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

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.
HYPAETH'ROS (ὕπαιθρος). Literally, under the sky, or in the open air; whence applied to a temple, or other edifice which had no roof over the central portion of its area, so that the interior was open to the sky. Hypaethral structures were generally the largest and most magnificent of their kind; indeed, the difficulty of roofing over a very large area may be regarded as a principal motive for adopting the expedient. The great temple at Paestum affords an existing specimen of this style; but no instance was to be found in Rome when Vitruvius wrote. Vitruv. iii. 2.
# Illustrated Companion to the Latin Dictionary/Hypaetrum

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.
HYPAE'TRUM. A latticed window constructed over the grand entrance of a temple (Vitruv. iv. 6. 1.), as in the annexed example (Hypaetrum/1.1), which represents the door of the Pantheon at Rome. One of the Xanthian marbles in the British Museum affords an example of the same contrivance, which possesses the double advantage of giving grandeur without, and admitting air within.
# Illustrated Companion to the Latin Dictionary/Hyperthyrum

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.
HYPER'THYRUM (ὑπέρθυρον). An ornamental member, consisting of a frieze and cornice supported upon trusses or consoles (ancones, parotides), usually placed above the lintel of a door-frame in temples and other great buildings (Vitruv. iv. 6. 4.); an example of which is given in the annexed engraving (Hyperthyrum/1.1), with one of the trusses in profile by its side, from the temple of Hercules at Cora, constructed precisely as Vitruvius directs in the passage cited; and the preceding woodcut affords an example of a similar ornament, but differently designed, placed over the hypaetrum, in the Pantheon at Rome. This member was intended to increase the apparent size of the doorway, in order to preserve the level of the horizontal line formed by the architrave of the pronaos and the antae; whence it is directed that the top of the cornice of the hyperthyrum should coincide with the tops of the capitals belonging to the columns and antae of the pronaos. If the doorcase itself were made thus high, the valves would be ill-proportioned, and cumbersome to open.
# Illustrated Companion to the Latin Dictionary/Hypocausis

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.
HYPOCAU'SIS (ὑπόκαυσις). A furnace with flues running underneath the pavement of an apartment in a private house or set of baths, for the purpose of increasing the temperature of the air in the chamber above. (Vitruv. v. 10. 1. and 2.) It is very plainly shown in the annexed engraving (Hypocausis/1.1), representing the sectional elevation of a bath-room, discovered in a Roman villa at Tusculum; the small arch on the left shows the mouth of the furnace (propnigeum), over which are placed the vessels (vasaria, Vitruv. l. c.), containing hot and tepid water, which it served to heat; and, on the right, under the floor of the room, which is supported upon a number of low and hollow tubes, is an offset from the hypocausis, which warmed the chamber above it.
# Illustrated Companion to the Latin Dictionary/Hypocaustum

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.
HYPOCAUS'TUM (ὑπόκαυστον). A room, of which the temperature is warmed by means of a furnace and flues (hypocausis) directed under it, as represented by the last engraving, Plin. Ep. ii. 17. 11. and 23. Compare Stat. Sylv. i. 5. 59., where the word seems to be applied to the flues under the chamber rather than to the chamber itself.
# Isometry/Decomposition/Section
## Lemma
### Proof

$\Box$

## Theorem
### Proof

$\Box$
In this decomposition, $G_{1}\oplus \cdots \oplus G_{p}$ is the eigenspace for the eigenvalue $1$, and $H_{1}\oplus \cdots \oplus H_{q}$ is the eigenspace for the eigenvalue $-1$; the decompositions are not unique. The Isometry is proper if and only if $q$ is even.
# Illustrated Companion to the Latin Dictionary/Hypocrita

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.
HYPOC'RITA, or -TES (ὑποκριτής). An actor or performer who plays a part upon the stage (Suet. Nero, 24. Compare Quint. xi. 3. 7.) The word is properly a Greek one; and corresponds with the Latin histrio.
# Illustrated Companion to the Latin Dictionary/Hypodidascalus

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.
HYPODIDAS'CALUS (ὑποδιδάσκαλος). A sub-master, or under teacher; at a school (Cic. Fam. ix. 18.); of a Greek chorus. Plat. Ion. 536. A.
# Illustrated Companion to the Latin Dictionary/Hypogaeum

