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

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.
CARCER (κάρκαρον). A prison or gaol. The Roman prisons were divided into three stories, one above the other, each of which was appropriated to distinct purposes. The lowermost (carcer inferior, γοργύρη) was a dark underground dungeon, having no other access but a small aperture through the floor of the cell above, and was used not for detention, but as the place of execution, into which the criminal was cast in order to undergo his sentence, if condemned to death. The middle one (carcer interior), constructed immediately over the condemned cell, and on a level with the ground, but having, like the preceding, its only access through an aperture in the roof, served as a place of confinement where the punishment of imprisonment in chains (custodia arcta) was expiated, or until the sentence, if a capital one, was about to be carried into effect. The upper one, forming a story above the ground, was provided as a place of detention for those convicted of minor offences, or who were only condemned to an ordinary term of imprisonment (custodia communis), in which the confinement was much less severe, the prisoners not being chained, nor excluded from the enjoyment of air and exercise. Thus we may understand with precision the sort of confinement to which Dolabella was subjected by Otho — neque arcta custodia, neque obscura (Tac. Hist. i. 88.); i. e. in the upper chamber of all, not in the close confinement of the carcer interior (the upper one in engraving), nor in the dark underground dungeon below. All these three divisions were apparent in the gaol of Herculaneum, when it was excavated; and the two lower ones still remain entire in the prisons constructed by Ancus and Servius, near the Roman Forum, a section of which (Carcer/1.1) is introduced above, showing their relative positions and plan of construction. The wall at the top, with the inscription, commemorating the person by whom it was repaired, faced the forum, and enclosed the upper story, now decayed.
2. The stalls in the Circus where the chariots were stationed before the commencement of a race, and to which they returned after its conclusion. (Ovid. Her. xviii. 166. Auct. ad Herenn. iv. 3.) These were vaults closed in front by large wooden gates, and usually twelve in number (Cassiodor. Var. Ep. iii. 51.), whence the word is mostly used in the plural (Cic. Brut. 47. Virg. G. i. 512.); one for each chariot, and situated at the flat end of the race course under the oppidum, six on each side of the porta pompae, through which the procession entered. Their relative position as regards the course is shown on the ground-plan of the CIRCUS (s. v.), on which they are marked A. A, and an elevation of four carceres, with their doors open (cancelli), is here given (Carcer/2.1), from a bas-relief in the British Museum.
* Carcer/1.1
* Carcer/2.1
# Illustrated Companion to the Latin Dictionary/Carchesium

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.
CARCHE'SIUM (καρχήσιον). A drinking-cup of Greek invention, having a tall figure, slightly contracted at its sides, with slender handles which reached from the rim to the bottom (Macrob. Sat. v. 21.), and used as a goblet for wine (Virg. Georg. iv. 380.), or milk. (Ovid, Met. vii. 247.) The figure in the engraving (Carchesium/1.1) is from a painting in the tomb of Caius Cestius, one of the Epulones or citizens who had the duty of providing a sumptuous banquet in honour of Jupiter. The locality where it is represented, and its perfect correspondence with the description of Macrobius, seem quite sufficient to identify the name and form.
2. An apparatus attached to the mast of a ship, just above the yard (Lucil. Sat. iii. 14. ed. Gerlach. Lucan. v. 418.), in which part of the tackle worked (Serv. ad Virg. Aen. v. 77. Non. s. v. p. 546.), and into which the seamen ascended to keep a look out, manage the sails, and discharge missiles, as seen in the illustration (Carchesium/2.1), from a painting in the Egyptian tombs. It thus answers in some respects to what our seamen call the "tops," but received its name from a real or fancied resemblance to the drinking-cup figure in the last wood-cut.
3. Carchesium versatile. The same apparatus, when made to revolve round the mast, and act as a crane for the loading and unloading of merchant vessels, by means of crossbar or crane-neck inserted horizontally into it. (Vitruv. x. 2. 10. Schneider, ad l.) Our seamen make use of the yard arm in a manner not dissimilar.
* Carchesium/1.1
* Carchesium/2.1
# Illustrated Companion to the Latin Dictionary/Bracae

