OpenLLM-France/wikimedia huggingface dataset:
# Latin III/Gerundive Lesson 1

Salvēte omnēs! Welcome back to Latin for Wikiversity. This lesson continues our study of the gerund and introduces its close relative, the gerundive. The gerundive is a verbal adjective expressing necessity or obligation, sometimes called the "future passive participle."

## The Gerundive

The gerundive is formed similarly to the gerund, but it is a true adjective that agrees with its noun in gender, number, and case. It expresses the idea of "needing to be done" or "ought to be done." For example:
liber legendus = a book needing to be read epistula scribenda = a letter that ought to be written rēs agenda = a thing that must be done
The gerundive is often used in place of the gerund when the verb would take a direct object. Compare:
* Gerund: cupidus librum legendī = eager of reading a book
* Gerundive: cupidus librī legendī = eager of the book needing to be read

## New Vocabulary

| Latin | English |
| cūra, cūrae, f. | care, concern |
| mora, morae, f. | delay |
| negōtium, negōtiī, n. | business, task |
| ratiō, ratiōnis, f. | method, system, reason |
| spēs, speī, f. | hope |

## New Sentences

| Latin | English | Notes |
| Epistula scribenda est. | The letter must be written. | Basic gerundive use |
| Cūra librōrum legendōrum. | The care of books needing to be read. | Gerundive agreeing with noun |
| Spēs pācis faciendae. | The hope of peace needing to be made. | |
| Tempus est urbis relinquendae. | It is time for the city to be left. | |
| Ratiō vīvendi. | A method of living. | Gerund with no object |
| Ars amīcōs parandī. | The art of acquiring friends. | Gerund with direct object |
| Ars amīcōrum parandōrum. | The art of friends needing to be acquired. | Gerundive construction |
| Negōtium conficiendum cūrāvit. | He saw to the task needing to be completed. | |
| Sine ūllā morā faciendā. | Without any delay needing to be made. | |
| Ad urbem videndam vēnimus. | We came to see the city (lit: for the city needing to be seen). | Purpose clause |
| Libri ad legendōs puerīs dantur. | Books are given to boys for reading. | Gerund |
| Librī legendī puerīs dantur. | Books needing to be read are given to boys. | Gerundive |
| Dē philosophiā discendā scrīpsit. | He wrote about philosophy needing to be learned. | |
| Haec agenda sunt. | These things must be done. | Common phrase |

## Practice

Practice and learn the words and phrases in this lesson :
| Step one | First learn the words using this lesson: Gerunds and Gerundives vocabulary |
| Step two | Next try learning and writing the sentencing using this: Gerunds and Gerundives Sentences |
| Note that the Memrise stage covers the content for all lessons in each stage. If you are skipping previous stages you may need to manually "ignore" the words in previous levels (use the 'select all' function) ||
Our next set of lessons will explore subordinate clauses in Latin. Valēte omnēs!
# Latin III/Gerundive Lesson 2

Salvēte omnēs! Welcome back to Latin for Wikiversity. This lesson introduces the passive periphrastic construction, which uses the gerundive with forms of sum to express necessity or obligation. This is one of Latin's most distinctive ways to express what "must" or "ought to" be done.

## The Passive Periphrastic

The passive periphrastic construction combines:
* A gerundive (-ndus, -nda, -ndum) agreeing with the subject
* A form of sum in any tense
This creates expressions of necessity, obligation, or duty. For example:
* Liber legendus est = The book must be read
* Pāx quaerenda erat = Peace had to be sought
* Hostēs vincendī erunt = The enemies will have to be conquered
When there is no clear subject, the neuter singular gerundive is used impersonally:
* Pugnandum est = It is necessary to fight
* Festīnandum erat = It was necessary to hurry
* Deliberandum erit = It will be necessary to deliberate

## New Vocabulary

| Latin | English |
| cōnsilium, cōnsiliī, n. | plan, advice, council |
| fidēs, fideī, f. | faith, trust, loyalty |
| memoria, memoriae, f. | memory, remembrance |
| officium, officiī, n. | duty, obligation |
| perīculum, perīculī, n. | danger, risk |

