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157-419: A gear or gearwheel is a rotating machine part typically used to transmit rotational motion and/or torque by means of a series of teeth that engage with compatible teeth of another gear or other part. The teeth can be integral saliences or cavities machined on the part, or separate pegs inserted into it. In the latter case, the gear is usually called a cogwheel . A cog may be one of those pegs or

314-420: A bevel gear , whose overall shape is like a slice ( frustum ) of a cone whose apex is the meeting point of the two axes. Bevel gears with equal numbers of teeth and shaft axes at 90 degrees are called miter (US) or mitre (UK) gears. Independently of the angle between the axes, the larger of two unequal matching bevel gears may be internal or external, depending the desired relative sense of rotation. If

471-415: A differential . Whereas a regular (nonhypoid) ring-and-pinion gear set is suitable for many applications, it is not ideal for vehicle drive trains because it generates more noise and vibration than a hypoid does. Bringing hypoid gears to market for mass-production applications was an engineering improvement of the 1920s. Rotating The special case of a rotation with an internal axis passing through

628-452: A rack , a straight bar with a row of compatible teeth. Gears are among the most common mechanical parts. They come in a great variety of shapes and materials, and are used for many different functions and applications. Diameters may range from a few μm in micromachines , to a few mm in watches and toys to over 10 metres in some mining equipment. Other types of parts that are somewhat similar in shape and function to gears include

785-399: A screw . It is possible for objects to have periodic circular trajectories without changing their orientation . These types of motion are treated under circular motion instead of rotation, more specifically as a curvilinear translation. Since translation involves displacement of rigid bodies while preserving the orientation of the body, in the case of curvilinear translation, all

942-633: A Greek polis founded by Alexander, became the capital city and a major center of Greek culture, learning, and trade for the next several centuries. Following the Syrian Wars with the Seleucid Empire , a rival Hellenistic state, the Ptolemaic Kingdom expanded its territory to include eastern Libya , the Sinai , and northern Nubia . To legitimize their rule and gain recognition from native Egyptians,

1099-408: A Ptolemaic queen deified Arsinoe II as Hera. Coins from this period also show Arsinoe II with a diadem that is solely worn by goddesses and deified royal women. The Statuette of Arsinoe II was created c. 150–100 BC, well after her death, as a part of her own specific posthumous cult which was started by her husband Ptolemy II. The figure also exemplifies the fusing of Greek and Egyptian art. Although

1256-751: A Roman woman). Their union produced three children; the twins Cleopatra Selene and Alexander Helios , and another son, Ptolemy Philadelphos . Mark Antony's alliance with Cleopatra angered Rome even more. Branded a power-hungry enchantress by the Romans, she was accused of seducing Antony to further her conquest of Rome. Further outrage followed at the donations of Alexandria ceremony in autumn of 34 BC in which Tarsus , Cyrene , Crete , Cyprus , and Judaea were all to be given as client monarchies to Antony's children by Cleopatra. In his will Antony expressed his desire to be buried in Alexandria, rather than taken to Rome in

1413-463: A chosen reference point. Hence, the distinction between rotation and circular motion can be made by requiring an instantaneous axis for rotation, a line passing through instantaneous center of circle and perpendicular to the plane of motion . In the example depicting curvilinear translation, the center of circles for the motion lie on a straight line but it is parallel to the plane of motion and hence does not resolve to an axis of rotation. In contrast,

1570-472: A continuation of developments based on Egyptian art tradition from the Thirtieth Dynasty . Such behavior expanded the rulers' social and political capital and demonstrated their loyalty toward Egyptian deities, to the satisfaction of the local people. Temples remained very New Kingdom and Late Period Egyptian in style though resources were oftentimes provided by foreign powers. Temples were models of

1727-601: A direction that limits the rotation angle to not exceed 180 degrees. (This can always be done because any rotation of more than 180 degrees about an axis m {\displaystyle m} can always be written as a rotation having 0 ≤ α ≤ 180 ∘ {\displaystyle 0\leq \alpha \leq 180^{\circ }} if the axis is replaced with n = − m {\displaystyle n=-m} .) Every proper rotation A {\displaystyle A} in 3D space has an axis of rotation, which

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1884-435: A fixed point is always equivalent to a rotation about an axis (which may be considered to be a rotation in the plane that is perpendicular to that axis). Similarly, the rotation rate of an object in three dimensions at any instant is about some axis, although this axis may be changing over time. In other than three dimensions, it does not make sense to describe a rotation as being around an axis, since more than one axis through

2041-609: A geared astrolabe was built in Isfahan showing the position of the moon in the zodiac and its phase , and the number of days since new moon. The worm gear was invented in the Indian subcontinent , for use in roller cotton gins , some time during the 13th–14th centuries. A complex astronomical clock, called the Astrarium , was built between 1348 and 1364 by Giovanni Dondi dell'Orologio . It had seven faces and 107 moving parts; it showed

2198-649: A long but minor presence in Egypt long before Alexander the Great founded the city of Alexandria. It began when Greek colonists, encouraged by many Pharaohs, set up the trading post of Naucratis . As Egypt came under foreign domination and decline, the Pharaohs depended on the Greeks as mercenaries and even advisors. When the Persians took over Egypt, Naucratis remained an important Greek port and

2355-447: A mechanism, so that in case of jamming they will fail first and thus avoid damage to more expensive parts. Such sacrificial gears may be a simpler alternative to other overload-protection devices such as clutches and torque- or current-limited motors. In spite of the advantages of metal and plastic, wood continued to be used for large gears until a couple of centuries ago, because of cost, weight, tradition, or other considerations. In 1967

2512-515: A motor communicates motion' is from 1814; specifically of a vehicle (bicycle, automobile, etc.) by 1888. A cog is a tooth on a wheel. From Middle English cogge, from Old Norse (compare Norwegian kugg ('cog'), Swedish kugg , kugge ('cog, tooth')), from Proto-Germanic * kuggō (compare Dutch kogge (' cogboat '), German Kock ), from Proto-Indo-European * gugā ('hump, ball') (compare Lithuanian gugà ('pommel, hump, hill'), from PIE * gēw- ('to bend, arch'). First used c. 1300 in

2669-464: A nonzero magnitude. This discussion applies to a proper rotation, and hence det A = 1 {\displaystyle \det A=1} . Any improper orthogonal 3x3 matrix B {\displaystyle B} may be written as B = − A {\displaystyle B=-A} , in which A {\displaystyle A} is proper orthogonal. That is, any improper orthogonal 3x3 matrix may be decomposed as

2826-447: A palace built by Caesar in their honor. In 44 BC, Caesar was murdered in Rome by several Senators . With his death, Rome split between supporters of Mark Antony and Octavian . When Mark Antony seemed to prevail, Cleopatra supported him and, shortly after, they too became lovers and eventually married in Egypt (though their marriage was never recognized by Roman law, as Antony was married to

2983-416: A plane), in which exactly one point is kept fixed; and also in three dimensions (in space), in which additional points may be kept fixed (as in rotation around a fixed axis, as infinite line). All rigid body movements are rotations, translations, or combinations of the two. A rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion. The axis

3140-430: A plane, which is perpendicular to the rotation axis, and which is left invariant by the rotation. The rotation, restricted to this plane, is an ordinary 2D rotation. The proof proceeds similarly to the above discussion. First, suppose that all eigenvalues of the 3D rotation matrix A are real. This means that there is an orthogonal basis, made by the corresponding eigenvectors (which are necessarily orthogonal), over which

