OpenLLM-France/wikimedia huggingface dataset:
# Encyclopædia Britannica, Ninth Edition/Togo-land

From volume XXIII of the work. See also the Project Disclaimer.
​TOGO-LAND, one of the portions of the African continent under the protection of the German empire. It forms part of the territory on the west coast formerly distinguished as the Slave Coast, and is bounded on the E. by Little Popo, on the S.W. by the British Gold Coast Colony, and on the N.W. by the still independent territory of the Anlo tribes. The coast-line is only 22 miles in length, and with an area of about 500 square miles Togo-land is estimated to have a population of 40,000 souls. The great physical feature of the country is the Togo, Hakko, or Avon lagoon, which is cut off from the ocean by the narrow belt of sandy soil on which are situated Bagida, Porto Seguro, and Bay. The town of Togo lies on the east side of the Togo lagoon, where it extends eastward to Little Popo Lake. The lagoon is fed by a stream from the north, the Haho or Hakko, first discovered by John Duncan in 1846.
See Hugo Zöller, Togoland; Petermann's Mittheilungen (1886); and Scott. Geogr. Mag. (1886), all of which give maps.
# Marriage of the Dead

* related portals: Marriage.
* sister projects: Wikidata item.
From Popular Science Monthly Volume 51, May 1897 Popular Miscellany
​Marriage of the Dead.—Among the many curious practices that Marco Polo came across in his travels in the far East, the Tartar custom of marrying the dead deserves notice. He says: "If any man have a daughter who dies before marriage, and another man have had a son also die before marriage, the parents of the two arrange a grand wedding between the dead lad and lass, and marry them they do, making a regular contract! And when the contract papers are made out they put them in the fire, in order that the parties in the other world may know the fact, and so look on each other as man and wife. And the parents thenceforward consider themselves sib to each other just as if their children had lived and married. Whatever may be agreed on between the parties as dowry, those who have to pay it cause to be painted on pieces of paper, and then put these in the fire, saying that in that way the dead person will get all the real articles in the other world." This custom is also noted by other writers, even as late as the beginning of the eighteenth century. It is said to have been adopted by Jenghis Khan, for political reasons, and is named in his Yasa, published in 1205 a. d.
# In the Forest (Wilde)

Out of the mid-wood's twilight
    Into the meadow's dawn,
Ivory limbed and brown-eyed,
    Flashes my Faun!
He skips through the copses singing,
    And his shadow dances along,
And I know not which I should follow,
    Shadow or song!
O Hunter, snare me his shadow!
    O Nightingale, catch me his strain,
Else moonstruck with music and madness
    I track him in vain!
# The Earth and its Inhabitants

* related portals: Geography.
* sister projects: Commons category, Wikidata item.

### Africa

* Volume 1: North-East Africa
* Volume 2: North-West Africa
* Volume 3: West Africa
* Volume 4: South and East Africa

### Asia

* Volume 1: Asiatic Russia
* Volume 2: East Asia
* Volume 3: India and Indo-China
* Volume 4: South Western Asia

### North America

* Volume 1: British North America
* Volume 2: Mexico, Central America, West Indies
* Volume 3: The United States

### South America

* Volume 1: The Andes Regions
* Volume 2: Amazonia and La Plata

### Europe

* Volume 1: Greece, Turkey in Europe, Rumania, Servia, Montenegro, Italy, Spain, and Portugal
* Volume 2: France and Switzerland
* Volume 3: Austria-Hungary, Germany, Belgium, Netherlands
* Volume 4: British Isles
* Volume 5: North-east Atlantic, Islands of the North Atlantic, Scandinavia, European Islands of the Arctic Ocean, Russia in Europe

