What does the mathematical sign mean. Complete Encyclopedia of Symbols

Continuation of the first part: Occult and mystical symbols and their meaning. Geometric symbols, Universal symbols-images and symbols-concepts. Emblems of modern religions. Crosses: the most common forms. Time images. Symbolism of the kingdom of plants and animals. Mythical creatures.

Encyclopedia of symbols

Swastika straight (left-handed)

The swastika as a solar symbol

A straight (left-handed) swastika is a cross with the ends bent to the left. Rotation is considered to be clockwise (opinions sometimes differ in determining the direction of movement).

A straight swastika is a symbol of blessing, good omen, prosperity, good luck and aversion to misfortune, as well as a symbol of fertility, longevity, health and life. It is also a symbol of the masculine principle, spirituality, which inhibits the flow of lower (physical) forces and allows the energies of a higher, divine nature to manifest.

Reverse swastika (right side)

Swastika on a Nazi military medal

The reverse (right-handed) swastika is a cross with the ends bent to the right. Rotation is considered to be counterclockwise.

The reverse swastika is usually associated with the feminine. Sometimes it is associated with the launch of negative (physical) energies that close the passage to the elevated forces of the spirit.

The Sumerian swastika, formed by four women and their hair, symbolizes the female generative power

Pentagram (pentacle): the general meaning of the symbol

pentagram sign

The pentagram, written in one line, is the most ancient of all the symbols that we own. It had different interpretations in different historical times of mankind. It became the Sumerian and Egyptian sign of the stars.

Later symbolism: five senses; masculine and feminine, expressed by five points; harmony, health and mystical powers. The pentagram is also a symbol of the victory of the spiritual over the material, a symbol of security, protection, a safe return home.

Pentagram as a magical symbol

Pentagrams of the White and Black Magicians

A pentacle with one end up and two down is a sign of white magic, known as the "foot of the druid"; with one end down and two up, it represents the so-called "goat's hoof" and the horns of the devil - a sign change characteristic of symbolism from positive to negative when it is turned over.

The pentagram of the White Magician is a symbol of magical influence and the dominance of a disciplined Will over the phenomena of the world. The will of the Black Magician is directed to destruction, to the refusal to perform a spiritual task, therefore the inverted pentagram is considered as a symbol of evil.

Pentagram as a symbol of a perfect person

Pentagram symbolizing the perfect man

The pentagram, a five-pointed star, is a symbol of a perfect man standing on two legs with outstretched arms. We can say that a person is a living pentagram. This is true both physically and spiritually - a person possesses five virtues and manifests them: love, wisdom, truth, justice and kindness.

Truth belongs to the spirit, love to the soul, wisdom to the intellect, kindness to the heart, justice to the will.

double pentagram

Double pentagram (man and the universe)

There is also a correspondence between the human body and the five elements (earth, water, air, fire and ether): will corresponds to earth, heart to water, intellect to air, soul to fire, spirit to ether. Thus, by his will, intellect, heart, soul, spirit, a person is connected with the five elements working in the cosmos, and he can consciously work in harmony with them. This is the meaning of the symbol of the double pentagram, in which the small one is inscribed in the large one: a person (microcosm) lives and acts inside the Universe (macrocosm).

Hexagram

Hexagram image

Hexagram - a figure made up of two polar triangles, a six-pointed star. It is a complex and solid symmetrical shape in which six small individual triangles are grouped around a large central hexagon. The result is a star, although the original triangles retain their individuality. Since the upward facing triangle is a heavenly symbol, and the downward facing triangle is a symbol of the earth, together they are a symbol of a person who unites these two worlds. It is a symbol of a perfect marriage that binds a man and a woman.

Seal of Solomon

Seal of Solomon, or Star of David

This is the famous magical seal of Solomon, or the star of David. The top triangle in her image is white and the bottom triangle is black. It symbolizes, first of all, the absolute law of analogy, expressed by the mystical formula: "What is below is similar to what is above."

The Seal of Solomon is also a symbol of human evolution: one must learn not only to take, but also to give, to absorb and radiate at the same time, to radiate for the Earth, to perceive from Heaven. We receive and are filled only when we give to others. This is the perfect union of spirit and matter in man - the union of the solar plexus and the brain.

five pointed star

five pointed star

star of bethlehem

The five-pointed star is interpreted in different ways, including it symbolizes joy and happiness. It is also the emblem of the Semitic goddess Ishtar in her martial incarnation, and in addition, the Star of Bethlehem. For Freemasons, the five-pointed star symbolizes the mystical center.

The Egyptians attached great importance to the five- and six-pointed stars, as is clear from the text preserved on the wall of the funerary temple of Hatshepsut.

seven-pointed star

Seven pointed star of magicians

In the seven-pointed star, the characteristic features of the five-pointed are repeated. The Gnostic star has seven rays.

Seven- and nine-pointed stars drawn in one line are mystical stars in astrology and magic.

The star of the magicians is read in two ways: sequentially along the rays (along the line of the star) and along the circumference. In the course of the rays, there are planets that control the days of the week: Sun - Sunday, Moon - Monday, Mars - Tuesday, Mercury - Wednesday, Jupiter - Thursday, Venus - Friday, Saturn - Saturday.

nine pointed star

Nine-pointed star of magicians

Nine-pointed stars, like seven-pointed ones, if they are drawn in one line, are mystical stars in astrology and magic.

The nine-pointed star, made up of three triangles, symbolizes the Holy Spirit.

Monad

The four constituent parts of a monad

It is a magical symbol called the monad by John Dee (1527–1608), advisor and astrologer to Queen Elizabeth I of England.

Dee presents the nature of magic symbols in terms of geometry and tests the monad in a series of theorems.

Dee explores the monad at such a deep level that he finds links to his theory with Pythagorean harmony, biblical knowledge, and mathematical proportions.

Spiral

Spiral structure of the Milky Way

Spiral shapes are very common in nature, from spiral galaxies to whirlpools and tornadoes, from mollusk shells to human finger prints, and even the DNA molecule has the shape of a double helix.

The spiral is a very complex and ambiguous symbol. But first of all, it is a symbol of the great creative (life) force both at the level of the cosmos and at the level of the microcosm. The spiral is a symbol of time, cyclic rhythms, the change of seasons, birth and death, the phases of "aging" and "growth" of the Moon, as well as the Sun itself.

Tree of Life

Tree of Life in a human being

Tree of Life

The Tree of Life does not belong to any culture - not even to the Egyptians. It is beyond race and religion. This image is an integral part of nature… Man himself is a miniature Tree of Life. He possessed immortality when he was associated with this tree. The Tree of Life can be thought of as the arteries of a large cosmic body. Through these arteries, as through channels, the life-giving forces of the cosmos flow, which nourish all forms of existence, and the cosmic pulse of life beats in them. The Tree of Life is a separate section, part of the scheme of the universal code of life.

Sphere

Armillary sphere (engraving from Tycho Brahe's book)

A symbol of fertility (like a circle), as well as integrity. In ancient Greece, the sign of the sphere was a cross in a circle - the ancient emblem of power. A sphere made up of several metal rings, illustrating the cosmogonic theory of Ptolemy, who believed that the Earth is at the center of the universe, is an ancient emblem of astronomy.

Platonic Solids

Platonic solids inscribed in a sphere

The Platonic solids are five unique shapes. Long before Plato, Pythagoras used them, calling them ideal geometric bodies. Ancient alchemists and such great minds as Pythagoras believed that these bodies are associated with certain elements: cube (A) - earth, tetrahedron (B) - fire, octahedron (C) - air, icosahedron (D) - water, dodecahedron ( E) - ether, and the sphere - emptiness. These six elements are the building blocks of the universe. They create the qualities of the universe.

Planet symbols

Planet symbols

The planets are depicted by a combination of the simplest geometric symbols. This is a circle, a cross, an arc.

Consider, for example, the symbol for Venus. The circle is located above the cross, which personifies a kind of "spiritual attraction" that pulls the cross up into the elevated areas belonging to the circle. The cross, subject to the laws of generation, decay and death, will find its redemption if it is raised within this great circle of spirituality. The symbol as a whole represents the feminine in the world, which is trying to spiritualize and protect the material sphere.

Pyramid

The Great Pyramids of Cheops, Khafre and Menkaure

The pyramid is a symbol of the hierarchy that exists in the universe. In any area, the pyramid symbol can help move from the lower plane of plurality and fragmentation to the higher plane of unity.

It is believed that the initiates chose the form of a pyramid for their shrines because they wanted the lines converging towards the top, rushing towards the Sun, to teach humanity the lesson of unity.

star tetrahedron

star tetrahedron

A star tetrahedron is a figure consisting of two mutually intersecting tetrahedra. This figure can also be perceived as a three-dimensional star of David.

Tetrahedra manifest as two opposite laws: the law of the spirit (radiation, bestowal, selflessness, selflessness) and the law of matter (drawing inward, cooling, freezing, paralysis). Only a person can consciously combine these two laws, since he is the link between the world of spirit and the world of matter.

The star tetrahedron thus represents the two poles of creation in perfect balance.

Universal symbols-images

A thing is not just because God wills it, but God wants it precisely because it is just.

Image symbols are often objects (things) or graphic images that imitate the shape of the creature or object with which they are associated. Their meanings are sometimes unexpected, but more often obvious, as they are based on some quality that these objects or creatures are originally inherent in: a lion - courage, a rock - stamina, etc.

arch, arc

Sacrifice to an astral deity (from a 13th-century Arabian manuscript)

The arch (arc), first of all, is a symbol of the vault of heaven, the god of Heaven. In the rites of initiation, passing through the arch means a new birth after the complete rejection of one's old nature. In ancient Rome, the army passed through the triumphal arch after defeating the enemy.

The arch and arc are common elements in the culture of Islam. Mosques often have arched entrances. It is believed that a person entering the mosque through the arched door will be protected by the symbolic forces of the spiritual (higher) sphere.

Ba-gua

Ba-gua and the Great Monad (charm against evil forces, China)

Ba-gua (in some sources, pa-kua) - eight trigrams and pairs of opposites, usually arranged in a circle, which symbolizes time and space.

Scales

Outweighing scales. The lung gives way. heavy overtights

Libra symbolizes justice, impartiality, judgment, evaluation of the merits and demerits of a person. A symbol of the balance of all opposites and complementary factors. Attribute of Nemesis - the goddess of fate.

Disk

Solar winged disk (Egypt)

The disk is a multifaceted symbol: a symbol of creation, the center of the Void, the Sun, Heaven, deity, spiritual and heavenly perfection. The disk of the rising Sun is a symbol of the renewal of life, life after death, resurrection. The disk of the Sun with the horned Moon or with horns means the union of solar and lunar deities, the unity of two in one.

The winged disk is a solar deity, the fire of Heaven, a combination of the solar disk and the wings of a falcon or eagle, the movement of the celestial sphere around the axis, transformation, immortality, the productive force of nature and its duality (protective and deadly aspects).

Wand, staff, scepter

Staff with hook and flail of Tutankhamen

The wand, staff, and scepter are ancient emblems of supernatural power.

The wand is a symbol of transformation associated with witchcraft and mysterious creatures. The staff is a symbol of male strength and power, often associated with the energy of trees, phallus, snake, hand (pointing finger). It is also an attribute of pilgrims and saints, but it can also mean knowledge, which is the only support of a person. The scepter is more ornate and is associated with higher deities and rulers, with spiritual power and at the same time compassionate wisdom.

Mirror

Divination scene depicted on the back of a bronze mirror (Greece)

It symbolizes truth, self-realization, wisdom, reason, soul, a reflection of the supernatural and divine intellect, reflected in the Sun, Moon and stars, the clearly shining surface of divine truth.

It is believed that the mirror has magical properties and is the entrance to the mirror world. If a mirror is hung with its reflective face down in a temple or over a tomb, it opens the way for the soul to ascend. In magic, mirrors serve to develop the gaze.

Snake Ouroboros (Oroboros, Ouroboros)

snake biting its own tail

The ring-shaped figure depicting a snake biting its own tail is a symbol of eternity, indivisibility, the cyclical nature of time, alchemy. The symbolism of this figure is interpreted in different ways, as it combines the creative symbolism of an egg (the space inside the figure), the earthly symbolism of a snake, and the heavenly symbolism of a circle. In addition, a snake biting its tail is a symbol of the law of karma, the wheels of samsara are the wheels of Incarnation.

Caduceus

Caduceus

Caduceus (Greek - "staff of the messenger") is often called the rod of Hermes (Mercury), the ancient god of wisdom. This is a “magic” wand with small wings, which is wrapped around two snakes, intertwined so that the bodies of the snakes form two circles around the wand, symbolizing the merging of two polarities: good - evil, right - left, light - darkness, etc., which corresponds to nature of the created world.

Caduceus is worn by all messengers as a sign of peace and protection, and it is their main attribute.

Key

Saint Peter with the keys to the gates of paradise (detail of a stone image, Notre Dame, Paris, 12th century)

The key is a very powerful symbol. This is power, the power of choice, breath, freedom of action, knowledge, initiation. The crossed golden and silver keys are the emblem of papal authority, the symbolic "keys to the Kingdom of Heaven" that Christ gave to the Apostle Peter. Although keys can both lock and unlock doors, they are almost always symbols of access, liberation, and (in initiatory rites) initiation, progression from one stage of life to another. In Japan, the keys to the rice vault are a symbol of prosperity.

Wheel

wheel of law

Wheel of existence (samsara)

The wheel is a symbol of solar energy. The sun is the center, the spokes of the wheel are the rays. The wheel is an attribute of all solar gods and earthly rulers. It also symbolizes the life cycle, rebirth and renewal, nobility, variability and changes in the material world (the circle is the limit of the material world, and the center is the “immobile engine”, the cosmic source of light and power).

The spinning wheel is associated with the cycles of manifestations (birth, death and rebirth) and the fate of man.

At the ordinary level, the wheel of Lady Luck (wheel of fortune) is a symbol of ups and downs and the unpredictability of fate.

Chariot

Antique hero on a chariot, symbolizing his readiness for battle

A dynamic symbol of power, power and speed of movement of gods, heroes or allegorical figures. The chariot is also a symbol of human essence: the charioteer (consciousness), using the reins (willpower and mind), controls the horses (vital forces) carrying the cart (body).

The chariot (in Hebrew - Merkaba) is also a symbol of the chain of descent from God through man into the world of phenomena and then the triumphant ascent of the spirit. The word Merkaba also means the body of light of a person.

Cauldron, bowl

Ritual cauldron (China, 800 BC)

Carl Jung sees the cup as a female symbol that takes and gives. On the other hand, the cup can be a symbol of a difficult fate ("bitter cup"). The so-called poisoned bowl promises hope, but brings trouble.

The cauldron is a more powerful symbol and is often associated with ritual and magic, representing transformative power. The cauldron is also a symbol of abundance, an inexhaustible source of life support, revival forces, the reproductive forces of the earth, the revival of warriors for a new battle.

Blood

Detail of the painting by Fey Pomeranes "The Sixth Palace of the Underworld": the last drops of blood, a symbol of life, flow out of a glass resembling an ankh in shape

Ritual symbol of vitality. In many cultures, the blood is believed to contain some of the divine energy, or more generally the spirit of the individual.

Blood is red solar energy. It embodies the principle of life, soul, strength, including rejuvenating. To drink someone's blood means to become related, but you can also absorb the strength of the enemy and thereby secure him after death. The mixing of blood is a symbol of union in folk customs (for example, blood brotherhood) or an agreement between people, as well as between a person and God.

labyrinth

Plan of a medieval labyrinth dance on the marble floor of the cathedral in Chartres (France)

The labyrinth symbolizes the world, the Universe, incomprehensibility, movement, a complex problem, an enchanted place. This is a symbol of mystery, mystery, which has many different interpretations, often contradictory, sometimes frightening.

Images of a labyrinth on houses are considered an amulet for protection from hostile forces and evil spirits.

Burials, burial caves and labyrinth-shaped burial mounds protect the dead and prevent them from returning.

Lotus

Vishnu and Lakshmi watching the creation: Brahma grows from a lotus flower originating from the navel of Vishnu

The amazing veneration of the lotus in various cultures is explained both by the extraordinary beauty of the flower, and by the analogy between it and the idealized form of the vulva as the divine source of life. Therefore, the lotus, first of all, is a symbol of fertility, birth and rebirth. Lotus is the source of cosmic life, a symbol of the gods who created the world, as well as the gods of the sun. The lotus symbolizes the past, present and future, as each plant has buds, flowers and seeds at the same time. This is a symbol of a noble person who has grown out of mud, but not stained with it.

Moon

Above - the growing moon and the full moon; below - the waning moon and the new moon

The moon is the ruler of the feminine. It symbolizes abundance, cyclical renewal, rebirth, immortality, occult power, volatility, intuition and emotion. The ancients measured time by the cycles of the moon; determined the timing of the onset of tides; predicted what the future harvest would be like.

Although the moon is usually symbolized in a positive way, in some cultures it is represented as an evil eye watching what is happening, associated with death and the ominous darkness of the night.

magic circle

Dr. Johann Faust and Mephistopheles (from The Tragic History of Dr. Faust by Christopher Mardlowe, 1631)

The magic circle is the basis of ceremonial magic. It serves as a symbol of the will of the magician and at the same time a protective barrier that protects the magician from the negative influence of the invisible world. In such a circle, all magical operations are performed. Different circles are used for different purposes. The drawing of a circle is a certain magical ritual that must be performed according to all established rules. In addition, it is believed that the inscription of magic circles and inscriptions contributes to the development of self-control and gait.

Mandala

The circle and square of the mandala represent the spherical shape of Heaven and the rectangular shape of the Earth. Together they symbolize the order of things in space and in the human world.

This is a geometric composition, symbolizing the spiritual, cosmic or psychic order. In Sanskrit, "mandala" means "circle". Even when this geometric composition is based on squares or triangles, it still has a concentric structure. The overall meaning of the composition remains unchanged and symbolizes the guiding mind, supernatural structures, the clarity of enlightenment.

Mandorla, or Vesica Piscis (surrounds the entire body of the person)

Mandorla, or Vesica Piscis

Image of an almond-shaped halo (radiance), which was used in medieval Christian art to highlight the figure of Christ ascending into heaven, and sometimes ascending saints.

In mysticism, "almond" (in Italian - mandorla) is a symbol of purity and chastity. The mandorla, due to its oval shape, was in antiquity the symbol of the vulva. It is also a graphic image of a flame, a symbol of spirituality. On the other hand, it symbolizes the dualistic unity of Heaven and Earth, depicted as two intersecting arcs.

