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3. ATOKVEW (aro, from, and kvw, I conceive, am pregnant), I
dr 2h 6c bear, I bring forth; Davaros, ov, ó, death.
to fractions having a common 3m 9
y 4. δοσις, εως, ή, a giving ; δωρημα, ατος, το, α gift και τελειος, a, ov, perfect; avwtev (ava), froin above, the termination Dev | denominator gives the idea of from, i compare in Number IX., odev and
dgry 6hmy 18cgm εντευθεν; καταβαινω (κατα, doum, and βαινω, I go), I come
3gmy' 3gmy' 3gmy down, soti kar, literally, is coming down, is constantly coming down~a beautiful description of the constancy of the
7+1 3. Reduce
and Heavenly Father's goodness ; pws, Autos, to, light.
to fractions having a dth
having a common 5. opyn, ns, a' (the root of opeyopai), desire, effort, a strong
denominator. emotion, anger; dekaloouvn, ns, i), justice, just designs ; katepyaSouai (kara, down, thoroughly, and epyov, a work), I accom
2dx+212 3ad73ah 3rx+32
3d.c+3h 3dx+3hx 3ds+-3hx 6. γινομαι (the old form of γιγνομαι, compare γενος, o race,
1 a kind), 1 become ; months, ov, ó, a doer, a maker, hence our
and to fractions having a common depoet, the great maker, that is, inventor; arpoarns, ov, ó, a
at-B a-b hearer.
nominator. 7. Opnokela, as, , service, God's service, religion; kalapos,
atb a, ov, pure; aulavtos (ulaivw, I spot), unspotted ; kai, éven,
and that is, ούτος, αύτη, τουτο, this ; επισκεπτομαι (επι, ουer, and
a2-62 a% -62 CKET TONAL, I survey) I overlook; from the same root is our
130. An integer and a fraction are easily reduced to fractions bishop,' that is, an overlooker, a superintendent; oppavos, oy, having a common denominator, by making the former a fraction ;
, our word orphan; xnpa, as, y, a widow ; Olyis, EWS, ~, see Art. 121. afliction ; avtwv, of them, their; aortos, ov, unstained (oridos,
7 a stain), tnpew, I keep, preserve,
5. Reduce a and to fractions having a common denominator. 8. åyvos, n, ov, chaste, holy; TowTov, in the first place, ETELTA, then, in the second place ; ELPnvikos (Elpnun, peace), peace
5 ful ; ET LELKIS, mild; EVTTĒLONS (TTELOW, I persuade), easy to be Here, a and
are equal to and which are equivalent to
c' entreated; LEOTOS, n, ov, full; adiakpiros (a, not, dia, through, kpivw, I distinguish), without partiality; avutropitos (a, not, ac 6 the v is interposed between the two vowels for the sake of and
the fractions having a common denominator. euphony; ůro, under, and kplvw, hence our word hypocrite), without hypocrisy; OTELOW, I sow; TOLS TTOLOVolv, for those doing,
h d that is, those who do or pursue.
6. Reduce a, b,
and to fractions having a common de9. modɛv, whence; EVTEVOɛv, thence; úpwv, of you, your ;
nominator. στρατευομαι, I war και των στρατ, which make μας και μελος, ους, To, a limb, member ; julv, in you.
and 10. Molgos, ov, o, an adulterer ; uolyahıs, idos, y, an adul
2782 my 9:24
my teress; OVK OLDATE, know ye not ? exopa, as, yj, hatred. 11. ůnepnpavos (Urtep, above, high, too much, and patvw, I
and show), haughty; avtiraooojai (avti, against, and tacow, I set),
to fractions having a common de. I array myself in opposition to; TATTELVOS, 17, ov, low, lowly, humble; didwol, he gives.
bde nominator. Ans.
and 12. vouoterns (vouos, a law, and Tionue, I place), a lawgwer ;
udf' bdf bdf δυναμαι, I am αθie; ο δυναμ. who is able; σωζω, I save ;
1 απολλυμι, I destroy; σωσαι and απολεσαι are infinitives
and to fractions having a common degoverned by ο δυναμενος. .
56' 2 nominator.
and LESSONS IN ALGEBRA.—No. VIII.
REDUCTION OF FRACTIONS.
