Dinosaur Theory Header

1. Galileo's Square-Cube Law

Introduction

Gulliver Standing Tall Above the Little People

Does size matter? More specifically, can a rock, a person, a car, a planet, an airplane, or a dinosaur exist at any arbitrary size? This is a fundamental scientific question, and yet for the most part the science community has sidestepped this simple question. Since the science community has not clarified why animals cannot be any size, science fiction writers have had fun playing with the idea that animals can be many times larger or many times smaller than their normal size. While many of us may enjoy the entertaining movies showing people or other animals the wrong size, this is not helping to clarify to the public that size really does matter. Galileo’s Square-Cube Law explains why size matters, and in fact Galileo’s Square-Cube Law is such a fundamental scientific truth that science is hardly science without it.

In 1638 Galileo published his book titled Dialogues Concerning Two New Sciences where he clarified the reasons why objects cannot be any arbitrary size. As he explained, when an object is scaled up its area increases by the square of the multiplier while the volume increases by the cube of the multiplier. Since the ratio between the area and the volume is changing with size the properties of the object are changing with size.

In Galileo’s time his Square-Cube Law was an advanced scientific idea, yet today it is a scientific concept that any grade school student can easily understand. Furthermore Galileo’s Square-Cube Law is not just another scientific concept but rather it is a scientific concept that lies at the foundation of every major science discipline. Specifically, in biology, physics, chemistry, astronomy, aerodynamics, nanotechnology, and other science disciplines there are many facts and less fundamental concepts that are difficult or impossible to understand without first recognizing how size matters.

Pyramids

So this begs the question: why is Galileo’s Square-Cube Law left out of science education? One source of confusion comes from the fact that even though changes in size always produce changes in properties, for many simple objects these changes in properties show no adverse effect and so they go unnoticed. For example, most rocks can withstand high compressive forces and so there is nothing preventing the construction of pyramids the size of mountains. In addition to rocks there are other simple objects that seem to be indifferent to scaling properties. It is only when we take a closer look at the more complicated objects such as airplanes and living organisms that it becomes apparent that size matters.

The second source of confusion is the discovery of the exceptionally large dinosaurs that appear to defy Galileo’s Square-Cube Law. With the discovery of dinosaurs, people have come to believe that there is nothing restricting the size of animals.


Explaining the Square-Cube Law

When we are comparing the size of objects it is common to make a statement such as one object is twice as large as another object, without clarifying if the size comparison is based on length, area, or volume. Usually this sloppiness does not cause mistakes since we can infer from the objects being compared whether the comparison is based on a one dimensional, two dimensional, or three dimensional attribute. Yet an important fact that is often overlooked is that the length, area, and volume of similar shaped objects do not scale proportionally.

Of great importance is the ratio between area and volume or, assuming equal densities, the ratio between area and weight. Two similar shaped yet different size objects will have a different ratio of area to volume: the larger object will have a lower area to volume ratio than the smaller object. This seemingly subtle point is actually fundamental to all of science and so it needs to be emphasized: because the ratio between area and volume changes with size it is incorrect to believe that two similar yet proportionally different objects can ever be the same. Size matters.

We can use a simple cube that is scaled up to be ten times taller to illustrate how the area to volume ratio changes with size. As shown in figure 1, if each dimension of a solid cube is made ten times larger than its original model, its area touching the ground will be 10 x 10 or one hundred times larger. Meanwhile the volume will increase by 10 x 10 x 10 or a thousand times greater than the original. If the larger cube is made of the same material as the original cube, the mass and the weight of the larger cube will be a thousand times greater than the smaller one.

Small and Large Cube

Figure 1.
Each dimension of cube B is ten times greater than the each dimension of cube A. Because of this the volume of cube B is a thousand times greater than cube A, while the area of one of cube B’s sides is only a hundred times greater than one of cube A’s sides. Thus, as objects are scaled up in size the volume grows at a faster pace than the area.

If our two cubes are made of the same material then they will have the same density, the density being the amount of mass per unit volume. Yet since the two cubes have different area to volume ratios they will likewise have different stress at the base of each cube. If too much stress is placed on an object then it will either break or collapse.

The Golden Gate Bridge

The stress s at the bottom of each cube is its weight divided by the area: s = F / A. Into this equation we can substitute the equation for weight F = g D V where g is the acceleration due to gravity, D is the density, and V is the volume to write the stress as s = g D V / A. When we then insert V = L3 and A = L2 into this equation and make cancellations we have s = g D L where L is the height of the cube.

