Dear Sir,
Your most affectionate Brother,
and very humble Servant,
D. P. Layard.
CII. _An Account of the Heat of the Weather in_ Georgia: _In a Letter
from his Excellency_ Henry Ellis, _Esq; Governor of_ Georgia, _and
F.R.S. to_ John Ellis, _Esq; F.R.S._
[Read Nov. 16, 1758.]
Georgia, 17 July, 1758.
Dear Sir,
THO’ some weeks have passed since I wrote to you, yet so little
alteration has happened in the state of our affairs, that nothing
occurs to me relative to them worth committing to paper. This indeed
I need not regret, as one cannot sit down to any thing, that requires
much application, but with extreme reluctance; for such is the
debilitating quality of our violent heats at this season, that an
inexpressible languor enervates every faculty, and renders even the
thought of exercising them painful.
’Tis now about three o’ clock; the sun bears nearly S. W. and I am
writing in a piazza, open at each end, on the north-east side of my
house, perfectly in the shade: a small breeze at S. E. blows freely
thro’ it; no buildings are nearer, to reflect the heat, than 60 yards:
yet in a thermometer hanging by me, made by Mr. Bird, and compared by
the late Mr. George Graham with an approved one of his own, the mercury
stands at 102. Twice it has risen this Summer to the same height;
_viz._ on the 28th of June, and the 11th of July. Several times it has
been at 100, and for many days successively at 98; and did not in the
nights sink below 89. I think it highly probable, that the inhabitants
of this town breathe a hotter air than any other people on the face of
the earth. The greatest heat we had last year was but 92, and that but
once: from 84 to 90 were the usual variations; but this is reckoned
an extraordinary hot summer. The weather-wise of this country say it
forebodes a hurricane; for it has always been remarked, that these
tempests have been preceded by continual and uncommon heats. I must
acquaint you, however, that the heats we are subject to here are more
intense than in any other parts of the province, the town of Savannah
being situated upon a sandy eminence, and sheltered all round with high
woods. But it is very sufficient, that the people actually breathe so
hot an air as I describe; and no less remarkable, that this very spot,
from its height and dryness, is reckoned equally healthy with any other
in the province.
I have frequently walked an hundred yards under an umbrella, with a
thermometer suspended from it by a thread to the height of my nostrils,
when the mercury has rose to 105; which is prodigious. At the same
time I have confined this instrument close to the hottest part of my
body, and have been astonished to observe, that it has subsided several
degrees. Indeed, I never could raise the mercury above 97 with the heat
of my body.
You know, dear Sir, that I have traversed a great part of this globe,
not without giving some attention to the peculiarities of each climate;
and I can fairly pronounce, that I never felt such heats any-where as
in Georgia. I know experiments on this subject are extremely liable to
error; but I presume I cannot now be mistaken, either in the goodness
of the instrument, or in the fairness of the trials, which I have
repeatedly made with it. This same thermometer I have had thrice in the
equatorial parts of Africa; as often at Jamaica, and the West India
islands; and, upon examination of my journals, I do not find, that the
quicksilver ever rose in those parts above the 87th degree, and to that
but seldom: its general station was between the 79th and 86th degree;
and yet I think I have felt those degrees, with a moist air, more
disagreeable than what I now feel.
In my relation of the late expedition to the north-west, if I
recollect right, I have observed, that all the changes and variety of
weather, that happen in the temperate zone throughout the year, may be
experienced at the Hudson’s Bay settlements in 24 hours. But I may now
extend this observation; for in my cellar the thermometer stands at 81,
in the next story at 102, and in the upper one at 105; and yet these
heats, violent as they are, would be tolerable, but for the sudden
changes that succeed them. On the 10th of December last the mercury was
at 86; on the 11th it was so low as 38 of the same instrument. What
havock must this make with an European constitution? Nevertheless, but
few people die here out of the ordinary course; tho’ indeed one can
scarce call it living, merely to breathe, and trail about a vigorless
body; yet such is generally our condition from the middle of June to
the middle of September. Dear Sir,
Yours most affectionately,
Henry Ellis.
CIII. _The Invention of a General Method for determining the Sum of
every 2d, 3d, 4th, or 5th_, &c. _Term of a Series, taken in order; the
Sum of the whole Series being known. By_ Thomas Simpson, _F.R.S._
[Read Nov. 16, 1758.]
AS the doctrine of Series’ is of very great use in the higher branches
of the mathematics, and their application to nature, every attempt
tending to extend that doctrine may justly merit some degree of regard.
