N. B. the side of the tower, as well as of the obelisk, is supposed to be perpendicular to the horizon. 12. What is the perpendicular height of a hill, whose angle of elevation taken at the bottom of it equals 46°, and 200 yards farther upon an horizontal plane equals 31°? 13. Wanting to know my distance from an inaccessible object on the other side of a river, I measure 100 yards from each of two stations, A and B, (which are 500 yards asunder) in a direct line from the object, and placing two marks C and D at the end of the 100 yards measured from A and B respectively, I find that from A to D there are 550 yards, and from B to C 560. Required the distance of the object from A and B. 14. Given the perimeter of a triangle and the three angles to construct it. PROPOSITIONS IN PLANE GEOMETRY. 1816, 1. If the exterior angle of a triangle be bisected by a straight line which also cuts the base produced, the segments between the bisecting line and the extremities of the base, have the same ratio which the other sides of the triangle have to one another. Shew that the converse is also true. 2. Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides. 3. The rectangle contained by the diagonals of any quadrilateral figure inscribed in a circle is equal to the sum of the rectangles contained by its opposite sides. 4. If the exterior angle of a triangle be bisected, and also one of the interior and opposite, the angle contained by the bisecting lines is equal to half the other interior and opposite angle of the triangle. 5. (1). If upon the sides BA, CA, of any triangle, any two pa. rallelograms be drawn, and their sides produced to meet in K 6. The square described upon the side of a regular pentagon inscribed in any circle is equal to the sum of the squares described upon the sides of a regular hexagon and decagon inscribed in the same circle. 7. If a straight line be drawn from C, the point of bisection of a given arc ACB, cutting the chord AB or the chord produced in any point E and the circumference of the circle in D, prove that in each case the rectangle contained by CD and CE is equal to the square described on CB. 8. The greatest of all straight lines passing through either of the points of intersection of two given circles which cut each other, and terminated both ways by the two circumferences, is that which is parallel to the line joining the centres of the two circles. 9. If the sides of a regular polygon of n sides be produced to meet, the sum of the angles made by the lines thus produced at the points of intersection is equal to 2n – 8 right angles. 10. Represent the arithmetic, geometric, and harmonic means, between two given lines geometrically. 11. The centre of the circle circumscribed about any triangle, the point of intersection of the perpendiculars let fall from the angular points of the saine triangle to the opposite sides, and the point of intersection of the lines joining the angular points with the middle of the opposite sides, all lie in the same right line. 12. If four circles touch each either internally or externally, three sides of any quadrilateral figure, the centres of these circles will lie in the circumference of the same circle. 13. Describe a circle passing through a given point which shall touch both a given circle and a given straight line. 14. If from the centre and angular points of a regular hexagon perpendiculars be drawn to any given right line, six times the perpendicular from the centre is equal to the sum of the perpendiculars froin the angular points. EUCLID, Book XI. 1817 1. DEFINE the five regular solid figures. 2. If a straight line stand at right angles to each of two straight lines in the point of their intersection, it shall also be at right angles to the plane which passes through them. 3. If two straight lines be parallel, and one of them be at right angles to a plane; the other also shall be at right angles to the same plane. 4. Draw a straight line perpendicular to a given plane from a given point without it. 5. Planes to which the same straight line is perpendicular are parallel to each other. 6. Every solid angle is contained by plane angles which together are less than four right angles. 7. The face of a regular octahedron and of a dodecahedron being given, it is required to construct the solids. x2 x2 2.c + y + Sy 3 2 Зу 5. A brewer, from a certain quantity of ingredients which cost £20. brews 500 gallons of ale, (on which there is a duty of 6d. a gallon,) and sells it at 2s. a gallon. Afterwards from the same quantity of ingredients, he brews a certain number of gallons of strong beer, (on which he pays the ale duty,) and the remainder small beer, making together the same number of gallons as before,—when by mixing them together, and selling the mixture as ale, he finds his gains increased in the proportion of 10:7. Determine the number of gallons of strong beer, supposing the duty on small beer of that on ale. 6. The number of deaths in a besieged garrison amounted to 6 daily, and allowing for this diminution, their stock of provisions was sufficient to last for 8 days. But on the evening of the sixth day 100 men were killed in a sally, and afterwards the mortality increased to 10 daily. Supposing the stock of provisions unconsumed at the end of the 6th day sufficient to support 6 men for 61 days. It is required to find how long it would support the garrison, and the number of men alive when the provisions were exhausted. 7. A man buys a guinea at the market price of standard gold, but an act of parliament passing which makes it illegal to sell the guinea in the same way that he bought it, he privately clips off one twenty-fifth part. He may now legally sell it as a light guinea, and he finds that in consequence of the rise of pure gold in the ratio of 239 : 249 he just gains the clippings by his purchase. It is required to find the ratio of pure gold and alloy in the guinea, and also ihe relative value of equal quantities of pure gold and alloy, it being known that the sum of the squares of the numbers which express the two ratios, exceeds eleven times their sum by the number 233 T&T 9 x+y+V x2 –yo (x+y) Sy (x2 + y)' + x - y = 2x.(x+y) + 506 5. At the review of an army, the troops were drawn up in a solid mass, 40 deep; when there were just one-fourth as many men in front as there were spectators. Had the depth however been increased by 5, and the spectators drawn up in the mass with the army, the number of men in front would have been 100 fewer than before. Determine the number of men of which the army consisted. 6. A number of persons purchased a field for £345. The youngest contributed a certain sum, the next £5, more, the third £5. more than the second, and so on to the oldest. For the greater accommodation of the seniors, the field was divided into two parts, the younger balf taking a portion proportional to the sum they had subscribed; and in order that each might have an equal share in this portion, they agreed to equalize their contributions, and each to pay £22. Required the number of persons, and the sums paid by each. 7. A and B travelled on the same road, and at the same rate, from Huntingdon to London. At the 50th mile-stone from London, A overtook a drove of geese, which were proceeding at the rate of three miles in two hours; and two hours afterwards met a stage waggon, which was moving at the rate of nine miles in four hours. Bovertook the same drove of geese at the 45th mile-stone, and met the same stage waggon exactly forty minutes before he came to the 31st mile-stone. Where was B when A reached London ? 9 Y a dy 1. A packet sailing from Dover with a fair wind, arrives at Calais in two hours; and on its return the wind being contrary, it proceeds six miles an hour slower than it went. Now when it is half way over, the wind changing, it sails two miles an hour faster, and reaches Dover sooner than it would have done had the wind not changed, in the proportion of 6 :7. Required the rates of sailing, and the distance between Dover and Calais. 2. From the middle of a town two streets branched off, and crossed a river that ran in a straight course, by two bridges A and B. From their junction, a sewer equally inclined to both streets led to a point in the river, at the distance of 6 chains from the bridge A, and a distance from B, less by 11 chains than the length of the sewer: the expense of making it amounting to as many pounds per chain as there were chains in the street leading to A. The sewer however being insufficient to carry off the water, an additional drain was made from a point in this street, distant 4 |