3297-678: A pointer on top of the chariot kept the direction of latter unchanged as the chariot turned. Another early surviving example of geared mechanism is a complex calendrical device showing the phase of the Moon, the day of the month and the places of the Sun and the Moon in the Zodiac was invented in the Byzantine empire in the early 6th century AD. Geared mechanical water clocks were built in China by 725 AD. Around 1221 AD,

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3454-658: A practice that, while pleasing to the Egyptian population, had serious consequences in later reigns. The material and literary splendour of the Alexandrian court was at its height under Ptolemy II. Callimachus , keeper of the Library of Alexandria , Theocritus , and a host of other poets, glorified the Ptolemaic family. Ptolemy himself was eager to increase the library and to patronise scientific research. He spent lavishly on making Alexandria

3611-428: A proper rotation (from which an axis of rotation can be found as described above) followed by an inversion (multiplication by −1). It follows that the rotation axis of A {\displaystyle A} is also the eigenvector of B {\displaystyle B} corresponding to an eigenvalue of −1. As much as every tridimensional rotation has a rotation axis, also every tridimensional rotation has

3768-403: A puppet king. Philometor's younger brother (later Ptolemy VIII Physcon ) was installed as king by the Ptolemaic court in Alexandria. When Antiochus withdrew, the brothers agreed to reign jointly with their sister Cleopatra II . They soon fell out, however, and quarrels between the two brothers allowed Rome to interfere and to steadily increase its influence in Egypt. Philometor eventually regained

3925-528: A range from idealistic to realistic. An example of realistic portrayal is the Berlin Green Head, which shows the non-idealistic facial features with vertical lines above the bridge of the nose, lines at the corners of the eyes and between the nose and the mouth. The influence of Greek art was shown in an emphasis on the face that was not previously present in Egyptian art and incorporation of Greek elements into an Egyptian setting: individualistic hairstyles,

4082-488: A relation of rate of change of unit vector by taking A → {\displaystyle {\vec {A}}} , to be such a vector: d A ^ d t ⋅ A ^ = 0 {\displaystyle {d{\hat {A}} \over dt}\cdot {\hat {A}}=0} showing that d A ^ d t {\textstyle {d{\hat {A}} \over dt}} vector

4239-476: A research centre located in the royal sector of the city. Its scholars were housed in the same sector and funded by Ptolemaic rulers. The chief librarian served also as the crown prince's tutor. For the first hundred and fifty years of its existence, the library drew the top Greek scholars from all over the Hellenistic world. It was a key academic, literary and scientific centre in antiquity. Greek culture had

4396-507: A rotating body will always have its instantaneous axis of zero velocity, perpendicular to the plane of motion. More generally, due to Chasles' theorem , any motion of rigid bodies can be treated as a composition of rotation and translation , called general plane motion. A simple example of pure rotation is considered in rotation around a fixed axis . The laws of physics are currently believed to be invariant under any fixed rotation . (Although they do appear to change when viewed from

4553-417: A rotating viewpoint: see rotating frame of reference .) In modern physical cosmology, the cosmological principle is the notion that the distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throughout the universe and have no preferred direction, and should, therefore, produce no observable irregularities in

4710-470: A rotation, e.g. a translation. Rotations around the x , y and z axes are called principal rotations . Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis, and followed by a rotation around the z axis. That is to say, any spatial rotation can be decomposed into a combination of principal rotations. The combination of any sequence of rotations of an object in three dimensions about

4867-461: A series of civil wars and feuds between the members of the Ptolemaic dynasty started and lasted for over a century. Philometor was succeeded by yet another infant, his son Ptolemy VII Neos Philopator . But Physcon soon returned, killed his young nephew, seized the throne and as Ptolemy VIII soon proved himself a cruel tyrant. On his death in 116 BC he left the kingdom to his wife Cleopatra III and her son Ptolemy IX Philometor Soter II . The young king

Gear - Misplaced Pages Continue

5024-605: A series of wooden pegs or cogs around the rim of a wheel. The cogs were often made of maple wood. Wooden gears have been gradually replaced by ones made or metal, such as cast iron at first, then steel and aluminum . Steel is most commonly used because of its high strength-to-weight ratio and low cost. Aluminum is not as strong as steel for the same geometry, but is lighter and easier to machine. powder metallurgy may be used with alloys that cannot be easily cast or machined. Still, because of cost or other considerations, some early metal gears had wooden cogs, each tooth forming

5181-436: A small gear drives a larger one, the mechanical advantage of this ideal lever causes the torque T to increase but the rotational speed ω to decrease. The opposite effect is obtained when a large gear drives a small one. The changes are proportional to the gear ratio r , the ratio of the tooth counts. namely, T 2 / T 1 = r = N 2 / N 1 , and ω 2 / ω 1 = 1/ r = N 1 / N 2 . Depending on

5338-418: A type of specialised 'through' mortise and tenon joint More recently engineering plastics and composite materials have been replacing metals in many applications, especially those with moderate speed and torque. They are not as strong as steel, but are cheaper, can be mass-manufactured by injection molding don't need lubrication. Plastic gears may even be intentionally designed to be the weakest part in

5495-471: A typical Egyptian Pharaoh and actively supported the Egyptian priestly elite through donations and temple construction. Ptolemy III had introduced an important innovation in 238 BC by holding a synod of all the priests of Egypt at Canopus . Ptolemy IV continued this tradition by holding his own synod at Memphis in 217 BC, after the victory celebrations of the Fourth Syrian War. The result of this synod

5652-481: Is an invariant, the rotation angle α {\displaystyle \alpha } for a proper orthogonal 3×3 rotation matrix A {\displaystyle A} is found by Using the principal arc-cosine, this formula gives a rotation angle satisfying 0 ≤ α ≤ 180 ∘ {\displaystyle 0\leq \alpha \leq 180^{\circ }} . The corresponding rotation axis must be defined to point in

5809-449: Is called tidal locking ; the Moon is tidal-locked to the Earth. This rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravitation the closer one is to the equator . Earth's gravity combines both mass effects such that an object weighs slightly less at the equator than at the poles. Another is that over time the Earth

5966-467: Is conserved . Euler rotations provide an alternative description of a rotation. It is a composition of three rotations defined as the movement obtained by changing one of the Euler angles while leaving the other two constant. Euler rotations are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. They constitute a mixed axes of rotation system, where

6123-431: Is defined such that any vector v {\displaystyle v} that is aligned with the rotation axis will not be affected by rotation. Accordingly, A v = v {\displaystyle Av=v} , and the rotation axis therefore corresponds to an eigenvector of the rotation matrix associated with an eigenvalue of 1. As long as the rotation angle α {\displaystyle \alpha }

6280-505: Is made of Cleopatra and Antony's sons in the known historical texts of that time, but their daughter Cleopatra Selene was eventually married through arrangement by Octavian into the Mauretanian royal line, one of Rome's many client monarchies. Through Cleopatra Selene's offspring the Ptolemaic line intermarried back into the Roman nobility for centuries. With the deaths of Cleopatra and Caesarion,

6437-439: Is no direction in the plane which is kept unchanged by a 2-dimensional rotation, except, of course, the identity. The question of the existence of such a direction is the question of existence of an eigenvector for the matrix A representing the rotation. Every 2D rotation around the origin through an angle θ {\displaystyle \theta } in counterclockwise direction can be quite simply represented by

