Calendar history and chronology

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STORY
CALENDAR
AND
CHRONOLOGY

This is the name of the book *), in which the author managed to tell a very interesting and clear story about the complex history of the calendar.

For a long time, people have perceived the passage of time, observing the constant change of day and night, the systematic repetition of natural phenomena. They learned to measure time much later. From the 19th century to the present day, and especially in the last fifty years, the “calendar problem” has attracted much attention in many countries around the world. Due to the imperfections of the Gregorian calendar, introduced in 1582 and still in force in most countries, the International Committee on the Form of the Calendar was organized back in 1923 in Geneva at the League of Nations. Its goal is to create a universal and unchanging calendar that meets the requirements of the economic and cultural life of society.

How was the system for counting long periods of time—the calendar—created and improved? What are lunar, lunisolar and solar calendars? What is the mathematical basis of calendars? What calendars existed in ancient countries - Egypt, Greece, Rome, Babylon? Is it true that the calendar of the Mayan Indians in Central

*) S.I. Seles and kov. History of the calendar and chronology, publishing house “Science”, M., 1970.

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New America, which existed many centuries ago, or rather our modern calendar? These questions are answered by “The History of the Calendar and Chropology.”

The first chapter of the book describes primitive forms of telling time, starting with notches on a stick or knots that were tied on special laces. Here you can also find interesting information about the counting of the beginning of the year among different peoples of our planet.

The astronomical foundations of the calendar form the content of the second chapter. It examines in detail the three main units of time: the solar day, the lunar month and the solar year, which form the foundation of any calendar system.

The third chapter, devoted to solar calendars, begins with their mathematical theory. It contains a detailed description of various calendars, starting with the ancient Egyptian calendar and ending with the calendar of the French Revolution of the late 18th century, developed by a special commission led by Gilbert Romm (1750-1785).

The book (chapters IV and V) gives a detailed introduction to the lunar, lunisolar calendars and the calendars of the peoples of Asia.

Until recently, it was believed that the birthplace of astronomy was the countries of the Middle East, Ancient China and India. However, in recent decades, archeology has discovered a new cultural center located in the New World. They found that in the late Yucatan, which was once inhabited by Mayan Indians, there was a distinctive culture; Astronomy, associated with practical needs, received special development among this people. Please, for the first time in literature, the author devotes such a significant place to the Mayan calendar and chronology (Chapter VI), introducing readers in detail to the calendar systems used.

A description of the history of the calendar in Russia and the USSR, starting with the ancient Slavs, is brought by the author to the present day. The origin of the seven-day week is no longer clogged here.

The book ends with a detailed story about the World Calendar. Analyzing the shortcomings of the current calendar, the author cites various projects for its reform, including the draft World Calendar, considered by the UN Social and Economic Council.

Chropology and some calendar eras are described in the last chapter. The appendix contains various calendars and a lot of interesting materials, including an “eternal” timesheet calendar for determining the day of the week of any calendar date.

Despite the small volume of the book, the author managed to highlight and systematize a large range of issues related to the “calendar gap”.

The task set by the author - to create “… a short encyclopedic reference book on letonics and calendars” - has been completed, and this book is of undoubted interest to a wide range of readers.

L. S. Khrenov

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To the article “Electrostatics in the language of power lines”

1. The picture of the field lines looks like that shown in figure. The left charge is twice as large as the right.
2. No, they can’t. If there were a closed line of force, then by crossing a charge along it, we would do non-zero work.
3. No. The direction of the tangent to the force line coincides with the direction of the force acting on the charge, and therefore with the direction of acceleration of the charge. The trajectory of the charge is a line, the direction of the tangent to which coincides with the direction of the charge’s velocity.
4. Let us draw a sphere of radius $r$ with the center coinciding with the center of the charged sphere. From symmetry it follows that the density of lines of force crossing the surface of this sphere is the same at all points. This means that the direction of the field, equal to the density of the field lines, depends on the distance to the center of the sphere. The total number of field lines intersecting the surface of the sphere we drew is $4\pi r^2 E$, where $E$ is the direction of the field a on the surface of the sphere. According to Gauss’s theorem
   $$4\pi r^2 E = 4\pi Q,$$
   or
   $$E = \frac{Q}{r^2},$$
   where $Q$ is the charge that is inside the sphere.
   If $r < R_1$, then $Q = 0$ and $E = 0$; if $R_1 \leq r < R$, then $Q = q_1$ and $E = \frac{q_1}{r^2}$; if $r \geq R_2$, then
   $$Q = q_1 + q_2 \text{ и } E = \frac{q_1 + q_2}{r^2}.$$
5. Drawing, as we did when solving the previous problem, a sphere of radius $r$ with a center at the center of the charged ball, we find that $E = \frac{Q}{r^2}$, where $Q$ is the charge that is inside the sphere. If the charge density (charge per unit volume) is $ρ$, and the radius of the charged ball is $R$, then at $r < R$ $Q = \frac{4}{3} \pi r^3 ρ$ and
   $$E = \frac{4}{3} \pi r ρ,$$
   that is, the direction of the field returns in proportion to the radius of the sphere we draw.
   At $r < R$ $Q = \frac{4}{3} \pi R^3 ρ$ and $E = \frac{4\pi R^3 ρ}{3r^2}$ - outside the ball, the direction of the field changes in the same way as it would change if the charge $Q$ was concentrated in the center of the ball.
6. This problem is solved in the same way as the previous ones, only in order to use Gauss’s theorem, you need to build not a spherical surface, but a cylindrical one with an axis coinciding with the axis of the thread. Since the thread is infinite and uniformly charged, the lines of force intersect the cylindrical surface along radii, and their density is the same at all points of the surface. This means that the direction of the field depends only on the distance to the thread. If the height of the constructed cylinder is 1, and the radius of its base is $r$, then the surface of the cylinder is intersected by $n = 2\pi r \cdot 1 \cdot E$ force lines. According to Gauss’s theorem
   $$2\pi r \cdot 1 \cdot E = ρ \cdot 1$$

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(ρ is the charge per unit length of the thread, ρ 1 is the charge that is inside the cylinder). Hence $E = \frac{\rho}{2\pi r}$.

