William Vincent.

The commerce and navigation of the ancients in the Indian Ocean online

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dates and circumstances agree together when the precession of
the equinoctial points is allowed for. You wish also to have a
popular explanation of the term Precession, in antecedentia, ani
an account of its application to, and effect on, the phenomena-
which have been explained above.

The two points where the ecliptic crosses the plane of the
earth's equator are called the Equinoctial Points. That which
the sun is in on the 20th or 21st of March, when he passes to
the northward of the plane of the earth's equator, is called the
Vernal Equinoctial Point ; and the other is called the Autumnal
Equinoctial Point.

The earth is not a perfect sphere, but is in the form of such a
bowl as is used on a bowling-green ] the two poles being in the
two flat sides, and its greatest diameters all in the plane of the
equator. Now, as all bodies attract each other, the protuberant
parts about 4hfc earth's equator are acted on by the sun and



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DISSERTATION I i. 549

moon, when they are out of the plane of that equator, ki such
a manner as to cause the two equinoctial points to be carried
backward, along the ecliptic, at the rate of 50i seconds of a
degree in a year ; and this motion of the equinoctial points is
called, though somewhat improperly perhaps,^ the Precession of
the Equinoctial Points.

As the vernal equinoctial point is carried backward by the
above-mentioned quantity yearly, while the fixed stains retain '
their places, and as we continue to reckon the longitudes of the
^tars from that point, it is manifest the longitudes of the stirs
will be increased every year by 50| seconds. But as the mo-
tion of these points is in the plane of the ecliptic, this apparent
motion of the stars will be parallel to the ecliptic ; and, conse-
({uently, their distance from the ecliptic, which is called their
latitude, will not be altered by it. It must be farther observed
that the year (as it relates to astronomy) always begins when
the sun is in the vernal equinoctial point; fmm which it will be
evident that it is later, by a small quantity^ every year than it
was the year before, when the sun comes ta the same longitude
with any particular star, or to that point of the ecliptic where it
rises or sets with it : and this is the cause why the Pleiades rise
as the sun sets, and set as the sun rises, later now than they
did formeriy.

It has been already said, that the longitude of the Lucida
Fldadum was « 26'' 38' 34", at the beginning of the year I76O ;
but in the I7I8 years which elapsed between the years 42 and
1760, the precession of the equinoxes j at the rate of 501 seconds
in a year^ amounts to 86,472f seconds, or 24** T 121", which
being, taken from 8. 26" 38' 34",. leaves » 2'37'2ir for the



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-$5«> 'APPENDIX.

longitude of iti Pleiadum in the year 42 after Clirist : and, as
the latitudes of the stars remain the same % the point of the
cchptic which then rose with this star was r 29*" 7' 9\ the ob-
liquity of the ecHptic being at that time 23'' 41' 24"^ Hence the
point which set as the star rose was ^ 29"* 7' 9" ; and this point,
I find by Mayer's Tables, the sun was in on the 19th of Oc-
tober. By a similar process, I find that the point of the echptic
which rose as the Pleiades set was n 4* 20', which point the sun
occupied on the 29th of October that year.

The former of these determinations differs nine days, and tlie
latter ten from the times assigned by Columella ; but it may be
remarked that the former of these errors is in defect, and the
latter in excess ; and as the stars rise and set sooner as the year
advances, it follows, that on the 10th of October the sun would
set a short time before the star would rise, an(J on the 8th of
November the star would set some time before the sun rose;
both which circumstances appear to be necessary if these phe-
nomena were determined by observation, as, most probably,
was the case. For it is manifest the star's rising cannot be ob-
served when it rises exactly a« the sun sets ; nor can its netting
be seen when it sets exactly as the sun rises, on account of the
daylight, as hath been already, observed : but, perliaps, the
one might be seen by a good eye, in the latitude of Rome, nine
or ten days before, and the other as much after the time when
the two circumstances happened together; and I have not a
doubt but that the differenc<3 between Colmnella's observation
and my calculation is to be attributed to this cause.

• I take np notice here of the very small fixed stars by the actions of the other planets
ehange which is caused in the places erf the «n the eartih.

