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\title[Greek Mathematics in Clavius]
{Greek Mathematics Recovered in Books 6 and 7 of Clavius' {\it Geometria Practica}}

\institute[College of the Holy Cross]
{
   John B. Little\\
   Department of Mathematics and CS\\
   College of the Holy Cross

}

\date[ ] 
{June 29, 2018}

\begin{document}
\begin{frame}
\maketitle
\end{frame}

\section{Introduction -- Clavius and {\it Geometria Practica}}

\begin{frame}
\begin{itemize}
\item I've always been interested in the history of mathematics 
(in addition to my nominal specialty in algebraic geometry/computational methods/coding theory, etc.)
\item Want to be able to engage with original texts on their own terms -- you might recall the talks on 
Apollonius's {\it Conics} I gave at the last Clavius Group meeting at Holy Cross (two years ago)
\item So, I've been taking Greek and Latin language courses in HC's Classics department
\item The subject for today relates to a Latin-to-English translation project I have recently 
begun -- working with the {\it Geometria Practica} of Christopher Clavius, S.J. (1538 - 1612, CE)
\end{itemize}
\end{frame}

\begin{frame}
{\bf Overview}
\tableofcontents
\end{frame}

\begin{frame}
{\bf Clavius' Principal Mathematical Textbooks}
\begin{columns}
\column{.5 \textwidth}
\begin{figure}
\includegraphics[height=1.8in, width=1.4in]{clavius.jpg}
\caption{source: ricci.bc.edu/people/christopher-clavius.html}
\end{figure}
\column{.5\textwidth}
\begin{itemize}
\item {\it Euclidis Elementorum, Libri XV } (first ed. 1574)
\item {\it Epitome artithmeticae practicae} (first ed. 1583)
\item {\it Geometria practica} (first ed. 1604)
\item {\it Algebra} (first ed. 1608)
\item Also: Commentary on Theodosius, The Sine, Plane Triangles, Spherical Triangles
\end{itemize}
\end{columns}
\end{frame}

\begin{frame}
{\bf ``Practical Geometry''}

\begin{itemize}
\item Part of a long tradition from the late Medieval period (e.g. Leonardo Pisano -- ``Fibonacci'') into the 
early 1800's;
\item Showing influence of both Islamic and Greek traditions;
\item Intended to be \emph{useful} for questions in surveying, architecture, engineering, 
military science, etc.  
\item Quite stable tables of contents:
\item Construction and use of actual geometrical instruments -- astrolabe, quadrants, proportional compasses, etc.
\item ``Practical'' methods for geometric constructions supplementing what Euclid gives
\item {\it Numerical measures of lengths and angles used}; numerical examples given instead of, or in addition to, Euclidean-style proofs
\end{itemize}
\end{frame}

\begin{frame}
{\bf ``Practical Geometry,'' cont.}

\begin{itemize}
\item Overtly ``practical'' problems: measuring heights of towers or mountains, 
depths of valleys, etc. via trigonometry
\item Mensuration -- area and volume formulas, often expressed in Archimedean terms: e.g.
\emph{The volume of a sphere equals the volume of the rectangular solid contained by the radius and
a base with area the third part of the surface area of the sphere.}
\item ``Geodesy'' -- problems about subdividing plane areas or solid volumes
into parts in given ratios
\end{itemize}
\end{frame}

\begin{frame}
{\bf Contents of Clavius' {\it Geometria Practica}}
\begin{itemize}
\item Book 1: Geometrical instruments, the quadrant, triangles (refers to 
his {\it Triangula Rectilinea} for proofs)
\item Books 2 and 3:  Measuring linear dimensions by the quadrant and pendulum (trigonometry)
\item Book 4: Areas of plane figures (with Archimedes' approximation
of $\pi$ from \emph{On the dimension of the circle} and a more accurate approx. to 19 decimal places(!))
\item Book 5: Volumes and surface areas of solids (including Archimedes' results on the sphere and the cylinder)
\end{itemize}
\end{frame}

