Sundial

Wall (vertical) sundial in the Solovetsky Monastery . Shooting time 13:40 Moscow time

Original sundial in Raifa Bogoroditsk monastery

Sundial. 640-650, Zvartnots , Armenia . Museum piece

Solar clock - a device for determining the time by changing the length of the shadow from the gnomon and its movement on the dial . The appearance of this watch is associated with the moment when a person realized the relationship between the length and position of the sun's shadow from certain objects and the position of the Sun in the sky.

The simplest sunshine shows local true, rather than local average solar time , and does not take into account the difference between official time and local solar time. It is possible to use the sundial only during the day with a clear sky or with light cloudiness, which does not interfere with the formation of a clear shadow. To improve the accuracy of the sundial should take into account the amendment to the formula of time .

Currently, the sundial for its intended purpose is practically not used, and have given way to various types of other watches .

Content

History

The most ancient tool for determining the time served as a gnomon . The change in the length of its shadow indicated the time of day. Such the simplest sundial is mentioned in the Bible (2 Kings, 20:10, Isaiah, 38: 8).

Ancient Egypt

The first known description of a sundial in ancient Egypt is the inscription in the tomb of Seti I , dated 1306-1290. BC e. It refers to a sundial that measures time along the length of a shadow and is a rectangular plate with divisions. At one end of it is attached a low bar with a long horizontal bar, which cast a shadow. The end of the plate with the plate was heading east, and the hour of the day was set by the marks on the rectangular plate, which in Ancient Egypt was defined as 1/12 of the time from sunrise to sunset. In the afternoon, the end of the plate was heading west. Tools made according to this principle were also found. One of them dates back to the reign of Thutmose III ^[1] and dates from 1479-1425. BC Oe., the second - from Sais ^[2] (according to other data - from Fayyum ^[3] ), it dates from 1000–600. BC. Both tools are stored in the Berlin Egyptian Museum , the first - under inventory number 19744, the second - under number 19743 ^[3] . At the end they have only a bar, without a horizontal bar, and there is also a groove for a plumb to give the device a horizontal position ^[2] .

The other two types of ancient Egyptian clocks, measuring time along the length of a shadow, were clocks in which a shadow fell on an inclined plane or on a step. They were deprived of the lack of hours with a smooth surface: in the morning and evening hours the shadow went beyond the plate. These types of watches were combined in a model of limestone, which is stored in the Cairo Egyptian Museum and dates from a somewhat later time than the clock from Sais. On the one side of the model - two inclined planes with steps, one of them was oriented to the east, while the other pointed to the west. Before noon, the shadow fell on the first plane, gradually descending the steps from top to bottom, and after noon - on the second plane, gradually rising from bottom to top, at noon there was no shadow ^[1] . On the other hand, the model has two inclined planes without steps, this type of clock acted similarly to a clock with steps. A concrete implementation of the type of sundial with an inclined plane was a portable watch from Kantara, created around 320 BC. e. with one inclined plane, on which divisions were made, and a plumb line. The plane was oriented on the Sun ^[2] .

In 2013, scientists at the University of Basel reported the discovery of a sundial, supposedly vertical, by an age of 3.3 thousand years, painted on limestone . They were discovered in the Valley of the Kings near the workers' dwelling between the tombs KV29 and KV61 ^[4] ^[5] .

Reconstruction of the sundial as described from the tomb of Seti I

Sundial from Saïs ^[2] (or Fayoum ), 1000–600 BC. ^[3]

Model of a sundial from the Cairo Egyptian Museum

Drawing hours with an inclined plane, something like the clock looked from Kantara

Ancient China

Equatorial sundial in the Forbidden City

The first mention of the sundial in China is probably the gnomon problem , given in the ancient Chinese problem book "Zhou bi Suan Jing" , compiled around 1100 BC. e. ^[1] In the Zhou epoch in China, equatorial sundials were used in the form of a stone disk, installed parallel to the celestial equator and piercing it in the center of the rod, installed parallel to the earth's axis . During the Qing era in China, a portable sundial with a compass was made : either equatorial - again with a rod in the center of the disk, installed parallel to the celestial equator, or horizontal - with a thread in the role of a gnomon above the horizontal dial ^[6] .

