The thing is that proportionality means a constant in the formula. $R$ is a constant, but $P$ isn't.

• Someone (Ohm) found out that $V$ is proportional to $I$: $$V \propto I$$ He found out by testing and experimenting. Everytime $V$ was doubled, $I$ was also doubled. This is what is called (direct) proportionality. We can write it with a proportionality constant, for example called $R$: $$V=RI$$ $R$ is constant, which causes the doubling of $V$ in our circuit to double $I$ as well.

• Someone then also found the relationship: $$P=VI$$ by testing and experimenting. And yes, we could write this as: $$V=P\frac1I$$ which at first sight looks like inverse proportionality. Double the $V$ should then half the $I$.
  But $P$ is not constant. If we change $V$ in our circuit, both $I$ and $P$ change as well. Doubling $V$ does not half the $I$.

Conclusion is that $R$ is a proportionality constant, while $P$ certainly isn't (because it isn't constant). It looks like inverse proportionality at first sight, but isn't.

The thing is that proportionality means a constant in the formula. $R$ is a constant, but $P$ isn't.

• Someone (Ohm) found out that $V$ is proportional to $I$: $$V \propto I$$ He found out by testing and experimenting. Everytime $V$ was doubled, $I$ doubled as well. This is what is called (direct) proportionality. We can write it with a proportionality constant, for example called $R$: $$V=RI$$ $R$ is constant, which causes the doubling of $V$ in our circuit to double $I$ as well.

• Someone then also found the relationship: $$P=VI$$ by testing and experimenting. And yes, we could write this as: $$V=P\frac1I$$ which at first sight looks like inverse proportionality. Double the $V$ should then half the $I$.
  But $P$ is not constant. If we change $V$ in our circuit, both $I$ and $P$ change as well. Doubling $V$ does not half the $I$.

Conclusion is that $R$ is a proportionality constant, while $P$ certainly isn't (because it isn't constant). It looks like inverse proportionality at first sight, but isn't.

The thing is that proportionality means that there is a constant in the formula. $R$ is a constant, but $P$ isn't.

• Someone (Ohm) found out that $V$ is proportional to $I$: $$V \propto I$$ That just means that when you double $V$, you also double $I$. Therefor you can write this with a proportionality constant, which we happen to call $R$: $$V=RI$$ This $R$ is then interpretted as a resistance. But $R$ is constant. Therefor you can double the $V$ in your circuit and that gives you the double $I$.

• Someone then also found the relationship: $$P=VI$$ And yes, we could also write this as: $$V=P\frac1I$$ which at first sight looks like inverse proportionality. That would mean that double the $V$ gives half the $I$.
  But $P$ is not constant. If I change $V$ in my circuit, I will se that both $V$ and $P$ changes. When doubling $V$, I don't get half the $I$.

Conclusion is that $R$ is a proportionality constant, while $P$ certainly isn't (because it isn't constant). The inversed proportionality relationship appeared to be not true.

The thing is that proportionality means a constant in the formula. $R$ is a constant, but $P$ isn't.

• Someone (Ohm) found out that $V$ is proportional to $I$: $$V \propto I$$ He found out by testing and experimenting. Everytime $V$ was doubled, $I$ was also doubled. This is what is called (direct) proportionality. We can write it with a proportionality constant, for example called $R$: $$V=RI$$ $R$ is constant, which causes the doubling of $V$ in our circuit to double $I$ as well.

• Someone then also found the relationship: $$P=VI$$ by testing and experimenting. And yes, we could write this as: $$V=P\frac1I$$ which at first sight looks like inverse proportionality. Double the $V$ should then half the $I$.
  But $P$ is not constant. If we change $V$ in our circuit, both $I$ and $P$ change as well. Doubling $V$ does not half the $I$.

Conclusion is that $R$ is a proportionality constant, while $P$ certainly isn't (because it isn't constant). It looks like inverse proportionality at first sight, but isn't.