By transferring the equation to the focus the author has endeavoured to point out the change which takes place when the curve approaches to a parabola; and on that account he has deduced the latus rectum and the co-ordinates of the vertex of the parabola from those of the ellipse in its transition state. The reduction of the equation to the focus led to the forms which have been determined for the elements of the curve; and it was afterwards found that most of them might be obtained more briefly by the polar equation from the centre. In the case of oblique co-ordinates the expressions are remarkably symmetrical, and are placed in such a form that the reader may perceive at once their connexion with the corresponding results when the co-ordinates are rectangular. In the latter part of the second appendix the author has endeavoured to trace a conic section geometrically as far as it appeared practicable, by the determination of a successive series of points when five points of the curve can in any manner be found. A remarkably simple construction for a tangent at one of the five given points has been obtained (Art. 89); a tangent has also been drawn from a given point without the curve; and the position of a point in the conic section in any proposed direction has been determined by its points of intersection with a given straight line. In a subject which for ages has exercised the skill and ingenuity of the most profound mathematicians, little can be expected which is really original; but as only a very small portion of the matter in the appendices has been met with by the author elsewhere, even if he may have been anticipated in any or all the properties which are inserted, it is extremely probable that the proofs now given will be widely different from any which have been hitherto published. A reference has been made in several places to an edition of Euclid lately published by Mr Potts, in which will be found much valuable information; it is well deserving the attention of every one who wishes to study Geometry. CAMBRIDGE, T. G. GEOMETRICAL PROBLEMS. 1. ST JOHN'S COLLEGE. DEC. 1830. (No. I.) PARALLELOGRAMS upon the same base and between the same parallels are equal to one another. 2. Of unequal magnitudes, the greater has a greater ratio to the same than the less. 3. If the diameter of a circle be one of the equal sides of an isosceles triangle, the base will be bisected by the circumference. 4. The line joining the centres of the inscribed and circumscribed circles of a triangle subtends at any one of the angular points an angle equal to the semidifference of the other two angles. 5. Find a point without a given circle, such that the sum of the two lines drawn from it touching the circle, shall be equal to the line drawn from it through the centre to meet the circumference. 6. If a circle roll within another of twice its size, any point in its circumference will trace out a diameter of the first. 7. If from any point in the circumference of a circle, a chord and tangent be drawn, the perpendiculars dropped upon them from the middle point of the subtended arc, are equal to one another. 8. If a, ß, y represent the distances of the angles of a triangle from the centre of the inscribed circle, and a, b, c the sides respectively opposite to them, then a2 a + ß3b + y2c = abc. 9. Describe a circle through a given point and touching a given straight line, so that the chord joining the given point A and point of contact, may cut off a segment capable of a given angle. 10. Shew that the perimeter of the triangle formed by joining the feet of the perpendiculars dropped from the angles upon the opposite sides of a triangle, is less than the perimeter of any other triangle whose angular points are on the sides of the first. 11. Explain what is meant by the equation to a curve; find the equation to a straight line, and state clearly the meaning of the constants involved. 12. Trace the circle whose equation is a (x2 + y2) + b2 (x + y) = 0 ; draw the lines represented by the equations and shew that the angle between them is a. 13. The portion of a straight line intercepted by two rectangular axes, and the perpendicular upon it from their intersection are each of given length; what is the equation to the line? 14. Find the equation to an ellipse, and deduce that to the parabola from it. 15. Find the co-ordinates of the point from which if three lines be drawn to the angles of a triangle, its area is trisected. 16. In the last question, the (distance) from the angle A 17. If the centre of the inscribed circle of a triangle be fixed, and a, ß, y represent the distances of its angles from any fixed point in space; then whatever position the triangle assume, the expression a2a + ß2h + y2c is invariable. 2 SOLUTIONS TO (No. I.) 1. EUCLID, Prop. 35, Book 1. 2. Euclid, Prop. 8. Book v. 3. Let AB, AC (fig. 1) be the equal sides of an isosceles triangle; upon AB describe a semicircle cutting the base BC in D; join AD: then ADB is a right angle = ▲ ADC; also ▲ ABD = ▲ ACD, and AD is common to the two triangles ABD, ADC, .. BD = DC. 4. Let ABC (fig. 2) be a triangle, d, D the centres of the inscribed and circumscribing circles; draw DE, de perpendicular to AB, and join AD, Ad: then In like manner it may be proved by joining DB, dB, DC, dC that shews that AD would in that case lie below Ad; and so of the rest. 5. Let D (fig. 3) be the required point in any diameter ACB produced; DE a tangent drawn from D; then |