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\markright{MATH 110 -- Mathematics Through Time, Final Exam Solutions}

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\noindent
I.  Identifications.  For any 5 of the following historical figures, give the approximate historical 
period in which they lived, where they were active, and the main contributions they made that we discussed.
(If you submit answers for more than 5, only the best 5 will be counted.)
\vskip 10pt
\noindent
(A) (10)  Eratosthenes -- about 276 - 195 BCE, active in Alexandria.  He made
a surprisingly accurate estimate of the radius of the Earth based on measurements
with a gnomon (sundial).  He also developed a method for finding all prime numbers
less than or equal to given number (the ``sieve method'').
\vskip 10pt
\noindent
(B) (10) Claudius Ptolemy -- about 90 - 168 CE, active in Alexandria.  He made some
contributions to geometry (in particular a theorem about the relation of the edges and
the diagonals in cyclic quadrilaterals).  He wrote on astronomy and developed the largest
geographical database known from the ancient world, allowing for construction of maps
of the known world at the time.  
\vskip 10pt
\noindent
(C) (10) Muhmammad ibn Musa al-Khwarizmi -- about 780 - 850 CE, active in Baghdad.  
He wrote a treatise on algebraic techniques (the word ``algebra'' comes from a 
corruption of the Arabic in the title).  He also wrote a text about the use
of Hindu-Arabic numerals (our base-10 positional system) that was very influential
\vskip 10pt
\noindent
(D) (10) Thabit ibn Qurra -- about 826 - 901 CE, active in Baghdad.  He was active
in the House of Wisdom set up by the Caliph al-Mansour and participated in the 
translation of many Greek mathematical texts into Arabic.  He extended the results
of Archimedes' {\it Quadrature of the Parabola}, made a critical reexamination
of the basis of Euclidean geometry (especially the role of Postulate 5) and extended Euclid's work
on perfect numbers to give a similar result for amicable pairs.  
\vskip 10pt
\noindent
(E) (10) Leonardo Pisano (``Fibonacci'') -- about 1170 - 1250 CE, active in Italy (various cities).  He wrote a book introducing the Hindu-Arabic numerals to scholars in western Europe and made various other algebraic and numerical contributions, including the definition of the well-known Fibonacci 
sequence.  
\vskip 10pt
\noindent
(F) (10) Filippo Brunelleschi -- about 1377 - 1446 CE, active in Florence.  He was an architect
and artist whose best-known work was the dome of the cathedral in Florence (still the largest masonry dome in existence in the world).  He studied geometry deeply for use in design, and 
also for basis of linear perspective in painting (methods for reproducing what the eye
sees in a 3D scene on a flat 2D canvas or piece of paper).  
\vskip 10pt
\noindent
II.
\begin{enumerate}
\item[(A)] (10) What are the 5 Common Notions (Axioms) and 5 Postulates that appear at the beginning
of Book I of Euclid's {\it Elements}?
\vskip 10pt
\noindent
\item[] {\it Answer:} The Five Common Notions are:
\begin{enumerate}
\item Things equal to the same thing are equal to one another
\item If equals are added to equals, the results are equal
\item If equals are subtracted from equals, the remainders are equal
\item Things that coincide with one another are equal to one another
\item The whole is greater than the part
\end{enumerate}
The Five Postulates are:
\begin{enumerate}
\item (It is possible) to connect any two points with a straight line
\item (It is possible) to extend a line indefinitely in both directions
\item (It is possible) to draw a circle with any given center and any given radius
\item All right angles are equal
\item If two lines falling on a third line make angles summing to less than
two right angles on one side, then the two lines, if extended, will meet on 
that side of the third line.
\end{enumerate}
\item[(B)] (15) Proposition 32 in Book I deals with the sum of the angles
in a triangle.  What is the exact statement that Euclid proves?  Outline the construction and proof
that establishes this Proposition.  (You don't need to quote earlier propositions by number, but
do give a reason for each of your assertions.)
\item[] {\it Answer:}  Proposition 32 says that (a) each exterior angle formed by extending
a side of a triangle is equal to the sum of the two opposite
interior angles, and (b) the sum of the interior angles in a triangle
is equal to 2 right angles (or a straight, 180-degree, angle in our language).  The proof 
is based on this construction:
\begin{center}
\includegraphics[scale=.4]{../EuclidI32.png}
\end{center}
Extend the side $AB$ to $D$ and construct a parallel $BE$ to $AC$ through the point $B$
(using Proposition 31 -- OK just to say this construction had been shown previously).  
