“Δός μοί ποῦ στῶ καὶ κινῶ τὴν γῆν,”  translated to something like “give me a place to stand on, and I will move the Earth”, is a quote attributed to Archimedes. But what did Archimedes mean by this? Archimedes was a Greek mathematician who lived in Sicily around 200 BCE. His personal life is relatively mysterious; we mainly know him through his inventions and mathematical feats. His works delve into a variety of topics, but they mainly deal with geometry and physics. There are many extraordinary things about Archimedes’ works: the rigor with which he proves things was exceptional for his time, rivaled only by other Greek mathematicians like Euclid and Diophantus; he was also a great physicist due to his strong intuition in geometry. One of his works, Περὶ ἐπιπέδων ἱσορροπιῶν, meaning “On the Equilibrium of Planes”, showcases both his intuition and rigor. The work proves from foundations how levers operate, and it still largely holds up to this day.

Useful Mathematical Terminology

The works in Archimedes’ On the Equilibrium of Planes that we will be analyzing are about the mechanics of levers. His work is mathematical in nature, so there are several components of mathematical proofs to define before we can proceed, which are given below:

• Proof: A proof is the process of showing something we don’t know to be true with something we do know to be true through careful logical steps.

• Postulate: Postulates are logical statements that are not proven, instead they are assumed for their usefulness. These assumptions can then be used to prove other mathematical statements called propositions.

• Proposition: Propositions are not assumed, rather they must be proven true from both the set of assumed postulates, and from propositions already proven to be true.

Archimedes’ Postulates

Archimedes uses two postulates to build up three propositions [1], which are then used to prove that, given two weights on a lever, the lever will balance at distances inversely proportional to the weights. The two postulates he initially assumes are as follows:

Postulate One: 

• Part 1: A lever that has two equal weights oppositely located equal distances from the fulcrum must be in equilibrium. 

• Part 2:  A lever that has two equal weights oppositely located at unequal distances from the fulcrum is not in equilibrium. Instead the lever is inclined toward the weight which has the greater distance. 

Postulate Two: Given two weights on a lever in equilibrium, if anything is taken away from one of the weights, the lever ceases being in equilibrium. Instead, the lever inclines toward the weight from which nothing was taken. 

Notice that Postulate One deals with equal weights and their equilibrium while Postulate Two deals with potentially unequal weights and what disturbs their equilibrium. The difference is subtle. However, mathematicians are quite nitpicky about which assumptions they make, so the logical difference between the two is important. Greek mathematicians carefully defining their assumptions is partly what makes their works so timeless. Starting with assumptions we observe experimentally, which we have in the form of postulates, that we can mathematically develop a model until it is able to predict unintuitive results. Thus, Archimedes’ initial postulates and propositions may seem obvious, but we end up finding the not so obvious after some exploration. 

It’s Proofin’ Time

As far as Archimedes' proofs go, they can be quite technical, so we will only prove a lesser proposition to develop intuition of what goes into the mathematics. Afterward we will explore the results of his big proofs. 

Proposition One: Unequal weights will balance at unequal distances, and the greater weight will be at a lesser distance from the lever.

Notice that both of the postulates we have don’t make a statement exactly the same as this proposition. In mathematics, if a statement is even just a little different than the ones we know to be true, we want to demonstrate that the new statement is still true; otherwise, we’re making an assumption when we don’t need to, and sometimes we’re just wrong! Archimedes’ proof goes something like this:

1. Take the given setup: unequal weights at unequal distances

2. Then, given the setup, we use a proof by contradiction to demonstrate the greater weight is at the lesser distance (from the fulcrum). 

Interlude: Proof by Contradiction

Proof by contradiction is a useful tool in logic that one can apply almost anywhere. Basically, a proof by contradiction goes like this:

1. We expect that the statement is true, so let’s make the assumption that it's false.

2. If that assumption leads to a contradiction, assuming we didn’t make any errors in the rest of our proof, the problem is the assumption itself.

3. A statement can only be true or false; therefore, if it’s not false, it’s true.

An example would be if you can’t find your much needed house keys. Let’s say you haven’t left the house since the last time you had your house keys on your person. You’ve looked everywhere but you still can’t find your house keys anywhere in the house. Then you might be thinking they’re outside. Let’s assume the keys are outside; then if the keys are outside, you must have taken the keys outside, but that couldn’t be, because you never went outside. Thus, the keys must still be inside (assuming there aren’t any key stealing gnomes living in your garden).