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DISSERTATIOrl^- I. i. SSt

I am next to inquire whether the effect, of the precession of
the. equinoctial points will reconcile Strabos account, which
states that Nearchus sailed at the time when the Pleiades rose
in the evening, that is to say, as the sun set, with the account
of Arrian, who says expressly, that he sailed on the 2d of Oc*
tober in the year before Christ 326. In the interval between the
year 42 after, and the year 326 before Christ, the precession
amounts to 5^ 8' 42f", which being taken from « 2^ 37' 21*', the
star's longitude in the year 42 after Christ, leaves t 27° 28' 38ft
; for the longitude of the Lticida Pleiadum in the year 326 before
Christ ; and the point of the ecliptic which rose with the star, in
this situation, at Rome, in the year 326 before Christ, the ob-
liquity of the ecliptic being then 23^ 44' 13", was t 19° 26' 41' :
but as the sun was setting when the star rose, it must have been
in '^ 19° 26' 41'^, the opposite point of the ecliptic, which point
the sun occupied on the 17th of October ; fifteen days after that
which is fixed by Arrian for the sailing of Nearchus^. Now if
nine or ten days were sufficient to render the rising of the
Pleiades visible at Rome, we are certain that more could not be
requisite to render their rising visible at the place Nearchus
sailed from, which is in a much lower latitude ; we are there-
fore led to suppose,- either that Strabo spake in. general terms,
(as indeed seems to be the case,) meaning only to jx)int out the
season, and not the day when Nearchus set out on his ex-
pedition, while Arrian gave the precise day on which it hap^
pened, or that some mistake has crept into one or the other
of these authors : to me, the former supposition seems most
natural



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55« APPENDIX

But notwithstanding it is highly probable that the apparent
difference between the two historians ought to be referred to one
or other of ttiese causes, it is by no means certsun that either one
or other of them must be resorted to. It is possible that it ought
to be attributed to another cause.

The preceding calculation is foimded <m a supposition that the
Julian calendar has been in use ever since the year 326 before
Christ ; but we know it was not established by law till about
^ years before Christ, and that before that aera different modes
of computation were used by different persons, who did not
always tell us what mode of computation they made use of.
Now, notwithstanding both Arrian and Strabo refer to the same
authority, it is possible the years by which that author reckoned
might differ from Julian years ; and if they did, a greater dif-
ference than that which exists between them might arise from
that circumstance.

There is a circumstance occurs in the foregoing ealculations
which may lead some persons to conclude I have committed a
mistake in them ; and which it is therefore necessary to obviate.
The quantity of the precession in tlie interval between the year
326 before, and the year 42 after Christ is 5"* 8' 421", a space
which the sun is more than five days passing over. It may
therefore be supposed, that the difference between the achrtv
^lycal risings of the same star, at these two times, ought to be
between five and six days, whereas I make it Kttle more than
*wo : but it must be considered, that near three of these five
4ays are ?inticipated by the excess of the Julian year above the
^e length of the solnr 3'ear in that intcrvaL



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DISSEIITATTON I U 553

This, Sir, is the plainest answer I can give to the questions
you have been pleased to propose. Tliey betray no ignorance
in a person who does not profess to be an astronomer, as the
circumstances are certainly sufficient, to create doubt, which
every rational mind must be anxious to clear up. If I have
contributed any thing toward this by what I have written, I
shall be very happy, and am,

Reverend Sir,

Your's very sincerely,

WILLIAM WALES.

Christ's Hospital,
Jan. i4ftb9 1796,



4B



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C 554 3



DISSERTATION I. ii.



On the Rising of the Constellations^

Dear Sir,

^^T'OU will receive, I hope, some satisfaction from the result
•*; of my calculations upon the different dates, assigned by
Arrian and Strabo, to the commencement of Nearchus's expedi-
tion ; which at first, I confess, I thought too discordant to be
reconciled by any probable conjecture, without tampering with
the text of Arrian, which, in my judgment, seemed to carry
some marks of corruption.

The method I have taken has been to go directly to the in-
vestigation of the time of the acronychal rising of the Pleiades,
in that part of the world where the voyage was undertaken, in
the year before Christ 326, which was the year of the voyage ;
and, for a reason which will presently appear, I have not con-
cerned myself at all with Columella's risings or settings.

Arrian says, that Nearchus sailed from the mouth of the Indus
as soon as the Etesise ceased, in the eleventh year of the reign
of Alexander, according to the reckoning of the Macedonians
and the Asiatics, and on the 20th of the Athenian month Boe-
dromion* This eleventh of the reign of Alexander, it is agreed,
was the year before the vulgar aera of our Lord 326 ; and the

4



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DISSERTATION L iu sss

20th of Boedromion in this year, upon the authority of eminent
chronologers, you take to have been the 1st of October, St. Jul.
And in this reduction, if there be any error, which, though I
suspect, I will not too confidently assert, it cannot be of more
than a single day.