Gear - Misplaced Pages Continue

6594-546: Is nonzero (i.e., the rotation is not the identity tensor), there is one and only one such direction. Because A has only real components, there is at least one real eigenvalue, and the remaining two eigenvalues must be complex conjugates of each other (see Eigenvalues and eigenvectors#Eigenvalues and the characteristic polynomial ). Knowing that 1 is an eigenvalue, it follows that the remaining two eigenvalues are complex conjugates of each other, but this does not imply that they are complex—they could be real with double multiplicity. In

6751-425: Is not only acceptable but desirable. For basic analysis purposes, each gear can be idealized as a perfectly rigid body that, in normal operation, turns around a rotation axis that is fixed in space, without sliding along it. Thus, each point of the gear can move only along a circle that is perpendicular to its axis and centered on it. At any moment t , all points of the gear will be rotating around that axis with

6908-620: Is not required to find the rotation axis. If n {\displaystyle n} denotes the unit eigenvector aligned with the rotation axis, and if α {\displaystyle \alpha } denotes the rotation angle, then it can be shown that 2 sin ⁡ ( α ) n = { A 32 − A 23 , A 13 − A 31 , A 21 − A 12 } {\displaystyle 2\sin(\alpha )n=\{A_{32}-A_{23},A_{13}-A_{31},A_{21}-A_{12}\}} . Consequently,

7065-411: Is perpendicular to the plane of the motion. If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results. The reverse ( inverse ) of a rotation is also a rotation. Thus, the rotations around a point/axis form a group . However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than

7222-741: Is perpendicular to the vector, A → {\displaystyle {\vec {A}}} . From: d A → d t = d ( | A → | A ^ ) d t = d | A → | d t A ^ + | A → | ( d A ^ d t ) {\displaystyle {d{\vec {A}} \over dt}={d(|{\vec {A}}|{\hat {A}}) \over dt}={d|{\vec {A}}| \over dt}{\hat {A}}+|{\vec {A}}|\left({d{\hat {A}} \over dt}\right)} , since

7379-511: Is produced by net shape molding. Molded gearing is usually powder metallurgy, plastic injection, or metal die casting. Gears produced by powder metallurgy often require a sintering step after they are removed from the mold. Cast gears require gear cutting or other machining to shape the teeth to the necessary precision. The most common form of gear cutting is hobbing , but gear shaping , milling , and broaching may be used instead. Metal gears intended for heavy duty operation, such as in

7536-856: Is real, it equals its complex conjugate v T v {\displaystyle v^{\text{T}}v} , and v ¯ T v {\displaystyle {\bar {v}}^{\text{T}}v} and v T v ¯ {\displaystyle v^{\text{T}}{\bar {v}}} are both representations of the same scalar product between v {\displaystyle v} and v ¯ {\displaystyle {\bar {v}}} . This means v + v ¯ {\displaystyle v+{\bar {v}}} and i ( v − v ¯ ) {\displaystyle i(v-{\bar {v}})} are orthogonal vectors. Also, they are both real vectors by construction. These vectors span

7693-484: Is separated from other scenes by two vertical columns of texts. The figures in the scenes are smooth, rounded, and high relief, a style continued throughout the 30th Dynasty. The relief represents the interaction between the Ptolemaic kings and the Egyptian deities, which legitimized their rule in Egypt . In Ptolemaic art, the idealism seen in the art of previous dynasties continues, with some alterations. Women are portrayed as more youthful, and men begin to be portrayed in

7850-544: Is slightly deformed into an oblate spheroid ; a similar equatorial bulge develops for other planets. Another consequence of the rotation of a planet are the phenomena of precession and nutation . Like a gyroscope , the overall effect is a slight "wobble" in the movement of the axis of a planet. Currently the tilt of the Earth 's axis to its orbital plane ( obliquity of the ecliptic ) is 23.44 degrees, but this angle changes slowly (over thousands of years). (See also Precession of

8007-417: The orbital poles . Either type of rotation is involved in a corresponding type of angular velocity (spin angular velocity and orbital angular velocity) and angular momentum (spin angular momentum and orbital angular momentum). Mathematically , a rotation is a rigid body movement which, unlike a translation , keeps at least one point fixed. This definition applies to rotations in two dimensions (in

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8164-453: The Antikythera mechanism an example of a very early and intricate geared device, designed to calculate astronomical positions of the sun, moon, and planets, and predict eclipses . Its time of construction is now estimated between 150 and 100 BC. The Chinese engineer Ma Jun (c. 200–265 AD) described a south-pointing chariot . A set of differential gears connected to the wheels and to

8321-656: The Hellenistic period . It was founded in 305 BC by the Macedonian general Ptolemy I Soter , a companion of Alexander the Great , and ruled by the Ptolemaic dynasty until the death of Cleopatra VII in 30 BC. Reigning for nearly three centuries, the Ptolemies were the longest and final dynasty of ancient Egypt , heralding a distinctly new era for religious and cultural syncretism between Greek and Egyptian culture. Alexander

8478-586: The Musaeum was supplemented by a second library built in the Serapeum . He was said to have had every book unloaded in the Alexandria docks seized and copied, returning the copies to their owners and keeping the originals for the Library. It is said that he borrowed the official manuscripts of Aeschylus , Sophocles , and Euripides from Athens and forfeited the considerable deposit he paid for them in order to keep them for

8635-464: The Roman Senate . The main Roman interest in Egypt was always the reliable delivery of grain to the city of Rome. To this end the Roman administration made no change to the Ptolemaic system of government, although Romans replaced Greeks in the highest offices. But Greeks continued to staff most of the administrative offices and Greek remained the language of government except at the highest levels. Unlike

8792-507: The Seleucid Empire when his sister, Queen Berenice , and her son were murdered in a dynastic dispute. Ptolemy marched into the heart of the Seleucid realm, as far as Babylonia , while his fleets in the Aegean Sea made fresh conquests as far north as Thrace . This victory marked the zenith of the Ptolemaic power. Seleucus II Callinicus kept his throne, but Egyptian fleets controlled most of

8949-498: The battle of Panium in 200 BC transferred Coele-Syria from Ptolemaic to Seleucid control. After this defeat Egypt formed an alliance with the rising power in the Mediterranean, Rome . Once he reached adulthood Epiphanes became a tyrant, before his early death in 180 BC. He was succeeded by his infant son Ptolemy VI Philometor . In 170 BC, Antiochus IV Epiphanes invaded Egypt and captured Philometor, installing him at Memphis as

9106-481: The sprocket , which is meant to engage with a link chain instead of another gear, and the timing pulley , meant to engage a timing belt . Most gears are round and have equal teeth, designed to operate as smoothly as possible; but there are several applications for non-circular gears , and the Geneva drive has an extremely uneven operation, by design. Gears can be seen as instances of the basic lever "machine". When

9263-431: The transmissions of cars and trucks, the teeth are heat treated to make them hard and more wear resistant while leaving the core soft but tough . For large gears that are prone to warp, a quench press is used. Gears can be made by 3D printing ; however, this alternative is typically used only for prototypes or very limited production quantities, because of its high cost, low accuracy, and relatively low strength of

9420-540: The Earth around the Sun); and stars slowly revolve about their galaxial centers . The motion of the components of galaxies is complex, but it usually includes a rotation component. Ptolemaic Egypt The Ptolemaic Kingdom ( / ˌ t ɒ l ɪ ˈ m eɪ . ɪ k / ; Koinē Greek : Πτολεμαϊκὴ βασιλεία , romanized:  Ptolemaïkḕ basileía ) or Ptolemaic Empire was an Ancient Greek polity based in Egypt during