## New Sentences

| Latin | English | Notes |
| Mihi legendum est. | I must read. | Impersonal construction with dative of agent |
| Tibi currendum est. | You must run. | Another dative of agent example |
| Pāx omnibus quaerenda est. | Peace must be sought by everyone. | Dative plural of agent |
| Memoria exercenda est. | Memory must be exercised. | Personal construction |
| Perīcula vītanda erant. | Dangers had to be avoided. | Past tense |
| Cōnsilium capiendum erit. | A plan will have to be made. | Future tense |
| Fides servanda est. | Faith/loyalty must be kept. | Common Roman saying |
| Officia nobīs agenda sunt. | Duties must be done by us. | Plural subject with dative |
| Magistrō audiendum est. | The teacher must listen. | Dative of agent |
| Librī discipulīs legendī sunt. | Books must be read by the students. | Plural subject with dative |
| Mihī scribendum erat. | I had to write. | Past tense impersonal |
| Epistula tibi mittenda erit. | The letter will have to be sent by you. | Future tense |
| Nōbīs festīnandum est. | We must hurry. | Impersonal with plural agent |
| Vincendum aut moriendum est. | We must conquer or die. | Famous Roman saying |

## Practice

Practice and learn the words and phrases in this lesson :
| Step one | First learn the words using this lesson: Passive Periphrastic vocabulary |
| Step two | Next try learning and writing the sentencing using this: Passive Periphrastic Sentences |
| Note that the Memrise stage covers the content for all lessons in each stage. If you are skipping previous stages you may need to manually "ignore" the words in previous levels (use the 'select all' function) ||
In our next set of lessons, we will begin exploring subordinate clauses in Latin. Valēte omnēs!
# Proper symmetry/Isometry on a geometric object/Definition/Remark

The set of all symmetries of a geometric object $T$ form a group, with the composition of mappings as binary operation. This group is called the (proper) symmetry group of $T$. It is a subgroup of the group of all (proper) isometries of the Euclidean vector space. If $T$ is sufficiently complicated, then there are no isometries beside the identity. In the cases $T=\emptyset$ and $T=\{0\}$, every isometry is a symmetry. The set of all symmetries allowing the improper isometries form also a group. The proper isometries are the symmetries that can be performed in the physical world; they are the mappings that preserve the orientation.
# Real finite-dimensional vector space/Oriented/Isomorphism/Orientation-preserving/Definition

Let $V$ and $W$ be finite-dimensional oriented real vector spaces. A bijective linear mapping
    $\varphi \colon V\longrightarrow W$
is called orientation-preserving, if for every basis $v_{1},\ldots ,v_{n}$ representing the orientation on $V$, the image vectors $\varphi (v_{1}),\ldots ,\varphi (v_{n})$ represent the orientation on $W$.
# Real vector space/Bijective/Orientation-preserving/Positive Determinant/Fact

Let $V$ be a finite-dimensional real vector space, and let $\varphi \colon V\rightarrow V$ denote a bijective linear mapping.
$\varphi$
orientation-preserving if and only if the determinant
$\varphi$
# Real vector space/Bijective/Orientation-preserving/Positive determinant/Fact

Let $V$ be a finite-dimensional real vector space, and let $\varphi \colon V\rightarrow V$ denote a bijective linear mapping.
$\varphi$
orientation-preserving if and only if the determinant
$\varphi$
# Orientation/Vector space/Orientation-preserving mapping/Introduction/Section
## Definition

Let $V$ and $W$ be finite-dimensional oriented real vector spaces. A bijective linear mapping
    $\varphi \colon V\longrightarrow W$
is called orientation-preserving, if for every basis $v_{1},\ldots ,v_{n}$ representing the orientation
$V$
$\varphi (v_{1}),\ldots ,\varphi (v_{n})$
$W$
It is enough to establish this property for just one basis representing the orientation; see exercise. For an automorphism $\varphi \colon V\rightarrow V$, we can directly talk about orientation-preserving, without declaring an orientation. We have an orientation-preserving linear mapping if and only if every basis $v_{1},\ldots ,v_{n}$ and its image basis $\varphi (v_{1}),\ldots ,\varphi (v_{n})$ have the same orientation. This follows from the following fact.

