πος. Του δε ομολογούντος, έπειθεν αρνείσθαι, λέγων Αιδέσθητί σου την ηλικίαν, και έτερα τούτοις ακόλουθα, ως έθος αυτοϊς λέγειν· "Ομοσον την καίσαρος τύχην, μετανόησον, είπέ: Αίρε τους αθέους. Ο δε Πολύκαρπος έμβριθεί τω προσώπω εις πάντα τον όχλον των εν τω σταδίω ανόμων εθνών εμβλέψας, και επισείσας αυτοίς την χείρα, στενάξας τε και αναβλέψας εις τον ουρανόν, είπεν: Αίρε τους αθέους. 'Εγκειμένου δε του ανθυπάτου και λέγοντας: "Όμoσον, και απολύω σε, λοιδόρησαν τον Χριστόν ο Πολύκαρπος έφη: 'Όγδοήκοντα και εξ έτη έχω δουλεύων αυτω, και ουδέν με ήδίκησε και πως δύναμαι βλασφημήσαι τον βασιλέα μου, τον σώσαντα με ; Where is this account of Polycarp’s martyrdom preserved ? JOSHUA VII. Add the vowels to the following passage: וימעלו בני-ישראל מעל בחרם ויקח עכן בן־כרמי ,Beginning קרא שם המקום ההוא עמק עכור עד היום הזה: ,Ending 1. TRANSLATE into ENGLISH, adding such notes as you think needful, Isaiah xlii. The Ordiuary B.A. Degree . Mathematical Examiners : Classical Examiners : Examiners in the Acts, Paley, &c. : EUCLID. FIRST DIVISION.-(A.) 1. To describe an equilateral triangle on a given finite straight line. 2. If two straight lines cut one another, the vertical, or opposite, angles shall be equal. If four straight lines meet in a point, so that the opposite angles are equal, these straight lines are, two and two, in the same straight line. 3. If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. the sides adjacent to equal angles in each; then shall the other sides be equal, each to each, and also the third angle of the one equal to the third angle of the other. 4. The opposite sides and angles of parallelograms are equal to one another, and the diameter bisects them. 5. To describe a square upon a given straight line. On a given straight line describe a rhombus, each of whose acute angles is half a right angle. 6. If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts, are together equal to the square on the whole line. 7. In every triangle, the square on the side subtending either of the acute angles is less than the squares on the sides containing that angle, by twice the rectangle contained by either of those sides and the straight line intercepted between the perpendicular let fall upon it from the opposite angle, and the acute angle. In any acute-angled triangle, lines are drawn through the angles perpendicular to the sides respectively opposite to them; prove that the rectangles contained by the segments into which the sides are respectively divided, are together less than half the sum of the squares on the sides of the triangle. 8. If two circles cut one another, they shall not have the same centre. 9. If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the touching line, the centre of the circle shall be in that line. 10. In equal circles, equal angles stand upon equal circumferences, whether they be at the centres or circumferences. In equal circles, straight lines equidistant from the centre subtend equal angles at the centre. 11. From a given circle to cut off a segment, which shall contain an angle equal to a given rectilineal angle. 12. The sides about the equal angles of equiangular triangles are proportionals; and those which are opposite to the equal angles are homologous sides. FIRST DIVISION.-(B.) 1. From the greater of two given straight lines to cut off a part equal to the less. 2. The angles which one straight line makes with another, on the same side of it, are either two right angles, or are together equal to two right angles. 3. If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. the sides opposite to equal angles in each, then shall the other sides be equal, each to each, and also the third angle of the one equal to the third angle of the other. The perpendiculars let fall on two sides of a triangle, from any point in the line bisecting the angle between them, are equal to each other. 4. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts, are also themselves equal and parallel. The straight lines which join the extremities of two equal and parallel straight lines, not towards the same parts, bisect each other, 5. To describe a square upon a given straight line. 6. If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square on the aforesaid part. 7. In every triangle, the square on the side subtending either of the acute angles is less than the squares on the sides containing that angle, by twice the rectangle contained by either of those sides and the straight line intercepted between the perpendicular let fall upon it from the opposite angle, and the acute angle. In any acute-angled triangle ABC, AD, BE, CF are drawn respectively perpendicular to the opposite sides; prove that twice the rectangles contained by BD and CD, CE and AE, AF and BF, are together less than the sum of the squares on the sides of the triangle. 8. If two circles touch one another internally, they shall not have the same centre. 9. If a straight line touch a circle, the straight line drawn from the centre to the point of contact, shall be perpendicular to the line touching the circle, 10. In equal circles, equal straight lines cut off equal circumferences, the greater equal to the greater, and the less to the less. In equal circles, straight lines subtending equal angles at the centre are equidistant from the centre. 11. From a given circle to cut off a segment, which shall contain an angle equal to a given rectilineal angle. 12. The sides about the equal angles of equiangular triangles are proportionals; and those which are opposite to the equal angles are homologous sides. THURSDAY, January 10, 1856. 121 to 3. SECOND DIVISION.-(A.) 1. If two angles of a triangle be equal to each other, the sides also which subtend the equal angles, shall be equal to each other. If one angle of a triangle be equal to the sum of the other two, the triangle can always be divided into two isosceles triangles. 2. The angles which one straight line makes with another upon one side of it, are either two right angles, or are together equal to two right angles. 3. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another. 4. Parallelograms upon the same base and between the same parallels are equal to one another. Of all parallelograms, that can be described on the same base and between the same parallels, which has the least perimeter? 5. To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a right angle. 6. If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts are together equal to the square of the whole line. 7. To divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts shall be equal to the square of the other part. 8. If a straight line drawn through the centre of a circle bisect a straight line in it, which does not pass through the centre, it shall cut it at right angles; and conversely, if it cut it at right angles, it shall bisect it. 9. If two circles touch each other externally in any point, the straight line which joins their centres, shall pass through that point. Describe a circle of given radius which shall touch externally a given circle in a given point. 10. Straight lines in a circle which are equally distant from the centre are equal to one another. 11. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles. No parallelogram except a rectangular one can be inscribed in a circle. 12. If the angle of a triangle be divided into two equal angles by a straight line which also cuts the base; the segments of the base shall have the same ratio which the other sides of the triangle have to one another. SECOND DIVISION.—(B.) 1. To draw a straight line at right angles to a given straight line, from a given point in the same. If two straight lines bisect each other at right angles, every point in either of them is equidistant from the extremities of the other. 2. Any two sides of a triangle are together greater than the third side. In any quadrilateral figure, the sum of the sides is greater than the sum of the diagonals. 3. If a straight line falling on two other straight lines, make the alternate angles equal to each other, these two straight lines shall be parallel. 4. Parallelograms upon equal bases, and between the same parallels, are equal to one another. 5. To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to half a right angle. |