Halo

halo buddha

A kind of halo: a luminous circle surrounding a person's head. The golden halo symbolizes the sanctity of the individual or confirms the fact that a person communicates directly with a higher plane.

The image of the halo is borrowed from the magical symbolism of the Egyptians, as evidenced by images from the ancient Egyptian Book of the Dead.

Nimbus

The halos and halos surrounding the heads of the saints symbolize the Light of God emanating from them.

Nimbus - a kind of halo: a luminous ring around the head. It symbolizes spiritual strength, as opposed to secular power, represented by the crown. Sometimes a halo is used as an attribute of the Phoenix bird as a symbol of solar power and immortality.

The halo can be blue, yellow or rainbow colored. In Greek mythology, a blue halo is an attribute of Zeus as the god of Heaven. The Romans have a blue halo - an attribute of Apollo and Jupiter. A triangular nimbus or a nimbus in the form of a rhombus means God the Father.

Sword

Inlaid swords found by Schliemann at Mycenae (Athens, National Museum)

The sword is one of the most complex and most common symbols. On the one hand, the sword is a formidable weapon that brings life or death, on the other hand, it is an ancient and powerful force that arose simultaneously with the Cosmic Balance and was its opposite. The sword is also a powerful magical symbol, the emblem of witchcraft. In addition, the sword is a symbol of power, justice, supreme justice, all-pervading reason, insight, phallic strength, light. The sword of Damocles is a symbol of fate. A broken sword is a defeat.

bird feather

Aztec feather headdress (drawing from the Codex Mendoza)

The bird feather symbolizes truth, lightness, Heaven, height, speed, space, soul, the element of wind and air, opposed to the principle of moisture, dryness, travel beyond the material world. In a broader sense, feathers worn by shamans, priests or rulers symbolized a magical connection with the spirit world or divine power and patronage. Wearing feathers or feathered hairstyles means taking the power of a bird into yourself. Two feathers symbolize light and air, two poles, resurrection. The white feather symbolizes clouds, sea foam and cowardice.

Horns

Depiction of a Persian king from the Sasanian period

The horns symbolize supernatural power, divinity, soul power, or the life principle arising from the head. Horns are both a solar and a lunar symbol. Being sharp and piercing, the horns are a phallic and masculine symbol; being hollow, they signify femininity and receptivity. Horned gods symbolize warriors, fertility for both people and animals. Horns with a long ribbon falling from them signify the god of the storm. In more recent times, horns have become a symbol of shame, contempt, depravity, and a deceived husband.

Hand

"Hand of Fatima" (Muslim carved pendant)

Power (worldly and spiritual), action, strength, domination, protection - this is the main symbolism that reflects the important role of the hand in human life and the belief that it is capable of transmitting spiritual and physical energy.

The hands of kings, religious leaders, and miracle workers are believed to have healing powers; hence the laying on of hands in religious blessing, confirmation and ordination. They bless with the right hand, they curse with the left. In Islam, the open palm of Fatima, the daughter of Muhammad, symbolizes the five pillars: faith, prayer, pilgrimage, fasting, mercy.

Sun

Variants of the image of the disk of the Sun

The sun is one of the twelve symbols of power, the main symbol of creative energy.

As a heat source, the Sun represents vitality, passion, courage, and eternal youth. As a source of light, it symbolizes knowledge, intelligence. In most traditions, the Sun is a symbol of the masculine. The sun is also life, vitality, the embodied character of the personality, the heart and its aspirations. Sun and Moon are gold and silver, king and queen, soul and body, etc.

tetramorphs

Image of Christ with tetramorphs in the corners (from a manuscript of the 12th-13th centuries)

Tetramorphs are considered a synthesis of the forces of the four elements. In some cults, these are four-headed guards of the four cardinal directions. In many traditions, they symbolize the universality of divine protection and protection from the return of primary chaos.

The four biblical tetramorphs have the heads of a man, a lion, an ox, and an eagle. Subsequently, in Christianity, these images began to be identified with the apostles - Saints Matthew, Mark, Luke and John, as well as with the incarnation of Jesus Christ, his resurrection and ascension.

Thyrsus

Thyrsus

Thyrsus is the rod of the Greek god of wine Dionysus (in Roman mythology Bacchus). It is a spear-shaped pole (originally from a hollow stem of dill) topped with a pine cone or bunch of grapes and entwined with vines or ivy. It symbolizes fertilizing, fertile power - both sexual and vegetative.

The bump is present on the thyrsus, probably because fermented pine resin was mixed with the wine that was drunk during bacchanalia - it was believed that this enhances sexual sensations.

Ax (axe)

Great Mother with a double ax in her hands (the ax here is a phallic symbol)

The ax is a symbol of power, thunder, fertility, rain brought by the heavenly gods, and stormy winds, correction of mistakes, sacrifice, support, help. It is also a common symbol of sovereignty associated with the ancient sun gods.

The double ax (double-sided ax) denotes the sacred union of the god of Heaven and the goddess of the Earth, thunder and lightning. Sometimes the blades of a double-sided ax, resembling crescents, symbolize the Moon or the unity of opposites. It is also a symbol of supreme power and strength.

Trident

Vishnu's trident as a symbol of his triune essence: creator, keeper and destroyer (from a painting from Rajasthan, 18th century)

The trident is the most famous symbol of power over the sea and an attribute of the ancient Greek god Poseidon (in Roman mythology - Neptune).

The trident symbolizes thunder and lightning, three flames, triple weapons - the forces of heaven, air and water. This is the weapon and attribute of all heavenly, thunder gods and goddesses of the storm, as well as all water gods, the strength and fertility of the waters. It can symbolize the Heavenly Triad, as well as the past, present and future.

Trigrams

Eight trigrams underlying the "Book of Changes"

Trigrams are a triple combination of continuous (yang) and broken (yin) lines. There are eight of them, and they formed the basis of the great Chinese book of predictions "The Book of Changes" ("I-Ching"). Trigrams symbolize the Taoist doctrine that the cosmos is based on constant flows of complementary forces: male (active, yang) and female (passive, yin).

Trigrams also personify the three essences of a person - his body, soul and spirit; irrational emotions, rational mind and suprarational intellect.

Trikvetra (three-pronged swastika)

Triquetra

The triquetra is largely swastika-inspired. This is also the movement of the Sun: at sunrise, at the zenith and at sunset. There have been suggestions about the connection of this symbol with the lunar phases and the renewal of life. Like the swastika, it is a symbol of good luck. He often appears with solar symbols; it can be seen on ancient coins, on Celtic crosses, where, as they say, this sign symbolizes the triad and is a symbol of the sea god Manannan. It is also present in Teutonic symbolism, where it is associated with Thor.

Triskelion

triskelion

The symbol of dynamic energy in the form of three legs connected together. It is similar to the swastika, but with three rather than four bent arms, creating a cyclic effect. As a motif in Celtic art and on Greek coins and shields, the triskelion has less to do with the solar and lunar phases (one of the suggested meanings) than with power and physical strength. In addition, the triskelion is a symbol of victory and progress.

Shamrock

Shamrock

Heraldic shamrock

The shamrock-clover symbolizes unification, balance, and also destruction. The sour shamrock, which the Arabs call shamrah, symbolizes the Persian triads. The shamrock is generally a symbol of triads, it is the Mystic Tree, the “solar wheel”. In Christianity, it is a symbol of the Trinity, as well as the emblem of St. Patrick and the coat of arms of Ireland.

To always be profitable, carry a dried shamrock with you.

Trimurti

Trimurti - the Indian Trinity (a sketch of a very ancient image on granite, India House Museum)

Holy Hindu Trinity - Brahma, Vishnu and Shiva. Symbolizes the three cycles of life: creation, preservation and destruction. Despite similarities to the Christian Trinity, the Trimurti is not a monotheistic concept of a "triune god".

Trimurti is sometimes depicted as a tortoise. She also symbolizes the Great Mother - both in her terrible manifestation (with symbols of flame and skulls), and in her beneficial (as Lotus, Sophia, Tara, as wisdom and sympathy).

Trinity

Symbol of the Trinity - Father, Son and Holy Spirit - as One God

The trinity differs from the triad in that it is unity, the union of three in one and one in three. It is a symbol of unity in diversity.

In Christianity, this is the Father, Son and Holy Spirit or Mary, Joseph and Jesus. The symbols of the Trinity are the hand (symbol of the Father), the lamb (symbol of the Son), and the dove (symbol of the Holy Spirit).

The Trinity is symbolized by the colors yellow, red and green; three qualities - Love, Faith and Hope.

Human

Symbolic representation of man as the Universe: a square in a circle (China)

The crown of all living things. A symbol of what is capable of improvement. Created in the image and likeness of God, it combines the material and the spiritual, the heavenly and the earthly. This is a microcosm, symbolically containing all the elements of the universe (macrocosm). The human body in the Pythagorean tradition is depicted as a pentagram consisting of arms, legs, and a head. In man, three principles are united together, which modern scientists call the body, life and will. Symbolically, this can be represented by three points (beginnings) enclosed in a circle.

Universal symbols-concepts

The knowledge of ideas reveals in temporal phenomena their timelessly eternal meaning.

Andrey Bely

Concept symbols are numbers or geometric shapes that reflect ideas, feelings, or abstract qualities of something directly related to the inner world of a person.

Duality of the world

Solomon's Double Triangle Diagram: God of Light and God of Reflection

The duality of the world - the interaction of the two polarities behind the created universe (light and darkness, good and evil, etc.) - is reflected in many symbols. The most famous of these is the yin-yang symbol. Also of interest are the symbols presented by the famous occultist Eliphas Levi, such as the "Double Triangle of Solomon" diagram.

The main symbol used by people far from the occult to depict duality is the most common number two, although, nevertheless, it also has a magical nature.

Yin-yang (principle)

Yin-yang sign

The Chinese call the symbol "yin-yang" Tai Shi - the circle of existence. The circle is divided into two equal parts by an S-shaped curve: dark, feminine (yin), and light, masculine (yang). The circle seems to rotate, the darkness is replaced by light, and then the light is replaced by darkness. The Chinese claim that even in the purest light there is an element of darkness, and vice versa. Therefore, in the center of each part, a small circle of the opposite color is depicted: black on a white background and white on a black one. This image symbolizes the balanced dynamism of opposing forces and principles in the cosmos.

Rays

Sun with zigzag rays (golden mask of the Incas)

It is a symbol of fertilizing power, holiness, spiritual enlightenment and creative energy, creative power. The rays can depict the hair of the sun god, a manifestation of the divine essence, or a radiance (halo) emanating from the saints. In solar symbolism, the seventh ray is the main path to heaven.

Wisdom

Ancient Greek goddess of wisdom Athena (in Roman mythology Minerva) with a coiled snake at her feet

The main symbols of wisdom are the snake (daytime, solar, but feminine flexible male sign) and the owl (night, lunar, acting imperceptibly, silently, but resolutely and quickly masculine female sign). It is the combination in each of them of the most important properties of the male and female principles that very accurately corresponds to wisdom. Other symbols of wisdom: dragon, griffin, peacock, sphinx, unicorn, bird, bee, rat, lotus, heart, number seven, scepter, scroll, ring, etc.

“Out of many roses, a drop of oil; out of many torments, a drop of wisdom” (Persian saying).

world axis

Tet of Osiris

In the esoteric tradition, the symbols of the axis of the world, the World Tree, are the spear, sword, key and scepter.

The Egyptians as a symbol of the world axis and the North Pole use Tat (or Tet) - the spine of Osiris, which, in addition, personifies stability, strength, immutability, preservation.

Light

Light emanating from the Buddha

Light is the first creation. It is associated with the beginning and the end. Light and darkness are two aspects of the Great Mother: life and love, death and burial, creation and destruction.

The light of the Sun personifies spiritual knowledge, and the reflected light of the Moon represents rational, analytical knowledge.

Light is usually depicted as straight or undulating rays, the disk of the Sun, or a halo. As a rule, a straight line represents light, and a wavy line represents heat. Light and heat symbolically complement each other and are the two poles of the element of Fire.

Death and rebirth

Death and rebirth of human beings. Detail of symbolism on a gravestone in Dieste (Belgium)

This image in Christianity is expressed by ancient complex symbols. The above composition combines two pairs of "circle-cross", each pair personifies death and rebirth. The lower pair is represented by crossbones and a rounded skull (a symbol of death). From the lower circle (skull) grows a cross similar to the one on which Christ died - the cross of resurrection, rebirth. This whole allegory is inscribed in a larger circle - a sign that the death and rebirth of human beings are within the great spiritual sphere of the cosmos.

Consciousness (three aspects)

Symbols representing the three aspects of consciousness

Usually the three aspects of consciousness are depicted as three animals: one of them lives underground, the other on the ground, and the third flies above the ground. The animal that lives underground represents the microcosm; that which flies in the air is the macrocosm; and the animal that walks the earth represents the middle step between the first two - like us, for example. The most common symbols are: in Egypt - a cobra, the right eye of Horus, a hawk; in Peru - rattlesnake, puma and condor; the American Indians have a rattlesnake, a mountain lion and an eagle; in Tibet - a snake, a pig and a rooster.

Dance

Dervish dance (the grace of God descends to the dancer through the raised hand, penetrates through his body and spirit and, leaving him, unites with the earth through the lowered hand)

The main symbolism of the dance: cosmic creative energy, the transformation of space into time, the rhythm of the universe, the imitation of the divine "game" of creation, the maintenance of strength, emotions, and activity.

Circle dances are an imitation of the movement of the Sun in the sky. Dancing in a chain is a symbol of the connection between a man and a woman, Heaven and Earth. When the dance is performed around an object, it is thereby closed, enclosing in a magic circle, protecting and giving strength.

Shadow

Priestly Esotericism: The Sign of Anathema (from Eliphas Levi's Transcendental Magic, 1896)

A symbol of the negative beginning, as opposed to the positive solar. In some primitive tribes, the shadow symbolizes the human soul, the same is in witchcraft and conspiracies. To fall into the shadow of another person is a bad omen.

The above engraving depicts a human hand performing an act of blessing. A strong beam of light casts a shadow from the blessing hand on the wall, and this shadow is the image of the horned head of the Devil. The main idea of ​​the allegory is this: evil and good are intertwined, and darkness and light oppose each other in a kind of moral duel.

Emblems of modern religions

It is difficult to find the Creator and Father of this universe, but even having found Him, it is impossible to express Him in a language understandable to everyone.

In our time, there are three world religions on the globe - Christianity, Islam (Muslim) and Buddhism. Each of them is accepted in many countries. They arose a long time ago: Christianity is 2000 years old, Islam is almost 1400 years old, and Buddhism is about 2500 years old.

There are other religions, which, although not global, are also widespread.

Christianity

Cup and cross

One of the symbols of Christ's love is the combination of the cup and the cross. The cup, or goblet, in this case indicates the great suffering that Jesus endured, calling it "the cup."

The image of the bowl indicates the prayer of Jesus in the Garden of Gethsemane: “Father! Oh, that You would be pleased to carry this cup past me! however, not my will, but yours be done.”

The cross is depicted as pointed. Its sharp ends, like swords of sorrow and pain, pierce the suffering soul.

Islam

Star and crescent of Islam

The main emblem of the youngest world religion, Islam, founded by the Prophet of Allah, Muhammad (570-632), is a crescent with a star inside. The emblem denotes divine patronage, growth, rebirth, and, together with the star, paradise. The star is a traditional symbol of independence and divinity. The crescent is one of the true forces capable of resisting evil, a powerful talisman.

The crescent in Islamic countries replaces the cross in Red Cross organizations.

Buddhism

Maitreya

In Buddhism, Maitreya is the name of the Buddha of the coming world order. This is the only Bodhisattva ("whose essence has become the mind"), which is recognized by all major branches of Buddhism. The essence of a Bodhisattva is the act of sacrifice: giving up the bliss of nirvana in order to help humanity within the limits allowed by karmic limitations.

Maitreya is depicted sitting on a throne in a "European pose" (with legs down), which indicates a sign of the haste of his arrival; it is golden in color. Next to Maitreya, it is customary to depict the wheel of dharma, a stupa and a vase.

Judaism

Mogendovid, or Shield of David

Judaism is the oldest of the monotheistic world religions (it arose in the 1st millennium BC in Palestine 4000 years ago). The main provisions of Judaism were later incorporated into Christianity and Islam.

The symbol of Judaism is Mogendovid, or the Shield of David. Most commonly associated with the six-pointed Star of David. A less common name is the Star of the Creator; each end of the star symbolizes one of the six days of creation, and the central hexagon symbolizes Shabbat (the holy day of rest).

Zoroastrianism

Ahura Mazda

Zoroastrianism is an ancient spiritual tradition, founded about 2500 years ago by the prophet Zoroaster, and now, unfortunately, forgotten. The supreme god is Ahura Mazda. The sacred canon is the Avesta ("law").

Zoroastrianism proceeds from the doctrine of the justice of the world order and the triumph of justice in the world struggle between good and evil, in which the free choice of a person, his active participation play a decisive role. Zoroastrian morality contains an ethical triad: a good thought, a good word, a good deed.

Hinduism

One of the symbols of the Trimurti

Hinduism combines elements of different creeds, rooted in the mists of time. Sacred books - Vedas (Rigveda, Samaveda, Yajurveda, Atharvaveda). The three main gods make up the Trimurti (triad): Brahma is the creator of the world, Vishnu is the keeper of the world and Shiva is the destroyer. Their images symbolize the fundamental processes of change in nature (prakriti).

The basis of Hinduism is the doctrine of the reincarnation of souls (samsara), which occurs in accordance with the law of retribution (karma) for virtuous or bad behavior.

Confucianism

The symbol of Confucianism is the figure of the “Highest Saint” himself

Confucianism and Taoism are the most famous of the philosophical currents that existed in China even before its unification (221 BC). Gradually intertwined with the traditions of Buddhists and Taoists, the teachings of Confucius acquired a religious connotation. According to Confucius, one must live in such a way that human behavior reflects the laws of the universe, which exists according to a certain order. “The master teaches his students the four disciplines: culture, behavior, loyalty and faith” (Lun Yu, 7.25).

Taoism

Tai Chi (yin-yang circle)

Taoism literally means "School of the Tao". (Tao means “way”). This is an integral part of the philosophical and religious triad (Buddhism, Confucianism, Taoism). The Chinese apply all three teachings in practice, depending on the life situation. As part of his personal life, the Chinese profess Taoism, but when it comes to social norms of behavior, he becomes a Confucian, and when faced with troubles and life's hardships, he turns to Mahayana Buddhism.

Graphically, the concept of Taoism is expressed by tai chi (in some sources - Tai Shi) - a symbol of a single limit.

