The way I understood this proposition is we have two different sets, say set

*A*of all natural numbers and set

*B*is a consequence of set

*A*where

*a*is any element in set

*A*and

*b*is any element in set

*B*. Euclid states "If any number of numbers are proportional" Which I believe to be the same as stating "the elements from each set must be corresponding such that

*a*:

*b*." I then found through experimenting that if

*a*:

*b*then there is a term in

*A*and a corresponding term in

*B*that is the exact same proportion of the sum of some terms in the set. In other words, if you take any numbers in a and add them, then add the corresponding terms in

*B*, the sum of the

*A*terms will be another term in the set

*A*and the sum of the

*B*terms will be the corresponding value in the set B. Some examples are posted below.

The new statement I came up with is: Let there be set

*A*of real numbers and set

*B*such that set

*B*is a consequence of set

*A*and

*a*is any element in the set

*A*and

*b*is any element in the set

*B*. If the elements from each set are corresponding such that

*a*:

*b,*then the sum of any elements in

*A,*say a_o, is an element in

*A*and the sum of the corresponding elements in

*B*, b_o, is in the set

*B*such that a:b=a_o:b_o.

The way I first thought of "four number proportional" was

*a*:

*b*:

*c*:d, but after I experimented and found a counter example I realized that was not it. Then I experimented with

*a*:

*b=*c:

*d*and found that this seems to be work. What Euclid is saying, is that if we have four numbers

*a,b,c,d*such that

*ad=bc*then

*a*:

*b=*c:

*d*and if

*a*:

*b=*c:

*d*then

*ad=bc.*If we think about the definition of proportional it would makes sense because when the proportions are equal that means we are multiplying the terms in one proportion to get the other proportion. In other words we have

*c=ax*and

*d=bx,*making

*a:b=ax:bx*where

*x*is any number. Therefore, when we multiply ad and bc we find

*abx:abx*which will always be 1:1. Going the other direction, where we start with

*ad=bc,*we use algebra to find a/b=c/d. Here are some examples.

The new statement I came up with is: Let

*a,b,c,*and

*d*be any number such that

*a*<

*c*and

*b*<

*d*,

*ad=bc*if and only if

*a:b=c:d.*

Some examples are posted below.