where D is the diameter of the circumcircle. (Here we
make use of the fact that any angle subtended from a
diameter is 90°.) Thus the three quantities expressed in
the law of sines each equal the diameter of the circumcircle of the triangle, thereby offering an alternative
proof of the law.
See also CIRCLE THEOREMS.
law of large numbers If one tosses a fair coin 10
times, one would expect, on average, five of those
tosses to be “heads.” Of course, in any single run of 10
tosses, any number of heads is possible, even a string of
10 heads in a row, but this is extremely unlikely. The
number of heads actually observed in an experiment is
likely to be four, five, or six, close to 50 percent. (The
probability of any particular count of heads appearing
is described by the BINOMIAL DISTRIBUTION.)
In a run of 100 tosses we would expect the effects
of excessive runs of heads, or tails, to “average out”
and the proportion of heads obtained to be even closer,
on average, to the “true value” of 50 percent—and
closer still if we run an experiment of 1,000 or 10,000
tosses, or more. This is the law of large numbers in
action. Precisely, this law states the following:
The more times a random phenomenon is performed, the closer the proportion of trials in
which a particular outcome occurs approximates the true probability of that outcome
occurring.
If, for example, a 1 never occurred when rolling a die
10 times, we can be assured, however, that the proportion of 1s appearing in another 100, 1,000, 10,000, …
tosses will approach the value one-sixth.
Many gamblers incorrectly interpret the law of
large numbers as a method for predicting outcomes of
random events. (See LAW OF AVERAGES.)
The law of large numbers is a mathematical consequence of CHEBYSHEV’S THEOREM. It can be interpreted
as saying that if a random phenomenon produces
numerical outcomes with mean value µ, then the mean
of N observed values of the phenomenon approaches
the value µ as N increases. Chebyshev’s theorem is
related, for it gives measures of how values are distributed about the mean.
See also MONTE CARLO METHOD; STATISTICS:
DESCRIPTIVE.
law of sines (sine rule) See LAW OF COSINES/LAW
OF SINES.
law of the lever ARCHIMEDES OF SYRACUSE (ca.
287–212 B.C.E.) recognized that two weights w 1 and w 2
placed at distances x 1 and x 2 , respectively, from the fulcrum (pivot point) of a simple lever will balance when
x 1 w 1 = x 2 w 2 . This principle is called the law of the
lever. For example, an adult weighting twice as much
as a child will balance on a seesaw if she sits half the
distance from the pivot point as the child.
See also CENTER OF GRAVITY.
laws of thought The Greek philosopher ARISTOTLE
(384–322 B.C.E.) identified three laws of logic, all tautologies (meaning that each has a TRUTH TABLE with
constant value T) that have since been deemed fundamental descriptions of the way we think. His three laws
of thought are:
1. Law of Noncontradiction: It is not the case that
something can be both true and not true.
Symbolically: ¬[p (¬p)]
2. Law of Excluded Middle: Each must either be true
or not true.
Symbolically: p ∨ (¬p)
3. Law of Identity: If something is true, then it is
true.
Symbolically: p → p
Mathematicians often rely on the law of excluded
middle to establish the validity of mathematical results:
an INDIRECT PROOF or a PROOF BY CONTRADICTION
proves that a statement p is true by showing that it
cannot be false. However, not all philosophers (and
mathematicians) agree with this approach and question
the validity of this second law. For example, as the
20th-century Austrian mathematician KURT GÖDEL
showed, there are some statements in mathematics that
can neither be proved nor disproved, and are consequently neither true nor false. The constructivist movement accepts results established by DIRECT PROOF only.
To move beyond the law of the excluded middle,
logicians have attempted to generalize FORMAL LOGIC
to include three possible values of truthhood: true,
∨
304 law of large numbers
make use of the fact that any angle subtended from a
diameter is 90°.) Thus the three quantities expressed in
the law of sines each equal the diameter of the circumcircle of the triangle, thereby offering an alternative
proof of the law.
See also CIRCLE THEOREMS.
law of large numbers If one tosses a fair coin 10
times, one would expect, on average, five of those
tosses to be “heads.” Of course, in any single run of 10
tosses, any number of heads is possible, even a string of
10 heads in a row, but this is extremely unlikely. The
number of heads actually observed in an experiment is
likely to be four, five, or six, close to 50 percent. (The
probability of any particular count of heads appearing
is described by the BINOMIAL DISTRIBUTION.)
In a run of 100 tosses we would expect the effects
of excessive runs of heads, or tails, to “average out”
and the proportion of heads obtained to be even closer,
on average, to the “true value” of 50 percent—and
closer still if we run an experiment of 1,000 or 10,000
tosses, or more. This is the law of large numbers in
action. Precisely, this law states the following:
The more times a random phenomenon is performed, the closer the proportion of trials in
which a particular outcome occurs approximates the true probability of that outcome
occurring.
If, for example, a 1 never occurred when rolling a die
10 times, we can be assured, however, that the proportion of 1s appearing in another 100, 1,000, 10,000, …
tosses will approach the value one-sixth.
Many gamblers incorrectly interpret the law of
large numbers as a method for predicting outcomes of
random events. (See LAW OF AVERAGES.)
The law of large numbers is a mathematical consequence of CHEBYSHEV’S THEOREM. It can be interpreted
as saying that if a random phenomenon produces
numerical outcomes with mean value µ, then the mean
of N observed values of the phenomenon approaches
the value µ as N increases. Chebyshev’s theorem is
related, for it gives measures of how values are distributed about the mean.
See also MONTE CARLO METHOD; STATISTICS:
DESCRIPTIVE.
law of sines (sine rule) See LAW OF COSINES/LAW
OF SINES.
law of the lever ARCHIMEDES OF SYRACUSE (ca.
287–212 B.C.E.) recognized that two weights w 1 and w 2
placed at distances x 1 and x 2 , respectively, from the fulcrum (pivot point) of a simple lever will balance when
x 1 w 1 = x 2 w 2 . This principle is called the law of the
lever. For example, an adult weighting twice as much
as a child will balance on a seesaw if she sits half the
distance from the pivot point as the child.
See also CENTER OF GRAVITY.
laws of thought The Greek philosopher ARISTOTLE
(384–322 B.C.E.) identified three laws of logic, all tautologies (meaning that each has a TRUTH TABLE with
constant value T) that have since been deemed fundamental descriptions of the way we think. His three laws
of thought are:
1. Law of Noncontradiction: It is not the case that
something can be both true and not true.
Symbolically: ¬[p (¬p)]
2. Law of Excluded Middle: Each must either be true
or not true.
Symbolically: p ∨ (¬p)
3. Law of Identity: If something is true, then it is
true.
Symbolically: p → p
Mathematicians often rely on the law of excluded
middle to establish the validity of mathematical results:
an INDIRECT PROOF or a PROOF BY CONTRADICTION
proves that a statement p is true by showing that it
cannot be false. However, not all philosophers (and
mathematicians) agree with this approach and question
the validity of this second law. For example, as the
20th-century Austrian mathematician KURT GÖDEL
showed, there are some statements in mathematics that
can neither be proved nor disproved, and are consequently neither true nor false. The constructivist movement accepts results established by DIRECT PROOF only.
To move beyond the law of the excluded middle,
logicians have attempted to generalize FORMAL LOGIC
to include three possible values of truthhood: true,
∨
304 law of large numbers