues, the disk becoming

more and more concave to the w e st, till, about the

e nd of anothe r seven days, it disappears altoge

ther.

After a few,days, the moon again appears, like

a fine c resc ent,to the eastw ard of the sun, w ith its

concavity turned tow ard the east , and continues

to increase on that, side till it become en tirely full

orbed, about 29 days from the time when it w as

last full.

The‘

line separating the light and the dark part of themoon is irregular and serrated, and its form va

ries while one is looking at it through the teles

c0pe : the light, as it advances, touches some

points, w hile they are yet at a distance from the

illuminated surface , and while all round them is

dark . The light on them spreads, till it be un ited

to the rest.

The Moon , w hen full, is opposite to the Sun when

she disappears, or w hen it is New Moon , she is in‘

conjunction w ith the sun : these two aspects of the

moon are called the Syzz'

gies . A t the time when

the moon appears as a semicircle , she is 90° distantfrom the sun on either side : she is then said to bein the quadratures .

“The moon , during all these changes, advances amongthe fixed stars, at the rate of 18° at an ave

rage

126 OUTL t s or Nam “. PHILOSOPHY.

l? & The position of.the uodcs rs found, by ob

se rving the long'itude ot

J the moon ,when the has

no latitude , an d it appears, of

such Obse rvatioris, that the fi not

fixed , but has a slow retrograde motion, at the

rate of in a day ; so as to complete a

volation relatively ro the fixed stars in “m a le

lam . 4 am Pigs . vol. u. p. 351 . an east

A s the momen t of the moon being irr the fi iifiticmay not be actually observed, yet, from eeveral

Observations, taken before and after, the exact

124 . By comparing places Of the moon ,

ved at ve ry remo te pe riods, it i

secular mean motion of the

the fixed stars, is 1336 c ircumfe

w hich give s for the t ime of a

revolution M 3251661 .

In the same w ay, the tropical revolution,

It also has been found, on making these

for very remote ages,

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As'raonomir. 129

a s(60 16

’

sin x (12'4 sin 2 3 sin 3Q:

a] be ing put for the mean anomaly, reckoned from

th e perigee .

This is the eqiration of the centre, as deduced fromD]:Lusan ’

s Tables, published in Vi nce’

s Astron.

vol. m . see p. &c.

The maximum of the equation is 3° 17

’and

takes place when the mean anomaly=86°5

’

(Y.

128 . The axis of the lunar orbit is not at rest;

but has a progressive motion , like that of the sun’s

orbit . This motion is 0° 6’

4 1"in a day, or 4 0

°

4 1'

33” in a year ; so that it makes an en tire revo

lution ,relatively to the fixed stars, in 8232

d.5807,

or in a little more than n ine years.

The Trop ical revolution of the perigee is shorter byl ‘1 4 056.

The motions hitherto enumerated, are similar m the

Moon and in the Sun . There are other inequalities peculiar to the moon .

129 . T he moon ’s longitude , calculated accord

ing to the law s of the elliptic motion, doe s notagree exactly w ith her true place , but require s to

be correc ted by an etch proportional to the sine of

VOL . II. I double

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132 OUT L INES or NA TURA L PHILOSOPHY.

plains them, but has led us to distinguish manymore, of which the accumulated effect was perce i

ved, by the disagreement of the calculated w ith the

observed places of the moon , though their law s

w ere unknown . These w ill be considered under

the head ofPhysical A stronomy.

Of the Moon’

s Revolution on her Ar id.

132. The moon, as she revolves in her orb it

1about the earth, prese rves alw ays, at least nearly ,

the same face turned tow ard the earth, and the re

fore must revolve on her axis, in the same d irce

t ion , and in the same time , that she revolves in her

orbit.

a . The moon’s surface, when view ed w ith a telescope ,

presents an'

object so strongly characterised, as to

leave no doubt of its being always the same.

The accurate examination of the spots, which the

tracing of the progress of the illumination has na

turally induced, has discovered some apparent ine

qualities ia the moon’

s revolution on her axis, to

which the name of Libration has been given . They

are optical appearances, and argue no real inequalic

ty in the moon’

s rotation.

133. If,

ASTRONOMY. 138

133. If the angular velocity w ith which the

moon revolves on her axis is un iform, that w ith

w h ich she revolves in her orb it, be ing, as w e have

see n , alternately accelerated and retarded, small

segments on the east and w e st sides ought alter

nataly to home in sight and to disappear. This is

conformable to observation .

Thus there is produced in the orb of the moon, an

apparent libration backw ard and forward, called

her L ibration in Long itude .

134 . If the axis on w hich the moon turns,w e re

pe rpendicular to the plane of her orbit, or if her

equator coinc ided w ith that plane , w e should perf

ce ive no other librat ion than that w hich has now

be en de scribed. But in fac t the spots on the

n orth and south of the plane of the moon’s orb it ,

alte rnately a dvance tow ard the middle of the

d isk , and recede from it by a very small quan .

t ity .

This is called the Lz’

bratz’

on in Latitude, and shew s

that the moon’s axis is not exactly, though nearly,

perpendicular to the plane Of her orbit.

135. A third libration arise s from this, that the

point to w h ich the moon presents always the same

face, is the centre of the earth, round w hich the

l 3 spec tator

134 our tms s or NATURAL PHILOSOPHY.

spec tator desc ribes a c ircle parallel to the equator,in the course of the diurnal revolution Of the

earth.

Hence, when the moon rises, a spectator sees some

Spots tow ard the upper limb of the moon , which hew ould not see at the centre . As the moon becomes

more elevated, these points approach more to the

upper edge, while others, on the inferior limb, come

in sight . This order is reversed as the moon de

scends to the horizon . This is called the Diurnal

or Parallactic L ibrarian.

136. From an attentive obse rvation of the lunar

spots, it has be en found, that the equator of the

moon is inclined to the plane of the ecliptic , at an

angle of 1 ° and that the line in w hich its

plane cuts the plane of the ecliptic , is parallel to

the line of the nodes, or to that in w hich the

moon ’s orbit cuts the same plane .

Bro-

r, tom. 1 1 . p. 4 11 . 72.

a . Suppose three planes to pass through the centre of

the moon , one representing the equator of the

moon , another the plane of her orbit, and let the

third be parallel to the ecliptic . This last w ill lie

betw een the two others, and w ill intersect them in

the same line in which they intersect one another.

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136 our tmas or murmur. surrosorn r .

tent from it'

hy more than certain limits, an eclipsealways takes plac e .

139. Hence it is evident , that an ec lipse hap

pens, ih consequence ofi

oue of the two Opaque

bodies, the Earth and the Moon , being so placed

as to prevent the Sun’s light from falling on the

other.

The in terposition of the moon betw een the sun and

the earth, produces an Eclipse of the Sun ; and the

interposition of the earth betw een the moon and

the sun , so that its shadow falls on the moon , or

on any part of the moon , produces an Eclipse of theMoon . The whole of the phenomena of eclipses

admit of explanation , 09 these principles.

140 . As the return of eclipses must depe nd on

the return of the line of the syzygies to the line

of the nodes and as the mean angular motion of

these lines is known , the periods at w hich eclipses

w ould return, w ere there no irregularity in the

motions of the earth and of the moon,may be easi.

ly calculated .

‘

a . T he time of a lunation, or of one revolution of the

line of the syl ygies, is as alre adystated ; and a revolutien of the line of the - odes,

relatively to the sun, is 3461 61968. If, by the

method of continued fractions, w e seek for smaller

numbers,

ASTRONOMY. i 87

numbers, that may nearly express this ratio, w e

shall find 19 and 223 ; so that after 223 lunations,

the node has nearly performed 19 revolutions. In

223 lunations, t herefore , or 18 Julian years 10 daysand 7 hours, the sun , the moon , and the node are

nearly in the same position w ith respec t to one ano

ther ; and the series of eclipses returns nearly in

the same order.

This period is thought to he the sauc e of the Chat

dean astronomers ; and their predictions of eclipses

w ere probably founded on it . It is particularlymentioned by PM NY . A period of 521 Julian

years, is considerably more exact. LA L anna,

Art. 1503.

b. The meanmotions of the lines of the syzygies, and

of the nodes, are also used in another way in the cal

culation of eclipses. Though the mean places of

those lines are different from theWe ; and though

it be on the latter that the phenomena of eclipses

depend, it is useful to have the mean places , in or .

der to know whether the circumstances are such,

that an eclipse can possibly happen or not at a gi.

ven new or full moon . The mean motions in

astronomical tables, afford the means of perform

ing this calculation, which is also much abridged

by a table of w hat are called the Epacts, that is, of

the ages of the moon , (reckon ing from the last

Mean conjunction , and supposing her motion uni

form), at the beginning of every year. L A Lu n e,

4stron. 1732 ; a oosson, Astron.

Eclipses of the Moon.

The length of the earth’s shadow varies,

according to the distance of the sun and earth, be

tw een the limits of and semi

diame ters of the earth ; its mean length being

Half the angle of the cone formed by the shadow of

the earth, = semid. G —parallax G) ; and therefore

if r be the radius of the earth, Sbeing the apps

rent semidiameter, and p the horizontal parallax

of the sub , the length of the shadow reckoned from

r

the earth’

s centresin

142. Hence half the angle subtended at the

earth’s centre , by the se c tion of the shadow at the

distance of the moon , is P S+p.

P is the horizontal parallax of the moon, S and p as

above.

The apparent semidiameter of the shadow , may there.

fore vary from 37'

42”

to

SeeWoonnousn, Astron. p. 340 .

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140 OUTLPNES or NATURAL PHILOSOPHY.

144k. Considering the small arches moved overby the moon, and the section of the shadow,

during the time of an eclipse , as Straight lines

given in posit ion , and de scribed by tw o points,

viz. the centre of the moon’

s disk , and the centre

of the sec tion of the shadow , moving w ith given

veloc ities, the determination of the phenomena of

a lunar eclipse is reduced to the solution of a

geometrical problem.

L et ST and MP, (fig. represent the portions of

the ecliptic, and of the moon’

s orbit traversed, da~

ring the time ofa lunar eclipse , the first by the centre

of the section»of the shadow , and the second by the

centre of the moon , considered as straight lines

L et S be the cen tre of the ear th’

s shadow , andHthe cen tre of the moon at the instant of the appaci.

tion, and let S‘and

'

M'

be any other cotemporary

positions of the se centres ; SS'

and MM’

be ing to

ken in opposite directions. Draw S’

N parallel to

SM ; and join M’N. Now if t be the time in

w hich SS’

and MM’ have been moved over, rec

koned from the moment of the opposition, in hours

and decimals of an hour ; let the horary motion of

the moon in longitude be as, so that M0 =m t,

her horary motion in la titude in, so that Pal/0 : A t ;

and le t the horary motion of the sun or of the she

clow be a, then SS’

: MN n t.

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M4 oue e s or h a rve s t. PHILOSOPHY.

ed through the shade the n eare r the moon hap

pens to be to the earth, or the farther she is

from the after of the shadow ; the darkness is

the greater.

The light that, by the refraction of the earth’s atmo:

sphere , is made to enter w ithin the limits of the

conical shadow , is no doubt the cause of this phes

nomenon .

In some instances the moon has disappeared entirely,

as in that mentioned by Karm a in June 1620.

Havas has taken notice Of another, where the

moon could not be seen even w ith a telescope;

though the night was remarkably clear .

146. A s an ec lipse of the mOon happens at the

same instant of absolute time to all Observers, it is

one Of the phenomena, from the Observation Of

w hich the longitudes of places may be most di

rec tly inferred,567.

On account Of the illsdefined boundary of the shadow,

this method Of ascertaining the longitude does not

admit of great precision . It is difficult to deter

mine the beginning or end to less than a minute

of time .

The arrival Of the boundary Of the shadow , at the dif

ferent spots, can be more accurately ascertained

than the'

beginning or end ; and, therefore , as manyobservations

ASTRONOMY. 146

W ef that kind shnuld be made as pon

sible .

When several such observations, made under two

fl uent meridians, are compared , the mean m yfi n ish a tolerably exact determination of the dif

fere nce of longitude .

The comparison of the beginning or end, w ith calcu.

lations previously made, may also serve for findingthe longitude, and may be useful for that purpose

at see .

147. The length of the moon ’s shadow is less

than that of the earth, in the same rat io that the

d iameter of the moon is less than the'

diameter of

the earth that is,in the ratio of l to

a . Hence , when the Earth is in the aphelion , the lengthof the Moon

’

s shadow is and if the moon.

is in the perigee , its distance fromthe earth is ouly so that the shadow may reach the earth,

’

and a total eclipse may take place . But if the moon

w ere in her apogee, when her distance is

Von H. K the

OUTL INES or NATURA L‘

YHILOSOPHY.

the « shadow bould not reach the e arth and the

eclipse could not anyw here be total.

6.When the earth is in the perihelion , the length -of the

moon'

s shadow is and if at the same time

the moon be in the perigee , or indeed nearer

than her mean distance, a total eclipse may hap

pen .

4

c. The moon’

s mean motion about the centre of the

earth 18 33’

in an hour ; and the shadow of the

moon , therefore , traverses the surface of the earth

w hen it falls on the surface perpendicularly, w ith

a velocity of about 380 miles in a minute . Whenthe shadow falls obliquely, its velocity appears

greater in the inverse ratio of the sine of the 0 i

quity.

Relatively to a point On the earth‘

s surface , the sha

dow may go much faster than this, as its motion

may be in an opposite direction to the diurnal ro e

tation .

The duration of a total eclipse, in any given place,

cannot . exceed 7’

58 7. LA Lawns , 1777 .

A n A nnular Eclipse, or one where the sun’

s disk'

appears like a ring all round the moon, may last

12m 245.

To have a partial eclipse of t he sun, it is not requi

site that the shadow should reach the earth ; it is

s ufficient that the d istance of the centres of the

sun and moon be less than the sum of their appa

rent semidiameters.

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ASTRONOMY. HQ

moon, aud that neuly in the ratio o( 8 to e ; but

few er eclipsos of the sun are observed in any given

place than of the moon , as a lunar eclipse is vis ible

te a whole hemisphere ; but a solar only to a

151 . The general phenomena of the ec lipse be

i ng calculated to the time of a given meridian, the

phenomena, as they w ill be obse rved at any parti

cular place , may also be dete rmined, by calcula

t ing the altitude s of the sun mid moon, and the

efl'

eets of the ir parallaxes for difl'

erent i naants of

t ime , and then employing theme thod o f in terpolati on , to de te rmine the t ime of the beginn ing and

and, and the quan tity of the greate st obscuration .

a . I.et the plaees of the sun and moon Be found for

an instant, far from the beginning of the eclipse,

and from thence let the ir altitudes for the giv

place be oomputed, as also the efl'

ec ts of parallax in

m a d am e .