### Oceania

* Oceania: Oceania
# 1911 Encyclopædia Britannica/Number/Complex Numbers

See also our 1911 Encyclopædia Britannica disclaimer.
19. Complex Numbers.—If $a$ is an assigned number, rational or irrational, and $n$ a natural number, it can be proved that there is a real number satisfying the equation $x^{n}=a$, except when $n$ is even and $a$ is negative: in this case the equation is not satisfied by any real number whatever. To remove the difficulty we construct an aggregate of polar couples $\{x,y\}$, where $x,y$ are any two real numbers, and define the addition and multiplication of such couples by the rules
    $\{x,y\}+\{x',y'\}=\{x+x',y+y'\}$;
    $\{x,y\}\times \{x',y'\}=\{xx'-yy',xy'+x'y\}$.
We also agree that $\{x,y\}<\{x',y'\},$ if $x<x'$ or if $x=x'$ and $y<y'$. It follows that the aggregate has the ground element $\{1,0\}$, which we may denote by $\sigma$; and that, if we write $\tau$ for the element $\{0,1\}$,
    $\tau ^{2}=\{-1,0\}=-\sigma$.
Whenever $m,n$ are rational, $\{m,n\}=m\sigma +n\tau$, and we are thus justified in writing, if we like, $x\sigma +y\tau$ for $\{x,y\}$, in all circumstances. A further simplification is gained by writing $x$ instead of $x\sigma$, and regarding $\tau$ as a symbol which is such that $\tau ^{2}=-1$, but in other respects obeys the ordinary laws of operation. It is usual to write $i$ instead of $\tau$; we thus have an aggregate ${\mathfrak {I}}$ of complex numbers $x+yi$. In this aggregate, which includes the real continuum as part of itself, not only the four rational operations (excluding division by $\{0,0\}$, the zero element), but also the extraction of roots, may be effected without any restriction. Moreover (as first proved by Gauss and Cauchy), if $a_{0},a_{1},\dots a_{n}$ are any assigned real or complex numbers, the equation
    $a_{0}z^{n}+a_{1}z^{n-1}+\dots +a_{n-1}z+a_{n}=0$, is always satisfied by precisely $n$ real or complex values of $z$, with a proper convention as to multiple roots. Thus any algebraic function of any finite number of elements of
${\mathfrak {I}}$ is also contained in ${\mathfrak {I}}$, which is, in this sense, a closed arithmetical field, just as ${\mathfrak {N}}$ is when we restrict ourselves to rational operations. The power of ${\mathfrak {I}}$ is the same as that of ${\mathfrak {N}}$.
# 1911 Encyclopædia Britannica/Number/Exponents, Primitive Roots, Indices

See also our 1911 Encyclopædia Britannica disclaimer.
29. Exponents, Primitive Roots, Indices. Let $p$ denote an odd prime, and $x$ any number prime to p. Among the powers $x,x^{2},\dots x^{p-1}$ there is certainly one, namely $x^{p-1}$, which $\equiv 1{\pmod {p}}$; let $x^{e}$ be the lowest power of $x$ such that $x^{e}\equiv 1{\pmod {p}}$. Then $e$ is said to be the exponent to which $x$ appertains ${\pmod {p}}$: it is always a factor of $(p-1)$ and can only be $1$ when $x\equiv 1$. The residues $x$ for which $e=p-1$ are said to be primitive roots of $p$. They always exist, their number is $\phi (p-1)$, and they can be found by a methodical, though tedious, process of exhaustion. If $g$ is any one of them, the complete set may be represented by $g,g^{a},g^{b},\dots \!\!\And \!\!\!\!\mathrm {c} .$ where $a,b,\!\!\And \!\!\!\!\mathrm {c} .$, are the numbers less than $(p-1)$ and prime to it, other than $1$. Every number $x$ which is prime to $p$ is congruent, ${\pmod {p}}$, to $g^{i}$, where $i$ is one of the numbers $1,2,3,...(p-1$); this number $i$ is called the index of $x$ to the base $g$. Indices are analogous to logarithms: thus
    $\operatorname {ind} _{g}(xy)\equiv \operatorname {ind} _{g}x+\operatorname {ind} _{g}y$. $\operatorname {ind} _{g}(x^{h})\equiv h\ \operatorname {ind} _{g}x{\pmod {\overline {p-1}}}$
Consequently tables of primitive roots and indices for different primes are of great value for arithmetical purposes. Jacobi's Canon Arithmeticus gives a primitive root, and a table of numbers and indices for all primes less than $1000$.
For moduli of the forms $2p,p^{m},2p^{m}$ there is an analogous theory (and also for $2$ and $4$); but for a composite modulus of other forms there are no primitive roots, and the nearest analogy is the representation of prime residues in the form $\alpha ^{x}\beta ^{y}\gamma ^{z}\dots$, where $\alpha ,\beta ,\gamma ,\dots$ are selected prime residues, and $x,y,z,\dots$ are indices of restricted range. For instance, all residues prime to $48$ can be exhibited in the form $5^{x}7^{y}13^{z}$, where $x=0,1,2,3$; $y=0,1$; $z=0,1$; the total number of distinct residues being $4.2.2=16=\phi (48)$, as it should be.
# 1911 Encyclopædia Britannica/Number/Linear Congruences

