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32X

UNIVERSny OF
WESTERN ONTARIO

LIBRAr^Y
select LIBRARY

^r^"-"'- -"' "r--

/ M E A S U II i; M K N I'

THE SUN'S DISTANCK.

-. "--v-.-f- -■•

JOHN IIAKRIS.

•!®

•^^^

ILflnbait :

N TRU13XER & CO., .)7 .^L- .)•». LVElWATE II ILL

. y..

■"PiMPiPMiVP*»^n ifi

<5 '-

ASTRONOMICAL LECTURES.

MEASUREMENT

THE SUN'S DISTANCE.

JOHN HARRIS.

N. TRUBNER & CO., 57 & 59, LUDGATE HILL.

SEPTEMBER, 1876.
AH rights reserved.

■A

II 86 V-

LECTURE FIFTH.

TMl E F A C E .

This fifth lecture, or fifth division of the serioa to which
it belongs, differs iu one respect from those preceding it,
inasmuch as we have not on this occasion to call in ques-
tion or to condemn the present doctrine on the particular
subject of which it treats.

It is true that, not long since, an error of about four
million miles in the estimate of the sun's distance which
had been accepted and agreed to by astronomers for many-
years previously, was discovered ; and that it is only quite
recently the correction of the error has been made and
adopted. But this error was of the nature of a mistake
which in combining the separate observations of several
different observers was almost unavoidable, if it happened
that any one of those observers, upon whose collective
reports the general computation was based, had from
misfortune, or want of due care, fallen into error and
furnished a report which, being accepted as trustworthy
and being in fact incorrect, vitiated the whole. Such a
circumstance does not necessarily prove, nor in any
degree evidence, the method itself to be unsound in
principle or unreliable in practice. But it does afford

'i^ ■

PREFACE.

evidence that the conditions necessitated b}'- the particular
method in question practically occasion a liability to error
in the collective result, and it suggests very pointedly
the desirableness of practical astronomy being in posses-
sion of some other method, or methods, equally reliable
as to ihe sound theoretical character of the basis, and of
such a nature in themselves that the individual observer,
usmg due care and diligence, will be independent^ of
the want tif care or correctness on the part of other
observers.

irticular
to error
jintedly
, posses-
reliable
, and of
bserver,
den*^ of
>f other

INDEX TO THE SEVERAL METHODS.

PAGE

Method First.— By observation of the relativo velocities of
earth's orbital motion round the sun, and of earth's equa-
torial surface in rotation on the polar axis 9

Mktiiod Seconb. — By observation of the angle of moon's
illumiui'tion 12

MetuoD Thiud. — By observation of the time occupied by
the entire disc of the moon in passing the sun's centre, as
seen from the centre of the earth 15

Method Fourt'II. — By observation of the time occupied by
the diameter of the earth in its orbital ascent or descent
through a definite aogle); or, observation of the vertical
angle subtended by earth's polar diameter, at the distance
of the suu, showu in its ascent or descent through the
equatorial plane of the dun 18

Method Fifth.— By observation of the angle of obliquity
in the path of a (so-called) solar spot in its motion upon
and with the surface of the sun 21

Method Sixth. — By comparative observation of the angle
of incidence of the sun's light on an object upon the
earth's surface, and the angle of incidence of th'i sun's
light bounding and containing the earth's shadow ... 25

Method Seventh.— By comparative observation of the angle
subtended by the sun's diameter as seen from a station on
the equatorial surface of the earth, and as seen at the
same time from the earth's ccutre 28

^xmr

PLATES

PAGE

FrontispifX'E. — The Transit of Venus according to the
perpendicular axis theory.

Fig. 1. — Solar illumination of the moon 12

Fig. 2. — The angle of moon's illumination 14

Fig. 3. — The moon's diameter passing over the sun's centre,

as seen from the centre of the earth ..... 1 G

Fig. 4. — The differential angle of moon's synodic period . . 17

Fig. 5. — The descent of a definite part ot the earth's polar

diameter through tlie equatorial plane of the sun . 20

Fig. 6. — The axial rotation of the sun, b} terrestrial observa-
tion of a solar spot 22

Fjg. 7. — Measurement of the sun's dictance by comparing the
angles of incidence of the sun's rays converging
(a) to the earth's centre, and (6) to the extremities
of the earth's diameter, respectively 25

Fig. 8. — Comparative angular value of the sun's diameter as
seen from the nearest point on the earth's surface
and, at the same time, from the earth's centre,
respectively 29

THE MEASUREMENT

THE SUN'S DISTANCE.

PAGE

o the

. . 12

. . 14

Dutro,

. . 17

polar
suu . 20

ierva-

. 22

g the

nities

ber as
irface
3utre,

The methods at present known in practice, and which
have been utilized with more or less success for the
purpose of ascertaining the distance of the sun from the
earth are two, that of geocentric parallax (or zenith
meridional observation), and the transit of the sun's disc
by the planet Venus.

