2.1 Overview of how to estimate the likelihood

First, for simplicity, we assume that the water molecules in the water Caesar drank on the day are distributed across the present world (the earth crust and atmosphere), mixed uniformly with all the other water molecules, with no loss in quantity since Caesar’s death (such as, transformation into other forms of molecules or escaping to space). Let us hereafter refer to it as the “uniform-distribution assumption”. This assumption will be evaluated later. If the assumption turns out to be incorrect to a small but significant extent, we will modify the conclusion accordingly, unless the assumption turns out to be completely off the mark.

Second, in this type of problems, one of the following types of answer should be suffice, rather than a certain fairly precise number.

1. Extremely unlikely
2. Unlikely
3. Neck and neck
4. Likely
5. Extremely likely

The first tactic to take should be an order estimate — so-called astronomers’ way.

Third, the numbers of the water molecules in your and Caesar’s glasses are critical. You can think of two extreme cases to prove the point intuitively. If a water molecule was as large as a glass of 300 ml, the probability the entire content of your 300-ml glass is identical to that drunk by Caesar would be hyper-extremely small, given that there is a huge amount of water on the Earth. Conversely, if a water molecule was infinitesimally small to the extent that a glass of 300 ml contains virtually an infinite number of water molecules, the probability that your 300-ml glass contains none of Caesar’s drink would be virtually nil; in other words, your water contains one or more of Caesar’s drink in (virtually) 100% confidence. In reality, the number is very large but finite, and therefore a quantitative, if rough, calculation is required to estimate the balance.

Fourth, I can think of two strategies to make an estimate.

Strategy A is to estimate the averaged number of Caesar’s water molecules contained in your glass, considering the volume of the water on the Earth. Then, you can estimate the probability in a statistical way.

Strategy B is to calculate or estimate the probability, using the standard theory of probability. Formulation is fairly straightforward, where you can apply some approximation, but the final computation may be problematic.

2.2 Solutions

3.1 Water molecules are not confined in Caesar’s body?

The water molecules a person drinks are discharged from her/his body in forms of bodily water discharges (sweat, urine, faeces, breathing out, shedding tears, spitting, bleeding etc), while s/he is alive.

Water a human takes can be retained in the body for a timescale of 10 days or so. This timescale is estimated as follows.

Roughly ∼60% in weight of a human body is water, which is translated into 36 kg for a person of 60 kg in body weight [³]. How much a person (needs to) takes a water greatly varies, depending on many factors, including the body weight, ambient temperature, exercise level, etc. The minimum amount for an adult is roughly 1.2 litres [⁴], which is a combined values of any forms of water intake, including breathing and that in foods. A couple to few litres of water per day is recommended to drink by health organisations, which is roughly 10% of the above-mentioned 36 kg. Therefore, though some water in a human body, such as that in bones, stays in the body for a much longer time, 10 days are a rough transit timescale of the water circulation in the human body.

Consequently, most of, say over 90% of, the 300-ml water Caesar took on the day of his death was retained in his body when he died.

If he had been buried or mummified, how much water escaped to the environment and then travelled across the globe would have been debatable. However, it is known that Julius Caesar was cremated (publicly in the Roman Forum [⁵]) shortly after his death in assassination on 15 March, 44 BC, roughly 2,100 years ago from the time of writing (2020).

Therefore, pretty much all the water molecules in Caesar’s body, including those Caesar took on the day of his death, were discharged into the environment — mostly into the air and possibly partly into sewage or on the ground in a form of urine and bleeding. Either way, any of these are easy ways for the water molecules to travel far since (see the following sections for the travel paths and timescales).

3.2 Bodily transformation of water molecules?

If a human body efficiently transformed water to different molecules (or single atoms), a significant portion of the water molecules Caesar took may no longer exist by the time of his death or cremation.

However, in general, an overwheming majority of the chemical reactions that happen in animals’ bodies do not transform water molecules to something else.