Shinto (Shinto)

Horin-rimbo - the wheel of the law (Japan)

Shinto is the Japanese national religion, its name comes from the Chinese word "shen-dao" ("sacred path" or "path of the gods"). At the heart of Shinto is the cult of the deities of nature and ancestors. The highest deities are Amaterasu (goddess of the Sun) and her descendant Jimmu. Jimmu is the first emperor of Japan, the mythical ancestor of Japanese emperors. Day February 11, when, according to myths, in 660 BC. e. Jimmu ascended the throne, is considered the founding day of the empire and is celebrated as a holiday.

Crosses: the most common forms

The cross is a cosmic symbol that should be studied and treated with the greatest respect.

"The Science of Initiation"

The common symbol of humanity is the cross. It can be found in the most ancient religions, among the most ancient civilizations: in Mesopotamia, Egypt, China, etc. Who invented the cross? Nobody - because it exists in nature. This is an ancient universal symbol and, above all, a symbol of the connection of micro- and macrocosm, spirit and matter in their combination. The cross symbolizes the involvement of the spirit (vertical line) in time (horizontal line).

The forms of the cross are various. They differ in the number of crossbars, and the number of ends of the cross, and proportions.

Greek cross

Greek cross

The cross of the simplest form: square, with ends of equal length, the horizontal crossbar is located in the middle of the vertical one. Cross of Saint George. This sign, also called crux quadrata, has been used since prehistoric times in a variety of meanings - as a symbol of the god of the sun, the god of rain, the elements from which the world is created: air, earth, fire and water. In early Christianity, the Greek cross symbolized Christ. It is also a symbol of secular, earthly power, but received from God. Used in medieval heraldry.

cross hammer

cross hammer

The hammer cross is a variation of the Greek cross. One of the main heraldic crosses, so named from the French potenee - "support", because its shape is similar to the supports used in antiquity.

latin cross

latin cross

Another name for the Latin cross is the long cross. Its horizontal bar is located above the middle of the vertical bar. This is the most common Christian symbol in the Western world. It is believed that it was from such a cross that Christ was taken down, hence his other names: the cross of the Crucifixion, the cross of the West, the cross of Life, the cross of Suffering. This form, so similar to a man with outstretched arms, symbolized God in Greece and China long before the advent of Christianity. For the Egyptians, the cross rising from the heart symbolized kindness.

Cross of Saint Peter

Cross of Saint Peter

The cross of St. Peter is an inverted Latin cross. Since the 4th century, it has been one of the symbols of Saint Peter, who is believed to have been crucified upside down on an inverted cross in 65 CE. e. during the reign of Emperor Nero in Rome.

An inverted Latin cross, that is, the cross of St. Peter, with pointed ends is the emblem of the Knights Templar.

St. Andrew's Cross (oblique cross)

St. Andrew's Cross (oblique cross)

It is also called diagonal or oblique. On such a cross, the apostle Saint Andrew was martyred. The Romans used this symbol to mark the border, the passage beyond which was forbidden. The oblique cross also symbolizes perfection, the number 10. In heraldry, this cross is called the saltire.

St. Andrew is the patron saint of Russia, and when Peter the Great created the Russian navy (in the 1690s), he adopted a blue oblique cross on a white background for the flag of the fleet.

Tau Cross (St. Anthony's Cross)

tau cross

Cross of Saint Anthony

The tau cross is so named because of its resemblance to the Greek letter "T" (tau). It symbolizes life, the key to supreme power, the phallus. In ancient Egypt - a sign of fertility and life. In biblical times - a symbol of protection. The Scandinavians have Thor's hammer. In Christian churches - the cross of St. Anthony (the founder of Christian monasticism, IV century). Since the beginning of the XIII century - the emblem of Francis of Assisi. In heraldry, this is the Almighty Cross. Also known as the "Gallows Cross" due to its resemblance to the gallows, as it was made in antiquity.

Ankh (Egyptian cross)

Ankh - the key to the gates of death

The ankh is the most significant symbol among the ancient Egyptians, also known as the "cross with a handle." This cross combines two symbols: a circle (as a symbol of eternity) and a tau-cross suspended from it (as a symbol of life); together they denote immortality, eternal life. Ankh also personifies "the life that will come", "the time that will come", hidden wisdom, the key to the secrets of life and knowledge, as well as the key that opens the gates of death. Perhaps it symbolizes the Tree of Life, as well as the sun rising over the horizon.

Maltese cross

Maltese cross

The Maltese cross is also called the eight-pointed. It symbolizes the four great gods of Assyria: Ra, Anu, Belus and Hea. Emblem of the Knights of Malta. The white cross of this form on a black background was from the very beginning the emblem of the military and religious order of the Hospitallers (Johnites), who moved their headquarters to Malta (in 1529) - hence the name.

In philately, the Maltese cross is the first postmark used to cancel postal items from 1840 to 1844.

Patriarchal Cross

Patriarchal Cross

The patriarchal cross is used by archbishops and cardinals. It is also called the cardinal's catholic cross and the two-bar cross. The upper crossbar is a titulary (board for writing a name), introduced by order of Pontius Pilate. Under the name of the archbishop's cross, it is often found on the coats of arms of archbishops.

This cross is widespread in Greece and is sometimes called Angevin or Lorraine. It is sometimes erroneously called the Lorraine cross.

papal cross

papal cross

The papal cross with three horizontal bars is also known as the triple cross. Used in processions in which the pope participates. Three cross lines symbolize power and the Tree of Life.

Russian cross

Russian cross (cross of St. Lazarus)

This eight-pointed cross is the cross of the Russian Orthodox Church. It is also called the eastern cross or the cross of St. Lazarus. Symbol of the Orthodox Church in the Eastern Mediterranean, Eastern Europe and Russia.

The upper of the three transverse crossbars is the titulary, where the name was written, as in the patriarchal cross, the lower crossbar is beveled.

Cross of Constantine (sign "Chi-Rho")

Cross of Constantine

Magic seal with the symbol "Chi-Rho" (Agrippa, 1533)

The Cross of Constantine is a monogram known as "Khi-Rho" ("chi" and "ro" are the first two letters of Christ's name in Greek). The legend says that the emperor Constantine saw this cross in the sky on the way to Rome, along with the cross he saw the inscription "Conquer this". According to another legend, he saw the cross in a dream the night before the battle and heard a voice: “With this sign you will win”). It is said that it was this prediction that converted Constantine to Christianity. And the monogram became the first generally accepted symbol of Christianity - as a sign of victory and salvation.

Rosicrucian cross

Cross with a rose (Rosicrucian)

Another name is the cross of the rose (five-petal). Emblem of the Rosicrucian Order. Symbol of harmony, center, heart. The rose and the cross also symbolize the Resurrection and Atonement of Christ. This sign is understood as the divine light of the Universe (rose) and the earthly world of suffering (cross), as the feminine and masculine, material and spiritual, spiritual and sensual love. The cross with a rose is a symbol of the initiate, who, thanks to the work on himself, has managed to develop in himself love, life-giving and transforming matter.

Masonic cross

Masonic cross (cross in a circle)

The Masonic cross is a cross inscribed in a circle. It means a holy place and a cosmic center. The four dimensions of space in the celestial circle symbolize the totality that includes the Great Spirit. This cross represents the Cosmic Tree extending horizontally over the Earth and touching the Heavens through the vertical central axis. Such a cross was either made in stone or depicted on the walls of Roman Gothic temples, symbolizing their sanctification.

Pacifist Cross

Pacifist cross (peace cross)

This symbol was designed by Gerald Holtom in 1958 for the then emerging movement for nuclear disarmament. To develop the symbol, he used the semaphore alphabet: he made a cross from its symbols - for "N" (nuclear, nuclear) and "D" (disarmament, disarmament) - and placed them in a circle, which symbolized a global agreement. Soon this cross became one of the most common signs of the 60s of the twentieth century, symbolizing both peace and anarchy.

images of time

The wise turn years into months, months into weeks, weeks into days.

Everything is perishable in this world.

The image of inexorable time is the road. The symbol of time is sand flowing through the fingers. Attributes of measured time - a clock, a burning candle; it is a symbol of the elusiveness of the present moment.

In the pantheon of gods of almost all ancient cultures, there is also the god of Time.

Abraxas

Abraxas - a symbol of time (gnostic gem)

Abraxas is the personification of the divine cycles of the solar year. This is the mystical image of the Supreme Being, the highest of the seven. It consists of five emanations (radiations): Nus (Mind), Logos (Word), Phronesis (Reason), Sophia (Wisdom), Dynamis (Power). The human body in the image represents God. The two serpent-supports emerging from it are Nus and Logos (intuition and quick understanding). The head of a rooster means foresight and vigilance (mind). Two hands hold the symbols of Sophia and Dynamis: the armor of wisdom and the whip of power.

Kalachakra

Namchu-vanden - emblem of Kalachakra

Kalachakra - literally "wheel of time", "course of time". Secret Doctrine in Vajrayana Buddhism. An astrological and astronomical system that entered Tibet from India. Kalachakra introduces the concept of the cyclical nature of time with periods of 12 and 60 years (Tibetan calendar). According to legend, the Kalachakra teaching was given by the Buddha Shakyamuni. According to other sources, this teaching was brought to Tibet by Pitop, or the Great Kalachakrapada, who, having miraculously found himself in Shambhala, was initiated there by the king of Kalki into the teachings of Kalachakra.

Kronos

Kronos (Rom. Saturn), XV century

The ancient Greek symbol of time - the titan Kronos - became the ancestor of many words in Russian (the particle “chrono” is part of compound words indicating their relationship to time): chronic, chronology, chronometer, etc.

Kronos (Rom. Saturn) - the god of Time, in the form of a fading autumn or the departing Sun, sometimes, along with his sickle, also has a hood, which symbolizes invisibility, death and retreat. Since the hood covers the head, it also signifies thought and spirit.

Ourobor (serpent that bites its own tail)

Ourobor as an Emblem of Death (from George Wheater's A Collection of Emblems, Ancient and Modern, 1635)

The most obvious meaning of the symbol is associated with the concept of time: the passage of time is accompanied by destruction, since the past seems to be irretrievably lost. This is reflected in the fact that the serpent "devours" its own tail, like time, which apparently consumes itself. It can be said that time has a cyclical nature (day follows night, the seasons repeat themselves, etc.), and this has found its expression in the form of a snake, in the fact that he curled up in a circle. The symbol of the emblem can be expressed by the phrase: "In my beginning lies my end" or "The end is in the beginning."

Tempus

Image of Time - Tempus (Rome)

The Romans depicted time in the form of a male winged figure with goat legs, with a scythe in his hands (“the inexorable scythe of time”) - this is Tempus (from Latin tempus - time).

The figure of Tempus personifies the frailty and transience of all living things, and therefore is associated with the symbol of Death.

"Clock" of our body

The “clock” of our body (the numbers in the inner circle are the time of day)

The Chinese consider it useful to influence the organs of the body in a strictly defined period of the day (stimulate - during activity, and vice versa).

Twelve major organs, according to medical practice, have two hours of activity (see figure). Designations: GB - gallbladder: (from 23:00 to 1:00); Liv - liver; Lu - lungs; Li - large intestine; St - stomach; Sp, spleen; H - heart; Si, small intestine; UB, urinary bladder; K, kidneys; P - brain; TW - spinal cord.

Symbolism of the plant kingdom

The beauty of plants is the common heritage of the world, that is, it is always macrocosmic, not microcosmic.

The symbol of the plant kingdom is the tree. Its branches, representing diversity, depart from a common trunk, which is a symbol of unity. A green, flowering tree is a symbol of life; dead, withered - a symbol of death. An old, gnarled tree can mean wisdom and strength.

The flower is the emblem of the cycle of birth, life, death and rebirth. Flowers represent beauty (especially feminine), innocence, divine blessing, spring, youth, but also the brevity of being. Everything in a flower can carry a certain symbolism: its shape, and the number of petals, and color, and smell ...

Vine

Ornament - vine motif

Grapes are one of the oldest symbols of fertility, abundance and vitality. The vine is one of the symbols of Christ. The importance of wine in many religious rituals is based on the symbolic association of grapes with divine blessings. The vine was the first plant Noah planted after the flood.

Grape juice resembles human blood. In some mysteries, grapes are a symbol of lust and debauchery, greed and drunkenness. The bunch of grapes is sometimes presented as a phallic symbol. But grapes are also seen as a symbol of the solar spirit.

Cherry

Sakura (19th century Japanese print by Utagawa Kunisada)

In Christian iconography, cherry is sometimes depicted instead of an apple as a fruit from the Tree of the Knowledge of good and evil; sometimes Christ is depicted with cherries in his hand. In China, the cherry tree is a symbol of good luck, spring (due to early flowering) and virginity; The vulva is called the "spring cherry". Cherry blossoms (sakura) - a symbol of Japan; it is cultivated as an ornamental tree; its fruits are inedible. The Japanese identify cherry blossoms with the rising sun. Cherry, in addition, is the emblem of the samurai.

Pomegranate

Popping pomegranate

The opening pomegranate (fruit) symbolizes the Easter Resurrection of Christ, gives Christians confidence in forgiveness, faith in the life to come and resurrection. Due to the abundance of seeds, pomegranates are a symbol of fertility. It is also the ancient eastern emblem of the sun god and the emblem of life, a divine symbol called the "forbidden secret."

The remains of a flower (thorn) on the upper part of the fruit serve as an image of a crown in heraldry. The pomegranate is always depicted as golden. And pomegranate seeds are always twelve - a number symbolizing perfection since ancient times.

Oak and acorn

Acorn

Oak is a symbol of power, endurance, longevity and nobility, as well as glory. In ancient Rome, a wreath of oak leaves was the highest award for a victorious commander.

As an emblem of valor and courage, oak (oak leaf, oak branch, oak wreath, oak garland) is used in military insignia in many countries.

Oak with acorns - the emblem of maturity, full of strength. An oak without acorns is the emblem of youthful valor. The acorn is a symbol of fertility, prosperity, spiritual energy growing from the grain of truth.

Kabbalistic Tree

Kabbalistic Tree (drawing from the book by R. Fludd, 1574–1637)

This is an inverted Cosmic Tree. Its crown touches the ground, and its roots are fixed in the spiritual world and feed on the spiritual energy of the sky, spreading it to the outer world and down. This is a favorite image in Kabbalism and other mystical and magical teachings. It testifies that human life is the descent of the spirit into the body and back. It is also a symbol of philosophical growth, growth inwards.

In the Bhagavad Gita, an inverted tree means the origin of everything from a single root, in Islam it is a symbol of happiness and good luck.

Cypress

Seven cypresses and twelve branches - the personification of the universe and its eternal truths (Istanbul, Turkey)

In the West, cypress is a mystical symbol of death and mourning, the personification of sadness and sorrow, as it was used for embalming the body and for making coffins. In Asia, it is a symbol of longevity and immortality. The Arabs call the cypress the Tree of Life. In Greece, the cypress has always had a dual reputation: it was a symbol of the gloomy god of the underworld, Hades, but at the same time, more cheerful gods - Zeus, Apollo, Aphrodite and Hermes. Therefore, it has become a symbol of rebirth and life after death. In China, the smoke of cypress branches is a symbol of light forces, a talisman against misfortune.

Clover

Clover with four petals

The three-leaf clover (shamrock) is a symbol of the Christian Trinity. The rare quatrefoil is a symbol of good luck; there is a belief that Eve took one quatrefoil as a memory of the lost paradise. But a five-leaf clover brings bad luck.

In China, clover is the emblem of spring. The Irish use clover leaves as their national emblem, which probably goes back to the veneration of this plant by the Celts for its active growth in spring.

Roots

Seed and roots

A symbol of connection with the earth, with the family.

"A man with roots" - they say about a man who stands firmly on his feet.

"Look at the root" - pay attention to the most essential, delve into the essence.

The "root of evil" is the source, the core of evil.

"Uproot" - take away life, cut off access to food, radically solve the problem.

laurel

Laurel wreath

Laurel symbolizes immortality, but also triumph, victory and success. He personifies peace, purification, protection, divinity, secret knowledge. According to ancient Greek myth, Apollo, the god of the Sun, dawn and poetry, pursued the nymph Daphne, who, running away from him, turned into a laurel bush (in Greek, "laurel" - "daphne"). In the arms of Apollo was a tree, with the branches of which he decorated his head and lyre. That is why in ancient Greece musicians, poets, dancers, whose patron was Apollo, were awarded with laurel wreaths. The Romans extended this tradition to military victors.

Lily

Fleur-de-lys, coat of arms of the French kings

One of the most versatile and even controversial characters. The triple lily is a symbol of the Trinity and three virtues: Faith, Hope and Mercy. Lily is an attribute of many saints, including the Archangel Gabriel. White lilies can sometimes symbolize death. The lily is also associated with fertility and erotic love because of its arrow-shaped or spear-shaped (phallus-like) pistil and specific strong fragrance. Lily - a sign of prosperity and royal power in Byzantium, later - the emblem of the French kings.

Palm branch

Palm branch

This is the main symbol of victory and triumph ("palm tree").

In ancient Greece, a palm branch was given along with a wreath to the winner of the Olympic Games as a personal wish for health and longevity. In ancient Rome, they were also awarded to victorious soldiers and gladiators. On Palm Sunday celebrations in Jerusalem, priests hand out consecrated palm leaves in the form of a cross. In Russia, they are replaced by willows. The palm branch is a symbol of longevity and one of the emblems of the world, and unlike the dove, it is a secular emblem.

Rose

ten petal rose

The rose has polar symbolism: it is heavenly perfection and earthly passion, time and eternity, life and death, fertility and virginity. It is also a symbol of the heart, the center of the universe, the cosmic wheel, divine, romantic and sensual love. Rose - completeness, the mystery of life, its focus, the unknown, beauty, grace, happiness, but also voluptuousness, passion, and in combination with wine - sensuality and seduction. The rosebud is a symbol of virginity; withered rose - the transience of life, death, sorrow; its thorns are pain, blood and martyrdom.

roses heraldic

Heraldic roses: 1 - Lancaster; 2 - Yorks; 3 - Tudors; 4 - England (badge); 5 - German Rose Rosenov; 6 - Russian stamp

The heraldic medieval rose has five or ten petals, which connects it with the Pythagorean pentad and decade. A rose with red petals and white stamens is the emblem of England, the most famous badge of the English kings. After the "War of the Scarlet and White Roses", named after the breastplates of the families that fought for the English crown, the scarlet rose of Lancaster and the white rose of York were combined in the form of the "Tudor Rose". The bright crimson rose is the unofficial emblem of Bulgaria. The famous tea rose is the emblem of Beijing. Nine white roses are in the coat of arms of Finland.

sprouts

Fern sprouts (four-part scheme)

Sprouts (heart-shaped scheme)

The sprout is a symbol of the awakening of life. The simplest species is a grain “hatching out of its shell”, a sprout resembling a folded fern leaf. These images are accompanied by a rounded or heart-shaped stripe. The heart-shaped scheme (point up) is a stable expression of the agrarian ornament. A four-part composition with fern sprouts (a sacred plant among many peoples) is widely used, the leaves of which are directed in all directions.

