Maximum distance to the horizon. Visible horizon and its range
What is the distance to the horizon for an observer standing on the ground? The answer—the approximate distance to the horizon—can be found using the Pythagorean theorem.
To carry out approximate calculations, we will make the assumption that the Earth has the shape of a sphere. Then a person standing vertically will be a continuation of the earth’s radius, and the line of sight directed towards the horizon will be a tangent to the sphere (the surface of the earth). Since the tangent is perpendicular to the radius drawn to the point of contact, the triangle (center of the Earth) - (point of contact) - (eye of the observer) is rectangular.
Two sides to it are known. The length of one of the legs (the side adjacent to the right angle) is equal to the radius of the Earth $R$, and the length of the hypotenuse (the side lying opposite the right angle) is equal to $R+h$, where $h$ is the distance from the earth to the observer’s eyes.
According to the Pythagorean theorem, the sum of the squares of the legs is equal to the square of the hypotenuse. This means that the distance to the horizon is
$$
d=\sqrt((R+h)^2-R^2) = \sqrt((R^2+2Rh+h^2)-R^2) =\sqrt(2Rh+h^2).
$$The quantity $h^2$ is very small compared to the term $2Rh$, so the approximate equality is true
$$
d\sqrt(2Rh).
$$
It is known that $R 6400$ km, or $R 64\cdot10^5$ m. We assume that $h 1(,)6$ m. Then
$$
d\sqrt(2\cdot64\cdot10^5\cdot 1(,)6)=8\cdot 10^3 \cdot \sqrt(0(,)32).
$$Using the approximate value $\sqrt(0(,)32) 0(,)566$, we find
$$
d 8\cdot10^3 \cdot 0(,)566=4528.
$$The answer received is in meters. If we convert the found approximate distance from the observer to the horizon into kilometers, we obtain $d 4.5$ km.
In addition, there are three microplots related to the problem considered and the calculations performed.
I. How is the distance to the horizon related to the change in altitude of the observation point? The formula $d \sqrt(2Rh)$ gives the answer: to double the distance $d$, the height $h$ must be quadrupled!
II. In the formula $d \sqrt(2Rh)$ we had to take the square root. Of course, the reader can take a smartphone with a built-in calculator, but, firstly, it is useful to think about how a calculator solves this problem, and secondly, it is worth experiencing mental freedom, independence from the “all-knowing” gadget.
There is an algorithm that reduces root extraction to simpler operations - addition, multiplication and division of numbers. To extract the root of the number $a>0$, consider the sequence
$$
x_(n+1)=\frac12 (x_n+\frac(a)(x_n)),
$$where $n=0$, 1, 2, …, and $x_0$ can be any positive number. The sequence $x_0$, $x_1$, $x_2$, … converges very quickly to $\sqrt(a)$.
For example, when calculating $\sqrt(0.32)$, you can take $x_0=0.5$. Then
$$
\eqalign(
x_1 &=\frac12 (0.5+\frac(0.32)(0.5))=0.57,\cr
x_2 &=\frac12 (0.57+\frac(0.32)(0.57)) 0.5657.\cr)
$$Already at the second step we received the answer, correct in the third decimal place ($\sqrt(0.32)=0.56568…$)!
III. Sometimes algebraic formulas can be so clearly represented as relationships between the elements of geometric figures that the entire “proof” lies in a drawing with the caption “Look!” (in the style of ancient Indian mathematicians).
The used “abbreviated multiplication” formula for the square of the sum can also be explained geometrically
$$
(a+b)^2=a^2+2ab+b^2.
$$Jean-Jacques Rousseau wrote in “Confessions”: “When I first discovered by calculation that the square of a binomial is equal to the sum of the squares of its members and their double product, I, despite the correctness of the multiplication I performed, did not want to believe it until until I drew the figures.”
Literature
- Perelman Ya. I. Entertaining geometry in the free air and at home. - L.: Time, 1925. - [And any edition of Ya. I. Perelman’s book “Entertaining Geometry”].
The spherical radius ВС of a small circle with с¹с²с³ is called the theoretical range of the visible horizon.
The value of the spherical radius depends on the height of the observer's eye above sea level.
So, if the observer's eye is at point A1 at a height BA¹ = e¹ above sea level, then the spherical radius Bc" will be greater than the spherical radius Bc.
To determine the relationship between the height of the observer’s eye and the theoretical range of his visible horizon, consider the right triangle AOC:
Ac² = AO² - Os²; AO = OB + e; OB = R,
Then AO = R + e; Os = R.
Due to the insignificance of the height of the observer's eye above sea level compared to the size of the Earth's radius, the length of the tangent Ac can be taken equal to the value of the spherical radius Bc and, denoting the theoretical range of the visible horizon through D T, we obtain
D 2T = (R + e)² - R² = R² + 2Re + e² - R² = 2Re + e²,
Rice. 8
Considering that the height of the observer's eye e on ships does not exceed 25 m, and 2R = 12,742,220 m, the ratio e/2R is so small that it can be neglected without compromising accuracy. Hence,
since e and R are expressed in meters, then Dt will also be in meters. However, the actual range of the visible horizon is always greater than the theoretical one, since the ray coming from the observer’s eye to a point on the earth’s surface is refracted due to the unequal density of the atmospheric layers in height.
