Part 14

A Postulate is a problem, the solution of which is self-evident.

Let it be granted--

I. That a straight line can be drawn from any one point to any other point;

II. That a straight line can be produced to any distance, or terminated at any point;

III. That the circumference of a circle can be described about any center, at any distance from that center.

The common algebraic signs are used in Geometry, and it is necessary that the student in geometry should understand some of the more simple operations of algebra. As the terms circle, angle, triangle, hypothesis, axiom, theorem, corollary and definition are constantly occurring in a course of geometry, they are abbreviated as shown in the following list:

Addition is expressed by + Subtraction is expressed by - Multiplication is expressed by × Equality and Equivalency are expressed by = Greater than, is expressed by > Less than, is expressed by < Thus B is greater than A, is written B > A B is less than A, is written B < A A circle is expressed by O An angle is expressed by L A right angle is expressed by R. L Degrees, minutes and seconds are expressed by ° ′ ″ A triangle is expressed by △ The term Hypothesis is expressed by (Hy.) The term Axiom is expressed by (Ax.) The term Theorem is expressed by (Th.) The term Corollary is expressed by (Cor.) The term Definition is expressed by (Def.) The term Perpendicular is expressed by ⊥ The difference of two quantities, when it is not known which is the greater, is expressed by the symbol ~ Thus, the difference between A and B is written A ~ B

GEOMETRICAL AXIOMS.

1. Things which are equal to the same thing are equal to each other.

2. When equals are added to equals the wholes are equal.

3. When equals are taken from equals the remainders are equal.

4. When equals are added to unequals the wholes are unequal.

5. When equals are taken from unequals the remainders are unequal.

6. Things which are double of the same thing, or equal things, are equal to each other.

7. Things which are halves of the same thing, or of equal things, are equal to each other.

8. The whole is greater than any of its parts.

9. Every whole is equal to all its parts taken together.

10. Things which coincide, or fill the same space, are identical, or mutually equal in all their parts.

11. All right angles are equal to one another.

12. A straight line is the shortest distance between two points.

13. Two straight lines cannot enclose a space.

Problems in Geometrical Drawing.

EXAMPLE 1.--To bisect (cut in two) a straight line or an arc of a circle, Fig. 99. From the ends of A B as centers, describe arcs cutting each other at C and D, and draw C D, which cuts the line at E or the arc at F.

EX. 2.--To draw a perpendicular to a straight line, or a radial line to a circular arc, Fig. 99. Operate as in the foregoing problem. The line C D is perpendicular to A B; the line C D is also radial to the arc A B.

EX. 3.--To draw a perpendicular to a straight line, from a given point in that line, Fig. 100. With any radius from any given point A in the line B C, cut the line at B and C. Next, with a longer radius, describe arcs from B and C, cutting each other at D, and draw the perpendicular D A.

Second Method, Fig. 101. From any center F above B C, describe a circle passing through the given point A, and cutting the given line at D; draw D F, and produce it to cut the circle at E; and draw the perpendicular A E.

Third Method, Fig. 102. From A describe an arc E C, and from E, with the same radius, the arc A C cutting the other at C; through C draw a line E C D and set off C D equal to C E, and through D draw the perpendicular A D.

EX. 4.--To draw a perpendicular to a straight line from any point without it, Fig. 103. From the point A with a sufficient radius cut the given line at F and G; and from these points describe arcs cutting at E. Draw the perpendicular A E.