4. The Solution to the Dinosaur Paradox

Summarizing what is known up to this point:

1. Size matters. It has been over 385 years since Galileo first explained how size affects the properties of objects. It is a major failure of science education that so few people are aware of Galileo’s Square-Cube Law, which shows why size matters.


2. Galileo's Square-Cube Law is especially important for biology. The Square-Cube Law becomes increasingly more important as we encounter objects of greater complexity. Advanced life forms such as highly evolved animals are at the top level of this complexity and so for these species size most definitely matters. This means that the highly evolved animals are the most limited in the range of size that they can have.


3. On a global scale, numerous terrestrial Mesozoic era species far exceeded the maximum size obtainable by modern day species. We can either see this as a general comparison of the typical dinosaur or pterosaur to that of present terrestrial animals or by comparing the largest or tallest examples of Mesozoic species to the present; for example we can compare Brachiosaurus to that of both the present-day elephant and giraffe.

Ground Rules for Finding Scientific Solutions

The solution to the paradoxes of large dinosaurs and large flying pterosaurs must not be an obvious one; otherwise, someone would have figured it out long ago. Therefore, we should keep an open mind to all remotely reasonable hypotheses rather than risk excluding the correct answer without giving it serious consideration. However, while we should be open-minded in exploring all possibilities, any promising hypothesis must meet extremely stringent requirements before it can be accepted as the correct solution.

The correct solution to any scientific problem should adhere to the following general guidelines:

1. It should be a simple solution. This principle is often stated as Occam's razor: "All things being equal, the simplest solution is most likely the correct one." Whether it is F = m a, E = m c², or the key principle of evolution — that only individuals who survive to reproductive maturity can pass on their genetic code — almost all correct ideas in science are simple at their core. Some of my colleagues complain that I make everything seem too simple, so for my fellow scientists: Non sunt entia multiplicanda praeter necessitatem.


2. Preferably, it should not require the discovery or invention of a new fundamental scientific principle. While groundbreaking discoveries like E = m c² do occasionally emerge, they are typically confined to fundamental sciences and should be considered a last resort. Furthermore, any hypothesis claiming a new law or property of science requires substantial additional evidence to justify its validity. A more common issue in science is that researchers in one discipline sometimes propose hypotheses that contradict well-established principles or evidence from another field. Greater interdisciplinary collaboration is needed to ensure that proposed solutions are consistent across scientific domains.


3. Most importantly, the solution needs to be supported by evidence. This is by far the most critical requirement. Physical evidence is what separates speculative opinions from legitimate science. Before Galileo's time, many believed that logic alone was sufficient to determine truth—a mistake that some still make today. While logic is invaluable for forming hypotheses, it is not infallible when it comes to understanding nature. Once a scientific hypothesis is proposed, it must be rigorously investigated and tested. A valid hypothesis must have unique characteristics that allow for the possibility of falsification. Stated positively, a strong hypothesis, when tested, will yield supporting evidence rather than expose a fatal flaw.

Before beginning our search for a solution, we must accept the fact that the only way Mesozoic species could have been substantially larger than present-day species is if there was a global change in their physical environment. While it is true that many previous investigators have failed to resolve this paradox, our belief in a rational reality must not waver. With confidence in science, we will proceed through a step-by-step process, systematically investigating all reasonable hypotheses until the solution is found.

Scaling Factor

In seeking a simple solution, there should be a single phenomenon that accounts for the larger size of terrestrial Mesozoic animals. Furthermore, it is not enough to merely acknowledge that terrestrial Mesozoic animals were larger; to evaluate different hypotheses, we need a quantifiable measure — a scaling factor that expresses the magnitude of size differences between past and present species.

The role of mathematics in science is not always clearly understood, so let us take a moment to clarify the necessity of numbers, or data, in applying reasoning to scientific questions. One of the primary reasons scientists collect data and work with numbers is to make logical comparisons.

