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LESSONS IN ALGEBRA.-XXI.

They are the basis or elements of what is called the Binomial

Theorem.
BINOMIAL THEOREM.

The Binomial Theorem may be, therefore, defined a general 189. To involve a binomial to a high power by actual multi- method of involving binomial quantities to any proposed pover. It plication is a long and tedious process. A much easier and more is comprised in the following general rule :expeditious way to obtain the required power, is by means of 1. SIGNS.- If both terms of the binomial have the sig t, all what is called the Binomial Theorem. This ingerious and the signs in every power will be + ; but if the given quantity is . beautiful method was invented by Sir Isaac Newton, and was residual, all the odd terms in each power, reckoning from the left, deemed of so great importance to mathematical investigation, will have the sign t, and the even terms that it was inscribed on his monument in Westminster Abbey. 2. INDICES. --The index of the first term or leading quantity

To illustrate this thereon, let the pupil involve the binomial must always be the index of the required power; and this decreases a +b, and the residual a - b, to the 2nd, 3rd, and 4th powers. regularly by 1 through the other terms. The index of the following Thus, (a + b)2=a+ 2ab + b2.

quantity begins with 1 in the second term, and increases regularly (a + b)3= a3 + 3a2b + 3abo + 63.

by 1 through the others. (a + b)= at + 4a3b + 6a*52 + 4ab3 + 6*.

3. CO-EFFICIENTS.The co-efficient of the first term is 1; that Also, (a - b)? = a’ – 2ab + 6.

of the second is equal to the index of the power; and, universally, (a - b)3 = 23 – 3aPb + 3ab? — 63.

if the co-efficient of any term be multiplied by the index of the (a - b)*= a* - 4a3b +6aPb2 — 4a63 +64.

leading quantity in that term, and divided by the index of the

following quantity increased by 1, it will give the co-efficient of By a careful inspection of the several parts of the preceding the succeeding term. operation, the following particulars will be observed to be 4. NUMBER OF TERMS. The number of terms will always be applicable to each power, especially if carried out to a greater one greater than the power required. number of powers.

In algebraic characters, the theorem is expressed thus1. By counting the terms, it will be found that their number

n-1 in each power, is greater by 1 than the index of that power ;

(a + b)n = an t nan

an_262 +1h. thus, in the 3rd power the number of terms is 4; in the 4th

-2

63 + etc. power it is 5, and so on.

3 2. If we examine the signs, we shall perceive, when both It is here supposed that the terms of the binomial have no terms of the binomial are positive, that all the signs in every other co-efficients or exponents than 1; but other binomials may power are +; but when the quantity is a residual, all the odd be reduced to this form by substitution. terms, reckoning from the left, have the sign +, and all the

EXAMPLES. even terms have the sign Thus in the 4th power, the signs of the first, third, and fifth terms are t, while those of the

1. What is the 6th power of x +y second and fourth are —.

Here, the terms without the co-efficients are mb, aby, aty?, anay, 3. As to the indices, it will be seen that the index of the first æ*y*, xy', ye. And the co-efficients, by the rule, are term, or the leading quantity* in each power, always begins with

1, 6,
6 X 5 15 X 4 20 X 3

6, 1. the index of the proposed power, and decreases by 1 in each suc.

3 cessive term towards the right, till we come to the last term,

1, 6, 15,

6, 1. from which the letter itself is excluded. Thus in (a + b)* the indices of the leading quantity a are 4, 3, 2, 1.

Now, prefixing these co-efficients to the several terms, and 4. The index of the following quantity begins with 1 in the observing the rule of signs, we have the power required as

follows:second term, and increases regularly by 1 to the last term, whose index, like that of the first, is the index of the required

206 + 6x+y + 15z4y2 + 20.r%y3 + 15zRy* + 6xy + yo. Ans. power. Thus, in (a + b)* the indices of the following quantity 2. What is the 5th power of 2% + 3ya ? b are 1, 2, 3, 4.

Here, substituting a for az, and 6 for 3y, we have (a + boja 5. We also perceive that the sum of the indices is the same = a + 5a*b + 10a%b2 + 10a%b3 + 50b+ + bs. in each term of any given power; and this sum is equal to the And restoring the values of a and b, we have (x2 + 3ya)ó = index of that power. Thus, the sum of the indices in each of the 210 + 15x9y2 + 90x@y* + 270x4y6 + 405x2y$ + 243yló. terms of the 4th power is 4.