9577-422: The Egyptian custom of marrying their sisters, with many of their line ruling jointly with their spouses, who were also of the royal house. This custom made Ptolemaic politics confusingly incestuous, and the later Ptolemies were increasingly feeble. The only basilissa - regnant or female Pharaohs to officially rule on their own were Cleopatra II , Berenice III and Berenice IV . Cleopatra V did co-rule, but it

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9734-569: The Egyptian or the Hellenistic style, but over time, these characteristics began to combine. The continuation of the Egyptian art style evidences the Ptolemies' commitment to maintaining Egyptian customs. This strategy not only helped to legitimize their rule, but also placated the general population. Greek-style art was also created during this time and existed in parallel to the more traditional Egyptian art, which could not be altered significantly without changing its intrinsic, primarily-religious function. Art found outside of Egypt itself, though within

9891-432: The Egyptian people and recognized the importance of their religion and traditions. During the reign of Ptolemies II and III, thousands of Macedonian veterans were rewarded with grants of farm lands, and Macedonians were planted in colonies and garrisons or settled themselves in villages throughout the country. Upper Egypt , farthest from the centre of government, was less immediately affected, even though Ptolemy I established

10048-540: The Egyptians by the respect he showed for their religion , but he appointed Macedonians to virtually all the senior posts in the country, and founded a new Greek city, Alexandria , to be the new capital. The wealth of Egypt could now be harnessed for Alexander's conquest of the rest of the Achaemenid Empire. Early in 331 BC he was ready to depart, and led his forces away to Phoenicia . He left Cleomenes of Naucratis as

10205-570: The Great conquered Persian-controlled Egypt in 332 BC during his campaigns against the Achaemenid Empire . His death in 323 BC was followed by rapid unraveling of the Macedonian Empire amid competing claims by the diadochi , his closest friends and companions. Ptolemy, a Macedonian who was one of Alexander's most trusted generals and confidants, won control of Egypt from his rivals and declared himself its ruler. Alexandria ,

10362-548: The Greek colony of Ptolemais Hermiou to be its capital. But within a century, Greek influence had spread through the country and intermarriage had produced a large Greco-Egyptian educated class. Nevertheless, the Greeks always remained a privileged minority in Ptolemaic Egypt. They lived under Greek law, received a Greek education, were tried in Greek courts, and were citizens of Greek cities. The first part of Ptolemy I 's reign

10519-470: The Greeks, the Romans did not settle in Egypt in large numbers. Culture, education and civic life largely remained Greek throughout the Roman period. The Romans, like the Ptolemies, respected and protected Egyptian religion and customs, although the cult of the Roman state and of the Emperor was gradually introduced. Ptolemy I, perhaps with advice from Demetrius of Phalerum , founded the Library of Alexandria ,

10676-512: The Hellenistic Greek ideology of the victorious king and his ruler cult. Misrule by the Pharaoh in Alexandria led to a nearly successful revolt, led by a priest named Hugronaphor . He proclaimed himself Pharaoh in 205 BC, and ruled upper Egypt until his death in 199 BC. He was succeeded by his son Ankhmakis , whose forces nearly drove the Ptolemies out of the country. The revolutionary dynasty

10833-426: The Library rather than returning them. The most distinguished scholar at Ptolemy III's court was the polymath and geographer Eratosthenes , most noted for his remarkably accurate calculation of the circumference of the world . Other prominent scholars include the mathematicians Conon of Samos and Apollonius of Perge . Ptolemy III financed construction projects at temples across Egypt. The most significant of these

10990-491: The Nile while attempting to flee with his remaining army. In the summer of 47 BC, having married her younger brother Ptolemy XIV , Cleopatra embarked with Caesar for a two-month trip along the Nile. Together, they visited Dendara , where Cleopatra was being worshiped as pharaoh, an honor beyond Caesar's reach. They became lovers and had a son, Caesarion . In 45 BC, Cleopatra and Caesarion left Alexandria for Rome, where they stayed in

11147-503: The Ptolemaic Kingdom, sometimes used Egyptian iconography as it had been used previously, and sometimes adapted it. For example, the faience sistrum inscribed with the name of Ptolemy has some deceptively Greek characteristics, such as the scrolls at the top. However, there are many examples of nearly identical sistrums and columns dating all the way to Dynasty 18 in the New Kingdom. It is, therefore, purely Egyptian in style. Aside from

11304-404: The Ptolemaic claim to rule the whole of Nubia. The aforementioned inscription regarding the priests of Mandulis shows that some Nubian leaders at least were paying tribute to the Ptolemaic treasury in this period. In order to secure the region, the strategos of Upper Egypt, Boethus , founded two new cities, named Philometris and Cleopatra in honour of the royal couple. After Ptolemy VI's death

11461-461: The Ptolemaic empire. During the reign of Ptolemy II, Arsinoe II was deified either as stand-alone goddesses or as a personification of another divine figure and given their own sanctuaries and festivals in association to both Egyptian and Hellenistic gods (such as Isis of Egypt and Hera of Greece). For example, Head Attributed to Arsinoe II deified her as an Egyptian goddess. However, the Marble head of

11618-649: The Ptolemies adopted the local title of pharaoh , alongside the Greek title of basileus , and had themselves portrayed on public monuments in Egyptian style and dress; however, the monarchy otherwise rigorously maintained its Hellenistic character and traditions. The kingdom had a complex government bureaucracy that exploited the country's vast economic resources to the benefit of a Greek ruling class, which dominated military, political, and economic affairs, and which rarely integrated into Egyptian society and culture. Native Egyptians maintained power over local and religious institutions, and only gradually accrued power in

11775-474: The Romans in order to regain and secure his throne following the rebellion and brief coup led by his older daughters, Tryphaena and Berenice IV . Both daughters were killed in Auletes' reclaiming of his throne; Tryphaena by assassination and Berenice by execution, leaving Cleopatra VII as the oldest surviving child of Ptolemy Auletes. Traditionally, Ptolemaic royal siblings were married to one another on ascension to

11932-558: The Thompson Manufacturing Company of Lancaster, New Hampshire still had a very active business in supplying tens of thousands of maple gear teeth per year, mostly for use in paper mills and grist mills , some dating back over 100 years. The most common techniques for gear manufacturing are dies , sand , and investment casting ; injection molding ; powder metallurgy ; blanking ; and gear cutting . As of 2014, an estimated 80% of all gearing produced worldwide

12089-423: The astronomer Aristarchus . Ptolemy is thought to have commissioned Manetho to compose his Aegyptiaca , an account of Egyptian history, perhaps intended to make Egyptian culture intelligible to its new rulers. Ptolemy's first wife, Arsinoe I , daughter of Lysimachus , was the mother of his legitimate children. After her repudiation he followed Egyptian custom and married his sister , Arsinoe II , beginning

12246-467: The attacks of Antiochus III the Great on Coele-Syria, and the great Egyptian victory of Raphia in 217 BC secured the kingdom. A sign of the domestic weakness of his reign was the rebellions by native Egyptians that took away over half the country for over 20 years. Philopator was devoted to orgiastic religions and to literature. He married his sister Arsinoë , but was ruled by his mistress Agathoclea. Like his predecessors, Ptolemy IV presented himself as

12403-450: The axes, each section of one gear will interact only with the corresponding section of the other gear. Thus the three-dimensional gear train can be understood as a stack of gears that are flat and infinitesimally thin — that is, essentially two-dimensional. In a crossed arrangement, the axes of rotation of the two gears are not parallel but cross at an arbitrary angle except zero or 180 degrees. For best operation, each wheel then must be