Ancient Greece and Ancient Rome

Scafis is the ancient sundial. On the spheroidal notch there are lines of clocks. A shadow cast a horizontal or vertical bar, or a ball in the center of the tool.

According to the story of Vitruvius , the Babylonian astronomer Beros , who settled in the VI. BC e. on the island of Kose, he introduced the Greeks to the Babylonian sundials, which took the form of a spherical bowl - the so-called scifi . This sundial has been refined by Anaximander and Anaximenes . In the middle of the XVIII century, during excavations in Italy, they found just such a tool, which is described in Vitruvius. The ancient Greeks and Romans, like the Egyptians, divided the time interval from sunrise to sunset to 12 hours, and therefore their hour (as a measure of time) was of various lengths depending on the time of year ^[7] . The surface of the notch in the sundial and the “watch” lines on them were selected so that the end of the twilight shadow indicates the hour. The angle at which the upper part of the stone is cut depends on the latitude of the place for which the watch is made. Subsequent geometers and astronomers ( Eudox , Apollonius , Aristarchus ) invented various forms of sundials. There have been descriptions of such instruments, bearing the strangest names according to their appearance. Sometimes the gnomon , casting a shadow, was located parallel to the axis of the earth.

From Greece, the sundial reached Rome . In 293 BC. er Papyri Cursor ordered to build a sundial in the Quirinal Temple, and in 263 BC. er another consul , Valery Messala , brought a sundial from Sicily . Arranged for a more southerly latitude, they showed an hour incorrectly. For the latitude of Rome, the first clock is arranged in 164 BC. er By Marcie Philipp ^[1] . Sundial met throughout the Roman Empire, in particular, at the villas, and both in Rome and in the province ^[8] . One of the most famous portable sundials is the clock in the form of a pork ham ^[8] , found on June 11, 1755 ^[9] during the excavations of Herculaneum and related to the end of the 1st c. n e. They are stored in the National Archaeological Museum of Naples under accession number 25494 ^[10] .

Sundial used in the ancient Greek colonies of the Northern Black Sea . Four copies of the clocks found at the sites of Pantikapaion and Kitia are kept in the Kerch Museum .

Ancient Russia and Russia

Old Russian chronicles often indicated the hour of some event, which suggested that at that time certain tools or objects were already used in Russia to measure time at least during the day. Chernigov artist Georgy Petrash drew attention to the patterns in the sun illuminating the niches of the north-western tower of the Transfiguration of the Savior Cathedral in Chernigov and to the strange pattern ("meanders") above them. Based on a more detailed study of them, he suggested that the tower is a sundial, in which the hour of the day is determined by the illumination of the corresponding niche, and the meanders are used to determine the five-minute interval ^[11] ^[12] . Similar features were noted in other temples of Chernigov, and it was concluded that the sundial in Ancient Rus was used in the 11th century ^[13] .

In the 16th century, a Western European portable sundial appeared in Russia. In 1980, there were seven such watches in Soviet museums. The earliest of them belong to the year 1556 and are stored in the Hermitage ; they were designed to be worn around the neck and represent a horizontal sundial with a sector gnomon to indicate time, a compass to orient the watch in the north-south direction and a plumb on the gnomon to give the watch a horizontal provisions. All these elements are installed on the board, which can deviate from the horizontal position, providing the possibility of using the clock not on one, but in the interval of latitudes : 47-57 degrees ^[13] .

