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The study of form is the basis of Geometry. A cubic foot of matter may be in the shape of a ball, or of a cubical block,

may be irregular. When thus regarding only the amount of material, no attention is paid to the outline; Geometry, however, considers the outline to the exclusion of the amount of matter which it encloses.

A geometrical sphere is not a sphere of iron or wood, but a sphere of empty space. It is therefore an imaginary solid, which cannot be perceived by the senses, and for which we must use some representative, as a ball or a diagram, in order to describe it. It is a type of one class of geometrical magnitudes-solids.

These ideal portions of space are bounded by surfaces without thickness. Here, again, Geometry deals with the form of the surface. The surface may be flat, so that a straightedged ruler, in whatever direction it be laid, will touch along its length, or it may be curved. It may, if flat, be limited by straight lines or curved, by lines of equal or unequal length, etc.

If, now, we suppose the edge of the ruler to be without breadth, we obtain an idea of a new kind of magnitude-a geometrical line. As solids are bounded by surfaces, so surfaces are bounded by lines. Lines are also imaginary, having neither breadth nor thickness. We use marks to represent

them in our books. The form of the line is again important, whether it be straight or curved, long or short. These remarks prepare the


for the more concise definitions which follow.

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1. Geometry is that science which treats of the properties, relations and measurement of magnitudes.

Any solid may be considered as having three dimensionslength; breadth and thickness. A geometrical solid is the portion of space enclosed within the boundaries of a physical solid. These boundaries are surfaces; the boundaries of surfaces are lines, and lines are limited by points. The term magnitudes applies to solids, surfaces and lines. A solid has extension in three directions, a surface in two, and a line in one. A point has no magnitude. 2. A proposition is a general statement. It

may theorem, problem, axiom or postulate.

3. A theorem is a statement which it is required to prove. 4. A problem is a question which it is required to solve.

5. A demonstration is the course of reasoning by which the theorem is proved or problem solved.

A theorem may be proved either directly or by showing that an absurdity would result if it were supposed untrue.

6. An axiom is a self-evident proposition requiring no proof.

Such as : The whole is greater than a part.

7. A postulate is something to be done which is so simple that no one will hesitate to allow it.

Such as: Two points may be joined by a straight line.

8. A corollary is a consequence drawn from a preceding proposition.

9. A scholium is a remark made upon a preceding proposition, showing its limits or application.

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10. An hypothesis is a proposition assumed to be true in order to argue from.

The following expressions show the meaning of signs used through the book :

A=B means that A is equal to B.

A is greater than B.
A< B

A is less than B. A +B

A is added to B.

B is subtracted from A.
A ~B, or A.BA is multiplied by B.

A is divided by B.

A is taken twice, or A+A. AP

A is taken twice as a factor, or A A. AABC means the triangle ABC, as distinguished from the angle ABC

... means therefore.

Figures in parenthesis through a demonstration, thus, (I. 15), (VI. 2), refer to the previous proposition in which the statement was proved, meaning (Book I. Prop. 15), (Book VI. Prop. 2).

(Hyp.) stands for Hypothesis. (Ax.)

Axiom. (Post.)

Postulate. (Cor.)

Corollary. (Sch.)

Scholium. (Constr.)






Point.— Line. — Surface. 1. A point is that which has position, but not magnitude. 2. A line is that which has length, but not breadth.

The extremities of lines are points. The intersection of one line with another is also a point.

3. If a line preserve the same direction throughout, it is a straight line. If it change its direction at every point, it is a curved line or curve.

Corollary. Two straight lines cannot enclose space; neither can they coincide in any two points without coinciding altogether.

4. A surface is that which has length and breadth only.

5. A plane surface is one in which, if any two points be taken, the straight line between them lies wholly in that surface.

6. Parallel straight lines are such as are in the same plane, and being produced ever so far both ways do not meet.

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