なるときは又
にして,且此二つは同一の關係なり.げにも先づ α : β = γ : δ {\displaystyle \alpha :\beta =\gamma :\delta } なるとき此相等しき二つの比の値を ξ {\displaystyle \xi } と名づくれば α = β ξ {\displaystyle \alpha =\beta \xi } , γ = δ ξ {\displaystyle \gamma =\delta \xi } よりて α ( δ ξ ) = ( β ξ ) γ {\displaystyle \alpha (\delta \xi )=(\beta \xi )\gamma } 乘法の組み合はせの法則により α δ ⋅ ξ = β γ ⋅ ξ {\displaystyle \alpha \delta \cdot \xi =\beta \gamma \cdot \xi } 隨て α δ = β γ {\displaystyle \alpha \delta =\beta \gamma } .又逆に α δ = β γ {\displaystyle \alpha \delta =\beta \gamma } なるときは α : β {\displaystyle \alpha :\beta } の値を ξ {\displaystyle \xi } と名づけ, α = β ξ {\displaystyle \alpha =\beta \xi } を得,隨て順次 β ξ δ = β γ {\displaystyle \beta \xi \delta =\beta \gamma } , ξ δ = γ {\displaystyle \xi \delta =\gamma } , γ : δ = ξ {\displaystyle \gamma :\delta =\xi } を經て α : β = γ : δ {\displaystyle \alpha :\beta =\gamma :\delta } に達す.是故に
は必ず相隨伴す.
比の兩項に 0 {\displaystyle 0} と異なる同一の數を乘ずるも,比の値變することなし,卽ち α : β = α γ : β γ {\displaystyle \alpha :\beta =\alpha \gamma :\beta \gamma } げにも α ( β γ ) = β ( α γ ) {\displaystyle \alpha (\beta \gamma )=\beta (\alpha \gamma )}