See also our 1911 Encyclopædia Britannica disclaimer.
30. Linear Congruences.—The congruence $a'x=b'{\pmod {m'}}$ has no solution unless $\operatorname {dv} (a',m')$ is a factor of $b'$. If this condition is satisfied, we may replace the given congruence by the equivalent one $ax\equiv b{\pmod {m}}$, where $a$ is prime to $b$ as well as to $m$. By residuation (§§ 24, 25) we can find integers $h,k$ such that $ah-mk=1$, and thence obtain $x\equiv bh{\pmod {m}}$ as the complete solution of the given congruence. To the modulus $m'$ there are $m'/m$ incongruent solutions. For example, $12x\equiv 30{\pmod {21}}$ reduces to $2x\equiv 5{\pmod {7}}$ whence $x\equiv 6{\pmod {7}}\equiv 6,13,20{\pmod {21}}$. There is a theory of simultaneous linear congruences in any number of variables, first developed with precision by Smith. In any particular case, it is best to replace as many as possible of the given congruences by an equivalent set obtained by successively eliminating the variables $x,y,z,\dots$ in order. An important problem is to find a number which has given residues with respect to a given set of moduli. When possible, the solution is of the form $x\equiv a{\pmod {m}}$, where $m$ is the least common multiple of the moduli. Supposing that $p$ is a prime, and that we have a corresponding table of indices, the solution of $ax\equiv b{\pmod {p}}$ can be found by observing that $\operatorname {ind} \ x=\operatorname {ind} \ b-\operatorname {ind} \ a{\pmod {\overline {p-1}}}$.
# 1911 Encyclopædia Britannica/Number/Class-Number

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54. Class-Number.—The number of ideal classes in the field $\Omega (\surd m)$ may be expressed in the following forms:—
(i.) $m<0$:
    $h={\frac {\tau }{2\Delta }}{\underset {n}{\Sigma }}\left({\frac {\Delta }{n}}\right)n\qquad (n=1,2,\dots ,-\Delta )$;
(ii.) $m>0$:
    $h={\frac {1}{2\log \epsilon }}\log {\frac {\Pi \sin {\frac {b\pi }{\Delta }}}{\Pi \sin {\frac {a\pi }{\Delta }}}}$
In the first of these formulae $\tau$ is the number of units contained in $\Omega$; thus $\tau =6$ for $\Delta =-3$, $\tau =4$ for $\Delta =-4$, $\tau =2$ in other cases. In the second formula, $\epsilon$ is the fundamental unit, and the products are taken for all the numbers of the set $(1,2,\dots ,\Delta )$ for which $\left({\frac {\Delta }{a}}\right)=+1,\left({\frac {\Delta }{b}}\right)=-1$ respectively. In the ideal theory the only way in which these formulae have been obtained is by a modification of Dirichlet's method; to prove them without the use of transcendental analysis would be a substantial advance in the theory.
55. Suppose that any ideal in $\Omega$ is expressed in the form $[\omega _{1},\omega _{2}]$; then any element of it is expressible as $x\omega _{1}+y\omega _{2}$, where $x,y$ are rational integers, and we shall have $\mathrm {N} (x\omega _{1}+y\omega _{2})=ax^{2}+bxy+cy^{2}$, where $a,b,c$ are rational numbers contained in the ideal. If we put $x=\alpha x'+\beta y',y=\gamma x'+\delta y'$, where $\alpha ,\beta ,\gamma ,\delta$ are rational numbers such that $\alpha \delta -\beta \gamma =\pm 1$, we shall have simultaneously $(a,b,c)(x,y)^{2}=(a',b',c')(x',y')^{2}$ as in § 32 and also $(a',b',c')(x',y')^{2}=\mathrm {N} \{x'(\alpha \omega _{1}+\gamma \omega _{2})+y'(\beta \omega _{1}+\delta \omega _{2})\}=\mathrm {N} (x'\omega '_{1}+y'\omega '_{2})$, where $[\omega '_{1},\omega '_{2}]$ is the same ideal as before. Thus all equivalent forms are associated with the same ideal, and the numbers representable by forms of a particular class are precisely those which are norms of numbers belonging to the associated ideal. Hence the class-number for ideals in $\Omega$ is also the class-number for a set of quadratic forms; and it can be shown that all these forms have the same determinant $\Delta$. Conversely, every class of forms of determinant $\Delta$ can be associated with a definite class of ideals in $\Omega (\surd m)$, where $m=\Delta$ or ${\tfrac {1}{4}}\Delta$ as the case may be. Composition of form-classes exactly corresponds to the multiplication of ideals: hence the complete analogy between the two theories, so long as they are really in contact. There is a corresponding theory of forms in connexion with a field of order $n$: the forms are of the order $n$, but are only very special forms of that order, because they are algebraically resolvable into the product of linear factors.
# Fish