Our present purpose is to suggest, explain and illus-
trate, so far as may be necessary for the elucidation oi
this preliminary explanation, certain methods which we
believe to be essentially new and hitherto unpractised.

(1.) If, by direct observation, the earth's velocity of
revolution in its orbit relatively to the velocity of its
rotation can be ascertained with certainty and precision,
such information, together with the data already known,
will enable the distance of the sun from the earth to be
readily computed.

The mode of observation by which we propose to
obtain this information is by utilizing the retrograde or
backward movement of a station on the equator in its
rotation round the earth's axis, when on that side of the
earth nearest to the sun, compared with the advance of
the earth itself in its orbital revolution during the same
time. To explain the proposed mode of proceeding, let

MEASUREMENT OF SUN 8 DISTANCE

,'K

US suppose the station of observation to bo situated on the
equator and that the earth's rotation through 00 degrees
is to be subjected to the comparative observation.

Knowing the time at which the meridian of the
observatory will pass the sun, the observation is to
commence two hours before that time. The situation
of the sun as seen from the earth's centre, and as
seen from the station on the equator, is to be recorded,
and also the exact time of the sun's centre transitting
the meridian of the observatory. And, again, the situa-
tion of the sun as seen from the earth's centre, and from
the observer's station, four hours later than the time of
the first observation, is to be carefully determined.

Having thus ascertained by direct observation the
parallactic displacement of the sun consequent upon the
compound motion of the observatory, we shall be able
to deduce the linear velocity in miles of the earth in its
orbital revolution around the sun, because we already
know the linear value in miles of the arc through which
the retrograde or reverse motion of the earth's rotation
has carried the observatory, and we have ascertained the
value of the chord of that arc as a part of the orbital
circle of the earth's annual revolution. "VVe know also in
time and in angular measurement the quantity of orbital
arc traversed by the earth during the observation. For
example : the observation being made for 60 degrees of
the earth's rotation, we will assume that 9' 43" is the
parallactic displacement of the sun between the first
observation and the last. Now the orbital arc moved
through by the earth in 24 hours is a little less than one
degree, viz., 59' 10", so that the one-sixth of this quan-

i -• 'i^il

BY FIRST METHOD.

tity, viz., 9' 52" is the displacement of the sun caused by
the earth's orbital progress, and which would be the
difference between the two observations, if both were made
from the earth's centre. But the actual displacement
shown by the direct observations from the station on the
equator is (by the supposition) 9' 43". The 9" of difference
is therefore due to the reverse or retrograde movement of
the station in consequence of the earth's rotation. This
9", however, represents the chord of the terrestrial arc,
and not the arc itself. Now it is the terrestrial arc
which we must compare with the solar arc to obtain the
linear velocity of the one from the known linear velocity
of the other. Therefore, as 9'43" : 9".427 : : 62 : 1.
That is, the linear velocity in the orbital revolution of the
earth itself is 62 times that of the station on the equator
due to the rotation of the earth. Now since if the linear
velocities were equal the angular velocities would inversely
measure the comparative lengths of the radii, we should
have accordingly S60° : 59' 10" * as the proportion of the
greater length which the sun's radial distance would have,
compared with the radius of the earth, if the linear veloci-
ties were equal. But the linear velocity of the earth's
orbital motion being determined as 62 times greater than
that of the equatorial surface due to the earth's rotation,
the computation, taking the radius of the earth at 4,000
miles, will be 4,000x365x62=90^ millions of miles as
the sun's distance from the earth.

The same observation also directly furnishes the geo-
centric horizontal parallax of the sun. For, taking the
preceding example, since the chord of the arc of 60° equals
the radius of the circle, the difference between the

* Or 3G5 : 1.

MEASUREMENT OF SUN's DISTANCE

observed angle of parallactic displacement and 9' 52*
{which diflference we have assumed in the foregoing as 9")
is the geocentric parallax of the sun. Therefore, from
this quantity', which results immediately from the observed
displacement of the sun, and from the known magnitude
of the earth and velocity of the earth's rotation, the
distance of the sun from the earth can be readily deter-
mined in the usual manner.*

(2.) By the angle of the moon's illumination.

