bay. A Steppe is a barren plain in South Russia. 6. Egypt extends about 500 miles, between the A Bight is the name applied in Africa to a large Libyan desert on the West and the Red Sea on the East. The delta, which is a triangular district running from the Mediterranean to Cairo, is most fertile and productive. The valley of the Nile is also a fertile district, but much limited in width; among the products are wheat and silk, with some sugar and cotton. Among the principal towns may be named Cairo, the capital, Alexandria, the chief port, Rosetta, Damietta, Suez, &c. The great physical feature of Egypt is the Nile, whose annual overflow leaves a rich deposit, furnishing an admirable fertilizer. 7. Cardinal Beaton.-The head of the Catholic party in Scotland at the time of the Reformation. He was assassinated in 1546. Anne Askew-An accomplished English lady, belonging to the court of Henry VIII. She Lord Seymour of Sudeley.-Brother of the Anthony Babington, a gentleman of good family and independent fortune, who engaged in a conspiracy to murder Elizabeth and release Mary Queen of Scots. He was executed in 1586. The Earl of Lennox. The leader of the party John Knox.-The celebrated Scottish Reformer. Sir Thomas Wyatt was executed for the part he 8. A Meridian is a line drawn north and south through any place, marking the position of the sun at noon and cutting the equator at right angles. Neap-tide is applied to the small tides which occur when the moon is at her quarters. Sirocco is a hot wind which blows over the deserts in tropical regions. A Delta is an alluvial land formed at the mouth of some rivers, triangular in shape and naturally most productive. MEN CLERKS, LOWER DIVISION. GEOGRAPHY. 1. Explain what is meant by longitude. What is the length of a degree of longitude at the equator and at St. Petersburgh? When it is midnight on December 31st at London, what is the time and date at Sidney, 150° E., and Hono-. What are meant by parallels lulu, 160° W.? of latitude, and the ecliptic? 2. A person starting from York intends to make a circular tour, taking the following places in order-Carlisle, Glasgow, Portree, Inverness, Oban, Edinburgh, York. Describe his most probable route, mentioning the counties through which he will pass, the rivers, towns, and other objects of interest which he will see. 3. Draw an outline map of the British dominions in North America, and mark upon it the principal rivers, lakes, and mountains; also the provinces into which it is divided, and a few of the largest towns. Give some approximate distances between places which will afford a notion of the size of the territory. 4. Select four large rivers, two in Europe, and two in Africa. State their approximate length, where they rise, into what seas, and through what countries they flow, and the principal towns on or near their banks. 5. What are the subjects treated of under the head of Physical Geography? Explain and give A Glacier is formed on the sides of mountains by examples of the following terms:-plateau, delta, the freezing of half-melted snow. watershed, lagoon, and monsoon. 6. Mention the chief seaports of France, Russia, and Spain, and the articles exported from each. 7. Name ten of the largest islands in the world, and compare them in magnitude with England and Scotland. Describe their geographical situation, and mention the names of the straits which separate them from the mainland if they are contiguous to it. 8. Fill up the accompanying outline map of Asia, by marking on it the names and courses of the chief Asiatic rivers, and shewing the position of the following towns:-Mecca, Tiflis, Teheran, Herat, Yarkand, Irkutsk, Tobolsk, Lassa, Khiva, Rangoon, eight towns in Hindostan and three in China. Insert in their right position the lakes Baikal, and Balkash, and the Himalaya, HindooKoosh, Thianshan, and Ural mountains. ANSWERS BY "PERSEVERANCE." 