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 straight lines meet at a point, and make the opposite vertical angles equal, each alternate pair of st. 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, cr to two thirds of one right angle.

5. The vertical angle of a triangle is right, acute, or obtuse, according as the st. 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 st. lines be drawn from the points of bisection to the opposite angles, these two st. 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 st. 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 st. 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 st. 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 st. line internally, so that the rectangle under its segments shall be of a given magnitude.

8. To cut a st. 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 st. 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; 4° the sum of the squares of the sides; and 5° 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 st. 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 st. 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 st. 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

SERIES II. -THOR. 15. II.

233

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 st. 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 st. 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 st. lines drawn from it to two opposite angles, is equal to the sum of the squares of the st. 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.

220

TABLE OF CONTENTS.

PREFACE.

Respecting the Gradations in Euclid's Plane Geometry,
Propositions, &c.

INTRODUCTION.

PAGES.

Skeleton

iii. - viii.

SECTION I. Gradual Growth of Geometry and of the Elements of

Euclid

SECTION II. Symbolical Notation and Abbreviations that may be

used

SECTION III. Explanation of some Geometrical Terms..
SECTION IV. Nature of Geometrical Reasoning......

SECTION V. Application of Arithmetic and Algebra to Geometry...
SECTION VI. On Incommensurable Quantities......
SECTION VII. On Written and Oral Examinations and Means of

Progress

1-5

5-7

7-10

J0-20

21-27 28-30

30-35

The Properties of Right-angled Parallelograms or Rectangles......... 149
Definitions and Axiom

150

I. Problems 5-16, for the Construction of Geometrical Figures
stated, and proved in Books I. and II...
Subsidiary Problems, 17-31, in Books I. and II.

191-197

197-201

II. Problems in Books III., IV., and VI. most intimately connected with Books I. and II.:

III. Principles of Construction :

SECTION 1. For Geometrical Instruments to Measure Distances

and Angles

SECTION 2. For Geometrical Figures to exhibit representative
values of Magnitude and Space...

211-212

212-215

IV. Principles for accurately calculating Distances, Magnitudes, and Areas:

SERIES 2° Propositions not fully proved, or not inserted in

the Gradations :

"Even the Sleeping Geometrician," says old RALPH CUDWORTH, p. 160, "hath at that time, all his Geometrical Theorems and Knowledges some way on him as also the Sleeping Musician, all his Musical Skill and Songs: and therefore why may it not be possible for the Soul to have likewise some Actual Energie on it, which it is not Expressly Conscious of ?"

John Heywood, Printer, 141 and 143, Deansgate, Manchester.