In the accompanying figure (fig. 1), the moon is repre-
sented at A, in quadrature ; that is, in the situation
relatively to the sun and earth which she occupies when
the one-fourth of her orbital revolution is completed.
Now in her position at A, when so situated, rather more
than the one-half of the moon's disc, as viewed from the
earth, is illuminated by the sun, which is obviously a
consequence of the sun's light striking the moon at an
angle, with a line joining the centres of the earth and moon,
rather less than a right angle. For if, as shown at B, on
the other side of the figure, the moon be so situated in
her orbit that the direction of the sun's rays forms a right
angle with the line joining the centres of the earth and
moon exactlj^ one half only of the moon's hemisphere
will be illuminated.f

* The number of seconds contained in the circle are 1,21)0,000,
which, divided by 9 = 144,000 ; which, multipHed by 4,000 miles
(the length of tlie earth's semi-diameter) = .576,000,000 as the
circle of the earth's orbital circle. The distance of thf' aun from
the earth, which is the radius ot that circle, equals, therefore,
91^ million railes.

t To simplify the explanation we are here leaving out of con-
sideration, for the moment, the great comparative magnitude of
the sun, and consequent extension of the illuminated surface of
the moon beneath the equator. — Hce page 14.

sun's distance

-By the An^le of Moom illumindLion.

H-i-

^mimm^mimmmmnm^mimmm^

k1 A'

BY SECOND METHOD. 13

The moon's distance from the earth has been long since
ascertained by means of geocentric parallax, and it may
be assumed that the distance is now known with an
approximation to precision. Since a ray of light from
the sun to the moon is equivalent to a line joining the
sun and moon, the careful astronomical observation of
the angle of the moon's illumination at quadrature (by
measuring the magnitude of that part of the moon's
hemisphere illuminated in excess of the semi-hemisphere),
will furnish the angular distance of the moon from the
earth in terms of the earth's orbital circle ; or, in other
words, it will determine an arc of that circle in linear
measurement equal to the distance between the earth and
the moon, of which distance the metrical value in miles is
clready known. But so soon as the value of a circle, or of
any definite fraction of the perimeter of a circle, in
terms of the linear metrical standard, is ascertained, the
length of the radius in terms of that standard becomes
known. Therefore the distance of the earth from the
sun, which is the radius of the earth's orbital circle, will
become correctly known so soon as the angular illumin-
ation of the moon at quadrature has been carefully
measured, and accurately determined.
' To measure the angle of illumination it is not necessary,
however, that the moon should be at the place of quadra-
ture. By observing with exactitude the angular situation
• at which 16' more than the one-half of the moon's hemi-
•' sphere is illuniinated, the difference between that angle
and quadrature will furnish the angle subtended by an
arc of the earth's orbital circle contained in the inter-
vening space, of known metrical value, between the moon

MEASUREMENT OF SUN S DISTANCE

and the earth.* For oxamplo, lot us suppose in fig. 2,
the moon to be at tliat place A of its orbit, where a lino
joining the centres of the earth and moon is exactly at
right angles to a line joining the centres of the moon and
sun. We have first to consider that the diameter of tho
sun as seen from the earth subtends an angle of 32' ; and
since the moon, at that place in her orbit, is at the same
distance as the earth from the sun, the angular magni-
tude of tho sun's diameter as there seen from the moon
will be the same. Consequently, since the moon's entire
diameter as seen from the sun subtends an angle of only
about 2 ", the sun's rays, impinging upon the moon's globe
in a converging cone, will strike about 16' beyond the
central circle which, posited horizontally to a line joining
the centres of the sun and moon, divides the moon's
globe into equal hemispheres. Viewed, therefore from
the earth, the moon will appear illuminated to an angular
distance of 16' below the equator, or, in other words, the
whole of the moon's upper hemisphere and 16' of tho
lower hemisphere will be illuminated by the sun's light.
Now let the moon move onwards to the place of quadra-
ture at B. We will assume that astronomical measure-
ment determines the angle A E B as 9'. It follows that
16' + 9 =25' of the moon's dark hemisphere will now be
illuminated. And, because the line S A is perpendicular
to the line A E, and the line S E perpendicular to the
line E B, it follows that the angle E S A also contains
9' of arc, which arc belongs to the circle of the earth's

** Other situations of the moon may be made available for the
same purpose, only that the more directly the required data are
furnished by the observation, the more correct, caieris jxiribus, will
be the result.

sun's distancf.

By tJiH Anglo of Moon.--: illuniinaliot.

i^qp

BY THIRD METHOn.

ir>

orbital revolution and in known to equal in metrical value
60 times the radius of the earth. Therefore, taking the
earth's radius as before at 4,000 miles, wo have 0' of the
earth's orbital circle, equal to 240,000 miles, which
furnishes the radial distance of the sun from tho earth aa
nearly 92 million miles.*

(3.) In consequence of the earth's progressive orbital
advance during tho time occupied in its diurnal rota-
tion, the earth having completed a siderial rotation
has to overtake the sun by a space which is a known
definite fraction of tho circle bounding the earth's sphere.
Similarly in the moon's revolution around the earth, the
arc of difference between the siderial and synodic revolu-
tion is 1 known definite fraction of the moon's orbital
circle (as the earth's satellite). By ascertaining the time
occupied by the earth with the velocity of its orbital
revolution in moving through the same arc, the distance
of the sun may be ascertained ; or, in other words, if we
can ascertain the linear value of this differential angle
compared with the similar angle of the earth's orbital
circle, of which it is a consequent (and with which angle
it is necessarily equal) the distance of the sun will become
known. Now the moon itself as seen from the earth sub-
tends an angle sufficiently large to admit of very accurate
appreciation as a definite fraction of its own orbital circle,
and of the difierential angle belonging to that circle, of
which (diff. angle) the value in terms of the earth's
orbital circle is required. If, therefore, we can measure
the value of the moon's diameter in terras of the earth's