However, the importance of using numerical values to process logic is often overlooked in science education. Typically, when a scientific problem involves calculations, it concludes with a numerical answer. Many science educators stop at this point, assuming that students — future scientists and engineers — understand the significance of these numerical results. Unfortunately, this is often not the case. To ensure comprehension, educators should conclude investigations by asking, “Comparing this numerical result to another known value, what does it mean?”

To demonstrate how numerical data facilitates good decision-making, consider the example of determining whether classes should be canceled due to winter weather conditions. Since snowfall can range from as little as half a centimeter to more than half a meter, simply being told that snow is forecasted provides insufficient information for making an informed decision. A numerical value—the projected amount of snowfall, expressed with units — is necessary so that it can be compared to a predetermined threshold established by officials to determine whether conditions are safe for students to attend class. Throughout science, it is often essential to express information numerically so that meaningful comparisons can be made, leading to well-founded conclusions.

When addressing the paradox of how dinosaurs grew so large, we need a scaling factor to quantify how much larger they were compared to present-day terrestrial animals. Without this numerical reference, we cannot assess whether a proposed hypothesis adequately accounts for the required change in size. Thus, our first task is to determine this scaling factor.

This scaling factor represents the relative size of Mesozoic species in comparison to modern species. As our research progresses, we should be able to construct a graph depicting the scaling factor as a function of geological time.

For now, to keep the discussion simple and focus on solving the immediate problem, let us examine the peak of dinosaur gigantism, which occurred around 150 million years ago. Our immediate goal is to determine the scaling factor corresponding to approximately the middle of the Mesozoic Era — the age of the dinosaurs.

There are several methods available for determining the scaling factor between Mesozoic and present-day species. However, as is often the case in science, some methods are much more precise than others. Because of this, it would be unwise to simply average the results of all methods. A better approach is to first evaluate which methods are likely to be the most precise.

One possible method for determining a scaling factor is to compare the present-day size of long-lived species such as the dragonfly, horsetail, and crocodile to their Mesozoic-era counterparts. While this may initially seem like a reliable approach, it is not.

Our goal is to determine a scaling factor that reflects a global change in the physical environment affecting all terrestrial species. However, the environment that influences a species’ size, form, and behavior consists of both physical and biological factors, and it can often be difficult to determine which has the greater impact. Ideally, we want to be confident that observed changes in species size are due solely to changes in the global physical environment.

Unfortunately, the biological environment — the interaction between species — complicates our ability to draw clear conclusions about the relationship between species size and changes in the global physical environment. However, we can remove these confounding biological factors by focusing on large, dominant species whose size would not be significantly limited by competition with other species.

Neither the dragonfly nor the horsetail meets this requirement of being dominant terrestrial species throughout their existence.

When dragonflies first appeared over 300 million years ago, they were gigantic, with wingspans reaching 70 cm, and they were the dominant aerial predators. However, by the early Mesozoic era, the first pterosaurs had appeared, and during the last third of the Mesozoic, birds had also evolved.

The respiratory system of insects is inefficient for larger body sizes. This was not a serious limitation for dragonflies and other insects when they had no competition from flying vertebrates. However, once birds evolved, they dominated the skies over dragonflies. Birds have a high-temperature metabolism combined with a highly efficient circulatory and respiratory system—one even superior to that of mammals. Their high metabolism, along with excellent aerodynamic form, gives birds both greater speed and maneuverability than flying insects. Today, dragonflies no longer grow as large because increasing in size would make them slower and more vulnerable to becoming a meal for birds.

The horsetail, a plant, followed a similar path to that of the dragonfly. Like the dragonfly, horsetails first appeared during the Carboniferous period, growing exceptionally large — up to 30 meters tall. However, since that time, other plant species have evolved that have proven to be superior competitors. With only a few exceptions, most horsetail species today grow no more than 1.5 meters tall.