V = L3 and A = L2
s = g D L

This equation tells us that within a constant gravitational field, while using the same material, the structural stress at the base of the cube increases in direct proportion to the change in the height of the object.

We can understand the relationship between size and stress if we think about why sand castles are rarely more than a couple feet tall. If too tall of a sand castle is attempted the stress becomes too great such that the sand castle crumbles.

In engineering calculating the stress placed on a structure is important to determine that it will be strong enough that it will not fail. For safety reasons, engineers will normally design structures so that the greatest stress expected within a structure is only a small fraction of the known ultimate strength of the supporting material. Table 1 shows the ultimate tensile and ultimate compression strengths of a few selected materials.

Table 1
The Ultimate Strengths of Some Materials

Material Tensile Strength (MN/m2 ) Compressive Strength (MN/m2 )
Steel 500 500
Concrete 2 20
Nylon 75 ###
Bone (limb) 130 170

Picture of Tall Building

Let us combine what we have learned from the example with the cubes with the fact that materials have a limit to the stress that they can withstand. Say that we want to construct two buildings that are to be made out of either the same materials or at least materials with close to the same density. Building A is to be just a one story model apartment while building B is to be a ten-story, thousand unit apartment building. From our example with the cubes, we know that for both buildings the stress will be the greatest near the bottom of each building. But since stress is related to height, the stress near the bottom of building B will be ten times greater than the stress near the bottom of building A.

In choosing our materials we could use concrete for the one story building. One story concrete buildings are fairly common since concrete is fairly inexpensive and concrete is strong in compression. In the table, the ultimate stress limit of concrete in compression listed as 20,000,000 N/m2. But for the building that is ten stories tall, and certainly any buildings that are taller, steel would be a better choice since its compression strength is twenty five times greater than the compression strength of the concrete.

In Jonathan Swift’s 1726 novel, Gulliver’s Travels, Gulliver visits other lands where in Lilliput the people are twelve times smaller than himself while at another location, Brobdingnag, the people are twelve times taller than himself. While this makes for a fun fictional story it does not fit in well with reality.

From the table we see that bone has a set ultimate strength just like the other materials. But unlike the example with buildings we are stuck with just the one support material for all vertebrates. The proportionally built Brobdingnagian would have twelve times greater stress on his or her supporting bones. This excessively high stress would cause the leg bones of these giants to snap after taking just a few steps. Clearly then Gulliver and the giants of Brobdingnag can not be proportional in their dimensions as it is claimed by the story.


Effect of Scaling Properties on Biology

Getting back to reality, there are species of animals such as the deer and the elk that are closely related but of different size. Galileo took notice that the bones of the elk are not just proportionally thicker to the bones of the deer but instead the elk’s bones are even much thicker. The elk’s bone has to be much thicker to lower the stress in the bone below the breaking point of the bone. Even so, elk and all the other large vertebrates are still more likely of breaking their bones than the more active smaller animals.

Picture of deer and elk

The bones and muscles of the larger animals tend to be disproportionally thicker and larger than the smaller animals. Yet the smaller animals will still have the greater relative strength that allows them to jump higher and fall greater distances with less chance of breaking their bones.

The Square-Cube Law applies to the bone supporting frame of vertebrates in the same way that it applies to non-living objects. If an individual is twice as tall as similar animals, its cross sectional areas of its leg bones are four times greater while its weight is eight times greater. This means that the stress on its bones is twice as much. and so the larger animal is much more likely of breaking its bones. Because of scaling properties, the larger terrestrial vertebrates are at a much higher risk of breaking their bones than the smaller vertebrates.

Picture of Cat up in a Tree

When an animal lands after a fall the stress on the leg bones is many times greater than when it is standing still. So all animals need to have bones that are many times stronger than what is needed when it is just standing still. Also, similar to the engineering design of man-made structures, for safety sake, an animal’s bones need to be many times stronger than what would be the bare minimum required to withstand the greatest expected impact.

It is because of scaling properties that larger animals run a greater risk that the stress applied to their bones will exceed the ultimate strength of bone material and thus cause a bone to break. This relationship that produces greater stress on the bone with the greater size of an animal is the reason why cats or other small animals are usually unharmed after falling from a tall tree, while conversely a prize racehorse can shatter a leg in the act of running a race.