An exception in a human body I know is the chemical reaction to convert starch to glucose in the human digestion system, where water is actively used and is converted (see Fig.4 of [⁶]). However, given the facts that (1) the reaction uses only one-sixth of H₂O in amount contained in the glucose generated and (2) the reaction does not give energy to the (human) body, the total amount of the water used for the reaction should be many orders of magnitude smaller than the water generated by oxidation in a human body (calorie burning or broadly speaking, digestion). After all, if anything significant in a living organism’s body, it must generate energy on balance.

Thus, pretty much all the water molecules Caesar took on the day of his death were discharged without transforming into other forms of molecules.

3.3 Water molecule transformation, loss, and generation on the Earth

Water molecules experience cycles on the Earth. Do they possibly transform or disintegrate into different molecules or atoms during the cycles? Conversely, are water molecules newly generated from some other molecules and/or atoms?

3.4 Spread of water molecules across the globe

We have already confirmed the water molecules were discharged from Caesar’s body to the ambient environment. Now, the question is whether these water molecules are distributed evenly across the globe since his death 2,100 years ago.

It is a reasonable assumption that the water molecules discharged from Caesar’s body while he was alive and before and at his cremation were evaporated quickly into the air in timescales of minutes to a day. Then, let us consider the fate of these water molecules since.

3.4.2 Life cycle of water in/on the oceans and ground

I now consider what happens with the water molecules when they come back to the Earth’s surface from the atmosphere.

Hydrological cycle on the earth. Estimates of the main water reservoirs, given in plain font in thousand (10³) km³, and the flow of moisture through the system, given in slant font in thousand (10³) km³/yr, equivalent to Exagrams (10¹⁸ g) per year. Text and figure adopted from [¹⁴].

3.4.2.1 Life cycle of water in the oceans

A majority of precipitation occurs over the ocean, or specifically, roughly 77% [¹⁴], the fact of which is unsurprising, considering that ∼71% of the surface of the Earth is the ocean (70.9%, more precisely [¹⁵]).

Water molecules in the ocean can stay for a long time with an average of ∼3,000 years [¹¹, ¹⁶], before evaporating into the air. Given that (1) the average residence time is 3,000 years, that (2) the water close to the deep ocean bed has a considerably longer lifetime, whereas that close to the surface should have a shorter lifetime, and that (3) the water that originated from Caesar’s drink should initially arrive at the ocean by one of rainfall, rivers, and ground water flows and hence should start at the surface of an ocean, the timescale of Caesar’s drink-origin water should be somewhat shorter than 3,000 years, though a quantitative discussion is difficult.

The water in the ocean, being liquid, moves around. There are oceanic currents near the surface (typically up to 400 metres in depth) with a characteristic surface speed ranging 0.05—0.5 metres per second [¹⁷]. These speeds translate into 1,600—16,000 km/yr, and thus technically, water takes up to 25 years to circle around the Earth (40,000 km for the equatorial circumference). In reality, most of the ocean currents bridge neither multiple oceans nor multiple hemispheres. Therefore, the spread of water molecules with this move is limited mostly in a hemisphere in the single ocean, where it started the travel, and accordingly the required travelling timescale is more like 10 years from one end to the other. In reality, not all the surface water in the ocean are in the current, and so the averaged travel time is probably longer, perhaps by a factor of a few, such as 30 years. In any case, the timescale for the water to spread is 2 orders of magnitude shorter than 2,100 years, though the spread is limited to a part of an ocean.

By contrast to the horizontal movement, vertical movement of the water in the ocean (often referred to as upwelling and downwelling) exhibits much lower speeds of a few metres per month or ∼1×10^-6 m/s [¹⁷]. With these speeds, a water molecule on the surface takes 300 years to reach the averaged depth of the ocean, ∼4,000 metres, or 30 years to reach 400 metres, which is the depth of the surface ocean currents. The latter timescale is comparable with that of the ocean-wide horizontal movement of the surface ocean currents. In this sense, the water that starts its journey from the ocean surface will be distributed in a considerable part across the surface of the same hemisphere of the ocean in a timescale of a few decades, though some of them can sink deep in the ocean and do not travel along the surface ocean currents any more.