In this case, the ray from point A to c does not go along the straight line Ac, but along the curve ASm" (see Fig. 8). Therefore, to the observer, point c appears visible in the direction of the tangent AT, i.e., raised by an angle r = L TAc , called the angle of terrestrial refraction. The angle d = L HAT is called the inclination of the visible horizon. And in fact, the visible horizon will be a small circle m", m" 2, tz", with a slightly larger spherical radius (Bm" > Вс).
The magnitude of the angle of terrestrial refraction is not constant and depends on the refractive properties of the atmosphere, which vary with temperature and humidity, and the amount of suspended particles in the air. Depending on the time of year and the date of day, it also changes, so the actual range of the visible horizon compared to the theoretical one can increase up to 15%.
In navigation, the increase in the actual range of the visible horizon compared to the theoretical one is assumed to be 8%.
Therefore, denoting the actual, or, as it is also called, geographical, range of the visible horizon through D e, we obtain:
To obtain De in nautical miles (taking R and e in meters), the radius of the earth R, as well as the height of the eye e, is divided by 1852 (1 nautical mile is equal to 1852 m). Then
To get the result in kilometers, enter the multiplier 1.852. Then
to facilitate calculations for determining the range of the visible horizon in table. 22-a (MT-63) gives the range of the visible horizon depending on e, ranging from 0.25 to 5100 m, calculated using formula (4a).
If the actual height of the eye does not coincide with the numerical values indicated in the table, then the range of the visible horizon can be determined by linear interpolation between two values close to the actual height of the eye.
Visibility range of objects and lights
The visibility range of an object Dn (Fig. 9) will be the sum of two ranges of the visible horizon, depending on the height of the observer’s eye (D e) and the height of the object (D h), i.e.It can be determined by the formula
where h is the height of the landmark above the water level, m.
To make it easier to determine the visibility range of objects, use the table. 22-v (MT-63), calculated according to formula (5a): To determine from this table at what distance an object will open, you need to know the height of the observer’s eye above the water level and the height of the object in meters.
The visibility range of an object can also be determined using a special nomogram (Fig. 10). For example, the height of the eye above the water level is 5.5 m, and the height h of the setting sign is 6.5 m. To determine D n, a ruler is applied to the nomogram so that it connects the points corresponding to h and e on the extreme scales. The point of intersection of the ruler with the middle scale of the nomogram will show the desired visibility range of the object D n (in Fig. 10 D n = 10.2 miles).
In navigation manuals - on maps, in directions, in descriptions of lights and signs - the visibility range of objects DK is indicated at an observer's eye height of 5 m (on English charts - 15 feet).
In the case when the actual height of the observer's eye is different, it is necessary to introduce the AD correction (see Fig. 9).
Rice. 9
Example. The visibility range of the object indicated on the map is DK = 20 miles, and the height of the observer’s eye is e = 9 m. Determine the actual visibility range of the object D n using the table. 22-a (MT -63). Solution.
At night, the visibility range of a fire depends not only on its height above the water level, but also on the strength of the light source and on the discharge of the lighting apparatus. Typically, the lighting apparatus and the strength of the light source are calculated in such a way that the visibility range of the fire at night corresponds to the actual visibility range of the horizon from the height of the fire above sea level, but there are exceptions.
Therefore, the lights have their own “optical” visibility range, which can be greater or less than the visibility range of the horizon from the height of the fire.
Navigation manuals indicate the actual (mathematical) visibility range of the lights, but if it is greater than the optical one, then the latter is indicated.
The visibility range of coastal navigation signs depends not only on the state of the atmosphere, but also on many other factors, which include:
A) topographical (determined by the nature of the surrounding area, in particular the predominance of a particular color in the surrounding landscape);
B) photometric (brightness and color of the observed sign and the background on which it is projected);
C) geometric (distance to the sign, its size and shape).
Chapter VII. Navigation.
Navigation is the basis of the science of navigation. The navigational method of navigation is to navigate a ship from one place to another in the most advantageous, shortest and safest way. This method solves two problems: how to direct the ship along the chosen path and how to determine its place in the sea based on the elements of the ship’s movement and observations of coastal objects, taking into account the influence of external forces on the ship - wind and current.
To be sure of the safe movement of your ship, you need to know the ship’s place on the map, which determines its position relative to the dangers in a given navigation area.
Navigation deals with the development of the fundamentals of navigation, it studies:
Dimensions and surface of the earth, methods of depicting the earth's surface on maps;
Methods for calculating and plotting a ship's path on nautical charts;
Methods for determining the position of a ship at sea by coastal objects.
§ 19. Basic information about navigation.
1. Basic points, circles, lines and planes
Our earth has the shape of a spheroid with a semi-major axis OE equal to 6378 km, and the minor axis OR 6356 km(Fig. 37).