Strabo's account is,, that ** the fleet sailed in autumn about
the season of the evening rising of the Pleiades, before the winds
were fair, the barbarians attacking them and forcing them to
sea."'

This claims great attention, for it is Nearchus's own account.
The words of Strabo import as much, and the thing speaks, in
some measure, for itself. The character, by which the time is
described, is of a sort to have been taken from the journals of
the mariners themselves; for any second-hand writer of the
voyage would have expressed it in a more popular manner, by
affixing to it, as we see Arrian has done, a precise date, or a
date at least pretending to precision, in some well-known civil
reckoning. But if this character of the time of the commence-
ment of the expedition came from the original journals of the
mariners ; it follows, that some two or three days before they
sailed, or two or three days after, (for in this sort of date no
greater accuracy is to be expected,) they saw the Pleiades risen
in the east, some short ^ space of time after the su;i was set in
the west ; or rather, since the star could not be seen when the
sun was yet upon the horizon, they saw the star about an hour
after sunset with that altitude, that they concluded it had risen
at the moment when the sun set.

We have to inquire, therefore, on what day of the year, in
the year of this voyage, namely before Christ 326, the Pleiades

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55<J . APPXN15IX-

rose acronychally in that part of the world, from which these
voyagers set out ; that is to say, at the mouth of the Indus.
If this should be found to agree with Arrian's date, all will be
welL If not, the phaenomena of the Roman horizon in the
time of Columella, even upon the supposition that Columella's
representation of them is exact, will throw no light upon our
subject.

Now J assume 24*" north for the latitude of the mouth of the
Indus. This is nearly the truth ; and I take the even number,
because the difference of one-half of a degree, more or less, will
not affect the result of the calculation*

By Dr. Bradley's observations, the longitude of Lucida
Pleiadum, in the beginning of the year I76O, was » 26"" 38' 34%
and tlie latitude 4!" 1' 36" north.

The interval of time, between the beginning of the year I76O
and the beginning of the year before Christ 326, is 2085 Julian
years ; and, in this time, the retrogradation of the equinoctial
points amounts to 29'' 7' 55\

Therefore, iu the beginning of the year before Christ 326, the
longitude of Lucida Pleiadum was r 27"" SCX 39", and the lati-
tude 4° 1' 36' north.

The obliquity of the ecliptic at this same time was 23'' 44' 14"*

In the figure annexed, let II A h, D A d, r O E, represent
the horizon, the equinoctial circle and the echptic, all projected
upon the plane of the meridian of the mouth of the Indus, at
the instant when Lucida Pleiadum is .rising. Let the ecliptic
intersect the horizon, on the eastern side in O ; and on the
western, in o. Let ^ be Lucida Pleiadum upon tlie eastern
horizon ; then O will be the point of the ecliptic, which comes



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DISSERTATION I. H. 557

to the eastern horizon, and rises with the star ; and the opposite
point in the west, o, will be the point of the ecliptic, which sets
when the star rises.

Through * draw a great circle of latitude * B S, meeting the
ecliptic in B, and thie equinoctial in S. Then, in the spherical




triangle v B S, we have the angle at B right. B r S, the obli-
quity of the ecliptic, = 23° 44' 14" ; the side v B, the longitude
of Lucida Pleiadum, = 27° 30' 39". Therefore, by the resolu-
tion of the triangle, we find the angle BS r = 69° 4' 57", and
the side BS= 11° 28' 56".

But the arc ♦ B is the latitude of Lucida Pleiadum, =«
4° 1' 36" ; and * S =- » B + B S = 15° 30' 32". Therefore,
in the spherical triangle « S A, we have the side « S =
15° 30' 32" ; the angle * S A (or B S r) = 69" 4' 57", and the
angle * A S, the complement of the latitude of the place, = 66\
Tlierefore, by the resolution of the triangle, we find the angle
S * A = 43° 24' 29'.

Before we proceed further in the calculation it is proper to
observe, to save unnecessary trouble, that it will not serve our



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558 APPENDIX.

