8. A line which is perpendicular to one parallel, is also perpendicular to the other.

9. If a line falling on two other lines make the interior angles on the same side less than two rt. angles, those two lines on being produced shall intersect.

10. A parallel to the base of a triangle through the point of bisection of one side, will bisect the other side.

11. The lines which join the middle points of the three sides of a triangle, divide it into four triangles which are equal in every respect.

12. The line joining the points of bisection of each pair of sides of a triangle, is equal to half the third side.

13. A trapezium is equal in area to a parallelogram of the same altitude, and of which the base is half the sum of the parallel sides.

14. The squares on equal lines are equal; and if the squares are equal, the lines are equal.

15. Every parallelogram having one rt. angle, has all its angles rt. angles. 16. If a perpendicular be drawn from the vertex of a triangle to the base, the difference of the squares of the sides is equal to the difference between the squares of the segments.

17. If a perpendicular be drawn from the vertex of a triangle to the base, or to the base produced, the sums of the squares of the sides and of the alternate angles are equal.

Problems in Book II.

1. From Propositions 1, 2, and 3, deduce various methods for the Multiplication of Numbers, and demonstrate the rule.

2. From Prop. 4, point out a practical way of extracting the Square root of a number, and prove the correctness of the formula.

3. To find the difference between the squares of two unequal numbers without squaring them.

4. To find Quantities in Arithmetical Progression.

5. To find the value of an Adfected Quadratic Equation in Algebra.

6. By aid of Prop. 6, to ascertain the diameter of the earth.

7. Given the sum and the difference of two magnitudes, to find the magnitudes themselves.

8. From the Area of a rectangle and one side given, to obtain the other side.

9. To divide a given Line a, so that its parts x and a-x may make a (a−x) = x2. Let the solution be given both algebraically and arithmetically.

10. To ascertain the Area of a triangle when the three sides are known. 11. From the three sides of a triangle given, to obtain the perpendicular;— 1o. when the perp. falls within the base; and 2o. when it falls without the base.

12. To find a mean proportional to two given lines.

13. To approximate to the square of any curve-lined figure.

14. To calculate the Area of any right-lined figure.

Theorems in Book II.

1. The difference of the squares of two quantities, equals the rectangle of their sum and difference.

2. The difference of the squares of two quantities is greater than the square of their difference, by twice the rectangle of the less and their difference.

3. The square of the sum of two lines is equal to four times the rectangle under them, together with the square of their difference.

4. Four times the square of half the sum is equal to four times the rectangle under the lines, together with four times the square of half the difference.

5. The sum of the squares of any two lines is equal to twice the square of half their sum, together with twice the square of half their difference.

6. The sum of the squares is equal to half the square of the sum, together with half the square of the difference.

SERIES II.

PROPOSITIONS NOT FULLY PROVED, OR NOT INSERTED IN THE GRADATIONS.

Problems.-Book I.

1. To find a point which is equidistant from the three vertical points of a triangle.

2. To bisect a triangle by a line drawn from a given point in one of its sides.

3. Describe a circle which shall pass through two given points, and have its centre in a given line.

4. Through a given point to draw a line that shall be equally inclined to two given lines.

5. Given a triangle ABC, and a point D in AB; to construct another triangle ADE equal to the former, and having the common angle A.

6. To change a triangle into another equal triangle of a given altitude. 7. To draw a line which, if produced, would bisect the angle between two given lines, without producing them to meet.

8. To trisect a right angle.

9. To trisect a given st. line.

10. Given the sum of the sides of a triangle, and the angles at the base, to

construct it.

11. Given the diagonal of a square, to construct the square of which it is

the diagonal.

12. Given the sum and difference of the hypotenuse and a side of a rightangled triangle, and also the remaining side, to construct it.

13. To find the locus of all points which are equidistant from two given points.

Theorems.-Book I.

1. In an isosceles triangle, the right line which bisects the vertical angle also bisects the base, and is perpendicular to the base.

2. If four lines meet at a point, and make the opposite vertical angles equal, each alternate pair of lines will be in the same st. line.

3. The difference of any two sides of a triangle is less than the remaining side.

4. Each angle of an equilateral triangle is equal to one-third of two right angles, or to two-thirds of one right angle.

5. The vertical angle of a triangle is right, acute, or obtuse, according as the line from the vertex bisecting the base is equal to, greater, or less than half the base.

6. If the opposite sides or opposite angles of a quadrilateral be equal, the figure is a parallelogram.

7. If the four sides of a quadrilateral are bisected, and the middle points of each pair of conterminous sides joined by st. lines, those joining lines will form a parallelogram the area of which is equal to half that of the given quadrilateral.