Rice. 37. Determining the coordinates of a point on the earth's surface
In practice, with some assumption, the earth can be considered a ball rotating around an axis occupying a certain position in space.
To determine points on the earth's surface, it is customary to mentally divide it into vertical and horizontal planes that form lines with the earth's surface - meridians and parallels. The ends of the earth's imaginary axis of rotation are called poles - north, or north, and south, or south.
Meridians are large circles passing through both poles. Parallels are small circles on the earth's surface parallel to the equator.
The equator is a large circle whose plane passes through the center of the earth perpendicular to its axis of rotation.
Both meridians and parallels on the earth's surface can be imagined in countless numbers. The equator, meridians and parallels form the earth's geographic coordinate grid.
Location of any point A on the earth's surface can be determined by its latitude (f) and longitude (l) .
The latitude of a place is the arc of the meridian from the equator to the parallel of a given place. Otherwise: the latitude of a place is measured by the central angle between the plane of the equator and the direction from the center of the earth to a given place. Latitude is measured in degrees from 0 to 90° in the direction from the equator to the poles. When calculating, it is assumed that northern latitude f N has a plus sign, southern latitude f S has a minus sign.
The latitude difference (f 1 - f 2) is the meridian arc enclosed between the parallels of these points (1 and 2).
The longitude of a place is the arc of the equator from the prime meridian to the meridian of a given place. Otherwise: the longitude of a place is measured by the arc of the equator, enclosed between the plane of the prime meridian and the plane of the meridian of a given place.
The difference in longitude (l 1 -l 2) is the arc of the equator, enclosed between the meridians of given points (1 and 2).
The prime meridian is the Greenwich meridian. From it, longitude is measured in both directions (east and west) from 0 to 180°. Western longitude is measured on the map to the left of the Greenwich meridian and is taken with a minus sign in calculations; eastern - to the right and has a plus sign.
The latitude and longitude of any point on earth are called the geographic coordinates of that point.
2. Division of the true horizon
A mentally imaginary horizontal plane passing through the observer’s eye is called the plane of the observer’s true horizon, or true horizon (Fig. 38).
Let us assume that at the point A is the observer's eye, line ZABC- vertical, HH 1 - the plane of the true horizon, and line P NP S - the axis of rotation of the earth.
Of the many vertical planes, only one plane in the drawing will coincide with the axis of rotation of the earth and the point A. The intersection of this vertical plane with the surface of the earth gives on it a great circle P N BEP SQ, called the true meridian of the place, or the meridian of the observer. The plane of the true meridian intersects with the plane of the true horizon and gives the north-south line on the latter N.S. Line O.W. perpendicular to the line of true north-south is called the line of true east and west (east and west).
Thus, the four main points of the true horizon - north, south, east and west - occupy a well-defined position anywhere on earth, except for the poles, thanks to which different directions along the horizon can be determined relative to these points.
Directions N(north), S (south), ABOUT(East), W(west) are called the main directions. The entire circumference of the horizon is divided into 360°. Division is made from the point N in a clockwise direction.
Intermediate directions between the main directions are called quarter directions and are called NO, SO, SW, NW. The main and quarter directions have the following values in degrees:
Rice. 38. Observer's true horizon
3. Visible horizon, visible horizon range
The expanse of water visible from a vessel is limited by a circle formed by the apparent intersection of the vault of heaven with the surface of the water. This circle is called the observer's apparent horizon. The range of the visible horizon depends not only on the height of the observer’s eyes above the water surface, but also on the state of the atmosphere.
Figure 39. Object visibility range
The boatmaster should always know how far he can see the horizon in different positions, for example, standing at the helm, on deck, sitting, etc.
The range of the visible horizon is determined by the formula:
d = 2.08
or, approximately, for an observer's eye height of less than 20 m by formula:
d = 2,
where d is the range of the visible horizon in miles;
h is the height of the observer's eye, m.
Example. If the height of the observer's eye is h = 4 m, then the range of the visible horizon is 4 miles.
The visibility range of the observed object (Fig. 39), or, as it is called, the geographic range D n , is the sum of the ranges of the visible horizon With the height of this object H and the height of the observer’s eye A.
Observer A (Fig. 39), located at a height h, from his ship can see the horizon only at a distance d 1, i.e. to point B of the water surface. If we place an observer at point B of the water surface, then he could see lighthouse C , located at a distance d 2 from it ; therefore the observer located at the point A, will see the beacon from a distance equal to D n :
D n= d 1+d 2.
The visibility range of objects located above the water level can be determined by the formula:
Dn = 2.08(+).
Example. Lighthouse height H = 1b.8 m, observer's eye height h = 4 m.
Solution. D n = l 2.6 miles, or 23.3 km.
The visibility range of an object is also determined approximately using the Struisky nomogram (Fig. 40). By applying a ruler so that one straight line connects the heights corresponding to the observer’s eye and the observed object, the visibility range is obtained on the middle scale.
Example. Find the visibility range of an object with an altitude of 26.2 above sea level m with an observer's eye height above sea level of 4.5 m.