191. When one of the terms of a binomial is a unit, it is 6. As to the co-efficients of the several terms, that of the first generally omitted in the power, except in the first or last term: and last terms in each power is 1 ; the co-efficient of the second because every power of 1 is 1; and this, when it is a factor, has and nect to the last terms is the index of the required power. no effect upon the quantity with which it is connected. Thus, in the 3rd power, the co-efficient of the second and next EXAMPLE.—Find the cube of (x + 1). Ans.a3 + 3x2 x 1+ 3r to the last terms is 3; and in the same terms in the 4th power, X 12 + 13, or x3 + 3x2 + 3x + 1. it is 4, eto.

192. The insertion of the powers of 1 is of no use, unless it It is to be observed, also, that the co-efficients increase in a be to preserve the exponents of both the leading and the follow. regular manner through the first half of the terms, and then ing quantity in each term, for the purpose of finding the codecrease at the same rate through the last half. Thus, efficients. But this will be unnecessary if we bear in mind In the 4th power they are 1, 4, 6, 4, 1,

that the sum of the two exponents in each term is equal to the In the 6th power they are 1, 6, 15, 20, 15, 6, 1.

index of the power. So that, if we have the exponent of the

leading quantity, we may know that of the following quantity, 7. The co-efficients of any two terms equally distant from the and vice versa. extremes, are equal to each other. Thus, in the 4th power, the 193. The binomial theorem may also be applied to quantities second co-efficient from each extreme is 4; in the 6th power, consisting of more than two terms. By substitution, several the second co-efficient from each extreme is 6; and the third terms may be reduced to two; and when the compound expresis 15.

sions are restored, such of them as have exponents may be 8. The sum of all the co-efficients in each power is equal to separately expanded. the number 2 raised to that power. Thus, (2)* = 16; also, the EXAMPLE.—What is the cube of a +b + c? sum of the co-efficients in the 4th power is 16, and (2)6 = 64 ; Here, substituting h for (b + c), we have a + (b+c) = a + he so the sum of the co-efficients in the 6th power is 64.

And, by the theorem, (a + 1)3= a + 3a'h + 3ahto h. 190. If we involve any other binomial, or residual, to any Now, restoring the value of h, we have (a +6+0)3 = + 302 required power whatever, we shall find the foregoing principles x (b + c) + 3a X (b + c)2 + (b + c)3. trile in all cases, and applicable to all examples. Hence we may The last two terms contain powers of (b + c); but these may safely conclude that they are universal principles, and may be be separately involved, and the whole expanded. employed in raising all binomials to any required power. 194. Binomials, in which one of the terms is a fraction, may

be involved by actual multiplication, or by reducing the given * The first letter of a binomial is called the leading quantity, and the quantity to an improper fraction, and then involving the fraca other the following quantity.

tion. It may also be done by substitution.

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EXAMPLES. Find the squares of x +}; and of 2 - 1. wrath to be fulfilled through Persian swords, is full of interest.

But the Bible record gives little or no information concerning Here, ti

Also, a

the actnal history of the kingdom, only stimulating the very x + 1

legitimate desire, which it also engenders, to study in other ce

places the story of the people who made so great a stir upon + 4x + 1

- dx +

the earth. 2 + x + 1

r x +.

The very remote history of Persia is involved in much Otherwise, reducing the mixed quantities to improper frac- independence, attached to a neighbouring empire, and certainly

obscurity. The country was most probably, in spite of semi2x + 1 tions, we have e +}= ; and æ

2x – 1
Whence,

in the year 900 B.C. we find it forming an integral part of the

Assyrian dominions, and when these fell to pieces Persia did not 4x2 + 4x + 1

4x2 - 4x +1 become free, but was incorporated in the kingdom of Media. 4

The union was not a happy one, and the Persians sought every +*+1, and ? -x+ }, as before.

opportunity to break it off. They found themselves in the

position of thralls to men of a civilisation inferior to their own, EXERCISE 34.

bound down strictly to religious rules and observances with 1. What is the 5th power of (d + h) ?

which they had no sympathy. Even when they had succeeded 2. What is the nth power of (b + y) ?

in inoculating the minds of their masters with their own 3. What is the 6th power of (3x + 2y) ?

religion, the magi, the priest-rulers of Media, took upon them. 4. What is the 2nd power of (a - b)?

selves the administration of the priestly duties, and asserted in 5. What is the 3rd power of (a - b) P 6. What is the 4th power of (a - b)?

the most tender places the right of the strongest to dominate. 7. What is the 6th power of (x - y)?