12560-437: The axis, meaning that it is congruent with itself when the gear rotates by 1/ N of a turn. If the gear is meant to transmit or receive torque with a definite sense only (clockwise or counterclockwise with respect to some reference viewpoint), the action surface consists of N separate patches, the tooth faces ; which have the same shape and are positioned in the same way relative to the axis, spaced 1/ N turn apart. If

12717-400: The backpillar and the goddess's striding pose is distinctively Egyptian, the cornucopia she holds and her hairstyle are both Greek in style. The rounded eyes, prominent lips, and overall youthful features show Greek influence as well. Despite the unification of Greek and Egyptian elements in the intermediate Ptolemaic period, the Ptolemaic Kingdom also featured prominent temple construction as

12874-446: The best shape for each pitch surface is neither cylindrical nor conical but a portion of a hyperboloid of revolution. Such gears are called hypoid for short. Hypoid gears are most commonly found with shafts at 90 degrees. Contact between hypoid gear teeth may be even smoother and more gradual than with spiral bevel gear teeth, but also have a sliding action along the meshing teeth as it rotates and therefore usually require some of

13031-482: The body's own center of mass is known as a spin (or autorotation ). In that case, the surface intersection of the internal spin axis can be called a pole ; for example, Earth's rotation defines the geographical poles . A rotation around an axis completely external to the moving body is called a revolution (or orbit ), e.g. Earth's orbit around the Sun . The ends of the external axis of revolution can be called

13188-510: The bureaucracy, provided they Hellenized . Beginning with Ptolemy I's son and successor, Ptolemy II Philadelphus , the Ptolemies began to adopt Egyptian customs, such as marrying their siblings per the Osiris myth and participating in Egyptian religious life . New temples were built, older ones restored, and royal patronage lavished on the priesthood. From the mid third century BC, Ptolemaic Egypt

13345-557: The city. Facing certain death at the hands of Octavian , Antony attempted suicide by falling on his own sword, but survived briefly. He was taken by his remaining soldiers to Cleopatra, who had barricaded herself in her mausoleum, where he died soon after. Knowing that she would be taken to Rome to be paraded in Octavian's triumph (and likely executed thereafter), Cleopatra and her handmaidens committed suicide on 12 August 30 BC. Legend and numerous ancient sources claim that she died by way of

13502-423: The coalition against him. In 312 BC, allied with Seleucus , the ruler of Babylonia , he defeated Demetrius , the son of Antigonus, in the battle of Gaza . In 311 BC, a peace was concluded between the combatants, but in 309 BC war broke out again, and Ptolemy occupied Corinth and other parts of Greece, although he lost Cyprus after a naval battle in 306 BC. Antigonus then tried to invade Egypt but Ptolemy held

13659-510: The coasts of Anatolia and Greece. After this triumph Ptolemy no longer engaged actively in war, although he supported the enemies of Macedon in Greek politics. His domestic policy differed from his father's in that he patronised the native Egyptian religion more liberally: he left larger traces among the Egyptian monuments. In this his reign marks the gradual Egyptianisation of the Ptolemies. Ptolemy III continued his predecessor's sponsorship of scholarship and literature. The Great Library in

13816-572: The colonist population were used as mercenaries by both the rebel Egyptian princes and the Persian kings, who later gave them land grants, spreading Greek culture into the valley of the Nile. When Alexander the Great arrived, he established Alexandria on the site of the Persian fort of Rhakortis. Following Alexander's death, control passed into the hands of the Lagid (Ptolemaic) Dynasty; they built Greek cities across their empire and gave land grants across Egypt to

13973-533: The cosmic world with basic plans retaining the pylon, open court, hypostyle halls , and dark and centrally located sanctuary. However, ways of presenting text on columns and reliefs became formal and rigid during the Ptolemaic Dynasty. Scenes were often framed with textual inscriptions, with a higher text to image ratio than seen previously during the New Kingdom . For example, a relief in the temple of Kom Ombo

14130-447: The country became a de facto protectorate of Rome, which had by now absorbed most of the Greek world. Ptolemy XI was succeeded by a son of Ptolemy IX, Ptolemy XII Neos Dionysos , nicknamed Auletes, the flute-player. By now Rome was the arbiter of Egyptian affairs, and annexed both Libya and Cyprus . In 58 BC Auletes was driven out by the Alexandrian mob, but the Romans restored him to power three years later. He died in 51 BC, leaving

14287-476: The death of their father, Cleopatra VII and her younger brother Ptolemy XIII inherited the throne and were married. Their marriage was only nominal, however, and their relationship soon degenerated. Cleopatra was stripped of authority and title by Ptolemy XIII's advisors, who held considerable influence over the young king. Fleeing into exile, Cleopatra attempted to raise an army to reclaim the throne. Julius Caesar left Rome for Alexandria in 48 BC in order to quell

14444-405: The degenerate case of a rotation angle α = 180 ∘ {\displaystyle \alpha =180^{\circ }} , the remaining two eigenvalues are both equal to −1. In the degenerate case of a zero rotation angle, the rotation matrix is the identity, and all three eigenvalues are 1 (which is the only case for which the rotation axis is arbitrary). A spectral analysis

14601-597: The direction of rotation is reversed when one gear wheel drives another gear wheel. Philon of Byzantium was one of the first who used gears in water raising devices. Gears appear in works connected to Hero of Alexandria , in Roman Egypt circa AD 50, but can be traced back to the mechanics of the Library of Alexandria in 3rd-century BC Ptolemaic Egypt , and were greatly developed by the Greek polymath Archimedes (287–212 BC). The earliest surviving gears in Europe were found in

14758-399: The direction of the axis of rotation. Similarly, the torque is an axial vector. The physics of the rotation around a fixed axis is mathematically described with the axis–angle representation of rotations. According to the right-hand rule , the direction away from the observer is associated with clockwise rotation and the direction towards the observer with counterclockwise rotation, like

14915-529: The discovery of a wealth of papyri and ostraca written in Koine Greek and Egyptian . In 332 BC, Alexander the Great , King of Macedon , conquered Egypt, which at the time was a satrapy of the Achaemenid Empire later called Egypt's Thirty-first Dynasty . He visited Memphis , and travelled to the oracle of Amun at the Siwa Oasis . The oracle declared him to be the son of Amun. Alexander conciliated

15072-604: The dynasty of Ptolemies and the entirety of pharaonic Egypt came to an end. Alexandria remained the capital of the country, but Egypt itself became a Roman province. Octavian became the sole ruler of Rome and began converting it into a monarchy, the Roman Empire . Under Roman rule, Egypt was governed by a prefect selected by the emperor from the Equestrian class and not a governor from the Senatorial order, to prevent interference by

15229-520: The economic, artistic and intellectual capital of the Hellenistic world. The academies and libraries of Alexandria proved vital in preserving much Greek literary heritage. Ptolemy III Euergetes ("the Benefactor") succeeded his father in 246 BC. He abandoned his predecessors' policy of keeping out of the wars of the other Macedonian successor kingdoms, and began the Third Syrian War (246–241 BC) with

15386-438: The effect of the rotation matrix is just stretching it. If we write A in this basis, it is diagonal; but a diagonal orthogonal matrix is made of just +1s and −1s in the diagonal entries. Therefore, we do not have a proper rotation, but either the identity or the result of a sequence of reflections. It follows, then, that a proper rotation has some complex eigenvalue. Let v be the corresponding eigenvector. Then, as we showed in

