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The end of the annular appearance 3 3 J 43

The end of the eclipse 4 44 51

At Hopeton-house, 9 miles west, and a little northwards from Edinburgh,
Lord Hope observed the annular appearance begin at 3^ 25*", the end of this
appearance at 3*' 31™, and the end of the eclipse at 4'" 444-'". His lordship was
obliged to observe the eclipse at a distance from the clock, and to determine
the times by a pocket watch, that had been adjusted by a very good dial that
day at 12 o'clock; but he says, that the duration of the annular appearance
was 6"", as near as could be judged by a watch that did not show the seconds.

At Crosby, on the west coast of Scotland, about 4 miles north from Air,
Mr. FuUarton observed the eclipse to begin at 2 o'clock. A distinct annulus
was formed about 20™ after 3, which continued exactly 7*", measured by a pen-
dulum vibrating seconds. It appeared rather broader on the lower verge of the
sun ; but the difference must have been very small, for it was but barely dis-
cernible in a species of the eclipse 6 inches over, cast on a piece of paper be-
hind the eye- piece of a telescope 6 feet long. He adds, that the day light was
not greatly obscured, appearing only so much dimmer than usual, as that of
the sun is, when seen through a very gentle mist in a fine morning in April or
May. Sir Thomas Wallace found that the annular appearance continued at his
house, near Lochryan in Galloway, 5™.

From the observation at Crosby, the centre of the annular penumbra seems
to have entered Scotland not far from Irwine. It next proceeded towards the
east, with a considerable inclination northwards ; and probably left Scotland not
far from Montrose on the east coast ; for the Rev. Mr. Auchterlony found,
that the annular appearance continued there J'", as near as he could judge by
an ordinary watch. The annulus also appeared to him of a uniform breadth,
through a common telescope. This observation, though not so exact as that at
Crosby, is however confirmed by that at St. Andrew's, mentioned below. These
two observations at Crosby and Montrose, were made nearer the path of the
centre, than any others that have been communicated.

As for the southern limit of this appearance, the eclipse was not annular at
Newcastle, and there wanted about 40° of the sun's limb to appear in order to
form an annulus, according to the observation of Mr. Isaac Thomson, com-
municated by Mr. Blake, a gentleman of the county of Durham, who was



VOL. XL.] PHILOSOPHICAL TRANSACTIONS, }f^

present with us at Edinburgh during our observation. The whole duration of
the eclipse was 5Cf less by his, than by our observation. Nor was the eclipse
annular at Morpeth, whence Mr. John Willson writes, that the body of the
moon appeared almost entirely on that of the sun; an4 that, to the naked eye,
the disk, of the sun seemed to be almost round.

But of all the observations that have been communicated, that of Mr. Long
at Longframlington, 7 computed miles north of Morpeth, determines the
southern limit with the greatest exactness. The annulus, he says, was very
small there on the upper part, and the duration 40 or 41 half seconds, mea-
sured by a pendulum 9.8 1 inches long; from which we may conclude, that the
limit was very near this place. This curious observation, with several others,
was communicated by Mr. Mark at Dunbar. At Alnwick, in Northumberland,
the eclipse was annular, but the time of its continuance was not measured.

At Berwick, the annular appearance continued between 4 and 5 minutes:
the end of the eclipse at Dunbar, by Mr. Mark's observation, was at 4*^ 48™
16^; but some mistake was committed in reckoning the vibrations of the pen-
dulum, in measuring the continuance of the annulus.

At St. Andrew's, this appearance was observed to continue precisely 6™, by
a pendulum clock, by Mr. Charles Gregory and Mr. David Young, professors
in the university. By a figure of the annulus taken from its image, projected
through a telescope on a paper screen, the breadth towards the south-east part
of the sun's disk, was rather more than double of its breadth towards the
opposite part.

The observation at Montrose has been already mentioned. At Aberdeen the
annulus was observed by Mr. John Stewart, math. prof, for 3"^ 2^ It was
almost central, when the clouds deprived him of any further view of it; he
thinks it probable, that it continued there about 6". Several gentlemen, resid-
ing on the coast northwards from Aberdeen, were desired to observe the coutir
nuance of the annulus; but I do not find that any of them saw this pheno-
menon from the beginning to its end.

At Elgin, the eclipse was observed annular at 3^ 29"", the larger part of the
ring being uppermost, by the Rev. Mr. Irvin, who had a view of it for about
3(y; but by reason of intervening clouds could not determine the beginning or
end of this appearance. At Castle Gordon, Mr. Gregory had one view of the
eclipse while it was annular, but could make no further observation for the same
reason.

