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II. POPE AND DRYDEN.

و ته

The grave defiance of thine elder eye,

likewise in pròse: but Pope did not borrow his prose from his The usurper | trembles / in his fastnesses.

predecessor. The style of Dryden is capricious and váried ; Oh! not yet!

that of Pope is cautious and uniform. Dryden obeys the May'st thou unbrace thy corslet, nor lay by

motions of his own mind; Pope constràins his mind to his own Thy sword ; nor yet, O'Freedom ! close thy lids!

rules of composition. Dryden is sometimes veherrent and In slumber; for thine enemy | never sleeps,

rápid; Pope is always smooth, úniform, and gentle. Dryden's And thou 'must watch and combat || till the day

page is a natural field, rising into inequalities, and diversified Of the new earth ' and heaven, But wouldst thou rest by the varied exuberance of abundant vegetation ; Pope's Awhile / from tumult ' and the frauds of men,

is a velvet làwn, shaven by the sýthe and levelled by the These old and friendly solitudes | invite

roller. Thy visit. They, while yet the forest trees |

Of génius, that power which constitutes a poet; that quality Were young' upon the unviolated earth,

without which judgment is cold, and knowledge is inert; that And yet the moss-stains ! on the rock / were new,

energy which collècts, combines, àmplifies, and animates ; the Beheld thy glorious childhood, and rejoiced.-Bryant. superiority must, with some hesitation, be allowed to Drýden.

It is not to be inferred, that of this poetical vigour Pope had
only a little, because Dryden had more; for every other writer

since Milton must give place to Pòpe; and even of Drýden [This piece is marked in application of the rules of Inflection.] it must be said, that if he has brighter paragraphs, he has not Pope professed to have learned his poetry from Drýden, better poems. Dryden's performances were always bàsty, whom, whenever an opportunity was presented, he praised either excited by some external occasion, or extorted by through his whole life with unvaried liberàlity; and, perhaps, domestic necessity; he composed without consideration, and his character may receive some illustration, if he be compared published without correction. What his mind could supply

at call, or gather in one excursion, was all that he sought, and with his màster,

Integrity of understànding, and nicety of discérnment, were all that he gave. The dilatory caution of Pope enabled him not allotted in a less proportion to Drýden than to Pòpe. The to condénse his sentiments, to múltiply his images, and to rectitude of Dryden's mind was sufficiently shown by the accùmulate all that study might prodúce, or chance might dismission of his poetical préjudices, and the rejection of supply: If the flights of Dryden, therefore, are hígher, Pope unnatural thoughts and rugged numbers. But Dryden never continues longer on the wing. If of Dryden's fire the blaze is desired to apply

all the judgment that he had. He wrote, and brighter, of Pope's the heat is more régular and constant. professed to write, merely for the people; and when he pleased below it. Dryden is read with frèquent astonishment, and

Dryden often surpasses expectation, and Pópe never falls others, he contented himself. He spent no time in struggles to rouse latent powers; he never attempted to make that better Pope with perpétual delight.-Johnson. which was already good, nor often to ménd what he must have known to be faulty. He wrote, as he tells us, with very litile consideration : when occasion or necessity called upon him, he poured out what the present moment happened to supplý, KEY TO THE LESSONS IN GREEK. and, when once it had passed the préss, ejected it from his mind; for, when he had no pecuniary interest, he had no further

By John R. BEARD, D.D. solicitude. Pope was not content to satisfy; he desired to excel, and

No. VIII, Vol. IV. p. 10.-Greek-EXOLISH. therefore always endeavoured to do his bèst; he did not court The ravens croak. Avoid flatterers. Avoid the deceiver. the cándour, but dared the judgment of his reader, and, expect. Men are delighted with harp and dance and song. Horses are ing no indulgence from others, he showed none to himself

. driven with whips. The harps delight the minds of men. A He examined lines and words with minute and punctilious grasshopper is friendly to a grasshopper, and an ant to an observation, and retouched every part with indefatigable díli- ant. (The) shepherds sing to the pipe (pipes), Among the gence, till he had left nothing to be forgiven.

