that the sum of the distances AP, BP, is the least possible. Wellesley.

128. Two circles are tangent internally, the ratio of their radii being 2:3. Compare their areas, and also the area left in the larger circle with each.-Sheffield S. S.

129. A kite-shaped racing-track is formed by a circular arc and two tangents at its extremities. The tangents meet at an angle of 60°. The riders are to go round the track, one on a line close to the inner edge, the other on a line everywhere 5 ft. outside the first line. Show that the second rider is handicapped by about 22 feet.—Harvard.

130. The diameters of two water-pipes are 6 and 8 inches respectively. What is the diameter of a pipe having a capacity equal to their sum ?-Rutgers S. S.

131. (a.). There are two gardens: one is a square and the other a circle; and they each contain a hectare. How much farther is it around one than the other?

(b.) If the area of each is 2 hectares, what will be the difference of their perimeters ?— Yale.

132. Inscribe a square in a scalene triangle.-Cornell. 133. A horse is tethered to a hook on the inner side of a fence which bounds a circular grass-plot. His tether is so long that he can just reach the centre of the plot. The area of so much of the plot as he can graze over is 98 (4 π 3√3) sq. rd.; find the length of the tether and the circumference of the plot.-Harvard.

3

134. If the apothem of a regular hexagon is 2, find the area of its circumscribed circle.-Wesleyan University.

135. Of all polygons formed of given sides the maximum may be inscribed in a circle.-Sheffield S. S.

136. If the radius of a circle is 6, what is the area of a segment whose arc is 60°? (Take T = 3.1416.)—Mass.

Inst. Tech.

137. A stone bridge 20 ft. wide has a circular arch of 140 ft. span at the water level. The crown of the arch is 140 (1√3) ft. above the surface of the water. How many square feet of surface must be gone over in cleaning so much of the under side of the arch as is above water ?— Harvard.

138. Of all isoperimetric figures the circle has the greatest area. Cornell.

139. Compute by logarithms the value of

3

(2.3456) × (.301456) 2
(4.02356) 4

-Yale.

SELECTED

EXAMINATION PAPERS IN PLANE

GEOMETRY SET FOR ADMISSION TO A NUMBER OF THE LEADING COLLEGES AND SCIENTIFIC SCHOOLS IN THE UNITED STATES.

Harvard, June, 1892.

[In solving problems use for π the approximate value 34.]

1. Prove that if two sides of a triangle are unequal, the angle opposite the greater side is greater than the angle opposite the less side.

In a certain right triangle one of the legs is half as long as the hypotenuse; what are the angles of the triangle?

2. Show how to find on a given indefinitely extended straight line in a plane, a point O which shall be equidistant from two given points A, B in the plane. If A and B lie on a straight line which cuts the given line at an angle of 45° at a point 7 inches distant from A and 17 inches from B, show that O A will be 13 inches.

3. Prove that an angle formed by a tangent and a chord drawn through its point of contact is the supplement of any angle inscribed in the segment cut off by the chord. What is the locus of the centre of a circumference of given radius which cuts at right angles a given circumference?

4. Show that the areas of similar triangles are to each other as the squares of the homologous sides.

5. Prove that the square described upon the altitude of an equilateral triangle has an area three times as great as that of a square described upon half of one side of the triangle.

6. Find the area included between a circumference of radius 7 and the square inscribed within it.

Harvard, June, 1893.

[In solving problems use for the approximate value 3.]

1. Prove that two oblique lines drawn from a given point to a given line are equal if they meet the latter at equal distances from the foot of the perpendicular dropped from the point upon it.

How many lines can be drawn through a given point in a plane so as to form in each case an isosceles triangle with two given lines in the plane?

2. Prove that in the same circle, or in equal circles, equal chords are equally distant from the centre, and that of two unequal chords the less is at the greater distance from the centre.

Two chords of a certain circle bisect each other. One of them is 10 inches long; how far is it from the centre of the circle? A variable chord passes, when produced, through a fixed point without a given circle. What is the locus of the middle point of the chord?

3. A common tangent of two circumferences which touch each other externally at A, touches the two circumferences at B and C respectively; show that B A is perpendicular to A C.

4. Assuming that the areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles, prove that the bisector of an angle of a triangle divides the opposite side into parts which are proportional to the sides adjacent to them.

5. Prove that the circumferences of two circles have the same ratio as their radii.

6. A quarter-mile running track consists of two parallel straight portions joined together at the ends by semicircumferences. The extreme length of the plot enclosed by the track is 180 yards. Find the cost of sodding this plot at a quarter of a dollar per square yard.

Harvard, June, 1894.

[In solving problems use for the approximate value 34.]

1. Prove that any quadrilateral the opposite sides of which are equal, is a parallelogram.

A certain parallelogram inscribed in a circumference has two sides 20 feet in length and two sides 15 feet in length; what are the lengths of the diagonals?

2. Prove that if one acute angle of a triangle is double another, the triangle can be divided into two isosceles triangles by a straight line drawn through the vertex of the third angle.

Upon a given base is constructed a triangle one of the base angles of which is double the other. The bisector of the larger base angle meets the opposite side at the point P. Find the locus of P.

3. Show how to find a mean proportional between two given straight lines, but do not prove that your construction is correct.

Prove that if from a point, O, in the base, BC, of a triangle, A B C, straight lines be drawn parallel to the sides, AB, A C, respectively, so as to meet AC in M and AB in N, the area of the triangle A M N is a mean proportional between the areas of the triangles B NO and C MO.

4. Assuming that the areas of two parallelograms which have an angle and a side common and two other sides unequal, but commensurable, are to each other as the unequal sides, prove that the same proportion holds good when these sides have no common measure.

5. Every cross-section of the train-house of a railway station has the form of a pointed arch made of two circular arcs the centres of which are on the ground. The radius of each arc is equal to the width of the building (210 feet); find the distance across the building measured over the roof, and show that the area of the cross-section is 3,675 (4π - 3√3) square feet.