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Analytic Geometry and Calculus
Modern mathematics is almost entirely algebraic: we trust
equations and the rules of algebra morethan we trust pictures. For
example, consider the expression (x + y)2 = x2 + 2xy + y2. We trust
thatthis follows from various laws (axioms) of algebra:
(x + y)2 = (x + y)(x + y) (definition of ‘square’)= x(x + y) +
y(x + y) (distributive law)
= x2 + xy + yx + y2 (distributive law twice more)
= x2 + 2xy + y2 (commutativity)
For most of mathematical history, this result would have been
purely geo-metric: indeed it is Proposition 4 of Book II of
Euclid’s Elements:
The square on two parts equals the square on each part plustwice
the rectangle on the parts.
The proof was geometric: staring at the picture should make it
clear.
As we’ve seen, since 1200, the adoption of algebra and algebraic
notation proceeded slowly. Whileits utility for efficient
calculation was noted, it was not initially considered acceptable
to prove state-ments algebraically. Any algebraic calculation would
be justified via a geometric proof. From ourmodern viewpoint this
seems completely backwards: if a modern student were asked to prove
Eu-clid’s proposition, they’d likely label the ‘parts’ x and y,
before using the algebraic formula at the topof the page! Of course
each of the lines in the algebraic proof has a geometric basis.
• Distributivity says that the rectangle on a side and two parts
equals the sum of the rectangleson the side and each of the parts
respectively.
• Commutativity says that a rectangle has the same area if
rotated 90°.
The point is that we have converted geometric rules into
algebraic ones and largely forgotten the ge-ometric origin: a
modern student will likely never have considered the geometric
basis of somethingas basic as commutativity. After the Greek
invention of axiomatics, this slow switch from geometryto using
algebra as the grammar of mathematics is arguably the second major
revolution of math-ematical history: it has completely changed the
way mathematicians think. More practically, theconversion to
algebra has allowed for easy generalization: how would one
geometrically justify anexpression such as
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 ?
Euclidean Geometry is often termed synthetic: it is based on
purely geometric axioms without for-mulæ or co-ordinates. The
revolution of analytic geometry was to marry algebra and geometry
usingaxes and co-ordinates. Modern geometry is almost entirely
analytic or, at an advanced level, describedusing modern algebra
such as group theory. Modern mathematicians working in synthetic
geome-try are exceptionally rare; algebra’s triumph over geometry
has been total. The critical step in thisrevolution was made almost
simultaneously by Descartes and Fermat.
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Pierre de Fermat (1601–1665) One of the most famous
mathematicians of history, Fermat madegreat strides in several
areas such as number theory, optics, probability, analytic geometry
and earlycalculus. He approached mathematics as something of a
dilettante: it was his pastime, not his profes-sion.1 Some of the
fame of Fermat comes from his enigma: he published very little in a
formal sense,most of what we know of his work comes from letters to
friends in which he rarely offers proofs.Indeed he would regularly
challenge friends to prove results; it is often unknown whether he
hadproofs himself, or merely suspected a general statement. Being
outside the scientific mainstream, hisideas were often ignored or
downplayed. When he died, his notes and letters contained many
un-proven claims which eventually came to light. In particular,
Leonhard Euler (1707–1783) expendedmuch effort proving several of
these.
Fermat’s approach to analytic geometry was not dissimilar to
that of Descartes which we shall de-scribe below: he introduced a
single axis which allowed the conversion of curves into algebraic
equa-tions. We shall also return to Fermat when we discuss the
beginnings of calculus since he introducedone of the earliest
notions of differentiation.
René Descartes (1596–1650) In his approach to mathematics and
philosophy, Descartes is the chalkto Fermat’s cheese, rigorously
writing up everything! His defining work is 1637’s Discours de
laméthode. . . 2 While enormously influential in philosophy,
Discours was intended to lay the ground-work for investigations of
mathematics and the sciences; indeed Descartes finishes Discours
with anobservation of the necessity of experiment in science and of
his reluctance to publish due to the envi-ronment of hostility
surrounding Galileo’s prosecution.3 The copious appendices to
Discours containDescartes’ scientific work. It is in one of these,
La Géométrie, that Descartes introduces axes and
co-ordinates.
We now think of Cartesian axes and co-ordinates as being plural.
Both Fermat and Descartes, however,only used one axis. Here is the
rough idea of their approach.
• Draw a straight line (the axis) containing two fixed points
(the origin and the location of 1).