Athenians (there) were contests of quails and cocks. The For this reason he kept his pieces very long in his hands, shepherds drive the flocks of (the) goats to the meadows. while he considered and reconsidered them. The only poems (The) life of ants and quails is laborious. Many have a fair which can be supposed to have been written with such regard countenance, but a foul (bad) voice. to the times as might hasten their publication, were the two satires of Thirty-eight : of which Dodsley told me, that they

ENGLISH-GREEK. were brought to him by the author, that they might be fairly còpied. Every line," said he, “was then written twice

Φενακα φευγω. Οι κορακες κρωζουσιν. Τη φορμιγγι τερπ). över ; I gave him a clean trànscript, which he sent some Opxnopol rous avpw tovs reprovor. Tovs ir tous elavovat time afterwards to me for the préss, with every line written twice over a second time."

μαστιξι. Οι των ανθρωπων θυμοι τη φορμιγγι αγονται. Οι His declaration, that his care for his works ceased at their κορακες κρωζουσιν. Αί συριγγες τους ποιμενας τερπουσι. Αι publication, was not strictly trùe. His parental attention alyes apos Tov letuwva ayovrat. 'O Toruny spos taç oupeyyas never abandoned them; what he found amiss in the first | αδει. Η θυγατηρ ωπα μεν αγαθην εχει, κακην δε οπα, edition, he silently corrected in those that followed. He appears to have revised the Niad, and freed it from some of its imperfections; and the Essay on Criticism received many im

GREEK-ENGLISU. provements, after its first appearance. It will seldom be found (The) birds sing. Favour begets favour, strife (begets

) that he altered without adding clèarness, élegance, or vigour. strife. We account youth happy. Need begets strifes. (The Pope had, perhaps, the judgment of Dryden ; but Dryden cer rich often hide their (the) baseness by (their) riches... tainly wanted the diligence of Pòpe.

beautiful boy, love thy good brother and thy fair sister. The In acquired knowledge, the superiority must be allowed to love of money is (the) mother of all baseness. The poor are Dryden, whose education was more scholastic, and who, before often happy. (The) wisdom excites in the minds of men he became an author, had been allowed more time for stúdy, wonderful loves (love) of (the) beautiful things. (The) death with better means of informàtion. His mind has a larger frees (the) men from cares. Friendship arises through (from) ránge, and he collects his images and illustrations from a more similarity. Wine awakes laughter. In (the) night counsel extensive circumference of science. Dryden knew more of wisdom) arises to the wise. The wise punish (the) baseness, man in his general náture, and Pope in his local mànners. (The) men are often delighted by slight hopes, "The notions of Dryden were formed by comprehensive specuJation, and those of Pope by minute attention. There is more dignity in the knowledge of Drýden, and more cértainty in

P. 11.--ENGLISK-GREEK. that of Pope.

Οι ορνιθες αδoυσιν. Χαρις χαριτι τικτεται, ερις εριδι. Τη Ρουν να μος the sole praise of either: for both excelled σοφια γιγνεται εν τοις των ανθωπων θυμους θαυμαστος των

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καλων ερως. Τη των ορνιθων φθη τερπομαι. Αί των ορνιθων

P. 41.--GREEK-ENGLISH. φδαι τους ποιμενας τερπουσι. Τοις ορνιθοις τερπομεθα. Οι ανθρωποι τους βασιλευσι έπονται. Οι ανθρωποι τω βασιλει The tragedies of Sophocles are beautiful. We admire

Socrates for his (the) wisdom. To Socrates (there) are many

pupils (Socrates has many scholars). India bears (produces) GREEK-ENGLISH.

many reeds (rushes) along both the rivers and the marshy In hard circumstances few companions are faithful. We do places.

Always speak truth (the true things), o boy. not exchange the wealth of virtue for property. (The) sup- Anaxagoras, the sophist

, was (a) teacher of Pericles. ©

the unfortunate. . pliants touch the knees. (The) death is a separation of the Hercules, afford safety (salvation) to soul and of the body. (The) wealth affords inen various aids.

Epaminondas was of an unknown father (the father of Do not listen to the words of bad men. Be not enslaved, 0 Epaminondas was not known). Pity the unfortunate man. boy, to the service of the body. The Greeks offer to the young men (reaviai), desire (strive after) true words. The Nymphs goblets of milk. Accustom and exercise thy (the) intemperate serve a disgraceful servitude. Have not inter. body with labours and sweat. (The) chatterers wear away

course with (to) an intemperate man. our (the) ears by repetition. Accustom thy (the) soul, O boy, to useful things. (The) wicked speeches touch not the ears.