The religion of the Persians waş that which their own prophet 8. What is the nth power of (a - b) ?

or philosopher, Zoroaster, had taught them more than a thousand 9. What is the 4th power of (a — 1) ?

years before the birth of Christ. It had in the course of that 10. What is the 6th power of (1-y)?

time become corrupted, insomuch that the original belief in one 11. What is the nth power of (1 + x) ?

supreme God had become almost lost. Upon a comparatively 2 12. Find the square of a +

pure system all sorts of gross superstitions, borrowed from the

nations with whom the Persians had to do, were engrafted, 13. Find the square of x

until the worship of the sun, moon, and stars-that is, of a host

of deities—became a leading feature of the religion. Fire, as b 14. Find the square of

symbolising the light of the world, was worshipped by the

disciples of Zoroaster, who did not however omit, as their 15. Find the square of + 2abc.

descendants did, the adoration of Him who was symbolised by 7

the fire. EXERCISE 35.

Zoroaster was the first, we might also say the last, who

endeavoured to reconcile in his creed the existence of moral 1. Expand (x + y)".

10. Expand (a - b)'.

and physical evil with the attributes of a beneficent Creator 2. Expand (a + b)*.

11. Expand (a + b).

and Governor of the world. “ The first and original Being, in 3. Expand (a - b).

12. Expand (2 + x)'. 4. Expand (x + y)".

13. Expand (a - bix + c)".

whom or by whom the universe exists, is denominated in the 5. Expand (x - y)*.

14. Expand (a + 3bc).
writings of Zoroaster, Time without bounds."

“From 6. Expand (m + 11).

15. Expand (2ab - x)*.

either the blind or intelligent operation of this infinito Time, 7. Expand (a + b)'.

16. Expand (4ab + 50°)

which bears but too near an affinity with the chaos of the 8. Expand (2 +

17. Expand (31 — 6u)".

Greeks, the two secondary but active principles of the universe 9. Expand (6 -- y)^3.

18. Expand (5a + 3d)".

were from all eternity produced, Ormuzd and Ahriman, each of them possessed of the powers of creation, but each disposed, by

his invariable nature, to exercise them with different designs. KEY TO EXERCISES IN LESSONS IN ALGEBRA. The principle of good is eternally absorbed in light; the prin. EXERCISE 31.

ciple of evil eternally buried in darkness. The wise benevolence

of Ormuzd formed man capable of virtue, and abundantly pro. 1 27x). 7. 616".

13. (a + b).
8. Chandn.
14. (a + b)?n.

vided his fair habitation with the materials of happiness. By 3. 123c?

9. 216moyo.
15. (<

his vigilant providence, the motion of the planets, the order of

ymn. 4. b*m???. 10. a?''. 16. (x + y)?n,

the seasons, and the temperate mixture of the elements are 5. andnyn. 11. 610% 17. a®

preserved. But the malice of Ahriman has long pierced 6. d'hy. 12. 1296a?ex. 18. aloh'?.

Ormuzd's eggs, or, in other words, has violated the harmony of

his works. Since that fatal irruption, the most minute particles EXERCISE 32.

of good and evil are alternately intermingled and agitated 1 1 a. (d + m)?

9 b' + 86% + 246% + together; the rankest poisons spring up amidst the most salu. as' (x + 1)

32b + 16.

tary plants ; deluges, earthquakes, and conflagrations attest the 5. a? - 2ab + b). 10.45 + 5x + 10x +

conflict of nature; and the little world of man is perpetually 8x*y

20x+ 5.2 + 1. 6. a' + 3a+ 3a + 1.

shaken by sin and misfortune. While the rest of human kind

11, 1- Cb + 156 27y3 7. a? + 2ab + 2ah + bo

are led away captive in the chains of their infernal enemy, the

2003 + 156---665 + 2016 + ho.

faithful Persian alone reserves his religious adoration for his gagn

+6° 3. 8. art. 4ad + 6a - 40

friend and protector Ormuzd, and fights under his banner of ay* + 120 + 9.

light in the full confidence that he shall in the last day share EXERCISE 33.

the glory of his triumph. At that decisive period, the 1, 4a" + b3 + 4ab.

4. 36y* + 9 + 36y.

enlightened wisdom of goodness will render the power of 2. ho + 1 + 2h.

5. 90' + h' -- Odh.

Ormuzd superior to the furious malice of his rival. Ahriman 3. a"' + c'd' + 2abcd.

6. a' +1- 2a.

and his followers, disarmed and subdued, will sink into their native darkness; and virtue will maintain the eternal peace and

harmony of the universe." HISTORIC SKETCHES.—XLI.