The subject of the paper, which I have now the honour to lay before the
Society, will be found an improvement of some consequence in that part
of science. And how far the business of finding fluents may, in some
cases, be facilitated thereby, will appear from the examples subjoined,
in illustration of the general method here delivered.
The series propounded, whose sum (_S_) is supposed to be given (either
in algebraic terms, or by the measures of angles and ratio’s, _&c._)
I shall here represent by _a_ + _bx_ + _cx_² + _dx_³ + _ex_⁴, &c. and
shall first give the solution of that case, where every third term is
required to be taken, or where the series to be summed is _a_ + _dx_³ +
_gx_⁶ + _kx_⁶, &c. By means whereof, the general method of proceeding,
and the resolution of every other case, will appear evident.
Here, then, every _third_ term being required to be taken, let the
series (_a_ + _dx_³ + _gx_⁶, &c.), whose value is sought, be conceived
to be composed of _three_ others.
⅓ × (_a_ + _b_ × (_px_) + _c_ × (_px_)² + _d_ × (_px_)³ + _e_ ×
(_px_)⁴, &c.)
⅓ × (_a_ + _b_ × (_qx_) + _c_ × (_qx_)² + _d_ × (_qx_)³ + _e_ ×
(_qx_)⁴, &c.)
⅓ × (_a_ + _b_ × (_rx_) + _c_ × (_rx_)² + _d_ × (_rx_)³ + _e_ ×
(_rx_)⁴, &c.)
having all the _same form_, and the _same coefficients_ with the series
first proposed, and wherein the converging quantities _px_, _qx_, _rx_,
are also in a determinate (tho’ yet unknown) ratio to the original
converging quantity _x_. Now, in order to determine the quantities
of these ratios, or the values of _p_, _q_, and _r_, let the terms
containing the same powers of _x_, in the two equal values, be equated
in the common way:
So shall,
⅓ _b_ × _px_ + ⅓ _b_ × _qx_ + ⅓ _b_ × _rx_ = 0
⅓ _c_ × _p_²_x_² + ⅓ _c_ × _q_²_x_² + ⅓ _c_ × _r_²_x_² = 0
⅓ _d_ × _p_³_x_³ + ⅓ _d_ × _q_³_x_³ + ⅓ _d_ × _r_³_x_³ = _dx_³
⅓ _e_ × _p_⁴_x_⁴ + ⅓ _e_ × _q_⁴_x_⁴ + ⅓ _e_ × _r_⁴_x_⁴ = 0
&c.
And consequently,
_p_ + _q_ + _r_ = 0
_p_² + _q_² + _r_² = 0
_p_³ + _q_³ + _r_³ = 3
_p_⁴ + _q_⁴ + _r_⁴ = 0, &c.
Make, now, _p_³ = 1, _q_³ = 1, and _r_³ = 1; that is, let _p_, _q_,
and _r_, be the three roots of the cubic equation _z_³ = 1, or _z_³ -
1 = 0: then, seeing both the second and third terms of this equation
are wanting, not only the sum of all the roots (_p_ + _q_ + _r_) but
the sum of all their squares (_p_² + _q_² + _r_²) will vanish, or be
equal to nothing (by common algebra), as they ought, to fulfil the
conditions of the two first equations. Moreover, since _p_³ = 1, _q_³
= 1, and _r_³ = 1, it is also evident, that _p_⁴ + _q_⁴ + _r_⁴ (= _p_
+ _q_ + _r_) = 0, _p_⁵ + _q_⁵ + _r_⁵ (= _p_² +_q_² + _r_²) = 0, _p_⁶
+ _q_⁶ + _r_⁶ (= _p_³ + _q_³ + _r_³) = 3. Which equations being, in
effect, nothing more than the first three repeated, the values of
_p_, _q_, _r_, above assigned, equally fulfil the conditions of these
also: so that the series arising from the addition of three assumed
ones will agree, in every term, with _that_ whose sum is required: but
those series’ (whereof the quantity in question is composed) having
all of them the _same form_ and the _same coefficients_ with the
original series _a_ + _bx_ + _cx_² + _dx_³, &c. (= _S_), their sums
will therefore be truly obtained, by substituting _px_, _qx_, and _rx_,
successively, for _x_, in the given value of _S_. And, by the very same
reasoning, and the process above laid down, it is evident, that, if
every _nᵗʰ_ term (instead of every third term) of the given series be
taken, the values of _p_, _q_, _r_, _s_, &c. will then be the roots of
the equation _zⁿ_ - 1 = 0[155]; and that, the sum of all the terms so
taken, will be truly obtained by substituting _px_, _qx_, _rx_, _sx_,
&c. successively for _x_, in the given value of _S_, and then dividing
the sum of all the quantities thence arising by the given number _n_.