9. Reduce 6, and to fractions having a common de
2 129. To reduce fractions of different denominators to fractions
nominator, having a common denominator.
and Multiply together each nurnerator and all the denominators
2y ' 2y'
2y except its oun, and the product will be the required numerator of each fraction; next, multiply together all the denominators, and the
1 product will be the required denominator of each fraction ; these
and to fractions having a common Y
3 properly arranged in order will give the answer.
3xyz 3aby gacz ay Ans.
3ayz' 3ayz' 3ayz 3ayz m 1. Reduce
and to fractions having a common de.
to fractions having a common deHere, axdxy=ady, CXbXy-cby,
nominator. are the three numerators.
60cx 5ab 4acat and, inxbxd=mbd,
20ac' 20ac 20ac Also, bxdxy=bdy, is the common denominator. ady bcy bdm
1 Hence, the reduced fractions are
to fractions having a common bdy
4 The reason of this rule is plain, for the reduction consists in denominator. multiplying the numerator and denominator of each fraction into
28ay 14064 22463 7by all the other denominators, a process which does not alter the
28by' 28by value of the fractions ; see Art. 120.
02-06 to an improper fraction. Ans.
16ax 5. What is the value of when the denominator is nultiplied
by 4. Ans. 24.
36x+46 18. Reduce atb- to an improper fraction.
14. Add 2ats,
Ans. 20–36+5%+9. 4m
135. For many purposes, it is sufficient to add fractions in the
Zaca 39. Reduce
same manner as integers are added, by writing them one after
3bdy another with their proper signs.
of of of of of to a simple 15. Find the sum of and
abcan fraction. Ans.
3 d Here, the sum is simply +
b ADDITION OF FRACTIONS.
136. To add fractions and integers together. To add fractional quantities together.
Write them one after another with their signs;; of convert the 134. Rule.- Reduce the given fractions to fractions having integers into fractions, reduce the fractions to a common denominator, o common denominator if necessary; then add their numerators, and then add as before. and place the suni over the common denominator.
1. What is the sum of a and ? Ans. at
A KEY TO THE EXERCISES IN THE
Vol. III., p. 113.-ENGLISH-LATEX.
Canem verberant; filius patrem amat; epistolam .crib:3 · Angli
vincent hostes; hostes victi sunt ab Anglis ; Carthage & Scipione By John R. BEARD, D.D.
deleta est; pax fortes decet; ducem Vellingtonuna fecerunt Angli;
Naprleonem nominârunt imperatorem Galli; vigentesimum ago (Continued froin page 75, Vol. IV.)
annum ; tricesimum annan ago [instead of age read ago); frater
multos menses caecus fuit; fossa di ceu pedes est lata; exercitum Vol. III., p. 113.-LATIX-EXGLISE.
transduxit fumeu; samnium suntiavit Josephus ; aleâ luduot;
illud duvitas ? illud mili concedunt; regem adeo; boni veritatem How great your labour ! alas! how wretched I am! O the nunquam egrediuntur; araratia carcos reddit homines ; Socrates deceptive hope of men, and their frail fortune and our empty sapiens est babitus; consul sententiam rogatus est; puellas strifes ; by the faith of gods and men! what great undertaking, o musicam docent; Latinam docendi sunt pueri ; ebeu me miserum! sacred 'Jupiter, 'was ever accomplished not only in this city but in rae tibi, marite! all lands ! alas! my labours have been undertaken in raia! O my fallacious hopes, and empty thoughts! lo! here is the reason why
Vol. III., p.