MEASUREMENT OF SUN S DISTANCE

orbital circle, we can therefrom compute the value of the
differential angle, and hence obtain the sun's distance.
The Conditions of this method mav also he stated as
follows : since we know the time occupied by the kood
in completing a revolution around the earth, and we
know the fraction of that orbital circle contained in the
moon's disc, if we ascertain the time required by the
earth with the velocity of its orbital revolution around
the sun to pass through the angle subtended, at the dis-
tance of the moon, by the moon's diameter, we shall
thereby obtain knowledge of the comparative linear
velocity of the earth around the sun, to that of the moon
in its orbit around the earth ; from which data we can
compute thv3 sun's distance. ' ........

To ascertain the time occupied by the earth in passing
through an arc equal to that fraction of the circle of the
moon's orbit, made apparent to us and defined by the appari-
tion of the illuminated moon, as the angle subtended by
the moon's diameter, an occultation of the sun by the moon
(fig. 3) affords the most favourable opportunity. The centre
of the sun, if the occultation be puch that the centre of
the moon will pass over the sun's centre, or any clearly
defined spot so situated on the sun's disc that the equator
of the moon will pass over it, will equally well answer
the purpose of the observation, which is in the first
place to ascertain the apparent time occupied by the
earth in passing through a fraction of its orbital
circle equal to the angular value of the moon's di-
ameter. Now the apparent time thus observed would
be the actual time of the earth's velocity if the
moon were at rest ; but, in fact, as the earth in its solar

[lie of the
distance,
stated as

tha E300D

and we
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we shall
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sun's distance

By Llfie iutiar differenLial MetJiod.

B^.4

BY THIRD METHOD.

orbit is moving with a certain velocity in one direction/
the moon in its (tcrestrial) orbit is moving, with a lesser
velocity, in the opposite direction. Therefore, since tho
moon, instead of remaining at rest for the earth to pass
it, has, in part, taken itself out of the way, the apparent
time required by the earth to pass through the arc of the
lunar c'rcle equal to tho moon's diametei is, by so much,
less than the actual time. The velocity of the moon's
motion in its orbit is known, and we can estimate with
precision the time required by the moon to pass through
an arc of its orbital circle equal to its own diameter. To
deduce the actual time of the earth's velocity in passing the
moon from the apparent time of the observation, we have,
therefore, to add to the observed time a fraction propor-
tional to the time which the moon requires to pass through
an arc of its orbit equal to its own diameter. As an illustra-
tion, let us assume the observed time of the passage of the
moon's dianeter across the sun's centre, or over the solar
spot, to be 1' 56". Now the time required by the moon to
pass through the arc of its own diameter is, by computa-
tion, 59 minutes. Therefore, to 1' 56" we have to add
1' 56"-=- (59 -^ 1'94) ; so that the actual time comes out very
nearl)'-2'. Referring to the figure (fig.4) — in which S repre-
sents the Sun, 8 e the solar radius-vector of the Earth,
m rii' the Moon's orbit, and E, or e^, the Earth — the
angular situation of the moon and earth in relation to
the sun after 28 days' orbital progress, is shown on the
right of the figure. The moon having then completed
one sidereal revolution, and the earth having completed
the thirteenth part of her orbital circle, we have the arcs
subtending respectively the two equal angles, E S e

MEASUREMENT OP SUN\s DISTANCE

' 't.

;;l

and Sem, proportional to each other in the ratios of the
distances each to the other of their respective centres of
revolution — to wit, the distances of the sun from the earth
and of the earth from the moon. Now since the earth's
orbital velocity carries it through 32 minutes of the moon's
orbit in 2 minutes of -time, the time occupied by it in
moving through the diflferential arc of 27° 41' (of the moon's
orbit) would be 104 minutes 5 seconds. But the time
occupied by the earth's velocity in moving through the
greater arc (27° 41' of its own orbit) is 28 days. There-
fore the greater arc is proportional to the lesser as 387^ to
1. Consequently the sun's distance is 387^ times that of
the moon from the earth, and, taking the moon's distance
as 238,800 miles, the sun's distance equals about 92i
million miles. The same result may be also arrived at by
inversely estimating from the relative velocities. For, taking
the proportion of a very little less than 2' to 59', as 29| to 1,
since the angular velocity of the moon is 13 times greater
than that of the earth, we shall obtain the proportion of the
greater distance if we multiply 29-j x 13, which gives
386 times the radius of the moon's orbit for the distance
of the sun from the earth (equal to about 92 million miles).