The same method of solution holds equally, when, in taking every _n_ᵗʰ
term of the series, the operation begins at some term after the first.
For all the terms preceding _that_ may be transposed, and the whole
equation divided by the power of _x_ in the first of the remaining
terms; and then the sum of every _nᵗʰ_ term (beginning at the first)
will be found by the preceding directions; which sum, multiplied by
the power of _x_ that before divided, will evidently give the true
value required to be determined. Thus, for example, let it be required
to find the sum of every third term of the given series _a_ + _bx_ +
_cx_² + _dx_³ + _ex_⁴, &c. (= _S_), beginning with _cx_². Then, by
transposing the two first terms, and dividing the whole by _x_², we
shall have _c_ + _dx_ + _ex_² + _fx_³, &c. = (_S_ - _a_ - _bx_) ⁄
(_xx_) (= _S´_). From whence having found the sum of every third term
of the series _c_ + _dx_ + _ex_² + _fx_³, &c. beginning at the first
(_c_), that sum, multiplied by _x_², will manifestly give the true
value sought in the present case.
And here it may be worth while to observe, that all the terms preceding
_that_ at which the operation (in any case) begins, may (provided they
exceed not in number the given interval _n_) be intirely disregarded,
as having no effect at all in the result. For if in that part ((-_a_ -
_bx_) ⁄ _xx_) of the value of _S´_, above exhibited, in which the first
terms, _a_ and _bx_, enter, there be substituted _px_, _qx_, _rx_,
successively, for _x_ (according to the _prescript_) the sum of the
quantities thence arising will be
- _a_ ⁄ (_p_²_x_²) - _a_ ⁄ (_q_²_x_²) - _a_ ⁄ (_r_²_x_²)
- _b_ ⁄ _px_ - _b_ ⁄ _qx_ - _b_ ⁄ _rx_
which, because _p_³ = 1, _q_³ = 1, &c. (or _p_² = 1 ⁄ _p_, _q_² = 1 ⁄
_q_, &c.) may be expressed thus;
- _a_ ⁄ _xx_ × (_p_ + _q_ + _r_)
- _b_ ⁄ _x_ × (_p_² + _q_² + _r_²)
But, that _p_ + _q_ + _r_ = 0, and _p_² + _q_² + _r_² = 0, hath been
already shewn; whence the truth of the general observation is manifest.
Hence it also appears, that the method of solution above delivered,
is not only general, but includes this singular beauty and advantage,
that in all series’ whatever, whereof the terms are to be taken
according to the same assigned order, the quantities (_p_, _q_, _r_,
&c.), whereby the resolution is performed, will remain invariably the
same. The greater part of these quantities are indeed _imaginary_ ones;
and so likewise will the quantities be that result from them, when
substitution is made in the given expression for the value of _S_. But
by adding, as is usual in like cases, every two corresponding values,
so resulting together, all marks of _impossibility_ will disappear.
If, in the series to be summed, the alternate terms (_viz._ the 2d,
4th, 6th, _&c._) should be required to be taken under signs contrary to
what they have in the original series given; the reasoning and result
will be no-ways different; only, instead of making _p_³ + _q_³ + _r_³
(or _pⁿ_ + _qⁿ_ + _rⁿ_, &c.) = +3 (or +_n_), the same quantity must,
here, be made = -3 (or -_n_). From whence, _pⁿ_ being = -1, _qⁿ_ = -1,
&c. the values of _p_, _q_, _r_, &c. will, in this case, be the roots
of the equation _zⁿ_ + 1 = 0.
It may be proper, now, to put down an example, or two, of the use and
application of the general conclusions above derived. First, then,
supposing the series, whose sum is given, to be _x_ + _x²_ ⁄ 2 + _x³_
⁄ 3 + _x⁴_ ⁄ 4 ... + _xᵐ_ ⁄ _m_ + _xᵐ ⁺ ¹_) ⁄ (_m_ + 1) + _xᵐ ⁺ ²_ ⁄
(_m_ + 2) ... + _xᵐ ⁺ ⁿ_ ⁄ (_m_ + _n_) + _xᵐ ⁺ ⁿ ⁺ ¹_ ⁄ (_m_ + _n_ + 1)
+, &c. = - H. Log.(1-_x_) (= _S_); let it be required, from hence, to
find the sum of the series (_xᵐ_ ⁄ _m_ + _xᵐ ⁺ ⁿ_ ⁄ (_m_ + _n_) + _xᵐ
⁺ ²ⁿ_ ⁄ (_m_ + 2_n_) &c.) arising by taking every _nᵗʰ_ term thereof,
beginning with that whose exponent (_m_) is any integer less than _n_.
