OpenLLM-France/wikimedia huggingface dataset:
# Illustrated Companion to the Latin Dictionary/Culcita

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CUL'CITA (τύλη, στρωμνή). A mattrass for a sofa, couch, or bed, stuffed with wadding, wool, or feathers (Varro, L. L. v. 167. Pet. Sat. 38. Cic. Tusc. iii. 19. Seneca, Ep. 87.); which, consequently, was sometimes very soft, like our feather beds, and at others, like our wool and hair mattrasses, sufficiently hard not to take an impression from the body resting upon it. (Seneca, Ep. 108.) The illustration (Culcita/1.1) is from a painting at Pompeii.
# Illustrated Companion to the Latin Dictionary/Culeus

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CU'LEUS or CUL'LEUS. A very large sack made of pig's-skin or leather, and employed by the Romans for the transport of wine or oil (Nepos, Eum. 8. Plin. H. N. vii. 19. Cato, R. R. xi. 1.), as represented by the annexed illustration (Culeus/1.1), from a painting at Pompeii, which shows the manner of transporting it on a cart frame, of emptying its contents into smaller vessels (amphorae), and how it was filled; viz. by the neck at the top, which was then tied up with a cord. A contrivance of precisely the same kind is still employed in Italy for the transport and sale of oil. The size of this will likewise account for another use to which it was applied by the ancient Romans, for sewing parricides in. Cic. Q. Fr. i. 2. 2.
2. Also a liquid measure; the largest used by the Romans, containing twenty amphorae, or 118 gallons, and particularly employed in estimating the produce of a vineyard or olive ground. Rhemn. Fann. de Pond. et Mens. 86. Varro, R. R. 1. 2. 7.
# Illustrated Companion to the Latin Dictionary/Culigna

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CULIG'NA (κυλίχνη). A vessel for wine, the exact nature of which is not ascertained. Cato, R. R. 132.
# Illustrated Companion to the Latin Dictionary/Culina

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CULI'NA. A kitchen. (Cic. Fam. xv. 18. Pet. Sat. 2. 1. Seneca, Ep. 114.) The illustration (Culina/1.1) represents a kitchen stove in the house of Pansa at Pompeii, with some cooking utensils upon it, as discovered when first excavated; viz. a strainer (colum), a kitchen knife (culter coquinaris), and an implement for dressing eggs (supposed apalare); below is the ground-plan of a kitchen in the same city, from the house of the Quaestor, distributed into the following parts. Immediately on the left hand of the entrance there is a semicircular sink (1) , and on the right a staircase (2), which probably led up to the store-rooms; fronting the entrance are the remains of the brickwork which formed the stove (3), similarly constructed to the elevation above; and adjoining this is another small chamber (4), which we might call the back kitchen, with a privy (5) at its furthest extremity; a convenience, which, singularly enough, is generally found adjacent to the kitchens in the houses of Pompeii.
# Illustrated Companion to the Latin Dictionary/Cultellus

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CULTEL'LUS (μαχαιρίς, μαχαίριον). Diminutive of CULTER; and employed in nearly the same senses, only designating a lesser description of each kind. But the cultellus is never so small as our pocket and pen-knife (scalprum); for Juvenal designates a carving-knife by the diminutive (Sat. v. 122.); Ulpian (Dig. 9. 2. 11.), a barber's razor; and the cultellus of Horace (Ep. i. 7. 51.), which people used to clean and pare their nails with, was the same as the barber's instrument, which is expressly name for that purpose by Valerius Maximus (iii. 2. 15.), cultellum tonsorium quasi unguium resecandorum causa poposcit.
2. Cultellus ligneus. A wedge of wood; which is sharper at the edge than at the back, like the blade of a culter. Vitruv. vii. 3. 2.
# Illustrated Companion to the Latin Dictionary/Cultrarius

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CULTRA'RIUS. The minister or servant of an officiating priest, who despatched the victim at a sacrifice, by cutting its throat with a knife (culter), as contradistinguished from popa, who knocked it down with a blow of the axe (securis) or mallet (malleus). (Suet. Cal. 32. Inscript. ap. Grut. 640. 11.) The illustration (Cultrarius/1.1), from a very beautiful marble bas-relief discovered at Pompeii, represents an old woman and a Faun about to offer up a pig in sacrifice, the former in the character of a priestess, the latter as a cultrarius, cutting its throat.
# Illustrated Companion to the Latin Dictionary/Culullus

