Angles in the same segment of a cirole are equal; and, if the line joining two points subtends equal angles at two other points on the same side of it, the four points lie on a circle. The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle. The opposite angles of any quadrilateral inscribed in a circle are supplementary; and the converse. If a straight line touch a circle, and from the point of contact a chord be drawn, the angles which this chord makes with the tangent are equal to the angles in the alternate segments. If two chords of a circle intersect either inside or outside the circle, the rectangle contained by the parts of the one is equal to the rectangle contained by the parts of the other; and the converse.' P. 40, insert “For 1904, 1905 and 1906" below the words “III.— MATHEMATICS-Two Papers." P. 40, insert the following after line 5 from the bottom : "For 1907 and subsequent years. PAPER I. Geometry. The questions in practical Geometry shall be set on the constructions contained in Schedule A together with easy extensions of them as riders if desired. A candidate should provide himself with a ruler graduated in inches and tenths of an inch and in centimetres and millimetres, a set square, a protractor, compasses and a hard pencil. All figures should be drawn accurately. The questions on theoretical Geometry shall consist of theorems contained in Schedule B, together with questions upon these theorems, easy deductions from them and arithmetical illustrations. Any proof of a proposition shall be accepted which forms a part of any systematic treatment of the subject; the order in which the theorems are stated in Schedule B is not imposed as the sequence of the treatment. Proofs which are only applicable to commensurable quantities shall be accepted. The use of intelligible abbreviations is recommended. SCHEDULE A. Division of straight lines into parts in any given proportions. Construction of a triangle or a square equal in area to a given polygon. Construction of common tangents to two circles. Simple cases of the construction of circles from sufficient data. Construction of a fourth proportional to three given straight lines and a mean proportional to two given straight lines. Construction of a regular pentagon. Description in a given triangle of a triangle similar and similarly placed. to another given triangle. Description of squares in a triangle and in or about a given quadrilateral. SCHEDULE B. The Circle. A straight line drawn from the centre of a circle to bisect a chord which is not a diameter is at right angles to the chord; conversely, the perpendicular to a chord from the centre bisects the chord. There is one circle, and one only, which passes through three given points not in a straight line. In equal circles (or in the same circle) (i) if two arcs subtend equal angles at the centres they are equal; (ii) conversely, if two arcs are equal, they subtend equal angles at the centres. In equal circles (or in the same circle) (i) if two chords are equal, they cut off equal arcs; (ii) conversely, if two arcs are equal, the chords of the arcs are equal. Equal chords of a circle are equidistant from the centre; and the con verse. The tangent at any point of a circle and the radius through the point are perpendicular to one another. If two circles touch, the point of contact lies on the straight line through the centres. The angle which an arc of a circle subtends at the contre is double that which it subtends at any point on the remaining part of the circumference. Angles in the same segment of a circle are equal; and if the line joining two points subtends equal angles at two other points on the same side of it, the four points lie on a circle. The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle. The opposite angles of any quadrilateral inscribed in a a circle are supplementary; and the converse. If a straight line touch a circle, and from the point of contact a chord be drawn, the angles which this chord makes with the tangent are equal to the angles in the alternate segments. If two chords of a circle intersect either inside or outside the circle, the rectangle contained by the parts of the one is equal to the rectangle contained by the parts of the other. Proportion: Similar Triangles. If a straight line is drawn parallel to one side of a triangle, the other two sides are divided proportionally; and the converse. If two triangles are equiangular their corresponding sides are proportional; and the converse. If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar. If two triangles have one angle of the one equal to one angle of the other and the sides about another angle of each proportional, the sides opposite the equal angles being homologous, the third angles of the triangles are either equal or supplementary. The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle, and likewise the external bisector externally. In a right angled triangle the perpendicular drawn from the right angle to the base will divide the triangle into two parts which are similar to the whole and to each other. If an angle of a triangle be bisected by a straight line which cuts the opposite side, the sum of the rectangle contained by the two segments of that side and the square on the bisecting line is equal to the rectangle contained by the other two sides of the triangle. If a perpendicular be drawn from a vertex of a triangle to the opposite side, the rectangle contained by the other sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle. The rectangle contained by the diagonals of a quadrilateral inscribed in a circle is equal to the sum of the rectangles contained by the two pairs of opposite sides. The ratio of the areas of similar triangles is equal to the ratio of the squares on corresponding sides. If two triangles (or parallelograms) have one angle of the one equal to one angle of the other, their areas are proportional to the areas of rectangles contained by the sides about the equal angles. Concurrency and Collinearity. If three concurrent straight lines are drawn from the angular points of a triangle to meet the opposite sides, the product of three alternate segments taken in order is equal to the product of the other three segments. If a transversal is drawn to cut the sides or the sides produced of a triangle, the product of three alternate segments taken in order is equal to the product of the other three segments. The three medians of a triangle meet in a point, and their common point is a point of trisection of each median. The three lines drawn through the angular points of a triangle perpendicular to the opposite sides are concurrent. The three lines which bisect the angles of a triangle are concurrent; and so also are the bisector of one of the interior angles of a triangle and the bisectors of the other two exterior angles. The three lines drawn through the middle points of the sides of a triangle perpendicular to those sides are concurrent. In any triangle the three middle points of the sides, the three feet of the perpendiculars drawn from the angular points on the sides, and the three middle points of the lines joining the orthocentre to the angular points all lie on a circle whose diameter is equal to the radius of the circumscribed circle and whose centre is the middle point of the line joining the orthocentre and circumcentre. If from any point on the circumference of a circle, perpendiculars be drawn to the sides of an inscribed triangle, the three feet of the perpendiculars lie on a straight line. Harmonic Section. Division of a given straight line internally and externally so that its segments may be a given ratio. The locus of a point whose distances from two fixed points have a constant ratio is a circle. Centre of Similitude. If any two unequal similar figures are placed so that their homologous sides are parallel, the lines joining corresponding points in the two figures meet in a point, whose distances from any two corresponding points are in the ratio of any pair of homologous sides. Every straight line which passes through the extremities of two parallel radii of two fixed circles passes through one or other of the fixed points. Pole and Polar. If a straight line be drawn through a given point to cut a given circle the intersection of the tangents at the two points of section always lies on a fixed straight line. If one point lie on the polar of another point, the second point lies on the polar of the first point. Radical Axis. Determination of the locus of points from which tangents drawn to two given circles are equal. The radical axes of three circles taken in pairs are concurrent. PAPER II. ax+b Theory of Indices, Elementary Surds, Theory of Quadratic Equations; Graphs of ax2+bx+c and ; easy Simultaneous Equations of two unknown quantities involving quadraties; Ratio, Proportion, Variation; Arithmetical, Geometrical and Harmonical Progressions and other simple series; Permutations and Combinations; Binomial Theorem for a positive integral index; Logarithms; Interest and Annuities; the Remainder Theorem and its applications. N.B.-The scope of the above subjects is in no way to exceed the limits of an elementary work on Algebra, such as the Elementary Algebra by Hall and Knight, Algebra for Beginners by Todhunter and Loney, or any other similar work." Page 112, under "1867" omit the following (January.) 34 Rao Bahadur Khanderao Chimanrao Bedarkar, B.A., LL.B. ... Page 208, under 1904 read "St. X" for "St. K." in the last column opposite the name "Takakhav, Nilkant Sadashiv.” Page 244, under "1897" read "Cordeiro, Peter Adrian," for "Crodeiro, Peter Adrian." Page 313, under the head "Dakshina Fellows" read "Miss. Nissim, Hannah" for "Miss Nissin Hannah." |