Boyle, Charles and Gay-Lussac

Before the complete equation, three partial laws were discovered, each obtained by keeping one quantity constant.

Boyle-Mariotte's law (at constant temperature). Compress a gas, its pressure rises, and the product P·V stays constant:

   T fixed :   P · V = constant        (if V is halved, P doubles)

   P
   |•
   |  •
   |     •._            curve P·V = const (hyperbola)
   |         •.._______
   +------------------> V

Charles's law (at constant pressure). Heat a gas, it expands; the volume is proportional to the absolute temperature:

   P fixed :   V / T = constant        (T in kelvins ; V increases with T)

Gay-Lussac's law (at constant volume). Heat a gas trapped in a fixed volume, its pressure climbs, proportional to temperature:

   V fixed :    P / T = constant        (T in kelvins ; P increases with T)

These three laws say the same thing from three angles: P rises when you compress, V rises when you heat at fixed pressure, P rises when you heat at fixed volume. Each time, temperature is in kelvins — that is what makes the proportions exact.

The decisive step is to notice that these three laws can be united into a single one, valid when everything varies at once. That is the ideal gas law, which contains them all.