Unlike the dragonfly and the horsetail, the crocodile presents a different set of challenges when used as a scaling tool. Crocodiles spend much of their lives in the water, making them a poor gauge for environmental changes affecting terrestrial species in general. Additionally, their behavior may have changed over time — it is possible that ancient crocodiles hunted on land as much as in water. While crocodiles have persisted as large, dominant species throughout history, their semi-aquatic lifestyle compromises their usefulness as indicators of changes in the scaling factor.

Our next approach is to compare different species that are closely related enough to share similar biological structures. While mammals and dinosaurs belong to different classifications, both are vertebrates, making them viable subjects for comparison. This should be an obvious point, yet some confusion may arise due to the difficulty paleontologists have had in explaining the immense size of dinosaurs and pterosaurs. Paleontologists have frequently suggested that these animals had lighter yet stronger bones, more powerful muscles, or some other exceptional biological advantage. However, when examined scientifically, these claims lack a solid foundation.

From an evolutionary standpoint, the idea that past species were biologically superior or inferior is flawed. Species evolve in response to their environment — adapting when possible or going extinct when they cannot. The vertebrates of the Mesozoic era were suited to their environment, just as modern species are adapted to today’s world. Despite differences in size and shape, all vertebrates — past and present — are composed of the same biological building materials.

At the core of vertebrate bone structure is hydroxyapatite, a mineral with a fixed chemical composition:

Ca₁₀ [PO₄]₆ [OH]₂

This structure is consistent across all vertebrates, from fish and amphibians to birds and mammals. There is no evidence that Mesozoic vertebrates had bones that were fundamentally stronger or weaker than those of modern animals. The material properties of bone have remained the same, making claims of biologically superior dinosaur bones unfounded.

Furthermore, the claim by paleontologists that dinosaurs were lighter and stronger because they had hollow bones is misleading. Dinosaurs are not unique in this regard, as all or nearly all vertebrates have hollow bones. Likewise, the follow-up claim that dinosaurs were significantly lighter or stronger due to having hollow or more extensively hollowed bones is also incorrect. Increasing the degree of hollowness does not significantly reduce a dinosaur’s weight; instead, it makes the bones weaker rather than stronger.

Likewise, the idea that dinosaur muscles were significantly stronger — or weaker — than those of modern vertebrates lacks any plausible biological basis. Muscle strength is dictated by cellular structure and biochemistry, which have not undergone radical changes across vertebrate evolution. Modern vertebrates — including birds, mammals, and reptiles — exhibit a wide range of muscle performance suited to their ecological niches. If prehistoric muscle tissue had been dramatically superior, we would expect to see evidence of this advantage persisting in modern species. Instead, we observe a variety of muscle adaptations, none of which suggest that Mesozoic vertebrates had an inherent strength advantage — or disadvantage — over their modern counterparts.

The clade known as sauropods, which includes species such as Apatosaurus, Brachiosaurus, Argentinosaurus, and Patagotitan may have filled an ecological niche similar to that of the present-day African elephant. This similarity is evident in the size gap between the largest terrestrial animal and the rest of the fauna in both cases. It is possible that both sauropods and elephants evolved to push the limits of physical size as a strategy to deter carnivores from attacking them. A comparison between the sizes of sauropods and elephants suggests a scaling factor ranging from 2.5 to 3.5.

However, comparing the tallest animals may provide a more precise scaling factor. In species that reach extreme heights, it is more apparent that their cardiovascular systems are being pushed to their limits.

Present-day giraffes and Brachiosaurus of the Late Jurassic period occupied similar ecological niches in their respective eras. Both species evolved long necks to extend their reach to foliage beyond the grasp of other herbivores. Maximizing mouth height — primarily through neck extension and, to a lesser extent, leg length — enhanced their survival. Brachiosaurus was unusual among dinosaurs in having longer forelimbs than hind limbs. A reasonable hypothesis for this trait is that it further extended Brachiosaurus’ vertical reach, allowing it to access the highest foliage. This suggests that its maximum height was constrained by the limits of its cardiovascular system.

