Similar polygons may be divided into the same number of similar triangles, having the same ratio to one another that the polygons have; and the polygons are to one another in the duplicate ratio of their homologous sides.

TRIGONOMETRY.

(Time allowed, 3 hours.)

1. Detine the tangent and versine of an angle, and express the tangent in terms of the versine.

Between what limits does the value of the versine lie, and for what angle is it greatest? 2. What are the units chiefly made use of in measuring angles ?

If the unit of angular measurement be the angle of an equilateral triangle, how many degrees are there in the angle represented by .6?

3. Prove geometrically that

(a) cos (90°+4)=-sin A.

(b) cos (A+B) = cos A cos B - sin A sin B.

From the second of these equations deduce the value of sin (A - B).

4. Show that if x = 45° sin (x + a) = cos(x - a).

5. Prove the identities:

2

6. Investigate a formula, adapted to logarithmic calculation, for finding the angles of a plane triangle, the sides of which are given.

In the triangle ABC, a = 97:6, b == 101.4, and c = 119.6, find the angle A.

7. The area of a quadrilateral field ABCD is 3,495 square yards. The sides AB, BC and diagonal AC are 75, 81, and 64 yards respectively, the remaining sides AD, DC are equal to one another, and the angle D is 107° 38′. Find the length of the equal sides 4D, DC.

8. Define the terms: sphere, spherical triangle.

Show that the angles of a spherical triangle are together greater than two and less than six right angles.

9. Prove that in any spherical triangle ABC

cos A

cos a cos b cos c

sin b sin o

What does this formula become (1) when the angle A is a right angle, (2) when the side (a) is a quadrant, and (3) when applied to the Polar Triangle?

10. In the spherical triangle ABC, given a = 86° 59′, b = 106 17', and A=90°, find the remaining parts.

CHART DRAWING.

(Time allowed, 3 hours.)

N.JB.-The meridional parts, as taken out of the Tables, and all the work must be

sent up.

1. Construct a chart, and, laying down the bearings and courses, find the latitude and longitude in, from the following:

Two headlands A (lat. 54° 41′ N., long. 19° 40′ W.) and B (lat. 54° 44′ N., long. 200

25' W.), bore respectively NNE. and NW. by W.

Afterwards sailed as under:

To be drawn on a scale of 1.22 inches to a degree of longitude, and to extend from latitude 51° to latitude 55 N., longitude 18o W. to 22° W.

2. Find the compass bearing and distance of the ship in her last position from a port in lat. 52° 26′ N. and long. 21° 15′ W.; variation, 30° W.; deviation, 3o E.

3. What is the true bearing of a rock which bears from the ship in her last position SW. W.; variation, 30° W.; deviation, 9° W.

Place the rock on your chart, supposing its distance 14 miles.

THEORETICAL NAVIGATION.

(Time allowed, 3 hours.)

1. Define the terms: Equator, Parallels of Latitude, Course, Distance, and Middle Latitude.

Prove the formula made use of in Middle Latitude Sailing.

2. A ship in lat. 27° 30′ S. sails west 550 miles and then due south; subsequently she alters course to east, and after sailing 420 miles reaches the meridian from which she started. How many miles has she sailed in the southerly direction?

3. Define the term Celestial Meridian. What do we require to know with regard to the sun in finding latitude by meridian altitude?

On a certain day, at two places on opposite sides of the equator, the sun's meridian zenith distance was double the latitude of the place. Having given the declination 10° N., find the respective latitudes.

4. Explain the terms: Rational Horizon, Circles of Altitude, Prime Vertical. At what times will the sun be on the prime vertical of a place in latitude 15° 30′ S. when its declination has the following values: 0°, 15° 30′ S. and 5° S. respectively 5. What are the causes of twilight ?

How long does twilight last at a place on the equator when the sun's declination is 12° 30′ Ν.?

6. Having given the distance between the sun and moon 86° 48', the declination of the sun 19° 10′ 30′′ S., of the moon 10° 8′ 45′′ N., and the right ascension of the sun 15h 32m 47, find the right ascension of the moon, which lies to the westward of the

sun.

7. What is meant by the Amplitude of a heavenly body?

Find the compass bearing of the sun when it rises at 6h 54m a. m. (apparent time) at a place in lat. 30° 10' N., Var. 16o W., Dev. 7o E.