1. Longitude is the distance of a place east or west from any fixed meridian, and is measured in degrees. At the equator a degree of longitude measures 60 geographical or 69 English miles, and at St. Petersburgh, 34 miles. The difference in time is 10 hours; therefore it is 10 o'clock, a.m, on January 1st in Sidney, and it is twenty minutes past one, p.m., on December 31, in Honolulu. Parallels of latitude are circles drawn round the earth parallel to the equator. The ecliptic is the circle which the sun appears to describe in the heavens, among the fixed stars in the course of a year. 2. Counties:-Cumberland, Dumfries, Lanark, Dumbarton, Argyle, Inverness (Skye), Ross, Stirling, Perth, Linlithgow, Edinburgh, Selkirk, Roxburgh, Northumberland, Durham and York. Rivers-Esk, Annan, Clyde, Ness, Forth, Leith, Esk, Tweed, Tyne, Wear, Tees, and Ouse. He will also cross the Lowther Hills, Loch Lomond, the Grampians, Loch Etive, Loch Linnhe, Loch Sheil, Sound of Sleat, Caledonian Canal (near Ben Cruachan and Ben Lomond), Lake Katrine, Firth of Forth, Moorsfoot Hills, Cheviots, and Pennine. Towns:-Moffat, Lanark, Glasgow, Dumbarton, Inverary, Armadale, Portree, Jeanstown, Inverness, Fort Augustus, Fort William, Oban, Stirling, Edinburgh, Selkirk, Hawick, Barnard Castle, Aldborough and York. 7. Papua or New Guinea, 4 times the size of Great Britain, lies north of Australia, from which it is separated by Torres Strait. Borneo, 33 times the size of Great Britain, lies south-east of Asia, from which it is separated by the China Sea. Madagascar, 2 times the size of Great Britain, lies east of Eastern Africa, from which it is separated by the Mozambique Channel. Sumatra, twice the size of Great Britain, lies west of the Malaya peninsula, from which it is separated by Malacca Strait. Newfoundland is half the size of England and Scotland. It is E. of the Gulf of St. Lawrence, and separated from Labrador by the Strait of Belleisle. 8 Java is times the size of England and Scotland. It lies south of Borneo, from which it is separated by Java Sea. INTERMEDIATE EDUCATION EXAMIN- SENIOR GRADE, EUCLID. TIME-3 HOURS. (Candidates are permitted to use all intelligible abbreviations and algebraic symbols.) 1. What is the sum of the angles of any convex rectilineal figure having a given number of sides? 2. Construct a square equal to a given rectangle? 3. Describe an isosceles triangle having each of the angles at the base double of the vertical angle. What is the problem in which Euclid makes use of this. 4. State Euclid's definition of proportion; or equality of ratios. Show from this definition that the ratio 3 4 is different from that of 4: 5. 5. Prove that similar triangles are to one another in the duplicate ratio of their homologous sides. required number of equal parts. 7. In an acute-angled triangle ABC, CD is the perpendicular from the vertex on the base. Show that the square on the tangent from C to a circle on A B as diameter is equal to CD-AD X DB 8. Show that a circle through the middle points of the sides of a triangle passes through the feet of the perpendiculars from the angles on the opposite sides. 9. Through a given point A within the angle formed by two given straight lines, draw a line Saghalien, times the size of England, is meeting both and bisected at A. east of Manchooria, from which it is separated by the Gulf of Tartary. Ceylon is only half the size of Scotland. It lies south of India, from which it is separated by the Gulf of Manaar and Palk Strait. Cuba, nearly as large as England, is one of the West Indies, and is separated from Florida by the Strait of Florida, Show that this line cuts off from the given angle a triangle whose area is less than that of any other cut off by a line through A. 10. At the vertex C a tangent is drawn to the circle circumscribed about the triangle ABC, meeting the base produced in D. Prove that AC: BC: AD: BD. Multiplying the 1st and 3rd by 4, and the 2nd and 4th by 3, we find the multiple of the 1st equal to that of the 2nd; but the multiple of the 3rd greater than that of the 4th; therefore Euclid V., Defn. 7, the ratio of 3 to 4 is less than that of 4 to 5. 5. This is Euclid VI. 19. 