ENGLISH-GREEκ. With the ears we hear. Do not hate a friend on account of a small sin. O boy, taste the milk. The soldiers bear spears.

Το Σωκρατει ην θαυμαστη σοφια, Ελεαιρε τους ατυχείς.

Ελεαιρομεν τους ατυχείς (ανθρωπους). Πολλοι νεανιαι
ENGLISH-GREEK.

Σωκρατους μαθηται ησαν. Πολλη σοφια τη Σωκρατει ην.
Ο νεανιαι, τα σωματα γυμναζετε συν πονοις και ιδρωτι. Των | Την του Σωκράτους σοφιαν θαυμαζουσι. Ο ακρατης αισχραν

δουλειαν δουλευει. καλων πραγματων ορεγομεθα. Τα χρυσω τερπονται πολλοι.

Τας του Σοφοκλους τραγωδιας καλας

Τον των ακρατών
Εκ του καλου πραγματος γιγνεται δοξη. Τους των σοφων θαυμαζομεν. Οι αληθείς λογοι πιστευονται.
λογους θαυμαζομεν. Τα των καλων καλα πραγματα θαυμαζεται. | βιον ελεαιρω. Μη τους ακρασι ομιλιαν εχε.
Οι στρατιωται τους δορασι μαχονται. Τον της αρετης πλουτον

No. X. Vol. IV. ου διαμειβομαι τους βασιλευσι. Μη τοις των κακων μυθοις

p. 55.-GREEK-ENGLISH. πειθου.

Homer sings of many heroes. We admire the valour of

heroes. Slaves lead a painful life. The garden of the uncle No. IX. Vol. IV. p. 40.-GREEK-ENGLISH.. is beautiful. O boy, desire modesty. Modesty attends on The same mind is not to all men (all men have not the same

good men. We admire Lysias for his persuasiveness and mind). With the teeth we masticate the food. Dolphins are

grace. Reverence belongs to modesty. Do not look at the friendly to men. It is (the part) of a good man to bear all face of the Gorgon.. O Echo, thou ofien deceivest men. All evils bravely. Many parts (places) of Libya abound in ivory. to have inodesty, Clio and Erato are Musee.

desire a good condition. It becomes a boy and a young man

Historianis
fight against the gods (the giants once fought against the propioypapor

) honour Clio, but the lyric poets (honour)
gods). We rejoice in the beams of the sun. It is the office
(work) of the nostrils (nose) to smell.

Ρ. 56.-ENGLISH-GREE κ.

"Ομηρος αδει τον ήρωα τον Αχιλλεα. Ο Αχιλλευς ήρως (or
ENGLISH-GREEK.

ο Αχιλλευς ο ήρως) αδεται υπο του Ομηρου. “Η του ήρωος
Εχομεν ελεφαντα. “ο ελεφας εν ταις της Λιβυης χωραις αρετη θαυμαστη εστιν. Την των ηρωων αρετην θαυμαζομεν.
γιγνεται. Οι αδελφοι και οι αδελφοι ταις του ήλιου ακτισι | Τοις δμωσι λυπηρος εστι βιος. Το πατρων εστι καλος κηπος.
τέρπονται. Καλη εστι ή αδελφη. Ελεφαντα τον καλον θαυ- Το ενεστοι αγαλλονται παντες. Ω νεανια, μετ' αιδούς θαυμαζε
μαζομεν. Πολλοι εν τη Αιβυη εισι ελεφαντες. To των οδοντων τα των αγαθων εργα. Τη ηχοι πολλακις ψευδομεθα.
έργον τα βρωματα λεαινειν εστιν. Το θειον σεβεσθαι εστι
παντος ανθρωπου. Τοις θεοις ην ποτε πολεμος προς τους

GREEK-ENGLISHT.
γίγαντας,

The gods send prodigies to men. Death is a cure for the

evils of age. GREEK-ENGLISH.