At Inverness, the eclipse was annular for some minutes, as observed by several
gentlemen; but they did not measure the precise time how long it continued.
By the accounts from Fort Augustus and Fort William, it is doubtful whether



iif4 PHILOSOPHICAL TRANSACTIONS. [aNNO 1738.

the eclipse was annular in those places or not. Fort Augustus is at the west
end of Lochness, and probably was not far from the northern limit of this
phenomenon.

Several gentlemen of very good credit, who are not in the least short-sighted,
assure Mr. M. that about the middle of the annular appearance, they were not
able to discern the moon on the sun, when they looked without a smoked glass,
or something equivalent,

Mr. Maclaurin remarks this, because it may contribute to account for what
at first sight appears surprising, that there are so few annular eclipses in the lists
collected by authors. Kepler, in his Astron. Optic, does not seem to acknow-
ledge, that any eclipse, truly annular, had ever been observed. There are none
mentioned by Ricciolus, from the year 334 till 1567, though there are 13 or 14
total eclipses recorded within that period ; yet it is allowed, that the extent and
duration of the annular appearance may be considerably greater in the former,
than of the darkness in the latter. It may have contributed to this, that annular
eclipses must have been rather incident in the winter season in the northern
hemisphere, and that eclipses have been more readily total in the summer, when
their chance of being visible was greater, and the season more favourable for
observing them. But perhaps the chief reason why few annular eclipses appear
upon record, is, that they have not been distinguished in most cases from ordi-
nary partial ones. The darkness distinguished total eclipses, or such as were
very nearly total ; and it is these chiefly, that historians mention. There are
two central eclipses of the sun still famous among the populace in this country :
that of March 29, l652, was total here, and that day is known among them
by the appellation of Mirk Monday. The memory of the eclipse of Feb. 25,
1598, is also preserved among them, and that day they term, in their way.
Black Saturday. There is a tradition, that some persons in the north lost their
way in the time of this eclipse, and perished in the snow.

There was a remarkable total eclipse of the sun in this country, June 1 7,
1433, the memory of which is now lost among the populace; but it appears
from a passage in a manuscript in our library, that it was formerly called by
them the Black Hour, after their usual manner. It is described thus : " Hoc
anno fuit mirabilis Eclipsis Solis, 17""° die mensis Junii, hora quasi tertia post
meridiem ; et per dimidium horae tenebrae tanquam in nocte supergressas sunt
superficiem terrae, ita ut nihil obtutibus humanis pervium fuit ; unde abhinc
vulgariter dicta fuit hora nigra." This eclipse is not in Ricciolus's Catalogue,
but is mentioned by him in another place, Schol. cap. 2, 1. 5. By a computa-
tion of this eclipse, the sun was within 2° of his apogeum, and the moon with-
in 13" of her perigeum ; so that this must have been a remarkable eclipse.



VOL. XL.] PHILOSOPHICAL TRANSACTIONS. 175

The progress of the shadow was towards the south-east ; and "Sethus Calvisius
cites the Turkish annals for its being total in some part of their dominions.

You will perceive by this account, that we have no observatory in this place;
but we are in hopes that some time or other we shall obtain one from the patrons
of the university. I doubt this last eclipse will not be distinguished by any par-
ticular appellation among the populace, as the former that were central in this
country. The remembrance of it however will be preserved by the curious,
who observed it with great pleasure, and agree that it was the most entertain-
ing spectacle of this kind they ever saw.

4. The same observed at Edinburgh. By Sir John Clerk, Bart. F.R.S. p. IQ5.

The eclipse began at 5"" 36^ after 2. The annular appearance began at 25™
55' after 3 ; its continuation was 5™ 48^ The end of the eclipse was at 44"* 50*
after 4 ; all reckoned by apparent time. We had half a score good reflecting
telescopes to make these observations, and our calculations perfectly agreed, so
that you may depend upon them as most exact.

This was not done by us as a matter of mere curiosity, but to assist in
ascertaining the motions of the moon, on Sir Isaac Newton's theory, on which
a good deal of the doctrine of the longitude will depend. Sir Isaac's calculation,
as to the beginning of this eclipse, was pretty right ; but not so well as to its
central appearance Two spots in the sun made a very distinct appearance to
us, as they entered under the moon's body ; one was a little above the central or
horizontal line of the sun ; the other was near the edge, on the east quarter.
The first, by comparison with the sun's diameter, was larger than the disk of
our earth ; it was dark in the middle, and certainly emitted no fire or light.
The edge of the moon appeared a little ragged or rough, but not mountainous,
because of the sun's light. There was no considerable darkness, but the ground
was covered with a kind of a dark greenish colour. Two stars appeared, the
planet Venus, and another farther eastward. This account is what you may
depend on.