## Lemma

Let $V$ be a finite-dimensional real vector space, and let $\varphi \colon V\rightarrow V$ denote a bijective linear mapping. Then $\varphi$ is orientation-preserving if and only if the determinant
$\varphi$

### Proof

$\Box$
If $V$ is endowed with an inner product, and $\varphi$ is an isometry, then positive determinant means, due to fact, that the determinant is $1$. Therefore, a proper isometry is the same as an orientation-preserving isometry.
# Real vector space/Bijective/Orientation-preserving/Positive determinant/Fact/Proof

Let $v_{1},\ldots ,v_{n}$ be a basis of $V$. Because of the bijectivity of $\varphi$, the images
    $\varphi (v_{1}),\ldots ,\varphi (v_{n})$
$V$
    $\varphi (v_{j})=\sum _{i=1}^{n}a_{ij}v_{i}\,,$
so that
    $M={\left(a_{ij}\right)}_{ij}\,$
is the describing matrix of the mapping with respect to the basis $v_{1},\ldots ,v_{n}$. This matrix is also the base change matrix $M_{\mathfrak {v}}^{\mathfrak {\varphi (v)}}$. The positivity of the determinant of this transformation matrix means by definition that the two matrices represent the same orientation.
# Proper symmetry/Isometry on a geometric object/Definition

Let $T\subseteq V$ be a subset of a euclidean vector space. A proper isometry
    $f\colon V\longrightarrow V$
fulfilling $f(T)=T$ is called a proper symmetry or a motion of $T$.
# Symmetries/Cube/Motivation/Example

We consider a cube $W\subset \mathbb {R} ^{3}$, with sides of length $2$, and the origin as its center. Thus, the vertices are
    $(\pm 1,\pm 1,\pm 1).$
We ask what are the possibilities to transform the cube into itself. By this, we mean that the cube shall not be deformed in any way, the cube is considered rigid, and the motion should be realizable in the physical world. We talk about a (proper) motion of the cube. During such a motion, the center of the cube does not change its position, and faces are mapped to faces, edges to edges, vertices to vertices. Also, the centers of the faces are mapped to centers of faces, and opposite centers of faces are mapped to opposite centers of faces. The centers of faces are the six points
    $(\pm 1,0,0),\,(0,\pm 1,0),\,(0,0,\pm 1).$
If the point $(1,0,0)$ is mapped to the face center $S$, then $(-1,0,0)$ is mapped to the opposite point, that is, $-S$. Every face center $S$ is allowed, but this does not determine the motion. For the face center $(0,1,0)$, there are four possible image points (only $S$ and $-S$ are excluded), because the cube can be rotated around the axis given by $S$ by a multiple of $90$ degree. These rotations correspond to the possibilities to move the point $(0,1,0)$ to one of the four remaining face centers. The choice of the second face center $T$ determines the motion uniquely. How clear is that?
To understand this, the following observations are useful.
* Motions can be performed after another, that is, if we have two motions of the cube $$ \varphi $$, then the composition $\psi \circ \varphi$ is defined, which performs first $\varphi$ and then $\psi$.
* The identical motion, which moves nothing, is a motion. If man zu of a arbitraryen Bewegung the identische Bewegung davor or danach durchführt, so ändert the the Bewegung not.
* For a motion $\varphi$, there exists the opposite motion (or "inverse motion“) $\varphi ^{-1}$, which has the property that the compositions $\varphi ^{-1}\circ \varphi$ and $\varphi \circ \varphi ^{-1}$ are just the identity.
With these observation, we can understand the principle mentioned above: assume that there are two motions $\varphi$ and $\psi$, both sending $(1,0,0)$ to $S$ and $(0,1,0)$ to $T$. Let $\psi ^{-1}$ denote the inverse motion of $\psi$. Then we consider the composed motion
    $\theta =\psi ^{-1}\circ \varphi \,.$
This motion hat the property that $(1,0,0)$ is mapped to $(1,0,0)$ and that $(0,1,0)$ is mapped to $(0,1,0)$, as $\varphi$ sends $(1,0,0)$ to $S$, and $\psi ^{-1}$ sends $S$ back to $(1,0,0)$ (and accordingly for $(0,1,0)$). This means that $\theta$ has the property that $(1,0,0)$ as well as $(0,1,0)$ are sent to themselves, that is, these points are fixed points of the motion. But this implies that the $x,y$-plane is fixed. The only physical motion of the cube that fixes this plane is the identical motion. Therefore, $\psi ^{-1}\circ \varphi =\operatorname {Id}$, and $\varphi =\psi$. Note that the reflection an the $x,y$-plane swaps the points $(0,0,1)$ and $(0,0,-1)$, but this is a so-called improper motion; it can not be performed in the physical world.
# Characteristic polynomial/Coprime factorization/Direct sum/Fact