Let the m mce of the apparen t longitude of the

two bodies thus found be called 3, and the latitude

of the moon A. If the sun s parallax is included,le t his parallax in latitude be applied to the moon,taking notice , whether it increases or diminishes

the difl'

erence of latitude . Then ACB, fig. 15. be

ing an arch of the ed iptig A the place of the sun, D

of the moon, as just computed , P the pole of the

ecliptic, and FDC a circle of longitude, AC z ),

K 3

150 our t mas or NATURA L r n xt osormr.

and CD z a. and AD the distance of the centres ,

w hich w e may call y=V because the triangle

AGB may be regarded as rectilineal. In practice ,

3 may be found by a construction ; or if great ac

curacy is required, w e may compute y from the

trigonometrical formula , cosy cos 3x cos A.

6. Let similar calculations be made for other tw o

instants, separated by equal in tervals of time a ; so

that one may be near the middle , and another near

the end of the eclipse . L et the distances of the

centres found for these times be A , A'

, A"; let

the differences of these distances be D and D‘;

and let the second difference, or D D’

A .

Then if y be the distance of the centres for any tinne

t, reckoned from the instant for which the first

computation is made, 3

D Q A

The distance of the centres is thus expressed in terms

of the time , and from this equation the time of the

beginning and end of the eclipse , and the quan tity

of greatest obscuration , maybe determined.

c. The time of the greatest obscuration is

m (D sa)

and this being substituted for t, the value ofy w ill

give the nearest approach of the centres.

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45-1. oun mr s or NA TURA L PHILOSOPHY.

Occultation d Stars.

152. The same method used in calculat ing

eclipses, may‘

be applied to compute the occul

tation of a fixed star by the moon ; only, when

the moon is distant from the ec liptic, the base of

the right- angled triangle in the former construc

tion , must not be supposed equal to the difi'

eren ce

of longitude , but to that diffe rence multiplied

in to the sine of the zen ith distance , or the cosine

of the latitude .

If S (fig. b e the star, D the moon, D the d ifl'

e

rence of latitude , SE is not to be taken as equal

to AC , the difference of longitude, but as equal to

AC x sin SP dif. long. x cm A

The distance SD being thus found, the reat’

of the

computation is as before .

If at the time of the mean conjunction of the moon

and a star, that is, when the moon’s mean longi

tude is the same w ith the longitude of the star,

their difference of latitude exceed 1 37"there can

be no occultation , but if the difference be less than

there must be an occultation somewhere on

the face of the earth. Betw een these limits there

is a doubt, which can only be removed by the cal

culation of the moon’

s true place . VINCE, vol. 1.591 .

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156 oun m s s or NATURA L Pnuosornr .

chm oritur srdens, postal radians,” Hist. Nat.

lib. n . cap. 16.

By Lucifer is understood Venus, when seen in

the morning before sunrise . By Vesper, the

same planet seen in the eyeping after sunset.

c. The other five planets are visible only through the te

lescope, and have been lately discovered ; Uranus llyHs nscn s n, in 1781 Ceres by Plum, in 1311 ;

Pallas hy OLBEa s, in 1802 ; Juno by Han n a,

in 1803 ; Vesta by Quil t s, in 1807.

d. The planets have also particular characters, bywhich they are distinguished ; these, in the order

in which they have been enumerated, are,

s s s t a s i s

It is best to begin w ith the inferior planets, l i d

w ith Venus, as that of which the phen omena are

most easily observed.

154 . Venus the most brilliant of the planets,

always accompanies the sun , never receding fromhim more than and becoming, as she is on

the east or w est side , alternately the Evening or

the Morn ing Star.

a . Venus is the only planet mentioned in the SacredWritings, and in the most ancient poets, such as

Hzstep and Hous n .

b. The

u n m mr . 151

b. ThEEvm ing n d Moming Star, or the Hesperusand Phowhos

'us of the Greeks, w ere at first mppOsad to be difl

'

erent . The discovery that they are

the same is ascribed to Pr 'rns oons s.

155. This M t n an emning star, and at

hum m m fnem the summ at what 18

called her Greatest Elongation, m ean , through

the telescope , to have a semic ircular disk, like the

moon in the last quarter, w ith its convexity tumed to the w est , From that time , during her apptsaeh to the sun , he r splendour increase s for a

while, though the quantity of the illuminttted

“diminishe s, like the moon in the wane ; and.

tt the same time , her diamete r, measured by the

M e of the bonus, increases.

a. A t the time of her greatest elongation, Venus is

stationary w ith respect to the sun , or has the same

motion in longitude . Afi ci that, her motion in

longitude becomes slow er than the sun’

s, and she

comes nearer to the sun , as just remarked. A t a

certain she becomes stationary w ith respect

to the stars, having no motion in longitude .

spect d the fixed M a d is direeted m t

5. Venus at last approaches the sun, so as to be lost

in his fight ; and nlter some tithe, appear-s outh

158 0 111 1 1 112 3 or 11 11 1 0 11 11 1. r n rt osornr .

w est side, and is seen in the morning, before the

c. Though Venus in general is not visible at the time

of her conjunction w ith the sun, she has sometimes

been seen as a dark spot passing over the bodyof

the sun . This is the phenomenon called the Tus

ss’

t q euus. Her diameter is then grentest, sld

measures nearly one minute .

156. A s V enus proceeds to the w estw ard, bet

d isk 13 seen as a cre scent continually increasing,

atflthe same time that the diameter 13 d iminish

ing. At the elongat ion of the d isk is again

a semicircle ; and from thence it inc reases, while

the distance from the sun diminishes,t ill the pla

ne t is lost in the sun ’s rays ; her orb be ing almost

ac ircle

,but its diame ter not more than one - sixth

of what it w as at the former conjunction.

a . The conjunction , which is preceded by the sp

proach to a full orb , or that w hich follows the wes

tern elongation of Venus, is called the sup erior con

junction , as she is then farthest from the earth. The

other the inferior .

b. The time of the greatest elongation of Venus is

about sixty- nine days before or after the inferior

conjunction , when she is betw een 39°and 40

°dis

tant from the sun, and comes to the meridian l b

either before hr after how“

; her disk is then

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The diurnal motion in his orbit 4 °5’ 32”

T he periodic time nearly .

1 60 . The pomts m w hich a plan e t has no lat i

tude, are c alled , as in the c ase of the Moon

,the

Nodes of f/16 Planet . The Plane t is then in th e

e cl iptic , and on e - half of its orb it lies on the north,

the other on the south side of that plane .

4 . The line of the nodes of every planet, or the com

mon section of the plane of its orbit, w ith the

plane of the ecliptic, passes through the Sun . This

w as discovered by Ka r e n a . See Dr Sm i t h’

s Ao

count of K r:m e n’

s Discoveries, p . 154 .

b . The node through w hich the planet passes into the

northern signs, is called its Ascending Node ; that

through w hich it passes into the southern , is call

ed the Descending A’ode .

c . T he Heh'

ocentr ic place of a planet , is its place as

it would be seen from the Sun : the Geocenm'

c, as it

is seen from the Earth.

When the Earth is in the line of a pla

n e t’s nodes, or, w hich is the same , w he n the Sun

is seen from the Earth in that line , if the plane t’

elongat ion from the Sun . and its geoc e n tric lat i

tude be observed , the inclinat ion of the orbit ma

b e found ; for, the sin e of the elongation is to th

VOL . II. L radius

162 e a r n in gs or NA TURA L pm t osoe nr.

radius, as the tangent of the geocentric latitude to

the tangent of the inclinat ion .

L A L iana , 1358 . Gnm onv’

s Astron. Book ur.

prop . 20 .

If the planet be 90° distant from the Sun , the lati

tude observed is just equal to the 1nelination . K nr

LER made use of this last method for determiningthe inclination of the orbit ofMars .

162 . If an infe rior planet, at the infe rior and

supe rior conjunc t ions, or a supe rior planet at theOpposit ion and conjunc tion , be also 90

°

distant

from the node ; from the obse rvation of its geo

c en tric place , the inclinat ion‘

of the orbit beingknow n

,the rat io of the plane t

’s d istance from the

Sun,to the Earth

’s d istance from the Sun

, mayb e found .

In the annexed figure, (fig. let S be the Sun,

VV’ the orbit ofVenus, E the Earth ; then joiningEV , EV

’, and draw ing the perpendiculars VD,

V’D’

, if ES a, SV b, and the angle BSV I,

SD z b cos I : SD’; so that ED a— b cos I,

and ED’a 6 cos I . If the angle VES a,

and V'ES a'

, then because VD

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164 our t m xs or NATURA L PHILOSOPHY.

therefore be nearly in the centre of the orbit of

Mars. The same is true of Jupiter, Saturn ,&c . o

c . Mars appears w ith his disk perfectly round, both

at the opposition and the conjunction . In the in

termediate positions, he is found to w ant somethingOf perfect rotundity on the side turned farthest

from the Sun .

Observations made at the oppq ition of a planet,

and , w hen possible , near the conjunction , are

valuable , because the planet is then seen from the

Earth in the same place it would be seen in from

the Sun ; that is, the geocentric and heliocentric

places, e ither coincide, or differ exactly by 180

degrees .

164 . The supe rior plane ts are not alw ays pro

gre ssive , or do not alw ays move from w e st to

cast ; but, like the infe rior, they become station

ary, and also re trograde . They are progressive

at the conjunc t ion , and for a conside rame d istanceon each side Of it they are re trograde at the 0 p

posit ion , and at ce rtain poin ts be tw een , are station

af’

y, passing gradually from one of these state s

to anothe r.

a . Thus Mars, when he emerges from the Sun’

s rays,

a few days after the conjunction , and is seen to

166 ourmmts or NATURA L Pa rt o'

sorrtr'

:

For the six principal planets, these ares are

bited in the annexed Table, according to the ir

mean quan tities. See L A L ANDE, 51 192. A lso

BIO L , 97 . vol. 111.

TABLE.

Elongation , Arch of Re Time of Re

when sta trogradafion.

Uranus,

1 65. T he appare nt motion of an Object is affect ed by the motion of the spec tator ; and if there is

not a ce rtain tv that he is at re st , w hen the mot ion

Of the forme r appears ex tremely irregular, it is na

tural to inquire , w hethe r any mot ion that can

Son-ably

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168 OUT L INES OF NA T URA L eurt osorl-nr.

This is evident, because CF 18 equal to DB, an!makes the same angle w ith CD.

Hence it is evident, that an object, w ithout beingreally at rest, may he apparen tly so, if the observer

is in motion , and may even acquire an apparent mo

tion , in a direction contrary to its real. Thus al

so the observer and the object both moving w ith

perfect regularity, both describing concentric

c ircles, for example , w ith uniform velocities, and

d irected the same w ay, the one may become sta

t ione ty in respect of the other, and even acquire a

motion in an opposite direction .

167. Suppose (I to be the d istance of a planet

(imagin ing it to desc ribe a c ircle round the sun )from the sun , or the radius of its orb it, and e the

e longation from t he sun,at w hich it appears sta

t ionary, the rad ius of the circle in w hich the Oh

serve r must move , in order to see the plane t sta

t ionary at that e longation , be ing called .z'

,is found

from the equation x“

d .t ' dz cot“c or

ga/1 + 4 cotz

e

a. This follow s from w hat K e rr. has demonstrated in

his Astronomy, Sect. L auoatan , Institutions

ASTRONOMY. 169

Astronomiqucs, p. 585. LA Lanna has slinplified‘

the demonstration , 1188. See also BroT, vol. 111 .

p . Note .

If, by help of the above formula, w e inquire, suppo

sing the Earth to revolve in a year round the Sun ,

and Mars in days, what must be the ratio

of the distance of the Earth to the distance ofMars

from the Sun , in order that the latter may be sta

tionary at the elongation of l36~

‘

w e shall find it

to be tha t of l to which is the ratio deduced

from other phenomena.

c. Not only are the stations thus explained, but the ex

ten t Of the arches of progression and retrograde

tion also.

°

This coincidence affords a strong pre

sumption in favour of the system Of the Earth’

s

motion , or that which, from the name of its dis

coverer, is called the Cop ern ican System.

The same holds of Jupiter, Saturn , Uranus. The

same motion of the Earth, and the same distance

from the Sun , w ill account for the phenomena in

all the cases so that whatever probability there is,

from the phe nomena Of one planet , in favour of the

Earth’

s motion , the same is increased in a quadru

plicate ratio, from considering the phenomena of all

these four superior plane ts.

The tw o inferior planets, give a similar increase of

evidence The four new planets are not takeninto

170 OUTLINxs or NATURA i. rn rt osor n i .

into account, as the times of their stations, ac

may not yet have been sufiiciently determined b yobservation .

j:On the strength of this evidence, w e shall assume th é

motion of the Earth as a fact , and try w hether it is

consistent w ith the other phenomena of the plan e- 3

tary motions.

Orbits of the Planets.

168. If a plane t be obse rved tw ice in the same

node , the node in the interval be ing supposed to

remain fixed,the position Of the line of the nodes

may be dete rmined, and also the distance of the

planet from the sun at the times of observation .

a. L et a superior planet be Observed m its hode

N, (fig. from the Earth at E , and after the

plane t has made an entire revolution , and returned

to the point N, let the Earth be at E'. Then,

from the time , and the construction of the Earth’

s

orbit, EE'

is given , and the angles SEE’

, SE’

E.

But the angles SEN, SE’

N, are know n by obser

fatiou therefore the angles EE'

N, E'

EN, as also

the"

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172 cou rs e s or NATURAL PHILOSOPHY.

or the planet’

s distance from the Sun, is foun d;

The position of PS, relatively to NS, is also th us

determined for, in the right- angled spherical t ri

a

angle, of which the base is the arch that mea

sures the angle OSN, and the perpendicular the

arch which measures the angle PSO, the hypothe

nuse is the measure of the angle PSN, which the

radius vector makes w ith the given line SN.

6. Thus also, EP, the planet’

s distance from the

Earth, is found . If, then , by Observations made

a little before and after the opposition , the diur.

nal motion of the planet in its ow n orbit, relative:

ly to E, be determined , the same may be found re

latively to S, being to the other in the inverse ta

tio of SP to EP.