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CULUL'LUS. According to the Scholiasts on Horace, an earthenware calix employed by the pontifices and Vestals in their sacrificial rites; but commonly used in a general sense for any kind of drinking-cup. Acron. and Porphyr. ad Hor. Od. i. 31. 11. Hor. A. P. 434.
# Illustrated Companion to the Latin Dictionary/Cumera

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CUM'ERA. A sort of tub, pan, or basket with a convex lid, used by the country people for keeping corn in. Festus, s. Cumerum. Hor. Sat. i. 1. 53. Id. Epist. i. 7. 30. Acrod. ad ll.
# Illustrated Companion to the Latin Dictionary/Cumerum

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CUM'ERUM. A covered vase, or, perhaps, basket, carried by the camillus in a marriage procession (Varro, L. L. vii. 34.), and containing the necessaries (utensilia) of the bride. Festus, s. v.
# Illustrated Companion to the Latin Dictionary/Cunabula

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CUNA'BULA. A child's cradle. (Cic. Div. i. 36. Plaut. Amph. v. 1. 55. Serv. ad Virg. Ecl. iv. 23. Arnob. adv. Gent. iv.) The example (Cunabula/1.1) is from a very ancient MS. of Genesis, published by Lambeccius (Comment. Bibl. Caes. iii. 29.); but ancient cradles were also commonly made in the shape of a trough or boat, as in the next illustration; whence a Greek name for the same is σκάφη. Athen. xiii. 85.
2. Hence the place in which any living thing is born: a birth-place (Prop. iii. 1. 27.); a bird's nest (Plin. H. N. x. 51.); a bee-hive. Virg. Georg. iv. 66.
# Euclidean vector space/Isometry/Section

We now discuss isometries of a Euclidean vector space in itself. These are always stets bijective. With respect to any orthonormal basis of $V$, they are described in the following way.

## Lemma
### Proof

$\Box$
The set of isometries on a Euclidean vector space form a group; in fact, it is a subgroup of the group of all bijective linear mappings. We recall briefly the corresponding definitions.
For a field $K$ and $n\in \mathbb {N} _{+}$, the set of all invertible $n\times n$-matrices with entries in $K$ is called the general linear group
$K$
$\operatorname {GL} _{n}\!{\left(K\right)}$
Matrix/K/Special linear group/Definition

## Definition

Orthogonal matrix/Orthogonal group/Field/Definition

## Definition

Matrix/C/Unitary group/Definition
# Euclidean vector space/Isometry/Orthogonal/Fact

Let $V$ be a Euclidean vector space, and let $u_{1},\ldots ,u_{n}$ denote an orthonormal basis of $V$. Let
    $\varphi \colon V\longrightarrow V$
be a linear mapping, and let $M$ be the describing matrix of $\varphi$ with respect to the given basis.
$\varphi$
isometry if and only if
    ${M^{\text{tr}}}M=E_{n}\,$
# Euclidean vector space/Isometry/Orthogonal/Fact/Proof

Suppose first that $\varphi$ is an isometry. Then, $v_{i}=\varphi (u_{i})$ is an orthonormal basis due to fact. The coordinates of $v_{i}$ with respect to $u_{i}$ constitute the columns of the describing matrix $M$. Therefore, using exercise, we have
    ${v_{i}^{\text{tr}}}v_{j}=\left\langle v_{i},v_{j}\right\rangle =\delta _{ij}\,.$
Read as a matrix equation, this means
    ${M^{\text{tr}}}M={\begin{pmatrix}1&0&\cdots &\cdots &0\\0&1&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&1&0\\0&\cdots &\cdots &0&1\end{pmatrix}}\,.$
This argument can be read backwards to get the reverse implication.
# Matrix/K/Special linear group/Definition

For a field $K$, and $n\in \mathbb {N} _{+}$, the set of all invertible $n\times n$-matrices over $K$ with
    $\det M=1\,$
is called the special linear group over $K$. It is denoted by $[[:Template:Op:SLG]]$.
# Orthogonal matrix/Orthogonal group/Field/Definition