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.
BRAC'AE or BRAC'CAE (ἀναξυρίδες). An article of dress which entirely covered the lower part of the person from the waist (see cut 2.) to the ankles, and was either made to fit the figure nearly tight, like our pantaloons, or to sit more loosely round the legs, like trowsers. The word contains the elements of the Scotch breeks, and English breeches; but answers more closely to the pantaloons and trowsers of the present day. The Romans included both kinds under the general term of bracae; but the Greeks distinguished each particular form by a characteristic name; as follows: —
1. ἀναξυρίδες. A pair of tight trowsers or pantaloons, more especially proper to the Eastern nations, and amongst these the Amazons and Persians (Ovid. Trist. v. 10. 34. Herod. i. 71.), as shown by the engraving annexed (Bracae/1.1), which represents a Persian prince at the battle of Issus, from the great mosaic at Pompeii.
2. Bracae laxae (θύλακοι). A pair of loose trowsers, worn in the same manner as the preceding, but more generally characteristic of the northern nations (Ovid. Trist. v. 7. 49. Lucan. i. 430.), as seen in the annexed figure (Bracae/2.1), representing one of the German auxiliaries in the army of Trajan; and of the Phrygians, amongst the Asiatics (Eur. Cycl. 182); consequently the usual costume of Paris.
3. Bracae virgatae (Propert. iv. 10. 43.), or pictae. (Val. Flacc. vi. 227.) Striped, checked, and embroidered trowsers, which were much worn by the inhabitants of Asia. See the next illustration.
* Bracae/1.1
* Bracae/2.1
# Illustrated Companion to the Latin Dictionary/Caligatus

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.
CALIGA'TUS. Wearing the caliga, or soldier's shoe (Juv. Sat. iii. 322.), as seen in the last cut but one; and thence by implication, a common soldier (Suet. Aug. 25. Id. Vitell. 7.), because its use was peculiar to the rank and file.
# Proportionality/Velocity/Exercise

Lucy Sonnenschein rides her bicycle with 10 meter per second.
a) How many kilometers does she ride per hour?
b) How long does it take for her to ride 100 kilometers?
# Vector space/K to V/Linear/Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space. Prove that, for $v\in V$, the map
    $K\longrightarrow V,\lambda \longmapsto \lambda v,$
is linear.
# Vector space/Scaling/Linear mapping/Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space. Prove that, for $a\in K$, the map
    $V\longrightarrow V,v\longmapsto av,$
is linear.
# Linear mapping/Basis/(0,1,2) to (3,-2) and (1,4,1) to (1,0) and (2,1,3) to (7,2)/(3,-5,4)/Exercise

Let a linear mapping
    $\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{2}$
satisfying
    $\varphi {\begin{pmatrix}0\\1\\2\end{pmatrix}}={\begin{pmatrix}3\\-2\end{pmatrix}},\,\varphi {\begin{pmatrix}1\\4\\1\end{pmatrix}}={\begin{pmatrix}1\\0\end{pmatrix}},{\text{ and }}\varphi {\begin{pmatrix}2\\1\\3\end{pmatrix}}={\begin{pmatrix}7\\2\end{pmatrix}}$
be given. Compute
    $\varphi {\begin{pmatrix}3\\-5\\4\end{pmatrix}}.$
# 4-number-problem/Ordered by size/Matrix/Exercise

We consider the mapping
    $\Psi \colon \mathbb {R} _{\geq 0}^{4}\longrightarrow \mathbb {R} _{\geq 0}^{4}$
that assigns to a four-tuple $(a,b,c,d)$ the four-tuple
    $(\vert {b-a}\vert ,\vert {c-b}\vert ,\vert {d-c}\vert ,\vert {a-d}\vert ).$
Describe this mapping by a matrix, under the condition
    $a\leq b\leq c\leq d\,.$
# Elementary-geometric mappings/Inverse mappings/Exercise

Describe the inverse mappings of the elementary-geometric mappings such as reflection at an axis, point reflection, rotation, homothety, translation.
# Complex numbers/Real and imaginary part/Conjugation/Linearity/Exercise