Then in the resulting figure $\angle CAB = \angle EBD$ since those are corresponding
angles for the two parallels and the transversal.  Similarly, $\angle EBC = \angle ACB$
since those are alternate interior angles.  From this, we get part (a) immediately from
Common Notion 2.
This also shows (again by Common Notion 2) part (b):  that the angle 
$\angle DBA$ is equal to the sum of the angles in the triangle.  But $\angle DBA$ is 
equal to 2 right angles since $D,B,A$ are all on the same line (previous proposition).  QED
\end{enumerate}
\vskip 10pt
\noindent
Choose either this problem (III) or III' on the next page.  (If you submit solutions for 
both, only the better of the two will be used.)
\begin{enumerate}
\item[(A)] (15)  What is true about the sum of the angles in a triangle under the alternate {\it hyperbolic}
form (H) of Postulate 5?  Outline the proof, assuming the needed
properties of Saccheri quadrilaterals.
\item[] {\it Answer:}  Under the assumption (H), the sum of the angles in any triangle
is strictly less than 2 right angles.  This is proved as follows:  
\begin{center}
\includegraphics[scale=.5]{../HTriAngSum.jpeg}
\end{center}
$E$ is constructed to be the midpoint of $AC$, so $AE = CE$.  Similarly $D$ is constructed
to be the midpoint of $AB$, so $AD = DB$.  Then we extend the line $ED$ and drop perpendiculars 
from $A$, $B$, and $C$.  By the AAS side criterion (using equality of vertical angles and Postulate 4), $\Delta CGE$ and $\Delta AHE$ are
congruent, as are $\Delta AHD$ and $\Delta BFD$.  This shows that $CG = HA = BF$, so 
$CBFG$ is a Saccheri quadrilateral by definition.  We know, however that the sum 
of the ``summit angles'' in such a quadrilateral is less than two right angles.  Here
the ``summit angles'' are $\angle GCB$ and $\angle FBC$.  But then by the congruences
proved before, the sum of the angles in the triangle $\Delta ABC$ is equal 
$\angle GCB + \angle FBC$, so the angle sum is less than two right angles.   
\item[(B)] (10)  Was the result in part A the contradiction that Girolamo Saccheri, S. J. was seeking under
the assumption (H)?  Explain.   What did Janos Bolyai and Nikolai Ivanovich Lobachevsky eventually
show about the search for a contradiction starting from (H)?
\item[] {\it Answer:}  No, this is not a contradiction because the angle sum theorem from
Euclid's Proposition 32 (see problem II) is only valid under his Postulate 5.  (The equality
of the corresponding angles and the alternate interior angles formed by 
a transversal on two parallel lines from Proposition 29 is 
proved using Postulate 5.)  Bolyai and Lobachevsky (and Gauss too) eventually realized
that no contradiction would result from (H).  There is another equally valid geometric
theory here, the one we now call ``hyperbolic geometry.'' 
\end{enumerate}
\vskip 10pt
\noindent
III'.  (If you submit solutions for both this problem and III on the last page, only the better of the two will be used.)
\begin{enumerate}
\item[(A)] (15) The following problem appears in a text by the Indian mathematician
Mahavira (about 850 C.E.).  Solve it using our modern notation
and algebraic methods:  Of a collection of mangoes, the king took $1/6$, the queen
$1/5$ of the remainder, the chief princes $1/4, 1/3, 1/2$ of the successive
remainders, and the youngest child took the remaining $3$ mangoes.  Oh you who
are clever in miscellaneous problems on fractions, give the measure (number) of the
original collection of mangoes.  
\item[] {\it Answer:}  The problem can be solved either by ``breaking down'' or 
``building up.''  If we build up from the 3 mangoes the youngest child got, then 
the last chief prince took $1/2$, so there were 6 mangoes left before he made his choice.  
Similarly the second chief prince took $1/3$, so there were 9 mangoes left before his choice.
Working back in this way, we can see the original pile contained 18 mangoes and 
each of the six people involved took equal shares -- 3 mangoes each.  