* Search for titles containing or beginning with: "Fish."
* sister projects: Wikipedia article, Wikidata item.
This is a disambiguation page. It lists works that share the same title. If an article link referred you here, please consider editing it to point directly to the intended page.
* The Fish, a story by Anton Chekhov
* The Fish, a poem by Rupert Brooke
* "Fish," in Encyclopædia Britannica (11th ed., 1911)
# Encyclopædia Britannica, Ninth Edition/Torre Del Greco

From volume XXIII of the work. See also the Project Disclaimer.
TORRE DEL GRECO, a town of Italy, in the province of Naples, 71/2 miles to the south-east of that city, lies at the south-west foot of Vesuvius, on the shore of the Bay of Naples. It is built chiefly of lava, and stands on the lava stream of 1631, which destroyed two-thirds of the older town. Great damage was done by the eruptions of 1737 and 1794, when immense streams of lava flowed through the town into the sea; the earthquake of 1857 and the eruption of December 8, 1861, were even more destructive. After each disaster the people have returned and repaired the ruin, the advantage de rived from the rich land on the flanks of the volcano and the proximity to the sea and to Naples being more than enough to overcome apprehensions of danger. In the outskirts are many beautiful villas and gardens. The inhabitants are largely employed in fishing (tunny, oyster, sardine, and especially coral), and the neighbourhood is famed for its fruit and wine. The population in 1881 was 21,586.
# The Sleeper Awakened

adapted from the 'One Thousand and One Nights'
* "The Sleeper Awakened" Edmund Dulac. Sindbad the sailor and other stories from the Arabian nights. London: Hodder and Stoughton. 1914. [five color plates]
* "The Sleeper Awakened" Edmund Dulac's picture-book for the French Red Cross. circa 1914. [abridged; one color plate]
# 1911 Encyclopædia Britannica/Number/Automorphs