Let $K$ be a field, and $E_{n}$ the identity matrix of length $n$. A matrix $M\in \operatorname {GL} _{n}\!{\left(K\right)}$ fulfilling
    ${M^{\text{tr}}}M=E_{n}\,$
is called an orthogonal matrix. The set of all orthogonal matrices is called orthogonal group; it is denoted by
    $[[:Template:Op:Orthogonalgroup]]={\left\{M\in \operatorname {GL} _{n}\!{\left(K\right)}\mid {M^{\text{tr}}}M=E_{n}\right\}}\,.$
# Matrix/C/Unitary group/Definition

A matrix $M\in \operatorname {GL} _{n}\!{\left(\mathbb {C} \right)}$ fulfilling
    ${{\overline {M}}^{\text{tr}}}M=E_{n}\,$
is called a unitary matrix. The set of all unitary matrices is called unitary group; it is denoted by
    $[[:Template:Op:Unitarygroup]]={\left\{M\in \operatorname {GL} _{n}\!{\left(\mathbb {C} \right)}\mid {{\overline {M}}^{\text{tr}}}M=E_{n}\right\}}\,.$
# Linear isometry/Determinant is 1 or -1/Fact/Proof2

Due to fact, we have
    ${M^{\text{tr}}}M={\begin{pmatrix}1&0&\cdots &\cdots &0\\0&1&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&1&0\\0&\cdots &\cdots &0&1\end{pmatrix}}\,.$
Therefore, the statement follows from the multiplication theorem for the determinant and from fact.
# Euclidean vector space/Proper Isometry/Introduction/Section
## Definition

Euclidean vector space/Proper isometry/Definition
An isometry that is not proper, that is, its determinant is $-1$, is also called an improper isometry.

## Definition

Matrix/Special orthogonal group/Field/Definition

## Definition

Matrix/C/Special unitary group/Definition
# Matrix/Special orthogonal group/Field/Definition

Let $K$ be a field, and $n\in \mathbb {N} _{+}$. An orthogonal $n\times n$-matrix $M$ fulfilling
    $\det M=1\,$
is called a special orthogonal matrix. The set of all special orthogonal matrices is called special orthogonal group; it is denoted by $[[:Template:Op:Specialorthogonalgroup]]$.
# Matrix/C/Special unitary group/Definition

A unitary $n\times n$-matrix $M\in \operatorname {GL} _{n}\!{\left(\mathbb {C} \right)}$ fulfilling
    $\det M=1\,$
is called a special unitary matrix. The set of all special unitary matrices is called Special unitary group; it is denoted by $[[:Template:Op:Specialunitarygroup]]$.
# Aircrafts and aerodynamics
## Airplane Parts

The main parts of an aircraft are:
* the fuselage;
* the wings;
* the tail section (empennage);
* the engine(s);
* the landing gear.
The fuselage is the main body of the aircraft, holding the crew, passengers, as well as cargo.
The wings are the main source of lift on an airplane. They work through forcing pressure from the air onto its bottom half, thus lifting the plane up.
The tail section (or empennage) is the structure at the rear of the aircraft that helps stabilize the plane during flight, similar to feathers on an arrow.
The engine is located in the front or on the wings of an aircraft. It creates mechanical energy through the use of aviation fuel.
The landing gear is the undercarriage of the aircraft and is used for taxiing, takeoff, and landing. Attached to it are wheels that allows it to maneuver when making contact with the ground.
The control surfaces include:
* the ailerons;
* the elevators and stabilators;
* the rudder; and
* the trim systems.

## Aircraft Forces
### Four Forces on an Airplane

The four forces which act upon an aircraft in flight are:
* Lift
* Drag
* Thrust
* Weight
    Lift acts upwards at right angles to the airflow through the center of pressure. In simple terms, Lift is perpendicular to the Relative Wind.
    Drag is the resistance opposing the forward motion of the aircraft. It acts parallel and in the reverse direction of the aircraft's movement. Drag opposes thrust.
    Thrust is the forward propulsive force provided by the engines.
    Weight is the force acting vertically downward due to gravity. Weight opposes lift.