• All points on the line are immediately identified with numbers
x.
• To describe a curve in the plane, one draws a family of
parallel lines intersecting the curve andthe axis.
• The curve can then be thought of as a function f , where f (x)
is the distance from the axis to thecurve measured along the
corresponding parallel line.
• While neither Descartes nor Fermat had a second axes, their
approach implicitly imagines one:through the origin, parallel to
the family of measuring lines. It therefore makes sense for us
tospeak of the co-ordinates4 (x, y) of a point, where y = f
(x).
1He was from a wealthy but not aristocratic family, attending
the University of Orléans for three years where he trainedas a
lawyer.
2. . . of rightly conducting one’s reason and of seeking truth
in the sciences. Quite the mouthful. The primary part of thiswork
is philosophical and contains the first use of his famous phrase
cogito egro sum (I think therefore I am).
3At this time, France was still Catholic. Descartes had moved to
Holland partly to be able to pursue his work. Eventu-ally, in 1649,
Descartes moved to Sweden where he died the next year.
4The term co-ordinates suggests a symmetry of view when
considering the point (x, y). The traditional terms are abcissa(for
x) and ordinate (for y), stressing the idea that x is the
independent variable and y is dependent on x.
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Example: the parabola The function f (x) = x2 describes the
standard parabola in the usual way,where f (x) measures the
perpendicular distance from the axis to the curve.
Here is an alternative description of a parabola. This time the
function is f (x) = x2 + 1. Noticethe slope: with only one axis,
Descartes and Fermat could measure the distance to the curve
usingparallels inclined at whatever angle they liked. In a modern
sense, this example has a second axis,drawn in green, inclined 30°
to the vertical.
−2 −1 0 1 2 3x
f (2)
f (x)
x, X axes
Y axisy axis
origin x
y Y
X
60◦
P
If this makes you nervous, you can perform a change of basis
calculation from linear algebra: thepoint P in the second picture
has co-ordinates (X, Y) relative to ‘usual’ orthogonal Cartesian
axes; itsco-ordinates are (x, y) relative to the slanted axes. It
is easy to see that{
X = x + y cos 60° = x + 12 yY = y sin 60° =
√3
2 y
For any point on the curve, we then have
√3X−Y =
√3x =⇒ (
√3X−Y)2 = 3x2 = 3(y− 1) = 3
(2√3
Y− 1)
=⇒ 3X2 − 2√
3XY + Y2 − 2√
3Y + 3 = 0
which recovers the implicit equation for the parabola relative
to the standard orthogonal axes. Incase this alarms you, a simple
calculation of the discriminant5 shows that this really is a
parabola inthe modern algebraic sense!
Other curves could be similarly described. Descartes was
comfortable with curves having implicitequations. The standardized
use of a second axis orthogonal to the first was instituted in 1649
by Fransvan Schooten; this immediately gives us the modern notion
of the co-ordinates.
5A non-degenerate quadratic curve aX2 + bXY + cY2 + linear terms
is a parabola if the discriminant ∆ = b2 − 4ac = 0.A hyperbola has
∆ > 0 and an ellipse ∆ < 0.
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Descartes used his method to solve several problems that had
proved much more difficult in syntheticgeometry, such as finding
complicated intersections. Given the novelty of his approach,
Descartestypically gave geometric proofs of all assertions to back
up his algebraic work (similarly to how Is-lamic mathematicians had
proceeded). He was not, however, shy regarding the discovery,
statingthat, once several examples were done, it really wasn’t
necessary to draw physical lines and providea geometric argument:
the algebra was the proof. This point of view was controversial at
the time, butover the following centuries it eventually won
out.
As an example of the power of analytic geometry, consider the
following result.
Theorem. The medians of a triangle meet at a common point (the
centroid), which lies a third of the wayalong each median.
This can be done using pure Euclidean geometry, though it is
somewhat involved. It is comparativelyeasy in analytic
geometry.
