# Proof: By Euclid

• Let any multitude whatsoever of numbers, $A$, $B$, $C$, $D$, $E$, $F$, be in continued proportion, (starting) from a unit.
• I say that the third from the unit, $B$, is square, and all those (numbers after that) which leave an interval of one (number). • And the fourth (from the unit), $C$, (is) cube, and all those (numbers after that) which leave an interval of two (numbers).
• And the seventh (from the unit), $F$, (is) both cube and square, and all those (numbers after that) which leave an interval of five (numbers).
• For since as the unit is to $A$, so $A$ (is) to $B$, the unit thus measures the number $A$ the same number of times as $A$ (measures) $B$ [Def. 7.20] .
• And the unit measures the number $A$ according to the units in it.
• Thus, $A$ also measures $B$ according to the units in $A$.
• $A$ has thus made $B$ (by) multiplying itself [Def. 7.15] .
• Thus, $B$ is square.
• And since $B$, $C$, $D$ are in continued proportion, and $B$ is square, $D$ is thus also square [Prop. 8.22].
• So, for the same (reasons), $F$ is also square.
• So, similarly, we can also show that all those (numbers after that) which leave an interval of one (number) are square.
• So I also say that the fourth (number) from the unit, $C$, is cube, and all those (numbers after that) which leave an interval of two (numbers).
• For since as the unit is to $A$, so $B$ (is) to $C$, the unit thus measures the number $A$ the same number of times that $B$ (measures) $C$.
• And the unit measures the number $A$ according to the units in $A$.
• And thus $B$ measures $C$ according to the units in $A$.
• $A$ has thus made $C$ (by) multiplying $B$.
• Therefore, since $A$ has made $B$ (by) multiplying itself, and has made $C$ (by) multiplying $B$, $C$ is thus cube.
• And since $C$, $D$, $E$, $F$ are in continued proportion, and $C$ is cube, $F$ is thus also cube [Prop. 8.23].
• And it was also shown (to be) square.
• Thus, the seventh (number) from the unit is (both) cube and square.
• So, similarly, we can show that all those (numbers after that) which leave an interval of five (numbers) are (both) cube and square.
• (Which is) the very thing it was required to show.

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