The vertical heart-to-head distance in Brachiosaurus was approximately 8.8 meters, compared to 2.8 meters in giraffes, yielding a scaling factor of 3.1.

Effective Gravity

Another reason to focus on the cardiovascular blood pressure problem rather than the overall size paradox is that paleontologists often present unrealistically low mass estimates for the largest dinosaurs, creating the impression that no paradox exists. As discussed in Chapter Two, this approach has been effective because many accept expert claims without scrutiny. Without greater demand for transparency in mass estimation methods, the illusion persists.

While strong scientific arguments support the idea that these dinosaurs were extraordinarily massive, challenging the established narrative is difficult once it has been widely accepted. However, the issue of how these dinosaurs pumped blood to their heads is far more difficult to dismiss.

As explained near the end of Chapter Two, The Paradox of Large Dinosaurs, the diastolic pressure inside arteries is a function of height, given by the equation:

P = ρ g h

The numerous cardiovascular adaptations of giraffes suggest they are at the upper limit of what an animal can tolerate in terms of blood pressure. Similarly, the distinctive shape of Brachiosaurus — with forelimbs noticeably longer than its hind limbs — suggests that it, too, was pushing its cardiovascular system to the limit. Thus, we can equate the pressure requirements of giraffes and Brachiosaurus as follows:

P_g = P_B

ρ g_g h_g = ρ g_B h_B

Since the density of Brachiosaurus blood is unlikely to be significantly different from that of a giraffe, the density variables cancel out. Solving for the effective gravitational acceleration experienced by Brachiosaurus, we obtain:

g_B = g_g (h_g / h_B)

Here, g_B​ represents the effective gravitational acceleration during the time of Brachiosaurus, while g_g is not just the gravitational constant for giraffes but rather the present-day gravitational acceleration of Earth. The term (h_g / h_B) corresponds to the inverse of the 3.1 scaling factor calculated earlier. Substituting the present value for gravitational acceleration, g = 9.8 m/s²​, yields an estimated Mesozoic acceleration of 3.2 m/s². This represents the effective gravitational acceleration during the Mesozoic era.

A Review of the Gravitational Constant g

The acceleration due to gravity, g, is the rate at which an unrestricted object accelerates toward the center of the Earth when released near the surface. Most objects around us do not experience free fall because they rest on the Earth’s surface or on some intermediate structure. The force pulling an object toward the Earth's center is known as the force of gravity, or simply, the object's weight.

The gravitational acceleration g is used to calculate an object's weight from its mass using the equation:

F_g = g m

It is important to note that g is not a fundamental constant of physics but a derived value. It originates from the general gravitational force equation:

\[ F_g = \frac{G\: M_1\: M_2}{R^2} \] where:

• F_g is the force of attraction between the objects,
• G is the universal gravitational constant (6.67×10^-11 N m² / kg²,
• M₁ and M₂​ are the masses of two objects, and
• R is the distance between their centers of mass.

This equation applies universally, from objects on your desk to planets and stars. When applied to an object on Earth's surface, it simplifies to:

\[ F_g = \frac{G\: M_E\: M}{R_E^2} \] where:

• M_E is Earth's mass,
• M is the object's mass, and
• R_E is Earth's radius.

Comparing this to our earlier equation F_g = g m, we see that the Earth's gravitational acceleration is given by:

\[ g = \frac{G\: M_E\:}{R_E^2} \]

Investigating the Possibilities

This gives us three possible variables that, if one or more were to change, could alter the acceleration due to gravity. However, it is difficult to imagine how the universal gravitational constant G, the mass of the Earth M_E, or the radius of the Earth R_E could have changed significantly between the Mesozoic era and the present. Both physical evidence and simple calculations of what is physically possible confirm that none of these values could have changed significantly over the last 150 million years. Still, in the spirit of keeping an open mind, let us take a moment to investigate why none of these values could have changed by a meaningful amount during that time.