Proof. Choose axes pointing along two sides of the triangle with
with the origin as one vertex.6
If the side lengths are a and b, then the third side has
equa-tion bx + ay = ab or y = b− ba x. The midpoints now
haveco-ordinates:( a
2, 0)
,(
0,b2
),(
a2
,b2
)Now compute the point 1/3 of the way along each median:for
instance
23
( a2
, 0)+
13(0, b) =
13(a, b)
One obtains the same result with the other medians.(0, 0) (a,
0)
(0, b)
( a2 , 0)
(0, b2) (a2 ,
b2)
G
With the assistance of his notation, Descartes made many other
mathematical breakthroughs. Forinstance, he was able to state a
critical part of the Fundamental Theorem of Algebra, the factor
theorem:if a is a root of a polynomial, then x− a is a factor. He
didn’t give a complete proof of this fact as hethought it to be
self-evident, perhaps because his notation made it so easy to work
with polynomials.Strictly the full theorem7 wasn’t proved until
Cauchy in 1821. The factor theorem is essentially acorollary of the
division algorithm for polynomials: if f (x), g(x) are polynomials,
then there existunique polynomials q(x), r(x) for which
f (x) = q(x)g(x) + r(x) deg r < deg g
If deg g = 1, then r is necessarily constant. Suppose g(x) = x−
a and f (a) = 0. Then r = 0.6This ability to choose axes to fit the
problem is a critical advantage of analytic geometry. In one
stroke, this dispenses with
all the tedious consideration of congruence in synthetic
geometry.7Every polynomial over C has may be factorized completely
over C. This needs some heavier analysis to show that a
root exists in the first place, then the factor theorem allows
you to pull these out one at a time.
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Calculus
At the heart of calculus is the relationship between four
concepts:8
• The instantaneous velocity of a particle is the rate of change
of its displacement.
• The net displacement of a particle is the net area under its
velocity-time graph.
To state such principles essentially requires the concept of a
graph and thus some form of analyticgeometry (rate of change means
slope. . . ). Once this appeared in the early 1600’s, it is
arguable thatthe development of calculus over the next 50 years was
inevitable. Perhaps this is true, thoughseveral of the basic ideas
of calculus were in place independently of analytic geometry, and
severalmathematicians made use of graph-like approaches prior to
Descartes and Fermat.
In the context of the above principles, the Fundamental Theorem
of Calculus is essentially a triviality;it merely states that
complete knowledge of displacement is equivalent to complete
knowledge ofvelocity. Of course, the modern statement is far more
daunting:
Theorem (FTC). 1. If f : [a, b] → R is continuous, then F(x) :=∫
x
a f (x)dx is continuous on [a, b],differentiable on (a, b), and
F′(x) = f (x).
2. Let F : [a, b] → R be continuous and differentiable on (a,
b). Then the net area under the curvey = F′(x) between a and b
is
∫ ba
F′(x)dx = F(b)− F(a)
The triumph of the Fundamental Theorem is its abstraction: no
longer must f (x) describe the velocityof a particle at time x, and
F(x) its displacement. The challenge of teaching9 and proving the
Funda-mental Theorem is entirely that of understanding what is
meant by continuous and differentiable. Thequest for good
definitions of these concepts is really the story of analysis in
the 17 and 1800’s, and isbeyond the scope of this course. We will
describe some of the story of how calculus came in to being:we
begin with some ancient considerations of the velocity and area
problems.
The Velocity Problem pre 1600
The concepts of uniform and average velocities are
straightforward:
Measure how far an object travels in a given time interval and
divide one by the other.
Several ancient Greek mathematicians (e.g. Autolycus) had
thought about uniform velocity and evenuniform acceleration, but
neither were considered quantities that could be measured. There
was es-sentially no progress with regard to the measurement of
velocity for over 1000 years. Around 1200,Gerard of Brussels tried
to define velocity as a ratio of two unlike quantities (distance :
time), thoughthis was not yet thought of as a numerical quantity in
its own right.
8Modern calculus abstracts these statements to form,
respectively, the subtopics of differentiation and
integration.9Introductory calculus students can easily be taught
the mechanics of calculus (e.g., the power law, the chain rule)
without having any idea of what it means: witness both the power
and the curse of analytic geometry and algebra!
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By contrast, the difficulty involved in defining instantaneous
velocity is profound: the foundationof differentiation is the
measurement of average velocity over smaller and smaller intervals
beforeinvoking the notion of a limit. Modern students are in good
company if they find this challenging:Zeno’s arrow paradox is
essentially an objection to the very idea of instantaneous
velocity! Even ifone accepts the concept, its direct measurement,
even in modern times, is essentially impossible.10 Itis important
therefore to appreciate that any pre-modern direct measurement of
velocity can only bea measurement of the average velocity over some
time interval.