(4.) By the ascent and descent of the earth in its orbital
revolutio.i. This method consists in choosing three sta-
tions, one of them on the equator, and of the others — one
in high latitude in the northern hemisphere, and one in
similar high latitude in the southern hemisphere. The
longitude of the stations to be respectively such that each,
when passing through the plane of the solar equator, will
have the sun on its meridian. The vertical distance

BY FOURTH METHOD.

between these stations (measured as the chord of the
vertical arc joining three places having the same longitude
in common, and, respectively, having the same latitude as
each of the stations of observation) being known, it is
required to determine, by observation, the exact place of
the sun in the ecliptic at the time when the meridian of
each station successively transits the centre of the sun.
We then have the vertical quantity contained in a definite
small angular section of the ecliptic measured in the
known metrical value of the vertical linear distance between
the terrestrial stations. Hence the linear value of the
earth's orbital circle and therefrom, of the sun's distance
becomes readily determinable. Such observations would be
preferably made near the time of the equinox, or when
the sun is not, at most, more than two months from one
of the nodes. And it is to be observed that, since the
earth's vertical velocity is greatest at and near to the time
of crossing the nodal plane of the sun, and for a brief
period at each of the solstices becomes nothing, the
quantity directly obtained by this method would hav6 to
be rectified accordingly, in order to get the average of the
vertical motion throughout the entire orbit or semi-orbit.
Such rectification, however, the angular velocity and
magnitude of the angle being known, and the nature of
the orbit, as a circle or ellipse posited obliquely, being
apprehended, would present no difficulty.

To illustrate this method, the following considerations
may be stated : — 45° of the earth's vertical motion com-
bine with and occupy the same time as 180° of the earth's
horizontal motion, i.e., the vertical motion is to the
horizontal as one to four. ^

■■'!'9*'

MEASUREMENT OF SUN S DISTANCE

The circle of the earth's rotation equals by time very
nearly one degree of its orbital horizontal motion.

One degree of the earth's orbital horizontal motion
equals by time \ degree of its vertical motion.

If, therefore, it be found that the entire vertical angle
between the two extreme stations of the northern and
southern hemispheres, which we will suppose to be 4,000
miles apart (measured as the chord of the vertical arc
joining the latitudes of the two stations), contains 9
seconds, then, since the earth's horizontal orbital motion
will occupy about 3"^ 40^ in moving through 9", the vertical
motion will occupy 14™ 40^* in moving through the same
angle. But 14™ 40* represents about one-hundredth of the
earth's circlp of rotation, therefore the longitude of the two
extreme stations would require to be such respectively that
they would be 3° 36' apart. The meridian of the equatorial
station would have such longitude as to be situated
equidistantly between the meridians of the two extreme
stations, 1° 48' from each.* (See fig. 5.)

Instead, however, of thus confining the linear measur-
ing distance to 4,000 miles of the earth's diameter,
12,000 miles might be made available for the purpose.
Reference to the figure (fig. 5) will, it is thought, make
the manner , of the intended application sufficiently in-
telligible for the present purpose — viz., by taking the
extreme stations c and e in such high latitudes, north
and south respectively, as to be 6,000 miles apart
(measured by the chord of the vertical arc joining the

'" In strictly computing the required difference of longitude, an
^ allowance would have to be made for the onward progress of the
«?f earth in its orbit.

* + *

'-C

■ J^

C/3

C/5

■rs

>*N,

i"'S|

8J, ' f "I

BY FIFTH METHOD.

latitudes, as before.) . . Commencing the compound ob-
servation when the southern station reaches the nodal
plane of the sun's equator, and completing the observation
when the northern station reaches the same plane. The
parallactic displacement of the sun would thus furnish
the measurement of a definite sectional arc of the ecliptic.
The distance of longitude between the meridians of the
stations, as determined by computation, may be rectified
by repeated experiment (i.e., by shifting the localities of
the stations as required) until an indefinite approxima-
tion to accuracy is obtained.