5. The same Eclipse observed at Cambridge, and Kettering; p. 197.

The beginning by the clock at 2*^ 36™ 40^

The end at 5 14 12 Exact.

Times observed at Kettering, as follow :

Beginning 2*' 21"

2 Digits 2 36



im



Centre....,.., , / ^

jaJtffl livJcn HUM »• I 4

End 4 5Q



7
4 22



176 PHILOSOPHICAL TRANSACTIONS. [ANNO 1738.

N. B. The observatory clock was I minute 50 seconds too slow, which being
added all the way, will give true time.

6. The same observed at the Institute of Bononia. p. igg.
The eclipse began at 3*^ 33" 35% being more than ^ minutes sooner than the
calculation made it.

1 . The same observed at the Aventine Hill at Rome. By the Abbe de

Revillas, p. 200.
The beginning there was at 3** 43'".

8. The same Eclipse observed at Wittemberg. By J. F. Weidler. p. 201.
Neither the beginning nor the end was seen ; only some digits were observed
on the decrease ; particularly 8 digits were eclipsed at 4'' 50" 31'.

A Proposal to make the Poles of a Globe of the Heavens move in a Circle
round the Poles of the Ecliptic. By the Rev. Ebenezer Latham, M. D. and
F.D.M. N°447, p. 201.

As we now have the globes of the heavens, they are only formed for the
present age, and do not serve the purposes of chronology and history, as they
might, if the poles on which they turn were contrived to move in a circle
round those of the ecliptic, according to the present obliquity of this. By
this means we might have a view of the heavens suited to every period, and
that would answer the ancient descriptions, those of Eudoxus, for instance,
who is supposed to borrow his from the most early observations ; and of Hip-
parchus, &c. Nor could any contrivance better enable the lowest reader to
judge of the merits of the controversy about the Argonautic expedition, as far
as it depends on this: for it will verify to the sight the path of the colours, &c.
at any time.

N. B. That globes, to answer the end here proposed, though differently
constructed, had long before been made and published by Mr, Senex, who at
the next meeting of the R. S. gave the following account of his contrivance.

A Contrivance to make the Poles of the Diurnal Motion in a Celestial Globe
pass round the Poles of the Ecliptic, Invented by John Senex,* F. R. S.
N"447, p. 203.

The poles of the diurnal motion do not enter into the globe, but are affixed

• Mr. Senex, F. R. S. was a bookseller, and a celebrated maker of globes and planispheres, &e.
He died Dec. 30, 1741.



ii



VOL. XL.] PHILOSOPHICAL TRANSACTIONS. 177

at one end, to two shoulders or arms of brass, at the distance of 23° and -4.
from the poles of the ecliptic. These shoulders at the other end are strongly
fastened on to an iron axis, which passes through the poles of the ecliptic,
and is made to move round, but with a very stiff motion; so that when it is
adjusted to any point of the ecliptic, which you desire the equator may
intersect, the diurnal motion of the globe on its axis will not be able to
disturb it.

When it is to be adjusted for any time, past or to come, bring one of the
brazen shoulders under the meridian, and holding it fast to the meridian with
one hand, turn the globe so about with the other, that the point of the ecliptic,
which you would have the equator to intersect, may pass under no degrees of
the brazen meridian : then holding a pencil perpendicular to that point, and
turning the globe about, it will describe the equator as it was posited at that
time; and transferring the pencil to 23-4-°, and 66-J° on the brazen meridian,
the tropics and polar circles will be described for the same time.

By this contrivance, the celestial globe may be so adjusted, as to exhibit not
only the risings and settings of the stars, in all ages, and in all latitudes, but
the other phaenomena likewise, that depend on the motion of the diurnal axis
round the annual axis.

The Solution of Kepler s Problem, by J. Machin, Sec. R. S. N° 447, p. 205.

Many attempts have been made at different times, but Mr. Machin thinks
never any yet with tolerable success, towards the solution of the problem pro-
posed by Kepler : to divide the area of a semicircle into given parts, by a line
from a given point of the diameter, in order to find a universal rule for the
motion of a body in an elliptic orbit. For among the several methods offered,
some are only true in speculation, but are really of no service. Others are not
different from his own, which he judged improper: and as to the rest, they
are all some way or other so limited and confined to particular conditions and
circumstances, as still to leave the problem in general untouched. To be more
particular; it is evident, that all constructions by mechanical curves are seem-
ing solutions only, but in reality unapplicable; that the roots of infinite serieses
are, on account of their known limitations in all respects, so far from affording
an appearance of being sufficient rules, that they cannot well be supposed as
offered for any thing more than exercises in a method of calculation. And
then, as to the universal method, which proceeds by a continued correction of
the errors of a false position, it is, when duly considered, no method of solu-
tion at all in itself; because, unless there be some antecedent rule or hypothe-
sis to begin the operation, (as suppose that of a uniform motion about the

VOL. vni. A A .