Prove that the functions
    $\mathbb {C} \longrightarrow \mathbb {R} ,z\longmapsto \operatorname {Re} \,{\left(z\right)},$
and
    $\mathbb {C} \longrightarrow \mathbb {R} ,z\longmapsto \operatorname {Im} \,{\left(z\right)},$
are $\mathbb {R}$-linear maps. Prove also that the complex conjugation is $\mathbb {R}$-linear, but not $\mathbb {C}$-linear. Is the absolute value
    $\mathbb {C} \longrightarrow \mathbb {R} ,z\longmapsto \vert {z}\vert ,$
$\mathbb {R}$-linear?
# Linear mappings/To product/Exercise

Let $K$ be a field, and let $U,V,W$ denote vector spaces over $K$. Let $\varphi \colon U\rightarrow V$ and $\psi \colon U\rightarrow W$ be linear mappings. Show that the mapping
    $U\longrightarrow V\times W,u\longmapsto \varphi (u)\times \psi (u)$
into the product space $V\times W$ is also a linear mapping.
# Linear system/Solution methods/Adaptivity/Remark

The methods described in fact, remark, and remark to solve a linear system differ with respect to speed, strategic conception, complexity of the coefficients, error-proneness. In the elimination method, the systematic reduction of the number of variables (reduction of dimension) is obvious, and it is unlikely to make mistakes (except for miscalculations). It is always clear how to continue. However, these advantages emerge starting with three variables. For two variables, it does not make a difference what method we choose.
The evaluation of the methods depends also on the features of the concrete system. Such features should be taken into account in order to find "short-cuts“ to the solution. The adequate choice of the solution method appropriate for the given problem is called adaptivity (a concept which is used in the didactic context with different meanings). If, for example, one row of the system has the form $x=3$, then one should recognize that a part of the solution can be read off immediately, and one should not add to this row any other row. Here, one should replace in the other rows everywhere $x$ by $3$, and continue then. Or: if four equations are given, where in two equations only the variables $x$ and $y$ appear, and where in the two other equations only the variables $z$ and $w$ appear, then one should realize that, in principle, two unrelated linear systems are given, each in two variables, and these should be solved independently. Or: it might be that a small subsystem of the system guarantees that there is no solution at all. Then only this has to be worked out, there is no need to consider the other equations. And: consider the exact question! If the question is whether a certain tuple is a solution, then we only have to plug this tuple into the equations, no manipulations are necessary.
# Endomorphismus/Finite dimensional/Change of basis/Fact

Let $K$ denote a field, and let $V$ denote a $K$-vector space of finite dimension. Let
    $\varphi \colon V\longrightarrow V$
be a linear mapping. Let ${\mathfrak {u}}$ and ${\mathfrak {v}}$ denote bases of $V$.
${\mathfrak {u}}$ and ${\mathfrak {v}}$ respectively (on both sides), fulfil the relation
    $M_{\mathfrak {u}}^{\mathfrak {u}}(\varphi )=M_{\mathfrak {u}}^{\mathfrak {v}}\circ M_{\mathfrak {v}}^{\mathfrak {v}}(\varphi )\circ (M_{\mathfrak {u}}^{\mathfrak {v}})^{-1}\,.$
# Linear mapping/Matrix and basis/Rank/Fact

Let $K$ denote a field, and let $V$ and $W$ denote $K$-vector spaces of dimensions $n$ and $m$. Let
    $\varphi \colon V\longrightarrow W$
be a linear mapping, which is described by the matrix $M\in \operatorname {Mat} _{m\times n}(K)$, with respect to bases of the spaces.
    $\operatorname {rk} \,\varphi =\operatorname {rk} \,M\,$
# Linear mapping/Linear subspaces/Kernel/Introduction/Section

A typical property of a linear mapping is that it maps lines to lines (or to a point). More general is following statement.

## Lemma

Let $K$ be a field, let $V$ and $W$ denote $K$-vector spaces and let
    $\varphi \colon V\longrightarrow W$
be a
$K$
* For a linear subspace $S\subseteq V$, the image $\varphi (S)$ is a linear subspace of $W$.
* In particular, the image $\operatorname {Im} \varphi =\varphi (V)$ of the mapping is a linear subspace of $W$.
* For a linear subspace $T\subseteq W$, the preimage $\varphi ^{-1}(T)$ is a linear subspace of $V$.
* In particular, $\varphi ^{-1}(0)$ is a linear subspace of $V$.