Pumpkin

Painted gourd, vessel and talisman (China, 19th century)

The gourd in Chinese culture is a symbol of health, wisdom, and even the entire universe.

In America, a pumpkin is the main attribute of the traditional holiday of evil spirits - Halloween. For this holiday, faces are carved on pumpkins, and candles are inserted inside the pumpkins and they go home with such “lamps”.

In humiliated symbolism, a pumpkin is called a head.

Thistle

Thistle

Emblem of Scotland

Thistle means challenge, asceticism, vindictiveness, misanthropy. Donkey food. It also symbolizes sin, sorrow, the curse of God when expelled from paradise; according to Genesis, Adam was punished with thistles. In Christian art, the thistle is the emblem of martyrdom.

But there is another side to the thistle symbolism. Like some other thorny plants, it is considered a talisman and is endowed with the ability to heal wounds. It is a plant with strong magical properties.

Apple tree, apple

Sovereign apple - one of the symbols of monarchical power

The apple tree is a symbol of fertility, one of the symbols of Mother Earth. Blooming apple tree - eternal youth, and in China - peace and beauty. An apple is a symbol of bliss, especially sexual, a symbol of restoration of potential, integrity, health and vitality. The apple represents love, marriage, spring, youth, longevity or immortality, in Christianity it is associated with temptation, the fall of man and his salvation. A bitten apple is a symbol of sin, anarchy, but also knowledge and hope. In art, an apple in the mouth of a monkey or a snake is a symbol of original sin.

Symbolism of the animal kingdom

The animal kingdom in its different breeds embodies the different impulses of the human psyche.

N. P. Rudnikova

In the human mind, animals (animals, birds, fish, insects, etc.) act as symbols, on the basis of which figurative pictures of certain aspects of being are compiled. The symbolism of animals also extends to the highest foundations of man himself (for example, ideas about the soul are expressed in the form of a bird).

The ancient Egyptians believed that certain animals could embody cosmic and divine energies. The twelve animals of the zodiac are archetypal symbols and represent a closed cycle of energies.

Stork

“He who has gained immortality flies on a stork into the sky” (stork and crane are symbols of immortality)

The stork symbolizes new life, the arrival of spring, good luck, daughter or son affection. In Christianity, the stork personifies purity, chastity, piety, vigilance. In the East, the stork is a symbol of immortality. Among the Slavs, the stork is an ancient totem bird, a symbol of the motherland, family well-being, home comfort, love for one's home. The punishment for destroying a nest or killing a stork is a fire that incinerates the killer's house or himself. There is a belief that a stork brings newborn babies. A stork carrying a baby is a symbol of christening.

Butterfly

Butterfly image

At present, the symbolism of the butterfly is dominated by the meaning of the anemone, a carefree creature, but also pure joy. In ancient times, it was presented as a symbol of transformation and immortality due to its life cycle: life (bright caterpillar) - death (dark chrysalis) - rebirth (free flight of the soul). The butterfly is a symbol of the soul in many regions of the world. In China, it is a symbol of light entertainment and a sign of lovers. In Japan, a butterfly is a symbol of a fickle and windy lover, as well as female fussiness and the craft of a geisha; two butterflies - marital happiness.

Ram (ram)

ram head

One of the most important symbols and one of the most widespread emblems in the world (variants: lamb, golden fleece, ram's head, ram's horns). The ram symbolizes fire, solar energy, passionate passion, courage, impulsiveness, stubbornness. In many cultures since ancient times it means male power and sexual potency. The symbol of the elements - both creative and destructive, requiring sacrifice.

In the modern everyday sense, the word "ram" is often synonymous with stupidity or stupid stubbornness.

Bull

Sacred bull Apis (Egypt)

Symbol of the fertility of the earth. The most common symbol of sexual power, as well as violence and rage. This is the embodiment of power, power, male fertility. A symbol of divinity, royalty, the elemental forces of nature, which changed meanings in different eras and in different cultures. The bull's horns are a sign of the full moon, its huge body is the pillar of the world in the Islamic and Vedic traditions; its abundant seed is nourished by the Moon in Iranian mythology; his bellowing, hoof-stomping, and horn-shaking are universally associated with thunder and earthquakes.

Wolf

She-wolf feeding Romulus and Remus, the founders of Rome (bronze figure, 6th–5th century BC)

The symbolism of the wolf is dual.

Negative symbolism: ferocity, deceit, greed, cruelty, evil, gluttony and sexuality. Stories of witches turned into wolves and men turned werewolves epitomize the fear of demonic possession and male violence.

Positive symbolism: courage, victory, concern for the food of the family. The wolf is a common symbol of knowledge through experience, the emblem of warriors.

AT heraldry the wolf is a symbol of anger, gluttony, greed.

Raven, crow

The raven on the shield is a symbol of purification

“Daughters of Jerusalem! I am black, but beautiful ”(alchemical symbol)

Raven and crow have similar symbolism. On the one hand, crows are associated with war, death, desolation, evil and misfortune. Because of their blackness, they are considered symbols of chaos and darkness that preceded the light of creation. On the other hand, the raven is a symbol of wisdom and justice. The raven has a connection with the world of the dead, he can get living and dead water. There is an opinion that the raven is a travel assistant and a fortune teller. There is a belief that when crows begin to leave their nests, this portends famine or other misfortunes.

Pigeon

Dove as a symbol of peace

Peace, purity, love, serenity, hope. Traditional Christian symbol of the Holy Spirit and baptism. There is a legend that the devil and witches can turn into any creature except a dove and a sheep. Pigeon cooing is associated with both sex and the birth of children. A pair of doves is a symbol of sexual harmony; so the dove became the personification of the tender wife. A dove with a laurel branch is a symbol of peace, a dove with a cornucopia is a happy accident. In the East, the dove is one of the many symbols of longevity.

Dolphin

"Boy with a Dolphin" (Andrea del Verrocchio, 1475. Sculpture for a fountain)

The dolphin symbolizes love, the power of the sea, speed, salvation, transformation. This is a friend of man in the sea element and its symbol. The dolphin is also a symbol of boundless joy, playfulness, unpredictability and even spiritual enlightenment. In ancient Greece, the lord of the waters, Poseidon, (the Roman counterpart is Neptune), was often depicted in a cart pulled by dolphins. As a symbol of the sacrifice of Christ, the dolphin is often depicted with a pierced trident or anchor (the secret symbol of the cross). Intertwined with an anchor, the dolphin is a symbol of caution, speed limits: "Hurry slowly."

Toad, frog

Stylized image of a frog

The toad is one of the attributes of witchcraft. According to European superstitions, this is a companion of witches, reminiscent of the death and torment of sinners. At the same time, the toad, which in the Middle Ages personified darkness and evil, greed and lust, is associated with birth and rebirth. A symbol of ugliness, behind which a beautiful soul can lurk. It also symbolizes longevity and wealth: it is believed that the toad, like a snake, carries a gem in its forehead that attracts good luck.

The frog is a widespread symbol of fertility, a harbinger of spring rains and the awakening of nature.

Crane

Dancing Cranes (bracelet from Kyiv)

In China and Japan, the crane symbolizes vigilance, longevity, wisdom, devotion, honor. The image of a crane flying towards the Sun is a symbol of social aspirations, its snow-white body is a symbol of purity, its red head is the fire of life. In India and in some Celtic regions, the crane is a symbol of betrayal, a harbinger of misfortune. In Russia, cranes, along with storks and nightingales, are considered "God's birds", their symbolism is associated with the Sun.

Throughout the world, the crane is a symbol of communication with the gods.

Snake: general symbolism

Python (Greece)

The snake is the most versatile and most complex of all animal symbols, as well as the most common and perhaps the most ancient of them. The snake signifies death and destruction, but also life and resurrection. This is both the solar and the lunar, light and darkness, good and evil, wisdom and blind passion, healing and poison, the keeper and the destroyer. Such a duality of symbolism forces one to balance between fear and worship, the snake appears either as a hero or as a monster.

Snake: positive symbolism

"Snake Power"

An example of the positive symbolism of a snake is the concept of kundalini: a symbol of inner strength, psychic energy, a snake-like ball of vital energy dormant at the base of the spine. The kundalini energy is called the "serpent power". Sometimes she is depicted as a coiled snake with heads at both ends. In India and other regions, snakes are often seen as guardians of shrines, water sources, and treasures. This tradition is associated with the symbolism of fertility inherent in the snake, and with the belief that precious stones are the frozen saliva of snakes.

Snake: negative symbolism

Illustration for the "Poem of Gilgamesh" (seal of the Sumero-Akkadian kingdom)

If we consider the frightening part of the symbolism of the snake, then it is a clear prototype of dragons and sea snakes or snake-like hybrids, symbolizing the many dangers that await a person in life. The snake is one of the worst omens, a symbol of darkness, evil, hatred, sin, temptation, deceit. The snake is blamed for the fact that because of it people lost God's gift of eternal life.

Snakes were an indispensable attribute of sorceresses, witches' potions included some parts of snakes.

Snake: cosmogonic symbolism

Snake and egg (the image of a snake that supports the world)

The snake is primarily a magical symbol of the forces that gave birth to life. A snake biting its own tail is a symbol not only of eternity, but also of divine self-sufficiency. The image of a snake guarding the eggs it lays is associated with a huge snake that wraps around the whole world and supports it or helps the earth's disk to swim in the surrounding Ocean. The snake is in constant contact with the forces of the earth, waters, darkness and the underworld - lonely, cold-blooded, secretive, able to rejuvenate by shedding its skin.

The snake as a symbol of wisdom

A snake wrapped around a wand

Totem symbolism, combined with the belief that snakes know the secrets of the earth and are able to see in the dark, endows snakes with wisdom or the gift of divination. “Be wise as serpents and simple as doves,” Christ said to his disciples (Matthew 10:16). The Greek word for "dragon" (which not only refers to a monster, but also means "snake with a piercing gaze") is etymologically related to vision. In the art of the snake - an attribute of the goddess of wisdom Athena (Minerva) and the allegorical figure of Prudence, meaning the gift of foresight.

Snake: Alchemy and Healing

Rod of Mercury (caduceus)

Staff of Asclepius (Aesculapius)

The snake coiled around the wand is the alchemical symbol of the Philosophical Mercury in its primary state.

According to mythology, Hermes (Mercury), the messenger of the gods, received a caduceus - a winged staff with the power to reconcile opponents. When he placed it between two fighting snakes, they peacefully coiled around the staff and calmed down. The snakes wrapped around the caduceus symbolize the interaction of opposing forces. The snake wrapped around a knotty staff is the emblem of the Greek god of healing Asclepius (Aesculapius), who, it is believed, could even resurrect the dead.

Ibis

Ibis (Egyptian papyrus from the 19th Dynasty, 1295-1186 BC)

The ibis is the sacred bird of the Egyptians. Symbol of wisdom. In ancient Egypt, the ibis was considered the incarnation of the lunar deity Thoth, the greatest god of Egypt, the patron of occult knowledge, who gave mankind writing. He is depicted as a man with the head of an Ibis. This bird is also called the guardian of the harvest. Killing an ibis, even by accident, was considered a heinous crime.

It is believed that the ibis can only live in Egypt and, transported to other countries, dies of boredom there.

Goat

Goat

The goat is a symbol of potency, vitality, masculinity, but also cunning, lust and stupidity; he personifies destructive tendencies in a man. In Western tradition, a goat is often referred to as an old, lustful man. In China and India, the goat is a positive male symbol. In Christianity, the goat is the personification of impurity and base lust.

The goat is often used for sacrifice ("scapegoat"). The goat is closely related to Dionysus (Bacchus).

Cow

Holy cow

For many peoples, this animal symbolizes fertility, prosperity, as well as patience and passive endurance. The cow is an ancient symbol of mother's milk and (like the bull) the cosmic forces that created the world. In many cults, from Ancient Egypt to China, the cow personifies Mother Earth. She also symbolizes the moon and the sky, because her horns resemble a crescent moon, her milk is associated with the Milky Way. The heads of the goddesses of the moon in various cultures are decorated with cow horns. The cow enjoys exceptional honor in India.

a lion

The lion is the symbol of the sun

The lion, commonly referred to as the king of beasts, has been one of the most frequently seen symbols of power and majesty for thousands of years. General symbolism: divine, solar energy (symbol of fire and the Sun), royal power, strength, courage, wisdom, justice, patronage, protection, but also cruelty, all-devouring ferocity and death. The lion is the image of all the great and terrifying forces of nature. He is considered both a destroyer and a savior, he is able to represent both evil and the fight against evil. The lion is one of the hypostases of the Sphinx.

Lions heraldic

heraldic lions

In heraldry - the most common and favorite image of an animal. Attributes of a heraldic lion: a bow and arrows, a saber, a sword, an ax, an ax, halberds, etc. The main heraldic form is a lion on its hind legs and in profile. In this case, one eye and one ear are indicated on the head. The bloody tongue sticks out of the mouth. This lion is a symbol of strength, courage, generosity. There are other image options. In state emblems, a crowned lion is an emblem of power over subjects.

Bear

heraldic bear

The bear is a symbol of good nature and rage, heroic strength and clumsiness, laziness and tender maternal feelings, gluttony and asceticism (albeit involuntary: he sleeps all winter without any food, “sucks his paw”). The bear personifies unpredictability, bad temper, evil, rudeness, greed, sinfulness, the devil, as well as cruel primitive force. Badge of warriors in Northern Europe and Asia.

In addition, the bear is a symbol of the moon and resurrection. K. Jung believes that the bear symbolizes the dark side of the subconscious.

Mouse, rat

mouse wedding

In Russia, the mouse is often called the "grey thief". The mouse is also a symbol of timidity, invisibility. The mouse helps to find the loss in the house: "Mouse, mouse, play and give it back." The mouse gives an increase. In China, the mouse is one of the popular deities of wealth.

The general symbolism of the rat is destruction, aggressiveness, greed; the rat is associated with disasters (pestilence) and death, but it is also the embodiment of perseverance, dexterity, cunning and fertility, and also has the gift of foresight (the legendary ability to foresee the death of ships).

A monkey

Hanuman, Monkey God Playing with the Peaches of Immortality (from a Chinese dish)

The symbolism of the monkey is controversial. Most often, the monkey personifies sin, in particular physical. She is also a symbol of cunning, deceit, the pursuit of luxury, spitefulness, laziness (due to her angular movements), drunkenness, sometimes a symbol of learning. The monkey (along with the white elephant and the cow) is the third sacred animal in India. Even now, insulting a monkey by action causes great resentment among religious people. In Japan, the cry of a monkey is a symbol of deep longing. Carvings of three monkeys are considered in the East as a talisman protecting from slander.

Deer

Stag (badge of Richard II, late 14th century)

A universal symbol associated with the East, sunrise, light, purity, renewal, creation and spirituality, but also with loneliness. Characteristic qualities of a deer: swiftness, grace and beauty. Deer are wonderful messengers and guides. They are credited with healing powers, especially the ability to search for medicinal herbs. The deer, in addition, is a symbol of caution and keen hearing. In China, the deer is associated with wealth (abundance) and good luck. The deer is a strong magical intercessor, one of the patron spirits of the Siberian peoples.

Eagle

The eagle as a symbol of the supreme power and solar nature of the lord of heaven and the head of all the gods Zeus (painting on a Greek bowl, 6th century BC)

The eagle is the lord of the air, the embodiment of power, speed. Solar symbol of the sun gods, rulers, warriors. Associated with greatness, power, dominance, courage, inspiration. It personifies the midday Sun, liberation from bonds, victory, pride, contemplation, royal origin, height. It is believed that the eagle is able to fly to the Sun, so it is called the messenger of heaven. Double-headed eagles can mean omniscience and dual power. An eagle with a snake in its claws symbolizes the victory of the spirit. In this fight, the eagle is the personification of the power of good, and the snake is the power of evil.

Eagles heraldic

Double-headed eagle (Russian embroidery)

Eagle - the emblem of the United States

In heraldry, the eagle is a symbol of power, dominance, generosity and insight. On coats of arms, the eagle is most often depicted flying chest forward, with wings raised up or soaring. It happens one- or two-headed. Since the time of the founders of Rome, Romulus and Remus, it has been depicted on the standards as the “bird of Jupiter”. After the Christians conquered Palestine, the double-headed eagle became the coat of arms of the Holy Roman Empire, and later of the Austrian (Austro-Hungarian) and Russian empires. The American bald eagle with outstretched wings has become the emblem of the United States.

Peacock

Peacock (medieval Persian drawing)

This is shining glory, immortality, greatness, incorruptibility. The magnificent tail of a peacock is a symbol of the all-seeing Sun and eternal cosmic cycles, as well as the starry firmament and, as a result, unity and interconnectedness. In ancient Rome, the peacock was considered the emblem of the empress and her daughters, while the eagle was the bird of the emperor. In Islamic decorative art, the unity of opposites (the Sun at its zenith next to the full Moon) is depicted as two peacocks under the World Tree. In Christianity, the peacock, on the one hand, is a symbol of eternal life, and on the other, a symbol of pride, luxury and vanity.

Spider

Spider depicted on an American Indian amulet

feminine beginning. The Great Mother, in her terrible form as weaver of fate, is sometimes depicted as a spider. All moon goddesses are spinners and weavers of fate. The web that the spider weaves is woven from the center in a spiral - a symbol of the creative forces of the Universe, a symbol of the universe. The spider in the center of the web symbolizes the center of the world; Sun surrounded by rays; The moon, personifying the cycles of life and death, spinning the web of time. The spider is often associated with good luck, wealth or rain. Killing a spider is bad luck.

Pelican

Plate of red stone mass depicting a pelican feeding its chicks with its blood (Staffordshire, circa 1660)

Pelican symbolizes self-sacrifice and parental love, as well as mercy. In heraldry, this bird, as a rule, is depicted as looking like an eagle or a crane, standing in a nest and trying to feed the chicks with its blood. Early Christian writers compared the pelican, feeding offspring with its flesh, with Jesus Christ, who donated his blood in the name of the salvation of mankind. The pelican is also a symbol of European occultism (primarily alchemists and Rosicrucians), expressing the feat of self-sacrifice and the eternal rebirth of life.