15543-1071: The eigenvectors of A . A vector is said to be rotating if it changes its orientation. This effect is generally only accompanied when its rate of change vector has non-zero perpendicular component to the original vector. This can be shown to be the case by considering a vector A → {\displaystyle {\vec {A}}} which is parameterized by some variable t {\textstyle t} for which: d | A → | 2 d t = d ( A → ⋅ A → ) d t ⇒ d | A → | d t = d A → d t ⋅ A ^ {\displaystyle {d|{\vec {A}}|^{2} \over dt}={d({\vec {A}}\cdot {\vec {A}}) \over dt}\Rightarrow {d|{\vec {A}}| \over dt}={d{\vec {A}} \over dt}\cdot {\hat {A}}} Which also gives

15700-639: The end of his reign as a result of the Second Syrian War . In the 270s BC, Ptolemy II defeated the Kingdom of Kush in war, gaining the Ptolemies free access to Kushite territory and control of important gold deposits south of Egypt known as Dodekasoinos. As a result, the Ptolemies established hunting stations and ports as far south as Port Sudan , from where raiding parties containing hundreds of men searched for war elephants. Hellenistic culture would acquire an important influence on Kush at this time. Ptolemy II

15857-408: The equinoxes and Pole Star .) While revolution is often used as a synonym for rotation , in many fields, particularly astronomy and related fields, revolution , often referred to as orbital revolution for clarity, is used when one body moves around another while rotation is used to mean the movement around an axis. Moons revolve around their planets, planets revolve about their stars (such as

16014-489: The event of his death, which Octavian used against Antony, sowing further dissent in the Roman populace. Octavian was quick to declare war on Antony and Cleopatra while public opinion of Antony was low. Their naval forces met at Actium , where the forces of Marcus Vipsanius Agrippa defeated the navy of Cleopatra and Antony. Octavian waited for a year before he claimed Egypt as a Roman province. He arrived in Alexandria and easily defeated Mark Antony's remaining forces outside

16171-541: The expense of an eigenvalue analysis can be avoided by simply normalizing this vector if it has a nonzero magnitude. On the other hand, if this vector has a zero magnitude, it means that sin ⁡ ( α ) = 0 {\displaystyle \sin(\alpha )=0} . In other words, this vector will be zero if and only if the rotation angle is 0 or 180 degrees, and the rotation axis may be assigned in this case by normalizing any column of A + I {\displaystyle A+I} that has

16328-513: The first angle moves the line of nodes around the external axis z , the second rotates around the line of nodes and the third one is an intrinsic rotation around an axis fixed in the body that moves. These rotations are called precession , nutation , and intrinsic rotation . In astronomy , rotation is a commonly observed phenomenon; it includes both spin (auto-rotation) and orbital revolution. Stars , planets and similar bodies may spin around on their axes. The rotation rate of planets in

16485-443: The first term is parallel to A → {\displaystyle {\vec {A}}} and the second perpendicular to it, we can conclude in general that the parallel and perpendicular components of rate of change of a vector independently influence only the magnitude or orientation of the vector respectively. Hence, a rotating vector always has a non-zero perpendicular component of its rate of change vector against

16642-403: The following matrix : A standard eigenvalue determination leads to the characteristic equation which has as its eigenvalues. Therefore, there is no real eigenvalue whenever cos ⁡ θ ≠ ± 1 {\displaystyle \cos \theta \neq \pm 1} , meaning that no real vector in the plane is kept unchanged by A . Knowing that the trace

16799-479: The forces of disorder led by the god Set . In return, the priests undertook to erect a statue group in each of their temples, depicting the god of the temple presenting a sword of victory to Ptolemy IV and Arsinoe III. A five-day festival was inaugurated in honour of the Theoi Philopatores and their victory. The decree thus seems to represent a successful marriage of Egyptian Pharaonic ideology and religion with

16956-473: The frontier against him. When the coalition was renewed against Antigonus in 302 BC, Ptolemy joined it, but neither he nor his army were present when Antigonus was defeated and killed at Ipsus . He had instead taken the opportunity to secure Coele-Syria and Palestine, in breach of the agreement assigning it to Seleucus, thereby setting the scene for the future Syrian Wars . Thereafter Ptolemy tried to stay out of land wars, but he retook Cyprus in 295 BC. Feeling

17113-531: The geometry of the pair, the sense of rotation may also be inverted (from clockwise to anti-clockwise , or vice-versa). Most vehicles have a transmission or "gearbox" containing a set of gears that can be meshed in multiple configurations. The gearbox lets the operator vary the torque that is applied to the wheels without changing the engine's speed. Gearboxes are used also in many other machines, such as lathes and conveyor belts . In all those cases, terms like "first gear", "high gear", and "reverse gear" refer to

17270-540: The growing dominance of the Roman Republic . With one empire after another falling to Macedon and the Seleucid empire, the Ptolemies had had little choice but to ally with the Romans, a pact that lasted over 150 years. By Ptolemy XII's time, Rome had achieved a massive amount of influence over Egyptian politics and finances to the point that he declared the Roman senate the guardian of the Ptolemaic Dynasty. He had paid vast sums of Egyptian wealth and resources in tribute to

17427-433: The kingdom to his ten-year-old son and seventeen-year-old daughter, Ptolemy XIII Theos Philopator and Cleopatra VII , who reigned jointly as husband and wife. Cleopatra VII ascended the Egyptian throne on 22 March 51 BC upon the death of her father, Ptolemy XII Neos Dionysos. She reigned as queen "philopator" and pharaoh with various male co-regents from 51 to 30 BC. The demise of the Ptolemies' power coincided with

17584-464: The kingdom was now secure, Ptolemy shared rule with his son Ptolemy II by Queen Berenice in 285 BC. He then may have devoted his retirement to writing a history of the campaigns of Alexander—which unfortunately was lost but was a principal source for the later work of Arrian . Ptolemy I died in 283 BC at the age of 84. He left a stable and well-governed kingdom to his son. Ptolemy II Philadelphus , who succeeded his father as pharaoh of Egypt in 283 BC,

17741-483: The large scale structuring over the course of evolution of the matter field that was initially laid down by the Big Bang. In particular, for a system which behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally invariant . According to Noether's theorem , if the action (the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum

17898-475: The last independent Hellenistic state . Roman Egypt became one of Rome's richest provinces and a center of Greek culture. Greek remained the language of government and trade until the Muslim conquest in 641 AD. Alexandria remained one of the leading cities of the Mediterranean well into the late Middle Ages . The Ptolemaic reign in Egypt is one of the best-documented time periods of the Hellenistic era , due to

18055-399: The looming civil war, as war in Egypt, which was one of Rome's greatest suppliers of grain and other expensive goods, would have had a detrimental effect on trade with Rome, especially on Rome's working-class citizens. During his stay in the Alexandrian palace, he received 22-year-old Cleopatra, allegedly carried to him in secret wrapped in a carpet. Caesar agreed to support Cleopatra's claim to

18212-434: The most common configuration, the axes of rotation of the two gears are parallel, and usually their sizes are such that they contact near a point between the two axes. In this configuration, the two gears turn in opposite senses. Occasionally the axes are parallel but one gear is nested inside the other. In this configuration, both gears turn in the same sense. If the two gears are cut by an imaginary plane perpendicular to

18369-522: The most efficient and compact way of transmitting torque between two non-parallel axes. On the other hand, gears are more expensive to manufacture, may require periodic lubrication, and may have greater mass and rotational inertia than the equivalent pulleys. More importantly, the distance between the axes of matched gears is limited and cannot be changed once they are manufactured. There are also applications where slippage under overload or transients (as occurs with belts, hydraulics, and friction wheels)