Let $\varphi \colon V\rightarrow V$ be a linear mapping on a finite-dimensional $K$-vector space $V$, and let
    $\chi _{\varphi }=P\cdot Q\,$
be a factorization of the characteristic polynomial in coprime polynomials $P,Q\in K[X]$.
direct sum decomposition
    $V=\operatorname {kern} P(\varphi )\oplus \operatorname {kern} Q(\varphi )\,,$
where these linear subspaces are $\varphi$-invariant. We have
    $\operatorname {kern} Q(\varphi )=\operatorname {Im} P(\varphi )\,,$
$P(\varphi )$
$\operatorname {kern} Q(\varphi )$
# Home Shop Machining/CAD
## Computer Aided Design

CAD systems are 2 dimensional, much like traditional drafting, or 3 dimensional modelling systems. With 3D, there are 2 approaches: parametric, in that work is started as a 2D sketch that is "defined" and then processed into 3D forms, or direct modelling, where 3D objects are manipulated. There is some convergence and the lines between these 2 approaches, and even between 2D and 3D, are getting blurry.

#### Home Shop Machinists use CAD to:

* Make shop drawings to work from.
* Model 3D objects to check for fit and function.
* Generate code to run CNC machines.
* As well, running structural analysis on designs is becoming possible.

#### Common Applications
##### Free for Hobby Use

* Dune 3D - Opensource parametric 3D modelling application.
* FreeCAD - Opensource parametric 3D modelling system with multiple workbenches for specific features or functionality.
* Fusion 360 - From Autodesk, parametric 3D modelling system. Requires online account but does allow some offline editing.
* OnShape - Fully web browser based parametric 3D modelling system. Free but models must be public domain.
* SketchUp - From Trimble (bought from Google). Very easy to use 3D modelling system. Intuitive inference engine.
* Solvespace - Opensource parametric 3D modelling application with a limited CAM feature.
Also free and opensource, but rarely used is the venerable BRL-CAD.

##### Expensive but Industry Standard

* AutoCAD
* Inventor
* SolidWorks
# Computer numerical controls

Computer numerical controls use computers for the numerical control of machine tool automation. Some coding used are called M & N codes for programming purposes. By using computer numerical controlled machines precise work can be achieved.
# Latin III/Present Subjunctive Lesson 2

Salvēte omnēs! Welcome back to Latin for Wikiversity. Here you can peruse a new lesson in Latin, in a simple format. If you would like to catch up, you can find a directory of lessons, a classified vocabulary list, and Memrise courses at the links on the right.