Rooster

Rooster - a solar bird (image-amulet, China, XX century)

The rooster is vigilance, courage, courage, foresight, reliability. Herald of the dawn, symbol of the Sun and spiritual rebirth. These qualities of his prevail over pride, arrogance, lust, also inherent in him. The Romans mean "the third watch of time": between midnight and dawn. The rooster is a protector from all kinds of evil. It is believed that night ghosts and evil spirits disappear with the first cock crow. The red rooster takes away the fire from the house, and the white one - ghosts. The Eastern Slavs, before moving into a new house, launched a rooster there. If he spent the night safely, then it was possible to move in.

Bee

Young woman collecting bee honey (15th century herbalist)

The bee personifies hard work, diligence, organizational and creative abilities, cleanliness, sociability, modesty, spirituality, courage, wisdom, selflessness, eloquence (“honey speeches”). In the Greek, Middle Eastern, Islamic traditions, the bee is an allegory of the soul. The Chinese associate the bee with the fickleness of "choosy brides." Among the ancient Slavs, the bee was a symbol of love, as it combined "the sweetness of honey and the bitterness of the sting." Queen bee, mother goddess, symbol of supreme power, fertility.

Scorpion

Scorpio (gnostic gem)

Scorpio is a symbol of evil, self-destruction, death, punishment, retribution, revenge, betrayal, but also a deep understanding of the world. Sometimes the scorpion serves as a talisman and an amulet - Paracelsus advised people suffering from diseases of the reproductive system to wear it. In Africa, it was believed that the scorpion itself allocates funds against its poison, so it was a symbol of not only murder, but also healing. The red star Antares on the "back" of the celestial constellation Scorpio was considered in Europe the worst fire in the sky.

Elephant

White elephant

At present, the huge mass and slowness of the elephant have become metaphorical. However, the elephant, first of all, is a symbol of strength: both gentle, loving, and furious, destructive. Elephants are considered vindictive, as they never forget the wrongs and abuse done to them. The thick skin of an elephant symbolizes spiritual invulnerability. The elephant is also a symbol of power, insight, prosperity, happiness, personifies the element of the Earth, memory, wisdom, longevity, fidelity, patience, compassion. The elephant is often depicted on good luck charms.

Dog

Neter Anubis (dog god)

In some countries, a dog is a sacred animal, in others it is considered an unclean, greedy, even vile creature and personifies evil. According to Islamic beliefs, angels will never visit a house where a dog lives. But most often the dog is a symbol of protection and self-sacrifice. As well as hunting (sometimes this symbol carries a negative connotation - persecution).

In ancient Egyptian mythology, dogs, as good guides and guards in the afterlife, were considered companions of Anubis, depicted with the head of a jackal or a dog.

Owl

Wise owl - attribute of Athena (Greece)

The owl is a traditional symbol of wisdom, an allegorical figure of Night and Sleep. In some ancient cultures, especially in China, the owl has an ominous symbolism, signifying darkness, personifying the yang principle with a negative, destructive connotation. Due to the noiselessness of night flight, glowing eyes and eerie screams, the owl is associated with death and occult forces. She is also credited with the gift of prophecy. At present, the owl is mainly a symbol of insight and book erudition. "Scientific owls" are called people of mental labor.

Falcon

Falcon - the image of the rising sun

The falcon, like the eagle, is a solar symbol of victory. The personification of superiority, strong spirit, light, freedom. In ancient Egypt, the falcon was a sacred symbol of the Sun, temples were dedicated to it, killing a falcon was considered a grave sin. In Western tradition, the falcon is a symbol of hunting. A falcon with a cap on its head is a symbol of hope for light and freedom. The falcon as a symbol of aggression is rare. Among the Slavs, this bird is a symbol of strength, courage, a good fellow. The falcon is opposed to the crow (as the embodiment of evil forces): "Where the falcons fly, they don't let the crow in."

Ostrich

Australian coat of arms

In ancient Egypt, the ostrich feather is an attribute of the goddess of truth and justice, Maat. This feather, according to legend, was placed on the scales when weighing the souls of the dead to determine the severity of their sins. Since ostrich feathers are the same length, they were used as a symbol of justice. The belief that an ostrich hides its head in the sand when danger appears (a symbol of avoiding problems) probably came from the threatening posture of the ostrich, when it bends its head to the ground itself.

In the Australian coat of arms, the emu is the shield-holder along with the kangaroo.

Tigers

“The tiger spring contains the tiger. Having mastered the contents of the tiger cave, the perfect man who subjugated yin and yang "

The tiger is a symbol of energy, strength, speed and talent. This image is both lunar and solar at the same time. He is both the creator and the destroyer. A tiger fighting a snake is a symbol of solar power. In a battle with a lion or dragon, he becomes a symbol of the moon, cruel and ferocious. In Europe, the tiger is a symbol of power and bloodlust. In the Far East, it is a symbol of nobility and happiness. In the cultures of Asia and India, it can be a symbol of aggression and protection, life and death, evil and good.

Turtle

Turtle wrapped in a snake

The turtle symbolizes strength, patience, endurance, constancy, slowness, fertility, longevity, senile strength, wisdom. In many cultures, the turtle is the most ancient symbol of cosmic order surrounded by special reverence. According to ancient beliefs, a turtle entwined with a snake is a symbol of the creation of the world. In India, the symbolism of stability is expressed in the idea that the Earth rests on four elephants, which stand on a huge turtle, slowly making its way through chaos. The turtle is also a symbol of protection from fire and water.

Lizard

Pumpkin with a lizard

This nimble, fast animal is a symbol of agility, elusiveness, and also rebirth (the latter) is associated with the ability of a lizard to leave its tail to the one who caught it, which then grows back. Lizards, because they hide in the shade during the heat of the day, are considered the guardians of the shadows, as well as the guardians of sleep and dreams. The lizard, in addition, can symbolize the subconscious and the shadows of our inner world.

The lizard was considered a good sign in Egypt and in the ancient world, where it was sometimes associated with wisdom. It has become an attribute of allegorical images of Logic. The symbol of Mercury, the messenger of the gods.

Mythical creatures

Imaginary animals are found throughout the world in myths and folklore ... They enable us to clearly characterize phenomena that would otherwise be difficult to identify.

J. Tresidder

Mythical creatures are, as a rule, a combination of several animals, which allows the human imagination to endow them with unusual abilities, including freedom from the usual principles of our world. Monsters, combining the appearance of several different animals, are a symbol of the initial chaos or the terrifying forces of nature, they also personify the evil forces in the nature of man himself. Fairytale animals are often depicted guarding treasures or hidden, secret knowledge.

Ba (bird)

The bird of the soul Ba, bent over the mummy before flying to another world (Egypt)

The Ba bird is an Egyptian symbol of the soul of a person, which flies away after his death to another world. This bird has the body of a falcon (according to some sources, a hawk) and a human head.

Basilisk (cockatrice)

Rooster head basilisk

Basilisk is one of the deadliest creatures of medieval symbolism. According to some sources, the basilisk looks like a simplicissimus, but with the head and legs of a rooster. In occult and magical symbolism, the basilisk is depicted as a crowned serpent. Since, as is commonly believed, the basilisk destroys everything with its gaze, it has been adopted as a magical symbol of wisdom, devouring a person with it symbolizes the process of initiation. It is believed that the only way to defeat a basilisk is to place a mirror in front of it.

Harpies

Harpy (XVI century)

These are half-woman, half-bird (female head and chest, and claws - a vulture) of a disgusting appearance. Associated with sudden death, whirlpools and storms. The feminine principle in its destructive aspect.

Garuda

Garuda (emblem of Thailand)

Bird of Life, Heaven, Sun, victory. Sometimes identified with the Phoenix. She is also the vehicle of the god Vishnu, the creator and destroyer of everything ("Vishnu's horse"). She emerges from the egg already an adult and nests in the Tree of Life, which fulfills all desires. The head, chest (female), torso, legs to the knees at the garuda are human, beak, wings, tail, hind legs (below the knees) are eagle.

The garuda is often depicted fighting nagas (serpents) who personify evil.

Hydra

Hydra (Greece, 16th century)

In Greek mythology, the Hydra is a dragon-serpent with seven heads. She symbolizes the difficulties in the fight against evil: as soon as one of her heads is cut off, a new one immediately grows. Blind, animal force of life.

Griffin

Griffin-protector (XVI century)

A solar hybrid creature that combines the head, wings and claws of an eagle with the body of a lion - these animals personify power over air and earth (the king of birds and the king of beasts), so the griffin is a symbol of strength and vigilance. In Greece, the griffin was dedicated to Apollo, whose chariot he drove through the sky, for Athena he personified wisdom, for Nemesis - retribution. Legends say that the griffins guarded the gold of India and the Scythians. There is also a legend that griffins living in the Far North guard the gold of Zeus, located in the country of the Hyperboreans.

The Dragon

Chinese dragon Chiao, a symbol of a happy occasion

The dragon - "winged serpent", but only with paws like an eagle - combines a snake and a bird, spirit and matter. This is one of the most versatile and most complex symbols. The dragon can be solar and lunar, good and evil. This is the keeper of treasures and secret knowledge. Longevity symbol. In the East, the dragon, as a rule, is the Power of Heaven, bringing good, in the West - a destructive and evil force. In Russia, the dragon is a sign of Satan, the devil. The victory over the dragon means the victory of light over darkness, over one's own nature.

Unicorn

Heraldic image of a unicorn

A unicorn is a mystical creature, an animal with the body of a horse or deer, which has a long, sharp horn. In the general case, it symbolizes the feminine, lunar principle, purity, purity, chastity. In China, it represents abundance and longevity. According to the legends, it can only be caught by a chaste maiden sitting alone in the forest: sensing her purity, the unicorn can come up to her, lay her head on her lap and fall asleep. On the basis of these legends, he became a symbol of purity, in particular feminine.

Centaur

Centaur, lore hunter

According to Greek myths, a centaur is a creature with the body of a horse and the torso of a man. This is a symbol of the lower nature of man (lust, violence, drunkenness), his animal nature, connected with a higher nature by human virtues and the ability to judge. It is a symbol of the conflict between the ferocious and good aspects of human nature.

There is also a version about morally impeccable centaurs (among them - Chiron), descended from Kronos. They symbolize the superiority of reason over instinct.

Makara

Makara

In the Western tradition, makara is a fantastic sea monster of enormous size (a fish with the head of a crocodile). Symbol of the power of the seas and oceans, rivers and lakes. In Hinduism, Makara has the appearance of a fish with the head and front legs of an antelope. This is one of the creatures on which Vishnu travels. It is a positive symbol associated with the rainbow and rain, with the lotus rising from the water, the return of the Sun after the winter solstice. Makara in a number of legends is associated with deities acting as guardians of the world - lokapal (Varuna, Soma, Indra, Kubera ...).

Medusa Gorgon

Medusa Gorgon (Greece) - horror

Medusa Gorgon - a female monster with snakes instead of hair, boar teeth, golden wings and bronze legs. This is the most blatant personification of hostile evil, the Great Mother in her terrible aspect of the destroyer, the embodiment of horror. One look at her turned people to stone, so her image later became a protective amulet. After Perseus cut off Medusa Gorgon's head, the giant Chrysaor and the winged horse Pegasus were born from her blood.

Nagas

Buddha sitting on a coiled naga, symbolizing the knowledge hidden in instinct (statue from the temple of Angkor)

In Hinduism, these are demigods depicted with a snake body and one or more human heads (sometimes they are just many-headed snakes). They, according to legend, own the underworld - Patala, where they guard the countless treasures of the earth. According to legend, the nagas washed Gautama Buddha at his birth, and also protected his remains after death. Nagas are the guardians of treasures and esoteric knowledge, serpent kings and queens, the vital forces of the waters, passionate nature. These are the guardians of natural forces that can be controlled.

Pegasus

Pegasus (XVI century)

This is the winged horse of Muses, which emerged from the neck of Medusa when Perseus cut off her head. Pegasus, on which Bellerophon defeated the Chimera, personifies the combination of the lower and higher nature, striving for the higher, and symbolizes the superiority of the spiritual over the material. It is also a symbol of eloquence, poetic inspiration and contemplation. In European heraldry, Pegasus is depicted on the coats of arms of thinkers. Today it is often used as an emblem of air transport.

Mermaid

Mermaid (XV century)

A fish woman capable of living in the human world and the supernatural world. The magical symbol of initiation. The mermaid is a marine version of the Centaur. However, it also has more positive symbolism, according to the sailors. In Slavic mythology, mermaids (bathing, vodonitsy, rags, pitchforks, undines) are harmful creatures, especially dangerous in the mermaid week (following the Trinity). Often mermaids are confused with such creatures of ancient Greek mythology as Nereids, naiads, water nymphs. But these eternally young maidens do not have, unlike mermaids, a fish tail.

Salamander

salamander on fire

Salamander is a mythical creature in the form of an ordinary animal, but with supernatural powers. The salamander is usually depicted as a small lizard or wingless dragon, sometimes with a human or dog-like figure amidst flames. These creatures are considered the most poisonous of creatures, their bite is deadly. The salamander is the element of fire and is able to live in fire because it has a very cold body. It is a symbol of the fight against sensual temptations. Since the salamander is considered a sexless creature, it also symbolizes chastity.

Simplicissimus

Harold's emblem

Simplicissimus is a fictional beast that looks like a dragon, but with two eagle legs and a tail in the shape of a spearhead twisted in a loop. Symbolizes war, envy, stench, disaster, Satan, but also vigilance.

Simplicissimus was the personal emblem of King Harold (on the French carpets from Bayeux, telling about the Battle of Hastings and the death of Harold in 1066, the simplicissimus is depicted twice).

Fo dog

Dog Fo (China)

“Fo” means “great luck” in Chinese. This is a symbol of valor and energy, a talisman for the home. Pho dogs should be purchased in pairs and placed side by side. If you put them (or hang their images) in front of the front door, they meet all incoming people and protect each family member from troubles and failures. Placed in the Wealth Zone (Southeast), Pho dogs contribute to the well-being and prosperity of the home. Located in the central sector, they will bring wealth to the house faster.

Sphinx

Egyptian coin depicting the Sphinx

The Sphinx is a creature with the body of a lion and a human head (male or female) or with the head of a ram. The oldest and largest is the Great Sphinx in Giza (Egypt). This is an ancient image, personifying a mysterious, solar power, a symbol of dignity, royalty, wisdom, power, a symbol of the union of physical power with the highest intellect.

The Egyptian sphinx has nothing to do with the later Greek legend of the "mystery of the Sphinx", which made it a symbol of mystery, a keeper of ancient wisdom, but Jung considered the sphinx a symbol of female greed, as well as the "Terrible Mother".

Scylla and Charybdis

Scylla (Greece) - danger

In Greek mythology, these are two monsters of the Sicilian Sea, who lived on both sides of a narrow strait and killed sailors passing between them. Ruthless manifestations of the forces of the sea. Once upon a time, from beautiful nymphs, they were turned into monsters with six heads, with three rows of teeth in each head, with ugly long necks. These roaring, rumbling monsters swallowed the sea and spat it back out (the image of an opening sea abyss). Being between Scylla and Charybdis means being in danger from different directions at the same time.

Triton

Triton (Greece) - wave calmer

Depicted as an old man or a young man with a fish tail instead of legs. In Greek mythology, it is considered a sea deity - the son of Poseidon and the mistress of the seas, Amphitrids. Triton blows a horn from a shell and rules over the powers of the waters. A marine version of a mermaid, but male.

Phoenix

Phoenix (XVI century)

Phoenix is ​​the most famous of all symbols of resurrection, an ancient symbol of immortality, the Sun. An animal that has a normal appearance, but with supernatural powers. This legendary bird is reborn every 500 years from the ashes in a fire. Phoenix has become an emblem of the rebirth of the human spirit in the eternal struggle with the difficulties of the material world. From Ancient Egypt, this symbol passed into Slavic mythology (Firebird, Finist-Clear Falcon) in full integrity.

Chimera

Chimera (Vatican)

According to Homer's description, this is a monster with the head of a lion, the body of a goat and the tail of a snake. Feeds on fire, was killed by Bellerophon, who rules the winged pegasus.

In heraldry, the chimera is sometimes depicted with the head and chest of a woman and the tail of a dragon.

The Chimera causes winds and storms on land and sea. Symbolizes danger, as well as delusion (may give rise to illusions). In addition, it is a symbol of non-existence.

“I have already said that science is the process of knowing the Truth.
It should not be a means to achieve power.”

Studying the history of the emergence of mathematics as a separate and isolated science, you can find many interesting facts. For example, the founders of modern mathematics, according to some, are ten people, according to others, twenty famous people. This information is open and available to anyone.

It is interesting to read the biography of each of these "founders" of mathematics. All these people were fond of and studied, to a greater or lesser extent, philosophy, religion, physics, astronomy, celestial mechanics and other sciences. They studied in Jesuit schools, belonged to certain orders, were members of various societies.

Information about the origin of symbolism in mathematics is posted in the public domain with approximately the following words: "a sign such and such was invented by a certain person."

Suggests the word thought up. But mathematics has always been considered the most exact science. These ten or twenty famous personalities lived in different eras, in different territories, and often never crossed paths. How could it happen that all of them suddenly come up with some signs and symbols to denote mathematical expressions and abstractions?

After reading the book by A. Novykh "Sensei 4", expanding the horizons of knowledge in various directions, observing, comparing and analyzing, a person understands how science is done and created, where the generally recognized authorities come from, whose opinion in subsequent centuries becomes generally recognized by the entire world community, without putting calling into question any of the "immutable" truths.

It is clear that none of the founders of mathematics invented anything himself. And at the same time, being familiar with the primordial knowledge, he, either himself or someone else, used this or that symbol in a way that was convenient or beneficial for him.

This can be traced to one of the patterns of the system: "divide and conquer." After inventing one's own interpretation of primordial knowledge, there is an invariable struggle and enmity for the universal recognition of a new idea. The report "PRIMORDIAL ALLATRA PHYSICS" outlines the concept of a holistic perception and knowledge of the world. Developed civilizations have never separated one science from another. The training took place in the understanding of the single grain of truth and indivisibility. In ancient times, this single science was known under the name "Belyao Dzy" - the science of the "White Lotus".

In the section on the origin of mathematical symbols and signs, one can get acquainted with the “general” opinion that their origin is unclear and most likely such symbols were previously used in trade, in buying and selling. However, delving into the biography of each individual, the founder of mathematics, one can come to the conclusion that all of them were inclined to perceive mathematics as a philosophy and, above all, as a reflection on the providence of God about the sensory perception of the world. But, apparently, it is beneficial for someone to fit any sensible thought into one standard of material thinking.

