1. Area is the product of two averages.

2. Volume is the product of three averages.

3. Hypervolume in n dimensions is the product of n averages.

1. A point is:

a. the idea of location or place.

b. the foundational geometric object.

c. the object from which all other geometric objects are defined.

2. Distance is the idea of proximity.

2. Path describes the distance between any two points.

3. Between any two points, there are infinitely many paths.

4. The measure of distance is called length.

5. The path with the least length between any two points is called a straight line.

6. Area is the idea of surface.

7. A plane is a flat area.

8. Given any magnitudes, an average is that value each of the magnitudes would

have if they were equal or made to be equal.

9. Straight lines in a plane which never meet even if extended indefinitely at either end,

are called parallel lines.

10. A geometric object whose opposite sides are equal and parallel is called a parallelogram.

11. For any parallelogram, the average length of the lengths of infinite parallel lines it contains, is

equal to one of its sides.

12. A triangle is a geometric object described by the shortest paths joining three points in a plane.

13. The point at which two planar lines meet or intersect in a plane is called a vertex or corner.

14. An angle is the idea of orientation or inclination between two intersecting planar lines from

their vertex in the plane.

15. If two planar lines cross each other in a plane forming four equal angles, then each of these

angles is called a right angle.

16. Any triangle that contains one right angle is called a right angled triangle.

17. The side of greatest length in a right angled triangle, is called a hypotenuse.

18. A diagonal is a line that cuts a parallelogram in two identical triangles.

19. Any parallelogram whose sides all meet at right angles is called a rectangle.

20. The diagonal of a rectangle is the hypotenuse of the two right angled triangles.

21. The shortest length between the diagonal end points of a rectangle is the diagonal.

22. Comparison of parallelogram areas is called rectangular area measure.

23. For any right-angled triangle it can easily be proved that the square on the hypotenuse is

equal to the sum of the squares formed on the remaining two sides of the same triangle.

(Pythagoras)

24. The shortest length between any two points in a plane can be indirectly described by the

square of the hypotenuse length, if these points are the endpoints of the hypotenuse.

1. A magnitude is the concept of size or dimension.

2. The comparison of any two magnitudes is called a ratio.

3. A ratio of two equal magnitudes is called a unit.

4. A magnitude that is completely measurable by another magnitude is a rational ratio of

magnitudes.

5. A rational ratio in which the antecedent and consequent magnitudes are related as

factor/multiple or multiple/factor, is called a number.

6. Any magnitude that is a multiple of a unit is called a natural number.

7. Zero is the idea of no magnitude.

8. A fraction is a ratio of two numbers.

9. If for any ratio, one magnitude is an exact multiple of the other, then such a ratio is a fraction if

and only if, each of the magnitudes are multiples of a unit.

10. If any magnitude or ratio of magnitudes cannot be completely measured, then it is called an

incommensurable magnitude or ratio of magnitudes.

1. A symbol represents a magnitude.

2. An expression is given in terms of magnitudes and symbols.

3. An algebraic difference is the comparison of two expressions.

4. Two expressions are equal if their algebraic difference is zero.

5. An equation is formed from two equal expressions.

6. An inequality is formed from two unequal expressions.

7. The difference of zero and any expression is the expression.

1. The *difference* (or subtraction) of two magnitudes is that magnitude which describes how

much the larger exceeds the smaller.

2. The difference of equal magnitudes is zero.

3. The sum (or addition) of two magnitudes is that magnitude whose *difference* with either of the

two magnitudes is either of the two magnitudes.

4. The quotient (or division) of two magnitudes is that magnitude that *measures* either magnitude

in terms of the other.

5. If a *unit* is divided by a magnitude into parts, then each of these parts of a unit, is called the

reciprocal of that magnitude.

6. Division by zero is undefined.

7. The product (or multiplication) of two magnitudes is the quotient of either magnitude

with the reciprocal of the other.

8. The difference of any magnitude and zero is the magnitude.

Observe that **all** the basic arithmetic operations are defined in terms of *difference*.

(*) It is only possible to *measure* magnitudes by comparing the same qualitatively or quantitatively, that is, the prior definition of *difference* is mandatory.