Here, the terms preceding _xᵐ_ ⁄ _m_ being transposed, and the whole
equation divided by _xᵐ_, we shall have 1 ⁄ _m_ + _x_ ⁄ (_m_ + 1) +
_x_² ⁄ (_m_ + 2) + _x_³ ⁄ (_m_ + 3), &c. = -(1 ⁄ _xᵐ_) × H. Log.(1 -
_x_) - (_x_ + ½_x_², &c.) ⁄ _xᵐ_. In which value, let _px_, _qx_, _rx_,
&c. be, successively, substituted for _x_ (according to prescript)
neglecting intirely the terms (_x_ + ½_x_²) ⁄ _xᵐ_, as having no effect
at all in the result: from whence we get -1 ⁄ (_(px)ᵐ_) × Log.(1 -
_px_) - (1 ⁄ _(qx)ᵐ_) × Log.(1 - _qx_) - (1 ⁄ _(rx)ᵐ_) × Log.(1 -
_rx_), &c. Which multiplied by _xᵐ_ (the quantity that before divided)
gives -1 ⁄ (_pᵐ_) × Log.(1 - _px_) - 1 ⁄ (_qᵐ_) × Log.(1 - _qx_) - 1 ⁄
(_rᵐ_) × Log.(1 - _rx_), &c. = _n_ times the quantity required to be
determined.
But now, to get rid of the imaginary quantities _q_, _r_, &c. by means
of their known values α + √(αα - 1), α - √(αα - 1), &c. it will be
necessary to observe, that, as the product of any two corresponding
ones (α + √(αα - 1) × (α - √(αα - 1)) is equal to unity, we may
therefore write (α - √(αα - 1))_ᵐ_ (= _rᵐ_) instead of its equal 1
⁄ (_qᵐ_), and (α + √(αα - 1))_ᵐ_ (= _qᵐ_) instead of its equal 1 ⁄
(_rᵐ_): by which means the two terms, wherein these two quantities
enter, will stand thus; -(α - √(αα - 1))_ⁿ_ × Log. (1 - _qx_) - (α +
√(αα - 1))_ᵐ_ × Log. (1 - _rx_).
But, if _A_ be assumed to express the co-sine of an arch (_Q_), _m_
times as great as that (360° ⁄ _n_) whose co-sine is here denoted by α;
then will _A_ - √(_AA_ - 1) = [156](α - √(αα - 1))_ᵐ_, and _A_ + √(_AA_
- 1) = (α + √(αα - 1))_ᵐ_: which values being substituted above, we
thence get
-_A_ × (log. (1 - _qx_) + log. (1 - _rx_)) + √(_AA_
- 1) × (log. (1 - _qx_) - log. (1 - _rx_));
whereof the former part (which, exclusive of the factor _A_, I shall
hereafter denote by _M_) is manifestly equal to -_A_ × log. ((1 - _qx_)
× (1 - _rx_)) (by the nature of logarithms) = -_A_ × log. 1 - (_q_ +
_r_)._x_ + _qrx_² = -_A_ × log. (1 - 2α_x_ + _xx_) (by substituting
the values of _q_ and _r_): which is now intirely free from imaginary
quantities. But, in order to exterminate them out of the latter part
also, put _y_ = log. (1 - _qx_) - log. (1 - _rx_); then will _ẏ_ =
-_qẋ_ ⁄ (1 - _qx_) + _rẋ_ ⁄ (1 - _rx_) = -((_q_ - _r_) × _ẋ_) ⁄ (1 -
(_q_ + _r_) × _x_ + _xx_) = -(2√(αα - 1) × _ẋ_) ⁄ (1 - 2α_x_ + _xx_)
= -(2√(-1) × √(1 - αα) × _ẋ_) ⁄ (1 - 2αx + xx); where (√(1 - αα) × ẋ)
⁄ (1 - 2α_x_ + _xx_) expresseth the fluxion of a circular arch (_N_)
whose radius is 1, and sine = (√(1 - αα) × _ẋ_) ⁄ (1 - 2α_x_ + _xx_);
consequently _y_ will be = -2√(-1) × _N_: which, multiplied by √(_AA_ -
1), or its equal √(-1) × √(1 - _AA_), gives 2√(1 - _AA_) × _N_; and,
this value being added to that of the former part (found above), and
the whole being divided by _n_, we thence obtain (-_AM_ + 2√(1 - _AA_)
× _N_) ⁄ _n_, or 1 ⁄_n_ × (-co-s. _Q_ × _M_ + sin. _Q_ × 2_N_) for that
part of the value sought depending on the two terms affected with _q_
and _r_. From whence the sum of any other two corresponding terms will
be had, by barely substituting one letter, or value, for another: So
that,
{ -log. (1 - _x_)
{ -co-s. _Q_ × _M_ + sin. _Q_ × 2_N_
(1 ⁄ _n_) × { -co-s. _Q´_ × _M´_ + sin. _Q´_ × 2_N´_
{ -co-s. _Q´´_ × _M´´_ + sin. _Q´´_ × 2_N´´_
{ -&c. + &c.