When many oppositions of a planet are thus oboer

ved, many different radii of the planetary orbits

are determined , as w ell as the angular motions

corresponding to them.

170 . It appears,. On laying down the radn de

te rmined as above , that the orbits of the planets

are e llipses, having the Sun in the ir common fo

cus ; and that the angular motions of a plane t

round the Sun , are inve rse ly as the squares of its

d istances from the Sun ; so that the sec tOrs de

sc ribed by the radius vec tor,are proportional to

the time .

ASTRONOMY. 1 73

These tw o propositions, w hich have already been

shew n to hold of the Earth’

s motion , are therefore

common to the motions of all the planets. Theyw ere discovered by KEPLEn , and w ere first found

out by him, w ith infinite ingenuity and labour,

w hen he was endeavouring to determine the orbit

171 .When the focus of an e llipse , and three

points in its c ircumfe renc e , are given , the e llipse

may be desc ribed and hence the plane tary orbits

may be de te rmined , that is, the axis, the cecen

tricities, and thence the equat ions to the centres,

&c.

The application of this to find the three E lement:

of an orbit , the Eccentricity, the place of the

Aphelion , and the Epoch, or radicalmean place, for

a given time , is in LA L ANDE ,1288, &c . A lso

V in c e’

s Astronomy , vol. 1 . §257 . See also Nzuo

TON"Prin . Math. lib . Imus, prop . 21 . Schol.

172 .When the mean d istances of the planets

are compared , and also the ir period ical times, it is

found that the squares of the pe riodical times are

as the cube s of the d istances.

This great general fact w as also discovered by K er

LER , and is the third of the law s that hear his

name .

173.When

174 OUT L INES or NATURA L rm t osornr .

173.When the elemen ts Of the orb it are foundfrom Observation , at pe riods considerably d istan t

from one another, the line of the apsides Of e ach

of the plan e ts is discove red to have a slow mot ion

forw ard .

L A L anna , 1309, Si c .

The Elements of the Orbits of the different Plan ets

are given in the annexed tables.

Inclination of Secular motionthe Orbit to the

Eclipt ic for

1801 .

Venus

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176 our t tn rzs or NA TURAL.

PHILOSOPHY.

Secular var.tude of the Pe

74°21

’

46"128 37 l .

32 17

The First of the above Tables gives the inclination of

the orbits, the position of the line Of the nodes, and

the secular motion of the nodes for all the plan e ts .

The sign minus, prefixed to the motions of the

nodes, sign ifies that they are retrograde . The in

clinations of the orbits of Vesta, C eres and Pallas,

are greater than those of the other planets ; and

the orbit of the last goes far beyond the zodiac.

T he

53 18 1 1249 43 o .

146 39 39121 14 1

1 1 s

89 s 58167 21 42

178 OUTL INES or NATURAL rmt osor nv.

Rotation cf the Planets.

1 76. Four of the plan ets, V enus, Mars, Jupiterand Saturn, w hen examined w ith the telescope ,

appear to revolve on axe s, in the same direction

in w hich thev revolve in the ir orbits ; the axis

of each remaining alw ays parallel, or nearly pa

rallel, to itself.

a . This conclusion is derived from the motion of certain

spots, w hich are distinguished, by the colour or ia

tensity of their light , from the other parts of the

planetary disk: In this w ay the time of rotation

is also determined, as in the case Of the Sun,

b. It is thus found, that Venus revolves in 2311 21m 90,

on an axis which makes a very small_angle w ith

the plane of the ecliptic: This w as first observed

by the elder CASSINI. L A L anna , 334-1 .

c . Mars revolves in 1 day 39 minutes, on an axis incli

ned at an angle Of to the ecliptic .

d . Jupitcr revolves in 9h56m, on an axis nearly perpene

dicular to the ecliptic .

Saturn revolves on his axis in 10h 16m.

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180 OUTL INES or NATURA L rurt osornr .

SECT . X .

OF THE SECONDA RY PLANETS.

1 79. JUPITER, w hen view ed through a telescope ,

is ~ found to be accompan ied by four small stars,

ranged nearly in a straight line , paralle l almost to

the plane of the ecliptic and Occasionally on the

same , or on opposite sides of the plane t .

a . Sometimes these small stars pass betw een us and J 11

piter, and their shadow s are then seen traversing his

disk. Hence , it is eviden t, that both Jupiter and

they are opaque bodies, w hich d erive their light

from the Sun .

6. Jupiter, accordingly, projects behind him . a con ical

shadow , in w hich the little stars just mentionedare often immersed ; so that they d isappear, and

are eclipsed , as the Moon is, byTelling in to the

shadow of the Earth.

0 . Hence these little stars are to .be considered as

moons, w hichc irculate round Jupiter and‘

as theyare alway s obseri'ed to move eastw ard, when theyare eclipsed , and w estward when they pass over the

d isk, it is evident, that their motion is progressive ,

182 0 0 1 1 1 8 1 3 or NA TURA L r n rt osorur .

credible labour, a complete theory of the ir motion ;has been established .

The t ime from the middle of one eclipse

Of a satellite , to the middle Of the next , is the

t ime of its synodic revolut ion , and is equal to the

t ime of its revolut ion round Jupite r, increased bythe t ime

i

w hich it takes to de sc ribe an arch of its

orbit , equal to the arch w hich Jupiter has describ ed in the same t ime round the Sun .

a . The synodic revolution of a satellite is subject to va

riation , as the rate of Jupiter’

s motion , in his 0 1‘

bit, is not uniform. The greatest equation of Ju

piter’

s orbit is and the time of the first

satellite’

s moving over an arch equal to this, is

39m 229 ; and so much, therefore , may the synodic

r evolutions of that satellite differ from the‘

mean .

Those of the 4 th may differ 6h from the

same cause . V ixe n, Ast. vol. 1 . 4-19. LA LAnnn,Ast. 2925.

1 82 The synodical revolutions of the same sa

tellite, are subje ct to an inequality, depending on

the distance of the Earth from Jupite r.

a .When the Earth 18 nearest to Jupiter, or at the .timeof Jupiter

’

s opposition to the Sun , if:the synodic

revolution be a , the successive eclipses should happen at the intervals 11, 2 11, 3 n, &c . reckoning from

the

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184 our t rnrs or 11 1111 13 11 1. rn 1 t osorar .

bxli s sm to belong to sll the n tellitu. 1t haalso been found , that the eccentrieity of dupiw

‘

s

orbit does afi'

ect the inequxlity in queetion . Mt »

11 .

of the m mmw Mam distances fromJa

piter, which connec ts thc periods of the plm ets.

and their mean d istances from the 81111 ; that is,

the 811mm of the former Quantities are as the

cubes Of the latte r.

Nam e s:M an y“ lib. un prop.

The mean mmion of thc firuun lfitg d ds

ed to tw ice the meanmotion pfto three time s the mean moti

on

L“, are the longitndes of tha e w

tellites, l .’

Celeste , tom. 1. 9 3432,

1 86. The orb its of the satellites are not in the

same plane w ith the orbit of Jupiter .

This appears fi'

om the duration of the eclipses of thesame satellite, when compared with one another .

The

186 OUT L INES or NATURA L rnrt osorn r :

From analog we may conclude, w ith considerable

probabil ity, that the orbits of. all the satellites are

e ither circles or ellipses.

6. Dr Heasc annn has observed, that the same satellite

ismore luminous at one time than another, and that'

the period of these changes is for each satellite the

samew ith the time of its revolution about Jupiter ;

hence he'

has inferred , that the satellite also revolves

on its axis, in the same time that it revolves about

Jwhen

This is also the law of the Moon’s rotation, 132.

190 . The beginning or end of an eclipse of a sa

tellite , marks the same instan t of absolute t ime to

all the inhabitants of the Earth , and w ay the re

fore be employe d for find ing the longitude , in the

same w ay w ith an eclipse of the Moon, S67. and14 6.

a . The immersion of a satellite into the shadow ofJu.

piter, and its emersion fi om it , are Histant'

s more

precisely defined than the beginning or end of a lo.

nar eclipse ; and ther fore the longitude is more

accurately found by the former . The comparison

may e ither be made betw een the observation and

the T ables ; or betw een it and another observation

made under a known meridian .

f

b. The first satellite is the most proper to be observed,

its motions being best known, and its eclipses re

curring most frequently.

A stronomers,

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188 cur t ms s or NATURA L rn rt osornr .

the third , and the fourth,

9 39"

192. The plane ts Saturn and Uranus have also

satellite s ; the forme r seven, and Still:latte r'

six ;

w h ich, w ith the Moon , make e ighteen secondary

plane ts in the solar system.

a. The fourth of the'

satellites of Saturn w as the first

discovered ; it w as seen by Hovoans in 1655.

The first, second , third and fifth, w ere.

discovered

by Cassius, be tw een the years 1671 and 1684.

Other two satellites, w hich w e i rnproperly call the

sixth and seventh, though they are nearer to Sa

turn than any of the rest, w ere discovered by Dr

HEe sca su. in 1789.

1 93 . The connec tion betw een the period ic times

and the mean distances of the satellites of Saturn,

is the same as in the satellites of Jupiter, and in

the primary plane ts.

a . The fifth satellite disappears regularly for about one

half of the time of its revolution round Saturn ;and hence Naw '

ron concluded, that its revolution

on its axis is of the same . duration w ith its re

volution round Saturn . Prindp . Mat/1. lib . 111.

prop:17.

1941. The

390 ourm’

rta s or fiATURA l. rmw soru’

v.

observed between the ring and the planet. Sn irn’

s

Op tics . DE LA Laun t , tom. m . §3353.

b. The plane of the ring i s in the plane of the‘

equator

ofSaturn ; it is inclined to the orbit of the planet,nearly at an angle of and remains always pa.

rallel to itself.

1 96.When Saturn is in the longitude of5°

or of 1 1“

the plan e of the ring passes through

the Sun , and the light then falling upon it edge s

Wise, it is no longer visible to us.

é . This disappearance of the ring has been often observed . LA L ANDE , 3354 .

The disappearance of the ring from this cause , lasts

only a few days ; for when Saturn has passed the

node of the ring three or four minutes, or the Sun

has risen above its plane by that quantity, the ringbecbmes visible .

1 97. The ring also disappears, w hen its plane

passes through the Earth ; for its edge , or its

th ickne ss, be ing then direc ted to the eye , and be

ing too fine to be se en,the plane t appears quite

round .

sfIn this case, the Earth requires to be elevated above

the plane of the ring, at least half a degree before

the ring is seen, which makes it continue invisibleseven

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198 c tr-

turn s or ua '

ruuu. ra osovn r .

200 . The ring revolves on an axis at right:

angles to its ow n plane nearly, in the same time

w ith the planet itself, or in a little more than ten

This observation w e also w e to Dr B l u e s t “ .

m The fime ot'

the uvalutioa at'

the fing ia AS'

l d n

6. It is ru b ble, that if a satellite at the meaa dis

tance of the middle of the ring, revolved round Sa

turn, and obeyed the law of Karat : in respect ofthe other satellites, it would revolve exactly in10“89“ Bin , Act . Phys. tom. m

’

. p. 96 .

“2 i sm oaomr . 193

SECT . XI.

br con e/rs.

COMET 1s a luminous body, which appears

i n t he heavens only fora limited t ime , seldom'

ex

(reading a few months duri ng w hich, beside the

diurnal motion , of w hich it partakes in common

with the other heavenly bodies, it hasalw ays a

motion pecuhar to itself, by which it change s itsiplaee among the fixe d stars. 1 ts appearance is

usually that of -a collection of vapbnrp in the

centre ofw hich is a nuc leus, for the most part, but

indistinc tly defined .

I n some Comets, the peculiar motion is progressive,in others retrograde . i n the same comet, the

motion is all nearly in one plane ; but in different

C omets, these planesmake all different angles w iththe e cliptic . A t the beginning and end of the ap

pearance of a Comet, it deviates from the plane'

m

which the middle part of its course lies. Naw 'rov

dc Systm ate Mandi , 59 . The Comets have no

parallax , and are therefore certainly beyond the

limits of our atmosphere .

V OL . II. 20 2 . A

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196 our t t s or NATURA L run osorfir .

m ay believed , that the solutton does not fallw ithin the limits of an elementary treatise . A t

th‘

e‘

sauie time, a very simple geometrical problemis the foundation of it. fi rt

'

th. The

det ermination of the orbit implies that of the five

quantities, w hich are its elements

1 . The inclination of the orbit . 2. The position of

the line of the nodes. 8 . The longitude of the pe ‘

rihelion . 4 . The perihelion distance from the

Sun . 5. The time whe n the‘

Comet‘

m in the pe

rihelion .

NEW-

run 3 solution being a laborious and indirect ap

proximation , the problem has been attempted bymany others. L A CA ILLE, Astron . 775, Ste.

Boscovrcn , Op era, tom. 1 11 . p . 14 . &c. The solu

tions of Boscok and L A PLACE are illustrated bySir H. ENGLEFIELD, in his Determination of the Ora‘

bits of Comets, Lond . 1793.

L A LANna has given a mechanical construction, that

serves for finding the orbit nearly, it is

follow ed by Vm c a , 653. L AMBERT has demon .

strated some remarkable properties of the orbits ofCome ts, in a w ork en titled Insig niorcs Orb

’

t'

tc'

Co

metaram p rop rietatcs , Aug. Vind . 1761 , 8vo. See

also, LA GRANGE , Mém. dc Berlin , 1783.

The most perfect solution of all is supposed to be that

of LA PLACE, Mcchanique Celeste.

A s'

raonomv. 197

The latest, and in practice one of the best, is

that of DE Luri ng, Abrégé

d’

Astron . Logan 21 .

524.

205. The only Come t w hich is know n with ah

solute certainty to have return ed, is that of 1682,

w h ich, conformably to the predic tion ofDrHA L

Lav, appeared in 1759 .

Dr HA v was led to this prediction by observing,that a Comet had appeared in 1607, and another

in 1531 , and that the elements of their orbits,

w hen calculated from the observations made on

them, agreed nearly w ith those of the Comet of

1682, the period being betw een seventy- five and

seventy- six years.

Though there can he no doubt that these four Co.

mets w ere the same , they w ere considerably unlikein appearance . The Comet of 1531 w as of a

bright gold colour ; that of 1607 dark and livid ;

in 1682 it w as bright ; in 1759 it w as obscure .

PmGaE'

, Cometog raphie, tom. 1 1 . p . 189 .

The return of some of the other Comets is probable,

thgugh not certain .