Vertical sundial on the "marble pyramid" at the corner of the Fontanka embankment and Moskovsky Prospect, St. Petersburg

During the reign of Anna Ioannovna , on August 23, 1739, a Senate decree was issued ^[14] , according to which wooden milestones-obelisks were installed on the road from St. Petersburg to Peterhof , in 1744 a decree was issued about the road from St Petersburg to Tsarskoye Selo . Instead of milestones, obelisks, subsequently, they installed “marble pyramids” with design according to the works of Antonio Rinaldi . Some of them had a sundial, and the traveler could recognize the distance and time from them. “Marble pyramids” with sundials are preserved in the following places: in St. Petersburg at the corner of the Fontanka River Embankment and Moskovsky Prospect (marking one mile from the Post Office building) and in Pushkin at the Orel Gate ^[15] , located on the southern border Catherine Park . On the "marble pyramid" at the Orlov gates the installation date is indicated - 1775 ^[16] .

Middle Ages

Arab astronomers ( Sabit ibn Korr , Ibn ash-Shatir , Abu-l-Hasan ibn Yunis ) left extensive treatises on the gnomonics , or the art of building a sundial. The basis was the rules of trigonometry. In addition to the "watch" lines, the direction of Mecca , the so-called Qibla , was also applied on the surface of the Arabic clocks. Particularly important was the moment of the day when the end of the shadow of a vertically set gnomon fell on a qibla line.

With the introduction of equal hours of the day and night (not dependent on the season), the gnomonic task was considerably simplified: instead of noticing the place of the end of the shadow on complex curves, it became sufficient to notice the direction of the shadow. If the pin is located in the direction of the earth's axis, then its shadow lies in the plane of the hour circle of the sun, and the angle between this plane and the plane of the meridian is the hour angle of the sun or true time. It remains only to find the intersection of successive planes with the surface of the “dial” of the clock. Most often it was a plane perpendicular to the pin, that is, parallel to the celestial equator (equatorial, or equinoctial hours); on it, the direction of the shadow changes by 15 ° every hour. In all other positions of the face of the dial, the angles formed by the direction of the shadow with the line of noon on it do not grow evenly.

Gnomonica was engaged in drawing up the rules for finding the various positions of the shadow on these surfaces. The sundial, as already said, does not give an average, but a true solar time. One of the special tasks of the gnomonica was to build a curve on the dial of a sundial, which would indicate an “average” noon at different times of the year. In medieval Europe, the gnomonics were engaged in: Apian , Albrecht Dürer , Kircher . He lived at the beginning of the XVI century. Münster was recognized as the "father of the gnomonica."

Equatorial sundial for the Northern Hemisphere

Types of sundials

There are sundials equatorial, horizontal, vertical (if the face of the dial is vertical and directed from west to east), morning or evening (the plane is vertical, from north to south). Conical, ball, cylindrical sundials were also built.

Equatorial

Equatorial sundials consist of a frame (a plane with hour divisions) and a gnomon. Hour divisions on the frame are applied at equal angular intervals, as on the dial of an ordinary watch, and the gnomon is usually a metal rod mounted on the frame perpendicular to its surface. Then the frame is oriented in a horizontal plane so that the straight line connecting the gnomon base and the hour division, corresponding to noon, is parallel to the midday line towards the south - for the Northern Hemisphere , or towards the North - for the Southern hemisphere, and inclines relative to the horizon plane, respectively , to the north or south side at an angle α = 90 ° –φ, where φ is the geographical latitude of the place where the sundial is installed ^[17] . The frame will be parallel to the celestial equator (hence the name of this type of sundial), and since the celestial sphere rotates uniformly during the day, the shadow of the gnomon at any hour of the day will describe equal angles (therefore, the hour divisions are carried out in the same way as on the dial normal hours).