Solution. Dn= 15.1 miles (dashed line in Fig. 40).
On maps, directions, in navigation manuals, in the descriptions of signs and lights, the visibility range is given for the height of the observer's eye 5 m from the water level. Since on a small boat the observer’s eye is located below 5 m, for him, the visibility range will be less than that indicated in manuals or on the map (see Table 1).
Example. The map indicates the visibility range of the lighthouse at 16 miles. This means that an observer will see this lighthouse from a distance of 16 miles if his eye is at a height of 5 m above sea level. If the observer's eye is at a height of 3 m, then the visibility will correspondingly decrease by the difference in the horizon visibility range for heights 5 and 3 m. Horizon visibility range for height 5 m equal to 4.7 miles; for height 3 m- 3.6 miles, difference 4.7 - 3.6=1.1 miles.
Consequently, the visibility range of the lighthouse will not be 16 miles, but only 16 - 1.1 = 14.9 miles.
Rice. 40. Struisky's nomogram
Synonyms: horizon, horizon, skyscape, skyscraper, sunset sky, eye, raymo, curtain, close, gaze, see, look around.
Distance to visible horizon
- If visible horizon defined as the boundary between heaven and earth, then calculate geometric range visible horizon using the Pythagorean theorem:
Where d And h in kilometers or
Where d in kilometers, and h in meters. Below is the distance to the horizon when observed from various heights:
| Height above the Earth's surface h | Distance to horizon d | Example of a surveillance location |
|---|---|---|
| 1.75 m | 4.7 km | standing on the ground |
| 25 m | 17.9 km | 9-storey house |
| 50 m | 25.3 km | Ferris wheel |
| 150 m | 43.8 km | balloon |
| 2 km | 159.8 km | mountain |
| 10 km | 357.3 km | airplane |
| 350 km | 2114.0 km | spaceship |
To facilitate calculations of the horizon range depending on the height of the observation point and taking into account refraction, tables and nomograms have been compiled. The actual range of the visible horizon can differ significantly from the table, especially at high latitudes, depending on the state of the atmosphere and the underlying surface. Raising (lowering) the horizon refers to phenomena related to refraction. At positive refraction the visible horizon rises (expands), geographical range the visible horizon increases compared to geometric range, objects usually hidden by the curvature of the Earth are visible. Under normal temperature conditions, the horizon rise is 6-7%. As the temperature inversion intensifies, the visible horizon can rise to the true (mathematical) horizon, the earth's surface will seem to straighten out, become flat, the visibility range will become infinitely large, and the radius of curvature of the beam will become equal to the radius of the globe. With an even stronger temperature inversion, the visible horizon will rise higher than the true one. It will seem to the observer that he is at the bottom of a huge basin. Because of the horizon, objects located far beyond the geodetic horizon will rise and become visible (as if floating in the air). In the presence of strong temperature inversions, conditions are created for the occurrence of upper mirages. Large temperature gradients are created when the earth's surface is strongly heated by the sun's rays, often in deserts and steppes. Large gradients can occur in middle and even high latitudes on summer days in sunny weather: over sandy beaches, over asphalt, over bare soil. Such conditions are favorable for the occurrence of inferior mirages. At negative refraction the visible horizon decreases (narrows), even those objects that are visible under normal conditions are not visible.
By the way: space horizon(particle horizon) is both a mentally imaginary sphere with a radius equal to the distance that light has traveled during the existence of the Universe, and the entire set of points in the Universe located at this distance. Visibility range
In the figure on the right, the visibility range of an object is determined by the formula
,
Where - visibility range in kilometers,
And - height of the observation point and object in meters.
For an approximate calculation of the visibility range of objects, the Struisky nomogram is used (see illustration): on the two extreme scales of the nomogram, points corresponding to the height of the observation point and the height of the object are marked, then a straight line is drawn through them and at the intersection of this straight line with the middle scale, the visibility range of the object is obtained.
On nautical charts, sailing directions and other navigation aids, the visibility range of beacons and lights is indicated for an observation point height of 5 m. If the height of the observation point is different, then a correction is introduced.
Horizon on the Moon
It must be said that distances on the Moon are very deceptive. Due to the absence of air, distant objects are seen more clearly on the Moon and therefore always seem closer.
Artificial horizon- a device used to determine the true horizon.
For example, the true horizon can be easily determined by holding a glass of water to your eyes so that the water level is visible as a straight line.
Horizon in philosophy
The concept of horizon is introduced into philosophy by Edmund Husserl, and Gadamer defines it as follows: “The horizon is a field of vision that embraces and embraces everything that can be seen from any point.”
see also
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Notes
- .
- Article "Horizon" in the Great Soviet Encyclopedia
- Ermolaev G. G., Andronov L. P., Zoteev E. S., Kirin Yu. P., Cherniev L. F. Marine navigation / under the general editorship of sea captain G. G. Ermolaev. - 3rd edition, revised. - M.: Transport, 1970. - 568 p.