The simplicity of the Persian worship is vouched for by

Herodotus; indeed, it seems to have impressed all who came in THE PERSIAN POWER.

contact with it. “That people,” says the Greek historian, The book of the prophet Daniel would alone bo sufficient to "rejects the use of temples, of altars, and of statues; and make one desire to know more of Persian history than is given smiles at the folly of those nations who imagine that the gods in the Bible. In many of the prophetical writings reference is are sprung

from, or bear any affinity with, the human nature. made to the kingdom of Persia, under the name of Elam; and The tops of the highest mountains are the places chosen for the prediction of things to come, through Persian agency, of sacrifices. Hymns and prayers are the principal worship; the

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supreme God, who fills the wide circle of the heaven, is the found wanting, was then and there given over to the Medes and object to whom they are addressed.”

Persians. The end had come. The Median kingdom was not of long duration. Itse.f ori. ginally a province of the Assyrian empire, it shook off the foreign

“Crownless and sceptreless Belshazzar lay,

A robe of purple round a form of clay." yoke when that empire collapsed, and sprang almost immediately into importance. Allied with the rising power of the Babylonians, Cyrus, under the direction of his uncle, Cyaxares II., who had it gave the finishing strokes to Assyrian existence, and included accompanied the army, took military possession of the famous within its borders the smaller but still strong province of Persia. city, and having made it as strong as possible went back to Not without the exercise of much cruelty, and the exhibition of Persia, laden with the almost fabulous wealth which successive a ferocity which betokened the barbarian, were the Persians Babylonian kings, notably Nebuchadnezzar, had accumulated. subdued ; and it is probable that at no time was the country Cyaxares, anxious to secure the benefits of so fine a city, and completely under subjection, unless it might be in the plains glad of an opportunity which gave him, a Persian, the means and lowlands, the warrior caste and the princes preserving in of eradicating from the Median mind that there was any actual the highlands the spirit and even the form of independence. necessity for governing from their capital, moved the seat of his The Medes were almost afraid—they had good reason to be so government to Babylon, a situation which also afforded & better -of the acquisition they had made. They saw in the superior base for those military operations which he contemplated against intellects and greater knowledge of their subjects the signs of a several other of the Eastern monarchies. Soon after this removal power that might one day prove fatal to their role, and they occurred the remarkable incident chronicled in the book of the endeavoured by all the means at their disposal to conciliate prophet Daniel. The Persian king, called in the prophet's them, though not till after they had made irreconcilable ene- writings Darius, a title common to all the Median princes, and mies of them. The Persians temporised, waited for their meaning simply " the king,” began to persecute the priesthood opportunity, and never ceased, while yielding nominal allegiance which he found in Babylon. The Persians, as the worshippers to the Mede, to look forward to the day when the tables should of one God, and as followers of the simple and pare faith of be turned, and when the one pure religion and the one Aryan Zoroaster, were extremely averse to the complicated and de(noble) race should be acknowledged as supreme.