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.
HYPOGAE'UM (ὑπόγαιον). (Inscript. ap. Donat. cl. 8. n. 14. ap. Grut. 1114. 3.) Same as HYPOGEUM.
# Illustrated Companion to the Latin Dictionary/Hypogeum

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.
HYPOGE'UM (ὑπόγειον). That part of a building which lies below the level of the ground (Vitruv. vi. 8.); whence a subterranean vault in which the Greeks buried their dead without burning the body (Pet. Sat. iii. 2.); consequently, corresponding with the Roman CONDITORIUM.
# Illustrated Companion to the Latin Dictionary/Hypotrachelium

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.
HYPOTRACHE'LIUM (ὑπροτραχήλιον). The uppermost part of the shaft of a column, where it is of the smallest diameter, immediately under the neck of the capital. Vitruv. iii. 3. 12. Id. iv. 7. 3.
# Illustrated Companion to the Latin Dictionary/Iatralipta

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.
IATRALIP'TA, or -TES (ἰατραλείπτης). A medical man who treated his patients upon what was called the iatraliptic system (Iatraliptice, Plin. H. N. xxix. 2.); i. e. by the external application of unguents and friction, combined with a regular gymnastic regimen. Plin. Ep. x. 4. Cels. i. 1.
# Illustrated Companion to the Latin Dictionary/Ichnographia

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.
ICHNOGRAPH'IA (ἰχνογραφία). A chart, map, or ground-plan, made in outline by architects and surveyors for the workmen to build by, or as a map of reference. (Vitruv. i. 2. 2.) The annexed engraving (Ichnographia/1.1) affords a specimen of Roman mapping, from a plan of the city engraved upon slabs of marble, originally forming the pavement of the temple of Romulus and Remus; many fragments of which are preserved in the Capitol. It is supposed to have been executed in the age of Septimius Severus; and when entire, afforded a complete guide to the city, in which every street, house, and public edifice was laid down in its proper place, and in sufficient detail to show its ground-plot and architectural design, together with the name of each inscribed upon it. The fragment here introduced shows the original plan of the portico of Octavia surrounding the temples of Jupiter and Juno; of all which buildings considerable remains are still standing near the present fish market. The dotted lines are only cracks in the marble. Other specimens from the same plan are presented at pp. 67. 248 and other parts of this work, some of which indicate the great skill with which the ancient draughtsmen contrived to express constructive forms by a few simple outlines.
# Illustrated Companion to the Latin Dictionary/Ignispicium

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.
IGNISPIC'IUM. A branch of the art of divination, which consisted in foretelling the secrets of futurity by the inspection of ignited matter. (Plin. H. N. vii. 57.) Compare Sen. Oed. 306 — 330., where the various appearances of the flames, and the results supposed to be indicated by them, are set out at length.
# Illustrated Companion to the Latin Dictionary/Illix

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.
ILLIX or INLEX sc. Avis (παλευτής). A decoy bird, employed by the ancient fowlers to entice others within reach of their nets and snares. For this purpose they made use both of birds which were of a kindred and of a hostile species, such as the owl and falcon, which was also trained to catch those which it had decoyed within its reach. (Plaut. As. i. 3. 68. Pallad. x. 12. Mart. xiv. 216. Oppian. Cyneg. i. 65.) The illustrations at p. 59. s. ARUNDO, 4. afford two examples of the use of a call-bird, from ancient works of art.
# Illustrated Companion to the Latin Dictionary/Imaginarii

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.
IMAGINA'RII. Standard bearers in the Roman Imperial armies, whose ensigns had an image of the emperor amongst the other devices (Veget. Mil. ii. 7.), as seen in the annexed woodcut (Imaginarii/1.1), from the Column of Trajan, in which the emperor's portrait occupies the top place, surrounded by a wreath of laurel.
# Illustrated Companion to the Latin Dictionary/Imagines majorum