Horizontal sundial in the city of Perth , Australia, with characteristic features of the Southern Hemisphere : the shadow at midday in the south and the Sun in the north; The visible path of the Sun passes right-to-left across the sky, so the clock numbering goes counterclockwise

Equal angular intervals (t = 15 °) between adjacent watch divisions, like on the dial of an ordinary watch, and perpendicularity of the gnomon to the cadran are the main advantages of the equatorial sundial over the horizontal and vertical. The main disadvantage of equatorial sundials is that they, unlike horizontal ones, will work only from the day of the spring equinox to the day of the autumn equinox (in the Northern Hemisphere the spring equinox in March, the autumn equinox in September, in the Southern hemisphere spring equinox in September ^[18] , autumn - in March). In the rest of the year, they will not work, because the Sun will be on the other side of the plane of the celestial equator, and the entire upper surface of the frame will be in the shade ^[17] . Of course, this disadvantage can be eliminated if the frame is made in the form of a plate, hour divisions are applied to both the upper and lower surfaces, and the gnomon is continued under the plate, but even then in the days close to the day of spring or autumn equinox the sundial will not work - The sun will shine on the plate not from above and not from below, but from the side.

Horizontal

The horizontal sundial, as well as the equatorial one, consists of a frame and a gnomon. However, in this case, the frame is set parallel to the horizon plane. Most often, the gnomon is a triangle perpendicular to the plane of the frame, and one of its sides is inclined to it at an angle equal to the geographical latitude of the place where the clock is installed ^[19] . The line of intersection of the gnomon and the frame is directed parallel to the midday line — the line along which the shadow of the vertical rod is directed at this place at true midday .

If we denote the geographical latitude of the place where the clock is set by φ, the number of hours before noon (after noon) by m, then the angle between the noon line and the corresponding hourly division on the clock α can be determined by the formula

\operatorname {tg} \alpha =\sin \varphi \cdot \operatorname {tg} (15^{\circ }\cdot m)

{\ displaystyle \ operatorname {tg} \ alpha = \ sin \ varphi \ cdot \ operatorname {tg} (15 ^ {\ circ} \ cdot m)}

Derivation of the formula

Figure to the conclusion of the formula for the angle of the tic of the sundial to the noon line

For definiteness, we will assume that clocks are set in the mid latitudes of the northern hemisphere. The figure shows a celestial sphere centered at point O. The horizon plane is designated NESW, NS is the midday line, Z is zenith, Z 'is nadir, PL is the axis of the world, ZNZ'S is the plane of the celestial meridian. As is known, the angle of inclination of the axis of the world to the horizon plane is equal to the geographical latitude of the place of observation (indicated by the letter φ in the figure), therefore the OP side of the sundown gnomon, indicated in the TPO figure, is shown coinciding with the axis of the world. The angle between the midday line and the edge of the gnomon shadow that we need to find is indicated by α. T denotes the angle between the circle of declination of the Sun PBL and the plane of the celestial meridian. Again, the position of the Sun is shown before noon for definiteness.

Thus, in the figure we have a spherical triangle PNB. Note that the plane of the celestial meridian is perpendicular to the plane of the horizon, that is, the angle PNB is straight, therefore our spherical triangle is rectangular. To determine any element of a rectangular spherical triangle, it is enough to know any two of its other elements. In the spherical triangle PNB, the side φ and the angle t are known; we must find the side α. By Naper's mnemonic rule, we have:

\cos(90^{\circ }-\varphi )=\operatorname {ctg} t\cdot \operatorname {ctg} (90^{\circ }-\alpha )

{\ displaystyle \ cos (90 ^ {\ circ} - \ varphi) = \ operatorname {ctg} t \ cdot \ operatorname {ctg} (90 ^ {\ circ} - \ alpha)}

From here

\operatorname {tg} \alpha =\sin \varphi \cdot \operatorname {tg} t

{\ displaystyle \ operatorname {tg} \ alpha = \ sin \ varphi \ cdot \ operatorname {tg} t}

It remains to be noted that

t={\frac {360^{\circ }}{24}}\cdot m=15^{\circ }\cdot m

{\ displaystyle t = {\ frac {360 ^ {\ circ}} {24}} \ cdot m = 15 ^ {\ circ} \ cdot m}