(5.) A method of measuring the sun's distance quite
distinct from the foregoing, but allied thereto, because
dependent upon the vertical motion of the earth in its
orbit, may be described as characteristically consisting in
the astronomical observation of the phenomena called
solar spots, which are sometimes seen to traverse the sun's
disc. It is well known that at two seasons of the year only,
namely at the summer and winter solstices, in June and De-
cember, are the spots seen to cross the sun's disc horizon-
tally. At other seasons of the year, the path of the spot in
crossing the sun is oblique, either ascending or descending
as the season is approaching the summer or the winter
solstice. It has been already pointed out elsewhere that
the apparent obliquity of tiie paths of the solar spots har-
monizes perfectly with the perpendicular axis theory, and
and may indeed be considered to constitute a part of the
demonstration of the truth of that theory. The apparent
obliquity is, we consider, certainly an effect of the vertical
orbital motion of the earth combined with its orbital

MKASURKMENT OF 8UN .S DISTANCE

horizontal motion. To apply this method, however, wo
must assumo that the velocity of the spots in traversing tho
circle of the sun's equator has been correctly ascertained,
or, to speak with more particularity, that tho velocity of
tho sun's rotation, by which the spots are carried around
it cquatorially, has been so ascertained. Then since such
rotation, assuming it to exist, must bo certainly in tho
same direction as that of the earth's orbital revolution,
the apparent motion of tho spots in circulating around
the sun, as seen from the earth's centre, must be the
resultant of the difference in the velocities of tho two
motions, for if the velocity of the sun's rotation were such
that a spot on the solar equator was carried around
with the same angular velocity as that of tho earth's
orbital revolution, the appearance to the terrestrial ob-
server (from the earth's centre) would ho that of a spot
motionless and constantly occupying tho same situation
on the sun's surface. Now if the angular velocity of the
sun's rotation were so grout as one complete rotation in
24 days, this velocity being 15 times greater than that
of the earth's orbital re^ olution, and the velocity of the
earth's vertical motion being only one-fourth of its hori-
zontal orbital motion, the obliquity produced in the appa-
rent motion of the spots across the sun's disc would be
scarcely appr . .'able, because the deviation from a per-
fectly horizoJLiTal plane would be so small in amount. But
if we reflect that in consequence of the circumferential cur-
vature of the sun's globe (as of every other globe) not more
than, at most, about 90 degrees of the hemispherical sur-
face would present the spot at such a visual angle as to bo
visible from the earth, it will become apparent that an

^evcr, wo

Tsiug the

ertainod,

Jlocity of

I around

ICO such

' ill the

'olution,
around
bo the

ho two

ro such

around

earth's

'al ob-

a spot

nation

3f the

ion in

1 that

f the

hori-

ippa-

Idbe

per-

Bnt

cur-

lore

3ur-

) be
an 1 ^

.^^S_^ROMr/^

Fig. 6.

'-i-J

BY FIFTH METHOD.

angular velocity of rotation by the sun, equal to about
six times that of the earth's orbital motion, would bring
the spot into the field at the one side of the sim's disc,
and take it out of the field at the opposite side, within
about twelve days ; because, beyond the '!mit of the 90 de-
grees, the foreshortening angular effect would become so
great as to present the spot almost edgewise, and so ren-
der it invisible, whether approaching on the one side or
receding at the opposite.

This interpretation of the facts may become more readily
appreciable by aid of the accompanying figure (fig. 6).
In twel''^ clays, which is about the average or most usual
time that a spot remains visible, the earth will have ad-
vanced nearly 12^ in its orbit, that is, in the same direc-
tion in whioh the rotation of the sun, or of the sun's
surface, carries the spot. This advance will evidently
occasion an extension of the solar arc throughout which
the spot will be visible ; so that if, supposing the earth
had remained stationary in its orbit, the visual solar arc
would have been about 78°, it will have extended to about
9C" io consequence of the orbital advance of the earth.
Hence, to complete the circuit of the sun's equator, the
spot will occupy 48 days (say about 50 days), which may
be considered to measure the angular velocity of the sun's
axial rotation. It is not, however, to be inferred in such
assumption that, supposing the spot to remain existent
and unchanged, it will reappear at the sun's eastern limb
at the expiration of about 36 days. Such inference would
overlook the continuance of the earth's orbital motion,
which, in that time, would add nearly '66° to the 270°,
making a total of 306°, throughout which arc, equivalent

MEASUREMENT OF SUN's DISTANCE

to about 41 days in time, the spot would be occulted by
the sun's globe.*

One of the most distinctly observed phenomena belong-
ing to the solar spots is the apparent obliquity of their
paths across the sun's disc, with exception of two semi-
annual periods in the year when those paths form straight
lines. Another observed phenomenon is that the direction
of the angle of obliquity during five months of the year
is inverted during the other five months. Now this appa-
rent obliquity is, we consider, certainly attributable to the
vertical ascent and descent of the earth's orbital path.
But if such be the cause, it follows (1) that the angle of
obliquity of the path will necessarily measure the angular
velocity of the spot in its revolution around t'l'^ i' ind
therefore of the sun's rotation ; and (2) that the obliquity
of the path of the spot, i.e., the amount of its deviation
from a horizontal line, may be utilized as a means of
measuring the sun's distance in diameters of the sun, for
the vertical amount of that deviation is parallax of the

** The most direct and probably the best method of deducing
the period of the sun's rotation from the observed velocity of a
solar spot would be simply to determine accurately the angular
progress of the spot, as seen from the earth's centre, when near
the central part of the sun's disc during one complete rotation of
the earth, i. c, during 24 hours. Now the earth, during the 24
hours, will have moved round the sun in the direction of its rota
tion nearly one degree ("9863 of a degree) ; therefore the observed
angle, with addition of chis terrestrial quantity, will be that part
of the circle of the sun's equator moved through in its rotation
during 24 hours. The time, therefore, occupied by a complete
rotation of the sun will be simply 24 hours multiplied by the
number of times the observed angular quantity increased by
addition of the angle moved through by the earth, is contained
in 360".