- . Interpretations of the expression “visible horizon”. .
- . Horizon. Space and astronomy. .
- Dal V.I. Explanatory dictionary of the living Great Russian language. - M.: OLMA Media Group, 2011. - 576 p. - ISBN 978-5-373-03764-8.
- Veryuzhsky N. A. Nautical astronomy: Theoretical course. - M.: RConsult, 2006. - 164 p. - ISBN 5-94976-802-7.
- Perelman Ya. I. Horizon // Entertaining geometry. - M.: Rimis, 2010. - 320 p. - ISBN 978-5-9650-0059-3.
- Calculated using the formula “distance = 113 roots of height”, thus the influence of the atmosphere on the propagation of light is not taken into account and the Earth is assumed to be spherical.
- Nautical tables (MT-2000). Adm. No. 9011 / editor-in-chief K. A. Emets. - St. Petersburg: GUN i O, 2002. - 576 p.
- . Calculate distance to horizon and line of sight online. .
- . Which horizon is next?. .
- Lukash V. N., Mikheeva E. V. Physical cosmology. - M.: Physico-mathematical literature, 2010. - 404 p. - ISBN 5922111614.
- Klimushkin D. Yu.; Grablevsky S.V. . space horizon (2001). .
- . Chapter VII. Navigation.
- . Visible horizon and visibility range. .
- . Have Americans been on the moon?. .
- . Interpretations of the expression “true horizon”. .
- Zaparenko Victor. Large encyclopedia of drawing by Viktor Zaparenko. - M.: AST, 2007. - 240 p. - ISBN 978-5-17-041243-3.
- Truth and Method. P.358
Literature
- Vitkovsky V.V.// Encyclopedic Dictionary of Brockhaus and Efron: in 86 volumes (82 volumes and 4 additional). - St. Petersburg. , 1890-1907.
- Horizon // Great Soviet Encyclopedia: [in 30 volumes] / ch. ed. A. M. Prokhorov. - 3rd ed. - M. : Soviet encyclopedia, 1969-1978.
Excerpt describing Horizon
- What's wrong with you, Masha?
“Nothing... I felt so sad... sad about Andrei,” she said, wiping her tears on her daughter-in-law’s knees. Several times throughout the morning, Princess Marya began to prepare her daughter-in-law, and each time she began to cry. These tears, the reason for which the little princess did not understand, alarmed her, no matter how little observant she was. She didn’t say anything, but looked around restlessly, looking for something. Before dinner, the old prince, whom she had always been afraid of, entered her room, now with a particularly restless, angry face, and without saying a word, he left. She looked at Princess Marya, then thought with that expression in her eyes of attention directed inward that pregnant women have, and suddenly began to cry.
– Did you receive anything from Andrey? - she said.
- No, you know that the news could not come yet, but mon pere is worried, and I’m scared.
- Oh nothing?
“Nothing,” said Princess Marya, looking firmly at her daughter-in-law with radiant eyes. She decided not to tell her and persuaded her father to hide the receipt of terrible news from her daughter-in-law until her permission, which was supposed to be the other day. Princess Marya and the old prince, each in their own way, wore and hid their grief. The old prince did not want to hope: he decided that Prince Andrei had been killed, and despite the fact that he sent an official to Austria to look for his son’s trace, he ordered a monument to him in Moscow, which he intended to erect in his garden, and told everyone that his son was killed. He tried to lead his previous lifestyle without changing, but his strength failed him: he walked less, ate less, slept less, and became weaker every day. Princess Marya hoped. She prayed for her brother as if he were alive and waited every minute for news of his return.“Ma bonne amie, [My good friend,”] said the little princess on the morning of March 19th after breakfast, and her sponge with mustache rose according to an old habit; but just as in all not only smiles, but the sounds of speeches, even the gaits in this house since the day the terrible news was received, there was sadness, so now the smile of the little princess, who succumbed to the general mood, although she did not know its reason, was such that she reminded me even more of general sadness.
- Ma bonne amie, je crains que le fruschtique (comme dit Foka - the cook) de ce matin ne m "aie pas fait du mal. [My friend, I'm afraid that the current frishtik (as the cook Foka calls it) will make me feel bad. ]
– What’s wrong with you, my soul? You're pale. “Oh, you are very pale,” said Princess Marya in fear, running up to her daughter-in-law with her heavy, soft steps.
- Your Excellency, should I send for Marya Bogdanovna? - said one of the maids who was here. (Marya Bogdanovna was a midwife from a district town who had been living in Bald Mountains for another week.)
“And indeed,” Princess Marya picked up, “perhaps for sure.” I will go. Courage, mon ange! [Don't be afraid, my angel.] She kissed Lisa and wanted to leave the room.
- Oh, no, no! - And besides the pallor, the little princess’s face expressed a childish fear of inevitable physical suffering.
- Non, c"est l"estomac... dites que c"est l"estomac, dites, Marie, dites..., [No, this is the stomach... tell me, Masha, that this is the stomach...] - and the princess began to cry childishly, painfully, capriciously and even somewhat feignedly, wringing his little hands. The princess ran out of the room after Marya Bogdanovna.
