Historical Topics: The Development of Integral Calculus

Historical Topics: The Development of Integral CalculusAuthor(s): D. R. GreenSource: Mathematics in School, Vol. 8, No. 3 (May, 1979), pp. 24-28Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30213467 .

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Introduction It is often stressed by historians of mathematics that new ideas or new branches of mathematics are not due to one man but rather are the result of the work of a whole succession of men. The history of the calculus is an excellent example of this, with the first steps being made around 500 BC, culminating in the generalising discoveries of Newton and Leibniz in the seventeenth century, and then the placing of the fundamentals of calculus on a rigorous basis by Cauchy and others in the nineteenth century. Whole books have been devoted to this topic and only a very brief sketch of the development of integral calculus will be given here. A later article will consider differential calculus.

Ancient Greek Thought One of the great conceptual difficulties of the calculus is that of the infinite and infinitesimal. How many points make up a line? How big is a point? What are the smallest mass, shortest distance and smallest unit of time?

The Pythagoreans held the view that a line was made up of a whole number of small indivisible units - but they discovered that this could not be the case for the diagonal of a square when they proved that V2 is irrational, i.e. not expressible as plq where p and q are whole numbers. This "incommensur- ability" problem was never to be satisfactorily overcome by the Greeks. The corresponding physical doctrine was that of atomism - all things are made up of hard indivisible particles. This was the view first formulated by Leucippus (c. 430 BC) and adopted by Democritus (c. 400 BC).

Whatever their intention, the paradoxes of Zeno (c. 400 BC) dealt a severe blow to these schools of thought and also to those who held the opposite views! They are well worth careful consideration today.

Zeno's Paradoxes The Arrow If time is made up of indivisible instants then an arrow is always

24

at rest, for at any instant the arrow is in one fixed position, and this is true for every instant. Since the arrow is at rest at every instant, it can never move.

The Dichotomy Before an object can travel a certain distance it must travel half, and before that one-quarter ... and so on ad infinitum. Since this is an infinite regression, motion is impossible as it would require the object to traverse an infinite number of divisions in finite time.

The Achilles Achilles gives a tortoise so many metres start in a race. By the time Achilles reaches the starting point of the tortoise, the tortoise has moved on a certain distance. By the time Achilles has covered that extra distance the tortoise will have got further ... and so on. Thus Achilles can never overtake the tortoise.

The arrow paradox argues against space and time being made up of small indivisible elements ("atomism"). The Dichotomy and the Achilles paradoxes argue against the opposite idea - against the infinite divisibility of space and time.

These paradoxes were to trouble philosophers, scientists and mathematicians through the centuries, and were only finally disposed of with the formulation of differential calculus, with its limit concept.

Quadrature The Greeks were well aware that any figure bounded by straight lines could be reduced to a square of equal area. The stages in this interesting geometrical exercise can be found in Euclid, Book 2. The question then naturally arose: "Can all figures be so reduced?" The process of finding the square whose area is equivalent to that of a given figure is known as quadrature. One of the three famous problems of antiquity, namely "Squaring the circle", is in fact a quadrature problem, which today would be a problem for the integral calculus.



The Method of Exhaustion - Description Early problems considered by the Greeks were concerned with finding areas and volumes enclosed by figures, and the method of exhaustion was devised by the Platonic School probably in the fourth century BC to prove results found by less rigorous means. It is often ascribed to Eudoxus (c. 370 BC) and can be found in Euclid, Book 12.

The method assumes that magnitudes can be infinitely sub- divided and takes as its basis the following proposition:

Proposition "If, from any magnitude, a part not less than one half is subtracted and then from that remainder a part, again not less than one half, is subtracted and so on ... then eventually that which remains will be less than any desired magnitude."

As an example of the use of this method we take the problem (solved by the Greeks) of proving that if a and A are the areas of circles with diameters d and D respectively, then a :A = d2 :D2 (a result which pupils often forget!). The reader can skip this example if he or she wishes and go straight to the discussion in the next section.

PROOF Assumptions: We assume the Proposition stated above and use the fact known to the Greeks (and proved by them) that for two similar polygons of areas p and P with diameters d and D, p : P = d2 : D2.

Stage 1 We show that we can inscribe in any given circle a regular polygon whose area is as close as we please to that of the circle.