J - ■

;od by

)long-
their
semi-
aight
jction
year
appa-
o the
path,
^le of
••ular
md
luity
ition
18 of
, for
' the

icing
of a
jular
neai'
mof
e 24
fota
rved
part
tion
)lete
the

incd

■p

-c

c .

,- 1 ■ '

i'- 1

■^ i

y'l

BY SIXTH METHOD.

'^D

spot as projected on the suii's disc, occasioned by the
ascent and descent of the earth in its orbital revolution.*
Let us suppose the observed parallax of the spot, takin;^
the extreme limit of displacement, north and soutli, on
the sun's disc to amount to 13 minutes : we shall then have
the proportion.. As 13' : 47° : : the Sun's radius: the
number of times that radius is contained in the radius of
the Earth's orbit. Fur the purpose of illustration we will
assume the sun's radius at 420,000 miles. We then have
420,000 X 217 = 91 millions of miles as the radius of
the earth's orbit or distance of the earth from the sun.

(6.) A method of measuring the sun's distance quite
distinct in character from the preceding — for the pre-
ceding methods are all dynamical in character, whereas
the method now about to be described is statical in
character — may be termed the geometrical method. In
it a knowledge of the sun's distance is attained by
observing the relative angles subtend, \ by the diameters
of the sun, the earth, and the moon, respectively, when
seen directly or indirectly from several points of view.

In Fig. 7, let ^ represent the sun ; E the earth ; and
MM' the moon at the two opposite extremities of its orbit,
viz., at conjunction and at opposition. The sun's light

•■' Supposing the spot to remain permanent or unchanged for
some considerable time, and that the angular velocity of the sun's
rotation were only the same as that of the eai th's orbital revolu-
tion, the effect would then be that the spot would, to the terrestrial
observer, appear to ascend and descend vertically on the sun's disc
throughout an angle of 2U° beneath and 231° above the solar
equator, i. c, throughout an angle of 47'' (45°).

MEASUREMENT OF SUn's DISTANCE

shining past the splierical earth, projects its shadow as a
(lurk cone to the point x. The angle axb h therefore
the angle subtended by the sun's diameter as seen
from the point x. Now since the linear value of the
earth's diameter is known, and the distance of the moon
from the earth at opposition is known, and the breadth
of the shadow, at g h, is ascertained from the time occu-
pied by the moon in passing through it,— the distance
E x~(i.e., the length of the earth's shadow)— becomes
also known ; consequently the value of the angle c x d, or
axh, which is the same, is known. Since the point /is
on the surface of the earth, from which the astronomer
views the sun, the angle a / ft is the observed angle
subtended by the sun's diameter. It is at once evident
that the angle afbk greater than the angle e x d-,
consequently if the lines a: c and a: ^ be produced in-
definitely, they must eventually intercept the lines /a and
fh in the points a and b, at the two extremities of the
sun's diameter. If, therefore, we ascertain by observation
the exact value of the angle a /ft, at the time the moon
is in opposition, we shall have the means of readily com-
puting the distance 8 E oi the sun from the earth. To
illustrate this by example, we have the distance ^ ^ of
the earth's shadow ascertained to be equal to 218 semi-
diameters of the earth, and, hence, the value of the angle
Exck determined as 15' 46^ (Now the angle of the
sun's diameter, as observed at different times of the year,
is supposed to vary from 31' 32".0 to 32' 36".2, but. with
respect to an actual difference in the distance of the sun,
we must either decline to accept this reported great
variation, as correct only in respect to an apparent

BY SIXTH METHOD.

variation in the sun's magnitude when viewed at different
seasons from the same locality, or from places situated
nearly in the same latitude, and presumable erroneous in
fact, in respect to observation made from the earth's centre
through an atmosphere at all seasons in the same condition
as to temperature, density, and humidity ; or, otherwise, we
must accept the assumption that the earth, in the course of
each annual revolution, approaches and recedes from the sun
through a space equal in linear extent to nearly six times
the diameter of the moon's orbit.) For the present purpose
we will assume the average angular value of the sun's
semi-diameter to be determined as 15' 55". The difference
between this angle (15' 55") and the angle Exc (15' 4G"),
which equals 9", is the angle subtended by the earth's
semi-diameter as seen from the sun, and, therefore, since
the actual length, or linear value in miles, of the earth's
semi-diameter is known, we have ascertained the value
of a definite fraction of the earth's orbital circle, and
hence the distance of the sun from the earth, which is the
radius of that circle, becomes known. For instance, if 9^
be the ascertained difference between the angles, then,
since 1 degree contains 400 times 9", multiplying 400 by
4,000 miles as the linear value of the earth's semi-
diameter, we obtain 92,800,000 miles as the distance of
the sun from the earth.*