Vertical

Vertical sundials are usually placed on the walls of buildings and various buildings. Therefore, their frame is vertical - perpendicular to the plane of the horizon, but can be rotated in different directions. From the side, in which the frame is turned, the location of the hour divisions on the frame depends. They will be symmetrical with respect to the midday division only with the quadrang facing strictly to the south (geographical, not magnetic!) - in the northern hemisphere, or to the north - in the southern hemisphere, in other words - with a quadrangle, perpendicular to the noonline. For a directional frame like this, the gnomon should lie in the plane of the celestial meridian, in other words, be perpendicular to both the frame plane and the horizon plane, and one of its sides should be parallel to the earth axis, which means that it should make an angle with the frame plane. 90 ° -φ, where φ is the latitude of the installation location ^[1] . The formula for the angle of inclination of the hour lines to the noon division is derived similarly to the horizontal clock - from consideration of a rectangular spherical triangle formed by the Sun's declination circle, the frame plane and the plane of the celestial meridian. It has the form ^[20] :

\operatorname {tg} \alpha =\cos \varphi \cdot \operatorname {tg} t=\cos \varphi \cdot \operatorname {tg} (15^{\circ }\cdot m)

{\ displaystyle \ operatorname {tg} \ alpha = \ cos \ varphi \ cdot \ operatorname {tg} t = \ cos \ varphi \ cdot \ operatorname {tg} (15 ^ {\ circ} \ cdot m)}

,

where φ is the latitude of the place where the clock is set, m is the number of hours before noon (afternoon) through, α is the angle between noon division and the corresponding hour division, t is the angle between the circle of declination of the Sun and the plane of the celestial meridian.

In the Moscow Planetarium installed vertical sundial , showing time and date.

Horizontal sundial 1611 on the brand of the GDR ( Sc # 2345)

Sundial in Philately

A series of postage stamps of the Dominican Republic of 1931—1933 was devoted to sundials in Santo Domingo , established in 1753 . ( Sc # C10 — C17) ^[21] . Four of the six stamps of the series devoted to the exhibits of the State Physics and Mathematics Salon in Dresden , released in 1983 in the GDR ( Sc # 2345—2348) , are devoted to various old tabletop sundials.

Interesting Facts

If you fly by plane at noon and fly parallel to the west as fast as the Earth rotates (that is, with the speed of the earth’s points on this parallel) ^[22] , then the sundial installed on the aircraft (both horizontal and vertical) , equatorial) will always show noon. However, when crossing the date line you will need to add one day.