- Mon Dieu! Mon Dieu! [My God! Oh my God!] Oh! – she heard behind her.
Rubbing her plump, small, white hands, the midwife was already walking towards her, with a significantly calm face.
- Marya Bogdanovna! It seems it has begun,” said Princess Marya, looking at her grandmother with frightened, open eyes.
“Well, thank God, princess,” said Marya Bogdanovna without increasing her pace. “You girls shouldn’t know about this.”
- But how come the doctor hasn’t arrived from Moscow yet? - said the princess. (At the request of Lisa and Prince Andrey, an obstetrician was sent to Moscow on time, and he was expected every minute.)
“It’s okay, princess, don’t worry,” said Marya Bogdanovna, “and without the doctor everything will be fine.”
Five minutes later, the princess heard from her room that they were carrying something heavy. She looked out - the waiters were carrying a leather sofa that was in Prince Andrei's office into the bedroom for some reason. There was something solemn and quiet on the faces of the people carrying them.
Princess Marya sat alone in her room, listening to the sounds of the house, occasionally opening the door when they passed by, and looking closely at what was happening in the corridor. Several women walked in and out with quiet steps, looked at the princess and turned away from her. She did not dare to ask, she closed the door, returned to her room, and then sat down in her chair, then took up her prayer book, then knelt down in front of the icon case. Unfortunately and to her surprise, she felt that prayer did not calm her anxiety. Suddenly the door of her room quietly opened and her old nanny Praskovya Savishna, tied with a scarf, appeared on the threshold; almost never, due to the prince’s prohibition, did not enter her room.
“I came to sit with you, Mashenka,” said the nanny, “but I brought the prince’s wedding candles to light in front of the saint, my angel,” she said with a sigh.
- Oh, I'm so glad, nanny.
- God is merciful, my dear. - The nanny lit candles entwined with gold in front of the icon case and sat down with the stocking by the door. Princess Marya took the book and began to read. Only when steps or voices were heard, the princess looked at each other in fear, questioningly, and the nanny. In all parts of the house the same feeling that Princess Marya experienced while sitting in her room was poured out and possessed everyone. According to the belief that the fewer people know about the suffering of a woman in labor, the less she suffers, everyone tried to pretend not to know; no one spoke about this, but in all the people, in addition to the usual sedateness and respect for good manners that reigned in the prince’s house, one could see one common concern, a softness of heart and an awareness of something great, incomprehensible, taking place at that moment.
No laughter could be heard in the big maid's room. In the waitress all the people sat and were silent, ready to do something. The servants burned torches and candles and did not sleep. The old prince, stepping on his heel, walked around the office and sent Tikhon to Marya Bogdanovna to ask: what? - Just tell me: the prince ordered me to ask what? and come tell me what she says.
“Report to the prince that labor has begun,” said Marya Bogdanovna, looking significantly at the messenger. Tikhon went and reported to the prince.
“Okay,” said the prince, closing the door behind him, and Tikhon no longer heard the slightest sound in the office. A little later, Tikhon entered the office, as if to adjust the candles. Seeing that the prince was lying on the sofa, Tikhon looked at the prince, at his upset face, shook his head, silently approached him and, kissing him on the shoulder, left without adjusting the candles or saying why he had come. The most solemn sacrament in the world continued to be performed. Evening passed, night came. And the feeling of expectation and softening of the heart in the face of the incomprehensible did not fall, but rose. Nobody was sleeping.It was one of those March nights when winter seems to want to take its toll and pours out its last snows and storms with desperate anger. To meet the German doctor from Moscow, who was expected every minute and for whom a support was sent to the main road, to the turn to the country road, horsemen with lanterns were sent to guide him through the potholes and jams.
Princess Marya had left the book long ago: she sat silently, fixing her radiant eyes on the wrinkled face of the nanny, familiar to the smallest detail: on a strand of gray hair that had escaped from under a scarf, on the hanging pouch of skin under her chin.
Nanny Savishna, with a stocking in her hands, in a quiet voice told, without hearing or understanding her own words, what had been told hundreds of times about how the late princess in Chisinau gave birth to Princess Marya, with a Moldavian peasant woman instead of her grandmother.
“God have mercy, you never need a doctor,” she said. Suddenly a gust of wind hit one of the exposed frames of the room (by the will of the prince, one frame was always displayed with larks in each room) and, knocking off the poorly closed bolt, fluttered the damask curtain, and, smelling cold and snow, blew out the candle. Princess Marya shuddered; The nanny, having put down the stocking, went to the window and leaned out and began to catch the folded frame. The cold wind ruffled the ends of her scarf and the gray, stray strands of hair.
- Princess, mother, someone is driving along the road ahead! - she said, holding the frame and not closing it. - With lanterns, it should be, doctor...
- Oh my god! God bless! - said Princess Marya, - we must go meet him: he doesn’t know Russian.