Rewards (gifts of honour) impel soldiers to

bravery. Milk and flesh come (are supplied by) from goats (The) kings have (take) care of the citizens. The flock and sheep for nutriment. The soldiers give signs by horns follows the shepherd. Hector is slain by Achilles. The and trumpets. We taste (eat) various kinds of flesh. A good priests sacrifice oxeu to the gods. Cyrus was a son of good condition of the body (constitution) in youth is a safe (fair) parents

. The ungrateful dishonour their (the) parents. O foundation for (of) old age. Stags have horns. Life in old
boy, obey thy (the) parents.

Telemachus was a son of age is peevish.
Ulysses. Wish before all to have (hold) thy parents in
honour (honours). The chatterings of old women wear away

ENGLISH-GREEK.
the ears

. O king, you rule well. (The) old women Τα τερα τους ανθρωπους πεμπεται υπο των θεων. Οι στρατιω.
talkative. The shepherds lead the herd of oxen to pasture: ται τοις κερασι και ταις σαλπιγξι τερπονται. Γαλακτος και
Homer likens the eyes of Hera (Juno) to the (those) of (the)
the king of the Persians, for both valour and wisdom.
BBIPatroclus was a son of Achilles. We admire Cyrus, | κρεως γευομεθα. Τα του γηρως κακα απολυει ο θανατος. Γερα

τοις στρατιωταις πεμπει ο βασιλευς. Τα γερα τους στρατιωτας
προτρέπει.

Oi

στρατιωται τοις χερασι προτρέπονται.
ENGLISH-GREEK.

GREEK-ENGLISH,
Αι αγελαι τη νομεί έπονται. ο βασιλευς του πολιτου
επιμελειαν εχει. Τα ωτα τοις των γραών ληροις τειρονται. The earth blooms with flowers. Do not keep from cold and
T'paüc coti foludoyos (in Vocab. read &, , instead of on). 'O heat. We judge beauty not by length of time but excellence.
νόμευς την βοών αγελης προς την πολιν αγει.

Every elevation in mortal kind is unsafe. Do not speak false-
Βοές τοις

hoods. Keep from wicked gains. Wicked gains always bring
θεοις θεονται υπο των ιερεων. θυετε, ω ιερείς, βούν τους θεοις. punishment. Brass 18 mirror of the countenance, and wine
δε παιδες τους γονεας στεργουσι. Οι γονείς στεργονται υπο | Phe mind. Men desire glory. Men Tejongiorg. The
των παιδων. Του αγαθου νομεως εστι την των αγελων επιμε- Brave long for noble deeds. We admire the achievements

(brave) men.

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Τους Αθηναιοις πολλαι ησαν νήες. Το Διϊ ησαν πολλοι
ENGLISH-GREEK.
Οι ιχθυες εκ του ύδατος ανακυπτουσιν,

ο κυβερνητης την Των πονηρων κερδων απεχου. Οι αγαθοι ανδρες των πονηρών ναυν ιθυνει. Η ναυς ιθυνεται υπο του κυβερνητου. Σεβεσθε κερδων απεχονται,

οι αγαθοι ανδρες των κλεων . ορέγονται. τον Δια και τον Απολλω.
Ω νεανια, μη ψυχους και θαλπους αλλα πονηρων κερδων απεχου.
Το ψευδει έπεται ζημια. Τους Ελληνας επι τοις κλεεσι θαυ-

GREEK-ENGLISH.
μαζομεν. Των πονηρων κερδων απεχομεθα, Τοις κλεεσι χαιρoυσιν

To drink much wine is an evil. Kings have large revenues. οι στρατιωται,

In Egypt there was much want of corn. The sea is great. We GnEEK-ENGLISH.

call a great suffering an Iliad of evils. Cræsus had much

wcalth. Often from short pleasure (there) arises great grief. The fish emerge from the river. Hunters chase wild boars. We willingly yield to soft words. The great gifts of fortune All (the) dead are equal, but God rules souls (Yuxwv). The have (are aitended by) fear. The manners of many men are vine bears bunches of grapes. The earth bears ears of corn mild, Labour greatly assists virtue. Children love their mild and bunches of grapes. To the mice (there) was once a fight fathers and their mild mothers. Have intercourse with mild against the frogs (the mice once fought against the frogs). (gentle) men. Mice are caught by traps,

The women are gentle. Many call Alexandet,
The Syrians worship fish

the king of the Macedonians, great.
divinities. We catch fish by hooks.
ENGLISH-GREEK. ,

ENGLISH-GREEK.
Τους ιχθυας (ιχθύς) αχρευουσιν αγκιστροις. Οι ιχθυες αγκισ-

Πολλου οινου απέχου.