will truly express the sum of the series proposed to be determined;
_M_, _M´_, _M´´_ &c. being the hyperbolical logarithms of 1 - 2α_x_ +
_xx_, 1 - 2β_x_ + _xx_, 1 - 2γ_x_ + _xx_, &c. _N_, _N´_, _N´´_ &c. the
arcs whose sines are _x_√(1 - αα) ⁄ √(1 - 2α_x_ + _xx_), _x_√(1 - ββ) ⁄
√(1 - 2β_x_ + _xx_), _x_√(1 - γγ) ⁄ √(1 - 2γ_x_ + _xx_), &c. and _Q_,
_Q´_, _Q´´_, &c. the measures of the angles expressed by (360° ⁄ _n_)
× _m_, 2 × (360° ⁄ _n_) × _m_, 3 × (360° ⁄ _n_) × _m_, &c. And here it
may not be amiss to take notice, that the series _xᵐ_ ⁄ _m_ + _xᵐ ⁺ ⁿ_
⁄ (_m_ + _n_) + _xᵐ ⁺ ²ⁿ_ ⁄ (_m_ + 2_n_) + &c. thus determined, is that
expressing the fluent of (_xᵐ ⁻ ¹ẋ_) ⁄ (1 - _xⁿ_); corresponding to one
of the two famous _Cotesian forms_. From whence, and the reasoning
above laid down, the fluent of the other _form_, _xᵐ ⁻ ¹ẋ_ ⁄ (1 +
_xⁿ_), may be very readily deduced. For, since the series (_xᵐ_ ⁄ _m_ -
_xᵐ ⁺ ⁿ_ ⁄ (_m_ + _n_) + _xᵐ ⁺ ²ⁿ_ ⁄ (_m_ + 2_n_) - _xᵐ ⁺ ³ⁿ_ ⁄ (_m_ +
3_n_) &c.) for this last fluent, is that which arises by changing the
signs of the alternate terms of the former; the quantities _p_, _q_,
_r_, &c. will here (agreeably to a preceding observation) be the roots
of the equation _zⁿ_ + 1 = 0; and, consequently, α, β, γ, δ, &c. the
co-sines of the arcs 180° ⁄ _n_, 3 × 180° ⁄ _n_, 5 × 180° ⁄ _n_, &c.
(as appears by the foregoing note). So that, making _Q_, _Q´_, _Q´´_,
&c. equal, here, to the measures of the angles (180° ⁄ _n_) × _m_, 3 ×
(180° ⁄ _n_) × _m_, 5 × (180° ⁄ _n_) × _m_, &c. the fluent sought will
be expressed in the very same manner as in the preceding case; except
that the first term, -log. (1 - _x_) (arising from the _rational_ root
_p_ = 1) will here have no place.
After the same manner, with a small increase of trouble, the fluent of
_xᵐ ⁻ ¹ẋ_ ⁄ (1 ± 2_lxⁿ_ + _x_²_ⁿ_) may be derived, _m_ and _n_ being
any integers whatever. But I shall now put down one example, wherein
the impossible quantities become exponents of the powers, in the terms
where they are concerned.