Gerard was credited in the 1330’s by the Oxford Thinkers group11
as influencing their investigationsof instantaneous speed. They
formulated the following definition and gave the first statement
ofthe ‘mean speed theorem’. Both are vague and logically dubious,
but they are at least an attempt toapproach this difficult
notion.
Definition. The instantaneous velocity of a particle at an
instant will be measured as the uniformvelocity along the path that
would have been taken by the particle if it continued with that
velocity.
This is really the idea of inertial motion, although they are
assuming without justification that thisexists! Newton eventually
gets round this problem by positing that a force is necessary to
alter (accel-erate) the motion of a body.
Theorem. If a particle is uniformly accelerated from rest to
some velocity, it will travel half the distance itwould have
traveled over the same interval with the final velocity.
For centuries it was thought that Galileo was the first to state
such ideas, but the Oxford groupwere 250 years earlier. They had no
algebra with which to prove their assertions. Indeed the bestthey
could manage was to assert the proportion of powers rule. E.g., if
two time intervals are inproportion 2 : 1 and if particles subject
to the same uniform acceleration are accelerated from restover
these intervals, then the resulting distances traveled will be in
the proportion 4 : 1. In modernnotation:
v1 = at, v2 = 2at =⇒ d1 =12
v1t =12
at2, d2 =12
v2 · 2t = 2at2 = 4d2
In the 1350’s, Nicolas Oresme (mostly working in Paris) went
further. He thought of velocity ge-ometrically by (essentially)
drawing graphs of speed against time. He could distinguish
betweenuniform velocity, uniformly changing velocity, and
non-uniformly changing velocity (accelerationzero, constant or
non-constant). As we’ve seen, this is essentially the approach
taken by Galileo.Indeed Oresme considered most of the mathematics
on velocity that Galileo would later discuss. Amajor difference for
Galileo is that he married mathematics to observation: uniform
acceleration forGalileo was precisely the motion of a falling
body.
As we’ll see shortly, further progress on the velocity problem
really depended on the advent of ana-lytic geometry.
10For instance, radar Doppler-shift (as used by the police to
catch speeding motorists) still requires a measurement of
thewavelength of a radar beam, which in turn requires a finite
(albeit miniscule) time interval. Indeed, given our modern
un-derstanding of quantum mechanics and the Heisenberg uncertainty
principle, instantaneous velocity and precise locationare perhaps
meaningless concepts. Thankfully mathematicians can choose to deal
with idealized models of the universerather than the real
thing!
11Or Merton Thinkers, based at Merton College, Oxford. They
included Thomas Bardwardine, William Heytesbury, etc.
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The Area Problem pre 1600
We have already discussed two situations in which mathematicians
used calculus-like methods todescribe areas.
• Archimedes computed the area bounded by a parabola by
constructing an infinite sequence oftriangles to fill up the space.
He also approximated the area/circumference of a circle by
ap-proximating the circle with decreasingly sized triangles. His
‘cross-section’ approach to com-puting area/volume also seems very
modern, though this work was unknown until 1899 whenmodern calculus
was already well-established.
• Kepler argued for his second law (equal areas in equal times)
using infinitessimally small tri-angles to approximate segments of
an ellipse. Indeed he applied this method to several otherproblems
(even crediting Archimedes with the approach), while acknowledging
that there werephilosophical problems with infinitessimals.
The modern idea of Riemann sums is just a special case of
approximating a large area using smallerones: the philosophical
challenge is again the notion of limit! Riemann sums use rectangles
with fixedbase widths, but there is nothing stopping us from using
more general widths or even other shapes.
In what appears to be the earliest antecedent of Rie-mann sums,
Oresme described how to compute the dis-tance travelled by a
particle whose speed was constanton a sequence of intervals. For
example:
Over the time interval[
12n+1
,12n
)a particle travels at
speed 1 + 3n. How far does it travel in 1 second?
Oresme drew boxes to compute areas and obtained
d =∞
∑n=0
(1 + 3n)2−n−1 = 4
The infinite sum was evaluated by spotting two pat-terns,
similarly to how Archimedes had done things:
0
10
20
30
y
0 1x
12+
122
+123
+124
+ · · ·+ 12n+1
= 1− 12n+1
02+
122
+223
+324
+ · · ·+ n2n+1
= 1− n + 22n+1
Of course Oresme had none of our notation, and certainly didn’t
have our modern (limit-dependent)definition of an infinite sum!