Οι κακοι πολλώ oινω τερπονται, τροις ενεδρευονται. ο θηρευτης ενεδρευσει τας αγριας σύς. Πολυς οινος τους ανθρωπους βλαπτει. Τοις βασιλευσι μεγαλαι Οι βοτρυες και οι σταχυες καλοι εισιν. Ο αμπελος τους βοτρυας

εισι προςοδοι. Η της βασιλείας προςοδος μεγαλη εστιν. Αιγυπτω (βοτρύς) φέρει. Τους βατραχοις ην ποτε μαχη προς τους μιας σιτος εστι πολυς. Πολλοις πλουτος μεν εστι πολυς, μικρος δε (μυς). Τους νεκύς προς βλεπομεν. Πολλους αμπελους φερει η νούς. Τον πραεων εθεων ορέγου. Τα των γυναικων εθη πραια γη. ο θεος βασιλευει των ιχθυων και των βατραχων.

Τοις πρασι εθεσι εστι κοσμος. Αλεξανδρος, ο Μακεδων

βασιλευς, πολλακις απαγορεύεται μεγας. No. XI, Vol. IV.

p. 71.-GREEK-ENGLISH.
Wantonness begets insult. In mcat and drink many are
friends, but in a serious undertaking few, Wealth frees men

LESSONS IN GEOMETRY.--No. XXXII.
from deficiency and need. Follow nature. Bodily desires
(desires springing from the body) occasion wars and insur-

LECTURES ON EUCLID.
rections and battles. In states the rulers are guardians of the
laws. O citizens, abstain from insurrections. O men, long

BOOK I.
for honourable actions. The natures of mortals are different.

(Continued from page 154.)
From insult many evils arise. The gifts of a bad man have
(bring) no profit. Glory and wealth without understanding

PROPOSITION XXXVI.
are not safe possessions. The fruits of figs are sweet. Only
the possessions of Virtuc are firm. Many towns have walls. Parallelograms upon equal bases and between the saine parolltls, are
The towers of the town are firm. The towers are an orna.

equal to one another. ment to the town. .

In fig. 36, let A c and eG be parallelograms upon equal bases uc

and yo, and between the same parallels A and BG. The pural. ENGLISH-GREEK,

lelogram A c is equal to the parallelogram Eo.
Join BE and C H. Because HQ

Fig. 36.
Ο πλουτος την ενδειαν λυει. Εν πάσει και βρωσει έταιρους | is equal to FG, (Hyp.) and FG

Η
εχομεν, αλλ' ουκ εν δυςτυχια. Εν τη πολει εστι ο βασιλευς το EH (Ι. 31), therefore, Bc is
φυλαξ των νόμων. ΙΙειθου, ω νεανια, ταις αρχαις. Ω παι, equal to EH (Α.Σ. 1). But HO
ορεγου των καλων πραξεων. Μονη και της αρετης κτησις εστι

and E are parallel, and joined

towards the same parts by the βεβαια. Τη πολει πολλοι εισι πυργοι. Οι καλοι νομοι δοξαν

straight lines B E and ch, And τη πολει παρεχουσιν. Τη φυσει επου. Οι στρατιωται περι straight lines which join the extremities of eq:al and parallel της του αστέος σωτηρίας μάχονται, ψευγε, ω πολιτα, την straight lines towards the same parts, are (I 33) iliensives

equal and parallel. Therefore the straight lines B E and cu P. 72.-GREEK-ENGLISH,

are both equal and parallel, Wherefore bu is a parallelo

grain (Def. 36). Because the parallelograms AC and B H, are Women rejoice in ornament. The Greeks worship Zeus upon the same base B C, and between the same parallels o c and (Jupiter), and Poseidon (Neptune), and Apollo, and other A u, the parallelogram A c is equal (I. 35) to the parallelogram . Modesty becomes women. Dogs guard the house.

