Eratosthenes: Using Induction and Deduction

Valid but not Sound?

Recall this argument from Chapter 1.


Only people who believe in the Christian God are moral.
John Smith is moral.
So, John Smith must believe in the Christian God.

Valid argument, but many people who are not Christians would be insulted by the first premise.   If true, then one must be a Christian to be a moral person, because the first premise is essentially saying, "All moral people are Christians," and "If people are not Christians, then they cannot be moral." Buddhists, Hindus, Muslims, Jews, and atheists do not believe one has to be a Christian to be a good person. For many people this statement (premise) is false

So the message in C1 from this example was that arguments can be valid, but contain questionable or false content (information).   Same message given by the south-of example in C1.   Also see 1-3 and 1-4 on abortion.   Both valid arguments but very controversial issue and different premises that lead to opposing conclusions. Remember: Valid arguments have true conclusions IF the premises are true, but valid arguments can have all false premises and a false conclusion; however, they can never have all true premises and a false conclusion.

Consider this argument:

Premise 1:

If the theory of natural selection (evolution) is true, then it is the only view that should be taught in science classes on the origin and development of life on Earth.

Premise 2:

The theory of natural selection (evolution) is true.


So, the theory of natural selection (evolution) is the only view that should be taught in science classes on the origin and development of life on Earth.

Valid deductive argument and modern science argues that this argument is sound as well - there is overwhelming evidence that Darwin's basic natural selection theory of evolution is true.   However, all scientific theories are like the cigarette example discussed in C3.   The conclusions are vast generalizations (albeit from lots of evidence) and hence based on the best inductive reasoning.

Remember - even the best inductive reasoning will have conclusions that are at best highly probable, and hence involve some risk of being false.   But it is also important to remember that from the point of view of modern science, the theory of evolution is supported as well as, if not more so by overwhelming evidence, the causal link between smoking cigarettes and lung cancer and that gravity is universal on Earth and anyone would fall from a 12th story building if they jumped from a window or lanai. All three generalizations -- evolution, the link between cigarette smoking and lung cancer, and the universality of gravity -- are considered very reliable beliefs and probably true.

Take Away Message

Valid deductive reasoning is an essential tool in being a critical thinker.   If the premises are true and an argument valid, we are certain the conclusion is true.   If the conclusion is false, we are certain that something is wrong in our premises (at least one must be false) and we have to reexamine what we think is true.   In using valid deductive reasoning, we always expand and grow our understanding. We either learn new truths by deducing them from known truths or learn what we thought was true is not true. BUT the truth of the premises used is often based on inductive reasoning.   Even the best inductive reasoning involves some risk because we are always generalizing beyond the evidence given with the premises of inductive arguments.   Life is uncertain and any decision, no matter how reasoned out, involves risk.   But this does not mean that anything goes and that all beliefs are equal.   Some beliefs are much better supported than others.

How We Reason in Science

Science uses deduction and induction to know how to test hypotheses, to learn new truths, and to learn that some hypotheses are false.

The famous example of Eratosthenes and his assertion and calculation of the circumference of a spherical Earth was given in Chapter 3.   His work is not only important in terms of how shockingly far ahead his conclusions were for his time (about 240 BC) - many people continued to believe the Earth was flat for many centuries after his time --   but it is an excellent example of using a combination of induction and deduction.

Study the narrative and figures in C3 on Eratosthenes' reasoning.

First, consider the generalizations and alleged information that he used as premises.

  1. The sun is very far away from Earth.
  2. If 1 is true, then the beams of light from the sun are virtually parallel by the time they strike the Earth.
  3. On the day of the summer solstice, at exactly noon, there were no shadows cast in Syene, but he measured a shadow with a 7 degree angle in Alexandria.
  4. The distance between the two cities was about 500 miles (using a modern measurement).

Second, from these premises he also generalized that it is most likely true that the surface of the Earth is curved and there existed corroborating evidence at the time that the Earth was probably a sphere.

Third, now we use deduction - some math and logic.

  1. A circumference of a sphere can be measured as 360 degrees.
  2. The distance between the two cities is a proportion of the 360 degree circumference.
  3. Using the geometry of Euclid, he knew that he could consider the beams of light striking objects in the two cities as parallel, creating imaginary parallel lines.
  4. Key geometric point = the angle created by the shadow in Alexandria (7 degrees) is equal to the angle created by a line (called a transversal) from Alexandria to the center of the Earth at the point where a line from Syene to the center of the Earth meet.   What are called the alternate interior angles are equal.   See:
  5. Look at figure 3-4. Important: angle a = angle b.
  6. Angle b thus tells us the proportion of the distance of Syene to Alexandria compared to the entire circumference of the Earth.

So, because angle a was measured to be 7 degrees, b is 7 degrees and we have 7/360 for the proportion of the distance between the two cities and the circumference of the entire Earth.

Next is a little algebra.

Eratosthenes' equation


d = distance between the two cities (Syene and Alexandria)
b = a = measurement of the angle of the shadow at Alexandria


  =   = x

  =   = 25,714

Not bad. The actual distance is 24,901 miles or 40,075 kilometers. His reasoning was mathematically and logically valid given his premises. But he was assuming that the two cities were on the same meridian of longitude. Because they are not exactly on the same line, his result was a little off.

Imagine Eratosthenes walking into a local bar and excitingly telling the patrons what he had discovered. What do you think the reaction would be about 2,250 years ago? Imagine what happens today when astronomers tell the average bar patron that the universe is about 14 billion years old and that the nearest star (other than the sun) to Earth is about 24 trillion miles away?

Keep in mind that Eratosthenes result is essentially a prediction. There were no pictures from space of the Earth over 2000 years ago. No one has the means then to circumnavigate the Earth. He has good reason to believe that his result was close to accurate IF all his information (premises) was true, but humankind was far from corroborating his basic result.

Notice that with slightly different measurements, Eratosthenes would have produced different results.

If he had measured the angle of the shadow in Syene to be 6 degrees:


  = 30,000

If he had measured the distance between the cities to be 600 miles:


  = 30,857

If he had measured the angle to be 6 degrees and the distance to the cities as 400 miles:


  = 24,000


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Eratosthenes used both inductive reasoning (to assume his premises were true) and deductive reasoning (logic and math) to come to a conclusion about the size of the Earth.

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Eratosthenes was certain that the Earth was about 25,000 miles in circumference.

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Eratosthenes created a mathemtical/logical valid argument, where if all his information was true, his conclusion would be true. But he had to use inductive reasoning to establish the reasonableness of his premises.

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