For example, Henri Poincaré in his books “Science and Hypothesis”, “The Value of Science”, “Science and Method” described his vision of mathematical creativity, in which, in his opinion, intuition plays the main role, and he assigns the role of substantiation of intuitive insights to logic. Poincaré created his own creative method. He presented it to the Parisian Psychological Society in the report "Mathematical creativity". In his creative method, he relied on the creation of an intuitive model of the problem. He always solved any problem in his head first, and then wrote down the solution. Poincaré never worked on one problem for a long time. He believed that the subconscious has already received the task and continues to work, even when he thinks about other things.

Descartes is also considered one of the founders of the science of mathematics. He formulated the main theses in his work "Principles of Philosophy": “God created the world and the laws of nature, and then the whole universe acts as an independent mechanism. There is nothing in the world but moving matter of various kinds. Matter consists of elementary particles, the local interaction of which produces all natural phenomena. Mathematics is a powerful and universal method of understanding nature, a model for other sciences.”

Based on scattered data provided on the Internet, we will review the most famous symbols of mathematics. It is worth noting that these symbols, according to archaeological finds, have been known to mankind since the Paleolithic. Moreover, the analysis of the extensive research presented in the book "AllatRa", shows that these symbols were used to transfer spiritual knowledge about man and the world to future generations.

Signs “+” and “-” (plus and minus) were “invented” by Johann Widmann.

The sign "x" (multiplication) was introduced by William Oughtred in 1631 in the form of an oblique cross.

The sign “≈” (approximately) was “invented” by the German mathematician S. Günther in 1882.

Signs “<”, “>” (comparisons) was “invented” and introduced by Thomas Harriot, an English astronomer, mathematician, ethnographer and translator. In 1585 - 1586. Thomas Harriot traveled to the New World with an expedition. There he became closely acquainted with the life of the Algonquin tribe. This tribe had its own pictographic writing. The legendary history of the Valam Olum tribe, discovered in 1820 and containing the most interesting legends and myths, was set out in such a letter. (“Valam olum” basically contains cosmogonic myths, legends about the universe, the struggle between good and evil spirits, about good and evil.)

Upon his return from the expedition, Thomas Harriot wrote a treatise in which he outlined the life of the native inhabitants of America with detailed maps of North Carolina. This expedition paved the way for the massive British colonization of North America.

The symbols were introduced by John Vallis. However, this symbol became widespread only after its support by the French mathematician Pierre Bouguer. In the biography of Buger it appears that he studied at the Jesuit Collegium.

The symbol of the nabla operator (vector differential operator, an equilateral triangle with its vertex down) was “invented” by William Hamilton. William Rowan Hamilton was interested in philosophy, especially Kant and Berkeley. He did not believe that the laws of nature, discovered by people, adequately reflect the real patterns. The scientific model of the world and reality, he wrote, are "intimately and miraculously connected by virtue of the ultimate unity, subjective and objective, in God, or, to put it less technically and more religiously, by virtue of the holiness of the discoveries which He Himself was pleased to make in the Universe for the human intellect" . Based on the teachings of Kant, Hamilton considered scientific ideas to be products of human intuition.

The infinity symbol was also "invented" and proposed by John Vallis. He was the son of a priest. Subsequently, he himself became a priest. According to his merits, he was invited to work at Oxford University, where he headed the Department of Geometry and at the same time acted as archive keeper.

You can get closer to unraveling the history of the origin of mathematical symbols by studying the biographies of each of its founders.

Hermann Weyl, for example, assessed the generally accepted definition of the subject of mathematics as follows: “the question of the foundation of mathematics and of what, ultimately, mathematics is, remains open m. We do not know any direction that will allow, in the end, to find a final answer to this question, and whether it is generally possible to expect that such a "final" answer will ever be received and recognized by all mathematicians. "Mathematization" may remain one of the manifestations of human creative activity, like music-making or literary creativity, bright and original, but forecasting its historical destinies cannot be rationalized and cannot be objective.

“It is impossible to know everything, but you need to strive for this.”

Anastasia Novykh

The AllatRa modern encyclopedia of primordial knowledge gives an answer to the question: where do the symbols and signs come from and that, first of all, the signs and symbols convey the idea of ​​the creation of the world, the Universe, reflect the energy structure of a person, as well as the general picture of the creation and transformation of matter, the dominance of the spiritual world over the material.

When people interact for a long time within a certain area of ​​activity, they begin to look for a way to optimize the communication process. The system of mathematical signs and symbols is an artificial language that was designed to reduce the amount of graphically transmitted information and at the same time fully preserve the meaning inherent in the message.

Any language requires learning, and the language of mathematics in this regard is no exception. To understand the meaning of formulas, equations and graphs, it is required to have certain information in advance, to understand the terms, notation, etc. In the absence of such knowledge, the text will be perceived as written in an unfamiliar foreign language.

In accordance with the demands of society, graphic symbols for simpler mathematical operations (for example, the notation of addition and subtraction) were developed earlier than for complex concepts like the integral or differential. The more complex the concept, the more complex sign it is usually denoted.

Models for the formation of graphic symbols

In the early stages of the development of civilization, people associated the simplest mathematical operations with their familiar concepts based on associations. For example, in ancient Egypt, addition and subtraction were indicated by a pattern of walking legs: lines directed in the direction of reading indicated “plus”, and in the opposite direction - “minus”.

Numbers, perhaps, in all cultures, were originally indicated by the corresponding number of dashes. Later, conventions began to be used for recording - this saved time, as well as space on tangible media. Often letters were used as symbols: this strategy has become widespread in Greek, Latin and many other languages ​​​​of the world.

The history of the emergence of mathematical symbols and signs knows the two most productive ways of forming graphic elements.

Word Representation Transformation

Initially, any mathematical concept is expressed by some word or phrase and does not have its own graphical representation (other than lexical). However, performing calculations and writing formulas in words is a lengthy procedure and takes up an unreasonably large amount of space on a material carrier.

A common way to create mathematical symbols is to transform the lexical representation of a concept into a graphic element. In other words, the word denoting a concept is shortened or transformed in some other way over time.

For example, the main hypothesis of the origin of the plus sign is its abbreviation from the Latin et, whose analogue in Russian is the union "and". Gradually, in cursive writing, the first letter ceased to be written, and t reduced to a cross.

Another example is the "x" sign for the unknown, which was originally an abbreviation for the Arabic word for "something". Similarly, there were signs for the square root, percent, integral, logarithm, etc. In the table of mathematical symbols and signs, you can find more than a dozen graphic elements that appeared in this way.

Arbitrary character assignment

The second common variant of the formation of mathematical signs and symbols is the assignment of a symbol in an arbitrary way. In this case, the word and the graphic designation are not related to each other - the sign is usually approved as a result of the recommendation of one of the members of the scientific community.

For example, the signs for multiplication, division, and equality were proposed by the mathematicians William Oughtred, Johann Rahn, and Robert Record. In some cases, several mathematical signs could be introduced into science by one scientist. In particular, Gottfried Wilhelm Leibniz proposed a number of symbols, including the integral, differential, and derivative.

The simplest operations

Signs such as "plus" and "minus", as well as symbols for multiplication and division, are known to every student, despite the fact that there are several possible graphic signs for the last two operations mentioned.

It is safe to say that people knew how to add and subtract many millennia BC, but standardized mathematical signs and symbols that denote these actions and are known to us today appeared only by the XIV-XV century.

However, despite the establishment of a certain agreement in the scientific community, multiplication in our time can be represented by three different signs (diagonal cross, dot, asterisk), and division by two (a horizontal line with dots above and below or a slash).

Letters

For many centuries, the scientific community has used Latin exclusively for the exchange of information, and many mathematical terms and signs find their origins in this language. In some cases, graphic elements have become the result of abbreviation of words, less often - their intentional or accidental transformation (for example, due to a typo).

The designation of the percentage ("%"), most likely, comes from the erroneous spelling of the abbreviation who(cento, i.e. "hundredth part"). In a similar way, the plus sign, the history of which is described above, occurred.

Much more was formed by intentionally shortening the word, although this is not always obvious. Not every person recognizes the letter in the square root sign R, i.e. the first character in the word Radix ("root"). The integral symbol also represents the first letter of the word Summa, but it is intuitively similar to a capital letter. f without a horizontal line. By the way, in the first publication, the publishers made just such a mistake by typing f instead of this character.

Greek letters

As graphic symbols for various concepts, not only Latin ones are used, but also in the table of mathematical symbols you can find a number of examples of such a name.

The number Pi, which is the ratio of the circumference of a circle to its diameter, comes from the first letter of the Greek word for circle. There are several lesser known irrational numbers, denoted by the letters of the Greek alphabet.

An extremely common sign in mathematics is the "delta", which reflects the amount of change in the value of variables. Another common sign is "sigma", which acts as a sum sign.

Moreover, almost all Greek letters are used in one way or another in mathematics. However, these mathematical signs and symbols and their meaning are known only to people who are engaged in science professionally. In everyday life and everyday life, this knowledge is not required for a person.

Signs of logic

Oddly enough, many intuitive symbols have been invented only recently.

In particular, the horizontal arrow, replacing the word "therefore", was proposed only in 1922. The quantifiers of existence and universality, i.e. signs read as: "exists ..." and "for any ..." were introduced in 1897 and 1935 respectively.

Symbols from the field of set theory were invented in 1888-1889. And the crossed out circle, which today is known to any high school student as a sign of an empty set, appeared in 1939.

Thus, the signs for such complex concepts as the integral or the logarithm were invented centuries earlier than some intuitive symbols that are easily perceived and assimilated even without prior preparation.

Mathematical symbols in English

Due to the fact that a significant part of the concepts was described in scientific works in Latin, a number of names of mathematical signs and symbols in English and Russian are the same. For example: Plus (“plus”), Integral (“integral”), Delta function (“delta function”), Perpendicular (“perpendicular”), Parallel (“parallel”), Null (“zero”).

Some of the concepts in the two languages ​​are called differently: for example, division is Division, multiplication is Multiplication. In rare cases, the English name for a mathematical sign gets some distribution in Russian: for example, a slash in recent years is often referred to as a "slash" (English Slash).

symbol table

The easiest and most convenient way to get acquainted with the list of mathematical symbols is to look at a special table that contains the symbols of operations, symbols of mathematical logic, set theory, geometry, combinatorics, mathematical analysis, linear algebra. This table shows the main mathematical signs in English.

Math symbols in a text editor

When performing various kinds of work, it is often necessary to use formulas that use characters that are not on the computer keyboard.

Like graphic elements from almost any field of knowledge, mathematical signs and symbols in Word can be found in the Insert tab. In the 2003 or 2007 versions of the program, there is the "Insert Symbol" option: when you click on the button on the right side of the panel, the user will see a table that contains all the necessary mathematical symbols, Greek lowercase and uppercase letters, various types of brackets and much more.

In versions of the program released after 2010, a more convenient option has been developed. When you click on the "Formula" button, you go to the formula designer, which provides for the use of fractions, entering data under the root, changing the register (to indicate degrees or ordinal numbers of variables). All signs from the table presented above can also be found here.

Is it worth learning math symbols

The system of mathematical notation is an artificial language that only simplifies the recording process, but cannot bring understanding of the subject to an outside observer. Thus, memorizing signs without studying terms, rules, logical connections between concepts will not lead to mastering this area of ​​knowledge.

The human brain easily learns signs, letters and abbreviations - mathematical notations are remembered by themselves when studying the subject. Understanding the meaning of each specific action creates so strong that the signs denoting the terms, and often the formulas associated with them, remain in memory for many years and even decades.

Finally

Since any language, including an artificial one, is open to changes and additions, the number of mathematical signs and symbols will certainly grow over time. It is possible that some elements will be replaced or adjusted, while others will be standardized in the only possible way, which is relevant, for example, for multiplication or division signs.

The ability to use mathematical symbols at the level of a full school course is practically necessary in the modern world. In the context of the rapid development of information technology and science, the widespread algorithmization and automation, the possession of a mathematical apparatus should be taken as a given, and the development of mathematical symbols as an integral part of it.

Since calculations are used in the humanitarian sphere, and in economics, and in the natural sciences, and, of course, in the field of engineering and high technology, understanding mathematical concepts and knowledge of symbols will be useful for any specialist.

Infinity.J. Wallis (1655).

For the first time it is found in the treatise of the English mathematician John Valis "On Conic Sections".

Base of natural logarithms. L. Euler (1736).

Mathematical constant, transcendental number. This number is sometimes called non-Perov in honor of the Scottish scientist Napier, author of the work "Description of the amazing table of logarithms" (1614). For the first time, the constant is tacitly present in the appendix to the English translation of the aforementioned work by Napier, published in 1618. The very same constant was first calculated by the Swiss mathematician Jacob Bernoulli in the course of solving the problem of the limiting value of interest income.

2,71828182845904523...

The first known use of this constant, where it was denoted by the letter b, found in Leibniz's letters to Huygens, 1690-1691. letter e started using Euler in 1727, and the first publication with this letter was his Mechanics, or the Science of Motion, Stated Analytically, 1736. Respectively, e commonly called Euler number. Why was the letter chosen? e, is not exactly known. Perhaps this is due to the fact that the word begins with it exponential("exponential", "exponential"). Another assumption is that the letters a, b, c and d already widely used for other purposes, and e was the first "free" letter.

The ratio of the circumference of a circle to its diameter. W. Jones (1706), L. Euler (1736).

Mathematical constant, irrational number. The number "pi", the old name is Ludolf's number. Like any irrational number, π is represented by an infinite non-periodic decimal fraction:

π=3.141592653589793...

For the first time, the designation of this number with the Greek letter π was used by the British mathematician William Jones in the book A New Introduction to Mathematics, and it became generally accepted after the work of Leonard Euler. This designation comes from the initial letter of the Greek words περιφερεια - circle, periphery and περιμετρος - perimeter. Johann Heinrich Lambert proved the irrationality of π in 1761, and Adrien Marie Legendre in 1774 proved the irrationality of π 2 . Legendre and Euler assumed that π could be transcendental, i.e. cannot satisfy any algebraic equation with integer coefficients, which was eventually proven in 1882 by Ferdinand von Lindemann.

imaginary unit. L. Euler (1777, in press - 1794).

It is known that the equation x 2 \u003d 1 has two roots: 1 and -1 . The imaginary unit is one of the two roots of the equation x 2 \u003d -1, denoted by the Latin letter i, another root: -i. This designation was proposed by Leonhard Euler, who took the first letter of the Latin word for this imaginarius(imaginary). He also extended all the standard functions to the complex domain, i.e. set of numbers representable in the form a+ib, where a and b are real numbers. The term "complex number" was introduced into wide use by the German mathematician Carl Gauss in 1831, although the term had previously been used in the same sense by the French mathematician Lazar Carnot in 1803.

Unit vectors. W. Hamilton (1853).

Unit vectors are often associated with the coordinate axes of the coordinate system (in particular, with the axes of the Cartesian coordinate system). Unit vector directed along the axis X, denoted i, a unit vector directed along the axis Y, denoted j, and the unit vector directed along the axis Z, denoted k. Vectors i, j, k are called orts, they have identity modules. The term "ort" was introduced by the English mathematician and engineer Oliver Heaviside (1892), and the notation i, j, k Irish mathematician William Hamilton.

The integer part of a number, antie. K. Gauss (1808).

The integer part of the number [x] of the number x is the largest integer not exceeding x. So, =5, [-3,6]=-4. The function [x] is also called "antier of x". The integer part function symbol was introduced by Carl Gauss in 1808. Some mathematicians prefer to use the notation E(x) proposed in 1798 by Legendre instead.

Angle of parallelism. N.I. Lobachevsky (1835).

On the Lobachevsky plane - the angle between the linebpassing through the pointOparallel to a straight linea, not containing a dotO, and perpendicular fromO on the a. α is the length of this perpendicular. As the point is removedO from straight athe angle of parallelism decreases from 90° to 0°. Lobachevsky gave a formula for the angle of parallelismP( α )=2arctg e - α /q , where q is some constant related to the curvature of the Lobachevsky space.

Unknown or variable quantities. R. Descartes (1637).

In mathematics, a variable is a quantity characterized by the set of values ​​that it can take. This can mean both a real physical quantity, temporarily considered in isolation from its physical context, and some abstract quantity that has no analogues in the real world. The concept of a variable arose in the 17th century. initially under the influence of the demands of natural science, which brought to the fore the study of movement, processes, and not just states. This concept required new forms for its expression. The literal algebra and analytic geometry of René Descartes were such new forms. For the first time, the rectangular coordinate system and the notation x, y were introduced by Rene Descartes in his work "Discourse on the method" in 1637. Pierre Fermat also contributed to the development of the coordinate method, but his work was first published after his death. Descartes and Fermat used the coordinate method only on the plane. The coordinate method for three-dimensional space was first applied by Leonhard Euler already in the 18th century.

Vector. O.Koshi (1853).

From the very beginning, a vector is understood as an object having a magnitude, a direction, and (optionally) an application point. The beginnings of vector calculus appeared along with the geometric model of complex numbers in Gauss (1831). Advanced operations on vectors were published by Hamilton as part of his quaternion calculus (the imaginary components of a quaternion formed a vector). Hamilton coined the term vector(from the Latin word vector, carrier) and described some vector analysis operations. This formalism was used by Maxwell in his works on electromagnetism, thereby drawing the attention of scientists to the new calculus. Gibbs' Elements of Vector Analysis (1880s) soon followed, and then Heaviside (1903) gave vector analysis its modern look. The vector sign itself was introduced by the French mathematician Augustin Louis Cauchy in 1853.

Addition, subtraction. J. Widman (1489).

The plus and minus signs were apparently invented in the German mathematical school of "kossists" (that is, algebraists). They are used in Jan (Johannes) Widmann's textbook A Quick and Pleasant Count for All Merchants, published in 1489. Prior to this, addition was denoted by the letter p(from Latin plus"more") or the Latin word et(conjunction "and"), and subtraction - by letter m(from Latin minus"less, less"). In Widman, the plus symbol replaces not only addition, but also the union "and". The origin of these symbols is unclear, but most likely they were previously used in trading as signs of profit and loss. Both symbols soon became common in Europe - with the exception of Italy, which used the old designations for about a century.

Multiplication. W. Outred (1631), G. Leibniz (1698).

The multiplication sign in the form of an oblique cross was introduced in 1631 by the Englishman William Outred. Before him, the most commonly used letter M, although other designations were also proposed: the symbol of a rectangle (French mathematician Erigon, 1634), an asterisk (Swiss mathematician Johann Rahn, 1659). Later, Gottfried Wilhelm Leibniz replaced the cross with a dot (end of the 17th century), so as not to be confused with the letter x; before him, such symbolism was found by the German astronomer and mathematician Regiomontanus (XV century) and the English scientist Thomas Harriot (1560 -1621).