130.--LATIN-ENGLISH, the law was passed ; abā! you have a talent of silver! I write å letter; I whip a dog; the son loves his father; the Romans Conon for the most part lived at Cyprus, Iphicrates in Thrace, conquered the Carthaginians; Scipio destroyed Carthage; the Timotheus at Lesbos; we were detained seven days at' Corcyra; very swift foot-soldiers were able in running to keep up with the Miltiades tarried in the Chersonesus; the Gauls went to their cavalry; serene peace befits men, fierce rage (befits) wild beasts; homes; great things were carried on abroad in those times; M. red hair is no discredit to a man among the Germans; let prudence Drusus was killed at his own house ; I spoke those same things at never fail [in the note for infinitive read imperative] the orator; my house; they carry gold and silver into the palace; they invite Themistocles did not escape from envy on the part of his fellow each other to their homes; I am not at my ease in another person's citizens; Ulysses wished to withdraw from warfare; fortune assists house ; at the house of Caesar there are no friends; many persons the brave; you think that I am a rival of Agamemnon; glory, like assemble at his house; the victory was announced at Corinth, at a shadow, follows valour ; brave and wise men are not accustomed Athens, and at Lacedemon; he was brought over from Apollonia, to seek so much the rewards of good deeds as good deeds them-a city of Pontus; the Romans came to the town Cirta; the selves ; Marius commanded in that part; he had formerly played strangers are in the town of Citium; Archias was born at Antioch, a similar game; I dreamed a wonderful dream; I am of opinion formerly a celebrated city; the soldiers stopped at Alba, a fortified that your fathers are alive; I enjoy that kind of life which alone city that was near; when I was directiog my way to Mutina; there deserves the name; I have sworn a very certain and very imposing was a look-down from Gergoria into the camp (castra); I intended oath ; Siccius Dentatus had nine triumphs; I cannot make the to go from Athens; ambassadors came from Ardea to Rome; a same boast; that I doubt not; in that I agree with you; Pytha vessel is ready for us at Cajeta; Socrates introduced philosophy goras visited the magi of the Persians; the consuls entered on even into homes; Antony committed impure deeds even in a chaste their consulship; Pythagoras went to many barbarous regions family; discipline flourished in that house; Alcibiades was educated on foot; I met no one; all fates surround us on every side; soon in the house of Pericles; he was said to celebrate mysterious rites the Roman legions besieged Carthage; Scipio did not refuse a in his own house; they went to their own rural properties ; Pompey conference; history ought not to depart from truth; the best plans being conquered by Caesar hastened to Alexandria; the emperor are forming; this town is under siege; a very large inheritance Gallienus was slain at Nilan; Caius Marius betook himself to comes into (your) possession; they declare that either by a letter Praeneste; many famous Romans are said to have gone to Rhodes; or a messenger they are approaching him as if king; Caesar went to the Romans were brave at home and abroad; at Delphi there was the colonies of his country; he wished to visit, and become a very famous temple of Apollo; wherefore both kings and peoples acquainted with those nations also; with great hope I enter on were accustomed to send ambassadors to Delphi, or to seek the remainder of my speech; the mother called her son a brother. oracular responses from Delphi; the Romans sent ambassadors to slayer; Ennius bas well said that anger_is the beginning of Athens; Demaratus had fled from Corinth to Tarquini in Etruria; madness; he nameä Sicily the nurse of the Roman populace; the we are punished by negligence in many things; they are in fault recollection of pleasures we have enjoyed makes life happy; who desert their duties through weakness of mind; I beg you to cupidity and avarice make men blind; the Carthaginians appointed think that I write to you more rarely than I was accustomed, not Hamilcar their commander-in-chief; the people created Ancus through forgetfulness of you, but through my bad health ; anxiety Martius king; Apollo judged Socrates to be the wisest of men; had taken possession of the senators test the people through fear Socrates accounted himself an inhabitant and citizen of the whole and anger should appoint military tribunes; I almost lost my world. Ariovistus replied to Caesar that he should consider him senses through excessive joy; the pilot's art is praised for its not as a friend but as an enemy; what is more foolish than to utility, one ought to grieve for a misdeed, and to rejoice at correcaccount ancertain things certain, and false things true ? Cato, by tion; I am wont to rejcice in nothing so much as in a consciousness himself, is as good to me as a hundred thousand men; Artaxerxes of performing my duties; being glad at your lot you will live asked the Athenians to let him have Iphicrates as a general; do wisely ; the Campanians were always proud of the excellence of you consider that nothing ? the Achaeans sought aid from Philip; their fields ; Greece formerly fourished in opulence, dominion and Verres (capital V) demanded from parents money to be allowed glory; the Roman state laboured under two viceș, avarice and 20 bury