\item[(B)] (10)  What was probably the single largest contribution of ancient Indian mathematics
to the way we do mathematics today?  How and when did this contribution enter our mathematical
tradition?
\item[] {\it Answer:} It was pretty definitely the use of the base-10 positional
number system containing digits $0,1,2,\ldots,9$ and methods for computing with that 
system.  Knowledge of this system reached Europe from India by way of the Islamic
world in the middle ages.  The book by al-Khwarizmi popularized this system in the
Islamic world, and European mathematicians such as Fibonacci learned it from Islamic
sources.  
\end{enumerate}
\vskip 10pt
\noindent
IV.  
\begin{enumerate}
\item[A)] (15)  Draw the diagram for the Chinese ``{\it go gou} theorem" (with a general right triangle, sides $a,b$ and hypotenuse $c$) and show how it provides an algebraic/geometric proof of the general ``Pythagorean theorem.''
\item[] {\it Answer:}  Here is the original Chinese figure we saw in class:
\begin{center}
\includegraphics[scale=.7]{../gogou.jpg}
\end{center}
If the longer sides of the triangles are all $b$ and the shorter sides are all $a$, then
the four triangles and the smaller central square fit together to make a large square
with side $c$, the hypotenuse of the triangles.  By adding areas, we see
$$c^2 = 4\cdot \frac{ab}{2} + (b - a)^2 = 2ab + b^2 - 2ab + a^2 = a^2 + b^2$$
This shows the algebraic form of the ``Pythagorean theorem.'' 
\item[B)] (10) How is this proof different from the one given by Euclid in Proposition 47 of 
Book I of the {\it Elements}?
\item[] {\it Answer:}  It is mainly different in that Euclid's proof uses no algebra
at all.  Euclid shows that in the figure
\begin{center}
\includegraphics[scale=.4]{../EuclidI47a.png}
\end{center}
The area of the small square on the left is equal to the area of the rectangle to 
the left of the blue line in the large square, and similarly, the area of the 
medium square on the right is equal to the area of the rectangle to the right of 
the blue line in the large square.  So ``the square on the hypotenuse equals the 
sum of the squares on the sides'' in terms of area.    
This is based entirely on comparisons between
areas of various triangles and parallelograms proved with no algebra.  
\end{enumerate}
\vskip 10pt
\noindent
V. 
\begin{enumerate}
\item[A)]  (15) What are {\it perfect numbers}?  What are
{\it amicable pairs of numbers}?  Give an example of a perfect number and an amicable pair 
and explain how they satisfy the definitions.
\item[] {\it Answer:}  A perfect number is a positive integer that is equal to the 
sum of its proper integer divisors.  For instance, $28 = 1 + 2 + 4 + 7 + 14$ is perfect.
An amicable pair of numbers is a pair $m,n$ of positive integers with the property 
that the sum of the proper divisors of $m$ is equal to $n$, and vice versa the sum
of the proper divisors of $n$ is equal to $m$.  The pair $m = 220$, $n = 284$
is an amicable pair since $220$ has proper divisors $1,2,4,5,10,20,11,22,44,55,110$ 
and these add up to $284$, while $284$ has proper divisors $1,2,4,71,142$, and these
add up to $220$.  
\item[B)] (10) How does the historical development of knowledge about these number-theoretical ideas
tie together the Greeks, the medieval Islamic mathematicians, and mathematics in the contemporary world?   
\item[] {\it Answer:}  The idea of perfect numbers and amicable pairs can be traced
back to the Pythagoreans and their number symbolism.  Euclid studied perfect numbers
in Book X of the {\it Elements}, showing that if $p$ is prime and $2^p - 1$ is also prime,
then $2^{p-1}(2^p - 1)$ is an even perfect number.  (We get can $28 = 4 \cdot 7 = 2^{3-1}(2^3 - 1)$ 
this way.)  The medieval Islamic mathematicians learned about these ideas from 
their study of Euclid and other Greek texts, and extended some of them -- for instance
the theorem of Thabit ibn-Qurra on amicable pairs.  These ideas lead to many 
questions whose answers are unknown even today and which are connected to research
going on in contemporary mathematics -- Are there any odd perfect numbers?
Are there infinitely many primes $p$ for which $2^p - 1$ is also prime?  Are
there infinitely many amicable pairs of numbers?