### Proof

$\Box$

## Definition

Let $K$ denote a field, let $V$ and $W$ denote $K$-vector spaces, and let
    $\varphi \colon V\longrightarrow W$
denote a $K$-linear mapping. Then
    $\operatorname {kern} \varphi :=\varphi ^{-1}(0)={\left\{v\in V\mid \varphi (v)=0\right\}}\,$
$\varphi$
Due to the statement above, the kernel is a linear subspace of $V$.

## Remark

For an $m\times n$-matrix $M$, the kernel of the linear mapping
    $K^{n}\longrightarrow K^{m},x\longmapsto Mx,$
given by $M$ is just the solution space of the homogeneous linear system
    $Mx=0\,.$
The following criterion for injectivity is important.

## Lemma

Let $K$ denote a field, let $V$ and $W$ denote $K$-vector spaces, and let
    $\varphi \colon V\longrightarrow W$
denote a $K$-linear mapping. Then $\varphi$ is injective if and only if $\operatorname {kern} \varphi =0$

### Proof

If the mapping is injective, then there can exist, apart from $0\in V$, no other vector $v\in V$ with $\varphi (v)=0$. Hence, $\varphi ^{-1}(0)=\{0\}$. So suppose that $\operatorname {kern} \varphi =0$, and let $v_{1},v_{2}\in V$ be given with $\varphi (v_{1})=\varphi (v_{2})$. Then, due to linearity,
    $\varphi (v_{1}-v_{2})=\varphi (v_{1})-\varphi (v_{2})=0\,.$
Therefore, $v_{1}-v_{2}\in \operatorname {kern} \varphi$, and so $v_{1}=v_{2}$.
$\Box$
# Illustrated Companion to the Latin Dictionary/Cardinalis

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.
CARDINA'LIS. See SCAPUS.
# Illustrated Companion to the Latin Dictionary/Cardinatus

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.
CARDINA'TUS. See CARDO 4.
# Illustrated Companion to the Latin Dictionary/Cardo

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.
CARDO. A pivot and socket, forming an apparatus by means of which the doors of the ancients were fixed in their places, and made to revolve in opening and shutting; thus answering the same purpose as the hinges more commonly in use amongst us, though the contrivance was entirely different in its character. (See GINGLYMUS.) The Greeks distinguished each of these parts by distinct names, using στρόφιγξ for the pivot, and στροφεύς for the socket in which the pivot worked; but the Latin writers commonly include the whole apparatus under the term cardo, though they sometimes apply it to each of the parts separately, and sometimes to the whole style of the door-leaf (scapus cardinalis), that formed the axle by which the contrivance acted. (Plin. H. N. xvi. 77. ib. 84. Id. xxxvi. 24. n. 8. Plaut. Asin. ii. 3. 38. Virg. Aen. ii. 480. Apul. Met. i. p. 9.) The figures in the annexed engraving (Cardo/1.1) will explain the nature of these objects, and the manner in which they were applied. The two top ones on the right hand exhibit a pair of bronze shoes from Egyptian originals in the British Museum, which were fastened on to the top and bottom of a door-leaf, to act as pivots (στρόφιγγες), for the wooden axles were cased with bronze to bear the wear and tear (Virg. Cir. 222. aeratus cardo); the two lower ones on the same side are two boxes which were let into the sill and lintel of the door case to act as sockets (στροφεῖς), in which the pivots turned; the left-hand one, which is Egyptian, and of very hard stone, is now in the British Museum, and was actually used with the pivot shoe drawn immediately above it: the right-hand one is of bronze, and was found in the sill of a door at Pompeii; the teeth or flutings round the sides are too keep it firm in its place, and prevent it from turning in its setting with the working of the door; the left-hand figure is an Egyptian door from Wilkinson, and shows the manner in which the apparatus was attached and worked. Compare the illustration s. ANTEPAGMENTUM.
2. The pin or pivot at each extremity of an axle in machinery, by means of which the axle revolves in the sockets which receive them, as in a wheel-barrow, roller, and similar contrivances. Vitruv. x. 14. 1.
3. A tenon in carpentry; i. e. the head of a timber cut into a particular form for the purpose of fitting into a cavity of the same size and shape in another piece, and so forming a joint (Vitruv. x. 14. 2.); hence cardo securiculatus, a tenon in the form of an axe, or as we call it "dove-tailed." Vitruv. x. 10. 3.
# Illustrated Companion to the Latin Dictionary/Carenum