Let PQ be the side of a regular inscribed polygon and let M

N

Q- L

Fig. 1

be the midpoint of the arc PMQ (Fig. 1). The area of triangle PMQ is half that of the rectangle PLNQ and so is more than half that of the segment PMQ of the circle. The segment represents the amount of area of the circle outside the polygonal side PQ. By doubling the number of sides (e.g. PQ replaced by PM and MQ) we reduce that excluded area to less than half its previous value. If we keep on doubling the number of sides we have the conditions described in the Proposition (where the "magnitude" is the region of the circle not included in the inscribing polygon). Thus we can conclude that the difference in area between a circle and an inscribed regular polygon can be made as small as desired by repeated doubling of the number of sides.

Stage 2 We suppose that a : A > d2 : D2 and hope to reach an absurd conclusion. We inscribe in the circle of area a, a polygon whose area p differs so little from a that p : A > d2 : D2 ... (i). Similarly we inscribe in the circle of area A a polygon of area P. From the well known theorem for similar polygons we have p:P=d2 :D2. .(ii).

Comparing (i) and (ii) we see that

p:A>p:P -p>A

But this implies that the polygon has an area greater than the circle in which it is inscribed, which is clearly absurd.

Hence we can conclude that

a:A > d2:D2

Stage 3 We suppose that a : A <d2 : D2 and hope to reach an absurd conclusion. (This is left as an exercise for the reader.)

This leads us to conclude that

a:A d2:D2.

This double reductio ad absurdum argument establishes the desired result

a :A d2 : D2.

The Method of Exhaustion - Discussion The above example shows how the method worked. A result was discovered by some other means and then rigorously proved by the method of exhaustion's double reductio ad absurdumi argument. Actually the word "exhaustion" is somewhat mis- leading. The difference in area between the circle and the polygon never comes down to zero - the area is never actually exhausted - and no "passage to the limit" is envisaged. The name "exhaustion" was first applied to the method only in the seventeenth century by Gregoire de Saint-Vincent and others who were inventing their own methods which really did exhaust the area, so leading to the integral calculus.

Of course the method of exhaustion did not find new results and other methods were used. Archimedes of Syracuse (c. 287- 212 BC), the greatest ancient mathematician, used ideas of levers and equilibrium and considered solids and surfaces to be made up of small "slices". His techniques were set down in his treatise The Method, a book lost for about 1000 years but fortunately rediscovered in 1906 in Constantinople.

The spirit of Archimedes' work, if not the technical detail, is well illustrated by the following example adapted, with permission, from Eves (Ref. 1, pages 320-321). For a good example of Archimedes' use of inequalities in similar problems the reader is referred to Baron (Ref. 2, pages 42-43).

Archimedes' Method of Equilibrium: Volume of Sphere The fundamental idea of Archimedes' method is this. To find a required area or volume, cut it up into a very large number of thin parallel plane strips, or thin parallel layers, and (mentally) hang these pieces at one end of a given lever in such a way as to be in equilibrium with a figure whose content and centroid are known. Let us illustrate the method by using it to discover the formula for the volume of a sphere.

C

A B

T N -

x-

Fig. 2

Let r be the radius of the sphere. Place the sphere with its polar diameter along a horizontal x-axis with the north pole N at the origin (see Fig. 2). Construct the cylinder and the cone of revolution obtained by rotating the rectangle NABS and the

25



triangle NCS about the x-axis. Now cut from the three solids thin vertical slices (assuming that they are flat cylinders) at distance x from N and of thickness Ax. The volumes of these slices are, approximately,

sphere: 7rx(2r - x)Ax cylinder: 7x2AxX

cone: 7rx2Ax.

Let us hang at T the slices from the sphere and the cone, where TN = 2r. Their combined moment about N is

2r. 7rx(2r- x)Ax + rx2Ax]= 47r2xAx = 4.x. wrr2Ax Slice of sphere Slice of Slice of

cone cylinder

This, we observe, is four times the moment of the slice cut from the cylinder when that slice is left where it is. Adding up all the slices we find

2r. [total volume of sphere + total volume of cone] = 4 x position of centroid of

cylinder x volume of cylinder = 4 x r x volume of cylinder

Now formulae for the volumes of the cone and cylinder are already known so

8irr3 2r.[volume of sphere + ] = 8irr4, 3

so volume of sphere= 3

The Middle Ages In the middle ages there was considerable interest in the nature of motion and its description. For example at Merton College, Oxford, in the period 1328-1350, Thomas Bradwardine and others clarified many concepts, including that of instantaneous velocity, uniform and difform (non-uniform) motion, and at

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about this time Nicole Oresme was undertaking similar important work in France which ultimately was to lead to the differential calculus.