There has been a suggestion that the universal gravitational constant G might have changed. However, as stated earlier, our preference is to avoid hypothesizing changes in fundamental physical constants as a means of resolving a scientific paradox. Yet, some may still argue that, without a time machine, we cannot be certain that G has remained constant. The problem with this argument is that, in effect, we do have a "time machine."

Light travels at a constant speed of 3.0 × 10⁸ m/s. While this is an exceptionally high speed compared to most objects on Earth, it is relatively slow in the context of traversing the universe. The size of the universe remains uncertain — whether finite or infinite — but it is so vast that, from our limited perspective, it might as well be infinite. As a result, light from distant stars and galaxies takes thousands, millions, or even billions of years to reach us, meaning that when we observe these celestial objects, we are also looking back in time.

If the gravitational constant G had changed significantly over cosmic timescales, it would create havoc in our understanding of stellar and galactic motion. Some scientists argue that such havoc is precisely what we are observing. Among other anomalies, the rotational speeds of distant galaxies do not match predictions unless there is significantly more mass than what is directly observed. To resolve this, many scientists propose that dark matter constitutes more than four-fifths of the universe. However, other cosmologists have suggested even more radical hypotheses to explain these gravitational discrepancies. One possibility is that G itself has slowly changed over billions of years.

At first, this idea may seem far-fetched. However, Einstein’s special relativity demonstrated that "constants" such as mass and time are not truly fixed—they change when an object's speed approaches the speed of light. In truth, we do not fully understand what gravity is, so how can we be certain that the gravitational constant G has remained forever unchanged?

Nevertheless, we must rule out the changing gravitational constant hypothesis as a realistic explanation for how dinosaurs grew so large. Changes in the gravitational constant G that may or may not be possible over billions of light-years will not account for the drastic increase in the size of terrestrial animals that occurred on Earth a mere hundred million years ago. Occam's razor directs us to discard this hypothesis because attempting to use it to explain the large size of dinosaurs introduces so many unsolvable problems that it completely muddles our understanding of the laws of reality.

The next possible variable is the total mass of the Earth. About 4.6 billion years ago, during the earliest stages of Earth's formation, the planet's mass increased rapidly as it, along with other planets, swept up debris from the early solar system. However, within the first several million years, nearly all possible collisions between objects that could have occurred did occur. For all practical purposes, the mass of the Earth has remained constant for billions of years. Physical evidence supporting this conclusion comes from the study of impact craters on planets and moons — particularly the Moon, which provides valuable insights into Earth's history.

The Earth and Moon revolve around their common center of mass while occupying the same orbit around the Sun. As these two bodies orbit the Sun, they are occasionally bombarded by asteroids. Small asteroids that approach Earth usually burn up in the atmosphere, and even when a large meteorite reaches the surface and creates a crater, the evidence is eventually erased by erosion. Consequently, only a partial record of meteorite impacts remains on Earth.

On the Moon, however, where there is no atmosphere or oceans, every meteorite impact leaves a lasting mark. As a result, we can study the Moon’s permanent crater record with the understanding that the rate of asteroid impacts on the Moon corresponds to the rate of impacts on Earth.

The vast majority of craters on the Moon are over three billion years old. This suggests that relatively few meteorite impacts have occurred on either the Moon or the Earth over the last three billion years. Consequently, for all practical purposes, we can conclude that the mass of the Earth has remained constant for at least that long. Therefore, the Earth's mass has not changed significantly between the Mesozoic era and the present.

The final variable in the equation is the Earth's radius. If the Earth were to contract significantly, its radius would decrease while the acceleration due to gravity, g, would increase. The Earth does have a mechanism for compacting, as both the Moon and the Sun exert tidal forces that cause not only ocean tides but also slight movements in the Earth's crust. These tidal forces may lead to a gradual reshuffling of rocks within the Earth's interior. In fact, planets experiencing the strongest tidal forces tend to have the highest densities, suggesting that tidal compaction has played a role in planetary formation. With the Moon exerting a strong tidal pull on Earth, our planet has the highest density of all the planets in the solar system.

