Equipped with only Requirement 2 (NOT axiom because there is no use of axioms in sound mathematics) of Euclid's Elements and Proposition 12 of Book V, the Ancient Greeks established all the theory of arithmetic with numbers.

Requirement 2 states that any shortest distance can be extended to a greater shortest distance or diminished to a shorter shortest distance. In more common language: A straight line can be extended or diminished.

Proposition 12 states that any ratio p:q is proportional to another ratio r:s if p is to r as q is to s. All of p, q, r and s are magnitudes (line segments or shortest distances are used throughout the Elements).

Thus, given _ : _ _ where the line segment _ appears before the colon and the line segment _ _ appears after the colon, we call them the antecedent and consequent line segments respectively.

Prop. 12 says that as long as we extend an antecedent by itself for every consequent extended by itself, the resulting ratio is proportional (NOT equal!) to the original ratio. Also, prop. 12 says that as long as we diminish an antecedent by equal parts of it for every consequent diminished by equal parts of itself, the resulting ration is proportional to the original ratio.

Subtraction and Addition:

Given the ratios _ _ : _ _ _ and _ : _ _ _ _ _ _, we first ensure that they are expressed in terms of the same consequent.

_ _ : _ _ _ is proportional to _ _ _ _ : _ _ _ _ _ _

Thus, we have ratios

_ _ _ _ : _ _ _ _ _ _ [A]

and

_ : _ _ _ _ _ _ [B]

The difference [A]-[B] is given by diminishing the antecedent of [A] by _ resulting in the ratio

_ _ _ : _ _ _ _ _ _ [C]

The measure of [A] is 2/3, the measure of [B] is 1/6 and the measure of [C] is 1/2. That is, 2/3 - 1/6 = 3/6 = 1/2.

The sum [A]+[B] is given by extending the antecedent of [A] by _ resulting in the ratio

_ _ _ _ _ : _ _ _ _ _ _ [D]

And the measure of [D] is 5/6.

Division is accomplished by finding a ratio that represents the quotient of the given ratios. So given the ratios _ _ : _ _ _ and _ : _ _ _ _ _ _, we have a quotient ratio _ _ : _ _ _ :: _ : _ _ _ _ _ _ but since _ _ : _ _ _ is proportional to _ _ _ _ : _ _ _ _ _ _ , we have the quotient ratio _ _ _ _ : _ _ _ _ _ _ :: _ : _ _ _ _ _ _

which is equivalent to _ _ _ _ : _ :: _ _ _ _ _ _ : _ _ _ _ _ _ or just _ _ _ _ : _ because _ _ _ _ _ _ : _ _ _ _ _ _ has a measure of the unit.

Thus, the measure of _ _ _ _ : _ is 4/1, that is, 2/3 -:- 1/6 = 4/1 as expected.

Multiplication is accomplished by forming a quotient where instead of one antecedent measuring the other, the second ratio's aliquot parts are switched so that the consequent becomes the antecedent and vice-versa. In other words, a reciprocal measure takes place.

_ _ : _ _ _ :: _ _ _ _ _ _ : _

is proportional to:

_ _ : _ _ _ :: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ : _ _ _

is proportional to:

_ _ : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

is proportional to:

_ : _ _ _ _ _ _ _ _ _ whose measure is 1/9, that is,

2/3 x 1/6 = 2/18 = 1/9.

I leave the second part of the multiplication as an exercise.

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****A crank is one who cannot be convinced in the face of overwhelming evidence. ****

The majority of mainstream mathematics academics are cranks.