Princess Marya threw on her shawl and ran towards those traveling. When she passed the front hall, she saw through the window that some kind of carriage and lanterns were standing at the entrance. She went out onto the stairs. There was a tallow candle on the railing post and it was flowing from the wind. The waiter Philip, with a frightened face and another candle in his hand, stood below, on the first landing of the stairs. Even lower, around the bend, along the stairs, moving footsteps in warm boots could be heard. And some familiar voice, as it seemed to Princess Marya, said something.
- God bless! - said the voice. - And father?
“They’ve gone to bed,” answered the voice of the butler Demyan, who was already downstairs.
Then the voice said something else, Demyan answered something, and footsteps in warm boots began to approach faster along the invisible bend of the stairs. "This is Andrey! - thought Princess Marya. No, this cannot be, it would be too unusual,” she thought, and at the same moment as she was thinking this, on the platform on which the waiter stood with a candle, the face and figure of Prince Andrei appeared in a fur coat with a collar sprinkled with snow. Yes, it was him, but pale and thin, and with a changed, strangely softened, but alarming expression on his face. He walked onto the stairs and hugged his sister.
-You didn’t receive my letter? - he asked, and without waiting for an answer, which he would not have received, because the princess could not speak, he returned, and with the obstetrician, who entered after him (he met with him at the last station), with quick steps he again entered the the stairs and hugged his sister again. - What fate! - he said, “Dear Masha,” and, throwing off his fur coat and boots, he went to the princess’s quarters.The little princess was lying on pillows, wearing a white cap. (Suffering had just released her.) Black hair curled in strands around her sore, sweaty cheeks; her rosy, lovely mouth with a sponge covered with black hairs was open, and she smiled joyfully. Prince Andrei entered the room and stopped in front of her, at the foot of the sofa on which she was lying. Brilliant eyes, looking childish, scared and excited, stopped at him without changing expression. “I love you all, I haven’t done harm to anyone, why am I suffering? help me,” her expression said. She saw her husband, but did not understand the significance of his appearance now before her. Prince Andrei walked around the sofa and kissed her on the forehead.
“My darling,” he said: a word he had never spoken to her. - God is merciful. “She looked at him questioningly, childishly and reproachfully.
“I expected help from you, and nothing, nothing, and you too!” - said her eyes. She wasn't surprised that he came; she did not understand that he had arrived. His arrival had nothing to do with her suffering and its relief. The torment began again, and Marya Bogdanovna advised Prince Andrei to leave the room.
The obstetrician entered the room. Prince Andrei went out and, meeting Princess Marya, again approached her. They started talking in a whisper, but every minute the conversation fell silent. They waited and listened.
“Allez, mon ami, [Go, my friend,” said Princess Marya. Prince Andrey again went to his wife and sat down in the next room, waiting. Some woman came out of her room with a frightened face and was embarrassed when she saw Prince Andrei. He covered his face with his hands and sat there for several minutes. Pathetic, helpless animal groans were heard from behind the door. Prince Andrei stood up, went to the door and wanted to open it. Someone was holding the door.
- You can’t, you can’t! – a frightened voice said from there. – He began to walk around the room. The screams stopped and a few seconds passed. Suddenly a terrible scream - not her scream, she could not scream like that - was heard in the next room. Prince Andrei ran to the door; the scream stopped, and the cry of a child was heard.
“Why did they bring the child there? thought Prince Andrei at the first second. Child? Which one?... Why is there a child there? Or was it a baby born? When he suddenly realized all the joyful meaning of this cry, tears choked him, and he, leaning with both hands on the windowsill, sobbed, began to cry, as children cry. The door opened. The doctor, with his shirt sleeves rolled up, without a frock coat, pale and with a shaking jaw, left the room. Prince Andrey turned to him, but the doctor looked at him in confusion and, without saying a word, walked past. The woman ran out and, seeing Prince Andrei, hesitated on the threshold. He entered his wife's room. She lay dead in the same position in which he had seen her five minutes ago, and the same expression, despite the fixed eyes and the paleness of her cheeks, was on that charming, childish face with a sponge covered with black hairs.
“I love you all and have never done anything bad to anyone, so what did you do to me?” her lovely, pitiful, dead face spoke. In the corner of the room, something small and red grunted and squeaked in Marya Bogdanovna’s white, shaking hands.Two hours after this, Prince Andrei entered his father’s office with quiet steps. The old man already knew everything. He stood right at the door, and as soon as it opened, the old man silently, with his senile, hard hands, like a vice, grabbed his son’s neck and sobbed like a child.
Three days later the funeral service was held for the little princess, and, bidding farewell to her, Prince Andrei ascended the steps of the coffin. And in the coffin was the same face, although with closed eyes. “Oh, what have you done to me?” it said everything, and Prince Andrei felt that something was torn away in his soul, that he was guilty of a guilt that he could not correct or forget. He couldn't cry. The old man also entered and kissed her wax hand, which lay calmly and high on the other, and her face said to him: “Oh, what and why did you do this to me?” And the old man turned away angrily when he saw this face.