And because the parallelograms Gx and B are upon

the same base E , and between the same parallels G B and we The steersman directs the ship. Drops of water hollow out the parallelogranı'eg is equal to the parallelogram Bu, Theten Treckee it is the woman's office to keep the house: It is the fore the parallelogram A c is equal

(Ar. 1) to the paralleloe, D. part of a good wife to preserve the home. The dice of Zeus always fall well. Dogs occasion men pleasure and profit. 'The G. Therefore, parallelograms upon equal bases, etc. Q. E. D. testimonies of witnesses are often incredible.

Looms, not
public meetings, are the part of wives. Carry the key of the

EXERCISE 1. TO PROPOSITION XXXVI.
Chest, O boy. o Zeus, receive the request of the unfortunate. If the base of a parallelogram be equal to half the sum of the two
Castor and llux were saviours of ships. Silence gives orna-
ment to every woman. The Æthiopians have black hair. O parallel sides of a trapezoid, between the same parallels, the paral-

lelogram is equal to the trapezoid.
woman, preserve thy home. We comb the hair with a comb.
Eacus keeps the keys of Hades.

In fig o, let A B C D be a trapezoid, of which the two sides

D and Bc are parallel; and let Á BLK be a parallelogram ENGLISH-GREEK,

haring its base el equal to half the sum of the sides a d and Κοσμος πρεπει τη γυναικι. Κοσμος πρεπει ταις γυναιξι. ; and let it be between the same parallels AD and Ac; then

the parallelogram a bl k is equal to the trapezoid å C D. Των γυναικων εστι την οικίανφυλάττειν. Τας της οικιας

Produce Bo to E, making ce equal to A D (1.2); completo κλείς κομιζουσι. Αί του οικου κλείς τη μητρι προσκομίζονται. the parallelogram ABEF, and through c and D draw od and

TIIEOREM.

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LESSONS IN GEOMETRY.

183 Dparallel to AB; then BG, GH, and u F, are parallelograms Produce 1 A to meet BC and Ki in Ñ and m; kB to meet (Const.).

GF in x; and c to meet Ed in o. Then each of the figures Because af is equal to bė, and ce to A D (Const.), there- 1 B and u c is a parallelogram (Const. and Def. 36.) fore Dr is equal to BC (Az. 3), and the parallelogram Bg to Because the parall grams A G and B are upon the same the parallelogram u (I. 36)." Because À Q is equal to BC base, A B, and between the same parallels A B and Q , there

fore they are equal (I. 35). For the same reason, the paralleloFig. 0.

grams A E and i c are equal. But u A is equal to x B (I. 34), and Bk is equal to H A (Const.); therefore x B is equal to BK (Ax. 1). Now, because the parallelograms # B and BM are upon equal bases x B and Bk, and between the same parallels HM and k x, therefore they are equal (I. 36). But the parallelograms I B and AG were proved equal; therefore the parallelogram A G is equal to the parallelogram BM (Ax. 1). In ihe same manner it may be shown that the parallelogram Am is equal to the parallelogram cm. Wherefore the parallelo

grams A G and a E are equal to the parallelograms BM and on (1. 34), and therefore to Dr (Ax. 1), it is also equal to 1 E

(Ax. 2). But the parallelograms Bn and om are together 1. 34, and Ax. 1); but Ad is equal io 0 B (Const.); therefore equal to the parallelogram B1; therefore the parallelograma a p is equal to c#, and the triangle GCD to the triangle cu ac and A E are together equal to the parallelogram BL (Ax. 1). (1. 34); now, adding the triangle GCD to the parallelogram Wherefore, the parallelograms described on any, etc. Q.E.D. BG, and the triangle Dou to the parallelogram u F, the tra

PROPOSITION XXXVII. THEOREM. pezoid ABCD is equal to the trapezoid DC E F (Ax. 2), and each is half of the parallelogram A B E F; but the parallelogram Triangles upon the same base, and between the same parallels, are, A BLX is equal to the parallelogram K LE F (I. 36) and each

equal to one another. is half of of the parallelogram A BEF; therefore, thé parallelogram A Blk is equal to the trapezoid a BCD (Ax. 1)." Where- base Bc, and between the same parallels A V and Bo.

In fig. 37, let the triangles A B C and D Bo be upon the same

The fore, if the base of a parallelogram, etc. * Q. E. D.

triangle A B C is equal to the triangle C. Fig. 37. EXERCISE II. TO PROPOSITION XXXVI.