Oresme also worked with similar problems for uniform
accelerationsover intervals. These are not Riemann sums, nor are
they physical, for a particle cannot suddenlychange speed!
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Calculus à la Fermat and Descartes
The advent of analytic geometry allowed Fermat and Descartes to
turn the computation of instan-taneous velocity and related
differentiation problems into algorithmic processes. In particular,
thevelocity of an object is identified with the slope of the
displacement-time graph, which can be com-puted using variations on
the modern method of secant lines. We disucss their competing
methods.
Fermat’s method of adequation Fermat first attacksthe question
of finding maxima and minima. In thepicture, the graph of p(x) = x3
− 12x + 19 is drawn,where the extreme value of p occurs at the
x-valuem = 2. The Fermat argues that if x1, x2 are located nearm,
in such a way that p(x1) = p(x2), then the polyno-mial p(x2)− p(x1)
(which equals zero!) is divisible byx2 − x1. Indeed
0 =p(x2)− p(x1)
x2 − x1=
x32 − 12x2 + 19− x31 + 12x1 − 19x2 − x1
=(x2 − x1)(x22 + x1x2 + x21 − 12)
x2 − x1= x22 + x1x2 + x
21 − 12
0
5
10
15
p(x)
0 1 2 3x
x1 x2
Fermat argues that since this holds for any x1, x2 near m such
that p(x1) = p(x2), that it must alsohold when x1 = x2 = m(!!), and
he concludes
3m3 − 12 = 0 =⇒ m = 2By considering values of x near to m, it is
clear to Fermat that he really has found a local minimum.We
recognize the idea that the slope of the tangent line is zero at
local extrema.
The above approach dates from around the 1620’s and is similar
to work done earlier by Viète. Fermatproceeds to alter the method
slightly: he considers the values p(x) and p(x + e) for a small
value e(he stated that x was ‘adequated’ by e). The difference p(x
+ e)− p(x) might be more easily dividedby e without nasty
factorizations. Compared with the above, we obtain
0 =p(x + e)− p(x)
e=
x3 + 3x2e + 3xe2 + e3 − 12x− 12e + 19− x3 + 12x− 19e
=3x2e + 3xe2 + e3 − 12e
e= 3x2 − 12 + 3xe + e2
He then sets e to zero and solves for x. Observe the derivative
p′(x) = 3x2 − 12 and that Fermat’s eis playing the same role as h
in the modern definition
p′(x) = limh→0
p(x + h)− p(x)h
If you recall elementary calculus, Fermat’s method is guaranteed
to work for any polynomial: theconcept of limit requires nothing
more for polynomials than simply evaluating at h = 0. Fermat
alsoextended his method to cover implicit curves and their
tangents.
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Descartes method of normals Descartes and Fermat are known to
have corresponded regardingtheir methods. Descartes indeed seems to
have felt somewhat challenged by Fermat, and engagedin some
criticism of Fermat’s method. Descartes’ approach (in La
Géométrie) involved a reliance oncircles and repeated roots of
polynomials in order to compute tangents. Here is an example
wherehe calculates the slope of the curve y = 14 (10x− x2) at the
point P = (4, 6).
0 4
PQ
R
t
N
n
ry
νx
Let N = (4 + ν, 0) be the point where the normal to the curve
intersects the x-axis.12 Draw a circleradius r centered at N. If r
is close to n, the circle intersects the curve in two points Q, R
near to P.The line joining Q, R is clearly an approximation to the
tangent line at P.
The co-ordinates of Q, R can be found by solving algebraic
equations: substituting y = 14 (10x− x2)into the equation for the
circle must result in an algebraic equation with two known roots,
namelythe x-values of Q and R. By the factor theorem, we have{(
x− (4 + ν))2
+ y2 = r2
y = 14 (10x− x2)=⇒ (x−Qx)(x− Rx) f (x) = 0
where f (x) is some polynomial. Rather than doing this
explicitly, Descartes observes that if r isadjusted until it equals
n, then Q and R coincide with P and the above equation has a
double-root:{(
x− (4 + ν))2
+ y2 = n2
y = 14 (10x− x2)=⇒ (x− Px)2 f (x) = (x− 4)2 f (x) = 0
The factorizing can be done by hand using long-division (note
that ν and n are currently unknown!):substituting as above, we
obtain
0 = x4 − 20x3 + 116x2 − 32(4 + ν)x + 16(4 + ν)2 − 16n2 = 0= (x−
4)2(x2 − 12x + 4) + 32(3− ν)x + 16(12 + 8ν + ν2 − n2)
It follows that the remainder 32(3− ν)x + 16(12 + 8ν + ν2 − n2)
must be zero, whence ν = 3. Bysimilar triangles, the slope of the
curve at P is therefore (phew!)
y√t2 − y2
=ν
y=
12
12At the time, ν was known as the subnormal and t the
tangent.