Notes

↑ ¹ ² ³ ⁴ ⁵ Stanislav Michal. Hodiny: Od gnómonu k atomovým hodinám. - 2nd ed. - Prague : Státní nakladatelství technické literatury (SNTL), 1987. - 269 p. The first edition of the book was translated into Russian: Stanislav Michal. Clock. From gnomon to atomic clocks. - M .: Knowledge , 1983. - 256 p. - ( Translated non-fiction ).
↑ ¹ ² ³ ⁴ Marshall Clagett. Ancient Egyptian science . - 2004. - V. 2. - ISBN 9780871692146 .
↑ ¹ ² ³ Vodolazhskaya, 2014 , p. 9.
↑ Ancient sundial discovered in Egypt
↑ Bickel S., Gautschy R. Eine ramessidische Sonnenuhr im Tal der Könige (Neopr.) // Zeitschrift für Ägyptische Sprache und Altertumskunde. - 2014. - Vol. 96 , No. 1 . - p . 3-14 . - DOI : 10.1515 / zaes-2014-0001 .
↑ The Spiritual Culture of China / Editors of the volume M. L. Titarenko, A. I. Kobzev, V. E. Yeremeyev, A. E. Lukyanov. - M .: Eastern literature, 2009. - T. 5. Science, technical and military thought, health care and education. - 1087 s. - ISBN 9785020363816 .
Shukhardin S. V., Laman N. K., Fedorov A. S. Use of devices // Technique in its historical development. From the advent of hand tools to the formation of machinery factory production . - M .: Science , 1979.
↑ ¹ ² Pipunyrov, 1982 , p. 52.
Pit Le pitture antiche d'Ercolano e contorni . - Napoli : Nella Regia Stamperia, 1762. - Vol. 3. - P. V.
↑ James Evans. Time and Cosmos in Greco-Roman Antiquity . - Princeton University Press, 2016. - p. 83.85.
↑ In Chernigov preserved sundial of the ancient city (photo)
↑ Yu. Yu. Shevchenko, T. G. Bogomazova “The most ancient preserved Christian church of Russia”
↑ ¹ ² V. Yu. Matveev. Sundial of 1556 from the Hermitage collection // Historical and astronomical research . - M .: Science , 1980. - Vol. Xv . - p . 177-180 .
Указ Decree text: The complete collection of laws of the Russian Empire, since 1649 . - 1830. - T. X. 1737-1739. - p. 880-881.
↑ Radchenko B.G. Shadow measures time // Hours of Leningrad. - L .: Lenizdat , 1975. - P. 59-69.
↑ Petersburg Highway. Tsarskoye Selo perspective. Milestones
↑ ¹ ² Moiseev A. Sundial in your garden (Rus.) // Science and Life . - 1992. - № 9 . - p . 65-66 .
↑ (Eng.) Article about the vernal equinox of 2008 on the site of the Sydney Observatory
↑ Trankovsky S. Sundial (rus.) // Science and Life . - 1983. - № 8 . - pp . 133-135 .
↑ Sundial // Big Soviet Encyclopedia / Ch. ed. A. M. Prokhorov . - 3rd ed. - M .: Soviet Encyclopedia , 1976. - T. 24, Vol. I: Dogs - String. - p. 148. - 631 000 copies.
↑ Space & Astronomy Stamps - Dominican Republic (República Dominicana)
↑ This speed is approximately equal to the length of the parallel divided by 24 hours (the height of the aircraft, if it is several kilometers, like the non-sphericity of the Earth, can be neglected in a rough approximation), for example, for a latitude of 55 degrees this ${\tfrac {2\pi \cdot 6370\cdot \cos 55^{\circ }}{24}}\approx 960$ ${\ displaystyle {\ tfrac {2 \ pi \ cdot 6370 \ cdot \ cos 55 ^ {\ circ}} {24}} \ approx 960}$ km / h

Literature

Pipunyrov V.N. The history of clocks from ancient times to the present day / Academy of Sciences of the USSR , Institute of the History of Science and Technology , responsible. ed. L. E. Maistrov. - M .: Science , 1982. - 25 000 copies.
Serafimov V.V. , Lermantov V.V. Clock // Encyclopedic dictionary of Brockhaus and Efron : in 86 tons (82 tons and 4 extras). - SPb. , 1890-1907.
Dmitriev V.I .; Walk through the sundial St. Petersburg // History of St. Petersburg. 2011. No. 5. P. 67–73.
Shestakov S. А. Sundial from the Kerch Museum // Bospor studies. Issue Ix. Simferopol - Kerch. 2005. C.360 - 373.
Vodolazhskaya L.N., Larenok P.A., Nevsky M.Yu. Ancient astronomical instrument from the log-house burial of the Tavriya-1 burial ground (Northern Black Sea Coast) // Archaeoastronomy and Ancient Technologies. - 2014. - V. 2 , no. 2 - p . 31-53 .
Vodolazhskaya L.N. Reconstruction of the ancient Egyptian sundials // Archaeoastronomy and Ancient Technologies. - 2014. - V. 2 , no. 2 - p . 1-18 .