6. This is a particular case of Euclid VI. 10, but may be done within the limits of the First Book thus: Through the extremities of the given line draw two lines running in opposite directions parallel to each other; then, commencing at the extremities of the given line, set off on each of these parallels a number of equal parts, less by one than the number into which it is required to divide the given line; then the lines joining the opposite points of division which are in the same direction shall divide the given line as required. (The figure, which the reader can easily draw, would render this description unnecessary). The proof, which is extremely easy, depends on the 33rd and 26th propositions of the First Book. 7. Let A B C be the given triangle, A B M the circle described on the base A B, O its centre, Then the angles CDO, CMO being right angles CM2+OM2 = CD + OD2 each being = CO2. But O M2 II. 5.) =OB AD. DB + OD (Euc. = Therefore C M2 + AD. DB + OD2 = CD' +OD; that is, rejecting O D' from each, and transposing the rectangle A D. D B. CM CD2 - A D. D B. Q. E. D. 8. Let L, M, N, be respectively the middle points of the sides A B, A C, B C, of the triangle A B C and D, E, F, the feet of the perpendiculars on these sides from the opposite angles; then the circle passing through the points L, M, N, shall also pass through the points D, E, F. As preliminary to the proof of this and the following exercise, we shall show that : The straight line drawn through the middle point of one side of a triangle parallel to the base bisects the other side. Let M be the middle point of BC, a side of the triangle ABC, draw MN parallel to AB, then AC is bisected in N. [Our readers can easily draw the figure]. Produce NM to 0, making MO=MN, and join OB. Then in the triangles BMO, CMN, BM, MO, are respectively equal to CM, MN and the contained angles equal (Euc. I. 15), therefore BO is equal, and parallel to CN. Hence the figure ABON is a parallelogram, and BO equal to AN. That is, AC is bisected in N. NOTE.-MN is equal to half of AB. Hence the straight line joining the middle points of two sides of a triangle is parallel to the third side, and equal to its half, Draw MP,LR, parallel to AF, and join ML, PR. Then from the property of the triangle just proved, CO is bisected in P, and BO in R., and, therefore, ML, PR, are each parallel to BC, and equal to its half; they are then equal and parallel to each other. Similarly MP, LR are equal and parallel. LMPR is therefore a parallelogram; and since MP, LR are parallel to AF, which is perpendicular to ML and PR; LMPR is a rectangle, and therefore inscribable in a circle, the centre of which is the point of intersection of its diagonals. This circle shall pass through D and E, because the angles CDL, BEM are right angles. If LN and NP are joined, these lines being respectively parallel to AC, BE, the angle LNP is a right angle, the circle will therefore pass through N. It will also pass through F, since the angles QFN, QMN are right angles. NOTE. This is called the Nine Point Circle of the Triangle; since it passes through the middle points of the sides, the feet of the perpendiculars, and the middle points of the segments of the perpendiculars between their common point of intersection and the angles. Through H draw HM parallel to BF; then the triangles AF K. A HM are equiangular and the sides AF, A H, equal; these triangles are therefore equal; adding to each the quadrilateral A HBK, the triangle BFH is equal to the quadrilateral B H M K, and therefore less than the triangle BL K by the triangle HL M. In the same manner BFH can be proved less than any other triangle cut off from DBE by a line passing through A. 10. Let CD be a tangent to the circle circumscribed about the triangle ABC, meeting the base A B produced in D, then (Euclid III. 32) the angle B C D=the angle BA C, and the angle D is common to the triangles A CD, BCD, which are therefore equiangular. Hence (Euclid VI. 19.) AC: BC: triangle A CD: triangle BCD. And (Euclid VI. 1) AD: BD:: triangle A CD: triangle BCD. And since ratios which are equal to the same ratio are equal to one another, AC: BC:: AD: BD. Note. The duplicate ratio of one quantity to another is identical with the ratio of the squares of their like dimensions. N.B.-In our next issue we shall give several of the papers set at the Intermediate Examinations commencing on 15th instant, with answers. |