Proof of Case One:

Now let’s apply proof by contradiction (also known as reductio ad absurdum) to make Archimedes happy. We assume for the sake of contradiction the opposite of Proposition One: given unequal weights on a lever in equilibrium, either the distance between the two weights is equal, or the distance to the greater weight is longer. Notice there are actually two cases we must prove to be false; we will only handle the first case, although the second case is similar in nature.

Case 1: Two unequal weights at equal distances are in equilibrium. 

Setup: We’re assuming the left weight, which we will call m1 (for mass #1), is heavier, while the lighter weight located on the right of the lever is called m2. These weights will be assumed to be in equilibrium (even though this looks cursed).

Step #1 (Manipulations): We will now manipulate our initial setup to arrive at the contradiction we expect. Because the two weights are at equal distances, let’s take away the weight (m1 – m2) from our heavier weight of  m1 on the left, thus giving us two equal weights.

Left weight = m1 – ( m1 – m2 ) = m2 = Right weight

Step #2 (Applying Postulates): Notice that one of our postulates actually applies here: Postulate Two says if we have two weights setting a lever in equilibrium, and we take away any weight from one end, then the lever inclines toward the other end. Thus, removing weight from the left side, the lever has inclined toward the right side.

Step #3 (Contradiction): The lever, having equal weights at equal distances, is inclined right. This doesn’t make sense, because equal weights at equal distances must be in equilibrium according to our first postulate, thus we’ve arrived at the contradiction we needed! 

Step #4 (Sealing the proof): Because this statement cannot be true, it must be false. Therefore, the distances on the lever cannot be equal. Thus, we’re left with one option: the distance to m2 is greater than the distance to m1. This is exactly Archimedes’ first proposition, so we’ve successfully proven an Archimedes’ proposition!

It’s important to reiterate there are three possible setups for the lever, two of which are incorrect. The other case, in which we assume the distance to the greater weight is longer, can similarly be shown to be a contradiction. Once both of these cases have been ruled out, this is when we know the first case, Archimedes’ actual proposition, is true.

Advice to the reader: when reading math proofs, it’s normal to not understand a math proof after reading it once or twice, even for experienced mathematicians and professors. This proof may require rereading several times to feel confident about one’s understanding.

Archimedes’ Law of the Lever

Archimedes goes on to use this statement to prove three other propositions, the final of which we’re interested in for its utility. 

The Law of the Lever:

Given a lever in equilibrium with masses m1 and m2, and given distances d1 and d2

m1 / m2 = d2 / d1

An important observation to make is that Archimedes chose to assume the dynamics of how levers physically behave. He never explained why the levers behave this way; rather he explained what we can understand just from the behavior of levers that he was able to observe (Postulate One and Postulate Two). We unlock a new level of understanding when we’re able to go even deeper, and the model of Newtonian mechanics (a model of forces and work) allows us to do that.

Newtonian Mechanics: Forces and Torque

We will prove several of the statements in Archimedes’ work using Newtonian mechanics, but first, we must define several of the following concepts:

• Force: Colloquially, a force is a “push” or “pull” that can act on an object; in physics, this is more precisely described by a vector. A vector is a magnitude (a positive quantity of something) and a direction.

• Gravitational Force: Gravitational force is a particular type of force produced by matter (matter pulls other matter toward it). In the case of the lever, the weights on either side of the lever are being gravitationally pulled toward the ground by the mass of the Earth. This pull can be mathematically described by the equation: 

F = m * g

Where m is the mass of a weight and g is the gravitational constant (g encapsulates the strength of the pull of the Earth on an object at the Earth’s surface).

• Torque [2]: Torque can be described as the rotational force that acts on an object. In the case of the lever, torque can be mathematically described by the equation:

T = F * d

Where F is a force and d is a distance from the point of rotation. Keep in mind that the distance d is positive or negative depending on whether we’re talking about the left or right side of the lever.

Newtonian Mechanics: Alternative Proofs for Levers

We will now prove Postulate One using Newtonian Mechanics (scroll back up to remind yourself of Postulate One if you have forgotten).