Also because of the Square-Cube Law, larger animals have less relative muscle strength than smaller animals. Both the muscle strength and bone strength are functions of the cross sectional area L2, while the weight of the animal is a function of volume L3. It is because of relative muscle strength that an ant can lift fifty times its weight while a human can lift an amount equal to its own weight, and an Asian elephant can only lift 25% of its own weight. The greater muscle to weight ratio of smaller animals is what allows them to jump higher than several times their own height. While at the other extreme an elephant can not even jump.

To summarize the ideas concerning strength in regards to animal size, the larger vertebrates will tend to have greater absolution muscle and bone strength but their relative muscle and bone strength is much less that the relative muscle and bone strength of the smaller animals.


The Effect of the Surface Area to Volume Ratio

Another important scaling property is the surface area to volume ratio. Again using the example of the cubes, the total surface area of the smaller cube is 6 L2 and its volume is L3. The surface area to volume ratio for the small cube is then 6 / L. Performing the same calculations with the larger cube produces a surface to volume ratio of 6 / (10 L). So that a comparison of the two ratios shows that the cube that is ten times larger has a surface to volume ratio that is one tenth of the smaller cube. From this we can draw the general conclusion that larger objects will have lower surface area to volume ratios than similar smaller objects.

Starting a Campfire

The surface area to volume ratio can be important to determining the rates of chemical reactions, diffusion rates, the rate of heat loss, and many other phenomena that are impacted by sizing variables. For example, one of the best ways of increasing the rate of a chemical reaction is to increase its surface area to mass ratio by pulverize the reactive substance into a fine powder. By breaking large pieces into smaller pieces the overall surface area available for reaction increases.

A campfire is a chemical reaction that changes solid wood into various gasses while giving off heat. Yet heat is first needed to get this chemical reaction to take place and when the fire is first started there is not much heat available. We overcome this problem of starting the fire by using the small twigs rather than the logs so that more wood is exposed at the surface to participate in the reaction. This allows the fire to grow from only the modest heat from the lit match.

Similar to what is best to speed up chemical reactions, to maximize the rate of diffusion it is desirable to have the largest surface area possible while keeping the separating membrane as thin as possible. For vertebrates, it is desirable to have a high diffusion rate within the lungs and the cardiovascular system’s capillaries, so it is not surprising that these diffusion systems have large surface areas with thin membranes.

Material Thermal Conductivity
(W/m*K)
Aluminum 200
Steel 20
Glass 1.1
Concrete 0.84
Human Tissue 0.2
Styrofoam 0.025

Diffusion rates are extremely important to the smallest recognized living unit, the single cell. Most eukaryotic animal cells are within a range between 10 and 30 micro meters. The size limitation of these and other cells is due to the diffusion process for transferring nutrients and waste across the cell’s membrane. A cell’s surface area to volume ratio, or likewise its surface area to mass ratio, decreases as the cell grows larger. This lower surface area to mass ratio slows down the diffusion rate and along with this the metabolism of the cell thus causing a larger cell to be less efficient than a smaller cell. So to keep an overall high efficiency throughout the body the larger cells will divide rather than continue growing. Thus multi-cell animals grow by increasing the number of cells rather than increasing the size of the cells.

The surface area to mass ratio is important to the transfer to thermal energy: heat. Whenever there is a difference in temperatures, thermal energy travels from one location to the other through conduction, convection, or radiation.

Conduction is the transfer of heat through a solid material such as when one side of a metal object is heated the heat easily distributes itself throughout the entire object. Different materials conduct heat at vastly different rates. So we need to choose the appropriate material when the transfer or lost of heat is an issue. For example, previously metal aluminum window frames were preferred over wood frame windows because unlike wood the aluminum metal did not rot. However as people became more energy conscious they recognized that the metal framed windows had their own undesirable attribute in that they allowed the heat to easily moved between the inside and the outside of the home. Because of this vinyl windows that neither rot nor conduct heat are now the preferred window frames.

Boys Staying Warm at a Campfire

Convection heating occurs through the movement of a warm fluid from one location to another. When there is a difference in temperature between two locations and there is a fluid that is free to flow between those locations a natural convection current will establish itself to transfer heat from the hot location to the cool location. If this transfer of thermal energy is undesirable then it is best to impede the flow of the fluid. For example, the primary way that clothing and coats keep us warm is due to the fibers in the clothing and insulation that get in the way of a free flowing current of air. Insulation that traps or slows down the movement of air is also used to keep homes warm and to keep hot beverages hot. Even though these insulating materials such as fiberglass and Styrofoam are rated as if they are preventing the conduction of heat they actually work by stopping the convection process by stopping the flow of air.