7. The problem is solved similarly to the previous one. If $R$ is the radius of the charged cylinder, ρ is the charge density and $r$ is the distance to the cylinder axis, then

$$E = \begin{cases} \frac{\pi r^2 \cdot 1 \cdot \rho}{2\pi r \cdot 1} = \frac{1}{2} r\rho & \text{при } r < R, \\ \frac{\pi R^2 \rho \cdot 1}{2\pi r \cdot 1} = \frac{R^2\rho}{2r} & \text{при } r \geq R. \end{cases}$$

8. Let us assume that the field in the cavity is not zero. Since there are no charges in the cavity on which the force lines could begin or end, the force lines must cross the entire cavity and not interrupt it. If a charge intersects along one of these lines, then we will do some non-zero work. This means that the points of the cavity boundaries lying on this line have different potentials. But this is impossible, since the cavity is inside the metal. This means that the field in the cavity must be equal to zero.

By the way, “Reducing algebraic fractions”

1. $\frac{x^2 - x - 1}{9(x^2 + x - 1)}$.
2. $\frac{x + a}{x^2 + ax + a^2}$.
3. $\frac{a - 1}{x^2 + 2x + 2}$.
4. $\frac{7}{15x^2 - 23}$.

By the way, “Maximum, minimum and the theorem of averages”

1. Let us denote the radius of the desired sector by $R$, the length of the arc by $L$ and the area of ​​the sector by $S$. Then $P = 2R + L$, $S = \frac{RL}{2}$. By theorem

$$4S = 2RL \leq \left(\frac{2R + L}{2}\right)^2 = \left(\frac{P}{2}\right)^2 = \frac{P^2}{4},$$

whence $S \leq \frac{P^2}{16}$. Equality is achieved at $2R = L = \frac{P}{2}$.

2. Let us denote by $R$ the radius of the circle and by $a$ the distance from the point $A$ to the center of the circle. Then the chord $MN$ passing through the point $A$ satisfies the correspondence $xy = (R + a)(R - a)$, where $x = MA, y = AN$. The length of chord $MN$ is equal to $x + y$. We get by theorem

$$x + y \geq 2\sqrt{xy} = 2\sqrt{(R + a)(R - a)},$$

equality is achieved at $x = y$.

Consequently, the length of the chord is minimal when at point $A$ it is divided in half, i.e., perpendicular to the diameter passing through point $A$.

3. Let us denote $\sqrt{x}$ by $z$. The expression will be rewritten as follows: $2z^6 + \frac{3}{z}$. From Example 3, the following assessment of the values ​​of this expression follows:

$$2z^6 + \frac{3}{z} \geq 7\sqrt{7\frac{2 \cdot 3^6}{1 \cdot 6^6}} = 7\sqrt{\frac{1}{2^5}} = \frac{7}{\sqrt{32}}.$$

The minimum is reached at

$$\sqrt{x} = \sqrt{7\frac{3 \cdot 1}{2 \cdot 6}} = 2^{\frac{2}{7}},$$ from where $x = 2^{\frac{4}{7}}$.

4. Let’s take the constant factor $c = k_1^1 k_2^2 \dots k_n^n$ out of brackets (you don’t have to write it) and rewrite the product like this:

$$\frac{x_1}{k_1} \cdot \frac{x_1}{k_1} \dots \frac{x_1}{k_1} \times \frac{x_2}{k_2} \cdot \frac{x_2}{k_2} \dots \frac{x_2}{k_2} \times \frac{x_n}{k_n} \cdot \frac{x_n}{k_n} \dots \frac{x_n}{k_n}$$

Note that

$$\frac{x_i}{k_i} + \frac{x_i}{k_i} + \dots + \frac{x_i}{k_i} = \frac{x_i}{k_i} k_i = x_i$$

($i = 1, 2, \dots, n$).

Consequently, the sum of all components in the new entry is equal to $x_1 + x_2 + \dots + x_n$, that is, constructed, and the product will be the smallest if the components are equal, that is, if

$$\frac{x_1}{k_1} = \frac{x_2}{k_2} = \dots = \frac{x_n}{k_n}.$$

BY radius*

1. [p] One of the characteristics of the matrix.
2. Unit of gamma radiation dose.
3. Unit of length.
4. The point of the radial orbit that is furthest from the Earth.
5. [p] Creator of one of the non-Euclidean geometries.
6. Ionized gas.
7. Outstanding French philosopher.
8. International system of units.
9. [p] Geometric figure.
10. An electrical apparatus that regulates voltage or current in an electrical circuit.
11. Land measure.
12. Polygon.
13. Unit of measurement of electrical embone.
14. [p] Symbol used when writing equations.
15. The place occupied by a digit when writing a number.
16. Ancient Greek mathematician.
17. A unit of measurement for electrical resistance.
18. [p] Product of condensed water vapor.
19. Star of the first magnitude [in the constellation Orla].
20. The ratio of the circumference to the diameter.
21. Part of a transmitting or receiving radio station.
22. Outstanding French astronomer, mathematician and physicist.
23. [p] Radioactive element.
24. Founder of the theory of polygamy.
25. The great ancient Greek materialist philosopher, atomist.

By circumference

3. Solid substance.
4. One of the electrodes.

• The symbol (p) means that the word is written starting from the center.