The great Comet of 1680 , w as supposed by Dr HAL

LEY to have a period of 575 y ears, and to be the

same w hich had appeared a little before the death

of Jan us CE SAR , i n the year 44 A . C . again , in

the reign of JUSTINIAN, in the year 531 P. C . and

N 3 in

198 OUTLNES or NATURAL PHILOSOPHY.

in 1 106, in the re ign of Hu nt 1 . A t all these

periods, appearances of a great and terrible Co

met are recorded, but no such observations as

can ascertain the identity of the elements . Synop

sis Astronaut!: Cometia s, subjoined to m v’

e

Astronomical Tables .

20 6. The C omet of 1680 , men tioned above , is

remarkable for having approached nearer to the

Sun than any other that is know n . At its pe ri

he lion ,its distance from the Sun w as on ly 757 th

part of the Earth’s. It descended to the Sun w ith

great ve loc ity,and almost pe rpend icularly, and

ascended in the same manner, remaining insight

for four mon ths.

When this Comet was in the perihelion , the diarrhe

ter of the Sun must have subtended an angle of

more than 1 12 degrees. See many interesting

particulars w ith respect to it, tom. 111 .

p rop . 4-1 . at the end.

The phenomena of the tails of Comets, shew the ce

lestial spaces to be void of resistance .

Some Comets have come very near the Earth. The

Comet of 1472, is said by RzorouoNrANus to have

moved over an arch of 120 degrees in one day .

The Comet 1760, moved over an arch of 4 1 de

grees in the same space of time . A s neither of

them could probably have described in its orbit

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202 ouruNas or NA TURA L r a osoruv.

If the star is in the pole of the ecliptic, it describes ac ircle, w ith the radius having the pole for

its centre .

If the star is in the ecliptic, it describes a straight

line in that plane, and extending on each

side of the star.

The purposes of practical astronomy require, that

the change made by the aberration on the longi

tude and latitude, and on the right ascension and

declination of a star, should be computed .

212 . If L be the longitude of the sun at anytime , and L

'

the longitude of a star, the abe rra

tion of the star in longitude is

x cos (L’

L)c

and the aberration in latitude is

x sin (L'

L) sin Lat .

These formulas w ere first given by C LAraAvT .

Acad. dc Scien . 1737.

See also CAGN0 L 1, Tr ig . g1529. LA LANDI , 2823.

and D3 L Aua , Ast. lecon . 19 . 20, 21 . &c .

2 13. If A be the right ascension, and D the de

clination ofa star, L be ing the sun

’s longitude , as

be fore,

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$04, OUT L INES or NATURAL PHILOSOPHY.

Though it is proved to demonstration , from facts

above enumerated, that the Earth is far from beingthe centre of the planetary motions, yet all the ap

pearances hitherto mention ed, are consistent w ith

w hat is called the Tvcnomc System:of the heavens,

(from its inventor Tvcno in which the

sun, accompanied by the planets, revolves in an or.

hit round the Earth. This system, which its want

of simplicity renders suspected , is entirely over

turned by the fact of the aberration , and the mo

tion of the Earth completely established.

When the aberration w as first discovered, it was

thought that the velocity of light, as inferred from

the eclipsea of the satellites of Jupiter, did not perfectly agree w ith it. It has, how ever, been found,

from more accurate comparisons, that they per

fectly coincide .

2 15. It appears, that the light of the, heave nly

bodies trave rses the space s be tw een them and the

earth w ith the same uniform veloc ity .

The reflected fight fmm the Sflflclh'

tes, travels w ith

them velocity with the 61t 1381“if the fixedstars ; and the velocity of this last in the same fromw hatever distance it comes .

There IS reason to think, that light w accelerated by

of the earth s motion, and proportional to the m

crease

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ASTRONOMY. 209

Sr cr . XIII.

D IMENSIONS OF THE SOLAR SYSTEM .

Hr'

rnaaro, the distance of the Sun from the Earth

has served as the unit, by which w e have measured

all other distances in the planetary system. It now

remains, (in order to have a precise idea of those

distances), to compare this unit w ith the diameter

Of the Earth, and Of consequence w ith the know nmeasures in which that diameter has already been

expressed . This depends on the parallax of the

Sim; w hich has already been shown to be less than

and, on account of its smallness, difiicult to be

ascertained . The method w hich first presents it

self, does not lead to any thing more precise than

the limit just mentioned .

222. Since the ratios Of the distances Of the pla

nets from the Sun, to the distance Of the Earth

from the Sun , are known , if the parallax Of anyOf the plane ts w e re d iscove red, that Of the Sun

would, Of consequence , become know n .

This follow s, readily, from the law s of the planetarymotions.

V9». 11 .

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8 12 our t s or na ruam. rm t osornr .

This construction suppoees the obsm ers O m '

to

be either exactly, or neu'ly in the plnne of the orWfi f i’mw . bn t it may be extended to cnm in

w hich that condition does not take ph ee . l t re

quires , too, that the lmghnde d me flnw d m .

“tion should be accurately known . To avoid

been prd'

erred , for ascertain ing the peullnx. If

w e suppose observers, situated in respect of one

another , so that the line w hich Venus is seen to de,

scribe on the Sun’

s disk, is longer at the oqe ste

tion than the other ; the duration of the transit

w ill evident ly depend-

on the distenoe of the obser

vens fi‘

om one another, estimated in the direction

perpendicular to the lines which Venus traces out

on the surfaeeof the Sun . The dM ences of dtb

ration, therefore , depend on the paralln of the

Sun, or on a functwn of it ; md therefw e whw

that function w know n, the parallax may be mfer

red, fi'

om the comparison of the durations of the

transit . SeeWoona oose , Ast1-on. p .m ate.

The transit of 1769, w as observed atWardhus or

the North Cape , and also at 0 tabcité in the South

Semand was found to be longer at the formcr thanat the latter by QS

’

HlO" The difl

'

a‘

ence , suppo

sing the parallax to bave been S’fiBfl, should hay e

amoun ted to kfl‘w26“95 and bence the parellax isVmon , Art Dr Mn

a u nt’

s

214 our tmr s or aut eu r. Pa t t osormr.

Ss c '

r. XIV

OF THE ANNUA L PA RA LLAX A ND DISTANCE Of

THE FIXED STARS.

226. THE fixed,stars, as has,

been already Shawn;

have no parallax with respect to this Earth, or

any line that can be measured on its surface ; and

their‘

ditmm is so great. that it is yet doubtfill

whether they have any parallax, even w ith re

spect to the orbit of the Earth round the Sun .

A fixed star not only occupies exactly the same place

in the heavens, from whatever point of the Earth’

s

surface it is observed, but it does so w ithin a quan

tity so small as to be hardly measurable , even

when view ed from opposite extremities of a dia

meter of the Earth’

s orbit.

On the supposition that the star does change its situa

tion , when so view ed, the angle whichmeasures that

change is called the Annual Parallax of the Star .

227. If

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or r Ll s or NATURA L PHI LOSOPHY.

which w e see in the heavens. A s it cannot he

doubted, that the fixed stars are luminous bodies

like the Sun, it is probable that they are not near

er to one another than the Sun is to the nearest of

tha n . When , therefore , two stars appear like a

shuttle star, or very near to one another, the one

must be placed far behind the other, but nearly in

the same straight line , w hen seen from the Earth.

The same must hold at least in a certain de

gree , w herever a great number of stars are seed

« unw an ted in a small spot. In the starry nebu

la». therefore, such as the MilkyWay, w hich derive their light from the number of small stars, ap

pearing as if in contact w ith one another, it is

plain, that the most distan t of these must be manyWetland times farther off than the nearest, and

light must, of course , require many thousand years

to come from them to theEarth. The poet, per

l-‘

ields of radiance , whose unfading light

“as travelled the profound six lbou nd years,

Nor ye t arr ived in sight of mortal things.

”

e t the fields w hich he describes, are far within

the circle to w hich the observations of the estrone

me r extend

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m oun txrs or rum “. ra tw sor a r .

naturally consists of a ser'ies od

’

termg prw eed’

n gaccording to the powm et

'

one of the vuhble

because it inserts a term in the midst d'

a nnw

ber of otherr.

232 . 1f r and y are two variable quant it ietg tf

which several values have been determined from

Observation ; it y be assumed equal to a serios of

the powers of x, beginning from O, and going on

to as many terms as the re are observations, viz.

& c . ; then , if for

y ae , be put the ir corresponding values, a de

termined by observation , as many. equations w ill

arisc as there are unkriown coefi cien ts A, B, C

ant o be found, from which they w ill becom

known.

one of the unknow n quantities, and whefi the ia

tervala betw een the observations are éqaal, u it

supposed in what tbllom :

233. Le t a d,

servations made at tbe times o, m, 9 m, 8m, 4 m,8 m.

82! our tms s or NA TURA L n ut osor nv.

6. If there are three observations, y as

k

Asa - 1

_ 2fl+ 7

c . A ttention must be paid to the signs of the difi'

nr

ences . The succeeding quan tity is here alw ays sub

tracted from the preceding, and therefore if the

former is the greater, the difference w ill be nega

tive . On the subject of interpolation , see New.

TON, as above quoted . on [fi nite Series ;

L A CA ILLB Astron . p, 69 Nautical Almanaclc, ex.

planstion at the end .

Interpolations of this kind, cannot be supposed to ex

tend far beyond the interval within w hich the oh

servations are contained , unless it shall appear, on

continuing the observations, that the formula ap

plies to them, w ithout requiring the addition of

new terms. A s long as the addition of new obser

vations, requires an alteration in the formula, it is

certain that the true law is not discovered .

234 . Le t it now-be supposed, that the form of

the function is known from theory, but that the

constant quantities that enter into it, are to be de

termined by observation ; required, consideringthat every Observation is liable to e rror, in w hat

w ay

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g m us e s or unruna r. e a osor nr .

g u m balance one another, so that a more

w fi e gmud formula to he

y : A sin s + B sin 2 x,

and that from observation w e have eight values of

a and y , viz .

Il ence,

64-28 A 9 848 B7071 A B7660 A 984-8 B8191 A 9337 B8660 A 8660 B9063 A 7660 B9897 A 6428 B9660 A 5000 B

By adding the first four into one, secondfour, w e get

A BA B

and also the

and therefore,

x x

or A

In like manner, B : l '.2 ; so that the equation be

comes,

y = (l°

sin x + sia .

Tins is nearly the equation of the centre in the

Earth’

s orbit.

In this w ay all the elements of any of the planetary

orbits may be de termined M ucous!” or cor

rected if they are already nearly kn ow n . In the

con struction ofA stronomical Tables, the number of

equations combin ed has amounted to many hun

In the example above , no method w as to be fol

low ed, but that of dividing the original equations

into tw o parcels or groups, from the sums of w hich

the new equations w ere to be deduced . But when

it happens in the given equations, that the terms

involving the same unknown quan tity, have differen t signs, the best w ay is to order all the equa

tion s so that one of the n uknown quantities, as

A , shall and then

to add them derivative

equations . Let the same be done w ith B, C , &c.

296 c ou rs e s or uam a t PHILOSOPHY.

coefficient ; and the oooflicientn of tbo sgme uqknow n quantity, in the differen t equations, w ill become by that means as pnequal as they can be rea

dered, w hich con tributes to make the divisor byw hich that quan tity is to he found , as large , and it?

self ot'

course, as accurate as the case will admit of,

Suppose, for example, that the formula'

sin x sin 2 x 2 3,

was reduced in to a table, and that by a comparn

son w ith observation , it w as required to correct the

quantities and that is, A and B, ae

cording to our former notation.

Suppose also that the observations w ere made when

the values of x w ere 80° 45°,and tha t the errors in the T ables w ere in

these instances found to be .318, — 334 , - 083,

+ 044 + 05— 021 — 084 ; than , ceiling s« andb the corrections on A and B, w e have

500 a

707 a

8 66 0 .866 6=

3 66 0

0 + .O5

966 a .500 6

866 9 3 66 6

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ASTRONOMY.

PART Ii.

PHYSICA L ASTRONOMY.

SECT . I;

br'

rn s roacs s wm ca RETA IN rm:PLANETS IN

THE IR onmrs.

236.IF a body gravitating to a fixed centre,

have a projec tile motion impressed on it, in a line

not passing through the centre , it w ill move in a

curve and the straight line draw n from the M

(ly to the cen tre, w ill describe areas proportional

to the times.

Princip . Math. lib. r. prop. I .

P 9 a. Conversely,

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282 our t w s s or NATURAL. rnrt osornr .

nitely small portion of time, or the momentary in“:

crement of AB, o 6 w ill be the velocity at the

end of that time, and 2 A b e D (o

Now if CB = x, B b z i , then , since the

a — xsquare of the velocity is as

xa constant quan

tity 111“may be found ; so that e

2

and therefore 2ABED m“ 11

For the same reason, 2A 61 D ma

a a

And therefore 2EB b e m2

or dividing a by x i , and rejecting the terms in

volving the higher pow ers of i ,

g , and f z gm‘x

The centripetal forcef is therefore inversely as the'

square of the distance .

6. If the point G bisect AC , then , iii—GLCG

if c be the velocity acquired by falling to G,

c’ m“x l , or c z m; us therefore is the veloci

if

” true s t am osrom . 288

ty aoquired by falling fi’

om A half -wsy to the

centre C .

c . Hence also, the force at G : l c"x

£ 39 . If to a tangent of an ellipsis, a perpendi

cular be draw n from either focus, the distance of

4 . M P(6g. be a point in the ellipm ADBE, AB

the transverse , DE the conjugate axis, C the centre ,

Q and F the foci ; GPH a tangent to the ellipsis

in P,89 the perpendicular on it from 8 . Draw

SP, PF, and make FH perpendicular to GP.

Be cause the angles SPG, FPH, are equd , the'

triangle: SPG, FPH are equiangular, and there

SP : PF FH x SG .

But EH x 8G . CD’

, therefore

SP z PF

SG‘ SPCs

PFM dCB

934: M W ? n am d nm w somn .

240 , If a body urged by a centripetal ferc t

must be inverse ly.

as the squai'

e (if the distano

F,

M ym d vmg b tge emmm n A Let P l

bther posifion of the body, aml wi1

radius SP let an afch

ing AB in L ; the frelocity of the'

falling h'

t

L , and of the revolving body at P, w iIl h’

é

If, then , the velocity'

of the revbb tihg body n

mean distance , or at the point D, he called '

its veloclty at e

Now 0 is the velocity of the falling

w ell as of the revolving body

LN be taken equal to FP, the

ing body at L is = Lfi'i

.

equal to SP,hndmic PF,SN Sis

to that the poin t N7is givem The

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249 opruruzs or NATURA L ens opa r.

mi'

m

from which the proposition above - follows rea

b. Bymeans of this theorem, themass of theSun, and

of any of the Planets which have satellites, maybe compared w ith the mass of the Earth.