## New Grammar

Last time we learned how to form the present subjunctive with first conjugation verbs by changing -ā- to -ē-. Second conjugation verbs have their own pattern, but it's just as regular! For these verbs, we change the characteristic -ē- to -ea- before adding the endings.
Let's see how this works with moneō (to warn, advise):
| Person | Present Indicative | Present Subjunctive |
| 1st person singular | moneō | moneam |
| 2nd person singular | monēs | moneās |
| 3rd person singular | monet | moneat |
| 1st person plural | monēmus | moneāmus |
| 2nd person plural | monētis | moneātis |
| 3rd person plural | monent | moneant |
Just remember: where first conjugation changed -ā- to -ē-, second conjugation changes -ē- to -ea-. Simple!

## Verbs in This Lesson

| Latin | English | Audio (Classical) |
| moneō, 2 | warn, advise | |
| doceō, 2 | teach | |
| videō, 2 | see | |
| habeō, 2 | have | |
| sedeō, 2 | sit | File:La-cls-sedeo.ogg |
| rideō, 2 | laugh, smile | File:La-cls-rideo.ogg |
| timeō, 2 | fear | |
| taceō, 2 | be silent | File:La-cls-taceo.ogg |

## New Words

| Latin | English | Audio (Classical) |
| nunc | now | |
| semper | always | |
| tandem | at last, finally | File:La-cls-tandem.ogg |

## New Sentences

| Latin | English |
| Nunc videam. | May I see now. |
| Doceāmus linguam Latīnam. | Let us teach Latin. |
| Moneās amīcum. | May you warn your friend. |
| Habeātis librōs bonōs. | May you (pl.) have good books. |
| Tandem sedeāmus. | Let us sit at last. |
| Puer taceat. | May the boy be silent. |
| Nunc rideāmus! | Let us laugh now! |
| Nē timeāmus. | Let us not fear. |
| Semper videant. | May they always see. |
| Bene doceant. | May they teach well. |
| Taceat et videat. | May he/she be silent and see. |

## Practice

Practice and learn the words and phrases in this lesson :
| Step one | First learn the words using this lesson: Present Subjunctive vocabulary |
| Step two | Next try learning and writing the sentences using this: Present Subjunctive Sentences |
| Note that the Memrise stage covers the content for all lessons in each stage. If you are skipping previous stages you may need to manually "ignore" the words in previous levels (use the 'select all' function) ||
Next time we'll learn how third conjugation verbs form their present subjunctive. Until then, valēte!
# Latin III/Present Subjunctive Lesson 3

Salvēte omnēs! Welcome back to Latin for Wikiversity. Here you can peruse a new lesson in Latin, in a simple format. If you would like to catch up, you can find a directory of lessons, a classified vocabulary list, and Memrise courses at the links on the right.

## New Grammar

We've learned how first conjugation verbs change -ā- to -ē- and second conjugation verbs change -ē- to -ea- in the present subjunctive. Third conjugation verbs have their own pattern - they add -a- to the stem. Let's see this with the verb regō (to rule):
| Person | Present Indicative | Present Subjunctive |
| 1st person singular | regō | regam |
| 2nd person singular | regis | regās |
| 3rd person singular | regit | regat |
| 1st person plural | regimus | regāmus |
| 2nd person plural | regitis | regātis |
| 3rd person plural | regunt | regant |
Notice how the vowel -a- appears throughout the subjunctive forms? This makes third conjugation quite straightforward once you know the pattern!

## Verbs in This Lesson

| Latin | English | Audio (Classical) |
| regō, 3 | rule, direct | File:La-cls-rego.ogg |
| dūcō, 3 | lead | File:La-cls-duco.ogg |
| scrībō, 3 | write | |
| legō, 3 | read | File:La-cls-lego.ogg |
| dīcō, 3 | say, speak | |
| mittō, 3 | send | File:La-cls-mitto.ogg |
| vīvō, 3 | live | File:La-cls-vivo.ogg |
| crēdō, 3 | believe, trust | |

## New Words

| Latin | English | Audio (Classical) |
| fēlīciter | happily | File:La-cls-feliciter.ogg |
| diū | for a long time | |
| rectē | rightly, correctly | File:La-cls-recte.ogg |