While there is evidence that compaction has occurred, the majority of it took place during the earliest stages of Earth's development. Billions of years ago, the Moon's tidal forces on Earth were much stronger than they are today. This is both supported by physical evidence and logically inferred from the fact that the Moon is gradually moving away from Earth. Since tidal force follows an inverse cube law (1/r³), when the Moon was half its current distance, the tidal force was eight times stronger. As a result, most of the Earth's compaction occurred billions of years ago, with relatively little occurring over the last 150 million years — the period relevant to our discussion.

Another reason why a changing radius cannot explain the observed differences in gravity is that it is not physically possible to compact the Earth enough to significantly affect g without encountering the problem of density inversion. Calculations show that if we attempt to reverse time and expand the Earth to a size where g was only 3.2 m/s², the planet’s average density would be lower than that of water — an impossibility. Unless movement of material is somehow restricted, layers of material must increase in density as they approach the Earth's center.

It may seem that we have examined all possible hypotheses without finding a solution. However, there are still a few possibilities left when we consider that forces other than gravity (weight) could be involved. Such forces could push upward on an object, effectively diminishing the strength of gravitational acceleration. A force acting in this way could produce an effective gg lower than the present gravitational constant.

On Earth's surface, two additional forces besides gravity act to reduce the net downward force on an object: the pseudo-centrifugal force due to Earth's rotation and the buoyant force. These forces slightly decrease an object's effective weight. Currently, the maximum centrifugal force accounts for only 0.34% of an object's weight, and for an object or animal with a density approximately equal to that of water (1000 kg/m³), the buoyant force of air reduces its weight by only 0.13%. Because these forces are so small, their effects are typically insignificant in most calculations and are usually neglected. However, if one of these forces had been significantly stronger in the past, it could have meaningfully reduced Earth's effective gravity.

The pseudo-centrifugal force is due to the Earth's rotation. This centrifugal force is calculated as:

\[ F_c = \frac{M\: v^2\:}{R} \]

where F_c is the centrifugal force, M is the mass of an object, v is the object's speed, and R is the distance to the rotational axis. This equation can also be written as:

F_c = m r ω²

where F_c​ is the centrifugal force, m is the mass of an object, R is the distance to the rotational axis, and ω is the angular speed of the Earth’s rotation. For computational purposes, the mass of the Earth is 5.98×10²⁴ kg, the Earth’s average radius is 6.38×10⁶ m, and the present angular speed of the Earth’s rotation is 7.272×10^-5 rad/s.

The pseudo-centrifugal force actually points away from the rotational axis, rather than directly upward against gravity. Therefore, the greatest effect of the centrifugal force occurs at the equator, while there is no effect at the poles. In our present world, the centrifugal force is so small that this change is unnoticeable. However, if we were to believe that the centrifugal force is responsible for reducing the effective weight of dinosaurs, the effect would be dramatic. Dinosaurs migrating from lower to higher latitudes would feel substantially heavier as they moved farther from the equator.

During the Mesozoic era, the Earth spun slightly faster. Physical evidence from coral samples and tidal rhythmites indicates that 150 million years ago, the sidereal year was 378 days long, compared to the present 365.25 days, while each solar day lasted only 23 hours and 11 minutes instead of the current 24 hours. Tidal rhythmites are sedimentary rocks from ancient shorelines, where alternating dark and light silty deposits can be counted to determine the number of days in a year. This physical evidence confirms the logical conclusion that tidal forces are gradually slowing the Earth's rotation.

The significance of this is that scientists have a well-established understanding of the Earth's rotational speed during the Mesozoic era. We know with certainty that 150 million years ago, at the approximate peak of dinosaur gigantism, the Earth's angular velocity ω was 7.526×10^-5 rad/s. However, this faster rotation would have resulted in only a negligible reduction in the effective gravity of the Mesozoic era. At best, the maximum centrifugal force at the equator would have been only 0.37% of the gravitational force, assuming the Earth was a perfect sphere.

