is there any proof of this shadow object ?
Yes. Although we cannot observe the Shadow Object directly due to the Sun's powerful rays, we can accurately predict the time, magnitude, and duration of the Lunar Eclipse by using the following formulas in
Chapter 10 of the book Earth Not a Globe by Dr. Samuel Birley Rowbotham:
---
Let A, B, R, (in the above diagram) be a section of the eclipse shadow at the distance of the moon; S,
n, the path described by its centre, S, on the ecliptic; M,
n, the relative orbit of the moon; M,
n, S,
n, being considered straight lines. Draw S,
o, perpendicular to S,
n, and S,
m, to M,
n; then
o, and
m, are in the places, with respect to S, of the moon in opposition, and at the middle of the eclipse.
Let α = S, B =
h + π - σ, the radius of the section of the shadow.
λ = S,
o, the moon's latitude in opposition.
f = the relative horary motion in longitude of the moon in the relative orbit, M,
n.
h = the moon's horary motion in the relative orbit.
g = the moon's horary motion in latitude.
μ = the moon's semi-diameter;
Let M, and N, be the place of the moon's centre at the time of the first and last contact; therefore
SM = SN = a + μ.
Now S
m = λ cos
n;
and
m,
o = λ sin
n.
If, therefore,
t, and
t´, be the times from opposition of the first and last contact,
The time from opposition, of the middle of the eclipse
The magnitude of the eclipse, or the part of the moon immersed,
= S u - S v.
= S u--S m + m, v.
=a - λ cos n + μ.
The moon's diameter is generally divided into twelve equal parts, called digits;
therefore the digits eclipsed = 12 :: α - λ,
n +μ : 2 μ
COR. 1.--If λ cos n, be greater than α + μ,
t and
t´ are impossible, and no eclipse can take place, as is also evident from the figure.
COR. 2.--In exactly the same manner it may be proved, if
t and
t´ be the times from opposition, of the centres of the shadow and moon being at any given distance c,
COR. 3.--If
c = h + μ + σ + μ = the radius of the penumbra, + the radius of the moon, the times of the moon entering and emerging from the penumbra are obtained.
The horary motion of the moon is about 32½´, and that of the sun 2½´; therefore the relative horary motion of the moon is 30´; and as the greatest diameter of the section at the distance of the moon is 1° 31´ 44″, a lunar eclipse may last more than three hours.