\end{enumerate}
\vskip 10pt
\noindent
VI.  Essay.  (50)  Here are two contrasting statements about the ultimate legacy of
Greek mathematics:
\begin{itemize}
\item ``The death of Archimedes by the hands of a Roman soldier is symbolical of a
world-change of the first magnitude: the Greeks, with their love of abstract science, were
superseded ... by the practical Romans.  The Romans were a great race, but they were cursed
with the sterility which waits upon practicality.  They did not improve upon the knowledge of their forefathers,
and all their advances were confined to the minor technical details of engineering. They were not
dreamers enough to arrive at new points of view ... No Roman lost his life because he was absorbed
in the contemplation of a mathematical diagram.''  (Alfred North Whitehead -- 20th century philosopher)
\item
``There is no denying that the Greek approach to mathematics produced remarkable
results, but it also hampered the subsequent development of the subject. ... The Greek preoccupation
with geometry ... was a serious constraint.  Great minds such as Pythagoras, Euclid, and Apollonius spent much of
their time creating what were essentially abstract idealized constructs; how they arrived at a conclusion
was in some way more important than any practical significance.'' (George G. Joseph, {\it The Crest of the
Peacock})
\end{itemize}
\noindent
Begin by summarizing and explaining your understanding of each statement. Then address
the following questions:  Have we seen examples in this class where practicality was {\it not sterile}, where 
concern with applications actually enriched the history of mathematics and made further progress possible?
Have we seen other examples where the ``abstract idealized constructs'' of pure mathematics led to major
advances?  Why do you suppose that Joseph doesn't mention Archimedes?  Does that omission weaken his point? 
\vskip 10pt
\noindent
{\it A possible response:}  Whitehead is saying that if a mathematical culture concentrates
only (or maybe too much) on practical applications (as he claims the Romans did), 
then their mathematics can be {\it sterile}, in the sense that they will not be ready 
to develop new ideas as needed.  He compares the Romans to the Greeks and finds
the Greeks to be more imaginative ``dreamers'' who ultimately contributed much 
more to the development of mathematics than the Romans did.

Joseph is saying, on the other hand, that if a mathematical culture focuses only (or too much)
on theoretical mathematics, with its characteristic preoccupations concerning methods
of proof (``how they arrived at results''), then their mathematics 
can also ``miss out'' by not developing practical applications.  He 
singles out the Greeks as a key example of this, saying that their 
theoretical orientation (for instance, the lack of notions of numerical
measures of lengths or angles) may in fact have constrained or limited the 
development of mathematics.    

In a sense, combining the underlying points here, one could argue that the ideal mix
is a healthy combination of pure and applied mathematics, each reinforcing the other.
But Whitehead is saying the Romans went too far in one direction and the Greeks
were more productive, while Joseph is saying the Greeks went too far in the other 
direction.  

As this applies to Greek mathematics in particular, one example that shows how
practicality can lead to great mathematical advances is, ironically considering 
Whitehead's point of view, the case
of Archimedes himself.  One could argue that some of Archimedes' most profound
and forward-looking pure mathematical work, the {\it Quadrature of the Parabola}, was
based on his understanding of the very practical physics of balances, centers of
mass, and so forth.  The way the Islamic mathematicians of the middle ages combined
Greek geometry with more practical Mesopotamian algebra and Indian arithmetic is 
another example of how (some) practicality can reinforce and enrich pure mathematics.

On the other hand, it was the ``abstract idealized concepts'' of pure mathematics,
in particular the basis from Euclid, the work of Apollonius on conics, etc. that
made it possible for Archimedes to do what he did.  Similarly, later, it was
the theoretical basis of Euclidean geometry that made it possible for 
Renaissance architects and painters to build things like the dome of the
cathedral in Florence and develop realistic perspective in painting.  

Mentioning Archimedes would certainly weaken Joseph's point since his work 
combined the theoretical and the practical in such fruitful ways.  But 
Joseph would also probably say that Archimedes belongs to the Hellenistic
period and marks the start of a different phase of Greek mathematics.  He
would say he's thinking mainly of the earlier period of Pythagoras, Euclid, 
and Apollonius.  By the time we get to the time of 
Heron, Claudius Ptolemy, etc. Greek mathematics has acquired a very 
practical side in addition to the theoretical preoccupations.  But
Archimedes is actually earlier than that too, so it's subtle!
\end{document}