See also our 1911 Encyclopædia Britannica disclaimer.
35. Automorphs. The Pellian Equation.—A primitive form $(a,b,c)$ is, by definition, equivalent to itself; but it may be so in more ways than one. In order that $(a,b,c)$ may be transformed into itself by the substitution ${\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}$, it is necessary and sufficient that
    ${\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}={\begin{pmatrix}{\tfrac {1}{2}}(t+bu),&-cu\\au,&{\tfrac {1}{2}}(t-bu)\end{pmatrix}}$
where $(t,u)$ is an integral solution of
    $t^{2}-\mathrm {D} u^{2}=4$.
If $\mathrm {D}$ is negative and $-\mathrm {D} >4$, the only solutions are $t=\pm 2,u=0$; $\mathrm {D} =-3$ gives $(\pm 2,0),(\pm 1,\pm 1)$; $\mathrm {D} =-4$ gives $(\pm 2,0),(0,\pm 1)$. On the other hand, if $\mathrm {D} >0$ the number of solutions is infinite, and if $(t_{1},u_{1})$ is the solution for which $t,u$ have their least positive values, all the other positive solutions may be found from
    ${\frac {t_{n}+u_{n}{\sqrt}\mathrm {D} }{2}}=\left({\frac {t_{1}+u_{1}{\sqrt}\mathrm {D} }{2}}\right)^{n}(n=2,3,4\dots )$.
The substitutions by which $(a,b,c)$ is transformed into itself are called its automorphs. In the case when $\mathrm {D} \equiv 0{\pmod {4}}$ we have $t=2\mathrm {T} ,u=2\mathrm {U} ,\mathrm {D} =4\mathrm {N}$, and $(\mathrm {T} ,\mathrm {U} )$ any solution of
    $\mathrm {T} ^{2}-\mathrm {N} \mathrm {U} ^{2}=1$.
This is usually called the Pellian equation, though it should properly be associated with Fermat, who first perceived its importance. The minimum solution can be found by converting ${\sqrt}\mathrm {N}$ into a periodic continued fraction.
The form $(a,b,c)$ may be improperly equivalent to itself; in this case all its improper automorphs can be expressed in the form
    ${\begin{pmatrix}\lambda ,&(\kappa +b\lambda )/2a\\(\kappa -b\lambda )/2a,&-\lambda \end{pmatrix}}$
where $\kappa ^{2}-\mathrm {D} \lambda ^{2}=4ac$. In particular, if $b\equiv 0{\pmod {a}}$ the form $(a,b,c)$ is improperly equivalent to itself. A form improperly equivalent to itself is said to be ambiguous.
# 1911 Encyclopædia Britannica/Number/Bilinear Forms

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40. Bilinear Forms.—A bilinear form means an expression of the type $\Sigma a_{ik}x_{i}y_{k}\,(i=1,2,\dots m;k=1,2,\dots n)$; the most important case is when $m=n$, and only this will be considered here. The invariants of a form are its determinant $[a_{nn}]$ and the elementary factors thereof. Two bilinear forms are equivalent when each can be transformed into the other by linear integral substitutions $x'=\Sigma \alpha x,y'=\Sigma \beta y$. Every bilinear form is equivalent to a reduced form ${\overset {r}{\underset {1}{\Sigma }}}e_{i}x_{i}y_{i}$, and $r=n$, unless $[a_{nn}]=0$. In order that two forms may be equivalent it is necessary and sufficient that their invariants should be the same. Moreover, if $a\sim b$ and $c\sim d$, and if the invariants of the forms $a+\lambda c,b+\lambda d$ are the same for all values of $\lambda$, we shall have $a+\lambda c\sim b+\lambda d$, and the transformation of one form to the other may be effected by a substitution which does not involve $\lambda$. The theory of bilinear forms practically includes that of quadratic forms, if we suppose $x_{i},y_{i}$ to be cogredient variables. Kronecker has developed the case when $n=2$, and deduced various class-relations for quadratic forms in a manner resembling that of Liouville. So far as the bilinear forms are concerned, the main result is that the number of classes for the positive determinant $a_{11}a_{22}-a_{12}a_{21}=\Delta$ is $12\{\Phi (\Delta )+\Psi (\Delta )\}+2\epsilon$, where $\epsilon$ is $1$ or $0$ according as $\Delta$ is or is not a square, and the symbols $\Phi ,\Psi$ have the meaning previously assigned to them (§ 39).
# 1911 Encyclopædia Britannica/Number/Characters of a form or class

