Past Def. Je soutins. And so on for other functions.

76

The above results can be verified by constructing a right- | being examined, but by dropping the perpendicular DG on to angle B (.. of 45° angled triangle, as in Fig. 3, with angle A = CA produced. each), where side a side b, and consequently tan. A = tan.

sin. DAG;

Sin. CAD is therefore

DG
AD

But

DG A D

is also

... sin. CAD = sin. D A G.

=

But since DAG CAB, and triangle A D G evidently tri angle A B C,

D G
A D

BC

=

;

AB

.. sin. C A D (an angle in second quadrant) sin. C A B (an angle in first quadrant).

DAG, CAB is the supplement of CAD;

Also A C =

But since CAB
therefore, generally,

√3
2

= = 0.86602,

As we know the ratios of 60°, we of course know the ratios of 30°, its complement.

VI. Supplemental Angles.-The supplement of an angle (less than two right angles) is the angle wanting to complete it to two right angles, or 180°. Thus the supplement of 30° = 180° 180° 175° 5°, and so - 30° 150°; supplement of 175° In sexagesimal measure, supplement of A = 180°- A. In circular measure, supplement of A = - - A.

VII. Trigonometrical Conception of an Angle-Functions of Angles exceeding 90°-Use of the Signs + and The trigonometrical idea of an angle being a quantity to bo calculated rather than, as in Geometry, a shape to be drawn, we find ourselves quite untrammelled by compass and pencil, and may therefore deal not only with angles exceeding 180°-which a geometer could only describe as angles turned inside out-but with angles of any number of degrees whatever, even exceeding 360°. We shall, however, find that the functions of every angle exceeding 90° are the functions of some angle below 90°, so that practically we have no need to calculate ratios for angles out of the first quadrant. Indeed, it is obvious that Fig. 2 cannot possibly be constructed for any angle not less than a right angle.

It is a conventional arrangement in this science that all positive angles (for definition of negative angles see Sect. IX.) are supposed to start from above a kind of horizontal base-line, which forms one side of the angle, the other being supposed free to revolve, in the direction of the arrows in Fig. 6, through an arc of any number of degrees, whether greater than an entire revolution or not. In Fig. 6 let A c be the "baseline" of the angle C A B (less than 90°, or "in the first quadrant"). Produce CA to G. Now let A B, the "free side," revolve to the position A D, making DAG = CA B, and AD = A B. Then CAD is more than 90° and less than 180°, or is "in the second quadNow there is clearly no way of constructing, for the angle CAD, the right-angled triangle which played so important a part in Fig. 2, in determining the ratios of the angle then

rant."

270

or, sin. (180° — A) = sin. A.S

(28)

From this it appears that the same ratio applies to more than one angle. A remedy for the confusion which might thus arise is found in the following arbitrary use of the signs + and

A perpendicular drawn upward from a given base is considered opposite in sign from a perpendicular drawn downward; and a line drawn to the right of a given point of opposite sign to a line drawn towards the left from the same point. Conventionally, lines measured to the right of a given point are regarded as +, therefore corresponding lines to the left are - ; and lines drawn upward are +, and downward

By this arrangement it appears that, in Fig. 6, BC, DG, and A C are positive, while C F, G E, and AG are negative quantities. As no negative quantities enter into the ratios of any angle in the first quadrant, its functions are all + or positive.

And the cosine of an angle in second quadrant is negative.
Let A D now revolve to the position A E, giving us the trigono-
metrical angle CAE, in the third quadrant-i.e., of more than
180°, and less than 270°. (This must not be mistaken for the
geometrical angle lying below the lines CA, AE, but is the
trigonometrical angle subtended by the arc CDE.) Making
EAG CAB, and noting that the lines AG and EG are both
negative, but equal in magnitude to a c and B C respectively, it
appears that

and the sine and cosine of an angle in the third quadrant are both negative.

If A E revolve further to A F in the fourth quadrant, making a (trigonometrical) angle C A F of more than 270°, but less than 360°, then, making CAF=CA B, and noting that E c is negative and AC positive, we find by precisely similar reasoning that sin. (360° A) = = sin. A; . (31) cos. (360° A) = cos. A. Thus the sine of an angle in the fourth quadrant is negative, and the cosine positive.

When, at 180°, A B (or A D) again coincides with AG, D G dis- line A C in Fig. 6, by the movement of its free side in a direction appears, and

If AB (represented now by A E) revolve further to 270°, EG quadrant of a negative angle the sine differs in sign from the coincides with A E, and A G disappears.

1. Such as append the person-endings immediately to the -és, do-01 = dos. In the second aorist of iarnμ, however, stem-vowels.