I

purpose to ascertain the longitudes of the points O and o, which
the resolution of one triangle more would give. But. the longi-
tude of the point o, which sets when the star really comes to
the horizon, would give us only the day, which would be the
day of the acronychal rising of Lucida Pleiadum, if the atmo-
sphere-possessed no refractive power. But when the star is
really upon the eastern horizon, it appears, by the effect of the
refraction of the atmosphere, at the height of about half a de-
gree above it. And if the sun at the same time were setting
upon the western horizon, he would appear, from the same
cause, at the height of about half a degree above it ; so that on
the day when tlie sun is really upon the western horizon, at the
same instant when the star is really upon the eastern, the star
by the effect of refraction will have risen, and the sun will no*
be set. What we want to find is the day when the star would
be seen rising, and the sun seen setting at the same instant, if
the star could be seen in the light of the setting sun ; which will
be an earlier day, than that whereon the rising star and set-
ting sun would come to the eastern and western horizon respec-
tively at the same time. To determine this day of the visible
acronychal rising of the star, we must estimate the effect of re-
fraction both upon the star and upon the sun. The effect of
refraction upon the star will easily be ascertained by means of
the angle S * A, the quantity of which we have already deter^
mined ; and this is the only "use of the calculation, so far as we
have yet carried it.

In figure 2, let O *, O B, * B, represent the same arcs of
the horizon, ecliptic, and circle of latitude passing through the
star, as in the former figure.



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DISSERTATION I. u. 559

Through ♦ draw a vertical cir-
cle ♦ V, and set oflf an arc * R
= to the horizontal refraction, i.e.
= 3Gf 5l\ Through R draw a
great circle of latitude, meeting
the ecUptic in C ; and through *
. draw a small circle parallel to the
ecliptic, and let this small circle
meet the great circle of latitude,
drawn through R, in a-.
Now since the light of the rising star upon the horizon is
thrown, by the effect of refraction, up to R, in the vertical
circle, so as to appear in the heavens in the point R ; the star,
which, without refraction, would be seen, where it really is, at
the point ♦ in the circle of latitude ♦ B, appears at the point
R in the circle of latitude R S. Both the latitude and longitude,
therefore, of the star are changed in appearance by refraction ;
the latitude being increased by the quantity of the arc R a-, and
the longitude diminished by B C.

In the triangle ♦ R (t, right-angled at <r, which for the small-
ness of its sides may be treated as a rectilinear triangle, the
side ♦ R = SCy 51" ; the angle R ♦ (t, which with a- ♦ o, makes
a right angle, must be equal to ^O ♦ B (A ♦ S of Figure 1.),
which with the same <r m o makes a right angle ; therefore,
R * (T = 43^ 24' 29". Therefore, by the resolution of the tri-
angle, the side R (t = 21' 12% and the side * (t = 22' 24\ This
is the length of ♦ a- in parts of a great circle ; whence B C will
be found 22' 28". Hence R C, the apparent latitude of the re-
fracted star, = R <r + <r C =r 4'' 22' 48", and r C, its apparent
longitude, = <r B ^ B C = r 27' 8' 11".



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56o APPENDIX




Now then, in Figure l,_we must resume the resolution of the
triangles, making use of the star's apparent longitude and lati-
tude, instead of the true. Thus in the spherical t B S, which is
right-angled at B, put v B = 27' 8' 11"; the angle B v S,
SS"" 44' 14" as before. Then, by the resolution of the triangle,
the angle B S v = 69' 0' 30", and the side B S = 11" 20' 29*.

But » B is to be taken as the star's refracted latitude »
4" 22' 48"; and * S = S B + B S = 15' 43' 17'. Therefore,
in the spherical triangle * S A, we have the side ♦ S ti?
15' 43' 17"; the angle m S A = 69' O'SO"; and the angle ♦AS,
the complement of the latitude of the place, = GG"". Therefore^
resolving the triangle, we find the angle S » A = 43' 29' 34^

Then in the spherical triangle ♦Bo, which is right-angled
at B, we have the angle B ♦ o (S * A) = 43' 29' 34"; and
tlie side m B, the star's refracted latitude, = 4' 22' 48". Whence,
resolving the triangle, we find the side O B = 4' 8' 39\

Now V B, tlie apparent longitude of the refracted star, =r
27' 8' 11"; and we have found O B ^ 4' 8' 39". Therefore,
vO = tB-OB^22' 59' 32"; and this is the longitude of
that point of the ecliptic, which comes to the eastern horizon,
at the same instant of time, with the refracted light of the star.

The point o opposite to this, which comes to the western
horizon, at the same instant of time, when the refracted light of
the star is upon the eastern horizon, is ia» 22*" 59' 32'',



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DIS^SERTATION L 5.