Links

[Hodiny-1] ¹ ² ³ ⁴ ⁵ Stanislav Michal. Hodiny: Od gnómonu k atomovým hodinám. - 2nd ed. - Prague : Státní nakladatelství technické literatury (SNTL), 1987. - 269 p. The first edition of the book was translated into Russian: Stanislav Michal. Clock. From gnomon to atomic clocks. - M .: Knowledge , 1983. - 256 p. - ( Translated non-fiction ).

[Sc-2] ¹ ² ³ ⁴ Marshall Clagett. Ancient Egyptian science . - 2004. - V. 2. - ISBN 9780871692146 .

[_b81c8eb0b8f8329e-3] ¹ ² ³ Vodolazhskaya, 2014 , p. 9.

[4] Ancient sundial discovered in Egypt

[5] Bickel S., Gautschy R. Eine ramessidische Sonnenuhr im Tal der Könige (Neopr.) // Zeitschrift für Ägyptische Sprache und Altertumskunde. - 2014. - Vol. 96 , No. 1 . - p . 3-14 . - DOI : 10.1515 / zaes-2014-0001 .

[6] The Spiritual Culture of China / Editors of the volume M. L. Titarenko, A. I. Kobzev, V. E. Yeremeyev, A. E. Lukyanov. - M .: Eastern literature, 2009. - T. 5. Science, technical and military thought, health care and education. - 1087 s. - ISBN 9785020363816 .

[7] Shukhardin S. V., Laman N. K., Fedorov A. S. Use of devices // Technique in its historical development. From the advent of hand tools to the formation of machinery factory production . - M .: Science , 1979.

[_96e4cf08231e3c02-8] ¹ ² Pipunyrov, 1982 , p. 52.

[9] Pit Le pitture antiche d'Ercolano e contorni . - Napoli : Nella Regia Stamperia, 1762. - Vol. 3. - P. V.

[10] James Evans. Time and Cosmos in Greco-Roman Antiquity . - Princeton University Press, 2016. - p. 83.85.

[11] In Chernigov preserved sundial of the ancient city (photo)

[12] Yu. Yu. Shevchenko, T. G. Bogomazova “The most ancient preserved Christian church of Russia”

[IAI_15-13] ¹ ² V. Yu. Matveev. Sundial of 1556 from the Hermitage collection // Historical and astronomical research . - M .: Science , 1980. - Vol. Xv . - p . 177-180 .

[14] Указ Decree text: The complete collection of laws of the Russian Empire, since 1649 . - 1830. - T. X. 1737-1739. - p. 880-881.

[15] Radchenko B.G. Shadow measures time // Hours of Leningrad. - L .: Lenizdat , 1975. - P. 59-69.

[16] Petersburg Highway. Tsarskoye Selo perspective. Milestones

[NiZ92-17] ¹ ² Moiseev A. Sundial in your garden (Rus.) // Science and Life . - 1992. - № 9 . - p . 65-66 .

[18] (Eng.) Article about the vernal equinox of 2008 on the site of the Sydney Observatory

[19] Trankovsky S. Sundial (rus.) // Science and Life . - 1983. - № 8 . - pp . 133-135 .

[GSE-20] Sundial // Big Soviet Encyclopedia / Ch. ed. A. M. Prokhorov . - 3rd ed. - M .: Soviet Encyclopedia , 1976. - T. 24, Vol. I: Dogs - String. - p. 148. - 631 000 copies.

[21] Space & Astronomy Stamps - Dominican Republic (República Dominicana)

[22] This speed is approximately equal to the length of the parallel divided by 24 hours (the height of the aircraft, if it is several kilometers, like the non-sphericity of the Earth, can be neglected in a rough approximation), for example, for a latitude of 55 degrees this ${\tfrac {2\pi \cdot 6370\cdot \cos 55^{\circ }}{24}}\approx 960$ ${\ displaystyle {\ tfrac {2 \ pi \ cdot 6370 \ cdot \ cos 55 ^ {\ circ}} {24}} \ approx 960}$ km / h