Division. I.Ran (1659), G.Leibniz (1684).

William Outred used the slash / as the division sign. Colon division began to denote Gottfried Leibniz. Before them, the letter was also often used D. Starting from Fibonacci, the horizontal line of the fraction is also used, which was used by Heron, Diophantus and in Arabic writings. In England and the United States, the ÷ (obelus) symbol, which was proposed by Johann Rahn (possibly with the participation of John Pell) in 1659, became widespread. An attempt by the American National Committee on Mathematical Standards ( National Committee on Mathematical Requirements) to remove the obelus from practice (1923) was inconclusive.

Percent. M. de la Porte (1685).

One hundredth of a whole, taken as a unit. The word "percent" itself comes from the Latin "pro centum", which means "one hundred". In 1685, the book Manual of Commercial Arithmetic by Mathieu de la Porte was published in Paris. In one place, it was about percentages, which then meant "cto" (short for cento). However, the typesetter mistook that "cto" for a fraction and typed "%". So because of a typo, this sign came into use.

Degrees. R. Descartes (1637), I. Newton (1676).

The modern notation for the exponent was introduced by René Descartes in his " geometries"(1637), however, only for natural powers with exponents greater than 2. Later, Isaac Newton extended this form of notation to negative and fractional exponents (1676), the interpretation of which had already been proposed by this time: the Flemish mathematician and engineer Simon Stevin, the English mathematician John Vallis and French mathematician Albert Girard.

arithmetic root n th power of a real number a≥0, - non-negative number n-th degree of which is equal to a. The arithmetic root of the 2nd degree is called the square root and can be written without indicating the degree: √. The arithmetic root of the 3rd degree is called the cube root. Medieval mathematicians (for example, Cardano) denoted the square root with the symbol R x (from the Latin Radix, root). The modern designation was first used by the German mathematician Christoph Rudolf, from the Cossist school, in 1525. This symbol comes from the stylized first letter of the same word radix. The line above the radical expression was absent at first; it was later introduced by Descartes (1637) for a different purpose (instead of brackets), and this feature soon merged with the sign of the root. The cube root in the 16th century was designated as follows: R x .u.cu (from lat. Radix universalis cubica). Albert Girard (1629) began to use the usual notation for the root of an arbitrary degree. This format was established thanks to Isaac Newton and Gottfried Leibniz.

Logarithm, Decimal Logarithm, Natural Logarithm. I. Kepler (1624), B. Cavalieri (1632), A. Prinsheim (1893).

The term "logarithm" belongs to the Scottish mathematician John Napier ( "Description of the amazing table of logarithms", 1614); it arose from a combination of the Greek words λογος (word, relation) and αριθμος (number). J. Napier's logarithm is an auxiliary number for measuring the ratio of two numbers. The modern definition of the logarithm was first given by the English mathematician William Gardiner (1742). By definition, the logarithm of a number b by reason a (a 1, a > 0) - exponent m, to which the number should be raised a(called the base of the logarithm) to get b. Denoted log a b. So, m = log a b, if a m = b.

The first tables of decimal logarithms were published in 1617 by Oxford mathematics professor Henry Briggs. Therefore, abroad, decimal logarithms are often called brigs. The term "natural logarithm" was introduced by Pietro Mengoli (1659) and Nicholas Mercator (1668), although the London mathematics teacher John Spidell compiled a table of natural logarithms as early as 1619.

Until the end of the 19th century, there was no generally accepted notation for the logarithm, the base a indicated to the left and above the symbol log, then over it. Ultimately, mathematicians came to the conclusion that the most convenient place for the base is below the line, after the symbol log. The sign of the logarithm - the result of the reduction of the word "logarithm" - occurs in various forms almost simultaneously with the appearance of the first tables of logarithms, for example Log- I. Kepler (1624) and G. Briggs (1631), log- B. Cavalieri (1632). Designation ln for the natural logarithm was introduced by the German mathematician Alfred Pringsheim (1893).

Sine, cosine, tangent, cotangent. W. Outred (middle of the 17th century), I. Bernoulli (18th century), L. Euler (1748, 1753).

Shorthand notation for sine and cosine was introduced by William Outred in the middle of the 17th century. Abbreviations for tangent and cotangent: tg, ctg introduced by Johann Bernoulli in the 18th century, they became widespread in Germany and Russia. In other countries, the names of these functions are used. tan, cot proposed by Albert Girard even earlier, at the beginning of the 17th century. Leonard Euler (1748, 1753) brought the theory of trigonometric functions into its modern form, and we also owe him the consolidation of real symbolism.The term "trigonometric functions" was introduced by the German mathematician and physicist Georg Simon Klugel in 1770.

The sine line of Indian mathematicians was originally called "arha jiva"("semi-string", that is, half of the chord), then the word "archa" was discarded and the sine line began to be called simply "jiva". Arabic translators did not translate the word "jiva" Arabic word "vatar", denoting the bowstring and chord, and transcribed in Arabic letters and began to call the sine line "jiba". Since short vowels are not indicated in Arabic, and long "and" in the word "jiba" denoted in the same way as the semivowel "y", the Arabs began to pronounce the name of the sine line "jibe", which literally means "hollow", "bosom". When translating Arabic works into Latin, European translators translated the word "jibe" Latin word sinus, having the same meaning.The term "tangent" (from lat.tangents- touching) was introduced by the Danish mathematician Thomas Fincke in his Geometry of the Round (1583).

Arcsine. K.Scherfer (1772), J.Lagrange (1772).

Inverse trigonometric functions are mathematical functions that are the inverse of trigonometric functions. The name of the inverse trigonometric function is formed from the name of the corresponding trigonometric function by adding the prefix "arc" (from lat. arc- arc).Inverse trigonometric functions usually include six functions: arcsine (arcsin), arccosine (arccos), arctangent (arctg), arccotangent (arcctg), arcsecant (arcsec) and arccosecant (arccosec). For the first time, special symbols for inverse trigonometric functions were used by Daniel Bernoulli (1729, 1736).Manner of notating inverse trigonometric functions with a prefix arc(from lat. arcus, arc) appeared at the Austrian mathematician Karl Scherfer and gained a foothold thanks to the French mathematician, astronomer and mechanic Joseph Louis Lagrange. It was meant that, for example, the usual sine allows you to find the chord subtending it along the arc of a circle, and the inverse function solves the opposite problem. Until the end of the 19th century, the English and German mathematical schools offered other notation: sin -1 and 1/sin, but they are not widely used.

Hyperbolic sine, hyperbolic cosine. W. Riccati (1757).

Historians discovered the first appearance of hyperbolic functions in the writings of the English mathematician Abraham de Moivre (1707, 1722). The modern definition and detailed study of them was carried out by the Italian Vincenzo Riccati in 1757 in the work "Opusculorum", he also proposed their designations: sh,ch. Riccati proceeded from the consideration of a single hyperbola. An independent discovery and further study of the properties of hyperbolic functions was carried out by the German mathematician, physicist and philosopher Johann Lambert (1768), who established a wide parallelism between the formulas of ordinary and hyperbolic trigonometry. N.I. Lobachevsky subsequently used this parallelism, trying to prove the consistency of non-Euclidean geometry, in which ordinary trigonometry is replaced by hyperbolic.

Just as the trigonometric sine and cosine are the coordinates of a point on a coordinate circle, the hyperbolic sine and cosine are the coordinates of a point on a hyperbola. Hyperbolic functions are expressed in terms of an exponent and are closely related to trigonometric functions: sh(x)=0.5(e x-e-x) , ch(x)=0.5(e x +e -x). By analogy with trigonometric functions, hyperbolic tangent and cotangent are defined as ratios of hyperbolic sine and cosine, cosine and sine, respectively.

Differential. G. Leibniz (1675, in press 1684).

The main, linear part of the function increment.If the function y=f(x) one variable x has at x=x0derivative, and incrementΔy \u003d f (x 0 +? x)-f (x 0)functions f(x) can be represented asΔy \u003d f "(x 0) Δx + R (Δx) , where member R infinitely small compared toΔx. First memberdy=f"(x 0 )Δxin this expansion is called the differential of the function f(x) at the pointx0. AT works of Gottfried Leibniz, Jacob and Johann Bernoulli word"differentia"was used in the sense of "increment", I. Bernoulli denoted it through Δ. G. Leibniz (1675, published in 1684) used the notation for "infinitely small difference"d- the first letter of the word"differential", formed by him from"differentia".

Indefinite integral. G. Leibniz (1675, in press 1686).

The word "integral" was first used in print by Jacob Bernoulli (1690). Perhaps the term is derived from the Latin integer- whole. According to another assumption, the basis was the Latin word integro- restore, restore. The sign ∫ is used to denote an integral in mathematics and is a stylized image of the first letter of a Latin word summa- sum. It was first used by the German mathematician Gottfried Leibniz, the founder of differential and integral calculus, at the end of the 17th century. Another of the founders of differential and integral calculus, Isaac Newton, did not offer an alternative symbolism of the integral in his works, although he tried various options: a vertical bar above a function or a square symbol that stands in front of a function or borders it. Indefinite integral for a function y=f(x) is the collection of all antiderivatives of the given function.

Definite integral. J. Fourier (1819-1822).

Definite integral of a function f(x) with lower limit a and upper limit b can be defined as the difference F(b) - F(a) = a ∫ b f(x)dx , where F(x)- some antiderivative function f(x) . Definite integral a ∫ b f(x)dx numerically equal to the area of ​​\u200b\u200bthe figure bounded by the x-axis, straight lines x=a and x=b and function graph f(x). The French mathematician and physicist Jean Baptiste Joseph Fourier proposed the design of a definite integral in the form we are used to at the beginning of the 19th century.

Derivative. G. Leibniz (1675), J. Lagrange (1770, 1779).

Derivative - the basic concept of differential calculus, characterizing the rate of change of a function f(x) when the argument changes x . It is defined as the limit of the ratio of the increment of a function to the increment of its argument as the increment of the argument tends to zero, if such a limit exists. A function that has a finite derivative at some point is called differentiable at that point. The process of calculating the derivative is called differentiation. The reverse process is integration. In classical differential calculus, the derivative is most often defined through the concepts of the theory of limits, however, historically, the theory of limits appeared later than differential calculus.

The term "derivative" was introduced by Joseph Louis Lagrange in 1797; dy/dx— Gottfried Leibniz in 1675. The manner of designating the derivative with respect to time with a dot above the letter comes from Newton (1691).The Russian term "derivative of a function" was first used by a Russian mathematicianVasily Ivanovich Viskovatov (1779-1812).

Private derivative. A. Legendre (1786), J. Lagrange (1797, 1801).

For functions of many variables, partial derivatives are defined - derivatives with respect to one of the arguments, calculated under the assumption that the remaining arguments are constant. Notation ∂f/ x, z/ y introduced by the French mathematician Adrien Marie Legendre in 1786; fx",zx"- Joseph Louis Lagrange (1797, 1801); 2z/ x2, 2z/ x y- second-order partial derivatives - German mathematician Carl Gustav Jacob Jacobi (1837).

Difference, increment. I. Bernoulli (late 17th century - first half of the 18th century), L. Euler (1755).

The designation of the increment by the letter Δ was first used by the Swiss mathematician Johann Bernoulli. The symbol "delta" entered into common practice after the work of Leonhard Euler in 1755.

Sum. L. Euler (1755).

The sum is the result of adding values ​​(numbers, functions, vectors, matrices, etc.). To denote the sum of n numbers a 1, a 2, ..., a n, the Greek letter "sigma" Σ is used: a 1 + a 2 + ... + a n = Σ n i=1 a i = Σ n 1 a i . The sign Σ for the sum was introduced by Leonhard Euler in 1755.

Work. K. Gauss (1812).

The product is the result of multiplication. To denote the product of n numbers a 1, a 2, ..., a n, the Greek letter "pi" Π is used: a 1 a 2 ... a n = Π n i=1 a i = Π n 1 a i . For example, 1 3 5 ... 97 99 = ? 50 1 (2i-1). The symbol Π for the product was introduced by the German mathematician Carl Gauss in 1812. In Russian mathematical literature, the term "work" was first encountered by Leonty Filippovich Magnitsky in 1703.

Factorial. K.Krump (1808).

The factorial of a number n (denoted n!, pronounced "en factorial") is the product of all natural numbers up to and including n: n! = 1 2 3 ... n. For example, 5! = 1 2 3 4 5 = 120. By definition, 0! = 1. The factorial is defined only for non-negative integers. The factorial of a number n is equal to the number of permutations of n elements. For example, 3! = 6, indeed,

♣ ♦

♦ ♣

♦ ♣

♦ ♣

All six and only six permutations of three elements.

The term "factorial" was introduced by the French mathematician and politician Louis Francois Antoine Arbogast (1800), the designation n! - French mathematician Christian Kramp (1808).

Module, absolute value. K. Weierstrass (1841).

Module, the absolute value of the real number x - a non-negative number defined as follows: |x| = x for x ≥ 0, and |x| = -x for x ≤ 0. For example, |7| = 7, |- 0.23| = -(-0.23) = 0.23. Modulus of a complex number z = a + ib is a real number equal to √(a 2 + b 2).

It is believed that the term "module" was proposed to be used by the English mathematician and philosopher, a student of Newton, Roger Cotes. Gottfried Leibniz also used this function, which he called "module" and denoted: mol x. The generally accepted notation for the absolute value was introduced in 1841 by the German mathematician Karl Weierstrass. For complex numbers, this concept was introduced by the French mathematicians Augustin Cauchy and Jean Robert Argan at the beginning of the 19th century. In 1903, the Austrian scientist Konrad Lorenz used the same symbolism for the length of a vector.

Norm. E. Schmidt (1908).

A norm is a functional defined on a vector space and generalizing the concept of the length of a vector or the modulus of a number. The sign "norm" (from the Latin word "norma" - "rule", "sample") was introduced by the German mathematician Erhard Schmidt in 1908.

Limit. S. Luillier (1786), W. Hamilton (1853), many mathematicians (until the beginning of the 20th century)

Limit - one of the basic concepts of mathematical analysis, meaning that some variable value in the process of its change under consideration approaches a certain constant value indefinitely. The concept of a limit was used intuitively as early as the second half of the 17th century by Isaac Newton, as well as by mathematicians of the 18th century, such as Leonhard Euler and Joseph Louis Lagrange. The first rigorous definitions of the limit of a sequence were given by Bernard Bolzano in 1816 and Augustin Cauchy in 1821. The symbol lim (the first 3 letters from the Latin word limes - border) appeared in 1787 with the Swiss mathematician Simon Antoine Jean Lhuillier, but its use did not yet resemble the modern one. The expression lim in a more familiar form for us was first used by the Irish mathematician William Hamilton in 1853.Weierstrass introduced a designation close to the modern one, but instead of the usual arrow, he used the equal sign. The arrow appeared at the beginning of the 20th century with several mathematicians at once - for example, with the English mathematician Godfried Hardy in 1908.

Zeta function, d Riemann zeta function. B. Riemann (1857).

Analytic function of the complex variable s = σ + it, for σ > 1, determined by the absolutely and uniformly convergent Dirichlet series:

ζ(s) = 1 -s + 2 -s + 3 -s + ... .

For σ > 1, the representation in the form of the Euler product is valid:

ζ(s) = Π p (1-p -s) -s ,

where the product is taken over all primes p. The zeta function plays a big role in number theory.As a function of a real variable, the zeta function was introduced in 1737 (published in 1744) by L. Euler, who indicated its decomposition into a product. Then this function was considered by the German mathematician L. Dirichlet and, especially successfully, by the Russian mathematician and mechanic P.L. Chebyshev in the study of the law of distribution of prime numbers. However, the most profound properties of the zeta function were discovered later, after the work of the German mathematician Georg Friedrich Bernhard Riemann (1859), where the zeta function was considered as a function of a complex variable; he also introduced the name "zeta function" and the notation ζ(s) in 1857.

Gamma function, Euler Γ-function. A. Legendre (1814).

The gamma function is a mathematical function that extends the notion of factorial to the field of complex numbers. Usually denoted by Γ(z). The z-function was first introduced by Leonhard Euler in 1729; it is defined by the formula:

Γ(z) = limn→∞ n! n z /z(z+1)...(z+n).

A large number of integrals, infinite products, and sums of series are expressed through the G-function. Widely used in analytic number theory. The name "Gamma function" and the notation Γ(z) were proposed by the French mathematician Adrien Marie Legendre in 1814.

Beta function, B function, Euler B function. J. Binet (1839).

A function of two variables p and q, defined for p>0, q>0 by the equality:

B(p, q) = 0 ∫ 1 x p-1 (1-x) q-1 dx.

The beta function can be expressed in terms of the Γ-function: В(p, q) = Γ(p)Г(q)/Г(p+q).Just as the gamma function for integers is a generalization of the factorial, the beta function is, in a sense, a generalization of the binomial coefficients.

Many properties are described using the beta function.elementary particles participating in strong interaction. This feature was noticed by the Italian theoretical physicistGabriele Veneziano in 1968. It started string theory.

The name "beta function" and the notation B(p, q) were introduced in 1839 by the French mathematician, mechanic and astronomer Jacques Philippe Marie Binet.

Laplace operator, Laplacian. R. Murphy (1833).

Linear differential operator Δ, which functions φ (x 1, x 2, ..., x n) from n variables x 1, x 2, ..., x n associates the function:

Δφ \u003d ∂ 2 φ / ∂x 1 2 + ∂ 2 φ / ∂x 2 2 + ... + ∂ 2 φ / ∂x n 2.

In particular, for a function φ(x) of one variable, the Laplace operator coincides with the operator of the 2nd derivative: Δφ = d 2 φ/dx 2 . The equation Δφ = 0 is usually called the Laplace equation; this is where the names "Laplace operator" or "Laplacian" come from. The notation Δ was introduced by the English physicist and mathematician Robert Murphy in 1833.

Hamiltonian operator, nabla operator, Hamiltonian. O. Heaviside (1892).

Vector differential operator of the form

∇ = ∂/∂x i+ ∂/∂y j+ ∂/∂z k,

where i, j, and k- coordinate vectors. Through the nabla operator, the basic operations of vector analysis, as well as the Laplace operator, are expressed in a natural way.

