AXIOMS ILLUSTRATED.
Axiom 1.
The triangle A is equal to the triangle C.
The triangle B is also equal to the triangle C.
What do you think of the two triangles A and B? Why?
If two things are separately equal to the same thing, they are equal to each other.
Axiom 2.
The square A is equal to the square B.
To the rectangle C add the square A, and we have an L pointing in what direction?
To the same rectangle C add the square B, and we have an L pointing in what direction?
Which is larger, the L pointing to the left, or that pointing to the right?
To what same thing did you add two equals?
What two equals did you add to it?
What was the first sum?
The second?
What do you think of the two sums?
If equals be added to the same thing, the sums will be equal.
Axiom 3.
The square A is equal to the square B.
From the inverted T take away the square A, and we have an L pointing in what direction?
From the same Fig. T take away the square B, and we have an L pointing in what direction?
Which is larger, the L pointing to the right, or that pointing to the left?
What two equal things did we take away from the same thing?
From what same thing did we take them away?
What did we find true of the two remainders?
If equals be taken from the same thing, the remainders will be equal.
Axiom 4.
The rectangle 1 2 is equal to the rectangle 1 3.
From the rectangle 1 2 take away the square A, and what rectangle remains?
From the rectangle 1 3 take away the same square A, and what rectangle remains?
Which is greater, the rectangle B, or the rectangle C?
What same thing did we take away from equals?
From what did we first take it?
What remained?
From what did we next take it?
What remained?
What did we find true of the two remainders?
If the same thing be taken from equals, the remainders will be equal.
Axiom 5.
If equals be added to equals, the sums will be equal.
Axiom 6.
If equals be subtracted from equals, the remainders will be equal.
Axiom 7.
If the halves of two things are equal, the wholes will be equal.
Axiom 8.
Every Whole is equal to the sum of all its parts.
Axiom 9.
From one point to another only one straight line can be drawn.
Axiom 10.
A straight line is the shortest distance between two points.
Axiom 11.
If two things coincide throughout their whole extent, they are equal.
