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.

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.

2

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. . .

11

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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