Radiation is the transfer of heat by electromagnetic waves, light, traveling through space from a warm location to a cooler location. Radiation is unique in that it transfers thermal energy from a hot location to a cool location without the need for there being anything in between the two locations. If not for radiation, the warmth of our Sun would never reach the Earth.


On the Earth, if we have an object that is isolated from other objects, then usually only the convection process of the flow of air or water or the radiation through space will allow heat to be transferred to or from the object. How much heat will be transferred through either the convection or the radiation process will depend on several variables and included among these variables is the surface area to mass ratio.

Since there are many variables that affect heat transfer but we just want to focus on how size affects the rate of heat transfer, let us imagine once again that we have two objects that are the same in all respects except that they are different in size. If the two objects are heated to the same temperature and then allowed the cool, the smaller object with its higher surface area to mass ratio will cool off faster than the larger object. Furthermore it does not matter if the objects are cooling off or heating up the smaller object will change its temperature faster than the larger object.

Picture of turkey and pizza

When preparing a banquet, a large turkey or ham is usually placed in the convection oven before the other food since these larger objects have a low surface area to mass ratio thus requiring more time for the heat to penetrate through to the interior. In contrast to a Thanksgiving turkey that requires 3-4 hours in the oven, a pizza has a large surface area to mass ratio and so it only needs about 14 minutes to cook.

Picture of Puppies

For another example, ice cubes in a glass are usually melted within a few minutes while the much larger iceberg will take several months to melt. The size of objects is often the most important factor determining the rate of heat transfer per unit mass.

Often, rather than heating up or cooling off, it is desirable for an object to be maintained at an elevated temperature above the temperature of its surroundings. To maintain this constant elevated temperature thermal energy must be added at the same rate that the object is losing heat at the surface. Since thermal energy is usually expensive we want to reduce this lost of energy as much as possible. One of the best ways to minimize this energy lost is to reduce the collective surface area to volume ratio of the group. For this reason, it will usually be much cheaper to heat a single apartment unit than to heat a house with the same amount of interior space. Conservation of thermal energy is also the reason many animals from bees to puppies huddle together to stay warm.


Applying Thermal and Scaling Properties to Understanding Mammals and Birds

Mammals and birds are more complex creatures than reptiles since maintaining a constant elevated temperature requires several unique adaptations. These adaptations allowing mammals this constant elevated body temperature gives mammals more active mobility. It also allows mammals to enter cold environmental niches such as living in the mid and upper latitudes of the Earth or to have nocturnal lifestyles. Yet these benefits require certain adaptations that come with a price.

Picture of Bat

First, since a mammal or bird’s body temperature is almost always greater than the surrounding temperature, heat is continuously being lost to the surroundings. This thermal energy is expensive meaning that mammals and birds must consume a large quantity of food to compensate for the lost of this thermal energy. For nearly all mammals or birds, the amount of food energy that goes towards keeping warm is greater than the amount of food energy that goes towards mobility. For the smaller mammals and birds, having a higher surface area to mass ratio, the constant consumption of food to maintain the required elevated body temperature is a primary concern of survival.

Small endothermic animals such as mice, bats, and humming birds have a high surface to volume ratio and so they have a high rate of heat loss relative to their total body mass. To limit the heat lost mice will huddle together, bats will hang from the ceiling where it is warmer and possible more secure, and none of these animals exists in cold habitats. To replace the thermal energy that they lose so quickly these small animals are committed to a life of being constantly on the move for the purpose of finding and consuming as much food as possible. Their daily intake of food can be more than one third of the animal’s total weight.

At the other end of the spectrum, elephants have a much lower surface-to-mass ratio so they have the opposite sizing problem in that they need a way to remove excess heat from their interior. The evolutionary solution is large earflaps whose purpose is more geared towards the radiation of heat rather than improvements in hearing. Elephants pump warm blood to the skin of their earflaps where through convection and radiation the excess heat is removed. These largest of mammals that live in warm climates are clearly the exception to almost all other mammals in regards to their desire to lose rather than maintain heat.