\begin{frame}
{\bf Contents, cont.}
\begin{itemize}
\item Book 6: ``\emph{In quo de Geodaesia, \& de figuris augendis, minuendisque in data proportione: 
Item de duarum mediarum proportionalium inter duas rectas inuentione : Ac denique de radicum
extractione agitur}''
\item Section on ``geodesy'' --
Clavius cites J. Dee and F. Commandino's translation of a book by ``Machometus Bagdadinus''  from Arabic, possibly 
partly derived from a Greek source -- Euclid's lost \emph{On Divisions of Figures}, and follow-up work by Simon Stevin
\item Book 7:  ``\emph{De figuris Isoperimetris disputans: cui Appendicis loco annectitur breuis de circulo
per lineas quadrando tractatiuncula}''
\item Book 8:  Advanced constructions
\end{itemize}
\end{frame}

\section{Book 6 and Greek approaches to duplication of the cube}

\begin{frame}
{\bf Some initial observations}

\begin{itemize}
\item Setting his work apart from other ``Practical Geometries,'' Clavius had (what seems to 
me to rise to the level of) a \emph{secondary goal}  -- to make known a number of potentially
applicable results from Greek geometry \emph{that did not ``make it into Euclid''} (e.g. because they 
were developed later).
\item The construction of two mean proportionals is one of these -- context 
is the \emph{duplication of the cube}
\item One of the three problems that spurred development of 
Greek geometry; still at forefront in Clavius' lifetime
due to Federigo Commandino (1509 - 1579) and others
\item Latin translations from Greek, esp. the {\it Synagoge} of Pappus of Alexandria (ca. 290 - ca. 350 CE).

\end{itemize}
\end{frame}

\begin{frame}
{\bf Relation of mean proportionals to duplication of the cube}

\begin{itemize}
\item Given $AB$ and $GH$,  $CD$ and $EF$ are \emph{two mean proportionals in 
continued proportion} if 
\begin{equation*}
\label{tmp}
AB : CD = CD : EF = EF : GH.
\end{equation*}
\item Hippocrates of Chios (ca. 470--ca. 410  BCE): if 
$GH = 2AB,$ then $CD^3 = 2AB^3.$
\item In other words, if $AB$ is the side of the original cube, then $CD$ is the side of the cube with 
twice the volume.  
\item  Geometric construction of the two mean proportionals still open, but all later work started from 
this reduction.  
\end{itemize}
\end{frame}

\begin{frame}
{\bf Clavius' Use of Eutocius' catalog}

\begin{itemize}
\item Today, the main thing most mathematicians know about this problem is that it cannot be 
solved with straightedge and compass--the Greeks also seem to have guessed this; Descartes
tried to prove it; first complete proof in 1800's using analytic geometry and algebra
\item  But the Greeks did not stop there and detailed accounts of other sorts of constructions have survived 
\item In Pappus and in a commentary on Archimedes' {\it On the sphere and cylinder} by 
Eutocius of Ascalon (ca. 480 -- ca. 540 CE. Note: $\sim$ 750 years after Archimedes!)
\item Clavius presented several solutions in this book because they apply to scaling solid 
objects up or down in a given proportion(!)
\end{itemize}
\end{frame}

\begin{frame}
{\bf What Clavius says}

\emph{Quo circa prius ... afferemus, quae antiqui Geometrae nobis hac de rescripta
relinquerunt.  Multorum enim ingenia res haec exercuit, atque torsit, quamuis
nemo ad hanc vsque diem, ver\`e, ac Geometric\`e duas medias proportionales
inter duas rectas datas inuenerit.  Praetermissis autem modis Eratosthenis; 
Platonis; Pappi Alexandrini; Spori; Menechmi tum beneficio Hyperbolae, ac
parabolae, tum ope duarum parabolarum; \& Architae Tarentini, quamuis
acutissimis, subtilissimisque; solum quatuor ab Herone, Apollonio Pergaeo,
Philone Bysantio, Philoppono, Diocle, \& Nicomede traditos explicabimus,
quos commodiores, facilioresque, \& \underline{errori minus obnoxios} iudicauimus.}  [my emphasis]
\end{frame}

