PART FIRST,

CONTAINING THE

GEOMETRICAL AND FINAL SOLUTION

OF THE

QUADRATURE OF THE CIRCLE,

BY AN ENTIRELY NEW METHOD, TOGETHER WITH AMPLE PROOFS OF

THE SAME.

LO

5

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QUADRATURE OF THE CIRCLE.

DEFINITIONS.

Of Lines, Angles, etc., see Plate 1.

1. Science is knowledge systematized.

2. Art is the skill with which the principles of a science are practically applied.

3. Quantity is anything which can be increased, diminished, or measured.

4. Mathematics is the science of quantity.

5. Geometry is that branch of mathematics which treats of the properties of extension and figure,

LINES.

6. A line has only one dimension, namely, length; without either breadth or thickness; lines are either straight or curved.

7. The extremities of a line are points.

8. A point has no dimensions, and therefore it has no size; and is usually represented to the eye by a dot, as at A4.

9. A straight line is the shortest distance between any two given points, as at B.

10. A curved line does not lie evenly between its extreme points; but constantly changes its course as at C.

11. A waved line is composed of curved lines as at E.

12. Parallel straight lines are in the same plane, but have no inclination towards one another; and being produced ever so far, both ways can not meet, as at D.

13. Parallel curved lines if produced would form two concentric circles; that is, having a common center, as at F.

14. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is a right angle, and the straight line which stands on the other is called a perpendicular to it; IKL are perpendiculars to the line GH.

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15. Inclined lines, as I and I if produced, would meet in a point as K forming an angle of which the point K is the vertex, and the lines H and I the legs of the angle.

16. When two straight lines cut each other so as to make the adjacent angles equal to one another, each of the angles is a right angle; and the straight lines which so cut each other are perpendicular to one another; thus QR is perpendicular to ST and XY to ZV; but perpendicular lines are not always vertical and horizontal; XY is a horizontal line, and ZV is a vertical line.

ANGLES.

17. If there is only one angle at a point it may be denoted by a letter placed at the vertex, as at K. But if several angles are at one point, any one of them is expressed by three letters, of which the middle one is at the vertex; thus the angle which is contained by the straight lines UWY is called the angle UWY or YWU.

REMARK.—Angles like other quantities may be added, subtracted, multiplied, and divided ; thus the angle UWY is the sum of the angles UWV, VWX, XWY, and the angle UWV is the difference between the two angles, YWU and YWV

18. An acute angle is less than a right angle; thus the angle at K is an acute angle.

19. An obtuse angle is greater than a right angle; thus the angle at P is an obtuse angle.

20. An oblique angle is formed by one straight line meeting another straight line on one side of it, so as to make the adjacent angles equal to two right angles.

21. A plane angle is the inclination of two lines to one another in a plane which meet together, but are not in the same direction.

22. A plane rectilineal angle is the inclination of two straight lines to one another which meet together, but are not in the same straight line.

23. A THEOREM is a truth requiring demonstration.

24. An AXIOM is a self-evident truth.

25. A PROBLEM is a question requiring a solution.

26. A POSTULATE is a self-evident problem.

Theorems, Axioms, Problems, and Postulates, are all called Propositions.