561



But if this were the true place of the sun, when the refracted
star is upon the eastern horizon ; the sun would not yet be set,
but would appear, by the effect of refraction, about half a de-
gree above the horizon. We must inquire, what the sun's true
place must be, in order that the horizontal refraction may throw
his light into the point o ; for the time when this happens will
be the true acronychal rising. And for this purpose we must
estimate the effect of the horizontal refraction upon the sun's
apparent longitude ; and this depends upon the angle which the
ecliptic, at sun-rise or sun-set, makes with the horizon ; that is
upon the angle v O A (Figure 1.) or its equal ♦OB.

The angle ♦ O B is easily found, by resolving the spherical
triangle ♦ O B, in which the angle at B is a right angle ; the
angle B * O = 45* 29' 34", and the side ♦ B = 4" 22' 48".
Hence the angle * O B comes out 46* 39^ 5T-

Now, to avoid confusion, draw the spherical triangle O * B
by itself in Figure 3. Through O draw a vertical circle O V,

and take the arc O R =
horizontal refraction =
30' 51". Through R
drav a great circle of
latitude, and let it ipeet
the ecUptic in the
point a.

Then, if the sun be
upon the horizon at O,
the horizontal refraction
will throw his light up
to R, and in that point
he will appear in the
heavens. He will appear
4c




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562 APPENDIX.

at R upon the circle of latitude R u; ta will be his apparent
place in the ecliptic ; and the arc of the ecliptic, O «, will be
the difference between his true and his apparent place j or the
effect of the horizontal refraction upon his apparent longitude.

In the triangle R o ii;, which, for the smallness of its sides,
may be treated as a rectilinear triangle, the angle at ea is a right
angle. The angle R O o^, being the complement of * O B, is
43"* 2Cf S". Hence O w comes out 22' 26" ; and this, as has been
said, is the effect of the horizontal refraction upon the rising
sun's apparent longitude, hi? true place being O, in the latitude
of 24"^ north. And the same will be the quantity of the effect
upon the setting sun, in the opposite point of the ecliptic o, in
the same latitude; for the quantity of the effect in any given
latitude, upon the rising sun, in any given point of the ecliptic,
and of the effect upon the setting sun in the opposite point of
the ecliptic, will be the same ; the angle which the ecliptic in
opposite points makes with the horizon, upon which the effect
depends, being equal : but the effect lies, in the two cases, in
opposite directions ; the refraction making the apparent longi-
tude of the rising sun west of his true place, and the apparent
longitude of the setting sun east of his true place.

Hence, that refraction might throw the light of the sun to the
point o in the western horizon, at the same instant of time when
refraction brought the light of Lucida Pleiadum to the eastern
horizon, in the climate in question, we must put the sun's true
place 22' 26" west of the point o. ^

The point o has been found to be •& 22'' 59^ 32" ; therefore the
sun's true place, in order that the required effect should be pro-
duced, must have been '^ 22^ 37' 6". By an accurate calcula-:
tion of the motions of the sun, (by Mayer's Tables,) I find that,
in the year before Christ 326, he came to this place October 19,



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DISSERTATION I. ii. 563

Ky*" 25' 9' St. Jul. mean time under the meridian of Green-
wich : but to this, to be exact, we must apply a correction for
the effect of the precession upon the longitude of Lucida Pleia-
dum, in the mterval between the commencement of the year
and October 19th, (since the sun's place is deduced from the
longitude of the star,) + 16' ; add also 4' 36", for the difference
between the meridians of Greenwich and the mouth of the In-
dus, and we have October 19lh^ 15**^* 17' mean time under the
meridian of the mouth of the Indus,

The 19th of October (St. Jul.) therefore was the day of the
acronychal rising of Lucida Pleiadum, upon the horizon of the
mouth of the Indus, in the year before Christ 326, i. e. in the
year of the Julian period 4388.

It may perhaps strike you as a difficulty, that the time that
our calculation gives for the appulse of the sun to the required
place, falling between three and four o'clock in the nu>ming of
the 20th, under the meridian of the mouth of the Indus, the
sun was actually set on the 19th many hours before he came to
that point of the ecUptic, which would have made a precise
acronychal rising of the star, had the instant of ,the sun s ap-
pulse, in his annual course, to that point coincided with the
instant of sunset : but this not being the case, you may wonder
how we can say there was an acronychal rising at all.

Now this is really the fact ; that, speaking with ^ometrical
precision, there was in this year no day of an exact acronychal
rising of Lucida Pleiadum ; and it very seldom happens, that



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