Produce a d both ways to the

points E and F. Through #draw Demonstrate the theorem of Pappus : the parallelograms described | B e parallel to ca (I." 31), and on any two sides of a triangle, are together equal to the parallelo- through c draw of parallel to gram described on the base, having its side equal and parallel to

Then each of the figures the straight line drawn from the point of intersection of the ec and BF, is a parallelogram exterior sides of the former, to the vertex of the triangle.

(Def. 36). In fig. p, let A B C be any triangle; and upon any two of its The parallelograms E c and t r are equal (1. 36), because sides A B and ac, let the parallelograms A E and AG be they are upon the same base B o, and between the sune paral

lels B c and EF. But the triangle A B e is half of the paralleloFig. p.

gram Ec (I 34), because the diagonal A B bisects it ; also, the triangle DBC is half of the parallelogram BF, because the diagonal Dc bisects it; and the halves of equal things are equal (Ax. 7). Therefore the triangle A Bo is equal to the triangla . D BC. Wherefore, triangles, etc. Q. E. D.

EXERCISE TO PROPOSITION XXXVII. To describe a triangle equal to any given roctilineal figure. N.B. In solving this problem, the learner may begin with a parallelogram; then proceed to a trapezium, a pentagon, a hexagon, and so on. He will then ultiinately find that, if a figure had a thousand sides, he could reduce it, by degrees, to an equivalent triangle.

In fig. 9, let ABCDEFG be any rectilineal figure; it is required to describe a triangle equal to it.

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described. Also, let their exterior sides E D and G f be produced till they meet in the point H (Ax. 12), and join A. Through the point B draw BK equal and parallel to H A (1, 31 | Take any side A B and produce it indefinitely in the direction and 3). Complete the parallelogram B L:

Then the parallelo- AI; join AF, and through g, draw q u parallel to AP (I. 31), Brams AG and a g together are equal to the parallelogram juin ru. Then the rectilineal figure u BCD Er is equal to the

rectilineal figure A B C D E F G.

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Best of red by Waris (East Dereha 1); E. J. Brenner (Carlisle); J. 1.

* Solved by WARIN (East Dereham); J. U. EastwoOD (Middleton); M, J. BREMNER (Carlisle); and others.

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Because the triangles AGF and an F are upon the same bisects it. And the triangle DEF is the half of the parallelo.
base A F, and between the same parallels GH and A F, there- gram e r, because the diagonal B r bisects it. But the halves of
fore the triangle Ak is equal to the triangle A G P (1. 37). equal things are equal (4.c. 7). Therefore the triangle A B C is
To these equals, add the rectilineal figure a BCDEF, and the equal to the triangle d'EF. Wherefore, triangles.upon equal
whole H BCDEF is equal to the whole ABCDEF G (Ar. 2); bases, etc. Q. E D.
that is, the former figure is equal to the latter, but it has one Corollary 1.-The straight line drawn from any angle of a
side fewer than it. Again, join u B, and through F, draw Fitriangle bisecting the opposite side, bisects the triangle.
parallel to KE (I. 31); join E l. Then the rectilineal figure Corollary 2.--If two triangles have two sides of the one equal
I B C D E is equal to the rectilineal figure I B C D E F.

to two sides of the other, each to each, and the angle contained
Because the triangles : fe and u r e are upon the same base by the two sides of the one, the supplement of the angle con.
H E, and between the same parallels FI and H E, therefore the tained by the two sides of the other, these triangles are equal.
triangle use is equal to the triangle u Fe (I. 37). To these
equals, add the rectilineal figure u Bode, and the whole I B C

EXERCISE I. TO PROPOSITION XXXVIII. DE is equal to the whole H B CD EF; that is, the former figure To bisect a triangle by drawing a straight line through any point in is equal to the larter, but it has one side fewer than it. Now the rectilineal figure HBCDEP was proved equal to the recti.

one of its sides. lineal figure ABCDEFG; therefore, the rectilineal figure IBC In fig. r, let A B C be a triangle, and D a point in one of its D) E is equal to the rectilineal figure ABCDEFG (Ax. 1); and sides BC; it is required to bisect the triangle A B C, by drawthe former has two sides fewer than the latter.

ing a straight line through D. In like manner, by joining id, and drawing through e, a parallel to ID, meeting A B produced as before, a recuilincal