9. In the Mecam'

qtte Celeste, they are calculated from

the most exact data, as below

Quantity ofmatter in the Sun

in Uranus,

d. A s the ratios of the Diameters of the planetsknown from observation

, the ratios of their Bulb ,

being the same w ith those of the cubes oftheir diameters, are also known ; and hence the Densities,which are proportional to the quantities of matter,divided by the bulks, are found.

PHYSICA L A'

STKONOMY.

Dtmity of the Sun ,

of the Earth,

ofJupiter,ofSaturn,

ofDrum,

946. The immoveable point to w hich the plat

ets gravitate , is not the centre (if the Sun , but

1c centre of gravity of the solar system.

From the equality of action and reaction , the gravii

tation of the planets to the Sun must be accompa

nied by the gravitation of the Sun to the planets;so that the quan tity of the n

ibtion of the former,

estimated in any direction , must be equal to that

of all the latter estimated in the Opposite . The

Sun, therefore, moves in an orbit, about the only

point of which the condition beuno‘t be disturbed,by the mutual fiction of the surrounding bodies,

viz. the centi e b’

f gravity of the whole.

if there were only one planet, the Sim and that pla

net would describe similar conicsections, of which

their common centre of gravity would be one of

the foci ; their distances from that point being al

w ays inversely as their masses . If there is a num

ber of planets, the path or the Sun w ill become a

more complicated curve , but w ill be such as to fur

nish a centrifugal force in respect of each planet;

just able to counteract the gravitation toward it .

i on. 11. 47 . Th ,

242 oun w ss or NATURA L enrt osomr .

24 7. The centre of the Sun is never distant byso much as his ow n dhmetér from the centre of

gravity of the system .

The diameter of the Sun is equal nearly to .009 of

the radius of the Earth’

s orbit. Now , if w e sup

pose:the Sun , and all

'

the great planets of the sys

tem, Jupiter, Saturn and Uranus, to be in a

straight line, and the planets all on one side of the

Sun , the centre of the Sun w ill be nearly t he far

thest possible from the centre of gravity of the

w hole ; yet w e shall find'

on computation , that the

distance is not greater than .0085of the radius of

the Earth’

s orbit. w 'ronr Princip . lib . 1 11 .

prop. 12.

24 8. Thus the existence of the princ iple ofgravi

tation, is e stablished by induction from the law s

ofK EPLER, and from it , by reasoning downw ard ,

conclusions have been obtained concern ing the

quant ity of matte r in the plane ts, to which obse r

vation, w ithout the assistance of theory, neve r

could have reached .

It yet remains to be shewn, that the same force which

occasions the descent of heavy bodies on the Earth’

s

surface , at the rate of feet per second, w he n

diminished in the inverse ratio of the square of the

distance,is just sufficien t to retain the Moon in her

orbit .

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244 commas or NA TURAL rmLosornr .

St ar . 11.

OF THE FORCESWHICH DISTURB THE ELLIPTICA L

MOT ION OF THE PLANETS.

Wan t there are only two bodies that gravitate to

one another, w ith forces inversely as the squares of

their distances, it appears from the last’

section

that theymove in con ic sections, and describe , about

their common centre ofgravity, equal areas in equal‘

times, that cen tre either nemaining at rest, or mo

ving uniformly in a straight line . But if there are

three bodies, the action of any one on the other

We , changes the nature of their orbits, so that thedetermination of their motions becomes a problem

of the greatest difficulty, distinguished'

by the name

ofm s:Paonu n or w e r an : BODIES:

The solution of this problem, in its utmost generali

ty, is not w ithin the pow er of the mathematical

sciences, as they now exist . Under certain limita

tions, how ever, and such as are quite consistent

w ith the condition of the heavenly bodies, it ad

mits of being resolved . These limitations are ,

that the force which one of the bodies exerts on the

other two, is, either from the smallness of that ho

dy,’

or its great distance, very inconsiderable, ia*

respect

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our LrNs s or Na'

rosu. gart osoenr.

tion of the line passing through the centres of theSun and Earth.

This force at the conjunction, exceeding that part atthe disturbing force which draws tha

'

Moon to the

Earth, tends to diminish the Moonls gravity to the

Earth. A t the oppositionB it does the same, bybecoming negatiye ; for the Earth is

.

then drawnmore than the Moon, and the difl

'

erence is nearlythe same as at the conjunotion.

b. IfMN be taken equal toHE , and NO made per-

7

pendicular to the radius vector, the force MN is

resolved into two, one directed fromM to 0 ,les

sening the gravity of the Moon to the Earth, and

the otherdirected from 0 toN, parallel to the tan

gent to the q n“s

‘orbit at M, and therefore acce

lerating theMoonfrom O to A , retarding her fromA to D, sod so alternately in the other two qua

drants .

A t the quadratures C andD, the forceHK van ishes,

and the only remaining force is directed to the

centre ofthe Earth, so that the areas are there pro

portional to the times.

f’ The analytical values of these foroes are next to befound. Draw CED the line of the quadratures,

put SE'

a, EM the radius vector of the Moon’

s

orbit r, the angle CEM x, and themass of the

Sun as. The force that retains the Earth in its

our tmzs or s n ou t. rmtosormr.

The mean quantity of the

‘

force (1 3 sin

6, when it boun cer;- 3 i sin1 a), the fluent

gx + gsin x x cos xy;

and this, when x is an entire circumference, be.

and it gives $227

for the mean disturbing force

acting on the Moon, m the direction of the radius

vector.

250 . Hence , at the quadratures, the gravity of

the Moon to the Earth is increased,

by aquant ityequal to the mass of the Sun mult iplied inito the

radius of the Moon’s orbit, and divided by the

w ho of the Sun ’s distance from the Eatth

'

p at the

opposition and conjunction it is dimin ished by

twice this quantity ; and the effect upon the

w hole

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our t m s s os -m ruaar. enrt osor nv.

,orbit in less time. As the area described by the

radius vector in the primitive and the disturbed on

bit is the same , it can be shewn, that in conse

quence (if t he mean disturbing force,vector is increased by a 858th part, and the angular vdocity diminished by a 179th part .

E xposition win Systems duNeeds, chap. v. p. 213.

9“edit.

952. The annual equation is an irregularity in

the Moon’s motion , arising from the variation of

the Sun’s distance from the Earth, and bearing a

given ratio to the equation of the Sun ’s cen tre .

Since the Sun’s disturbing force is inversely as the

cube of his distance, when he approaches the

Earth at the perihelion , theMoon’

smotion is slow er ,

and, for the same reason , at the aphelion , it is

quicker than the mean . This produces w hat is

called the annual equation, equal nearly to (l

x sin Sun’

s mean anomaly.

This has a contrary sign to the equation of the Sun’s

253. An inequality in the same disturbing force

depending on the position of the transverse axis of

the Moon ’s orbit, in respect of the line drawn

from the Earth to the Sun, produces the equa

tion

254 our tw s s or s a t-

vas t rnuosoi a'

i '.

254 . The Variation is an equation derived fromthe force at right angles to the radius Vector,which, from the quadratures to the syzygies, s e

calcretes the motion of the Moon, and from the‘

(

syzygies to thq quadratures re tards it. This force

w as found to be proportional to the sine of tw ice‘

the Moon’s angular distance from the Sun , and

the equation itself is nearly prOportional to the

same quantity.

If the angular distance of the Moon frbm the Sm!

A, the variation , according to Mar i a, is

sin 2 A

(20) sin 3

‘

A

sin 4 A

Beside the above inequalities, first found out by OHservation, and now explained by the theory of gra

vity, there are several others of smaller amoun t ,which theory alone has discovered ; all that oh

servation could do, being to ascertain that some

unknown inequalities existed, which introduced

an uncertainty into all the calculations of the

Moon’

s plac e . These have been separated by the

theory, and the Tables, by that means, brought to

agree very nearly with observation.

955. One of the equations w hich theory has discovered, is remarkable for the great length of its

pe riod,

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256 onrunns or NATURAL rriiLosor'

flY.

of the transverse axis of the orbit, hr of the line of

the apsides.

957. The motion of the line of the nodes is pro

duced by that part of the Sun’s disturbing forcewhich is in the direction of the straight line

joining the centres of the Sun and Earth, and

proportional to the distance of the Moon from a

plane passing through the centre of the Earth atright angles to the line joining the c entres of the

Sun and Earth.

Suppose the Moon to be on the same side of the lastmentioned plane that the Sun is. Then , if in the

direction of her motion for the instant ju'

st past,

there be taken a line equal to the space passed over

in that instant, and if in the line drawn through

the Moon, perpendicular to the above plane , on

the side opposite to the plane , there be taken a

part equal to the space which the force urging the

Moon from the plane would have made her de

scribe ih the same time, then the trtre path of

the Moon w ill be the diagonal of the parallelogram

under these two lines ; and the momentarychange in the place of the no

'

de will be the distance

between the point where an arch having the dirce

tion of this diagonal, and another ha'

ving the dirce

tion of its side , meet the plane of the ecliptic . The

same w ill happen in other situations of the Moon ,

and the line of the nodes w ill thus have a motion

in

258 our t w s s or NA TURAL rm nosorn r .

260:From the same cause arises a variat ion it !

the inclination of the Moon’s orbit to the ec lipt ic

confined w ithin ve ry narrow limits. A con strue

t ion for de te rmin ing the inc lination for any givef l

t ime , is given ibid. prop . and is found to agree

w ith obse rvation .

What respects the motion of the nodes is thus com

pletely explained ; and it is here that the indirec t

method of de termining the Moon’s inequalities has

been most successful . It has not been equallyosc ’

in ascertaining the motion of the apside s.

To conceive , in general, the cause w hich renders the‘

apsides of the Moon’

s orbit more than 180 degrees

distant from one another, w e must begin w ith sup

posing the Moon at the low er apsis ; then , if that

planet w ere acted on only by the force of gravity,

the radius vector, after it had described

w ould arrive at the upp er apsis, or w ould be inter

sooted by the orbit at right angles.

But as the mean disturbing force , in the direction of

the radius vector, may be considered as a quanti

ty constantly taken from the Moon’

s gravity, the

portion of her path de scribed in any instant, w ill

fall be tw een the tangent and the arch of the ellipt ic orbit w hich w ould have been described if the

Moon had been acted on by gravity alone . The

actual path of the Moon , therefore , w ill be less

bent than the elliptic orbit would have been in the

same

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260 ovum ” or NATURAL rnxt osorn r .

elegance. Dr Suwn r has demonstrated this

26 1. If r be the radius of the Moon’s orbit

,

supposing it to be a circle , and ac ted on on ly by F,

her gravity to the Earth . Then if a mean disturb

ing force a:f be supposed to diminish the Moon’s

gravity, the greatest distance she w ill gO to from

and the cube of this

distance w ill be to the cube of r in the duplicate

ratio of the angle desc ribed by the Moon from

one apsis to the n ext to two right angles.

Tracts Hath. dfr Phys . Tract w . prop. 27 ; also

Hence the angle described from one apsis to the

x(F— 5f)

Farsi has adOptcd the principle of MAcnm, includingalso the action of the forces perpendicular to the

radius vector . Op era, tom. m . p . 380.

Vmcn, in the 2d volume of his Astronomy , has com .

pated the motion of the apsides according to this»

lost method, and makes the quantity of it in a sy.

PHYSICA L ASTRONOMY. 261

derisl revolution of the and

M us e’s Tables make the latter Dr

Su wul 'r’

s Th erm gives W8 4

§d§ ’ the value sssigned to it, §250.

The result of these inve stigations, therefore, agrees

nearly with observation ; but it cannot be denied

that the principle on which they are fmmded is li .

able to some object ions, so that if it wm not for

the information derived from the direct solution

of the problem of the three bodies, it might still be

doubted, w hether the principle of gravity account

ed exactly for the motion of the Moon’s spsides.

CM l RAU’

l‘

, who first compared the result of that

solution w ith observation, met w ith the same didi

culty that Nsw roa trad done , and found that his for .

mula gave only half the true motion . He therefore M agined that gravity is not inversely as the

squares of the distances, but follow s a more compli.

cated law ,such as can only be expressed by g for

mula of two terms . In seeking for the eoefi cien t

262 OUTL INES or NA TURA L PHILOSOPHY.

position , that the force of gravity is inversely as the

square Of the distance .

262 . Since , by the Moon’s inequalities, the ra

tio of the Sun is d isturb ing force tothe force w ith

w hich the Moon gravitate s to the Earth become s

know n ; if the rat io of the forme r, to the w hole

force re taining the Earth in its orb it, could be

fouhd, the ratio of this last to the force Of the

Earth on the Moon w ould also be found , and from

thenc e the rat io of the d istance of the Moon to

the d istance of the Sun from the Earth.

The latter ratio w ould be given, because ifE andfare the forces that retain the Earth and the Moon

in their orbits if a and r are the radii of those or

bits, P and p the periodic times,

a r0F F

P‘P‘

and so if the ratio of F to F’ be given , P andp bemg also given, the ratio Of a to r is found .

In this w ay the Sun’

s distance might be found fromhis ow n f orce to disturb the Moon , the idea of

Which appears first to have Occurred to Dr MA

THEWST EWART . The principal difficulty is to

‘fitld‘

the ratio of the disturbing force of the“

Sun to the force w hich the Sun exerts on the

Earth. If the expression of that force was carried

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u n me t. “n arrows . 266

Ssc'r. III.

usruansuc e s 1N THE MOT IONS or THE PRIMA

ar PLANET: PRODUCED BY rnm a ACT ION ON

ONE s s or n aa.

Ir is necessary, in this inquiry, to know-

the quanti.

ties of matter in the different planets ; and these

have been she edy calculated for the planets which

have satellites. The masses of Venus and Mars

have been computed by M . LA Pu cn from some

disturbances which they appear to produce on the

Earth’

s motion . The mass of Mercury has been

estimated, from supposing his density, and that of

the Earth, to be inversely as their mean distances

from the Sun. This law holds w ith respect to

the Earth, Jupiter and Saturn , and analogy autho

rises the extension of it to Mercury. From know

ing the density and the bulk, the quantity of mat

ter is inferred. The mass of the Sun being 1 ,

that of Mercury is M of Venus

of Mars those of the others being as

266 . ourms rs or NATURAL r a t t osor nv.

