I was going through a copy of Leonardo da Vinci’s notebooks, and the following short passage caught my attention.

To heat the water for the stove of the Duchess, take four parts of cold water and 3 parts of hot water.

“Cold water” and “hot water” are both vague terms. But given that this was written in the late 15th century, over 100 years before Santorini invented the first scaled thermometer, we can’t blame da Vinci for being imprecise. Yet a little basic calorimetry and simple Cartesian graphs can help us come up with a range of temperatures that he was actually referring to.

First the simple calorimetry— we will assume that no heat is lost to the surroundings. If the transfer is done quickly enough, most of the heat lost by the hot water is gained by the cold water. The quantity of heat either gained or lost is equal to the mass of water(m) × specific heat of water(c) × change in temperature ( final temperature (z) – initial temperature). We will let y = initial temperature of the hot water and x = initial one for the cold water. Since the heat is lost by the hot water, we have to use a negative sign in front of its quantity of heat. Thus using the ratio of 4 parts of cold to 3 parts of hot water, we obtain:

### -3mc(z – y) = 4mc(z – x)

The mc terms cancel, so solving for z, we obtain:

### z = (4x + 3y) /7

If we assume that the Duchess, like most people, preferred her bathwater to be at or slightly above body temperature, say between 37 and 40 °C, we can then replace z with these values, one at a time, and obtain these two inequalities:

### y ≤ 7(40)/3 – (4/3) x   and  y ≥ 7(37)/3 – (4/3) x

We will assume that by cold water, da Vinci meant anything between 0 and 20°C. Hot water could have been from 40 to 95°C.

### 0 ≤ x ≤ 20 and 40 ≤ y ≤ 95

But as we shall see, the constraint of the final temperatures will narrow that range of possibilities. We plot all 4 inequalities and come up with the overlapped region of all four inequations, the so called polygon of constraints—in this case a parallelogram with the four vertices shown.

The graph reveals that to have bath water in between 37 and 40°C and to use 4 parts of cold water and 3 parts of hot water, the hot water could not have been cooler than 59.7°C. If the cold water was at 5°C, the range of possible temperatures for hot water could only be between 79.7 and 86.7°C.  Then of course, each cold water-value between 5 and 20 has its own possible range of hot water values within the possible extremes of 59.7 and 86.7°C.

I don’t think da Vinci would be too impressed with any of the above! He may very well have told me that I would be better off spending time drawing, painting or inventing something, the pastimes that he cherished the most.