The series here given is 1 - _x_ + _x_² ⁄ 2 + _x_³ ⁄ (2.3) + _x_⁴
⁄ (2.3.4) - _x_⁵ ⁄ (2.3.4.5), &c. = the number whose hyp. log. is
-_x_, and it is required to find the sum of every _nᵗʰ_ term thereof,
beginning at the first. Here the quantity sought will (according to the
general rule) be truly defined by the _nᵗʰ_ part of the sum of all the
numbers whose respective logarithms are -_px_, -_qx_, -_rx_, &c.; which
numbers, if _N_ be taken to denote the number whose hyp. log. = 1,
will be truly expressed by _N⁻ᵖˣ_, _N⁻⒬ˣ_, _N⁻ʳˣ_, &c. From whence, by
writing for _p_, _q_, _r_, &c. their equals 1, α + √(αα - 1), α - √(αα
- 1), β + √(ββ - 1), β - √(ββ - 1), &c. and putting α´ = √(1 - αα), β´
= √(1 - ββ), &c. we shall have 1 ⁄ _n_ × (_N⁻ᵖˣ_ + _N⁻⒬ˣ_ + _N⁻ʳˣ_),
&c. = 1 ⁄ _n_ into _N⁻ˣ_ + _N⁻ᵃˣ_ × (_N⁻ᵃ‘ˣ_√⁻¹) + (_Nᵃ‘ˣ_√⁻¹) + _N⁻ᵝˣ_
× (_N⁻ᵝ‘ˣ√⁻¹_) + _Nᵝ‘ˣ√⁻¹_) + &c. But _N⁻ᵃ‘ˣ√⁻¹_ + _Nᵃ‘ˣ√⁻¹_ is known
to express the double of the co-sine of the arch whose measure (to the
radius 1) is α´_x_. Therefore we have 1 ⁄ _n_ into _N⁻ˣ_ + _N⁻ᵃˣ_ × 2
co-s. α´_x_ + _N⁻ᵝˣ_ × 2 co-s. β´_x_, &c. for the true sum, or value
proposed to be determined.
The solution of this case, in a manner a little different, I have
given some time since, in another place; where the principles of the
general method, here extended and illustrated, are pointed out. I shall
put an end to this paper with observing, that if, in the series given,
the even powers of _x_, or any other terms whatever, be wanting, their
places must be supplied with cyphers; which, in the order of numbering
off, must be reckoned as real terms.
CIV. _Observatio Eclipsis Lunæ Die_ 30 Julii 1757. _habita Olissipone
à_ Joanne Chevalier, _Congregationis Oratorii Presbytero, é Regia_
Londinensi _Societate. Communicated by_ Jacob de Castro Sarmiento,
_M.D. F.R.S._
Tubo optico 8 pedum.
[Read Nov. 16, 1758.]
h ´ ´´
Initium penumbræ 9 15 18
Initium dubium eclipsis 9 22 24
Certo jam incœperat 9 23 34
Umbra ad mare humorum observata vitro plano cæruleo 9 31 2
Solo tubo optico observata 9 31 29
Vitro flavo observata 9 31 48
Umbra tangit Grimaldum observata vitro plano cæruleo 9 31 20
Solo tubo optico 9 31 50
Vitro plano flavo 9 32 8
Totus Grimaldus tegitur observatus vitro plano cæruleo 9 34 4
Solo tubo optico 9 34 28
Vitro flavo 9 34 47
Umbra ad Tychonem observata vitro plano cæruleo 9 38 25
Solo tubo optico 9 38 42
Vitro flavo 9 38 59
Umbra ad Harpalum vitro cæruleo observata 9 55 6
Solo tubo optico 9 55 35
Umbra ad Fracastorium 9 59 57
Umbra ad Mare Nectaris 10 00 50
Observata vitro flavo 10 1 8
Umbra ad Dionysium 10 5 2
Umbra tangit Mare Tranquillitatis 10 5 50
Umbra ad Mare Serenitatis 10 10 16
Umbra tegit Menelaum observata vitro cæruleo 10 11 4
Solo tubo optico 10 11 29
Vitro flavo 10 11 50
Totum Mare Fœcunditatis tegitur 10 18 39
Umbra tangit Mare Crisium vitro cæruleo observata 10 22 52
Solo tubo optico 10 23 12
Vitro flavo 10 23 29
Umbra ad Proclum 10 23 33
Possidonius totus tegitur 10 23 50
Totum Mare Serenitatis tegitur 10 24 36
Totum Mare Crisium ab umbra tegitur 10 30 27
Plato tegitur vitro cæruleo observatus 10 31 26
Solo tubo optico 10 31 48
Vitro flavo 10 32 4
Obscuratio maxima 10 55 40