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.
CARE'NUM. The must of new wine inspissated by boiling down to two-thirds of its original quantity. Pallad. Oct. 18.
# Illustrated Companion to the Latin Dictionary/Carina

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.
CARI'NA (τρόπις). The keel, or lowest piece of timber in the framework of a ship, running the whole length from stem to stern, and serving as a foundation for the entire fabric (Cic. de Orat. iii. 46.); including also the false keel or "keelson." Liv. xxii. 20. Caes. B. G. iii. 13.
# Illustrated Companion to the Latin Dictionary/Carnarium

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.
CARNA'RIUM. A frame suspended from the ceiling, and furnished with hooks and nails, for the purpose of hanging up cured provisions dried fruits, herbs, &c., similar to those still used in our kitchens. (Plaut. Capt. iv. 4. 6. Pet. Sat. 135. 4. Id. 136. 1. Plin. H. N. xviii. 60.) The illustration (Carnarium/1.1) is from a painting at Pompeii, in which it is suspended from the ceiling of a tavern, and shows sausages, vegetables, and such things hanging by strings or in nets.
2. In a more general sense, a safe or larder for the preservation of fresh viands. Plaut. Curc. ii. 3. 45. Plin. H. N. xix. 19. n. 3.
# Illustrated Companion to the Latin Dictionary/Carnifex

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.
CAR'NIFEX. The public executioner, who inflicted torture and scourging upon criminals, and executed the condemned by strangling them with a rope. Plaut. Capt. v. 4. 22. Suet. Nero, 54.
# Illustrated Companion to the Latin Dictionary/Carnificina

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.
CARNIFICI'NA. The place in which criminals were tortured and executed (Liv. ii. 23. Suet. Tib. 62.); viz. an underground dungeon beneath all the other cells of the gaol. The illustration (Carnificina/1.1) represents the interior of the carnificina in the state prisons at Rome, constructed by Servius Tullius, after whom it was called the Tullianum, and the identical spot in which the friends and accomplices of Catiline were executed by order of Cicero. The criminal was let down into it by a rope through the aperture in the ceiling, and his body dragged up again by an iron hook (uncus) after the execution. The small door-way on the left hand, though ancient, does not belong to the original construction; it gives admission to a low subterranean gallery, now filled with rubbish, but which takes a direction towards the Tiber, and was, perhaps, intended for carrying the dead bodies to the river, when they were not dragged out of the prison for exposure on the Gemonian stairs.
# Illustrated Companion to the Latin Dictionary/Carpentum

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.
CARPEN'TUM. A two-wheeled carriage, with an awning over it, and curtains by which it might be closed in front (Prop. iv. 8. 23. Apul. Met. x. p. 224.); capable of containing two or three persons, usually drawn by a pair of mules (Lamprid. Heliog. 4.), and used by the Roman matrons and ladies of distinction from remote antiquity. (Ov. Fast. i. 619. Liv. v. 25.) The illustration (Carpentum/1.1), which belongs to the earliest times is copied from an Etruscan painting (Micali, Italia avanti i Romani, tav. 27.), and represents a bride and bridegroom, or a married pair, as Livy describes Lucumo and his wife on their arrival at Rome (sedens carpento cum uxore. Liv. i. 34.).
2. Carpentum funebre or pompaticum. A state carpentum or carriage, in which the urn containing the ashes of the great, or their statues, were carried in the funeral procession. (Suet. Cal. 15. Id. Claud. 11. Isidor. Orig. xx. 12. 3.) These were likewise covered carriages, constructed upon the same principle as the preceding, but more showy and imposing in character; as may be seen by the example (Carpentum/2.1), from a medal struck in commemoration of one of the Roman empresses, its use being further implied by the form, which, it will be observed, is made in imitation of a tomb.
3. A cart employed for agricultural purposes, and apparently of very common and general use; for the same word is frequently applied in the sense of cart-load, as of dung, &c., to indicate a certain quantity, which every one would immediately recognise, as in the English phrase, "a load." (Pallad. x. 1. Veget. Mul. Med. iv. 3. Praef.) It was probably built like the first of the two specimens, but of coarser workmanship, and without the awning.
* Carpentum/1.1
* Carpentum/2.1