## Resources

NASA's Beginners Guide to Aerodynamics - Supplement your of study aerodynamics at your own pace and to your own level of interest. Some of the topics included are: Newton's basic equations of motion; the motion of a free falling object, that neglects the effects of aerodynamics; the terminal velocity of a falling object subject to both weight and air resistance; the three forces (lift, drag, and weight) that act on a glider; and finally, the four forces that act on a powered airplane. There are many interactive pages designed to illustrate aerodynamics.
Aerodynamics Lesson - Gold Seal Master CFI trainer Robert Still video presents basics of Aerodynamics in an easy to understand format.
See How It Flies - Free online book (in HTML format) running through aerodynamics and techniques for flight.
Wikipedia Aircraft - A definition of Aircraft and examples of various types of aircraft.
Wikipedia Aerodynamics - Technical overview of aerodynamics and the physics of flight.
FAA Aircraft - FAA information on Registration and Certification of Aircraft, Advisories and Guidance, Regulations and Policy, Forms, and Technical Information. Includes a searchable database of aircraft registered in the US.
# Isometry/C/Diagonalizability/Section
## Lemma
### Proof

$\Box$
The following statement is called spectral theorem or, more specific, spectral theorem for complex isometries.

## Theorem
### Proof

$\Box$
# Unitary vector space/Isometry/Orthogonal complement/Fact

Let $\varphi \colon V\rightarrow V$ be a linear isometry on a finite-dimensional ${\mathbb {K} }$-vector space $V$, endowed with an inner product. Let $U\subseteq V$ denote an invariant linear subspace.
orthogonal complement
$U^{\perp }$
In particular, $\varphi$ can be written as a direct sum
    $\varphi =\varphi _{U}\oplus \varphi _{U^{\perp }}\,,$
where the restrictions $\varphi _{U}$ and $\varphi _{U^{\perp }}$ are also isometries.
# Unitary vector space/Isometry/Orthogonal complement/Fact/Proof

We have
    $U^{\perp }={\left\{v\in V\mid \left\langle v,u\right\rangle =0{\text{ for all }}u\in U\right\}}\,.$
For such a $v\in U^{\perp }$ and an arbitrary $u\in U$, we have
    $\left\langle \varphi (v),u\right\rangle =\left\langle \varphi ^{-1}(\varphi (v)),\varphi ^{-1}(u)\right\rangle =\left\langle v,u'\right\rangle =0\,,$
since $u'=\varphi ^{-1}(u)\in U$ due to the invariance of $U$. Therefore, $\varphi (v)\in U^{\perp }$.
# Isometry/C/Diagonalizable/Fact

Let $V$ be a finite-dimensional $\mathbb {C}$-vector space, endowed with an inner product, and let
    $\varphi \colon V\longrightarrow V$
be an isometry.
$V$
orthonormal basis of eigenvectors
$\varphi$
In particular, $\varphi$ is diagonalizable.
# Angle/Introduction/Section
## Remark

For an affine space $E$ over a Euclidean vector space $V$, and three given points $P,Q,R\in E$ (a triangle) with $Q,R\neq P$, the angle $\angle (Q,P,R)$ of the triangle at $P$ is the angle $\angle ({\overrightarrow {PQ}},{\overrightarrow {PR}})$.
# Illustrated Companion to the Latin Dictionary/Cunae

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CUNAE. Same as CUNABULA. Cic. Div. i. 36.
# Illustrated Companion to the Latin Dictionary/Cunaria

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CUNA'RIA. A nurse, who rocked an infant in its cradle, washed it at its birth, wrapped it in swaddling clothes, &c. (Inscript. ap. Grut. 311. 7. Compare Mart. Ep. xi. 39.) The illustration (Cunaria/1.1) is from a marble bas-relief at Rome.
# Illustrated Companion to the Latin Dictionary/Cuneus