Five days later, the young Prince Nikolai Andreich was baptized. The mother held the diapers with her chin while the priest smeared the boy’s wrinkled red palms and steps with a goose feather.
The godfather grandfather, afraid to drop him, shuddering, carried the baby around the dented tin font and handed him over to his godmother, Princess Marya. Prince Andrei, frozen with fear that the child would not be drowned, sat in another room, waiting for the end of the sacrament. He looked joyfully at the child when the nanny carried him out to him, and nodded his head approvingly when the nanny told him that a piece of wax with hairs thrown into the font did not sink, but floated along the font.Rostov's participation in Dolokhov's duel with Bezukhov was hushed up through the efforts of the old count, and Rostov, instead of being demoted, as he expected, was appointed adjutant to the Moscow governor general. As a result, he could not go to the village with his entire family, but remained in his new position all summer in Moscow. Dolokhov recovered, and Rostov became especially friendly with him during this time of his recovery. Dolokhov lay sick with his mother, who loved him passionately and tenderly. The old woman Marya Ivanovna, who fell in love with Rostov for his friendship with Fedya, often told him about her son.
Under conditions of ideal visibility, that is, standing in an open area, an absolutely flat plain, without grass and trees, in the absence of fog and other atmospheric phenomena, a person of average height sees the horizon at a distance of about 4-5 kilometers. If you rise higher, the horizon line will move away; if, on the contrary, you go down to the lowland, the horizon will become much closer. There is a special formula that allows you to calculate the distance to the horizon, but I don’t think it’s worth doing, because in each specific case it will be different. The shortest distance to the horizon will be in the city - usually to the wall of the nearest house.
In fact, how subjectively the horizon is from us depends on what kind of landscape, mountains, desert, or even water, as well as conditions such as precipitation, fog, and so on. But nevertheless, there is a formula that is designed to calculate the distance to the horizon. However, the formula only works correctly under conditions of a completely flat surface, such as a water surface.
Formula for calculating the distance to the horizon:
S = (R+h)2 - R21/2
In this formula:
Letter S indicates the height of the observer's eyes in meters
Letter R indicates the radius of the Earth, usually: 6367250 m
Letter h indicates the height of the observer's eyes above the surface in meters
Using this formula, you can get a similar table.
The visible horizon is often called the line along which the sky is seen bordering the surface of the Earth. Also called the visible horizon is the celestial space above this boundary, and the surface of the Earth visible to humans, and the entire space visible to humans, to its ultimate limits.
The distance to the visible horizon is calculated depending on the height of the observer above the earth's surface; the radius of the earth is also taken into account in the calculation. The table shows the calculation results.
There is even a special formula for calculating the distance to the horizon. And approximately we can say that if a person is of average height, then the horizon line from him is at a distance of approximately 5 kilometers. The higher you rise, the farther the horizon line will be. So, for example, if you climb a lighthouse 20 meters high, you will be able to observe the water surface at a distance of 17 kilometers. But on the Moon, a person of average height will be at a distance of 3.3 kilometers from the horizon, and on Saturn already at 14.4 kilometers.
The apparent distance to the horizon depends on the terrain, but if you keep in mind that no objects block the horizon, for example in the steppe or at sea, then objects 5 kilometers away are visible. This is if you look at it from the height of the average person.
If a sailor climbs an eight-meter mast, he will be able to look at objects at a distance of 10 kilometers.
From the television tower in Ostankino the horizon will expand to 80 km; it is at this distance that there is a stable radio signal.
From an airplane flying at an altitude of 10 kilometers, a distance of 350 kilometers can already be seen, and astronauts from a space station in orbit can see up to 2 thousand kilometers.
The horizon can be visible and true, so the distance will be different if people are placed at different points.
If a person looks while standing, then approximately the distance is 5 km.
If you climb a mountain 8 km high, the distance to the horizon will be approximately 10 km.
At an altitude of 10 thousand meters the distance increases to 350 km.
That is, everyone has a different distance to the horizon they see.
On a flat surface (water surface) about 6 km. The higher the viewpoint, the further the horizon.
If we mean the line of the visible horizon, then the distance to does not depend on the height of the observer’s eyes. From the bridge of the ship on which I served, the horizon line was 5 miles away (1852 x 5 meters). Through the navigator's periscope raised on the surface, the distance to the horizon was already 11 miles...
Nothing at all. An hour's walk. It is very interesting to sit on the horizon, dangling your legs and dangling them. You can, of course, climb onto a rainbow, but for this you need a ladder. And the horizon is here, nearby. And you don’t need to take anything with you)))
The visible horizon line also depends on observation conditions (weather, atmospheric phenomena, etc.). So, from the same point of view (for me, for example, an embankment on the high bank of the Volga), depending on visibility, a certain horizon is visible in the direction of flood meadows, either 8-9 or more than 30 kilometers away.
The distance to the horizon depends on many parameters. For example, from your vision. And even more important is the height at which you are. So, from Everest the horizon will be visible at a distance of 336 kilometers. But from the lowlands you can see it even after 5 kilometers.