8. A ship sails from lat. 51° 26′ S., long. 17° 25′ W., to lat. 39° 22′ S., long. 21° 55' E. Find the highest latitude reached.

PRACTICAL NAVIGATION.

(Time allowed, 3 hours.)

October 27, 1866, at noon, a point of land in lat. 25° 10' S., long. 46° 30′ E., bore by compass N. by W. W. (ship's head being SSE.) distant 12 miles; afterwards sailed as by the following log account; work up the reckoning to noon, October 28.

October 27, at 6o 25m p. m., the sun set by compass W. 12° 30′ N. (Deviation as for

ship's head), find the Variation.

October 28, at 8h 30m a. m., the following sights were taken to find the longitude:

Variation
allowed.

18° W.

September 26, at G. M. noon, the chronometer was fast on G.M.T. 2h 27m 38.5o, gaining daily 6.5 seconds.

October 28, at noon, the obs. mer. alt. of the sun's L.L. was 76° 48′ 10′′ (zenith S.), index error and height of eye as above, find the latitude.

PHYSICS.

(Time allowed, 3 hours.)

1. Explain carefully how a Centigrade thermometer is made and graduated.

What temperature will be represented by the same number in the Centigrade and Fahrenheit thermometers ?

2. We require to measure accurately the changes in the length of a rod of metal due to changes of temperature. Explain how this can be done.

3. If ice at 0°C. is placed in a vessel to which heat is continuously applied trace the changes in size, state, and temperature which the heat will produce.

4. Water may be frozen in consequence of the evaporation from its surface. Explain under what circumstances this may be made to take place.

5. How has the velocity of light been determined ?

6. What is a photometer? Describe some form of photometer.

7. A candle is placed in front of a concave mirror whose radius is 2 feet. Where must we stand in order to see the images when the distance of the candle from the mirror is first 4 feet, then 1 foot 6 inches, and finally 6 inches? Say in each of the three cases whether the images are larger or smaller than the candle, and whether they are erect or inverted.

8. Describe the eye as an optical instrument. What defects of the eye are corrected by eye-glasses? When should the eye-glass be convex and when concave?

9. What is the fundamental law of magnetism? What differences as regards magnetic properties are there between soft iron and steel?

10. How may the intensity of the earth's magnetism in two different places be compared?

11. Describe the electrophorus, and show how by its means we may charge a Leyden

jar.

12. What properties of an electric current are utilized in telegraphy?

LATIN.

(Time allowed, 3 hours.)

1. Translate into English the following passages, parsing the words in italics :

(a.) Et conversus ad simulacrum Jovis, "Audi, Juppiter, hæc scelera," inquit; "audite jus Fasque. Peregrinos consules et peregrinum senatum in tuo, Juppiter, augurato templo, captus ipse atque oppressus, visurus es? Hæcine fœdera Tullus, Romanus rex, cum Albanis, patribus vestris, Latini, hæc L. Tarquinius vobiscum postea fecit? Non venit in mentem pugna apud Regillum lacum Adeo et cladium veterum vestrarum et beneficiorum nostrorum erga vos obliti estis?" Quum consulis vocem subsecuta patrum indignatio esset, proditur memoriæ, adversus crebram implorationem deum quos testes fœderum sæpius invocabant consules, vocem Annii spernentis numina Jovis Romani auditam.

(b.) Vicit tamen pars, quæ in præsentia videri potuit majoris animi quam consilii; sed eventus docuit, fortes fortunam juvare. Bellum ex auctoritate patrum populus adversus Vestinos jussit. Provincia ea Bruto, Samnium Camillo sorte evenit. Exercitus utroque ducti, et cura tuendorum finium hostes prohibiti conjungere arma. Ceterum alterum consulem L. Furium, cui major moles rerum imposita erat, morbo gravi implicitum fortuna bello subtraxit; jussusque dictatorem dicere rei gerendæ causa longe clarissimum bello ea tempestate dixit, L. Papirium Cursorem, a quo Q. Fabius Maximus Rullianus magister equitum est dictus, par nobile rebus in eo magistratu gestis, discordia tamen, qua prope ad ultimum dimicationis ventum est, nobilius.