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Fermat and Area The previous methods essentially allow
differentiation, albeit very inefficiently!Fermat also approached
the area problem in a manner not dissimilar to Oresme. Here is an
examplewhereby we can discover the power law for integration: we
find the area under the curve y = x3
between x = 0 and x = a.
Let 0 < r < 1 be constant. The area of the rectangle on
theinterval [arn+1, arn] touching the curve at its upper right
is
An = (arn − arn+1) · (arn)3 = a4(1− r)r4n
The sum of the areas is
∞
∑n=0
An = a4(1− r)∞
∑n=0
r4n =a4(1− r)
1− r4
=a4(1− r)
(1− r)(1 + r + r2 + r3) =a4
1 + r + r2 + r3
Setting r = 1 recovers the area under the curve: 14 a4.
y
xaarar2ar3· · ·
a3
(ar)3
(ar2)3
A0
A1A2
There are several dubious moments in Fermat’s approach. His
approach to the geometric seriesformula was not rigorous by modern
standards (he certainly didn’t use the above notation), and heis
again implicitly invoking limits at the end by setting r = 1. More
philosophically challengingis the idea that a finite area can be
written as an infinite sum of finite areas: we again run into
theproblem of infinitessimals. Regardless, through Fermat’s method
one easily obtains the power law∫ a
0 xn dx = 1n+1 a
n+1 for any positive integer n.
Italian Calculus in the 17th Century
In early 17th century Italy, three scholars made great use of
the infinitessimal method. Galileo was oneof the first. Is his
classic ‘soup bowl’ problem, he compares the volume of the ‘bowl’
lying between ahemi-sphere and a cylinder to that of a cone.
Galileo’s approach, like that of Archimedes,13 was to compare
the cross-sections: in modern lan-guage, the cross-sectional areas
on both sides are πy2. Galileo argues that since the cross-sections
areequal, so must be the volumes. The volume of the bowl is
therefore that of a cone V = 13 πr
3.
13Archimedes did essentially the same problem 1900 years
earlier, though as his work was not uncovered until 1899,Galileo
was unaware of it.
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While Galileo thought the method nice, he couldn’t properly get
round two philosophical objections:
The zero-measure problem If the cross-sections are equal,14 then
at the top doesn’t this mean that acircle ‘equals’ a point?
Indivisibles sum to the whole? Can we really claim that (the
volumes of) the bowl and the cone areequal just because their
cross-sections are?
Indeed it was Galileo’s advocacy on these points that first
gained him notoriety within the Church.His later evangelism for the
Copernican theory was therefore a rekindling of old
animosities.
Bonaventura Cavalieri (1598–1647) Cavalieri, a student of
Galileo and a Jesuat scholar, gave a morethorough discussion of
indivisibles in 1635. In particular he is remembered for
Cavalieri’s principle:
If two geometric figures have proportional cross-sectional
measure at every point relativeto some line, then the two objects
have measure in the same proportion.
Galileo’s soup bowl is an example of thisreasoning; the ‘line’
is any vertical (saythe axis of the bowl).Another classic example
involves slid-ing a stack of coins or a deck of cards.
Extending his principle, Cavalieri managed to infer the power
law∫ 1
0 xndx = 1n+1 a
n+1, giving rea-sonable arguments for n = 1 and 2.
Here is a sketch of the approach for n = 2.
Draw a cube of side x inside a cube of side a.Consider the
pyramid with apex O and whose base isthe square face nearest the
viewer. The green squarewith area x2 is a cross-section of this
pyramid.
In Cavalieri’s language, the pyramid is ‘all the squares’.In
modern notation we would say that the pyramid hasvolume
∫ a0 x
2 dx.
The remaining faces of the small cube should convinceyou that
the large cube consists of three copies of thepyramid: thus∫ a
0x2 dx =
13
a3
14The words area and volume weren’t really used at this time:
the word equal could therefore mean deveral differentthings. .
.
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Example Cavalieri also used his method (book IV, prop 19 of his
work Geometria Indivisibilis) to cal-culate the area enclosed in an
Archimidean spiral.