In 1853, the Irish mathematician William Rowan Hamilton introduced this operator and coined the symbol ∇ for it in the form of an inverted Greek letter Δ (delta). At Hamilton, the point of the symbol pointed to the left; later, in the works of the Scottish mathematician and physicist Peter Guthrie Tate, the symbol acquired a modern look. Hamilton called this symbol the word "atled" (the word "delta" read backwards). Later, English scholars, including Oliver Heaviside, began to call this symbol "nabla", after the name of the letter ∇ in the Phoenician alphabet, where it occurs. The origin of the letter is associated with a musical instrument such as the harp, ναβλα (nabla) in ancient Greek means "harp". The operator was called the Hamilton operator, or the nabla operator.

Function. I. Bernoulli (1718), L. Euler (1734).

A mathematical concept that reflects the relationship between elements of sets. We can say that a function is a "law", a "rule" according to which each element of one set (called the domain of definition) is associated with some element of another set (called the domain of values). The mathematical concept of a function expresses an intuitive idea of ​​how one quantity completely determines the value of another quantity. Often the term "function" means a numerical function; that is, a function that puts some numbers in line with others. For a long time, mathematicians gave arguments without brackets, for example, like this - φх. This notation was first used by the Swiss mathematician Johann Bernoulli in 1718.Parentheses were only used if there were many arguments, or if the argument was a complex expression. Echoes of those times are common and now recordssin x, lg xetc. But gradually the use of parentheses, f(x) , became the general rule. And the main merit in this belongs to Leonhard Euler.

Equality. R. Record (1557).

The equal sign was proposed by the Welsh physician and mathematician Robert Record in 1557; the character's outline was much longer than the current one, as it imitated the image of two parallel segments. The author explained that there is nothing more equal in the world than two parallel segments of the same length. Before that, in ancient and medieval mathematics, equality was denoted verbally (for example, est egale). Rene Descartes in the 17th century began to use æ (from lat. aequalis), and he used the modern equals sign to indicate that the coefficient could be negative. François Viète denoted subtraction with an equals sign. The symbol of the Record did not spread immediately. The spread of the Record symbol was hindered by the fact that since ancient times the same symbol has been used to indicate the parallelism of lines; in the end, it was decided to make the symbol of parallelism vertical. In continental Europe, the sign "=" was introduced by Gottfried Leibniz only at the turn of the 17th-18th centuries, that is, more than 100 years after the death of Robert Record, who first used it for this.

About the same, about the same. A. Günther (1882).

Sign " ≈" was introduced by German mathematician and physicist Adam Wilhelm Sigmund Günther in 1882 as a symbol for the relationship "about equal".

More less. T. Harriot (1631).

These two signs were introduced into use by the English astronomer, mathematician, ethnographer and translator Thomas Harriot in 1631, before that the words "more" and "less" were used.

Comparability. K. Gauss (1801).

Comparison - the ratio between two integers n and m, meaning that the difference n-m of these numbers is divided by a given integer a, called the modulus of comparison; it is written: n≡m(mod a) and reads "numbers n and m are comparable modulo a". For example, 3≡11(mod 4) since 3-11 is divisible by 4; the numbers 3 and 11 are congruent modulo 4. Comparisons have many properties similar to those of equalities. So, the term in one part of the comparison can be transferred with the opposite sign to another part, and comparisons with the same module can be added, subtracted, multiplied, both parts of the comparison can be multiplied by the same number, etc. For example,

3≡9+2(mod 4) and 3-2≡9(mod 4)

At the same time true comparisons. And from a pair of true comparisons 3≡11(mod 4) and 1≡5(mod 4) the correctness of the following follows:

3+1≡11+5(mod 4)

3-1≡11-5(mod 4)

3 1≡11 5(mod 4)

3 2 ≡11 2 (mod 4)

3 23≡11 23(mod 4)

In number theory, methods for solving various comparisons are considered, i.e. methods for finding integers that satisfy comparisons of one kind or another. Modulo comparisons were first used by the German mathematician Carl Gauss in his 1801 book Arithmetical Investigations. He also proposed the symbolism established in mathematics for comparison.

Identity. B. Riemann (1857).

Identity - the equality of two analytical expressions, valid for any admissible values ​​of the letters included in it. The equality a+b = b+a is valid for all numerical values ​​of a and b, and therefore is an identity. To record identities, in some cases, since 1857, the sign "≡" (read "identically equal") has been used, the author of which in this use is the German mathematician Georg Friedrich Bernhard Riemann. Can be written a+b ≡ b+a.

Perpendicularity. P.Erigon (1634).

Perpendicularity - the mutual arrangement of two straight lines, planes or a straight line and a plane, in which these figures make a right angle. The sign ⊥ to denote perpendicularity was introduced in 1634 by the French mathematician and astronomer Pierre Erigon. The concept of perpendicularity has a number of generalizations, but all of them, as a rule, are accompanied by the sign ⊥ .

Parallelism. W. Outred (1677 posthumous edition).

Parallelism - the relationship between some geometric shapes; for example, straight lines. Defined differently depending on different geometries; for example, in the geometry of Euclid and in the geometry of Lobachevsky. The sign of parallelism has been known since ancient times, it was used by Heron and Pappus of Alexandria. At first, the symbol was similar to the current equals sign (only more extended), but with the advent of the latter, to avoid confusion, the symbol was turned vertically ||. It appeared in this form for the first time in a posthumous edition of the works of the English mathematician William Outred in 1677.

Intersection, union. J. Peano (1888).

An intersection of sets is a set that contains those and only those elements that simultaneously belong to all given sets. The union of sets is a set that contains all the elements of the original sets. Intersection and union are also called operations on sets that assign new sets to certain sets according to the above rules. Denoted ∩ and ∪, respectively. For example, if

A= (♠ ♣ ) and B= (♣ ♦ ),

That

A∩B= {♣ }

A∪B= {♠ ♣ ♦ } .

Contains, contains. E. Schroeder (1890).

If A and B are two sets and there are no elements in A that do not belong to B, then they say that A is contained in B. They write A⊂B or B⊃A (B contains A). For example,

{♠}⊂{♠ ♣}⊂{♠ ♣ ♦ }

{♠ ♣ ♦ }⊃{ ♦ }⊃{♦ }

The symbols "contains" and "contains" appeared in 1890 with the German mathematician and logician Ernst Schroeder.

Affiliation. J. Peano (1895).

If a is an element of the set A, then write a∈A and read "a belongs to A". If a is not an element of A, write a∉A and read "a does not belong to A". Initially, the relations "contained" and "belongs" ("is an element") were not distinguished, but over time, these concepts required a distinction. The membership sign ∈ was first used by the Italian mathematician Giuseppe Peano in 1895. The symbol ∈ comes from the first letter of the Greek word εστι - to be.

The universal quantifier, the existential quantifier. G. Gentzen (1935), C. Pierce (1885).

A quantifier is a general name for logical operations that indicate the area of ​​truth of a predicate (mathematical statement). Philosophers have long paid attention to logical operations that limit the scope of the truth of a predicate, but did not single them out as a separate class of operations. Although quantifier-logical constructions are widely used both in scientific and everyday speech, their formalization took place only in 1879, in the book of the German logician, mathematician and philosopher Friedrich Ludwig Gottlob Frege "The Calculus of Concepts". Frege's notation looked like cumbersome graphic constructions and was not accepted. Subsequently, many more successful symbols were proposed, but the notation ∃ for the existential quantifier (read "exists", "there is"), proposed by the American philosopher, logician and mathematician Charles Pierce in 1885, and ∀ for the universal quantifier (read "any" , "every", "every"), formed by the German mathematician and logician Gerhard Karl Erich Gentzen in 1935 by analogy with the existential quantifier symbol (the reversed first letters of the English words Existence (existence) and Any (any)). For example, the entry

(∀ε>0) (∃δ>0) (∀x≠x 0 , |x-x 0 |<δ) (|f(x)-A|<ε)

reads as follows: "for any ε>0 there exists δ>0 such that for all x not equal to x 0 and satisfying the inequality |x-x 0 |<δ, выполняется неравенство |f(x)-A|<ε".

Empty set. N. Bourbaki (1939).

A set that does not contain any element. The empty set sign was introduced in the books of Nicolas Bourbaki in 1939. Bourbaki is the collective pseudonym of a group of French mathematicians formed in 1935. One of the members of the Bourbaki group was Andre Weil, the author of the Ø symbol.

Q.E.D. D. Knuth (1978).

In mathematics, a proof is understood as a sequence of reasoning based on certain rules, showing that a certain statement is true. Since the Renaissance, the end of a proof has been denoted by mathematicians as "Q.E.D.", from the Latin expression "Quod Erat Demonstrandum" - "What was required to be proved." When creating the computer layout system ΤΕΧ in 1978, the American computer science professor Donald Edwin Knuth used a symbol: a filled square, the so-called "Halmos symbol", named after the American mathematician of Hungarian origin Paul Richard Halmos. Today, the completion of a proof is usually denoted by the Halmos Symbol. As an alternative, other signs are used: an empty square, a right triangle, // (two slashes), as well as the Russian abbreviation "ch.t.d.".

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Chlorine Cl2 Hydrogen chloride HCl, aka hydrochloric acid. Refrigerants (refrigerants). Refrigerant (Refrigerant) R11 - Fluorotrichloromethane (CFCI3) Refrigerant (Refrigerant) R12 - Difluorodichloromethane (CF2CCl2) Refrigerant (Refrigerant) R125 - Pentafluoroethane (CF2HCF3). Refrigerant (Refrigerant) R134a - 1,1,1,2-Tetrafluoroethane (CF3CFH2). Refrigerant (Refrigerant) R22 - Difluorochloromethane (CF2ClH) Refrigerant (Refrigerant) R32 - Difluoromethane (CH2F2). Refrigerant (Refrigerant) R407C - R-32 (23%) / R-125 (25%) / R-134a (52%) / Percent by mass. other Materials - thermal properties Abrasives - grit, fineness, grinding equipment. Soil, earth, sand and other rocks. Indicators of loosening, shrinkage and density of soils and rocks. Shrinkage and loosening, loads. Slope angles. Heights of ledges, dumps. Wood. Lumber. Timber. Logs. Firewood… Ceramics. 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Corrosion resistance. Sealing materials - joint sealants. PTFE (fluoroplast-4) and derivative materials. FUM tape. Anaerobic adhesives Non-drying (non-hardening) sealants. Silicone sealants (organosilicon). Graphite, asbestos, paronites and derived materials Paronite. Thermally expanded graphite (TRG, TMG), compositions. Properties. Application. Production. Flax sanitary Seals of rubber elastomers Insulators and heat-insulating materials. (link to the project section) Engineering techniques and concepts Explosion protection. Environmental protection. Corrosion. Climatic modifications (Material Compatibility Tables) Classes of pressure, temperature, tightness Drop (loss) of pressure. — Engineering concept. Fire protection. Fires. Theory of automatic control (regulation). TAU Mathematical Handbook Arithmetic, Geometric progressions and sums of some numerical series. Geometric figures. Properties, formulas: perimeters, areas, volumes, lengths. Triangles, Rectangles, etc. Degrees to radians. flat figures. Properties, sides, angles, signs, perimeters, equalities, similarities, chords, sectors, areas, etc. Areas of irregular figures, volumes of irregular bodies. The average value of the signal. Formulas and methods for calculating the area. Graphs. Construction of graphs. Reading charts. Integral and differential calculus. Tabular derivatives and integrals. Derivative table. Table of integrals. Table of primitives. Find derivative. Find the integral. Diffury. Complex numbers. imaginary unit. Linear algebra. (Vectors, matrices) Mathematics for the little ones. Kindergarten - 7th grade. Mathematical logic. Solution of equations. Quadratic and biquadratic equations. Formulas. Methods. Solution of differential equations Examples of solutions to ordinary differential equations of order higher than the first. Examples of solutions to the simplest = analytically solvable ordinary differential equations of the first order. Coordinate systems. Rectangular Cartesian, polar, cylindrical and spherical. Two-dimensional and three-dimensional. Number systems. Numbers and digits (real, complex, ....). Tables of number systems. Power series of Taylor, Maclaurin (=McLaren) and periodic Fourier series. Decomposition of functions into series. Tables of logarithms and basic formulas Tables of numerical values ​​Tables of Bradys. Probability theory and statistics Trigonometric functions, formulas and graphs. sin, cos, tg, ctg….Values ​​of trigonometric functions. Formulas for reducing trigonometric functions. Trigonometric identities. Numerical methods Equipment - standards, dimensions Household appliances, home equipment. Drainage and drainage systems. Capacities, tanks, reservoirs, tanks. Instrumentation and control Instrumentation and automation. Temperature measurement. Conveyors, belt conveyors. Containers (link) Laboratory equipment. Pumps and pumping stations Pumps for liquids and pulps. Engineering jargon. Dictionary. Screening. Filtration. Separation of particles through grids and sieves. Approximate strength of ropes, cables, cords, ropes made of various plastics. Rubber products. Joints and attachments. Diameters conditional, nominal, Du, DN, NPS and NB. Metric and inch diameters. SDR. Keys and keyways. Communication standards. Signals in automation systems (I&C) Analog input and output signals of instruments, sensors, flow meters and automation devices. connection interfaces. Communication protocols (communications) Telephony. Pipeline accessories. Cranes, valves, gate valves…. Building lengths. Flanges and threads. Standards. Connecting dimensions. threads. Designations, dimensions, use, types ... (reference link) Connections ("hygienic", "aseptic") pipelines in the food, dairy and pharmaceutical industries. Pipes, pipelines. Pipe diameters and other characteristics. Choice of pipeline diameter. Flow rates. Expenses. Strength. Selection tables, Pressure drop. Copper pipes. Pipe diameters and other characteristics. Polyvinyl chloride pipes (PVC). Pipe diameters and other characteristics. Pipes are polyethylene. Pipe diameters and other characteristics. Pipes polyethylene PND. Pipe diameters and other characteristics. Steel pipes (including stainless steel). Pipe diameters and other characteristics. The pipe is steel. The pipe is stainless. Stainless steel pipes. Pipe diameters and other characteristics. The pipe is stainless. Carbon steel pipes. Pipe diameters and other characteristics. The pipe is steel. Fitting. Flanges according to GOST, DIN (EN 1092-1) and ANSI (ASME). Flange connection. Flange connections. Flange connection. Elements of pipelines. Electric lamps Electrical connectors and wires (cables) Electric motors. Electric motors. Electrical switching devices. (Link to section) Standards for the personal life of engineers Geography for engineers. Distances, routes, maps….. Engineers in everyday life. Family, children, recreation, clothing and housing. Children of engineers. Engineers in offices. Engineers and other people. Socialization of engineers. Curiosities. Resting engineers. This shocked us. Engineers and food. Recipes, utility. Tricks for restaurants. International trade for engineers. We learn to think in a huckster way. Transport and travel. Private cars, bicycles…. Physics and chemistry of man. Economics for engineers. Bormotologiya financiers - human language. Technological concepts and drawings Paper writing, drawing, office and envelopes. Standard photo sizes. Ventilation and air conditioning. Water supply and sewerage Hot water supply (DHW). Drinking water supply Waste water. Cold water supply Galvanic industry Refrigeration Steam lines / systems. Condensate lines / systems. Steam lines. Condensate pipelines. Food industry Supply of natural gas Welding metals Symbols and designations of equipment on drawings and diagrams. Symbolic graphic representations in projects of heating, ventilation, air conditioning and heat and cold supply, according to ANSI / ASHRAE Standard 134-2005. Sterilization of equipment and materials Heat supply Electronic industry Power supply Physical reference Alphabets. Accepted designations. Basic physical constants. Humidity is absolute, relative and specific. Air humidity. Psychrometric tables. Ramzin diagrams. Time Viscosity, Reynolds number (Re). Viscosity units. Gases. Properties of gases. Individual gas constants. Pressure and Vacuum Vacuum Length, distance, linear dimension Sound. Ultrasound. Sound absorption coefficients (link to another section) Climate. climate data. natural data. SNiP 23-01-99. Building climatology. (Statistics of climatic data) SNIP 23-01-99. Table 3 - Average monthly and annual air temperature, ° С. Former USSR. SNIP 23-01-99 Table 1. Climatic parameters of the cold period of the year. RF. SNIP 23-01-99 Table 2. Climatic parameters of the warm season. Former USSR. SNIP 23-01-99 Table 2. Climatic parameters of the warm season. RF. SNIP 23-01-99 Table 3. Average monthly and annual air temperature, °С. RF. SNiP 23-01-99. Table 5a* - Average monthly and annual partial pressure of water vapor, hPa = 10^2 Pa. RF. SNiP 23-01-99. Table 1. Climatic parameters of the cold season. Former USSR. Density. Weight. Specific gravity. Bulk density. Surface tension. Solubility. Solubility of gases and solids. Light and color. Reflection, absorption and refraction coefficients Color alphabet:) - Designations (codings) of color (colors). Properties of cryogenic materials and media. Tables. Friction coefficients for various materials. Thermal quantities, including temperatures of boiling, melting, flame, etc…… for more information, see: Adiabatic coefficients (indicators). Convection and full heat exchange. Coefficients of thermal linear expansion, thermal volumetric expansion. Temperatures, boiling, melting, other… Conversion of temperature units. Flammability. softening temperature. Boiling points Melting points Thermal conductivity. Thermal conductivity coefficients. Thermodynamics. Specific heat of vaporization (condensation). Enthalpy of vaporization. Specific heat of combustion (calorific value). The need for oxygen. Electric and magnetic quantities Electric dipole moments. The dielectric constant. Electrical constant. Lengths of electromagnetic waves (a reference book of another section) Magnetic field strengths Concepts and formulas for electricity and magnetism. Electrostatics. Piezoelectric modules. Electrical strength of materials Electrical current Electrical resistance and conductivity. Electronic potentials Chemical reference book "Chemical alphabet (dictionary)" - names, abbreviations, prefixes, designations of substances and compounds. Aqueous solutions and mixtures for metal processing. Aqueous solutions for the application and removal of metal coatings Aqueous solutions for removing carbon deposits (tar deposits, carbon deposits from internal combustion engines ...) Aqueous solutions for passivation. Aqueous solutions for etching - removing oxides from the surface Aqueous solutions for phosphating Aqueous solutions and mixtures for chemical oxidation and coloring of metals. Aqueous solutions and mixtures for chemical polishing Degreasing aqueous solutions and organic solvents pH. pH tables. Burning and explosions. Oxidation and reduction. Classes, categories, designations of danger (toxicity) of chemical substances Periodic system of chemical elements of DI Mendeleev. Periodic table. Density of organic solvents (g/cm3) depending on temperature. 0-100 °С. Properties of solutions. Dissociation constants, acidity, basicity. Solubility. Mixes. Thermal constants of substances. Enthalpy. entropy. Gibbs energy… (link to the chemical reference book of the project) Electrical engineering Regulators Uninterrupted power supply systems. Dispatch and control systems Structured cabling systems Data centers