grading superstitions which were common in all the countries So powerful had they become, and so threatening had grown around them. It had been the most galling part of their the position of external enemies in the time of Astyages (called bondage to the Mede, that they had to submit to the interference Ahasuerus in the book of Daniel), who reigned in Media about of a powerful priesthood, which dominated to the exclusion of 565 B.C., that the Medes thought it advisable to conciliate the all that was noble and admirable in the national mind, and Persians in every possible way. Astyages gave his daughter to which sought only to establish its own power at the expense of be married to Cambyses, one of the chief of Persian princes, whatever else might come in its way. Cyrus and Cyaxares, for and a member of the royal house. The issue of this marriage the latter now associated his nephew in the government which was Cyrus, immortal in human history, and specially famous as that nephew had originally handed over to him, never lost an the saviour of his country, the man who made the Medes opportunity of showing their contempt and hatred for profesexchange with the Persians the supremacy on the throne. This sional priesthoods and for the superstitions they taught. In young man, seeing as he grew up the exact position of things, Babylon they found a superstition and a priesthood worse than and ever mindful of what his countrymen had suffered at those of the Median magi. They determined, both as a matter Median hands in the old time, conceived the scheme of over- of policy and of morals, to insult the power which held the throwing the dynasty and of seating a Persian upon the throne people in awe, a power which, as they well knew, might at any of the two kingdoms. Though scarcely arrived at maturity, he time cause an insurrectionary spirit to spring up among the went through the land, inflaming the minds of the Persians people, and which from their hearts they despised as being by the remembrance of ancient wrongs; and making an oppor. based upon imposture, ignorance, and falsehood. Among the tunity, he unfurled his standard and marched against his grand prisoners at Babylon was a man, one out of thousands, to whom father Astyages, who was overthrown and flung into prison. the Persian princes were drawn at once by the force of % Cyaxares II., a kinsman of Cyrus, was seated on the throne, religious and intellectual sympathy, as well as by his personal while Cyrus pursued both against the Medes and Babylonians a merits. Daniel, the prophet of the one God, the man who had series of brilliant conquests which made the Persian arms dared even Belshazzar's wrath in testifying against the wickedsupreme in Asia. The Babylonian power he completely sub. ness of Babylon, and in asserting the only adorable Jehorab, verted, giving it the coup de grace when he captured the city of was the man whom Cyaxares singled out to help him in governBabylon under circumstances which must be familiar to all ing the new kingdom and in overthrowing the priesthood. The readers of the Old Testament. “Belshazzar the king made a Persian and the Hebrew worshipped one God, though in different great feast to a thousand of his lords, and drank wine before ways; and though the latter deemed it essential to proper por the thousand.” Relying on the enormous strength of the city ship that the service of God should be splendid and served by walls and on the power of his army, contemptuous of the host an exclusive priesthood, while the former held simplicity of of former subjects who had come to invade him, and careless in worship without the intervention of priests to be the more his supposed security, the Babylonian king took no military acceptable sacrifice, yet the conditions under which the two precaution to guard against the enemy that was encamped met in Babylon prevented any clashing in this regard. Daniel before his gates. Cyrus, recognising the great strength of the was an exile, a fugitive, singing the Lord's song in a strange defences, gave them the go-by, unwilling to hurl his men to land, remote from Jerusalem, "where God ought to be wor certain death when no advantage could be derived from the shipped,” away from the possibility of partaking in those cere sacrifice. Whilst the Babylonians revelled and drank, whilst monies and ritualistic performances which the Jews had been on their walls were appearing those dreadful and mysterious taught to look npon as so well-pleasing to God. Whatever be characters which none could decipher save the prophet of God, may have longed for, he could not at Babylon either celebrate the Perso-Median troops were diverting the course of the river or partake in any ceremonial of Jewish worship which might which ran through the city into a canal that had been dug for affront his new master. His prayers, his way of making his it, and which drained the river-bed. Along the bed the men of wants known to God, and his mode of worship, must have been Cyrus marched, and coming into the torror-stricken city found as simple and unaffected as those of Cyrus himself. He was no resistance worth speaking about. From point to point they an alien, it is true; but so were the Persian princes themselves went till they came to the royal palace, where Belshazzar was aliens, not only among the Babylonians whom they had con giving a dinner to a thousand of his lords. What happened quered, but also among the Medes by whose arms they had there all must know. Even as the words of interpretation were conquered. Herein was another bond of union. So Daniel was being uttered by the prophet Daniel, the Persian warriors rushed promoted to honour, apparently to the rank of grand vizier, into the hall; vain was the desperate resistance of the guards, in the Persian court. Cyrus was gone on military expeditions useless the valour with which Belshazzar himself and his com- which took him to Egypt and to Syria ; Cyaxares ruled alone, panions at the feast drew their swords and stood at bay. In a with the help of such assistance as the Hebrew prophet gare, few minutes the place was won; the prophecy, which even get We may reasonably suppose that some popular outburst of was discernible upon the wall, was dreadfully fulfilled; and the feeling on account of the priesthood, somo fanatical piece of Babylonian kingdom, having been weighed in the balance and enthusiasm of the priests themselves, led him in a moment of contemptuous anger to issue the famous decree that "whosoever From that time the Persian power toppled on the brink of shall ask a petition of any God or man for thirty days," savo ruin. Artaxerxes, the successor of Xerxes, who was murdered of the king, should be cast into the den of lions. The sequel by his guards, for a few years revived the fading splendour of is well known; the sorrow of Darius when he found where the the empire ; but he likewise in the end passed under the waters panishment fell; the endeavours he made" till the going down of adversity, and was compelled to sign treaties which Cyrus of the sun, to deliver him;" the envious insistance of the would not have touched with the tip of his sword. Another Median and Persian princes upon the law of the Medes and hundred years of fitful existence, and then the end came. Persians which altereth not; and how " the king commanded, Alexander of Macedon, gathering the reins of all Greek governand they brought Daniel, and cast him into the den of lions."ment into his own hands, was the incarnation of all that was The religious sympathy between the king and his great subject, strongest and wisest in the counsels of nations. At Issus and the common belief they had in the might and goodness of a Arbela he completed the work which Miltiades began; and God who was King of kings and Lord of lords, was distinctly three hundred and thirty years before Christ, the Persian power, and clearly shown in the speech of Darius : " Thy God whom which had been all bat universal, was laid low by those who in thou servest continually, he will deliver thee.”

turn suocumbed to the Latin race they once affected to despise. Soon after the delivery of Daniel from the lions' den, Cyaxares, who was sixty-two years old at the time of the capture of Babylon, died, and was succeeded in the sovereignty by

