Definition:Similar Numbers

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Definition

Similar Plane Numbers

Let $m$ and $n$ be plane numbers.

Let:

$m = p_1 \times p_2$ where $p_1 \le p_2$
$n = q_1 \times q_2$ where $q_1 \le q_2$

Then $m$ and $n$ are similar if and only if:

$p_1 : q_1 = p_2 : q_2$


Similar Solid Numbers

Let $m$ and $n$ be solid numbers.

Let:

$m = p_1 \times p_2 \times p_3$ where $p_1 \le p_2 \le p_3$
$n = q_1 \times q_2 \times q_3$ where $q_1 \le q_2 \le q_3$

Then $m$ and $n$ are similar iff:

$p_1 : q_1 = p_2 : q_2 = p_3 : q_3$


In the words of Euclid:

Similar plane and solid numbers are those which have their sides proportional.

(The Elements: Book $\text{VII}$: Definition $21$)