8. If two opposite sides of a parallelogram be bisected, and two lines be drawn from the points of bisection to the opposite angles, these two lines trisect the diagonal.

9. In any right-angled triangle, the middle point of the hypotenuse is equally distant from the three angles.

10. The square of a line is equal to four times the square of its half. 11. The st. line which bisects two sides of a triangle, is parallel to the third side, and equal to one-half of it.

12. If two sides of a triangle be given, its area will be greatest when they contain a rt. angle.

13. Of equal parallelograms that which has the least perimeter is the square. 14. The area of any two parallelograms described on the two sides of a

triangle, is equal to that of a parallelogram on the base, whose side is equal and parallel to the line drawn from the vertex of the triangle to the intersection of the two sides of the former parallelograms produced to meet.

15. The vertical angle of a triangle is acute, rt. angled, or obtuse, according as the square of the base is less than, equal to, or greater than, the sum of the squares of the sides.

Problems.-Book II.

1. The sum and difference of two magnitudes being given, to find the magnitudes themselves.

2. To describe a square equal to the difference of two given squares. 3. To divide a given line into two parts, such that the squares of the whole line and of one of the parts shall be equal to twice the square of the other part.

4. To divide a given line into two such parts that the rectangle contained by them may be three-fourths of the greatest of which the case admits.

5. Given the area of a right-angled triangle, and its altitude or perpendicular from the vertex of the rt. angle to the opposite side, to find the sides.

6. Given the segments of the hypotenuse made by the perp. from the rt. angle, to find the sides.

7. To divide a line internally, so that the rectangle under its segments shall be of a given magnitude.

8. To cut a line externally, so that the rectangle under the segments shall be equal to a given magnitude, as the square on A.

9. Given the difference of the squares of two lines and the rectangle under them, to find the lines.

10. There are five quantities depending on a rectangle, -1o. the sum of the sides; 2°. the difference of the sides; 3°. the area; 4o. the sum of the squares of the sides; and 5o. the difference of the squares of the sides-by combining any two of these five quantities, find the sides of the rectangle.

Theorems.-Book II.

1. The square of the perpendicular upon the hypotenuse of a right-angled triangle drawn from the opposite angle, is equal to the rectangle under the segments of the hypotenuse.

2. The squares of the sum and of the difference of two lines, are together double of the squares of these lines.

3. In any triangle the squares of the two sides are together double of the squares of half the base, and of the line joining its middle point with the opposite angle.

4. The square of the excess of one st. line above another, is less than the squares of the two st. lines by twice their rectangle.

5. The squares of the diagonals of a parallelogram are together equal to the squares of the four sides.

6. If a st. line be divided into two equal and also into two unequal parts, the squares of the two unequal parts are together equal to twice the rectangle contained by these parts, together with four times the square of the line between the points of section.

7. If a st. line be drawn from the vertex of a triangle to the middle point of the opposite side, the sum of the squares of the other sides is equal

to twice the sum of the squares of the bisector and half of the bisected side.

8. The sum of the squares of the sides of a quadrilateral figure is equal to the sum of the squares of the diagonals, together with four times the square of the line joining their points of bisection.

9. If st. lines be drawn from each angle of a triangle bisecting the opposite side, four times the sum of the squares of these lines is equal to three times the sum of the squares of the side of the triangle.

10. The square of either of the sides of the rt. angle of a rt. angled triangle, is equal to the rectangle contained by the sum and difference of the hypotenuse and the other side.

11. If from the middle point C, of a st. line AB, a circle be described, the sums of the squares of the distances of all points in this circle from the ends of the st. line AB, are the same; and those sums are equal to twice the sum of the squares of the radius and of half the given line.

12. Prove that the sum of the squares of two lines is never less than twice their rectangle; and that the difference of their squares is equal to the rectangle of their sum and difference.

13. If, within or without a rectangle, a point be assumed, the sum of the squares of lines drawn from it to two opposite angles, is equal to the sum of the squares of the lines drawn to the other two opposite angles.

14. If the sides of a triangle be as 4, 8, and 10, the angle which the side 10 subtends will be obtuse.

15. If in a rt. angled triangle a perpendicular be drawn from the rt. angle to the hypotenuse, the rectangle of one side and of the non-adjacent segment of the hypotenuse, shall equal the rectangle of the other side and of the other non-adjacent segment of the hypotenuse.

N.B.-A KEY to the GEOMETRICAL EXERCISES of the APPENDIX will be published immediately. The APPENDIX will also be sold separately, to be used as a Book of Exercises.

FINIS.