The Metric System on Goldilocks Earth.

Scuba divers know that the pressure experienced at depths increases by one atmosphere every 10 meters down. That’s just serendipitous; an easy conversion that isn’t actually part of the metric system and is due to specific physical properties of Earth that are “just right”.

TL; DR:

Some units are linked by definition: a litre of water IS a Kilogram.
Herein lie 3 (and a half) examples where non-definitions also offer nice ¹ , decimal conversions.

On Earth:

• A 1 meter long pendulum takes ~1 second to swing.
• Every 10 meters (down) in ocean water, is ~1 Atmosphere.

And slightly less accurately:

• A ball dropped, will be moving at ~10 m/s after 1 second.
• The atmospheric pressure is ~1Kg/cm²

All of these are convenient coincidences and not features inherent to the metric system.

The term “Goldilocks Earth” refers to the fact that Earth is “just right” for life to develop (a bit hotter or colder temperature, shorter or longer days, slightly different chemical composition would not have supported the development and evolution of life as we know it.)
These same “perfect conditions” give rise to some very nice conversions.

The Metric System

The metric system involve a lot of prefixes, “Kilo” “Mega” “centi” etc.
They are just shorthand for “scientific notation.” a.k.a just add zeros. Let’s skip over these, and jump straight to the physical properties themselves.
Length, Weight, Pressure, and Time. How did these come about?

Length — One Meter

Definition: Length & Volume
One Meter : The distance from the equator to the pole is 10,000,000 meters,

One Litre: A ⅒ meter cube, is a litre. (also a definition.)

If Earth were slightly bigger or smaller, 1 meter and thus 1 litre would be too.

Weight — One Kilogram

Definition: A litre of (fresh) water weighs a kilogram ².
The density of water is almost constant(under pressure);
The size and the density of the planet affect the gravitational pull at the surface, and so the relationship between the volume and the weight of water.
If Earth had a bigger or smaller iron core, a kilogram would be different
(but since this is how we defined it, it would still be a kilogram.)

Pressure — One Atmosphere

Definition: weight/Area (Kg/cm²)
Definition: 1 Atmosphere is the air pressure at sea level.(at 0 °C)
Besides the size and density of the Earth (which we encountered above), air pressure is also dependant on the thickness and composition of the atmosphere.
The chemical composition (Oxygen, Nitrogen, etc.) might have been different, and thus the density and the thickness would be affected.
Any one of the five properties bolded in the above paragraph could have been different to break this coincidence:

1 Atmosphere is ~1 Kg/cm² (well, 1.033 to be exact)

Pressure at depth:

When diving in the ocean the ambient pressure increases with the depth.
This is dependent on the density of water, the level of salinity, the weight of the salt (and the gravitational pull). In the past the salinity was different and this wasn’t as nice.

The coincidence:

Every 10 meters of depth (seawater) increases the pressure by ~1 Atm.
(well, 1.003 to be exact)

As noted above, an atmosphere is actually 1.033 Kg/cm² but seawater is 3.6% heavier than freshwater (and a Kg was defined for freshwater.) so they line up nice.

Time — One Second

Earth rotates around itself at a certain speed, it is what defines days.
There are 24 hours in a day (originally day and night were split into 12 equal parts.) each has 60 minutes, each of those has 60 seconds. That’s how a second is defined.³

The coincidence:

The time it takes for a pendulum that is 1 meter long to swing from one side to the other is 1 second. (actually: 1.003 seconds)

This is dependant on the size of the planet (which gave us the meter), the density of the planet (affecting how heavy a Kg is) and the rotational speed. None of them are forced by the others, and might have been different.
This is the same “coincidence” as the fact that the gravitational pull(g) of the Earth at its surface is very close to π² m/sec².

The velocity of a dropped object

A ball dropping for 1 second.will be moving at ~10 m/s (~9.8)

Given the previous coincidence (g ~ π²); This is derived from more of a mathematical coincidence: π² is ~10.
The actual numbers are :
π² = 9.87
g is between 9.7639 and 9.833 m/sec² (location dependent)

Conclusion

The definitions of the metric units are based off various properties of Earth.

• The size of the Earth. (Circumference.)
• The density of the Earth and its iron core. (The gravitational pull.)
• The rotation speed of the Earth.
• The chemical composition on Earth (Elements present and their change over time, e.g. the salt in the oceans, the composition of the atmosphere.)
• The density of the air and the water (current values, salinity, gas mix)
• The height of the atmosphere (today)

If another Earth existed, call it Earth` (Earth prime) with slightly different properties(more potassium, less sodium, more nitrogen, a bigger iron core, smaller circumference, etc.) the metric system would be the same as we know it (though 1 meter might not be 1 meter` … ).
But on Earth`, scuba divers will not have an easy mnemonic, and the distance from the equator to the pole through Paris` is not 10 million times as much as the length of the 1 second` pendulum.

[1] This kind of nice relationship sparks joy in the physics department, but makes some mathematicians squirm at the liberal use of the concept of equality

[2] If you’ve heard weight and mass are not the same but not sure this is the right one, don’t worry about it (we’re talking about the weight of a 1Kg mass on the surface of our Earth.)

[3] — Ancient numerical systems liked 12, and 60 a lot. The Mesopotamian system⁴ was in base 60. 60 is nice because it is divisible by 2, 3, 4, and 5 (and subsequently 6, 10, 15).
One definition for a meter that was considered at the time was that a meter would be the length of a 1 second pendulum, but since that was not constant everywhere on earth it was dumped.
The coincidence is still the same; ¼ of the circumference of the earth being ~10 million such pendulums is very lucky.

[4] — Ancient finger counting methods involved the thumb touching each of the 12 segments of the remaining 4 fingers (the proximal, medial and distal phalanges.) Now, we count on all 10 figures so Modern counting systems like base 10.

Bonus :

Other non metric system coincidences that I like too much to leave out, and leave here as a gift for making it all the way down

• The half-life of Carbon 14 (which is used to date biological samples) is 5,730 ± 40 years; Willard Libby got a Nobel for the method in 1960.
  The Jewish year was 5,720.
• There are ~π 𝗑 10⁷ seconds in a year.