LESSONS IN GERMAN.-LI. Cyrus, B.C. 534. Cyrus, who had carried the Persian arms to

§ 8.-GENDER OF COMPOUNDS AND FOREIGN WORDS. the borders of Egypt, and but for the need of his army for refreshment might have done as his successors did to that

(1.) Compounds in general adopt the gender of their last country, devoted several years to cousolidating at home the component; as :power he had won. Among the very first of his governmental Die Hoffirme (from Hof, court or yard, and Kirche, church), court acts was the issue of a decree that the Jews whom Nebuchad.

church. nezzar had brought into captivity should be free to return to Der Kirchhof, the churchyard. their native land and to rebuild the walls of Jerusalem. This Der Gichbaum (from die Eiche, the oak, and der Baum, tree), the magnanimous act was due not only to gratitude for the help he

oak tree. had received from Daniel, but to an appreciation of the valuo Die Windmühle (from der Wind, the wind, and die Mühle, mill), which the Jewish civilisation had been to the half-barbarous

the windmill. people with which it had been so long in contact; it was the Das Rathhaus (from der Rath, council, and das Haus, house), the generous act of a truly liberal mind, which could agree to forogo council-house. the manifest advantages of the Jews' presence out of deference to a belief that their compulsory residence in the land of their German languago, retain their original gonder. Thoso, however,

(2.) Foreign words, for the most part, when taken into the whilom conquerors was an oppressive and undesirable thing that have become fairly Germanised often take a different gonFor seven years he ruled with much satisfaction to his people der, as they take a different form; thus, Corpus (the body), and with glory to himself, and is said then to have been slain which in Latin is neuter, becomes in German ter Körper, which is with a large number of his warriors in an expedition against masculine. the Scythians.'

$ 9.-DERIVATION OF NOUNS. Cambyses succeeding in the year B.C. 529, invaded Egypt, overran it, and, inheriting the rancour of Cyrus against the

(1.) To what has been already said (§ 2. [3]) concerning the sacerdotal castes, almost annihilated the Egyptian priesthood, derivation of nouns, we add here, before entering upon the subat the same time that he restrained the priesthood in other ject of Declension, a brief view of the secondary derivatives, parts of his dominions. Trying to penetrate too far into the which are made by significant suffixes. For the sake of the African continent, he met with some grievous disasters, and learner we subjoin a list of the leading suffixes of this class ; returned home to find that, though he had put his own brother putting in brackets the equivalent English terminations, explainto death to prevent his being troublesome, there was a usurper ing severally their force and use, and illustrating the whole by in the field who claimed the crown. He died of an accidental suitable examples. wound from his own sword, and the usurper, a nominee of the § 10.-SUFFIXES USED IN FORMING NOUNS. magi and of the old Median party, was destroyed by the Per.

ENGLISH EQUIVALENTS. sians, who raised Darius Hystaspes, one of their own nobles, to

designates (male) persons ; the throne. Darius carried out the old policy of Cyrus and set

[er, ier or yer, zen]

also agents or instruments. Cyaxares so far as the home government was concerned, and

denotes (often contemptu. crushed severely all attempts on the part of the Babylonians to -ing, or -ling [ling, aster]

ously) persons, animals, and regain their freedom. His foroign wars were carried on by

things. himself in person, and were extended across the Danube into in [ess, ir]

designates (female) persons. Scythia, where Cyrus had been killed, into Greece, and beyond

indicates the act, practice, or the confines of India. It was the nascent period of the Greek -ci

[y, ry, ary, ery, ory]

place of business. dominion. Even on the return of Darius from his Soythian

signifies the act or the concampaign, which nearly proved as fatal to himself as Cyrus' -ung [ing, ure, ion]

tinuing to act. expedition into Scythia had been to him, the Greeks, under

[ness, ity, th] some of their ablest early generals, ventured to cross swords -heit [ness, ity, th] denote qualities or attributes. with him, and found, though defeated, that they had hope of feit [ness, ity, th] learning from him how

to conquer. The policy of the Greeks was to split up so overshadowing a power as that wielded by -íchaft [ship, hood, ity]

express rank, grade, office; the Persian king, and they used every opportunity of carrying thum [dom, hood, ity]

also, a number of things

taken collectively. out this policy. Wars frequent and bloody were the conse- / -sal [ude, cy] quence, and the strength of Persia, crippled as it was by

denote the state or condition ; Tel

[ude, cy] Miltiades at Marathon (B.C. 490), was gradually undermined.

sometimes the result. -niñ

[ness, cy] As the inferior civilisations had given way to the Persian, so-chen [kin, ule, et, let] that was now to give way to the superior civilisation of the ein

indicate diminutiveness.