In addition to surface oceanic currents, there is a global oceanic circulation called the Global Conveyor Belt [¹⁸] or the thermohaline circulation [¹⁹]. The circulation completes the entire global cycle in 1,000 years, travelling over all the major oceans. Thus, in 2,100 years, the water molecules in an ocean should complete 2 cycles in average, with which water molecules are spread in the oceans world-wide more or less evenly.

More specific history for this case would be like this. Given that Caesar’s drink-origin water started its journey from the surface of an ocean when it first reached it, and considering the fact that vertical movement of the water molecules in oceans is slow, a considerable portion of it must have remained in the ocean surface layer till the molecules got somewhat evenly distributed across the same ocean in an timescale of the initial few decades. Then, eventually (a timescale of a few hundred years), a significant part of the water molecules reached deeper in the oceans and eventually get on the Global Conveyor Belt, which spread the molecules world wide in a timescale of 1,000 years.

Consequently, the oceanic water move alone should achieve a fairly uniform world-wide distribution of Caesar’s water molecules across all the oceans. Given that the transit time of water molecules in the ocean is 3,000 years, which is longer than the timescale of interest of 2,100 years, this fact is critical.

3.4.2.2 Life cycle of water on/in the ground

Globally, roughly 23% of the world precipitation occurs over the land [¹⁴].

The fate of the water that falls on the land highly depends where it falls on; the numbers in the following sentences in this paragraph are taken from the article [¹¹] unless otherwise noted. More than a half of the water molecules that fall on the land evaporate back to the air fairly quickly with timescales ranging from minutes to 2 months, the timescale heavily depending on exactly where they fall. Water that falls on tropical land will evaporate quickly like a matter of minutes, unless it permeates the soil, whereas that in freezing environments would take much longer than 2 months before it finally melts and then evaporates. For example, it takes 6 months with the Arctic snow, unless it permeates glaciers or large ice sheets, in which case it can stay even greatly longer like a century for glaciers or even million years in extreme cases for ice sheets. Water that stays in soil takes 1 to 2 months before it is evaporated (or transpired). The residence times of the water in rivers and reservoirs/lakes are typically 2 weeks and 10 years, respectively [²⁰].

However, in reality, evaporation (plus transpiration) from the land accounts for only ∼65% of the precipitation on the land [¹⁴]. In other words, ∼35% of the water falling on the land does not evaporate back to the atmosphere (in the above-mentioned relatively short timescales). It either flows into a river/lake or sinks into groundwater, both of which eventually end up in an ocean/sea unless the molecules of interest evaporate back to the air during the process (n.b., the evaporation process accounts for a part of the ∼65%). The residence time (also called transit time) of the water in groundwater varies greatly from 2 weeks to over a million years [²¹].

To summarise, wherever water ends up on the land, except for the rare ice-sheet case and some deep groundwater, the timescale for it to evaporate to the atmosphere or to move to the ocean is by an order of 4 (or 2 for reservoirs) shorter than 2,100 years we are considering. In other words, we can interpret that the water that falls onto a land move to either back to the atmosphere or to the ocean in no time, compared with the timescale we are considering, 2,100 years.

3.5 Summary of the evaluation of the assumption

First, I have confirmed that the water molecules in the water drunk by Caesar 2,100 years ago remain basically on the Earth without transforming into other molecules or escaping to space. Also, there is no significant amount of water generated on the Earth during the period.

Second, the water molecules in the water drunk by Caesar were almost completely discharged into the environment, mostly into the air, shortly after his death at latest (stage 1).

Third, on and immediately after the day of Caesar’s death and cremation, the discharged water molecules travel in the air for typically 2,000 km in 9 days before they fall back down to the Earth surface, mostly on an ocean but some on land, in a form of rainfall (stage 2).

Fourth, those that have landed on land move either back to the atmosphere (hence repeating stage 2) or to an ocean in no time.

Fifth, the water that landed on an ocean tend to stay in the ocean for a long time of typically 3,000 years. It will initially in large part flow in the surface current and spread across the same hemisphere of the ocean which it has landed on in a couple of decades, while some evaporate to the air (back to stage 2) and some sink deeper in the ocean. In the first iteration since the discharge from Caesar’s body, few of their water molecules reached Pacific Ocean for this reason, and spread to this area must wait for either further iterations or the process explained in the next item.