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36. Characters of a form or class. Genera.—Let $(a,b,c)$ be any primitive form; we have seen above (§ 32) that if $\alpha ,\beta ,\gamma ,\delta$ are any integers
    $4(a\alpha ^{2}+b\alpha \gamma +c\gamma ^{2})(a\beta ^{2}+b\beta \delta +c\delta ^{2})=b'^{2}-(\alpha \delta -\beta \gamma )^{2}\mathrm {D}$
where $b'=2a\alpha \beta +b(\alpha \delta +\beta \gamma )+2c\gamma \delta$. Now the expressions in brackets on the left hand may denote any two numbers $m,n$ representable by the form $(a,b,c)$; the formula shows that $4mn$ is a residue of $\mathrm {D}$, and hence $mn$ is a residue of every odd prime factor of $\mathrm {D}$, and if $p$ is any such factor the symbols $\left({\frac {m}{p}}\right)$ and $\left({\frac {n}{p}}\right)$ will have the same value. Putting $(a,b,c)=f$, this common value is denoted by $\left({\frac {f}{p}}\right)$ and called a quadratic character (or simply character) of $f$ with respect to $p$. Since $a$ is representable by $f$ ($x=1,y=0$) the value $\left({\frac {f}{p}}\right)$ is the same as $\left({\frac {a}{p}}\right)$. For example, if $\mathrm {D} =-140$, the scheme of characters for the six reduced primitive forms, and therefore for the classes they represent, is
    ${\begin{array}{lccc }&\left({\frac {f}{5}}\right)&\left({\frac {f}{7}}\right)&\\(1,0,35)&+&+\\(4,\pm 2,9)&&\\\hline (5,0,7)&-&-\\(3,\pm 2,12)&&\\\hline \end{array}}$.
In certain cases there are supplementary characters of the type $\left({\frac {-1}{f}}\right)$ and $\left({\frac {2}{f}}\right)$, and the characters $\left({\frac {f}{p}}\right)$ are discriminated according as an odd or even power of $p$ is contained in $\mathrm {D}$; but in every case there are certain combinations of characters (in number one-half of all possible combinations) which form the total characters of actually existing classes. Classes which have the same total character are said to belong to the same genus. Each genus of the same order contains the same number of classes.
For any determinant $\mathrm {D}$ we have a principal primitive class for which all the characters are $+$; this is represented by the principal form $(1,0,-n)$ or $(1,1,-n)$ according as $\mathrm {D}$ is of the form $4n$ or $4n+1$. The corresponding genus is called the principal genus. Thus, when $\mathrm {D} =-140$, it appears from the table above that in the primitive order there are two genera, each containing three classes; and the non-existent total characters are $+-$ and $-+$.
# In the quiet of the morning I heard a knock at my door

From A Hundred and Seventy Chinese Poems (Alfred A. Knopf, 1919).
In the quiet of the morning I heard a knock at my door: I threw on my clothes and opened it myself. I asked who it was who had come so early to see me: He said he was a peasant, coming with good intent. He brought a present of wine and rice-soup, Believing that I had fallen on evil days. "You live in rags under a thatched roof And seem to have no desire for a better lot. The rest of mankind have all the same ambitions: You, too, must learn to wallow in their mire." "Old man, I am impressed by what you say, But my soul is not fashioned like other men's. To drive in their rut I might perhaps learn: To be untrue to myself could only lead to muddle. Let us drink and enjoy together the wine you have brought; For my course is set and cannot now be altered."
This work is a translation and has a separate copyright status to the applicable copyright protections of the original content.
# To G. P. Morris

A poem written to poet and songwriter George Pope Morris
Apollo once had leave to travel;  He sought our Yankee land, And he lionized it through,  With his golden lyre in hand.
Once, at "a cottage near a wood,"  Which promised a welcome's smile, He thought, by general invitation,  To rusticate awhile.
One morn he woke, - he yawned, - he turned, -  Sprang up with fright and grief, And cried, "By George! my lyre is stolen:  Without there, ho! stop thief!"
But vainly sought he east and west,  Half mad, - all broken-hearted; O, a most ungodlike look he wore,  With his glory all departed!
At last he turned Olympus-ward,  Thus lyreless, - woe's the day! For Juno frowned, and Venus wept,  And cupid ran away!
Those ennuied gods and goddesses,  Upon their mount sublime, O, had they not a weary lot,  A dull and dozing time!
One more there rose upon the air  Most sweet, though mortal song, By Zephyrus' glad wing upborne  To charm that heavenly throng.
Fair Venue bent her pearly ear,  Then earthward fixed her gaze, And smiled a curious kind of smile,  Half pleasure, - half amaze.
"I see a mortal bard, his hand  Across a lyre's strings flinging, And mortal lips catch up the strains,  Till all the land is ringing!
"About him throng the fair and young, -  They crown him! - I declare, Fast by him stands my truant boy! -  Apollo, dear, look there!"
The god rose from his cloud-divan:  "Ha! by my thundering sire, I understand that game of Morris.  There's the thief that stole my lyre!"