[kin, ule, et, let] Greeks. Prodigious as the efforts of Xerxes, the successor of Darius, were, enormous as were the cost and equipment of his

§ 11.-EXAMPLES. fleets and armies, they failed to make an impression upon tho SUFFIXES. tock-founded states of Greece. Xerxes himself, after collecting

Sånger, a singer ; Bürger, a citizen; Såger, a sawyor ; snch armies as had never before been heard of, after three years -cr Schneider, a tailor; Römer, a Roman ; Leipziger, a resi. spent in preparations against the inevitable, returned homo

dent of Leipzig ; Wiener, a Viennese. covered with disgrace; and the army he had left to cover his

Häuptling, a captain; Flüchtling, a fugitive; Miethling, a retreat, and to make a show of military dignity in retiring, was -ling hireling ; Dichterling, a poetaster; Hanfling, a linnet; completely destroyed at the battle of Platea.

Süßling, a shoot or sprig.

SUFFIXES.

MEANING.

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NOUNS.

-Fal

Gräfin, a countess; Heldin, a heroine ; Königin, a queen ; Köniy, to the king ; and is never used when a preposition stands -in

Professorir, í rofessor's wife; Lönin, a lioness. before the substantive without an article, pronoun, or adjective -ei

Dieberei, thievery; Heuchelei, hypocrisy; fischerei, fishery; preceding; as :-Mit Beifall, with approbation; mit Sturm, by
Brauerei, brewery.

storm; mit Fleiß, on purpose ; von Silber, of silver.
Belehrung, teaching, i.e., the act of teaching; Erbauung,

EXAMPLES. the building, or erecting, edification ; Krönung, the -ung

Singular.

Plural. crowning, or coronation ; Sipung, the sitting, or ses

Nom. Der Berg, the mountain. Die Berge, the mountains. sion.

Gen. Des Berges, of the moun. Der Berge, of the mountains. Güte, goodness ; Starfe, strength ; Krankheit, sickness ;

tain. -heit

Dummheit, stupidity; Heiligkeit, holiness; Feuchtigkeit, Dat. Dem Berge, to the moun- Den Bergen, to the mountains. -teit humidity.

tain. Freuntschaft, friendship; Priesterschaft, priesthood, that is, Acc. Den Verg, the mountain. Die Berge, the mountains. -schaft the body of the priests; Bereitschaft, readiness; Heiden. -thum thum, heathendom, heathenism; Christenthum, Chris

(1.) Some nouns of this declension take the letter « after e in tianity; Gigenthum, property.

all cases of the plural, and assume the Umlaut, if the radical Mühsal, the state of being in trouble, distress ; Hüdtiel, vowel be capable of it. (Sect. II. 12. Ne, etc.) that which has resulted from hacking and cutting,

EXAMPLES. - Tel

that is, cuttings (of straw); Betürfnis, the state of -nis

Singular.

Plural. being in want, necessity; Gleichniß, simile, parable. Nom. Das Dorf, the village. Die Dörfer, the villages. Icin Büchlein, a little book ; Knäblein, a little boy ; Stühlchen, Gen. Des Dorfes, of the village. Der Dürfer, of the villages. .chen a little chair; Gichen, a little egg.

Dat. Dem Dorfe, to the village. Den Dörfern, to the villages. Appellatives derived from the names of places and countries Acc. Das Dorf, the village. Die Dörfer, the villages. have the termination er, as :- Der Contoner, the Londoner. Names of countries, like Sachsen, Saxony; Prcußen, Prussia, etc., being Nom. Das Lied, the song. Die lieber, the songs. originally the names of the people, not of the countries, cannot Gen. Des Licdes, of the song. Der Lieder, of the songs. take er.

Dat. Dem Liebe, to the song. Den Liedern, to the songs. Nouns derived from the name of a city or town are often used Acc. Das Lied, the song. Die Lieter, the songs. indeclinably as adjectives ; as :-Dus Leipziger Bicr, the Leipzig The nouns of this form (e +r) in the plural, among which are beer. Gen. Des Leipziger Biers, of the Leipzig beer.

all substantives ending in thum (as, ter Reicbthum, riches), are (1.) It must be observed, in forming derivatives of the order for the most part neuters ;, as :illustrated above, that when a, o, u, or au, is contained in the radical part, it is modified into i, o, ů, or åu, upon receiving a Amt, office.

Aas, carrion.
Gespenst, spectre.

Liet, song.
Gewand, garment.