Almost all mammals and birds wish to slow down the lost of thermal energy. This is the reason mammals are covered with hair so as to trap a thin layer of air near their body as a means of insulating their body from the cold exterior. This is also the same reason why feathers surround the body of a bird so as to trap air and thus insulate the bird’s body from the cold. Generally feathers are a slightly better insulating material than hair and because of this birds are able to maintain a higher body temperature.

Picture of elephant and mouse

Notice that the shape of the mouse is as close as it can be to that of a sphere, a shape that produces the minimum amount of surface area. Now notice that the elephant’s skin is covered with wrinkles thus increasing the elephant’s surface area. Also notice that the mouse is like nearly all mammals in being covered with a thick fur coat while the elephant is nearly void of all hair. The small mouse is trying to maintain heat while the large elephant is trying to lose it.

Mammals and birds that spend a large amount of time in the water can not use air as an insulator so they use body fat. Also because of the problem of heat lost, most mammals living in the ocean or in the Polar Regions will tend to be large animals with a layer of fat just under their skin. In these challenging environments these animals need the lower surface area to mass ratio and the layer of insulating fat to prevent the cold from removing too much heat from their body. Baby mammals will also use thick layer a fat as a means of compensating for their greater surface area to mass ratio, thus allowing them to maintain their constant warm body temperature.


Warm-Blooded Babies

Unlike man-made objects, animals grow in size as they transition from new-born babies to adults. This transition is different for vertebrates depending on if they are cold or warm blooded. Reptiles do not internally maintain an elevated temperature so heat lost is not a concern for cold-blooded vertebrates. So the transition from baby to adult is simple for reptiles as babies look like miniature adults.

Picture of Baby Alligators

But mammals and birds must maintain their elevated temperature through out their lives as they grow in size from baby to adult. Because they are growing in size, these vertebrates start life with bodies having a high surface area to mass ratio and finish as adults with bodies having a low surface area to mass ratio.

A related problem for warm-blooded babies is that their small mass is less capable of storing thermal energy. Because of this warm-blooded babies have difficulty maintaining a constant body temperature when the exterior temperature changes. Clearly parents who fuss over their baby being snugly wrapped up in blankets are doing the right thing.

With the higher surface area to mass ratio and having far less mass the warm-blooded babies have greater difficulty than adults in maintaining and regulating a constant elevated body temperature. Because of these problems for warm-blooded babies in maintaining their required constant elevated body temperature birds and mammals have developed several adaptations. Through these adaptations warm blooded animals are better able to maintain their body temperature within a safe range of temperatures as they grow.

Relatively speaking, the warm-blooded or endothermic babies need to be much larger at birth than the cold–blooded ectothermic reptiles. This is necessary because of the difficulties the warm-blooded vertebrates have in maintaining their constant elevated body temperature, a problem that does not apply to the reptiles. The larger birth size of each individual that is required of the warm-blooded vertebrates means that there are fewer offspring from each gestation period of the endothermic vertebrates. Thus, while each mammalian gestation period will produce roughly one to ten offspring the typical reptilian gestation period will produce roughly ten to a hundred offspring. The problem of meeting the minimum size required to maintain constant body temperature is especially acute for the smallest warm-blooded vertebrates. A new-born bat can be 25% of the mass of the adult mother.

The next set of adaptations of endothermic vertebrates is that the warm-blooded babies do not grow proportionally as they transition from baby to adult. In other words, while reptiles appear as miniature adults most baby mammals can not be mistaken for an adult. Endothermic babies have shorted extremities, rounded features, and relative to their size human babies have larger brains than adults. These adaptations help the warm-blooded baby reduce its heat lost and maintain its constant elevated body temperature.

The importance to mammals to maintain their constant body temperature is illustrated by the outdoor guide’s survival rule of three’s. A person can survive for three weeks without food, or they can survive for three days without water, yet if a person does not have adequate insulation from the cold they can be dead within three hours. Physics and scaling properties are fundamental to understanding the vast majority of biology observations.

This is just a short introduction to some of the important scaling properties that are general to all objects. For many simple objects these important scaling properties may go unnoticed. This is because many structurally simple objects can be scaled up or down over a large range before reaching a size limitation that will cause a failure. However with more complex objects, such as aircraft and living organisms, the scaling range is much more limited before problems become apparent.