Fig. 5. figure may be found equal to the rectilineal figure ABCDEFG, and having three sides fewer than the latter; and so on, may the operation be continued, until a figure having only three sides, that is, a triangle, may be found, which shall be equal to the given rectilineal ABCDEFG. Q. E. F. •

Scholium.-This proposition is of extensive use in the practice of land-surveying; for whatever may be the form of the field to be measured, if the lengths of all the sides be correctly ascertained, as well as their relative positions to one another, that is, the angles which every two adjacent sides form between them; then, by laying down a plan of the field

E D upon paper, so that in the plan, the length of every side shall

If the point D were in the middle of the base pc, as at F, have to the length of every other the same relative proportions then by joining EA, the thing required is done (I. 38, Cor. 2); which they have in the field, the angles being laid down care that is, the triangle A B I is equal to the triangle a x C.

But fully with the protractor, and the sides measured to nicety by if the point o be not in the middle of the base, join D a; bisect means of the plane scale, the figure of the field may be reduced vc in E (1. 9), join ea, to that of a triangle, as shown in the proposition. When this parallel to AD (*. 31); and join pr. Thin de bisects the

and through the point e, draw ef is done, the area of the triangle can be found by measuring its triangle ABC. base and perpendicular from the same plane scale used in lay.

Because the two triangles A FE and D F E are upon the same ing down the sides, and applying the following Rule : Multiply base pe, and between the same parallels A D and re, the the length of the base in numbers

, by the length of the per- triangle are is equal to the triangle d r e 11. 37). To these pendicular in numbers, and divide the duct by 2; the equals, add the triangle A B e, and the whole triangle a B e is quotient will be the area of the triangle in square units, these equal to the whole triangle FRD being the squares of the units employed in measuring the is half of the triangle avc (I. 38, Cor. 1); therefore the

But, the triangle ABE lengths of the sides,

triangle FB D is also half of the triangle A B C (Ax. I); and the Definitions. The base of a triangle is any one of its sides, to triangle ABC is bisected by the straight line DF. Q. E. F.* which a straight line is drawn at right angles, from the vertex of the angle opposite to that side. This angle, in reference to

EXERCISE II. TO PROPOSITION XXXVIII. the base, is called the vertical angle ; and the straight line drawn at right angles to the base is called the perpendicular of the tri. The straight lines drawn from the opposile angles of a parallelo. angle. When one of the angles at the base of the triangle is gram to any point in a diagonal, divide it into equal triangles on obtuse, the perpendicular will fall without the triangle, upon the

opposite sides of the segments of the diagonal. base produced; otherwisu (except in the case of the right-angled In fig. s, let AB D C be a parallelogram, bc its diagonal, and triangle having its right angle at the base), the perpendicular E a point in it; join E a and ed. Then the triangle ACE will fall within the triangle, upon the base itself.

is equal to the triangle DC E, and the triangle A, B e to the

triangle DBE. PROPOSITION XXXVIII.

Fig. 8. Triangles upon equal bases, and between the same parallels, are equal to one another,

Fig. 38. In fig.38,let the triangles ABC and D E F, be upon equal bases Bc and E F, and between the same parallels b F and a D. The triangle Alc is equal to the triangle DEP.

.D Produce ad both ways to the points o and 1. Through B draw B G parallel to CA (I. 31),

Draw the diagonal A D, and let it meet the diagonal Bc in F.

Then each of the and through y draw fi parallel to ED.

Because AF is equal to PD (Exercise 2, Prop. xxxiv.) there. figures a cand e , is a parallelogram.

The parallelograms ac and Eu are equal to one another triangle afe to the triangle DPE (1, 38, Cor. 1); therefore the (1. 36), because they are upon equal bases Bc and EP, and whole triangle

Ace is equal to the whole triangle D C B (41; 2; half

of the parallelogram a C (1. 34), because the diagonal a B therefore the remaining triangle's be is equal to the remain• solved by J. H. EASTWOOD (Middleton); 'Q. PRINGLE (Glasgow); • Solved by J. 1. EASTWOOD (Middleton); J. JENKINS (Pembroke WARIN (East Dereham); and others.

Dock); WARIN (East Dereham); L. J. Brenner (Carlisle); and others

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