178 • PHILOSOPHICAL TRANSACTIONS. [aNNO 1738.

upper focus, for the orbit of a planet; or that of a motion in a parabola for
the perihelion part of the orbit of a comet, or some other such) it would be
impossible to proceed one step in it. But as no general rule has ever yet been
laid down, to assist this method, so as to make it always operate, it is the
same in effect as if there were no method at all. And accordingly in experi-
ence it is found, that there is no rule now subsisting but what is absolutely
useless in the elliptic orbits of comets ; for in such cases there is no other way
to proceed, but that which was used by Kepler : to compute a table for some
part of the orbit, and therein examine if the time to which the place is required,
will fall out anywhere in that part. So that, on the whole, it appears evident,
that this problem, contrary to the received opinion, has never yet been ad-
vanced one step towards its true solution : a consideration which will furnish a
sufficient plea for meddling with a subject so frequently handled ; especially if
what is offered shall at the same time appear, as he trusts it will, to contribute
towards supplying the main defect.

Lemma I.— The Tangent of an jirch being given, to find the Tangent of its
Multiple. — Let r be the radius of the circle, t the tangent of a given arch a,
and n a given number. And let t be the tangent of the multiple arch nX a,
to be found.

Then if f j be put for — rr, and tt for — tt ;

~~, — iB In

The tangent t will be ' \p :

r+r\ +r—r\

Which binomials being raised according to Sir Isaac Newton's rule, the ficti-
tious quantities t and j will disappear, and the tangent t will become equal to

», n— t. «— 2. ^' , B. n— 1. «— 2. n— 3. n— 4. ^5 o

" 1 ~2~ 3 j^"^! ~2 3 4~ 5 r*~



». »— 1 . ^ _i » . n^l.n— 2. n— 3. *■* » -

r 2 rr ' 1 2 3~ ~4~ H

This theorem, which he formerly found for the quadrature of the circle, at
a time when it was not known here to have been invented before, has now been
common for many years ; for which reason it is premised at present, without
any proof; only for the sake of some uses that have not yet been made of it.

Carol. 1. From this theorem for the tangent, the sine, suppose y, and
cosine z of the multiple arch n X a, may be readily found.

For if y be the sine, and z the cosine of the given arch a, then putting w

TT

for — WW, and substituting -^ for Y, and — for t, and - , , - for y:



The sine y will be !± F I -'-^I f.
2r«_

mu • -11 u 2 + '^l" + 2 - "I"

The cosme z will be — '



2r»-'



VOL. XL.] PHILOSOPHICAL TRANSACTIONS. 1^0

Each of these may be expressed differently in a series, either by the sine
and cosine conjointly, or by either of them separately.

Thus Y the sine of the multiple arch n X a, may be in either of these two

forms, viz.

z»-' . . n— 1. n-2 y' , «— 3. n— 4 y* « „

= — rV into n — ■ — -— A. •^ H r— — - B.~ — OfC.

r—i^ 2 3 z' ' 4 5 z*

nn—l 3 nn— 9 . nn— 25 , .

o"- = "i/ - T3;7 *^ - T^ "^y - tttt ""y - *^"-

Where the letters a, b, c, &c. stand, as usual, for the co-efficients of the
preceding terms.

The first of these theorems terminates when n is any integer number; the
other, which is Sir Isaac Newton's rule, and is derived from the former by sub-
stituting Vrr — yy for z, terminates when n is any odd number.

The cosine z may, in like manner, be in either of these two forms, viz.

2» • , ".'»— 1. y' 1 n. n— 1. n— 2, n— 3, y* »

nn 3 «»— 4 4 n»— 16 g .

2rr " 3.4rr " 5.6rr "

The latter of which terminates when the number n is even, and the other
as before, when it is any integer.

Corol. 2. Hence the sine, cosine, and tangent of any submultiple part of

an arch, suppose - a, may be determined thus:

_i^ j^

The tangentof i- a will be r+TK^IZlhf .

r+TJ" 4- r— t|



I



The sine of ^a will be i+ifcinlCj.

2r"

For these equations will arise from the transposition and reduction of the
former, for the tangent and sine of the multiple arch, on the substitution of
/, y, z and A; for t, y, z and n X a.