Proof of Postulate One: Note that a lever in equilibrium has zero net torque acting on it (it is not actively rotating), and a lever inclining in one direction is undergoing a nonzero net torque.

Case 1: We have two equal weights on either end of equally distanced arms of a lever. If the weights are equal

m1 = m2

Then, using the equation we gave for gravitational force, the gravitational forces applied by either weight must also be equal:

F1 = g * m1 = g * m2 = F2

If the forces being applied by either weight are equal, then the torques being applied must also have the same magnitude. But notice the distances d1 and d2  extend equally in opposite directions. In other words,  -d1 = d2. This causes rotating forces to oppose one another on the lever:

T1 +  T2 =  F1 * d1  + F2 * d2 = F1 * d1  + F1 * (-d1) =  F1 * d1  - F1 * d1 = 0

Thus, summing the two torques together yields a net torque of zero. So we have two rotational forces canceling each other, leaving us with a situation where essentially no torque is being imposed on the lever.

Case 2: If we reduced the masses of one of the weights, it can be shown with the same equations that this reduces the torque for one of the arms of the lever, thus causing the lever to incline in the other direction.

We can similarly prove Archimedes’ other postulate. Thus, knowledge of Newtonian mechanics and the law of gravitation supplies us with enough power to prove both of his postulates, and all of his other works follow from these postulates. So, what we’ve really done is expand on his work; we didn’t have to touch his actual mathematical justifications, rather, we demonstrated that they stem from even more fundamental assumptions. In mathematics and physics, it’s always important to ask whether our foundational assumptions really need to be assumptions, or whether they are caused by something else.

Newtonian mechanics also provides us with an alternative proof of the Law of the Lever.

Newtonian Proof of the Law of the Lever 

Setup: On the left side, we have a distance of d1 from the fulcrum to the weight m1, on the right side a distance of d2 from the fulcrum to the weight m2. Due to the Law of Gravity, there must be a force F1 acting on the weight m1, and a force F2 acting on the weight m2.

Applying our Equations: We can now define the torque acting on each side of the lever from our setup:

T1 =   F1 * d1

T2 =   F2 * d2

Summing the torques together, and assuming we have a net zero torque:

F1 * d1 + F2 * d2 = 0

Subtracting the product F2 * d2, we obtain:

F1 * d1 = -F2 * d2

A tricky detail is our torque still has a direction, but we are concerned only with the ratios of the magnitudes of our force and distances, so we’re going to take the magnitude, or absolute value, of both sides:

F1 * d1 = |F1 * d1 | = | -F2 * d2 | = F2 * d2

Now we can divide both sides by F2 * d1, giving us:

F1 / F2 = d2 / d1 

We’re actually done now, as F1 = m1 * g and  F2 = m2 * g:

F1 / F2 = m1 * g / m2 * g = m1 /m2

Substituting m1/m2 for F1/F2, we have our canonical expression: 

m1 / m2 = d2 / d1

Because of Archimedes’ mathematical rigor, we were able to directly expand onto Archimedes’ works with Newtonian theory. Newton’s law of gravitational force (which is a foundational assumption here) allows us to introduce more sophistication and more generalization to Archimedes’ already robust framework. Generally speaking, the best set of assumptions is the simplest possible set of assumptions that still give us what we want.

Big Stick

Now to get back to our original question, what did Archimedes mean when he said he could move the Earth with a lever? We can explain it something like this: take a lever with Archimedes on one end (he is mass m1) and Earth on the other end (mass m2). Then if he weighs, for example, a septillion times less than the Earth, a lever of a septillion times greater distance from the fulcrum (if any such lever exists) could hypothetically move the Earth! (assuming the lever doesn’t break).eak). 

Notes from the author:

[1] There are a total of seven postulate and fifteen propositions in Book 1 of the work

[2] The general relationship for torque is a cross product:  τ = F x d, where the bolded letters are vectors; in this particular setup, the force is perpendicular to the lever, thus simplifying the cross product: F x d = F * d

Resources

Archimedes. (2009). ON THE EQUILIBRIUM OF PLANES, BOOK I. In T. L. Heath (Ed.), The Works of Archimedes: Edited in Modern Notation with Introductory Chapters (pp. 189–202). Cambridge: Cambridge University Press.