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CUN'EUS (σφήν). A wedge; a body of wood, iron, or other substances, with a thin edge gradually thickening upwards, employed for splitting (Virg. Georg. i. 144.), tightening, and fastening. Cic. Tusc. ii. 10.
2. When applied to ships (Ovid, Met. xi. 514.), the exact meaning of the term is doubtful. Some suppose that it is used to designate projecting pieces of timber fastened to the sides and bottom of a vessel to protect it from rocks; others, the timbers themselves put together in the form of a wedge, like what is now called "diagonal trussing;" or thin wedges of wood driven in together with the tow, by which the seams are caulked. Scheffer, Mil. Nav. i. 6.
3. (κερκίς). A compartment of seats (gradus, sedilia, subsellia) in a theatre or amphitheatre (Vitruv. v. 6. 2. Suet. Aug. 44.), comprising the several rows contained in each tier (maenianum) between a pair of staircases (scalae). The illustration (Cuneus/3.1), which represents a portion of the interior of the larger theatre at Pompeii, shows six of these cunei, or compartments of seats, three in the lower tier, and three in the one above, with two flights of stairs in each, down which the spectator walked when he entered the theatre through either of the doors (vomitoria) at the top, until he arrived at the particular row in the cuneus on which his seat was situated. These compartments of seats were termed wedges on account of their cuneiform appearance, being narrowest at the bottom, and gradually expanding upwards as the circuit of the theatre increases; see the parts marked B on the general plan s. THEATRUM, 1., where the form is more characteristically displayed.
4. A wine bin, constructed with rows of shelves rising one over the other, like the seats of a theatre, and upon which the wine was deposited to ripen, after it had been drawn off from the bulk into amphorae, or, as we should say, bottled. Cato, R. R. ii. 3. 2. Pontedera, Curae Posth. ad l.
5. A body of soldiers drawn up in the shape of a wedge. Liv. xxii. 47. Veg. Mil. iii. 19.
# Illustrated Companion to the Latin Dictionary/Cunicularii

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CUNICULA'RII. Sappers and miners; or soldiers who effect an entrance into a town from a mine (cuniculus). Veg. Mil. ii. 11. Ammian. xxiv. 4. 22.
# Illustrated Companion to the Latin Dictionary/Cuniculatores

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CUNICULATO'RES. Same as the preceding. Luctat. in Stat. Theb. ii. 418.
# Illustrated Companion to the Latin Dictionary/Cuniculus

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CUNIC'ULUS (ὑπονόμος). Any subterranean passage, but more especially a mine in military operations. Veget. i. 6. Liv. v. 21. Ammian. xxiv. 4. 21.
# Illustrated Companion to the Latin Dictionary/Cunulae

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CU'NULAE. Diminutive of CUNAE; a small or common sort of cradle. Prudent. Cathem. vii. 164. Id. xi. 98.
# Illustrated Companion to the Latin Dictionary/Cupa

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
CU'PA (γαῦλος). A cask, or butt; made with wooden staves (tabulae, Pallad. i. 38. 1.), and bound round with iron hoops (circuli, Pet. Sat. 60. 3. Plin. H. N. xiv. 27.), in which wine, vinegar, and other articles were kept and transported from place to place; whence vinum de cupa (Cic. Pis. 27.) is equivalent to our expression out of the wood. The example (Cupa/1.1) is copied from the Column of Trajan.
2. (κώπη). An oblong block of wood, forming one of the component parts in a trapetum, or machine for bruising olives. It was made of elm or beech, and perforated through its centre, in order to be slipped on to a thick iron pivot (columella ferrea), which projected from the top of the stone cylinder (miliarium) in that machine. The object of it was twofold: to form a block for receiving the ends of the axles, which are inserted in it in the engraving, and on which the wheels (orbes) were suspended, while at the same time it enabled them to move in a circular direction round the bruising vat (mortarium) by turning round the pivot passing through its centre from the top of the upright stone cylinder on which it was placed. It was, therefore, cased with plates of metal, to prevent friction. (Cato, R. R. xxi. 1 — 4). The specimen (Cupa/2.1) here introduced is restored from the fragments of a trapetum discovered at the ancient Stabia, the wood-work of which had perished, but the iron plates remained entire, as well as the portions of the two axles inserted in it. The figure, however, sufficiently explains the meaning of the name, and why it was so called; for the word, in its literal sense, signifies the handle of an oar (Diodor. Sic. iii. 3. and Agath. quoted by Wesseling ad l.), to which the cupa of a trapetum, as shown by the engraving, bears a close resemblance. The situation occupied by it on the machine, and the manner in which it acted will be better understood by referring to the illustration s. TRAPETUM, where it is marked 5.
* Cupa/1.1
* Cupa/2.1