their children; Caesar daily required corn from the Aedui; luxury; oxen protect themselves by horns, boars by tusks, lions by who taught Epaminondas music? I have not concealed the con- jars (bites); Lycurgus confirmed his lairs by the authority of the Tersation from thee; know that my opinion has not been asked ; ( Delphian Apollo; Atticus gratuitously supplied all the Athenians Marius had learnt all military arts; am I even not to be taught with corn; we are supplied and adorned with the gifts of the gods; to speak either Greek or Latin ? in that you give me escellent it is the duty of the senate to assist the state by its counsel; many advice; this one counsel I give my pupils; Apollo is asked to old men have found pleasure in cultivating the land; being supply words; you must inquire your way; Epaminondas is taught oppressed with food and wine, we, in the hours of rest, see troubled music; Cato was asked his opinion; the legions were conveyed and confused things; Varus (dele comma) is a man possessed of through Italy, Rhegium, and Sicily, and then from Sicily across the highest religion and the highest authority; that belongs to into Africa ; Miltiades reached the Chersonesus; all hurry together each person which each enjoys and uses; the Helots among the into the houses; Mithridates sent ambassadors as far as Spain to Spartans performed the office of slaves; Caesar obtained the comCneius Pompey; Caesar's soldiers go up under the mountain; hemand of all Gaul; the Numidians for the most part lived on milk sent his forces across the Hellespont; he commanded the army to and the flesh of wild beasts; the crocodile is protected by his skin, pass over the Rhine; they pass the river on a bridge; we con- which is very hard, against all bloms; elephants breathe, drink, versed from eight o'clock till erening; he will sleep till sunrise; and smell by means of their trunk; some people eat locusts; the he walked seventy miles; Ariovistus proceeded from his own teeth are vom away by use, but are not burnt with fire; in siience boundaries three daya' march; the plain of Marathon is distant he led his forces out of the camp; Miltiades restored order in from the town of the Athenians about ten miles ; the towers stand Chersonesus with the greatest equity; let us always venerate the eighty feet apart; the Arabians have their swords about four ells gods with a pure, sound and uncorrupt mind and voice; they went 'long; the soldiers raised a mound three hundred and thirty feet over the forests of the Alps not without loss; Dolabella (not Dolabroad, and eighty feet high; Atticus for thirty years wanted no bellam) provided a fleet with the intention of going to Italy; allow medicine; Appius was many years blind; Saguntum has now me to say so; the stars accomplish their courses with the greatest been three years in the power of the enemy; the name of Pytha- speed; Iphicrates had a great soul; Caesar is reported to have goras fourished for many ages; Nestor had outlived two been tall, light-complexioned, slender of.limb, with a muuth somegenerations of men; I am in my eighty-fourth year; I sleep the what large, black and brilliant eyes, and excellent health; all the winter through ; like Mercury in all things, both voice and com- Britons colour themselves with Foad (blue), and on that account plexion
they have a more frightful appearance in battle, they also have
long hair, being shaved in every part of the body except the head This question was also solved by T, F., Ayrshire; and and upper lip; Cato in all things possessed singular foresight and X PLUS Y. industry; Dionysius ordered boys of rare beauty to stand at his Answer to D. F. F. C. H.'s query, to bisect a cone by a plane table; Caesar sent to Ariovistus Valerius Procillus, a youth of very high virtue and humanity; there was between Labienus and
parallel to its base. the enemy a river difficult to be passed and with broken banks; part will be a cone similar to the whole ; and similar solids
If a cone be divided by a plane parallel to its base, the upper Chrysogonus purchased a Corinthian vase at a great price; Caesar are to each other as the cubes of their like dimensions. shows the soldiers how much a victory must cost; the
conquered Hence, in the case proposed, we have this rule. purchased peace at a great cost; peace of mind is purchasable for neither gold nor gems; of old, boys were immolated to the gods at dicular height, or base) of the whole cone ; divide this cube
Cube any given dimension (either slant height, perpenThebes, Theocritus at Syracuse; in winter boars sleep in caves; 1 by 2, and the quotient will be the cube of the like dimension no mortal is wise on all occasions ; Iphigenia, the daughter of of the upper cone, or half.
X PLUS Y. Agamemnon, was slain at Aulis; Dione was put to death inter- This question was solved also by Peter Simple, Fleet-street, fecta est) at Syracuse; Dionysius got possession of Syracuse again; and others. pleasure having dominion (dominante) all virtues of necessity lie low (are necessarily depressed), if all things were lost, yet virtue could uphold itself; calling God to witness, he promised many
CORRESPONDENCE. things; in the civil war nothing happened which I did not foretell; it is not advantageous to quit the banks of the Rhine, when now
THE GIFT OF ORATORY, hostile nations are about to make irruptions,
Though I speak with the tongues of men and of angels !!'-Cor. xiii. 1.