A point moves at a constant speed along a line while the line
rotates at a fixed speed: this producesthe red curve; in polar
co-ordinates it is essentially r = θ for 0 ≤ θ ≤ 2π.Consider the
part of the blue circle through B which lies in-side the spiral. If
OA = r, then the blue circle has circumfer-ence 2πr. The point B
has polar co-ordinates (r, r), whencethe blue arc inside the spiral
has length
2πr · 2π − r2π
= r(2π − r)
We can imagine this as if the blue arc is a noodle which,when
cut at B and allowed to fall straight down, forms thedashed blue
line from A to the green parabola with equationy = −r(2π − r).The
area inside the spiral is therefore the same as that withinthe
parabola! It was well-known (e.g. Archimedes or usingCavalieri’s
work) that the area inside the parabola is 43 thatof the largest
triangle that can fit inside: we conclude thatthe area inside the
spiral is
43· 12 · π2 · π = 4
3π3
OA
B
r
y = −r(2π − r)
Cavalieri actually did this slightly differently, his parabola
was actually drawn in a rectangle, and thedifference subtracted
away from a triangle, but the above picture is easier to
visualize.
Galileo strongly encouraged Cavalieri in his investigations.
Unlike Galileo, Cavalieri did not courtcontroversy: his tome
(Geometria Indivisibilis) was very dense and difficult; moreover,
he was aware ofthe philosophical difficulty of indivisibles and
took great pains to never claim that the cross-sectionsequaled the
solid, etc. As such, Cavalieri was relatively safe, even as the
political rivals (the Jesuits) ofhis order (the Jesuats) within the
Church, worked hard to stamp out the study of indivisibles.
Evangalista Torricelli (1608–1647) Another contemporary of
Galileo and Cavalieri. He made manyuses of Cavalieri’s principle,
in particular arguing for its careful use.
Example 1 The sides of a rectangle are in the ratio 2 : 1; the
ratio ofthe red and blue segments is also 2 : 1. In Cavalieri’s
language, ‘allthe lines’ of the red triangle are twice ‘all the
lines’ of the blue triangle:Toricelli asks us to question whether
the red triangle has twice the areaof the blue.
Of course this is absurd for the triangles are congruent!
Torricelli points out that Cavalieri’s principlehas been
misapplied; the cross-sections aren’t measured with reference to a
common line.
In modern language, we would write∫ 2
012 x dx =
∫ 10 2y dy which are equal via the substitution x =
2y: the point is that the infinitessimals must also have ratio
dx : dy = 2 : 1!
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Example 2 Another of Torricelli’s examples offers a seeming
paradox.
A hyperbola with equation z = 1x is rotated aroundthe z-axis. A
(green) cylinder (cookie-cutter) centeredon the z-axis with radius
x lying under the surface willhave surface area
A = circumference · height = 2πxz = 2π
Underneath the graph at x, Torricelli draws a green cir-cular
disk with area 2π. Since the area of this disk isindependent of x,
Torricelli argues that the volume un-der the original blue surface
out to a radius a is given bythe volume of the solid orange
cylinder at the bottom ofthe picture:
V = 2πa
Torricelli argues that this is a realistic use of
Cavalieri’sprinciple since the cylindrical ‘cross-sections’ and
thecircular cross-sections are both measured with respectto the
same line (the x-axis).This is precisely the method of volume by
cylindricalshells that we learn in modern calculus:
V =∫ a
02πx · 1
xdx = 2πa
The conundrum is that the surface is infinitely tall! How can we
justify the idea that it lies above afinite volume?
The ideas of Galileo, Cavalieri and Torricelli were at once
useful and dangerous. The active perse-cution of these arguments
meant that they had few successors in Italy (the center of the
Church’spower). Italian science and mathematics somewhat stagnated
after this moment: some have arguedthat, were it not for the
Church’s disapproval, Rome could have rivalled London and Paris as
ma-jor cities of the world in the expansion to come. The center of
European science therefore movednorthwards: the English and French
reformations of the 1500’s together with developing ideas
ofreformed government15 meant that Northern Europe proved more
fertile ground for the flourishingof new ideas.
15For instance Hobbes’ Leviathan written during the English
Civil War (1642–1651) was a plea for the constraint of abso-lute
monarchical power. The Civil War itself proved to be a less subtle,
yet decapitatingly effective, strategy for reining ina King. .
.
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