(c.) Fabius contione extemplo advocata obtestatus milites est, ut, qua virtute rem publicam ab infestissimis hostibus defendissent, eadem se, cujus ductu auspicioque vicissent, ab impotenti crudelitate dictatoris tutarentur; venire amentem invidia, iratum virtuti alienæ felicitatique; furere, quod se absente respublica egregie gesta esset; malle, si mutare fortunam posset, apud Samnites quam Romanos victoriam esse; imperium dictitare spretum, tanquam non eadem mente pugnari vetuerit, qua pugnatum doleat.

2. (a.) Give the derivations of the words: simulacrum, numen, prorincia, auspicium, (b.) Give some particulars of the battle at the lake Regillus.

egregius.

(N. N.-Dictionaries are permitted for the remainder of the paper.)

3. Translate into English:

Cum in omnibus locis, consumpta jam reliqua parte noctis, pugnaretur, semperque hostibus spes victoriæ redintegraretur, eo magis quod denstos pluteos turrium videbant, nec facile adire apertos ad auxiliandum animum advertebant, semperque ipsi recentes defessis succederent, omnemque Galliæ salutem in illo vestigio temporis positam arbitrarentur, accidit inspectantibus nobis, quod, dignum memoria visum, prætermittendum non existimavimus. Quidam ante portam oppidi Gallus, qui per manus sevi ac picis traditas glebas in ignem e regione turris projiciebat, scorpione ab latere dextro trajectus exanimatusque concidit. Hunc ex proximis unus jacentem transgressus, eodem ille munere fungebatur; eadem ratione ictu scorpionis exanimato altero, successit tertius, et tertio quartus; nec prius ille est a propugnatoribus vacuus relictus locus, quam, restincto aggere atque omni ex parte submotis hostibus, finis est pugnandi factus.

4. Translate into Latin:

(a.) The love of money is so great that not even the fear of death can altogether quell it.

(b.) After that battle they sent Pausanius to Cyprus, to drive out the barbarians from that island.

(c.) Perseus showed his enemies the head of the Medusa, by the sight of which

all were changed into stones.

(d.) Cicero warned the conspirators to conceal nothing from the judges. (e.) We pity our fellow-citizens, who have lost all in the conflagration.

The first corps raised in England in accordance with our present system, and in fact the first germ of an English standing army, was the Coldstream Guards, raised by General Monk at Coldstream. In the course of a few years several others were added, and by 1665 the British infantry consisted of four regiments besides the Guards. Before the close of the century, a grenadier company, furnished with hand grenades, had been added to each regiment; bayonets had been introduced; several regiments of fusiliers, originally intended to protect artillery, had been raised; and the principle of a standing army of considerable numbers fairly established. Light horse were introduced in 1745, and lancers in the reign of George III. It is within the last few years, however, that the greatest changes have taken place in the British army. But the advancement and elevation of the soldier himself only render him more capable of appreciating the traditions of his corps,

II. Translate into English:

UN ÉPISODE DEVANT SÉBASTOPOL.

Je m'etais arrêté, et je regardais avec une émotion profonde tous ces hommes qui, un instant auparavant, étaient pleins d'audace et de courage; tous étaient immobiles. Cependant, sur l'un des brancards les plus rapprochés de moi se soulevait faiblement The capote, et le bras du blessé cherchait à atteindre le brancard que l'on avait placé à côté du sien; un instant après, deux mains se touchaient. Celui qui le premier avait cherché cette étreinte fraternelle, rejeta tout-à-coup la capote dont on l'avait couvert, et aux premières lueurs du jour je le vis lever la tête, essayer de se soulever, puis retomber. Je me penchai sur lui; le pauvre soldat était mort. Cette main étendue, qui voulait presser une autre main, avait été le dernier adieu du mourant à un frere d'armes. CINQ MOIS AU CAMP DEVANT SÉBASTOPOL, PAR LE BARON DE BAZANCOURT.

III. GRAMMAR:

1. Write down the present participle and past participle of each of the following verbs: Savoir, voir, mouvoir, pouvoir; rire, dire, écrire; manger, placer, régler, mener. Give also the present subjunctive 2d person singular and plural of each. Give the future, first person singular, of savoir, voir, pouvoir.