\begin{frame}
{\bf My translation}

We will first report what the ancient geometers have left to us in their writings
concerning this.  For this problem drove and tormented the talents of many, although
up to this day, no one will have truly and geometrically constructed two mean proportionals
between two given lines.  Although they are most acute and subtle, the solutions
of Eratosthenes, Plato, Pappus of Alexandria, Sporus, Menaechmus by means of the 
hyperbola and parabola, then with the help of two parabolas, and Archytas of
Tarentum will be omitted; we will explain only the four solutions from Heron, 
Apollonius of Perga, Philo of Byzantium, Philopponus, Diocles, and Nicomedes.
We have judged these to be more convenient, easier, and less susceptible
to error.
\end{frame}

\begin{frame}
{\bf Hero's method -- \`a la Clavius}

\href{https://www.e-rara.ch/zuz/content/pageview/14258691}{\beamergotobutton{Link}}
\end{frame}

\begin{frame}
{\bf The construction -- refer to Clavius' figure}
\begin{itemize}
\item The given segments are $AB$, $BC$; with a right angle at $B$, the 
rectangle $ABCD$ is constructed, with diagonals intersecting at $E$.  Perpendiculars
are dropped from $E$ to $CD$ at $H$ and $AD$ at $K$.  
\item With the lines $DA$ and $DC$ extended to meet a line through B in $F$ and $G$,
\item The line is ``moved,'' i.e. rotated, about $B$ until a position is reached where 
$EG = EF$.    
\end{itemize}
\end{frame}

\begin{frame}
{\bf Claim and proof -- in modern semi-algebraic language}

\begin{itemize}
\item Claim:  $AF$ and $CG$ are the mean proportionals, i.e.
$$\frac{AB}{AF} = \frac{AF}{CG} = \frac{CG}{BC}.$$
\item Proposition II.6 of Euclid implies
$$AF\cdot DF + AK^2 = KF^2$$
\item If $EK^2$ is added to both sides and we use the Pythagorean theorem (Proposition I.47),
$$AF \cdot DF + AE^2 = EF^2$$
\item Similarly on the vertical side,
$$CG\cdot DG + CE^2 = EG^2$$
\end{itemize}
\end{frame}

\begin{frame}
{\bf Proof, concluded}

\begin{itemize}
\item $EF = EG$ by construction, and $AE = CE$ since the diagonals bisect each other.
\item Hence $AF \cdot DF = CG \cdot DG$, or
$$\frac{AF}{CG} = \frac{DG}{DF}$$
\item But 
$$\frac{DG}{DF} = \frac{AB}{AF} = \frac{GC}{BC}$$
by similar triangles.
\item This shows 
$$\frac{AB}{AF} = \frac{AF}{CG} = \frac{CG}{BC},$$
which is what we wanted.
\end{itemize}
\end{frame}

\begin{frame}
{\bf Observations}

\begin{itemize}
\item Technically a \emph{neusis} to find the line through $B$ with $EF = EG$ -- involves 
taking limits in modern language, and that's why Clavius qualifies this as ``{\it inuenire prope verum}''
\item Earlier in Book 1, Clavius refered to another work for proofs and just gave
a procedure; here, it's clear he thinks the proof is important because it justifies the 
construction. So, he gives a complete argument, 
\item buttressed with 
marginal references to (his version of) Euclid to justify 
almost all of the steps.  
\item Clavius is often described as more of teacher than a researcher, and
this discussion actually shows a typical example of how he reworked texts for greater clarity. 
\end{itemize}
\end{frame}

\begin{frame}
{\bf Observations, cont.}

\begin{itemize}
\item In Eutocius and Pappus, the methods of Heron and Apollonius are presented
separately with the connection claimed -- Clavius makes
that clear by putting the diagrams together.
\item Similar to the approach in his version of Euclid -- he's always rethinking, making
connections, ... 
\item After this discussion and propositions about rescaling geometric figures,
Clavius concludes Book 6 of the \emph{Geometria practica} with an extensive
discussion of numerical algorithms for extracting approximate square and cube roots(!)
\end{itemize}
\end{frame}

