PART SECOND.

DEMONSTRATIONS IN GEOMETRY.

REMARK. Before entering upon this part of our work, it may be well to apprise the critical reader, that it has not been deemed at all necessary to adopt the rule apparently followed by Euclid, viz., never to suppose anything done, till the manner of doing it has been shown or explained. The rule was a very safe one, in the early progress of the science, to prevent the possibility of error, or the danger of resting on unwarrantable assumptions; but it also led to much unnecessary labor and tedious prolixity. The rule which I have rather endeavored to follow in these demonstrations, is, to give under each proposition all that is necessary to produce perfect conviction of the truth stated, and not to encumber the demonstration with anything more. So that if it should appear to the critical geometer, that links are sometimes omitted which he think ought to be brought into the chain of reasoning, he may understand the reason of the omission. He may sometimes miss the repetition of an axiom or a well-known and established principle of geometry, which might have served to lengthen out a demon

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stration, but would not make the truth any more apparent. So also in these demonstrations many figures are required to be drawn when no rule or mode of drawing them has been given. But, says Sir Isaac Newton, "geometry does not teach us to draw these figures, but requires them to be drawn." The construction of them is entirely a mechanical operation.

DEFINITIONS.

1. Numbers are the signs or representatives of things, or of whatever has existence.

2. Arithmetic is the science of numbers, and regards things only as they are numerable, or may be counted.

3. Geometry is the science of magnitude, and measures and compares extension and forms.

4. Arithmetic has but one language or mode of expression, which is by numbers.

5. Geometry has two languages or modes of expression, one by numbers, and one by material substances, or pictures representing material substances.

6. The unit in arithmetic is the sign or representative of anything considered as one and indivisible, without regard to form or magnitude.

7. The unit in geometry is the sign or representative of any assumed magnitude, considered as one and indivisible, and in the form of a cube. A unit in geometry, therefore, is always one in length, one in breadth, and one in thickness.

8. A unit in geometry may be of any positive magnitude, from magnitude infinitely diminished, to magnitude infinitely extended.

9. A straight line is composed of a succession of single and equal units. A line therefore always has a breadth of one.

10. A line, or length, is measured by the application of one dimension only of the unit, viz., its linear edge.

11. A surface or plane is composed of a succession of single lines. A surface therefore, always has a thickness of one.

12. Plane figures, or forms, are those in which extension is measured in two directions only, length and breadth, without regard to thickness.

13. The elements of plane figures consist of area, perimeter, circumference, and diameter.

14. The area of a plane figure is the quantity of extension or space inclosed by its circumference; and is measured by the application of two dimensions of the unit, its length and breadth.

15. The perimeter of a plane figure is the distance around it, measured upon the extreme limits of the figure.

16. The circumference of a plane figure is a line, or lines, touching and inclosing it, having a breadth equal to one, and a length equal to the perimeter of the figure.

17. The diameter of a plane figure is the diameter of its inscribed circle.

18. A circle is a plane figure, which has an equal extension in every direction from its center to its circumference.

19. The diameter of a circle is a straight line passing through its center, and extending in length to the extreme limits of the circle.

20. A circle is said to be inscribed in any plane figure, when the circle touches every side or line of the circumference of the figure; and circumscribed when the circumference of the circle touches every corner or angle of the figure.

21. A plane figure is said to be circumscribed about a circle, when every side of its circumference touches the circle; and inscribed when every corner or angle touches the circumference of the circle.

22. The base of a plane figure is the side on which it is sup'posed to rest, when considered in a vertical position.

23. The perpendicular, or height, of a plane figure is the shortest distance from any point in the base to a line drawn parallel to the base and resting on the highest point of the figure.

24. Parallel lines are those which everywhere preserve an equal distance between them. Parallel lines therefore can never meet each other, however far they may be produced.

25. Solid figures, or bodies, are those in which extension is measured in three directions, length, breadth, and thickness.

26. The elements of solid figures consist of solidity, face, surface, and diameter.

27. The solidity of a solid, in geometry, is the quantity of extension, or the amount of bulk, inclosed by the surface. The solidity is measured by the application of the unit in its three dimensions, length, breadth, and thickness.

28. The faces of a solid are the planes by which its extension is terminated.

29. The surface, [super-facies,] of a solid is the sum of all the planes supposed to perfectly cover all its faces, and everywhere having a thickness of one.

30. The diameter of a solid, with plane faces, is the diameter of its inscribed sphere.

31. A sphere is a solid figure which has an equal extension in every direction from its center to its surface. Its surface therefore is a perfect curve, everywhere returning into itself.

32. The diameter of a sphere is a straight line passing through its center, and extending in length to the extreme limits of the sphere.

33. A sphere is said to be inscribed in a solid with plane faces, when the sphere touches every plane of the surface; and circumscribed when the surface of the sphere touches every solid angle.

34. A solid is said to be circumscribed about a sphere, when every plane of its surface touches the sphere; and inscribed when every solid angle touches the surface of the sphere.

Lines are of two kinds, straight and curved.