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.
IMA'GINES MAJO'RUM. Family portraits, or likenesses, consisting of waxen masks, expressing the lineaments of deceased persons, which their surviving relatives preserved with studious care in cases or armoires placed round the atrium of their mansions, regarding them as the honoured representatives of their ancestral line. (Liv. iii. 58. Sall. Jug. 85. Suet. Vesp. 1.) The mask in the annexed woodcut (Imagines_majorum/1.1), from a sepulchral bas-relief, which represents a female bewailing the death of her husband, is probably intended for one of these images in its case. The honorary distinction of handing themselves down to posterity by these representations, was only permitted to certain persons amongst the Romans; viz. those who had passed through either of the high offices of aedile, praetor, or consul; and when the funeral of any individual of the above rank and ancient lineage took place, the masks were taken out of their cases, and worn by persons who walked in front of the bier, in a similar costume, and with the same insignia as had belonged to the personages they represented during their lives. (Eichstädt. Dissertt. de Imagg. Rom.) These were called the effigies (effigies) of the family; and they personated characters even as far back as traditional history, Aeneas, the Alban kings, Romulus, &c. (Tac. Ann. iv. 9. Compare Polyb. vi. 53. Hor. Epod. 8. 2.) It will be self-evident that no authentic or contemporary likeness of any individual ascribed to such remote antiquity could ever have been in existence, even though we should admit that the original was a real historical person: but there is no doubt that the great Roman families preserved characteristic representations of their early, and even fabulous, ancestors, modelled in lineament and costume after some traditionary type, well known to, and immediately recognized by, the people at large, which are met with on coins, medals, and engraved gems (e. g. the head of Numa s. BARBATUS); precisely as all modern representations of the Saviour exhibit a particular identity of character, style, and features, which, though not professing to be genuine likenesses, are still formed after a traditionary model of very great antiquity.
# Illustrated Companion to the Latin Dictionary/Imaginiferi

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.
IMAGINIF'ERI. (Veget. Mil. ii. 7. Inscript. ap. Grut. 1107. 1.) Same as the preceding.
# Illustrated Companion to the Latin Dictionary/Imbrex

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.
IMBREX (καλυπτήρ). A ridge-tile made to receive the shower (imber), and of a semi-cylindrical form, as contradistinguished from tegula, which was flat. (Isidor. Orig. xix. 10. 15. Plaut. Most. i. 2. 26.) The imbrex was intended to cover the juncture of two flat tiles, and, consequently, was made narrower at one end, so as to lap over one another and form a continuous ridge down the sides of the roof (woodcut s. IMBRICATUS), which threw off the rain water from its hog's back into the channel formed by the tegulae, between each row of imbrices. The modern Italian architects use tiles of the same description; two of which are represented by the annexed engraving (Imbrex/1.1), which shows their form, and the manner in which they were fitted to one another.
2. Imbrex supinus. A gutter formed by a series of ridge-tiles fitted into one another, and laid upon their backs (Columell. ix. 13. 6. Compare ii. 2. 9.), as in the annexed example (Imbrex/2.1), which shows a water conduit in the ruin, commonly known as the grotto of Egeria near Rome.
* Imbrex/1.1
* Imbrex/2.1
# Illustrated Companion to the Latin Dictionary/Imbricatim

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.
IMBRICA'TIM. Formed in undulations like the imbrices of a roof. Plin. H.N. ix. 52., and next woodcut.
# Illustrated Companion to the Latin Dictionary/Imbricatus

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.
IMBRICA'TUS. (From imbrico, καλυπτηρίζω). Imbricated, in architecture; that is, having the roof covered with a series of flat and ridge-tiles (tegulae and imbrices); the usual manner in which the Greeks and Romans protected the timber-work in the roofs of their buildings, and of which a specimen is afforded by the annexed engraving (Imbricatus/1.1), representing the roof of the portico of Octavia at Rome, the tiles of which are made of white marble.
# Illustrated Companion to the Latin Dictionary/Immissarium

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.
IMMISSA'RIUM. A basin, trough, or other contrivance built upon the ground, of stone or brick, and intended as a cistern to contain a body of water flowing from the reservoir (castellum) of an aqueduct, for the accommodation of the adjacent neighbourhood. (Vitruv. viii. 6. 1.) It differs from cisterna, which was underground; and is shown by the annexed engraving (Immissarium/1.1), from a specimen at Pompeii. The high vaulted building is the reservoir, from which the water flowed through the small dark aperture at its bottom, into the square stone trough (immissarium) on the level of the pavement. The city of Pompeii is furnished with several other conveniences of this description.
# Illustrated Companion to the Latin Dictionary/Immolatus