\section{Book 7 and squaring the circle via the quadratrix}

\begin{frame}
{\bf ``Squaring the circle''}

\begin{itemize}
\item Recall the following statement:  
``The volume of a sphere equals the volume of the rectangular solid contained by the radius and
a base equal to the third part of the surface area of the sphere. ''
\item Related:
``The area of a circle is equal to the area of the rectangle contained by the radius and
the semiperimeter.''
\item Question:  How could you construct such a rectangular solid or rectangle?   
\item Question:  Suppose you wanted the base of the rectangular solid 
to be a \emph{square} with area equal to the area of the given circle?   
\end{itemize}
\end{frame}

\begin{frame}
{\bf Book 7 -- Main topic}

\begin{itemize}
\item Begins with a discussion of \emph{isoperimetric problems}, following the approach of 
Zenodorus (ca. 200 - ca. 140 BCE), as preserved by Pappus and by Theon of Alexandria, 335 - 405 CE, father of Hypatia
\item Key fact: both circles and regular polygons satisfy the fact mentioned before:
$${\rm Area} = {\rm apothem} \cdot {\rm semiperimeter}$$
(an apothem is a line dropped perpendicular from the center to one of the sides $\leftrightarrow$ a radius of a circle)
\item Hence, if perimeter is fixed, more sides $\Rightarrow$ greater area, and circle with that perimeter 
has greater area than all the polygons.
\end{itemize}
\end{frame}

\begin{frame}
{\bf The connection}

\begin{itemize}
\item Note that this gives a way to describe a rectangle with the same area
as a given regular polygon or circle:
\item Take one side as an apothem, the other as the semiperimeter(!)
\item But how do you get a straight line equal to the semicircumference of a circle?
\item And if we want a square with this area, we need a \emph{mean proportional} between
the apothem and the semiperimeter, and this reduces to finding $\sqrt{\pi}$.
\item Hence this cannot be done by straightedge and compass either.  But the Greeks were
happy to ask: What other tools, auxiliary curves, 
etc. would yield a construction?
\end{itemize}
\end{frame}

\begin{frame}
{\bf Clavius' discussion}

\begin{itemize}
\item As in the discussion in Book 6, he starts off with a summary of previous work
\item Dismisses some approximate modern quadratures by ``Arabs,'' Durer, as only approximate
and Campano \emph{scathingly} as ``not worthy of geometry and plainly ridiculous''
\item Turns to Greek work, especially an attempt by Hippocrates (same figure we have seen before
in this talk in connection with the duplication of the cube)
\end{itemize}
\end{frame}
\begin{frame}
{\bf A partial success(?)}

\begin{columns}
\column{.5 \textwidth}
\begin{figure}
\vskip -50pt
\includegraphics[height=1.5in, width=1.5in]{Hippocrates.png}
\caption{Lunes of Hippocrates}
\end{figure}
\column{.5\textwidth}
\begin{itemize}
\item $\Delta ABC$ has right angle at $C$; with circumscribed circle and 
circles on the legs.
\item The {\it lunes} are the two regions between the circles.  
\item Claim is: \emph{The area of the lunes together equals the area of the triangle}
\end{itemize}
\end{columns}
\end{frame}

\begin{frame}
{\bf A partial success(?)}

\begin{columns}
\column{.5 \textwidth}
\begin{figure}
\vskip -50pt
\includegraphics[height=1.5in, width=1.5in]{Hippocrates.png}
\caption{Lunes of Hippocrates}
\end{figure}
\column{.5\textwidth}
\begin{itemize}
\item Let $a,b,c$ be the sides in the triangle, labeled as usual.
\item We have from semicircles
\begin{align*}
{\rm area}(ABC) &+ \frac{\pi a^2}{8} + \frac{\pi b^2}{8} =\\
{\rm area(lunes)} &+ \frac{\pi c^2}{8}
\end{align*}
\item But also $a^2 + b^2 = c^2$, so the claim follows.
\end{itemize}
\end{columns}
\end{frame}