The gravitation of one plane t to another, is expres.

sed by the quantity of matter in each, divided by

the square of the distance ; and therefore the tw o

bodies tend to come together w ith a force that is as

the sum of their masses divided by the square of the

distance ; so that w hen the motion of both is re

ferred to one only, the force must be expressed by

the sum of the masses divided by the square of the

distance .

The disturbances produced by the action of the pri

mary planets on one another, are of more difficult

investigation than‘

those produced by the Sun on

the motions of the Moon , because the disturbingbody

'

18 not at an immense distance, as in the lat

ter case . The only sure wayof subj ecting them to

calculation , is by a direct'

solution'

of the Problem

of t he Three Bodies ; the part of w hich that may ,

be accounted quite elementary 18 nowto be cousi

dered.

264 . The force s w hich act upon a body to

how ever many centres they tend,and w hat ever

law they may obey,may be resolved lino th

d irec tion s of three lines or as es, glven 1n pos1t1on,

at righ t angle s to one anothe r,

This is eviden t from Dynamics, vol. 1 . 70 .

Theadvan tage of this resolution of fbrces for deter

mining the motion of a body attracted to several

centres,

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PHYSICA L ASTRONOMY. 263

parallel to .r or SM, that afi'

ects the relative mo

tion ofP to S, 1s

For the action of the Sun on P being resolved as

above, the part of it that is in a direction parallel

to x or to SM is and the action of P'

on P

being resolved in like manner, the part of it paral

lel to SM ism

'

in) Now,supposing x to

vary by the momentary incremen t or fluxion 5, 3’

remaining constant , the increment of q, or q w ill

be such that 5! and thereforea,

m'

(xw hich, therefore, is the force

by which P'acts directly on P ; and if to this be

added the Sun’

s force in the same direction , viz.

the amount of the direct action of S and P’

on P, in a line parallel to SM, is

976 OUT Lnts s Or NA TURA L rut t osorm'

r.

Now, the Sun, by the united action of the plan ets P

and P', is draw n in the direction opposite to th'

m

last by a force and as the Sun is

here considered as immoveable , w e must conceive

this force to be transferred to Pin the opposite di

rection .

Thus the whole action on P, or the force F

By w riting in this formula y, y’and 31, instead ofx;

lx

’

and x, w e have F’ tr

oy

and in like manner

m 2

The substitution of these values of F, &c . in the

three formulas of the last article , w ill give three

fluxionary equat ions, On‘

which the mot ion of P

depends.

The same being done for P'

, there w ill come out six

fluxionary equations, from the in tegration of

w hich

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274 ouumr s or NA-

rua a r. rm nosor nv.

a. or the motio’n'

ot‘ in; apsidt-s, s

"a due to Venus,l i t t oMm-

s, and 52 5to Jupiter

'

nearly.

b. In"the secular diminution of the fi ltration to the

centre , 4 m s is an;area of Venus, v .94 of

Mars, of fiupiter, and the rest is produ’

edd byMeretn'

y and Saturn .

A s it is not the cen tre of the Earth , but

the cen tre of gravity of the Moon and'

Earth

which desc ribe s equal areas in equal time s, about

the centre of the Sun , the regularity of the Earth’s

mot ion is disturbed on that account , and the

Earth is forc ed out of the plane of the ecliptic .

a . The irregularities thus communicated to the Earth

are, by observers on its surface , transferred to the

Sun ; the Sun, therefore, has a motion in longitude,

by which he alternat ely advances before the point

that describes the elliptical orbit in the heavens,

andlfalls behind it ; and in like manner alternately

ascen'

ds above the plane of the ecliptic, and descends

below it .

6. These inequalities are small. The -mass of the

Moon is about {3 th of that of the Earth ; the

distan ce , therefore, of the centre of gravity of the

Moon and Earth, from the centre of the latter,must be less than a semidiameter, and therefore the

in equality in the Sun’

s place must be h as than his

horizontal parallax.

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1 76 ouruus s or run -

nu t. rn rt osorar .

273. The inequalit ie s of the small planets Ju

no, Vesta, Ceres and Pallas, have ne t yet been

computed ; the disturbances wh ich they must suf

fer from Mars and Jupiter are‘

no doubt con side o

table , and, on account of the ir vicinity , though

their masses are small, they may somewhat d i

sturb the motions of one anothe r. The lr act ion

on the othe r bod ies in the system is probaby ia

sensible .

A s two of these planets have nearly the se ine perim

dic time , they must preserve nearly the same dis

tance, and the some aspect w ith regard to one ano

ther. This offers a new . case in the computation

of disturbing forces, and may produce equations

of longer periods than are yet know n in our sys.

tem.

274. The ac t ion of Jupiter and Saturn on one

another, produces an inequality in the mot ion of

e ach, of considerable amoun t , and of a long pe

riod, viz. years.

a . If n express a number of years reckoned from the

beginning of 1750 , S the mean longitude of Sa

turn , and I of Jupiter, reckoned from the same

time , then the great equation that must be applied

to the‘

mean longitude of Jupiter, or to I, is

5 a x 02042733) xsin (5s e 1 n x

THYSICAL ASTRONOMY.

and that which must be applied to S is

n x 0". l ) x

sin (5S 2 I 11 x 58”.88)

These equations are to one another nearly in the ra

tio of 3 to 7. As the quantity 58 — 2 I n xrequires years to increase from 0 to

360 degrees, therefore the above equations require

that period to run through all their changes. See

LA PLAC E, Mem. Acad. dcs Sciences, 1785, 1786.

Also LA Lanna, 4 st. tom. 111. 3670.

275. Besides these two great inequalities, the re

are ten others, arising from the action ofSaturn,

to w hich Jupiter is subj ec t , and w hich may

amount w hen greatest to l l’.56 the re are also

six to which Saturn is subjec t from the act ion of

Jupiter, and these may amount to

For the part icular forms of these equations, see La

Lu na , ibid, and fi ner , Ast. vol. 111 . p. 94 . and

109 .

276 . The motion of the apsides, and the change

of eccentric ity in the orb its of Jupite r and Saturn ,are chiefly produced by the ir action on one ano

ther ; but in the d 1sturbance w hich the planes of

their orbits sufi'

e r, the other planets have”

a sen.

sible effec t.

Jumrnh.

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280 oue Ns s or NATURAL r

'

n l t osorur .

time ,‘

by considering the Orb it as an elltpSis, the

elements of w hich are cont inually changing.

T his is the method of LA Gu tte rs, and is followedinthe Mecam

'

quc Celeste, Part . u. chap. 9.

Dr Hu s s y , when he predicted the return of the co.

met of 1682, took into considerat ion the actiqn of

Jupiter, and concluded that it w ould increase the

periodic time of the Comet a little more than a

year ; he therefore fixed the time of the re - appear

ance to the end Of the year 1758:PT the beginningof 1759.

He professed, however, to have made this calculationhastily, or, as he expresses it, led calayw . Synop,

sis qf the Astronomy qf Comets .

279. GLA InAUT , on calculating w ith great care

and labour the effec ts both of Jupiter and Saturn,found that the re turn of the Comet w ould be re

tarded 51 1 days by the former,and 100 by the

latte r in consequence ofw hich hefore told that its

re turn to its perihe lion would be on the i5th of

April 1759 .

He said at the same time , that he might be out a

month in his calculation. The Comet actually

reached its perihelion on the 13th of March, just

days gerlier than was predicted ; thus affordinga

anvst eA L A sr xouomv. 281

‘

a veryremarkable va ification of the theory of Gra,

vity, and'

the caleulat ion ofDisturbing Forces,

This Comet may be expected again about the year

1835. The investigations of L A PLAC E w ill re nder it much easier to calculate the quan tity bywhich its arrival may be anticipated or retarded bythe notion of the planets .

A Comet, which w as observed in 1770, had a motion,when carefully examin ed, wh ich could not

’

be we

conciled w ith a parabolic orbit, but w hich might be

represented by an elliptic orbit of moderate eccen .

tricity, in which it revolved in the space of five

years and e ight months . This Comet, how ever,

had ne ver been seen in any former revolution, nor

has it bee n seen in any subsequen t one .

980 .

'

M rBtmxm n'

r, on trac ing the path of this

C ome t , found that bet w e en the years 1767 and

1 770 , it had been ve ry near to Jupite r, and again

had come ve ry near to that planet in 1779 ; he

there fore conjec tured , that the disturbance ofJu

p ite r might have so alte red its original orbit, as to

re nde r the Come t for a t ime visible from the

Earth ; and may have so changed it'

again , hitter

one revolution ,as to restore the Comet to the same

region in w hich it had formerly moved . This

conjec ture has been confirmed by a careful appli

(sailor) of the tbtmulas of the M ecaniquc Celeste.

ourmnas or n e w e s t. rnrnosorn r .

Mr Bnm m r found that the Comet had come so

near to Jupiter between 1767 and 1770, that it

may have been brought from an orbit of which the

semitransverse w as (that of the Earth‘

s or

bit being l ), and in which it revolved in a period

of years, to one in which the semitrans

verse w as and in which it revolved in five

years and eight months, as it was at that time oh

served to do. While revolving in this orbit, it

came near to Jupiter again ; and its time of revolu

tion, and its distance w ere so changed, that the lat

ter became and the former 16 years. In

this orbit it cannot, any more than in its first,

come so new the Earth as to be visible .

The preceding is the greatest instance of disturbance

that has yet been discovered among the bodies of

our system, and furnishes a very happy and un

expected application of the theory of Gravitation.

281 . Though the Come ts are disturbed in so

great a degree by the ac tion of the Planets, they

do not appear by the ir reac tion to produce any

sensible effects.

This must no doubt arise from the small quantity of

matter which a Comet contains.

The Comet of 1770 came so near to the Earth, as to

have its periodic time increased by days, ac

cording to LA Pu c n’s computation , and if it had

been equal in mass to the Earth, it would have

made

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.286 oun mzs or NATURAL ratw sornv.

Se er . IV .

or THE m sruaamvcas WHICH THE sa rumrss

or JUPITER surran FROM THE ACTION or'

os i‘

.

ANOTHER.

282. Tm; application of the same principles to

the satellites ofIupiter, has fully explained all the

irregularit ies which had been observed in their

motions, and has reduced under know n law s seve

ral others, of which the existence had been indi

stinc tly perce ived.

A very remarkable relation takes place betw een themean motions of the first three satellites, as re

marked 185; the mean motion of the first satel

lite tw ice that of the third, being equal to three

times the mean motion of the second, reckoningfrom any instant of time . L a y m an has shewn,

M ecam'

quc Celeste, liv. I t . chap . 8. that if the primi

t ive mean motion of these satellites w as near this

proportion, their mutual action on one another,

must in time have brought about an accurate con°

fom ity to it.

r a rsacA L ASTRONOMY. 285

It follows, that Long. 1a Sat. 3 Long. 2d Sat .

8 Long. 3d Sat. a constant quantity ; and it has

been found , since ever the sate llites w ere observed ,

that this con stan t quantity has been nearly equal to

This last must be the result of original con

stitution .

288: The fi rst satellite move s nearly i n the

pianoof Jupite r s equator, and has no ecbent ric i

ty exc ep tWhat is communicated to it from the

third and fourth , the irregularitie s of one of

these small planet s produc ing similar irregulari

ties i n the re st . It has heside an inequality chief.

ly produced by the action of the se cond ,’and

‘

e ir

cumsc ribed by the pe riod of d ays.

984 . T he orb it ofthe second satellite moves on a

fixed plane, to which it is inc lined at an angle of

. i3", and on which its nod e s have a ret rograde

motion , so that they comple te a revolution in

I4 years.

The motion of the nodes of this satellite 18 one of the

principal data that have been used for determiningthe masses bf the satellite s, w hich are so necessary

to be known m computing theirdisturbances . This

satelli te has no eccentricity but what it de rives

from t he action of the third and l‘

ourth.

2 85. The

ourmN58 or sta t us.“rati osdp nv.

985. The third sate llit e moves on a fixed plane

that isb e tw een the equator and the orb it ofJupiter

, and is inc lined to that plane at an angle of

its node s making a tropical revolution, (re

t rograde ), in 14 l .739 years.

The equator of Jupiter rs inclined to the plane ofhisorbit, at an angle of3° the fixed [11111108 011

which the planes of the orbits move, are determi

ned by theory, and probably could never be disco

vered by observation alon e .

286. The orbit of the third satellite is ecc entric ;but appears to have tw o d istinct equat ions of the

centre one which really a rises fromits ow n ec

centric ity and another, w hich theory shew s to be

an emanation from the equat ion of the centre of

th e fourth satellite . The first equat ion is refer

able to an apsis, w hich has an annual motion of

2° 36' forw ard in respect of the fixed stars ;

the 2d equation is refe rable to the apside s’

of the

4 th sate llite .

These two equations may be considered as formingone equation of the centre, referable to an apsis

that has an irregular motion . The twa equations

coincided in 1682, and the sum of their maxim was‘

In 1777, the equations were opposed, and

the ir difference was 5’

Observation

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9290 our t mas or NATURAL rnrt osornv

II. That the planets all move in the same di'rection, as both

“

primary and secondary do

from w est to east

III. That the planes of their orbits are but littleinclined to one another.

But for these three conditions, terms of the kind

mentioned above w ould come into the expres

sion of the inequalities, which might therefore

increase w ithout limit.

These three conditions do not accesso

out of the. nature of motion or of gravitation,or from the action of any physical ca

'

use w ith

w hich w e are acquainted . Neither can theybe considered as arising from chance ; for the

probability is almost infinite to one , that,w ithout a cause particularly directed to that

object,"such aconformity could not have ari

sen in the motions of thirty- one different bo

dies scattered over such a vast extent .

The only explanation , therefore, that remains,

is, that all this is the work of intelligence and

design, directing the orig ina l constitution of thesy stem, and impressing such motions cm the parts

as were calculated tog ive stability to the whole.

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396

298.

OUTLINE: NATURAL rumosornr .

Snc '

r . VI.

FIGURE OF THE EA RTH.

FROM'

observation it has already been infer

re d, that the Figure of the Earth is nearly that of

an oblate spheroid, of w hich the greateraxis, the

diameter of the equator, is to the less, the axis of

revolution , as 312 to 3 11 .

The strict mean ing of the phrase, the Figure ‘!f theEarth, has already been defined, and must be care

fully kept in view , in searching into the causes

which have determined it .