18526-444: The most viscous types of gear oil to avoid it being extruded from the mating tooth faces, the oil is normally designated HP (for hypoid) followed by a number denoting the viscosity. Also, the pinion can be designed with fewer teeth than a spiral bevel pinion, with the result that gear ratios of 60:1 and higher are feasible using a single set of hypoid gears. This style of gear is most common in motor vehicle drive trains, in concert with

18683-407: The name of the king, there are other features that specifically date this to the Ptolemaic period. Most distinctively is the color of the faience. Apple green, deep blue, and lavender-blue are the three colors most frequently used during this period, a shift from the characteristic blue of the earlier kingdoms. This sistrum appears to be an intermediate hue, which fits with its date at the beginning of

18840-512: The nymphs of the planthopper insect Issus coleoptratus . The word gear is probably from Old Norse gørvi (plural gørvar ) 'apparel, gear,' related to gøra , gørva 'to make, construct, build; set in order, prepare,' a common verb in Old Norse, "used in a wide range of situations from writing a book to dressing meat". In this context, the meaning of 'toothed wheel in machinery' first attested 1520s; specific mechanical sense of 'parts by which

18997-423: The object may be kept fixed; instead, simple rotations are described as being in a plane. In four or more dimensions, a combination of two or more rotations about a plane is not in general a rotation in a single plane. 2-dimensional rotations, unlike the 3-dimensional ones, possess no axis of rotation, only a point about which the rotation occurs. This is equivalent, for linear transformations, with saying that there

19154-433: The oval face, "round [and] deeply set" eyes, and the small, tucked mouth closer to the nose. Early portraits of the Ptolemies featured large and radiant eyes in association to the rulers' divinity as well as general notions of abundance. When Ptolemy I Soter made himself king of Egypt, he created a new god, Serapis , to garner support from both Greeks and Egyptians. Serapis was the patron god of Ptolemaic Egypt, combining

19311-629: The overall torque ratios of different meshing configurations, rather than to specific physical gears. These terms may be applied even when the vehicle does not actually contain gears, as in a continuously variable transmission . The earliest surviving gears date from the 4th century BC in China (Zhan Guo times – Late East Zhou dynasty ), which have been preserved at the Luoyang Museum of Henan Province, China . In Europe, Aristotle mentions gears around 330 BC, as wheel drives in windlasses. He observed that

19468-481: The points p and q are moving along different circles; therefore, the contact cannot last more than one instant, and p will then either slide across the other face, or stop contacting it altogether. On the other hand, at any given moment there is at least one such pair of contact points; usually more than one, even a whole line or surface of contact. Actual gears deviate from this model in many ways: they are not perfectly rigid, their mounting does not ensure that

19625-416: The points have the same instantaneous velocity whereas relative motion can only be observed in motions involving rotation. In rotation, the orientation of the object changes and the change in orientation is independent of the observers whose frames of reference have constant relative orientation over time. By Euler's theorem , any change in orientation can be described by rotation about an axis through

19782-453: The positions of the sun, the moon and the five planets then known, as well as religious feast days. The Salisbury Cathedral clock , built in 1386, it is the world's oldest still working geared mechanical clock. Differential gears were used by the British clock maker Joseph Williamson in 1720. However, the oldest functioning gears by far were created by Nature, and are seen in the hind legs of

19939-539: The previous topic, v ¯ {\displaystyle {\bar {v}}} is also an eigenvector, and v + v ¯ {\displaystyle v+{\bar {v}}} and i ( v − v ¯ ) {\displaystyle i(v-{\bar {v}})} are such that their scalar product vanishes: because, since v ¯ T v ¯ {\displaystyle {\bar {v}}^{\text{T}}{\bar {v}}}

20096-455: The reliefs on the great pylon were finished in the reign of Ptolemy XII . In 221 BC, Ptolemy III died and was succeeded by his son Ptolemy IV Philopator , a weak king whose rule precipitated the decline of the Ptolemaic Kingdom. His reign was inaugurated by the murder of his mother, and he was always under the influence of royal favourites , who controlled the government. Nevertheless, his ministers were able to make serious preparations to meet

20253-672: The resulting part. Besides gear trains, other alternative methods of transmitting torque between non-coaxial parts include link chains driven by sprockets, friction drives , belts and pulleys , hydraulic couplings , and timing belts . One major advantage of gears is that their rigid body and the snug interlocking of the teeth ensure precise tracking of the rotation across the gear train, limited only by backlash and other mechanical defects. For this reason they are favored in precision applications such as watches. Gear trains also can have fewer separate parts (only two) and have minimal power loss, minimal wear, and long life. Gears are also often

20410-425: The rotation axis will be perfectly fixed in space, the teeth may have slightly different shapes and spacing, the tooth faces are not perfectly smooth, and so on. Yet, these deviations from the ideal model can be ignored for a basic analysis of the operation of a gear set. One criterion for classifying gears is the relative position and direction of the axes or rotation of the gears that are to be meshed together. In

20567-491: The ruling nomarch to control Egypt in his absence. Alexander would never return to Egypt. Following Alexander's death in Babylon in 323 BC, a succession crisis erupted among his generals. Initially, Perdiccas ruled the empire as regent for Alexander's half-brother Arrhidaeus, who became Philip III of Macedon , and then as regent for both Philip III and Alexander's infant son Alexander IV of Macedon , who had not been born at

20724-456: The same angular speed ω ( t ), in the same sense. The speed need not be constant over time. The action surface of the gear consists of all points of its surface that, in normal operation, may contact the matching gear with positive pressure . All other parts of the surface are irrelevant (except that they cannot be crossed by any part of the matching gear). In a gear with N teeth, the working surface has N -fold rotational symmetry about

20881-409: The same subspace as v {\displaystyle v} and v ¯ {\displaystyle {\bar {v}}} , which is an invariant subspace under the application of A . Therefore, they span an invariant plane. This plane is orthogonal to the invariant axis, which corresponds to the remaining eigenvector of A , with eigenvalue 1, because of the orthogonality of

21038-523: The sense of 'a wheel having teeth or cogs; late 14c., 'tooth on a wheel'; cog-wheel, early 15c. The gears of the Antikythera mechanism are made of bronze , and the earliest surviving Chinese gears are made of iron, These metals, as well as tin , have been generally used for clocks and similar mechanisms to this day. Historically, large gears, such as used in flour mills , were commonly made of wood rather than metal. They were cogwheels, made by inserting

21195-409: The solar system was first measured by tracking visual features. Stellar rotation is measured through Doppler shift or by tracking active surface features. An example is sunspots , which rotate around the Sun at the same velocity as the outer gases that make up the Sun. Under some circumstances orbiting bodies may lock their spin rotation to their orbital rotation around a larger body. This effect

21352-524: The surrounding areas during the Wars of the Diadochi (322–301 BC). In 305 BC, Ptolemy took the title of basileus and pharaoh . As Ptolemy I Soter ("Saviour"), he founded the Ptolemaic dynasty that was to rule Egypt for nearly 300 years. All the male rulers of the dynasty took the name Ptolemy, while princesses and female rulers preferred the names Cleopatra , Arsinoë and Berenice. The Ptolemies also adopted

21509-562: The throne. In 145 BC, he was killed in the Battle of Antioch . Throughout the 160s and 150s BC, Ptolemy VI has also reasserted Ptolemaic control over the northern part of Nubia . This achievement is heavily advertised at the Temple of Isis at Philae , which was granted the tax revenues of the Dodecaschoenus region in 157 BC. Decorations on the first pylon of the Temple of Isis at Philae emphasise