Corol. 3. Hence regular polygons of any given number of sides may be
inscribed within, or circumscribed without, a given arch of a circle. For if
the number n express the double of the number of sides to be inscribed within,
or circumscribed about, the given arch A; then one of the sides inscribed will
be the double of the sine, and one of the sides circumscribed the double of
the tangent of the sub-multiple part of the arch, viz. -a.

Lemma Il.—-ToJind the Length of the Arch of a Circle within certain Limits,
by means of the Tangent and Sine of the Arch. — Let t be the tangent, y the
sine, and 2 the cosine of the arch a, whose length is to be determined; and

A A 2



160 PHILOSOPHICAL TRANSACTIONS. [aNNO 1738.

let f, T, V be expounded as before; then, if any number n be taken, the arch

of the circle will be

1 I

, . , 7*4- T|" r — t1 "

always less than \ '— X «f,

and greater than ^±^rjzliz3l — x wj .

For if, by the preceding Corollaries, a regular rectilinear polygon be in-
scribed within, and another without, the arch a, each having half as many
sides as is expressed by the number n ; then will the former of these quantities
be the length of the bow of the circumscribed polygon, or the sum of all its
sides, which is always greater, and the latter will be the length of the bow of
the inscribed polygon, which is always less, than the arch of the circle : how
great soever the number n be taken.

Carol. 1 . Hence the serieses for the rectification of the arch of a circle
may be derived.

For by converting the binomials into the form of a series, that the fictitious
quantities, j , t, v may be destroyed ; it will appear, that no number n can be
taken so large as to make the inscribed polygon so great, or the circumscribed
so little, as the series

"I" ~ ^ + 1» ~ ^ + ^*^" '" °"^ ^^^^' °^ ^^^ ^1"^^
t' <* f

t — T'i'h T7 — rrr -\- &c. in the other case.

Therefore, since the quantity denoted by the sum of the terms, in either of
these serieses, is always greater than any inscribed polygon, and always less
than any circumscribed, it must therefore be equal to the arch of the circle.

Carol. 2. If, in the first of the above serieses, the root \^rr — yy, be ex-
tracted, and substituted for z, there will arise the other series of Sir Isaac
Newton, for giving the arch from the sine; namely,

y + & + 157 + "iS^ + ^^- °' otherwise,

— V^ 1.2.3 ^ r^ ^ 1.2.3.4.5. ^ r* ^ 1.2.3.4.5.6.7. r" ^ ^^'

Schol. In like manner, as the arches of the polygons serve to determine
the arch of the circle, so by comparing the areas of the circumscribed and in-
scribed polygons, -^nrT and ^n\z, the area of the sector of a circle may be
found. For if t, y and z be the tangent, sine and cosine of the arch a; then,

by the second Lemma, the area of the circumscribed polygon

^ 1^

will be found to be -j-nrj X ^+''1" — r— t|" _ ,^^.j,_



VOL. XL.} PHILOSOPHICAL TRANSACTIONS. Iftl

and the area of the inscribed will appear to be



But on the expansion of these binomials it will appear, that no number n
can be taken so large as to make the one so large, or the other so small, as
the area denoted by the series.

Xr n\ t ~ h T-: — :r-s + «C'

^ Zrr ' 5H 7r°

So that this area being larger than any inscribed, and smaller than any cir-
cumscribed polygon, must be equal to the area of the sector.

It may further be observed, that as the arch or area is found from the sine,
cosine, or tangent of the arch, by means of the limiting polygons, so may the
sine, cosine or tangent be found from the length of the arch, by the same
method.

Thus, if A be the arch, whose tangent t, sine y, and cosine z, are to be
determined, then will the

A - -i- X - + ^ X — - &c.

1.2.3. '^ r^ ~ 1.2.3.4.5 r<



Tangent T be = ■- .. , .4

Cosine z = r - -jL X ^ + j^- X -^ - &c.

For it may be made to appear, from the first Lemma, and its Corollaries,
that if in any of these theorems, as suppose in the first, the quantity a stand
for the bow of the circumscribed polygon, then will the quantity t, exhibited
by the theorem, be always larger; but if for the bow of the inscribed, always
less than the tangent of the arch, how great soever the number n be taken ;
and consequently, if a stand for the length of the arch itself, the quantity x
must be equal to the tangent ; and the like may be shown for the sine, and,
mutatis mutandis, for the cosine.

These principles, from whence he has here derived the quadrature of the
circle, which is wanted in the solution of the problem in hand, happen to be,
on another account, absolutely requisite for the reduction of it to a manage-
able equation. But he has enlarged, more than was necessary to the problem



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