Fundamental Forces of Nature

Science has now identified four fundamental forces of nature and these forces of nature dominate at different ranges of size. So from a completely new perspective from Galileo’s scaling properties, once again, size matters. The four forces of nature are the strong and weak nuclear forces, the electrostatic / electromagnetic force, and the gravitational force.

The strong and weak nuclear forces operate within a very small distance of only slightly larger than the typical nucleus. The strong force holds the nucleus together while the weak force is believed to play a role in radioactivity decay.

Picture of Lighting Strike

The electrostatic / electromagnetism force operates primarily from the small size of the atom to the gigantic size of the magnetic field lines that produce the Sun spots. Electrostatic attractive and repulsive forces are present when there is an imbalance of positive charged protons or negative charged electrons. If there are only positive charges within a location then all of these positive charges will try to get away from each other. Likewise if all the charges at a location are negative then they will repel each other. Yet if there is a mixture of positive and negative charges then the opposite charged particles will be attracted to each other.

An example of opposite charges coming together can be seen in dramatic lighting strikes. Yet the electrostatic force is most noted for its effect at the atomic or microscopic level because at this level it is much easier for there to be one or two extra or one or two fewer electrons to create an unbalance charge. These small electrostatic forces are the basis for chemical bonds. So far from being insignificant, these electrostatic forces acting at the microscopic level ultimately determine the properties of almost everything in our world.

Gravitational force is a weak attractive force between every bit of matter. The gravitational force is so weak that it is not even noticed unless there is at least one extremely large object near by. For those of us that live on the surface of the Earth, that most important large nearby object is the Earth. Besides attracting us to the Earth’s surface, the gravitational force is what holds the Moon in its orbit about the Earth and the Earth in its orbit around the Sun.

The point of this discussion on fundamental forces is that each type of force tends to dominate at a different range of size. At the level of planets and stars we are interested in the attractive gravitational forces. When we come down to the size of a small insect, electrostatic forces can be every bit as important as the gravitational force. At the even smaller size of bacteria, electrostatic forces are all that matter. In the microscopic world of bacteria gravity no longer has any meaning. We develop a better understanding of our reality when we consider what forces are important at what size.

But while size is important to the fundamental forces, the effect of changing the size of an object is more readily apparent with Galileo’s Square-Cube Law. With fundamental forces we usually have to consider a huge change of scale, such as one object being millions or billions of times larger or smaller than another object before it actually makes a difference. Whereas the affect of the Square-Cubed Law may cause a failure when one object is not even ten times larger or smaller than another object.


Fundamental Science

Picture of Galileo

The two most important books of Galileo were Dialogues Concerning Two Chief World Systems and Dialogue Concerning Two New Sciences. In Dialogues Concerning Two Chief World Systems he showed the compelling evidence in support of Copernicus’ heliocentric model of the solar system while in Dialogue Concerning Two New Sciences he presented the arguments regarding the importance of size.

Currently every science and engineer discipline, from aerodynamics to biology to nanotechnology, are under the false belief that the scaling problems that they encounter are unique to just their discipline. But for real understanding it needs to be recognized that the Square-Cube Law is universal to all of science. Galileo's Square-Cube Law embodies a simple concept that brings amazing insight to understanding our reality.

But now that the importance of Galileo’s Square-Cube Law is settled, that still leaves the question of what about those exceptionally large dinosaurs. How do we explain them? In the next two chapters we will explore the logical conflicts presented by large dinosaurs and flying pterosaurs. Then in chapter four we will search for the resolution of these scientific paradoxes. We exist in a rational reality and there is a solution to the dinosaur paradox.



External Links / References

Physics of Size

Strength of Materials

Physics, Biology, and Scaling Properties

Scaling Properties and their Application to Biology

Size of Biological Cells

Surface Area to Volume Ratio

Heat Transfer

Heat Transfer in Regards to Mammals

Fundamental Forces of Nature

Scaling Properties Aplied to Nanoscience

Galileo


Send an e-mail if you have questions or wish to be informed as to when to book will be available.

Sending mail

THIS MATERIAL IS COPYRIGHTED.
PERMISSION IS GRANTED FOR TEACHERS AND OTHER EDUCATORS TO COPY
AND DISTRIBUTE THIS MATERIAL, UNALTERED OR CHANGED IN ANY WAY,
TO STUDENTS IN BOTH PUBLIC AND PRIVATE EDUCATIONAL INSTITUTIONS.