21666-481: The throne. Ptolemy XIII and his advisors fled the palace, turning the Egyptian forces loyal to the throne against Caesar and Cleopatra, who barricaded themselves in the palace complex until Roman reinforcements could arrive to combat the rebellion, known afterward as the Siege of Alexandria . Ptolemy XIII's forces were ultimately defeated at the Battle of the Nile and the king was killed in the conflict, reportedly drowning in

21823-399: The throne. These marriages sometimes produced children, and other times were only a ceremonial union to consolidate political power. Ptolemy Auletes expressed his wish for Cleopatra and her brother Ptolemy XIII to marry and rule jointly in his will, in which the Roman senate was named as executor, giving Rome further control over the Ptolemies and, thereby, the fate of Egypt as a nation. After

21980-468: The time of his father's death. Perdiccas appointed Ptolemy , one of Alexander's closest companions, to be satrap of Egypt. Ptolemy ruled Egypt from 323 BC, nominally in the name of the joint kings Philip III and Alexander IV. However, as Alexander the Great's empire disintegrated, Ptolemy soon established himself as ruler in his own right. Ptolemy successfully defended Egypt against an invasion by Perdiccas in 321 BC, and consolidated his position in Egypt and

22137-416: The torque on each gear may have both senses, the action surface will have two sets of N tooth faces; each set will be effective only while the torque has one specific sense, and the two sets can be analyzed independently of the other. However, in this case the gear usually has also "flip over" symmetry, so that the two sets of tooth faces are congruent after the gear is flipped. This arrangement ensures that

22294-416: The two gears are firmly locked together, at all times, with no backlash . During operation, each point p of each tooth face will at some moment contact a tooth face of the matching gear at some point q of one of its tooth faces. At that moment and at those points, the two faces must have the same perpendicular direction but opposite orientation. But since the two gears are rotating around different axes,

22451-533: The two gears are sliced by an imaginary sphere whose center is the point where the two axes cross, each section will remain on the surface of that sphere as the gear rotates, and the section of one gear will interact only with the corresponding section of the other gear. In this way, a pair of meshed 3D gears can be understood as a stack of nested infinitely thin cup-like gears. The gears in a matching pair are said to be skew if their axes of rotation are skew lines -- neither parallel nor intersecting. In this case,

22608-436: The vector itself. As dimensions increase the number of rotation vectors increases. Along a four dimensional space (a hypervolume ), rotations occur along x, y, z, and w axis. An object rotated on a w axis intersects through various volumes , where each intersection is equal to a self contained volume at an angle. This gives way to a new axis of rotation in a 4d hypervolume, were a 3d object can be rotated perpendicular to

22765-437: The venomous bite of an asp , though others state that she used poison, or that Octavian ordered her death himself. Caesarion, her son by Julius Caesar, nominally succeeded Cleopatra until his capture and supposed execution in the weeks after his mother's death. Cleopatra's children by Antony were spared by Octavian and given to his sister (and Antony's Roman wife) Octavia Minor , to be raised in her household. No further mention

22922-577: The veterans of their many military conflicts. Hellenistic civilization continued to thrive even after Rome annexed Egypt after the battle of Actium and did not decline until the Islamic conquests . Ptolemaic art was produced during the reign of the Ptolemaic Rulers (304–30 BC), and was concentrated primarily within the bounds of the Ptolemaic Empire. At first, artworks existed separately in either

23079-434: The whole gear. Two or more meshing gears are called a gear train . The smaller member of a pair of meshing gears is often called pinion . Most commonly, gears and gear trains can be used to trade torque for rotational speed between two axles or other rotating parts and/or to change the axis of rotation and/or to invert the sense of rotation. A gear may also be used to transmit linear force and/or linear motion to

23236-406: The z axis. The speed of rotation is given by the angular frequency (rad/s) or frequency ( turns per time), or period (seconds, days, etc.). The time-rate of change of angular frequency is angular acceleration (rad/s ), caused by torque . The ratio of torque to the angular acceleration is given by the moment of inertia . The angular velocity vector (an axial vector ) also describes

23393-598: Was a peaceful and cultured pharaoh, though unlike his father was no great warrior. Fortunately, Ptolemy I had left Egypt strong and prosperous; three years of campaigning in the First Syrian War made the Ptolemies masters of the eastern Mediterranean, controlling the Aegean islands (the Nesiotic League ) and the coastal districts of Cilicia , Pamphylia , Lycia and Caria . However, some of these territories were lost near

23550-422: Was an eager patron of scholarship, funding the expansion of the Library of Alexandria and patronising scientific research. Poets like Callimachus , Theocritus , Apollonius of Rhodes , Posidippus were provided with stipends and produced masterpieces of Hellenistic poetry, including panegyrics in honour of the Ptolemaic family. Other scholars operating under Ptolemy's aegis included the mathematician Euclid and

23707-401: Was dominated by the Wars of the Diadochi between the various successor states to the empire of Alexander. His first objective was to hold his position in Egypt securely, and secondly to increase his domain. Within a few years he had gained control of Libya , Coele-Syria (including Judea ), and Cyprus . When Antigonus , ruler of Syria , tried to reunite Alexander's empire, Ptolemy joined

23864-444: Was driven out by his mother in 107 BC, who reigned jointly with Euergetes's youngest son Ptolemy X Alexander I . In 88 BC Ptolemy IX again returned to the throne, and retained it until his death in 80 BC. He was succeeded by Ptolemy XI Alexander II , the son of Ptolemy X. He was lynched by the Alexandrian mob after murdering his stepmother, who was also his cousin, aunt and wife. These sordid dynastic quarrels left Egypt so weakened that

24021-508: Was finally defeated in 186, and a stele celebrating this event was historically significant as the famous Rosetta Stone . Ptolemy V Epiphanes , son of Philopator and Arsinoë, was a child when he came to the throne, and a series of regents ran the kingdom. Antiochus III the Great of The Seleucid Empire and Philip V of Macedon made a compact to seize the Ptolemaic possessions. Philip seized several islands and places in Caria and Thrace , while

24178-474: Was the Raphia Decree , issued on 15 November 217 BC and preserved in three copies. Like other Ptolemaic decrees , the decree was inscribed in hieroglyphs , Demotic , and Koine Greek . The decree records the military success of Ptolemy IV and Arsinoe III and their benefactions to the Egyptian priestly elite. Throughout, Ptolemy IV is presented as taking on the role of Horus who avenges his father by defeating

24335-480: Was the Temple of Horus at Edfu , one of the masterpieces of ancient Egyptian temple architecture and now the best-preserved of all Egyptian temples. Ptolemy III initiated construction on it on 23 August 237 BC. Work continued for most of the Ptolemaic dynasty; the main temple was finished in the reign of his son, Ptolemy IV, in 212 BC, and the full complex was only completed in 142 BC, during the reign of Ptolemy VIII , while

24492-445: Was the wealthiest and most powerful of Alexander's successor states , and the leading example of Greek civilization. Beginning in the mid second century BC, dynastic strife and a series of foreign wars weakened the kingdom, and it became increasingly reliant on the Roman Republic . Under Cleopatra VII , who sought to restore Ptolemaic power, Egypt became entangled in a Roman civil war , which ultimately led to its conquest by Rome as

24649-484: Was with another female, Berenice IV. Cleopatra VII officially co-ruled with Ptolemy XIII Theos Philopator , Ptolemy XIV , and Ptolemy XV , but effectively, she ruled Egypt alone. The early Ptolemies did not disturb the religion or the customs of the Egyptians . They built magnificent new temples for the Egyptian gods and soon adopted the outward display of the pharaohs of old. Rulers such as Ptolemy I Soter respected

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