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.
IMMOLA'TUS. Accurately speaking, means sprinkled with flour (mola salsa), in reference to a victim intended for the sacrifice, this being one of the usual ceremonies before it was slain. (Cato ap. Serv. Aen. x. 541.); whence the word came to be used in the less special sense of our term immolated, or killed in sacrifice. Hor. Od. iv. 11. 7.
# Illustrated Companion to the Latin Dictionary/Impages

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.
IMPA'GES. The broad transverse band in a door, which stretches from stile to stile, and divides the pannels horizontally from one another, technically called by our carpenters, the rail. (Vitruv. 6. iv. 5., and JANUA), where the component parts which form the leaf of a door are illustrated and explained.
# Illustrated Companion to the Latin Dictionary/Impedimentum

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.
IMPEDIMEN'TUM (τὰ σκεύη). The baggage of an army which was transported in waggons, or on beasts of burden (Caes. B. G. i. 26. Liv. xliv. 27.); including also the baggage waggons and the beasts which drew them. Caes. B. G. vii. 45. Front. Strateg. ii. 1. 11.
# Illustrated Companion to the Latin Dictionary/Impediti

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.
IMPEDI'TI. In military phraseology soldiers who marched with a heavy load of arms, provisions, and personal baggage (sarcina), as was the ordinary practice in the Roman armies (Caes. B. G. i. 12.), and shown by the annexed example (Impediti/1.1) from the column of Trajan. The soldier wears his heavy armour; his shield on the left arm, and helmet slung in front, from the right shoulder, while his personal necessaries, implements for cooking, and vessels for eating and drinking are made into a pack and carried on the top of a pole. The men thus loaded are opposed to EXPEDITI; which compare.
# Illustrated Companion to the Latin Dictionary/Impilia

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.
IMPIL'IA (ἐυπίλια, Hesych.). Thick and warm coverings for the feet, made of a felted fabric (Plin. H. N. xix. 10., and Ulp. Dig. 34. 2. 25., in which passage they are distinguished from fasciae crurales, and from pedules), but whether in the nature of stockings, socks, or shoes, there are not sufficient data to determine.
# Illustrated Companion to the Latin Dictionary/Impluviatus

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.
IMPLUVIA'TUS. A term used to designate some particular kind of garments worn by females (Plaut. Epid. ii. 2. 39.); but as it only occurs in reference to a temporary fashion, it is impossible to say from what caprice the term may have sprung, or what peculiarity it was intended to describe. Some refer it to the form, viz. square, like the impluvium of a house (Turneb. Advers. xiv. 19.); others to the colour, vey dark and dingy, like the water which drips down from the roof of a house into the impluvium (Non. Marc. s. v. p. 548.); both conjectures little to be depended on.
# Illustrated Companion to the Latin Dictionary/Impluvium

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.
IMPLUV'IUM. A large square basin sunk in the floor of the atrium in private houses, intended as a receptacle for the rain water which flowed in through the compluvium, or opening in the roof of the same. Varro, L. L. v. 161. Festus s. v. Plaut. Amph. v. i. 59. Liv. xliii. 13. Cic. Verr. ii. 1. 23.) The illustration (Impluvium/1.1) represents the impluvium as now seen in the house of Sallust at Pompeii; a roof is restored to the apartment in order to show the manner in which the rain would enter through the compluvium above.
2. In some passages the word appears to be used in the same sense as compluvium (Plaut. Mil. ii. 2. 4. Ter. Eun. iii. 5. 41. Vitruv. vi. 3. 6.); but there is good reason for doubting the accuracy of these readings, and most of the best editions have adopted compluvium in its place.
# Illustrated Companion to the Latin Dictionary/Inarculum

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.
INAR'CULUM. Same as ARCULUM. 1. Festus s. v.
# Endomorphism/Real/Two-dimensional invariant/Fact

Let $V\neq 0$ be a real finite-dimensional vector space, and let
    $\varphi \colon V\longrightarrow V$
denote an endomorphism.
$\varphi$-invariant linear subspace $U\subseteq V$ of dimension
$1$
$2$