ELEMENTS OF GEOMETRY.

BOOK I.

GEOMETRY is the science which treats of the properties, relations, and measurement of magnitude in general. Magnitude can have but three dimensions, length, breadth, and thickness, all of which are necessary to constitute a body, or solid. It is important, however, to consider magnitude under the three distinct denominations of lines, surfaces, and solids, and thus the science of Geometry becomes divided into three principal branches: the first part treating of lines described upon the same plane, and of the surfaces which they enclose; the second of lines situated in different planes, and of the relations of these planes to each other; and the third part contemplating body under its several dimensions of length, breadth, and thickness. Lines are obviously the boundaries of surfaces, and surfaces are the boundaries of solids: it is equally obvious that a line, being mere length, without either breadth or thickness, can exist only as the boundary of a surface, and that a surface being absolutely without thickness, can exist only as an attribute of body. Although, therefore, it cannot be supposed that a line, or a surface, can have separate or independent existence, the fact will not in the smallest degree interrupt or embarrass our reasonings in considering these several attributes of body or space, each apart from the others, nothing more being requisite than the abstracting these others from our inquiry; so that in considering lines, length only is recognized, and in contemplating surfaces, length and breadth are combined, and thickness excluded. Having made these preliminary remarks, which were deemed essential to the student, we may proceed with the definitions.

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DEFINITIONS.

1. Straight lines are those of which but one can be drawn from one point to another.

Two straight lines, therefore, cannot include space.

2. When two straight lines meet, the opening between them is called an angle; the point of meeting is called the vertex, and the lines themselves, which are said to contain the angle, are called the sides of the angle.

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An angle is referred to simply, by means of the letter, at its vertex. Thus the angle contained by the straight lines AB, BC, is designated as the angle B. When, however, two or more angles have the same vertex, then, in order to denote any one in particular, it becomes necessary to specify its sides by employing the three letters at their extremities; that at the vertex being always placed in the middle.

Thus the angle CBA or ABC, denotes that particular angle having the vertex Band contained by the sides AB, CB, and by the angle DBC or CBD, is in like manner meant the angle whose vertex is B, and whose sides are BD, BC.

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It is obvious that the quantity of an angle depends not upon the length, but entirely upon the position of its containing sides; for the opening between the sides AB, CB, must remain the same, however these lines be increased or diminished.

3. One straight line is said to be perpendicular to another, when it makes with it equal adjacent angles. A perpendicular at the extremity of a line, is that which makes an angle with it equal to the adjacent angle, which would be formed by prolonging the line beyond that extremity.

4. A right angle is the angle formed by a straight line and a perpendicular to it.

5. An acute angle is less than a right angle.

6. An obtuse angle is greater than a right angle.

7. A plane surface, or simply a plane, is that in which, if any two points whatever be taken, the straight line which joins them will lie wholly in it.

8. A straight line is said to be parallel to another when they are in the same plane, and can never meet, however far they may be produced.

If, for example, the straight line AB, A how far soever it be produced, can never meet the prolongation of CD, which is in the same plane, it is said to be parallel to it.

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9. By the distance of a point from a straight line, is meant the perpendicular from that point to the line; and one line is said to be equi-distant from another, when every point therein is equally distant from it.

10. A plane figure is an enclosed plane surface.

11. If it be bounded by straight lines only, it is called a rectilineal figure.

12. A polygon is a name used to comprehend every rectilineal figure, without regard to the number of its sides. The boundary of the figure is called its perimeter.

13. Among polygons, however, are more particularly distinguished the figure of three sides, called a triangle, and that of four sides called a quadrilateral.

14. An isosceles triangle is one, which has two equal sides.

15. An equilateral triangle is one which has all its sides equal.

16. When no two sides are equal the triangle is called scalene.

17. A right angled triangle is one which has a right angle.

18. In a right angled triangle the side opposite the right angle is called the hypothenuse. If, for example, the angle A is right, the side BC is the hypothenuse.

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Any side of a triangle may be considered as its base, but it is usual, in the case of the isosceles triangle, to confine this term to that side which is not equal to either of the others.

19. A rhomboid or parallelogram is a quadrilateral whose opposite sides are parallel.

20. If only two of the opposite sides are parallel, the figure is a trapezium.

21. A rhombus is a rhomboid, two of whose adjacent sides are equal.

22. A rectangle is a rhomboid having a right angle.

23. And a square is a rhombus having a right

angle.

24. The straight line which joins the vertices of two opposite angles of a quadrilateral, is called a diagonal.

Thus the line AC joining the vertices of the opposite angles DAB,DCB of the quadrilateral ABCD, is a diagonal.

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25. Plane figures are equal when, by supposing them to be applied to each other, they would coincide throughout; and they are said to be equivalent when they enclose equal portions space, and are at the same time incapable of such coincidence.

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AXIOMS-POSTULATES.

An Axiom is a self-evident truth.

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A Postulate requires us to admit the possibility of an operation.

A Theorem is a truth, the evidence of which depends upon a train of reasoning.

The reasoning by which a truth is established, is called a demonstration. It is a direct demonstration when the truth is inferred directly from the premises as the conclusion of a regular series of inductions. The demonstration is indirect when the conclusion shows that the introduction of any supposition, contrary to the truth advanced, necessarily leads to an absurdity.

A Problem proposes an operation to be performed.

A Lemma is a subsidiary truth, the evidence of which must be established preparatory to the demonstration of a succeeding theorem.

A Proposition is a general term for either a theorem, a problem, or a lemma.

A Corollary is an obvious consequence, resulting from a demonstration.

An Hypothesis is a supposition, and may be either true or false.

A Scholium is a remark subjoined to a demonstration.

AXIOMS.

1. Magnitudes which are equal to the same, are equal to each other.

2. Magnitudes which are double, triple, &c., of the same, or of equal magnitudes, are equal to each other.

3. Magnitudes which are each one half, one third, &c., of the same or of equal magnitudes, are equal to each other.

4. If equals be either added to, or taken from, equals, the results will be equal.

5. But if equals be either added to, or taken from, unequals, the results will be unequal.

6. The whole is greater than a part.

7. The whole is equal to the sum of the parts into which it is divided.

POSTULATES.

1. Grant that a straight line may be drawn from one point to another.