Of this second class only the verb oße-vvõ-μi (ΣBE), I extinguish, and imperative of the present and imperfect they retain rai and

2. THE SECOND CLASS OF THE VERBS IN μ. The tense-formation of the second class of the verbs in has no difficulty. After cutting off the termination vvu and vui, In the optative second aorist of the verbs iσrnμi, Tienui, add the tense-forms to the stem. The verbs in o which lengthen

this o into w in the present, retain the w in all the tenses, as στρωννύμι, ῥω-ννύμι, ἑω-ννύμι, χω-ννϋ-μι; future στρωσω, ¿w-ow, św-ow, xw-ow, and so on.

But the verbs whose stem ends in a liquid take for the formation of some tenses a theme ending in a vowel, as oμ-vu-μi, aorist wu-o-oa, from the theme OMON. The second aorist and second future passive are found in only a few verbs, as Čevy-vuμι, aor. 2 pass. εζύγην, fut. 2 pass. ζυγησομαι.

REMARKS ON THE MODELS.

In the dual and plural of the indicative, and in the other moods and the participle, for the first aorist active, the second aorist active is used.

Instead of the forms e-on-a-juny, e-dw-ka-uny, first aorist indicative middle, the Attic forms are used.

The middle optative forms of the imperfect and second aorist of the verbs in e, namely, or, as Tilouny, bouny, are preferred to those in et, ας τίθειμην, θείμην.

The perfect and pluperfect, έστηκα, ἑστηκειν (but not είστη KELY), form the dual and the plural immediately from the stem, as perfect, ἑ-στα-τον, ἑ-στά-μεν, ἑστᾶτε, ἑστᾶσι(ν); pluperfect, ἑστατον, ἑστάτην, ἑστᾶ-μεν, ἑστᾶτε, ἑστᾶσαν; instead of ἑστηκεναι, ἑστᾶναι is usually employed. The participle runs ¿σTws, wσa, ws, gen. ŵros, wons, as well as eστηkws, via, os, gen. OTOS, Vias. With éσTaTov compare тeтλаμеv (TAA), and тevaμev, τεθνατε, τεθνάσι(ν), inf. τεθνάναι, from τεθνηκα, θνησκω (ΕΝΑ).

KEY TO EXERCISES IN LESSONS IN GREEK.-XLIII.
EXERCISE 129.-GREEK-ENGLISH.

1. The soldiers will defend themselves against the enemy. 2. Do
not be vexed when blamed for your faults. 3. The shepherd will feed
the flock of goats on the mountains. 4. The soldiers wished to march
against the enemy.
5. The soldiers will be in want of provisions in
the enemy's country. 6. He is not the rich man who has much, but
he who wants little. 7. Pollux did not wish to be even a god by him-
self, but preferred rather to be a demigod with his brother. 8. The
barbarians, being pursued by the Greeks, were caught at the river.

EXERCISE 130.-ENGLISH-GREEK.

1. Η λεια ενεμηθη. 2. Νεμω την λειαν. 3. Η πολις τιμωρήσει τους πολεμίους. 4. Ω παί, μη αχθεσθητι ύπερ ὧν ημάρτανες ελεγχομενος. 5. Αγαθος παιδες ουκ άχθονται ὑπὲρ ὧν ἡμαρτανον ελεγχόμενοι. 6. Στρατευσομαι επι

τας Αθήνας. 7. Οζουσι μύρων. 8. Η ψυχή εις ουρανόν αναπτήσεται. 9. Αγαθοι επ' αγαθώ χαιρουσιν. 10. Τους στρατιώταις επιτηδείων δει. 11. Αγαθος των παιδων επι μελήσεται, αγαθοι δε παιδες των τοκέων επιμελήσονται,

EXERCISE 131.-GREEK-ENGLISH.

1. Even a slow man who is well advised can in pursuit catch a swiftfooted man. 2. The Athenians chose Themistocles general in the Persian war. 3. Ulysses came to the great hall of Hades. 4. What ever lot you may have taken, bear it and chafe not at it. 5. Do not trust very quickly before you exactly see the end. 6. Do not consider whether I am somewhat young to speak, but whether I speak the words of prudent men. 7. Mourn with moderation for friends who are dead, for they are not really dead, but they have gone before on the same road by which all must go.

EXERCISE 132.-ENGLISH-GREEK.

the Arab caravans from Gerrha. Their vessels in the Red Sea coasted Arabia Felix and Ethiopia, exchanging the produce of both these countries at Elath and Ezion-geber, in return for the commodities brought overland through Edom. The rich countries just referred to were the ancient Ophir, with which the Jews also traded, and whence were obtained gold, silver, ivory, apes, and peacocks. Meanwhile the Mediterranean was being slowly explored. Eventually the persistence of the Phoenicians extorted from the rulers of Egypt limited rights to the navigation of the Nile, and they were assigned a part of Memphis for warehouses and offices.

