sides, 'DE, FG, HP, the lines joining the adjacent corners; it is required to prove that the squares of DE, FG, HP, are together equal to three times the squares of AB, AC, BC.

Draw EL, FN, HM, perpendicular respectively to DA, GC, PB, produced; and AK, CI, perpendiculars from A and C on the opposite sides of the triangle.

The angles EAL, CAI, are equal, each being the supplement of DAE, the angles at L and I are right angles, and the sides AC, AE, are equal, therefore AL AI, and EL=CI.

Similarly, CN=CK, and BM=BI. DE=A D'+A E2 + 2 A D. A L (Euc. II. 12).

Similarly,

=AB2+AC+2AB. AI

FG-AC+CB2+2BC. CK

and HP-AB+CB2+2AB. BI

.. DE+FG2+HP2=2AB+2AC2 + 2CB+2(AB. AI+AB. BI)+ 2BC. CK.

Q. E. D.

1. Explain the term Equinox. What is meant by a Solar, and a Sidereal day? Why is there any difference between the length of them? What is the difference? What is the cause of the tides, and the difference between "spring" and "neap" tides.

2. Name in order the counties, rivers, and chief stations one would pass on the LondonNorth-Western Railway from London to Carlisle, thence to Newcastle, and thence back to London ? 3. Describe the course of the great European watershed from the Pyrenees to the Ural Mountains, which separates the rivers that flow to the South and South-east from those that flow into the Atlantic and North seas; and name in order the rivers whose sources are passed upon the right hand and upon the left.

4. Describe the course of a vessel sailing from Riga to Rangoon? Say what her probable cargo would be (1) on her outward, (2) on her homeward voyage?

5. Explain the following terms and give some examples:-Monsoon, Oasis, Plateau, Trade Winds, Delta, Volcano, Sirocco, and Fauna?

of the Mediterranean Sea from Port Said to the 6. Draw an outline map of the Eastern coast Damascus, Aleppo, Jerusalem, Mt. Ida, Tarsus, Bosphorus, marking Latakia, Smyrna, Scutari, Tripoli, and trace the Islands of Cyprus, Rhodes, and Mitylene.

give the approximate dimensions of three principal 7. Give the exact geographical position, and rivers in Ireland; three mountain ranges in North America, and three lakes in Asia.

8. Fill up the accompanying map of India by marking on it the principal rivers, chains of mountains, and twelve of the chief towns.

Cambridge Local Examination.

1. Draw on a rough sheet a map of Australia and Tasmania. Mark the principal Colonies, shewing their boundaries, capitals, and one river or mountain range in each.

2. Name the chief places in the United Kingdom where iron, lead, tin, rock-salt, slate, granite, respectively are found. Name the counties which touch Kildare, Shropshire, and Stirling respectively.

3. Shew by a sketch, on a separate sheet, the course of the St. Lawrence, giving the chain of lakes from which it flows, and the principal towns on its banks.

4. Name in order from west to east the principal islands in the Mediterranean, stating the country to which each belongs.

5. Name in order the countries on the coast of Africa which you would pass in circumnavigating the continent, starting southwards from the Suez Canal. Name their capitals, and state what you know of the Governments of each country.

SENIOR GRADE.

EUCLID.

Time, 3 hours.

(Candidates are permitted to use all intelligible

abbreviations and algebraic symbols.)

1. Prove that the square described on the hypotenuse of a right-angled triangle is equal to the sum of the squares described on the other sides.

2. Prove by a geometrical construction that the square on the sum of two lines is equal to the square on their difference, together with four times the rectangle contained by the lines.

3. Prove that angles in the same segment of a circle are equal to one another, and equal to the angle which the chord of the segment makes with the tangent at its extremity.

4. Inscribe in a given circle a triangle which shall have its angles equal to those of a given triangle.

5. Give Euclid's definition of Proportion, and show from this definition that angles in different segments of the same circle are proportional to the lengths of the arcs on which they stand.

6. If two triangles have all their corresponding sides proportional, show that all their corresponding angles are equal.

7. Construct a rectilineal figure which shall be similar to one given rectilineal figure and equal in area to another.

8. Prove that, if through a point in the diagonal of a parallelogram straight lines be drawn. parallel to the sides, the parallelograms thus formed about that diagonal are similar to the original parallelogram and to one another.

9. Through a given point draw a chord of a given circle which shall be equal to a given line.

10. Prove that the rectangle contained by the diagonals of a quadrilateral can never exceed the sum of the rectangles contained by the pairs of opposite sides.

11. Prove that, if a system of circles all pass through the same two points, all straight lines through either of the points are divided similarly by the circles.

12. Find the locus of the point which is such that the distance between the feet of the perpendiculars drawn from it to two given lines is constant.

2. Show how to draw a straight line perpendicular to a given straight line from a given point upon it.

Show that this is a particular case of the problem of bisecting a given rectilineal angle.

3. A straight line is drawn so as to meet two parallel straight lines. Show that it makes the alternate angles between the parallel lines equal.

State the other relations between the angles which the line drawn across makes with the two parallel straight lines.

4. Prove that, if one side of a triangle be produced, the exterior angle is equal to the sum of the two interior and opposite angles, and that the three angles of every triangle are together equal to two right angles.

The angles ABC, ACB of a triangle are bisected by two straight lines meeting in O. With centre B and radius BO, a circle is described cutting CO again in D. Prove that the angles DBO and B A C are equal.

5. Prove that the square on that side of an obtuse-angled triangle which subtends the obtuse angle, is greater than the sum of the squares on the two sides which contain the obtuse angle, by twice a certain rectangle.

If the base of a triangle be bisected, the sum of the squares on the other sides of the triangle is double of the sum of the square on half the base and the square on the line joining the middle point of the base with the opposite angle.

6. Show that if PQ be any chord of a circle, and a line be drawn perpendicular to P Q through its middle point, the centre of the circle must lie in this line.

Hence show how to describe a circle of which a portion only is given.