Since the Earth revolves on its axis , it is evident,

that its parts are all under the influence of a cen

trifugal force, proportional to their distances from

that axis, and that if the mass w ere fluid , the

columns toward the equator, being composed of

parts that are lighter, must extend in length, in

order to balance the columns in the direction of the

axis. By this means an oblateness or elevation at

the equator w ould be produced, similar, in some de

gree at least, to that w hich the Earth has been

found to possess . Though it is not evident how

the centrifugal force would produce such an effect

on:

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300 our tmas or NATURAL rurt osornr .

But though it w as thus demonstrated that the parts

of a homogeneous fluid, on which the figure of the

oblate spheroid just described was any how indu

ccd, would be in equilibn‘

o, yet it w as not shewnconversely, that, w henever an equilibrium takes

place in such a fluid mass, the figure of the mass

must be the oblate spheroid in question . D’

A uu

ne ar indeed shew ed, that there are more sphe

roids than one in which the state of equilibrium

may be main tained ; and this result, though it was

not observed by MACLAuam , might have been in

ferred from his solution . L n Gaunaa afterw ards

proved , that the solids of equilibrium must always

be elliptic spheroids, and that in general there are

two spheroids which satisfy the conditions.

In the case of a homogeneous mass of the mean den

sity of the Earth, revolving in the space of 231156’

M, one of the spheroids is that which has been

mentioned ; the other, is one in which the equato

rial diameter is to the polar, as 681 to 1 . Mb».

Acad des Sciences, 1784 . L A Pnacn has added

the limitation which follow s.

30 3. A fluid and homogeneous mass, of the

mean density of the Earth , cannot be in squill

brium w ith an elliptic figure , if the time of its rm

tation be less than eh 25m if the time of t e

volut ion is greater than this, the re w ill always be

two

r nr srcA L ASTRONOMY. 30 1

two elliptic spheroids, and not more , in which an

equilibrium may be main tained .

If the density Of the fluid is greater than the mean

density of the Earth, the time of rotation w ith

which the equilibrium ceases to be possible , is bad,

by dividing 2“25m 1 7980 by the square root of the

density Of the fluid, that of the Ear th being unity ,

L A PLACE , Theor ie da Mous ement cc de In Figure°

des Plund er, Paris, 1784 , p . 126.

30 4 . If the fluid mass, supposed to revolve on its

axis, be not homogeneous, but be composed of

strata that increase in den sity tow ard the centre ;

the solid Of equilibrium w ill still be an e llipt ic

spheroid, but of le ss oblatene ss than if it w ere ho

mogeneous.

This w as demonstrated by C LA l nau'r , Theorie, 8m.

NEWTON fell into the mistake Of supposing the

con trary to be the case , or that the greater densi

ty tow ard the centre , w ould be accompanied w ith

greater oblateness . If the density increase , so as

at the centre to be infinite , the ellipticity is

$289 s— ‘is which is the case Of the least el

is the case Of the greatest .

296. Hence

cos OUT LrNas or NATURAL rnrt osorar .

305. Henc e , as the elliptic ity Of the Earth has

1 1been shew n to be less than

230 3 12near

ly), it is eviden t, that if the Earth is a spheroid Of

equilibrium,it is denser toward the interior.

The greater density of the Earth tow ard the centre ,

is in itself probable, and has been put beyond all

doubt by very accurate experiments, made on the

sides Of the mountain Schehallien in Perthshire , by

the late Dr MA ss vNE .

By Observations of the zenith distances Of stars,

made on the south and north sides Of that moun

tain , the difference Of the latitude of tw o stations

was de termined. A trigonometrical survey Of the

mountain , ascertained the distan ce betw een the

same tw o poin ts, and thence , from the know nlength Of a degree Of the meridian , under that pa

rallel, the differe nce of the latitude Of the stations

w as again inferred, and w as found less by l l".6

than by the astronomical Observations.

The zeniths Of the stations had therefore been sepa

rated from one another more than in the usual pro

portion Of the meridian distance ; and this could

only arise from the plummet on each side , he

ing attracted toward the body Of the mountain

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303. cou rs es or NA TURAL enrt osoe nr .

ourement of degrees, and from experiments with

the pendulum.

As, in the actual figure of the Earth, the compression,

or the ellipticity, is nearly .0032, 60 . if w e take

this from .008095, the remainder, .005495, or

1is the diminution of gravity from the pole to

the equator . A nd the gravitation at any other

point of the spheroid, is g (l .0054-95sin 2 A),

g being the gravity at the equator. The length of

an isocronous pendulum is expressed by the same

formula. This agrees nearly w ith the observations

on the lengthof the pendulum in different latitudes.

See a Table of them, VINCE, Act. vol. u. p . 105.

L A L anna , Ast. 2712. Bro'r, Ast. tom. m.

p . 14-8.

The lengths of the pendulum in different latitudes, are

less subject to irregularities than the lengths of de

grees ; the intensity of gravity being, as might be

expected , less affected by local variations than its

direction .

807. The inequalities on the surface of the

Earth, and the unequal distribution of the rocks

which compose it, w ith respect to density, must

produce great local irregularities in the direc t ion

of the plumb- lin e , and are probably the causes of

the inequalities observed in the measurement of

contiguous

i n i sxcs t ASTRONOMY.“ 805I

contiguous arches of the me ridian; even where the’

work has been conduc ted w ith the greatest sk ill

and accuracy.

This is exemplified in the great areaof themeridian

measured across France ; and in thosemeasured inEngland and Hindustan . The cause may be some

thing concealed under the surface, which can at

present only be a subject of hypothet ical; or, at

best, of analogical‘

reasoning.

These irregularities are so considerable, that the’

spheroid which agrees best w ith the degrees in

France , is one having an ellipticity offile"near

ly double of what may be accounted the mean xelv

lipticity.

308. The apprmdimati'

on w hich, notw ithstand

mg these irregula rities, the figure of the Earth'

ha'

s

made to the sphe roid of equilibrium, cannot,in

a con sistency w ith other appearance s, be ascri

bed to its having been once in a fluid state .

Though the action of water may be evidently traced

in the formation of those stratified rocks whichVOL . II. U compose

0 0 7 1.1 l 0 7 NATORAL PmLOSOTHIY.

“ pose so h rge a m a th - i m m ane

fl ee s i t is of water, tunaporfing or depod tmgt be

fiugments and dem'

m d solid bodies. With re s

gud to those roch that cont’

ain no sueh detritua,

but have the eharaeter of a '

p td lil ofion in aw t

er or leu degreefi t is not eviden t that they are of

aqueous formation. The only action of watn '

,

M af which w e have pay distinct evidenee in the

roidal figune of the Earth.

let in its primit ive form, the p rominent parts

are subjec t to be worn down , and the detritus to

be carried to the low e r parts, occupied by water,

w here they acquire a horizon tal st ratification , and

are, by certain mineral ope rations, afte rwards con

solidated in to stone ; such a body, in the course

of ages, must acquire a surface every w here . at

right angle s to the direc tion of gravity, and com

sequen tly more or le ss approximating to a spin:

mid of equilibrium .

The natural history of the Earth gives great conn

tenan ce to the suppositions here in troduced ; which

therefore seem tb furnish fine most rational caplm

nation of the ellipticity belongingw the lamb, andto the planets mat ure known to revolve on their

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808 ourrmns or NATURA L rurt osornr ,

Sac'r. VII .

0 1 THE PRECI SSION OF EQ UINOXES:

31 1 . THE precession of the equi-

norms, is the slowangular motion by which the intersection of the

equator and ec liptic goes backw ard, at the rate of

annually while the inclination of these

planes continues nearly the same ; so that the

pole of the equator desc ribes a c ircle about the

pole of the ecliptic in the space of 2574 8 years

nearly .

In seeking for the cause of this phenomenon, it is natural to inquire how the gravitation toward distant

bodies, such as the Sun and Moon , may afl‘

ect the

Earth’s rotation on its axis.

319 . From what has been proved of the force

w ith w hich the Sun disturbs the motion of the

Moon, it is evident, that every partic le in that he

misphere of the Earth w hich is turned tow ard the

Sun , is drawn toward that body, w hile every par

t iele in the other hemisphere, is draw n in the 0p

posite direc tion, the force that acts on any particle

be ing,

PHYSICAL a s'

raortour . 309

b eing as its distance from the plane that separates

t hese hemisphe res .

that if the Earth were a perfect

orces acting on the opposite he .

would exactly balance one another,

could produce no motion in the Earth or its

e Earth may be con sidered as a

'

sphere

by a spheroidal she ll orm ,

equator. The tendency of the Sun’3

meniscus, except at the time of the

alw ays to make it turn round the

of the equator w ith the ec liptic , to

plan e of this latter c ircle .

matter of the meniscus msy be regarded as

ring round the Earth, in the plane of the

Now , the solar foree acting on the part of

that is above the ecliptic , may at every

be resolved into two ; one of w hieh is in the

of the equator , and the other perpendicular

T he result of all the latter, must be a fores

ring a motion round itseclip tic . The same holds of

the ring that is under the ecliptic .

illthe equator had no other motion,

ound its inte rse c tion w ith the ec lipU 3 tic,

810 c ou rs es or NATURAL ra osornr .

t ic, till it coinc ided w ith that plane ; the line of

their in tersect ion remain ing all the w hile at rest.

The mean quantity of the solar force which it thus

decomposed 2; r ecs ), where as is the mass

of the Sun, a the mean distance of the Sun from

the Earth, r the radius of the equator, and 3 the

declination of the Sun .

The part of this force which is perpendicular to the

1plane of the equator, and w hich tends to make it

move round the line of its nodes, is

3m

Fr e

’

ofi X sin ) .

3 15. As the ring w hich‘

surrounds the equator,

at the same t ime that it has the tendency Just de

scribed, revolves ou an axis perpendicular to its

p lane in t wenty- four hours, it will not revolve on

either ofthese axes, but on one in the same plane

wh ich“

divides the angle b etw een them, so that

‘

the'

sine of its angular distance from each axis, is

m the inverse ratio of the angular Velocity rahad

that axis.

If the arch, round the intersection of the equator

and ecliptic, w hich the solar force acting upon the

ring or the meniscus, would make the Earth describe

,

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f our tms s or us ruas t rul t osorn r .

Let ABC (fig. be the ecliptic ; the order of the

letters A , B, C , marking the order of the signs ; so

that if A be the vernal , B is the autumnal equi

no: AGB a circle perpendicular to the ecliptic

G the pole of the ecliptic ; P the pole of the equa

tor ; and APB the equinoxial colure . HG'L

‘ GK

the obliquity of the ecliptic ; HPG a parallel to

the ecliptic passing through the pole of the equator.

The equator is not represented in this figure ; but

the half of it, on the side turned tow ard the eye,

is supposed to be under the ecliptic ; its plane beingat right angles to PP.

T he change w hich the motion c makes on the pole

P, tends to bring it nearer to A . For the diurnal

revolution about the axis PD, being in the direc

tion ABC , and the motion'

about the axis AB be

ing in the direction PG, it will be in the quadrant

PA that these motions w ill be opposed, so as at a

certain point P’

(and in all the points of the line

P’D) to destroy one another. PP

’

is the arch, of

w hich the tangent has already been shewn to

fv

.

If now a great circle he described from the pole P4,it w ill be . the new equator, and the point in which

it w ill intersect the ecliptic, w ill be the new equi

nox . The relation of these small variations to one

PHYSI CA L a sraouomr . 8“

begb est seen by

’

psojeefing a pm or

aplsne touchislg the Earth ia A.

N°2.) be a small part of the ecliptic

straight line ; AP’

a like portioncolure and AQ of the equator.

H‘

AA'

he talccn equal to PP'

, in the former

and if A 'O be drawn perpendicular to AP ,

represent the new equa tor ; and AC , w hich is

{a antecedeplia, w ill represent the preces

1

however, cannot permanently revolve

diameter that is oblique to the plane of

because the centrifugal forces on Opposite

such a diameter w ould not balance one

and must therefore tend to bring the

the ring to be perpencficular to the new

to make the primitive axis ofcoincide w ith the latter.

change , therefore , w ill begin

in the former case , and the pole of tzhe

ill con stantlymove in anteceden t“round

the curve in w hich it will mm is the

K ; for as the great circle APB touches

and as the arch PP’

is veryM P’

the circumference of the latter

enlyWa quantity tha t is m nescetit ia re

cf PP’

. P'

will therd ore deacribe a circle

OM our t mxs or NATURAL rurnosopur .

round G, moving in antecedents!; at the same rate

with the line of the equiuoxes.

3 17. If, therefore, the equator had rece ived no

motion but from the solar force , its inclination to

the e cliptic would have changed con tinually ; but

the line of its intersection w ith that c ircle w ould

have remained at rest . In consequence of the d inr.

nal revolution , this efl'

ec t is entirely reve rsed 5the

inclination of the two planes remains constant , and

their intersec tion continually revolves.

This paradox is remarked by LA PLACE, Systems dz;Maude, p. 210. His solution of it is not so eleg

mentary that it could be adopted here .

318. In order to calculate the precise measures

of the d ifferen t actions that have now been traced,the momentum w ith w hich the solar force w ould

make the spheroidal mmniscus already referred to,

begin to revolve'

about the line of the equinoxes,supposing it to have no other motion , must be de

terminad ; the w hole of the terrestrial spheroid

must tend to revolve about the said axis w ith a

momentum just equal to this quantity.

Nawron, who first resolved the problem of the pre

cession, was in an error wi th respect to this part

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6 16 ourme or h arve s t rurt osorur .

319. The Moon produces a similar re trograde

t ion in the intersection of the equator w ith the

plane of the lunar orbit, w hich, being proportional

to the cosine of the inclination of these planes, is

subject to continual variation ; but the mean

quantity of the precession w hich is thus produced,

relatively to the ecliptic , is the same as if the

Moon , moved in the plane of that c ircle .

LA Pu s s , Systems dag Meade,liv. 4am , chap . 13.

p . 271. 2de edit .

If the precession due to the Sun’

s force be that

which is produced by the Moon is which is

to the former nearly as 7 to 3. If the effect of

the Sun w ere reduced to that of the Moon

would be triple of it, which is agreeable to the la

test results deduced from the theory of the tides,as will be seen in the next section .