DEAR SIR,--As one of your pupils, allow me to ask your advice,
through the medium of the P.E., viz.: What course of study must SOLUTIONS.
I pursue in order to become an orator, beginning at the very first
rudiments of the art?
good in this manner, for I possess (I am not writing under any
knowledge! We would strongly recommend to his perusal Mason's LZT A EFD be a section of the globe, WAL the wall, and c) "Treatise on Self-Knowledge, and to his reflections the celebrated the place of the lighted taper. Let ca be drawn through the adage of antiquity, uttered by Solon, yvū be geavrov, nosce teipcentre o, and cw and cL tangents to the section A E FD at D sum, know thyself. We would also recommend the perusal of the
Join De and Do, and let De meet ca in g. Then book once printed separately from the Bible, and used as a classwl is the diameter of the shadow of the globe on the wall
, book in the parochial schools of Scotland,
namely, " The Proverbs and Dre is a central section of the segment of the globe of Solomon," who was even wiser than Solon. Lastly, we would whose surface is enlightened, and GF is its height. Let recommend him to begin at the first volume of the P.É., and study the radius Ao or fora, and the distance cax; then every word of it, in order to gain that knowledge which he himself
seems to feel the want of. We give him a year to make himself Now, by the similar triangles, COD and master of it. After that the second volume awaits him; then the DOG, we have co:00::00:06, or t- a :@::8:06; third and the fourth. Having thus got something into his
a2 whence O G equals
O G equals cranium, he may try to spout or make an oration to the public, ,
with advantage to himself and instruction to his hearers.] a2
ANSWERS TO CORRESPONDENTS. enlightened segment, whose arc is EFD, is by mensuration,
W. FOLLETT (Bognor) should study well all Mr. Cassell's publications, a x2a3 ; that is, the product of its height by the especially the Popular Educator, vols. 1, 2 and 3, and 4 as far as published:
J. LOVETT (Ashfordby) and A WELL-WISHBR (Grahamston) are right. circumference of the globe, * being 3:141592, &c., Again, we Wrong on the wolf question, and right on the taper question.-F. GOLD
T. HILL (Manchester) : Lessons on Elocution are almost ready.-BENE: have, by the 47th Prop. of Euclid, Book I., cd=V (x2 - 2ax); ING (Abingdon): Received, and under consideration.-A STUDENT or and by the similar triangles CDO and CAW, CD:D0::CA: KING's : The omission has been noticed.-SUBSCRIBER (Wigtonshire): Ifa
hall be thrown at an angle of 45" it will have the greatest range. Thestature AW; or v (22 - 2ax): a:&:AW; whence Aw=
of an individual has nothing to do with the angle of elevation. W. TAYLOR v (za 2a1) (Cowbridge): We must defer the subject
of annuities for some time. Lingua
in Latin means both the instrument of speech, the tongue, and that which and the area of the shadow on the wall = 7
is spoken by it, language. In French, this word becomes La Langue, and
derived from this, which signities only that which is spoken, language.
guage. The English word language, borrowed from the French, has the same a--2a3
; and 2c 20
8a2. meaning, namely, that which is spoken by the tongue. Hence, it appears
that tongue and language are synonymous when applied to that which is
-J. W. TAPPER (Kilkenny); Covers for the P. E, may be had at this office,
1s. common, 18. 6d. fine edition. The cheapest cases of Mathematical
Instruments are said to be sold at the "Society of Arts, ." Adelphi, London, 1 inch, we have, by taking the upper sign, c=5.56155, &c. -INDOCTUS (Lubenham): Spirit-lamps may be had at all prices, frona 18.84. inches, the distance of the taper from the wall. We hope some and upwards according to their uses; see Knight and Son's Catalogus of our correspondents will give us an explanation of the case variety of other chemicals. Here also you will get chemical apparatus of when the under sign is taken, that is, when 4 = 1.4384. &c. every description.WARIN (East Dereham): Recelved,