This method furnishes the angular value of the earth's
diameter as seen from the sun, which is already obtained
with an approximation to correctness by means of the geo-

* Taking the value of tae earth's radius at .3,950 miles, which is
very nearly the actual estimate, the distance will be 91,640,000 mi es-
These computations arc, however, mainly intended to illustrate the
methods.

MEASIJUF.MKNT OF SUN S niSTANCK

centric parallax of tho sun. It is a question whether or
not this method now proposed is susceptible of a greater
degree of exactitude. As an entirely independent and
distinct method, however, it cannot fail to possess some
considerable degree of interest and utility.

Instead of measuring the angle of the shadow behind

the earth, the same angle may be indirectly obtained by

measuring the breadth of the sun's light at any known

definite distance between the earth and the sun. This

may be readily explained by reference to the figure ; for

instance, if the moment when tho limb of the advancing

moon commences to interpose itself between the sun and

earth on the one side be exactly determined, and also tho

moment of the conclusion of the egress of the moon on

the other side of tho orbit, so as to ascertain the exact

time occupied by the moon in traversing the angular

space of the sun's diameter (viewed from the earth), tho

required value of the angle at the distance of the moon's

semi-orbit would become known. More favourable for

this purpose would be a transit of the sun by one of the

inferior planets. Records of carefully observed transits

of Venus might be made available perhaps to determine

in this manner the precise value of the angle of the earth's

shadow, supposing the same records to include the angular

valu ~ of the sun's diameter as seen at the time of the transit.

(7.) A method, which may be considered as allied to
the preceding, coL/ats in measuring by astronomical
observation the apparent value of the sun's diameter, as
seen, on the one hand, from a station on the earth's surface
when the sun's centre is over the meridian of that station ;

SUNS DISTANCE

By the differenUal An^e al'liartiis Had

lUS.

Eg. 8.

BY SEVKNTH METHOD.

and, on the other hand, as Hoon, at the same timo, from the
«arth'a contro.

The manner in which it is proposed to obtain the com-
parative angles, is by means of the transit instrument,
by which, having first ascertained with precision the
angle subtended by the semi-diameter of the sun when
the sun's centre is over the meridian of the station, the
timo elapsing until the extremity of the sun's diameter
(i. c, the edge of the sun's limb) is over tho station, is to
be carefully observed, which time will measure the angle
subtended by the semi-diameter of the sun as viewed from
the earth's centre. This last angle must be evidently less
than tho former, and by the dliforonco the distance of tho
sun may bo determined, because the metrical value in
miles of the earth's semi-diameter is already known, and
if the two lines, tho inner of w^ Ich has a greater obli-
quity than the outer, be produced until they eventually
meet, tho point of interception will be the sun's distance,
and must be directly proportional to the semi-diameter of
the earth, in a ratio determined by the observed angle.*

The principle of this last method is fundamentally the
same as that explained in the case of the earth's shadow,
and it is possible that the convenience, directness and
simplicity of this method, notwithstanding the delicacy
and extreme accuracy of observation requisite, may
render it preferable and practically more advantageous

• Instead of tho semi-diameter, the eutire diameter of the snu
may of course be observed. A slight correction would be tlieo-
retically required as an allowance for orbital motion of the earth ;
the effect of which would tend to increase tlie time, and which
would be accordingly corrected by deduction. The quantity,
however, would be extremely minute (about the 1400th of a
degree of orbit,) and insufficient to be appreciable in the practical
application of the method.

! !l

30 MEASUREMENT OF SUN's DISTANCE BY SEVENTH METHOD,

than either of the methods previously described or
hitherto practised.

To illustrate this method : let the angle of the sun's
semi-diameter viewed from the station be assumed as 16',
and viewed from the earth's centre as .15' 59"-96, the
dilference of one-twenty-fifth of a second on the earth's
radius, taking the metrical value of that radius, as before,
at 4,000 mileo; would give about 96 million miles as the
sun's distance.* In the figure (fig. 8) the enormous ex-
aggeration of the ratio of the earth's radius to the sun's
distance (by representing the sun near to the earth)
renders the basis of the method more distinctly apparent.

* The computation is . . GO x 2 5 X 16 = 24,000 ; whioh x 4,000 =
96 millions.

WEiiTHEiMEn, Lea & Co., Circus Place, Finsbury Circui.