820. The

tHYSICAL a sr aosom . 847

action of the Moon prod

hich diminishes the

tional to the siga of the dlsa nce

asc ending node from theWe al

besides this, a diminution in the

e ec liptic , proportional to the co

d istance .

two inequalities constitute the Nutstion ; and

result here stated from the theory of gravita

is conformable to tha t w hich was before gi

from observat ion , 218 . Conformfi ly to what

is said there , these two inequalities may be ex

pressed by the revolution of the extremity of the'

Earth’

s axis produced to the Heavens, and describing an ellipse , as there represented ; of which the

greater axis is to the less, as the cosine of the obli

quity of the ecliptic to the cosine of twice that

39 1. The prec ession , on account of variations

in the solar ac tion,asWell as in the lunar force , is

subjec t to some inequalities, not included in the

p rec eding theorem s.

The ameunt ot‘

the pt ecessiorlfincluding all these in e

qunflties, may be calmdated for any peri od of tima,

by a formula given in the M ecenique Celeste, which»reduced to the sexagesimal notation , is

50".41.2 x t12954

'

sin 1". t298 x t . .

0 is the number of years reckoned from 1750 .

918 our tmas or naruaar. r‘

nt t osorn r .

As the annual precession is not always the same, the

length of the tropical year, in remote ages, has been

somewhat different from what it is at present . In

the age of e ancnus, it was about 10' longer.

The syderial year, as already observed, remains ia

variable .

Diurnal Rotation .

322 . The veloc ity of the Earth’s rotation on its

axis, or the length of the day, is not afi’

ec ted by

the action of the Sun or Moon , in such a de

gree as can ever become sensible, even to the ni

cest observat ion .

Systems doMaude, p . 271. The small inequalities so

produced, do not accumulate by time, but quicklycompensate one another.

323. The motions of bodies near the surface of

the Earth, tend, in some cases, to alter the veloci

ty of the diurnal rotation ; but if these motions

are only such as w e at pre sen t perce ive , their ef

fects, like the preceding, must for ever remain iri

sensible .

A body which, by descending froma height, or bymoving from the equator toward the poles, comes

nearer to the Earth’s axis, tends to accelerate the

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enr stcxr. asr aonomr . 821

the form of horizontal strata, the axis of the

Earth’

s rotation may have been very diiiierent from

what it n ow is ; it may have gone through a long

series of changes, and may have’

carried the equa

tor, and the accumulation of w aters which ecoemo

paniad it, over regions from which they are now far

distant .

Many facts in the natural history of the Earth, and

of the min eral kin gdom, give countenance to these

suppositions ; and if it be true that the m an

c ient strata have been se t on edge , and that conti

nents have been raised up by the action of an ex

pe nsive force in the in te rior of the Earth, such se

t ion may have materially assisted in changing the

position of the Earth’

s axis.

Obliquity (f the Ecliptic.

325. The position of the ec lipt ic is subjec t tochange by the action of the plane ts each of them

produc ing a ret rograde mot ion in the in tersectionof the plane

'

of the ec liptic w ith the plane of its

ow n orbit . This does not affe c t the inc lination ofthe se two plane s, nor doe s it afi

'

ec t the p lane of the

equator, but it nevertheless changes the inclinaVoL. 11. X tion

622 ove ns or N ATURAL e n n osor nv.

t ion of the eclipt ic to the equator, an d also the

line of the ir inte rsec tion .

This change in the inclination , and in the position

of the line of the equinoxes, is easily dedc by

spherical trigonome try from the retrogradation of

the intersection of the two planes, and from the

constancy of the ir inclination . See L A L anna,

Aslron . 2751, &c .

826. The variat ions in the obliquity of the

e cliptic , thus produced , are among the number of

the secular inequalit ie s w hich have long periods,

and, after reaching a max imum

,re turn in a con

trary dire c t ion .

A s far back as observation goes, the obliquity of the

ecliptic has bee n diminishing, and is doing so at

present, by 52”

in a century ; it w ill not, how ever,

always continue to dimin ish, but in the course of

ages w ill again increase , oscillating backw ards andforw ards 0 11 each side of a mean , from w hich it no

ver can depart far.

The secular variation of the obliquity w as less 111 an

c ien t times than it is at presen t ; it is now n ear its

maximum, and w ill begin to decrease l n the 22d

century of our‘

era .

L A Gas s e r:has shew n , that the total change of the

obliquity, reckoning from that in 1700, must be

less than 5° 23' Mem. Acad. dc Berlin, 1782.

P. 2841.

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ourmnxs or NATURAL rm LOSOPHY,

(its Equinox“, Paris, 1749. IA solution - equally cor.

‘

rect and original, w as given about the same time by"Ea t e n, Mew. Aoad. dc Berlin, 174-9.

Tw o solutions, in the Philosophical Transactions for

1 754 and 1756, continued to follow the method of

NEWTON . The first of these w as by Svnvxas nw ,

the second byWALMESLEY and this last, though it

re tained both the defects just mentioned, is remarkable for the elegance of the demonstrations . It ex

t ended the problem to the nutation of the Earth’

s

ax is, and it treated of the d iminution of the obli

quity of the ecliptic by the w h e n of the plane ts .

A memoir by L A GRANGE, on the Libratz'

on of the

Moon , w hich w as crow ned by the A cademy of Sci

ences at Paris, in 1769, con tained an excellent solu

t ion of the problem of the Precession .

Sm rsox , in his .Mz’

scellaneous Tracts, has given the

solution already referred to, w hich is one of great

simplicity and correctness. Its only defect is, that it

does not clearly point out the means by w hich the

un iform inclination of the Earth's axis is maintain

ed .

Another very elegant solution , is that of Fa 1s1 above

referred to ; Theoria Diurm'

Mates , Op era, tom. 111.

p . 292.

LA L anna has follow ed Sm r sON, as has also Vmc s , in

his Astronomy . The latest solution is that of Profes

sor Roszn'rson of Oxford, Phil. Traps. 1807 ; it is

also

enrsxcxl'

. s sraonomr. 325

also after the method of SIMPSON, and the investiga .

tion is accorate and concise .

The solutionof L A (Pub s , in the Mecam

'

que Celeste,

must be considered as the most perfect, and that

w hich can most certainly be said to include, and to

estimate w ith accuracy, all the causes w hich have a

share in this phenomenon . There is, how ever, one

defe ct it may be said to have , that as it proceeds en

tirely by the calculus, it does not sufficiently carrythe imagination along w ith it . . 1

327. Tm: alternate rise and fall of the surface

of the sea tw ice in the course of a lunar day , or of

$24,h 50

mof mean solar time , is the phenome

non known by the namc of the Tides.

The time from one high- water to the next , is, at a

mean , 12h 25m 24-00 . The in stant of low - water is

nearly, but not exactly, in the middle of this inter

val i the tide, in general, taking nine or ten mi

nutes more in ebbing than in flowing.

A t the time of new and full Moon , the tide is the

highest, and the interval betw een the consecutive

tides is the least, viz . 12h 19mm . A t the qua

dratures, or when the Sun and Moon are 90°die

stant, the tides are the least, and the in terval be

tw een them is the greatest, viz. 125301!

328. The time of high- wat er is mosdy regula

d by the Moon , and in general, in the Open sea,

from two to three hours after that plane t has

been

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OUT L INES OF NA TURAL PHILO SOPHY.

At Brest, the tides Of the syzigies rise to

feet ; and those Of the quadratures to not

quite the half Of the former quantity.

In the Pacific Ocean, the rise , in the first case ,’

is 5

feet ; in the second, 2 or

The greater the rise'

of high- w ater above the level of

a fixed point , the greater the depression Of the cord

responding low- w ater relatively to the same point.

TO estimate.

the height Of the tide , it seems best to

take the excess Of the mean of the two consecutive

high- w ater marks, above the intermediate low - wa

ter. This is the method Of LA PLACE.

330 . The he ight Of the tide is affec ted by the

vic in ity Of the Moon to the Earth, and inc reases,

ca teris paribus, w hen the parallax and apparent

diamete r of the Moon inc rease , but in a higher

ratio.

The greatest variation Of the Moon’

s semidiameter

above or below the mean is about T

'

Tth of the

w hole , and the corresponding variation Of the tide

at the syzigies is 95th Of its mean quan tity . Sys

tems da Monde, p . 77 .

33 1. The rise Of the t ide is affec ted by the dc

clination of the luminaries ; it is greate st , ce teris

paribus, at the equinoxes, and least at the solsti

c es.

V Vhan.

‘

830 oun mzs or NATURAL rn t t osornv.

preaches the shores, little or no progressive ind -f

In all this, no regard is had to the operation of local

causes, w inds, currents, are . by which these general

law s are modified, or overruled .

833. The dependence of the phenomena justenume rated on the motion of the Sun and Moon,

naturally suggests an inquiry into the effec ts

w hich the act ion of the se bod ies may produce on

the w ate rs w hich cover so large a proportion of

the Earth’s surface.

884 . Ifm‘

be the mass, and a the d istance of the

Sun , 9‘

the mean rad ius of the Earth , 7. the dis:

tance of the Sun from the zen ith of any place (or

the d istance of that plac e from the poin t to w h ich

the Sun is vertical) a partic le of matte r at that

place is drawn tow ard the Suri by a forc e equal

cos 3 ; be sides having l tS gravity inst eav

sed by another force

This is derived from the resolution of forces!, in the’

same way as when the Sun’

s action on the Moon

was investigated.

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PHYSLGAL m aonomr .

draw ing the water horizontallg is not

here ; its tendency 18 to increase the efst calculated .

by w hich the solar force everywhere

3 gravity, n eed not be taken into accoun t,

does not.

affect the equilibrium of the wa

eigh t.mwhiéh

'

the m w as tha s ’rise at high

level which i t ‘m h ld stand at if

as its

This is

the ‘conten t

’

of the sphe

einingw ays the smile .

The preceding is suffic ienvto shew , that

t ides are efi'

ect s that might

ed from the princ iple ofgravitat ion . This,

is an approx imation from w hich exac t

cannot b e Obtain ed ,‘

since a material

has be en le ft out, namely, the'

motion of

on w hich the forces of the Sun andexe rt ed .

The rapid motion of the w ate rs,in cousez

quence of the diurnal rotat ion, prevents them from

assuming, at every instant, the figure w hich the

equilibrium

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536 our t w as or NATURAL pnrt osoeny .

For since are as the forces of the Sun and

Moon on the waters of the Sea,

and therefore m’

m’

is thus found

the mass of the Earth being 1.

842 . The tw o t ide s immediately follow ing one

mothe r, or the t ides of the day and of the n ight,

should be very unequal w hen the Sun and Moon

are distant from the equator, if the theory of the

sphe roid , 336, w ere just . They are , how ever,

nearly e qual 3 and t his has been shewn , by LA

PLA C E, to be what -must necessarily happen in the

osc illat ion s of a sea‘

ofun iform depth .

The depth of the Sea is therefore nearly the same

throughout, or, though not exactly the same , there

is a certain mean depth, from which the deviations

are not considerable , if w e take in a large extent of

ocean . If this w ere not the case, the consecutive

tides w ould not approach so near to equality 88

Great

rursrczu. asraouon r . 357

643. Great ex ten t is nec essary, in order that the

Sea should be se nsibly affected by the ac t ion of the

Sun and Moon ; for it is only by the inequality of

that ac t ion ,on d ifl

'

e rent part s'

of the ma ss rof w a

ters, that their‘

equilibr ium -is disturbed and such’

inequal ity cannot take place ;unless a great ex tenthf surface be inc luded .

The'

value‘

of cos’z; in the precedingdbr

'

mulas , be

longinglto difl

'

erent parts of the same sea, must be

oond derably differ-

ant. in order that an M llation

of the waters may be prodnc ed The saine fis true

of the'

horizon tal force , of which sin g s . is the

multiplie r. T his las t 18 0 at its maxirhum at the dis

tance'

of 46 ° from the pomt where the attract ingbody is in the zenith:

The tides which are experienced tn nar

row seas, andonshores far from the mainbody of

the ocean , are not p roduced i t) those seas by the

direct act ion of the lumin arie s, but ar e wave s pro

pagated from the great d ihrnal . undulatson ,and

moving w ith much less ve loc ity .

Of this’

, th e tide s in the German Sea, and on the

coasts of Britain , are remarkable examples .

The high- w ater transmitted from the tide in the A t

lan tic, reaches Ushan t betw een three and four

hours utter the Moon has passed the meridian , and

its ridge stretchesNW, so as to fall a little south of

Yes . H. Y the

388 0 0 1 1 18 2 8 OF NA TURA L PHILOSOPHY.

M into three ; one part pauing up the Britith

Channel, another ranging along the w est side d

Ir'

uh Chnnnel, The first ot'

tbese flow s throufllthe Chann el at the rate ot

'

about w miles an hom,

so as to pass through the Straits of Dover, and to

reach the Nore about tw elve at night. The i1a

cond , being in a more open aea, mmes w itll more

rapidity ; by six it has reached the nm'th extremi

ty ot‘

the lrish oon t ; about nine it hu got to tlieOrkney Islands, and forms a ridgepor wm extend

ing due north ; at tw elve, the summit o£ the same

w ave exteads from the cout om eastwud

to the Naze ofNorway ; and inm 1n boun mom,

it reaches the Nore , where it meets the mm ingtide, that left the mouth of the Channel only eight

hours before . Thus these two tides travel round

Britain in about tw enty - eight hom in which time

the primitive tide has gone round the whoie é fl

cumference of the Earth, and near ly w eep ”

vol. 1. pl!“xm iii. fig.581.

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oun m s s or ram -

vain . rnrt osornv.

other than the inverse ratio of the squares of the dis.

tances . The expression for the force w hich this law

affords, viz. the quan tity of matter, d ivided by thesquare of the distance , is a line , or a magnitude of

one d imension , as the expression of a force must al»

w ays be w hen reduced to the utmost simplicity.

L astly, The lines described by two bodies attracting

one another according to this law , are always of the

second order . No other law could give the same sim

plicity to tbe'

celestial motions, nor is it likely that

any one could produce the same stability.

Gravitation , nevertheless, is not conceived by us as a

property essential to matter ; there may be manyother law s equally possible , an d the above considera

tions poin t out the law actually existing, as one that

has been w isely selected out of an infin ite number .

.

I’ there,then, any physical ca

’

use, yet more general,

intowhich Gravitation maybe resolved or isit an ui

f‘a

l

c

rt, beyond w hich our know ledge cannot ex.

tend we look at the ill success of the attornpts

hitherto made to explain gravitation, w e shall be dis

posed to embrace the latter opin ion, and to apply the

max im of Bacon Est eaten; e quc fmper r‘

t i ct leci

O

gi"viii

i ihow man

Q5

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