\begin{frame}
{\bf The quadratrix}
\begin{columns}
\column{.5 \textwidth}
\begin{figure}
\includegraphics[height=2.0in, width=2.0in]{Quadratrix.png}
\caption{source: en.wikipedia.org}
\end{figure}
\column{.5\textwidth}
\begin{itemize}
\item Attributed to Dinostratus (died ca. 320 BCE), and a Hippias (probably \emph{not} the sophist by same name)
\item Mechanical description: $DC$ drops at a constant rate, while $AD$ rotates
\item The red curve is the quadratrix -- locus of intersections $S$
\item Used to trisect angles \emph{and} to square circles(!) 
\end{itemize}
\end{columns}
\end{frame}

\begin{frame}
{\bf Squaring a circle with the quadratrix}

\begin{itemize}
\item  Assuming the point S tends to some $S'$ along the line AB (this was controversial 
even in Clavius' period -- it really requires understanding of limits):
\item Clavius gives proof to show $AS' : AB  = AB : {\rm arc}\ DEB$
\item Since there is a well-known construction for a line segment in this proportion
to $AS', AB$ too, we can find a straight line $GH$ equal to the arc of the circle in this quadrant 
\item Area of quadrant is equal to one half the rectangle contained by $AB$ and $GH$ (by the fact
we noted above in connection with the isoperimetric problem)
\item To find a square with same area find \emph{mean proportional} between $AB$ and $GH$.
\end{itemize}
\end{frame}

\begin{frame}
{\bf Clavius' point of view here}

\begin{itemize}
\item Taken from an earlier pamphlet on the quadratrix, from 1589, included
also in some editions of his Euclid
\item Clavius \emph{claimed at first}: the quadratrix can be constructed by Euclidean
geometry, so it gives an ``orthodox'' quadrature of a circle
\item  His idea:  you could construct arbitrarily many points 
by bisection of the angle and the length $AD$, \emph{ad infinitum}.  
\item In modern language -- gives a dense set of points on the curve, not the whole curve;
\emph{not an exact construction of the line $AS'$} -- just a way to get approximations
\item By 1604, he had modified his claim to recognize this.
\end{itemize}
\end{frame}

\section{Conclusions} 

\begin{frame}
{\bf Overall conclusions}
\begin{itemize}
\item Clavius' textbooks were developed to support his teaching activities at the {\it Collegio Romano}, 
and then used all over the world in Jesuit schools, and other institutions
\item \emph{Prolegomena} to his Euclid -- Clavius justified the study of mathematics as
\emph{both noble} -- an abstract way of discovering universal truths (hence preparation for philosophy and theology) --
\emph{and useful} as a tool for answering concrete questions about the physical world, especially for future leaders in society who were (still are) a ``target audience'' for the Jesuit schools
\item \emph{Geometria practica} a perfect illustration of this view, and of
\item his abiding interest in the mathematics of ``the ancients''
\end{itemize}
\end{frame}

\begin{frame}
{\bf References}
\begin{enumerate}
\item[]\hskip-20pt [1] C. Clavius, S.J. {\it Geometria practica} (1606 edition), online at {\tt http://www.e-rara.ch}
\item[]\hskip-20pt [2] T. L. Heath, {\it A History of Greek Mathematics}, vol. I, Dover, NY 1981; reprint of the original ed., Oxford University Press, 1921.
\item[]\hskip-20pt [3] Wilbur Knorr,  {\it The Ancient Tradition of Geometric Problems}, Dover, NY, 1993; corrected reprint of original ed., Birkh\"auser, Boston, 1986.
\item[]\hskip-20pt [4] Reviel Netz,  {\it The Works of Archimedes, Volume 1:  The Two Books On the Sphere and the Cylinder}, Cambridge University 
Press NY, 2004.  
\end{enumerate}
\end{frame}


\end{document}