Poch, hole.
suffix containing the vowel i or e; as:-r, el, ling, in, ig, ing,
chen, leix, e, niß, and sel; as in the case of Hündin (from Hund), Vand, ribbon.

Bad, bath.
Glas, glass.

Mabl, meal.

Glied, member. Maul, mouth. Burger (from Burg), and others of the like kind.

Wild, picture. Grab, grave.

Neft, nest. (2.) Often, moreover, in forming secondary derivatives certain Blatt, leaf.

Gras, grass. euphonic letters are inserted between the suffix and the word to

Pfand, pledge. Vrett, board.

Gut, good, estate. which it is added ; as, ig in Feuchtigfeit, humidity.

Rad, wheel.
Other letters

Haurt, head.
Buch, book.

Reis, twig.
employed in this way are en, n, and t.
These euphonic parts Dady, roof.

Haus, house.

Rint, cattle. aro easily distinguished from those having an influence on the Ding, thing (little). Holz, wood. meaning, by merely resolving the derivative into its elements.

Schilt, sign-board.
Derf, village.
Horn, horn.

Sdlog, castle. (3.) Here, too, may be noted the particle ge, which, being Gi, egg.

Huhn, hen. prefixed to certain primary words, form a class of nouns de

Schwert, sword. Kalb, calf.

Fach, compartment. noting either frequency of action, or a collection of things. Fab, vat, tub.

Thal, valley.

Kind, child.
These words, also, most commonly suffix the letter e; as :-

Tuch, cloth, shawl.
Felt, field.
Kleid, dress.

Volf, people. Gerede, constant talk; Geheul, continuous howling ; Gebirge, a

Geld, money.
Korn, grain.

Wamms, doublet. range of mountains, etc.

Gemacy, apartment. Kraut, herb.

Weib, woman. $ 12.-DECLENSION OF COMMON NOUNS.

Gemüth, mind.
Camm, lamb.

Wort, word (in a dic(1.) In German there are two declensions, distinguished as the Geschlecht, species.

Land, land.

tionary). Old and the New. The characteristic of each is the termination Gesicht, face. Licht, light. of the genitive singular. In the former, the genitive is formed The following nouns of this form are masculine :from the nominative by adding es or ; when the genitive is Pösewicht, villain Mann, man.

Strauß, nosegar. otherwise formed, the noun is of the New Declension. (2.) To the Old Declension belong almost all masculine and Geist, spirit.

Dorn, thorn.

Mund, mouth, Vormunt, guardian.
Ort, place.

Wald, forest. neuter nouns; that is, by far the greater part of all the nouns Gott, God.

Rand, margin. Wurm, worm. in the language.

Leib, body, (3.) In both declensions, the nominative, genitive, and accusative plural are alike in form ; while the dative terminates

(2.) Most nouns of this declension, whose radical vowel is always in the letter n. Unless, therefore, the word declineda, o, u, or au, assume, in the plural, the Umlaut. already ends in that letter, it is, in the dative, uniformly Sohn, Buch, Baus, make their plurals as follows :assumed.

EXAMPLES (4.) All feminine nouns are invariable in the singular; in the

Feminine.

Masculine. plural, they are, for the most part, inflected according to the Nom. Die Hante, the hands. Die Söhne, the sons. New Declension.

Gen. Der Hände, of the hands. Der Söbre, of the sons. (5.) In compounds, the last word only is subjected to the Dat. Den Händen, to the hands. Den Söhnen, to the sons variations of declension.

Aco. Die Hände, the hands. Die Söhne, the sons. $ 13. —THE OLD DECLENSION.

Neuter.

Neuter.
TERMINATIONS,

Nom. Die Bücher, the books. Die Häuser, the houses.
Singular.

Plural.

Gen. Der Vücher, of the books. Der Väuser, of the houses. Nom.

Dat. Den Blichern, to the books. Den Häusern, to the houses.
Gen. -es or 8.

Acc. Die Bücher, the books. Die Häuser, the houses.
Dat. - (or like Nominative).

The words in which the Umlaut thus occurs are, chiefly–1,
Acc.
In the genitive singular the e is frequently omitted before 6, have their plural in c, as also Mutter and Joc ter ; 3, neuter pris

:

primitive nouns of the masculino gender ; 2, feminínes which when the elision does not cause an unpleasant sound; as :-Des mitives having their plural in (e + r) er ; 4, and lastly, nouns Könige, of the king ; des Monate, of the month ; des Jahrs, of the adding the diminutive terminations chen and lein. year. Also, the è of the dative is often dropped ; as, tem (3.) Nouns ending in el, en, cr, chon, and lein, reject the vowel e

Thus, Fant,

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