## New Sentences

| Latin | English |
| Rectē regam. | May I rule correctly. |
| Diū vīvāmus. | May we live for a long time. |
| Epistulam scrībam. | Let me write a letter. |
| Librum legāmus. | Let us read a book. |
| Fēlīciter vīvās. | May you live happily. |
| Mihi crēdās. | May you believe me. |
| Dūcat exercitum. | May he lead the army. |
| Rectē dīcant. | May they speak correctly. |
| Litterās mittāmus. | Let us send the letter. |
| Legant et scrībant. | May they read and write. |
| Nē crēdant. | May they not believe. |

## Practice

Practice and learn the words and phrases in this lesson :
| Step one | First learn the words using this lesson: Present Subjunctive vocabulary |
| Step two | Next try learning and writing the sentences using this: Present Subjunctive Sentences |
| Note that the Memrise stage covers the content for all lessons in each stage. If you are skipping previous stages you may need to manually "ignore" the words in previous levels (use the 'select all' function) ||
Next time we'll complete our present subjunctive forms with fourth conjugation verbs. Until then, valēte!
# Latin III/Present Subjunctive Lesson 4

Salvēte omnēs! Welcome back to Latin for Wikiversity. Here you can peruse a new lesson in Latin, in a simple format. If you would like to catch up, you can find a directory of lessons, a classified vocabulary list, and Memrise courses at the links on the right.

## New Grammar

Today we'll complete our study of present subjunctive forms with fourth conjugation verbs. Like the other conjugations, fourth conjugation has its own pattern - we add -a- to the stem, but keep the -i- that's characteristic of fourth conjugation verbs. Let's see this with audiō (to hear):
| Person | Present Indicative | Present Subjunctive |
| 1st person singular | audiō | audiam |
| 2nd person singular | audīs | audiās |
| 3rd person singular | audit | audiat |
| 1st person plural | audīmus | audiāmus |
| 2nd person plural | audītis | audiātis |
| 3rd person plural | audiunt | audiant |
See how we keep the -i- and add the -a-? Once you've learned this pattern, fourth conjugation subjunctive is quite straightforward!

## Verbs in This Lesson

| Latin | English | Audio (Classical) |
| audiō, 4 | hear | File:La-cls-audio.ogg |
| veniō, 4 | come | |
| dormiō, 4 | sleep | |
| sciō, 4 | know | File:La-cls-scio.ogg |
| aperiō, 4 | open | File:La-cls-aperio.ogg |
| inveniō, 4 | find | File:La-cls-invenio.ogg |
| sentiō, 4 | feel, perceive | |
| nesciō, 4 | not know | File:La-cls-nescio.ogg |

## New Words

| Latin | English | Audio (Classical) |
| statim | immediately | File:La-cls-statim.ogg |
| clārē | clearly | File:La-cls-clare.ogg |
| fortasse | perhaps | File:La-cls-fortasse.ogg |

## New Sentences

| Latin | English |
| Clārē audiam. | May I hear clearly. |
| Statim veniāmus. | Let us come immediately. |
| Bene dormiās. | May you sleep well. |
| Fortasse sciāmus. | May we perhaps know. |
| Fenestram aperiāmus. | Let us open the window. |
| Librum inveniās. | May you find the book. |
| Clārē sentiant. | May they feel clearly. |
| Nē nesciāmus. | May we not be ignorant. |
| Mox veniat. | May he/she come soon. |
| Bene dormiant. | May they sleep well. |
| Vērum sentiāmus. | May we perceive the truth. |

## Practice

Practice and learn the words and phrases in this lesson :
| Step one | First learn the words using this lesson: Present Subjunctive vocabulary |
| Step two | Next try learning and writing the sentences using this: Present Subjunctive Sentences |
| Note that the Memrise stage covers the content for all lessons in each stage. If you are skipping previous stages you may need to manually "ignore" the words in previous levels (use the 'select all' function) ||
Now you know how to form the present subjunctive in all four conjugations! In future lessons, we'll learn more about when and how to use these forms. Valēte!