Adding a slight complication, the centrifugal force causes the Earth to bulge slightly at the equator, making the equatorial radius slightly greater than the polar radius. While this bulging slightly increases the effect of the centrifugal force, the total effect remains less than one percent of the gravitational force. Therefore, the centrifugal force would not have been strong enough to significantly reduce the effective gravity during the Mesozoic era.

Let us take a moment to understand why the Earth's rotation is slowing down.

Most people associate tidal forces with the twice-daily ocean tides, but the solid Earth also bulges twice a day due to the Moon’s pull. These two tidal bulges form because, while the Moon’s gravity pulls one side of the Earth toward it, a pseudo-centrifugal force acts on the opposite side. This force arises from the Earth and Moon orbiting their shared center of mass. According to Newton’s laws, these opposing forces must be equal and opposite. As a result, the Moon’s pull effectively stretches the Earth.

The Moon’s gravitational pull causes the Earth to stretch almost in line with the Moon. The key phrase here is almost in line rather than directly in line. This misalignment occurs because the Earth is rotating, causing the tidal bulge to lead the Earth-Moon axis by approximately three degrees. Since the bulge is not perfectly aligned with the Moon, it generates a torque that gradually slows the Earth's rotation.

This tidal lag also exerts a forward force on the Moon, pulling it into a higher orbit. In other words, the Moon is slowly moving away from the Earth at a rate of 3.8 cm per year.

The Earth's rotational slowdown and the Moon’s outward drift exemplify another fundamental principle of physics: conservation of angular momentum. As the Earth's rotation slows, its angular momentum is transferred to the Earth-Moon system, causing the Moon to move farther away. Eventually — billions of years from now — the Earth will become tidally locked, with one side always facing the Moon. At the same time, the Moon’s outward movement will cease.

It was necessary to present the arguments that eliminate incorrect hypotheses to prepare the reader for the solution.

The buoyancy force is best described by Archimedes' principle, which states that when an object is partially or fully submerged in a fluid, it experiences an upward buoyant force equal to the weight of the displaced fluid. Buoyancy is commonly associated with objects submerged in water or floating on its surface, but it also applies to air, as seen in the lifting force of hot air balloons. The key difference between buoyancy in water and air is the volume of fluid that must be displaced to achieve flotation. This is due to the significantly lower density of air at sea level (1.29 kg/m³) compared to water (1000 kg/m³). A denser fluid provides more buoyancy than a less dense one.

For terrestrial vertebrates, the net force produced by their weight often limits their size. However, this is not true for species that live in water. For aquatic species, factors such as food availability, rather than weight, may limit their size. Without weight restrictions, some aquatic species grow to display gigantism. The buoyancy of water allows whales—the largest animals today—to reach enormous sizes. Without this buoyancy to counteract gravity, a stranded whale soon suffers broken bones under its own weight.

For buoyancy to significantly reduce the effective weight of dinosaurs, Earth's atmosphere would need to be dense enough to approach the density of water. By summing the forces acting on a typical dinosaur, such as a Brachiosaurus, the necessary atmospheric density can be calculated as:

\[ ρ_F = ρ_S\:(1 - \frac{1}{S.F.}) \]

where ρ_F is the density of the fluid, ρ_S is the density of the submerged substance (such as the dinosaur), and S.F. is the scaling factor. Using a scaling factor of 3.1 and an overall vertebrate density of 980 kg/m³, Earth's atmospheric density during the Late Jurassic is calculated to be 660 kg/m³. This implies that for dinosaurs to grow to their exceptional sizes, Earth's near-surface air density would need to be about two-thirds that of water.

Some may find it difficult to imagine how Earth could have once had such a dense atmosphere. However, reality often surpasses the limits of human imagination. Esker’s Thick Atmosphere Theory violates no known scientific principles — it is the correct solution.

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How Dinosaurs "Evaded" the Square-Cube Law: David Esker's Answer

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