Sixth, in the longer timescale, the water that landed on an ocean joins the Global Conveyor Belt and travels across all the oceans typically twice in 2,100 years. Combined with the facts that some are transported by surface ocean currents and that some evaporate to repeat stage 2, this process distributes the water molecules across the globe fairly uniformly. In particular, given the iteration starts from the ocean surface, there is much greater chance of evaporation for the water molecules to repeat stage 2, with which the process to establish uniform distribution must be enhanced.

Seventh, in the timescale of 2,100 years, the distribution of Caesar’s water molecules across the existing water in the Earth crust and atmosphere should not yet be entirely uniform in some localised places as follows. Given the fact that the residence times in polar ice sheets, deep-underground groundwater, and deepest sea bed exceed several thousand years to even a million years in some cases, Caesar’s water molecules are unlikely to permeate them to the extent to achieve the number equilibrium with the existing molecules. In other words, the distribution is more patchy than complete uniformity, biased towards the accessible water sources for humans. This biased distribution of Caesar’s water molecules works as the tendency that the glass of water you drink is likely to have more of Caesar’s water molecules than in the case with the original ideal assumption.

Eighth, the fact that the number of the cycles with the Global Conveyor Belt, which is the most major driving force towards uniform distribution of Caesar’s water molecules across the globe, is limited to only 2 suggests that there may remain some deviation from the uniform distribution, namely patchiness, in the distribution of Caesar’s water molecules across the oceans of the Earth. As a conservative estimate, let us assume only 50% of Caesar’s water molecules have been distributed uniformly across the water on the Earth surface.

Finally, the drinking (fresh) water humans use is usually collected from one of reservoirs or lakes inland, surface flows (rivers), and shallow groundwater (wells). Their residence times are up to 10 years only. Therefore, the drinking water basically originates from recent evaporation, a majority (∼85%, [¹⁴]) of which comes from an ocean. Consequently, the uniformity of the water molecules in your drinking water basically reflects that of the ocean waters, regardless of the averaged residence time of the water in the ocean.

To summarise, the “uniform-distribution assumption” is mostly justified. In a conservative estimate, one can assume only 50% of Caesar’s original water molecules are distributed uniformly across the globe.

A.1 Total amount of the water on the Earth surface

It is possible to make a rough (but precise enough) estimate of the total amount of the water on the Earth from some basic knowledge of science.

1. The radius of the Earth is 6,400 km, or the circumference of the earth is 40,000 km. This could be derived alternatively from one of the following facts, depending on what knowledge one has:
   • Light can travel around the Earth 7.5 times in a second, and the speed of light is 300,000 km/s.
   • The speed of the spin of the Earth is 1700 km/h or 500m/s, and the Earth rotates every 24 hours.
   • A trans-continental aeroplane flies at a speed of roughly 850 km/h (as you see in the monitor on a seat onboard) or a Mach number of 0.8, and it takes roughly two 12-hours flights (namely 24 hours) to get to the opposite side of the Earth.
2. The averaged depth of the ocean is 4 km (or more precisely 3.68 km [²²]). This could be guessed from the following facts combined.
   • Each continent has a geological structure called a continental shelf at the boundary with the oceans that surround the continent. Beyond the continental shelf, the seabed steeply drops to deep sea.
   • The continent shelves cover only a tiny portion in terms of the area of the oceans.
   • The depth of a continental shelf is about 200 metres (as referenced in the Convention on the Continental Shelf (1964, Geneva), though the scientific definition is different.
   • From these facts, one can surmise the average depth of the ocean is an order of magnitude larger than 200 metres, that is, a few km.
   • The fact that the depths of the deepest oceanic trenches are more than 10 km are consistent with it.
3. The oceans and seas cover roughly 70% (70.9%, more precisely [¹⁵]) of the Earth’s surface.
4. Almost all the water (96.5%, precisely) on the Earth’s surface (crust and atmosphere) in all forms (solid, liquid, and gas) combined exists in the oceans and seas.