It is recorded that they were the first who rounded the Cape of Good Hope about 600 B.C., having started from the Red Sea at the instance of Pharaoh Necho, and in three years circumnavigated Africa. This event is involved in considerable obscurity, though there seems little reason to doubt its occur rence; but, whether true or not, the discovery was turned to no account for many centuries.

That the Phoenicians first passed the Pillars of Hercules is undisputed. Before Saul, the first King of Israel, had begun to reign, they had already ventured out into the Atlantic ; and the tin mines of Britain, and the amber lands of the Baltic, were probably visited by them before the days of Solomon. Long before this they had begun to frequent, and even to settle upon the isles of the Levant and the Egean. Cyprus in particular (the ancient Chittim) could be seen from their shores, and to reach it was one of their earliest efforts. Keeping near the shore, and guided at night by the stars, they gradually extended the length of their voyages. In the course of time they improved their skill in navigation and ship-building. The acquisition of wealth, whether by just or unjust means, appears to have been the sole object of their traffic. It was the universal custom to sell as slaves prisoners taken in war. The Phoenicians were ever ready to purchase any number of captives, and they would, it is said, when the chance offered, kidnap Greek and Hebrew children. The Greeks, amongst whom the Phoenicians at the first had settlements, suffered from their piratical habits, but they afterwards became powerful rivals to Phoenician commerce. The Greek ports, and the isles of the Egean, were closed against the Phoenicians, and, in alliance with the Etruscans, the Greeks expelled them from Southern Italy (Magna Græcia). But the desire for Oriental luxuries lessened the jealousies of trade, and Greece could not consistently deny herself Phoenician wares. In Sicily, Sardinia, and the Balearic Isles, the Phoenicians planted colonies, and successfully competed with the Etruscans. Colonies were gradually formed along the Mediterranean coasts. In Asia Minor and the Euxine, as well as Africa and the islands, the native tribes were taught husbandry, and thus to produce commodities valuable to Phoenician commerce. Carthage and Adramyttium, Great and Little Leptis, with several hundred smaller stations, arose in Africa.

Spain was literally a mine of wealth; for gold, lead, and iron abounded; and silver was so plentiful that the merchants are said to have ballasted their vessels with it, and to have made all their utensils, and even their anchors, of it. The profit was beyond compute. The natives gladly accepted Tyrian ornaments and glass trinkets for that upon which they set no value, and the Phoenicians disposed of this beautiful metal in the East, where it was held, comparatively, in higher estimation than gold. When the supply thus procured failed, the Phoenicians became AND POLITICAL HISTORY the taskmasters of the natives, whom they enslaved and comOF COMMERCE.

1. Οι Αθηναίοι πολλους στρατιωτας είλον. 2. Ἡ πολις Επαμεινονδαν είλετο στρατηγον. 3. Θεμιστοκλής ύπο των Αθηναίων στρατηγός ᾑρεθη. 4. Ελθε, ω φίλε. 5. Ω' αγαθοι φίλοι, έλθετε δευρο. 6. Εαν πεινης, τούτο ήδέως εδεί. 7. Ο παις όσον είχεν εδηδοκεν.

INDUSTRIAL

CHAPTER V.-FIRST COMMERCIAL PERIOD (continued)PHOENICIA-MARITIME OR COASTING TRADE. TRANSPORT by sea, in lessening the labour, time, and cost of procuring commodities from distant countries, gave a new life to commerce, and indefinitely widened its scope. It was chiefly as carriers that the Phoenicians distinguished themselves. They were the earliest recorded sailors. They were already a nation when the Israelites entered the Promised Land. Homer refers to their seafaring habits, and their daring as traders and pirates, as facts established at a date of about 1,000 years before the Christian era. As we have already seen, they navigated the Arabian and Indian Seas, and brought to the ports of the Persian Gulf the products of Ceylon and Malabar, of the Indus and the Ganges, thus linking the elephant traffic of Hindostan with the caravan commerce through Babylon and Palmyra, and with

Spain

pelled to work in the silver mines. Thus these poor aborigines
were the prototypes of the gentle Indians whom the Spaniards
in after days forced to labour in the mines of Mexico.
also possessed rich resources in animal and vegetable produce.
Fine wool, wax, and salt fish; corn, wine, oil, and luscious fruits
were only second in value to the precious metals. A Phoenician
origin can be assigned to nearly 200 Spanish towns. Cadiz,
"on the most remote point of the world," is to this day an
important commercial city. Hispalis (Seville), Carteïa, (Car-
taijo), near Gibraltar, and Malaka (Malaga), are other examples.
The Tarshish of Scripture was probably South Spain (Tartessus).
From Spain, the Phoenicians set out for still more distant enter-
prises. It appears hardly possible that their vessels could have
weathered the rough waters of the Bay of Biscay, yet it is certain
that they obtained tin and lead from the Scilly Isles and Corn-
wall, and amber from the shores of the Baltic. They also
visited, sooner or later, the Azores and the Madeira Isles. The