As part of Oresme's work he was much concerned with (effectively) finding the area under a curve of a velocity-time graph, which of course represents the distance covered. For example, he considered the following problem:

A body moves in a straight line with speed v, 2v, 3v, 4v,... etc., in successive intervals of time of lengths t, t/2, t/4, t/8 ... etc. What is the total distance travelled? (Fig. 3).

Fig. 3

1/2 11 l 1/8 1

The solution is

t t t t d= v.t + 2v.t +3v.- + 4v. +5v. +. 2 4 8 16

= 1++ +++ + +....

which Oresme summed to give

d= 4vt.

Summation of series had been carried out in simple cases by Archimedes and others, but Oresme's geometrical representa- tion did much to give impetus to the further study of series summation which was important in the development of the limit concept and the integral calculus.

AD 1500 to 1650 It was in the sixteenth century that great progress was made. The Greek works, especially those of Archimedes, became widely known through translations and were greatly admired. The scientists and mathematicians of the day sought a way round Archimedes' cumbersome use of the double reductio ad absurdum in his proof "by exhaustion" as they attempted to extend his results, particularly on problems of centroids.

Stevin (1548-1620), Valerio (c. 1552-1618) and Kepler (1571- 1630) were three prominent figures. Being an engineer, Stevin was less concerned about rigour of proof and so was prepared to dispense with the reductio ad absurdum part of the method of exhaustion and was content to find suitable inscribed and/or circumscribed figures to approximate the requisite area or volume.

To prove that the centre of gravity of a triangle lies on the median he proceeded as follows:

A

Fig. 4

Inscribe a number of parallelograms of equal height with bases parallel to BC and sides parallel to median AD (Fig. 4). Since the centre of gravity of each parallelogram lies on AD then the centre of gravity of the whole inscribed figure (shaded) must lie on AD, as the shaded area inscribed in ADC exactly balances that in ADB.

By increasing the number of parallelograms the two shaded parts, which are always equal to each other, more and more

26



closely approximate the area of the two triangles ADB and ADC. (Doubling the number of parallelograms halves the missed out area, and this doubling can be repeated ad infinitum.) Thus the difference between ADB and ADC is smaller than any amount we care to consider. Thus the centre of gravity of ABC must lie on the median AD.

Kepler's main preoccupation was with astronomy which led him to the evaluation of trigonometric integrals, in effect. One such problem (published 1609) concerned a theory of planetary magnetism where a planet was considered a giant magnet with N and S poles. It led to what we would write as I' sin x dx. This he proceeded to solve by summation and from this answer he was able to conclude inductively that the general result was really 1 - cos oa.

In 1615 Kepler published his Stereometria in which he brought together a collection of methods for estimating volumes of solids of revolution, ostensibly for the purpose of enabling wine-gaugers to estimate the volumes of wine cases. The Stereometria had a considerable influence in promoting later mathematical work. Whether or not it reduced the price of wine is unrecorded!

Another very important figure in the history of calculus was Bonaventura Cavalieri (1598-1647). He considered "a surface as made up of an indefinite number of equidistant parallel lines and of a solid as composed of parallel equidistant planes, these elements being designated the indivisibles of the surface and volume respectively" (Ref 3, page 117).

Typical of his approach was his proof that a parallelogram has twice the area of each of the two triangles formed by a diagonal (Fig. 5).

A B

S R

D C Fig. 5

He argued that since if one takes points P and R such that DP = BR and then draws PQ and RS parallel to AB then lines PQ and RS are equal, so all the lines in triangle ABD taken together are exactly matched by lines in triangle BCD. Thus AABD = ABCD and each is half the parallelogram ABCD.

By much more complicated arguments of this nature, Cavalieri went on to more advanced results: e.g. the sum of the squares of the lines in a parallelogram is three times the sum of the squares of the lines in each triangle. He also went on to consider higher powers of the lines, which enabled him to solve a problem proposed by Kepler in his Stereometria, namely to find the volume of revolution of a segment of a parabola rotated about its chord. Cavalieri's results were equivalent to

a an+l

0 n+1

in modern notation. His method was different in that he considered an infinity of indivisibles rather than small but finite elements. It attracted considerable criticism but was adopted by many mathematicians, including Wallis.

John Wallis John Wallis (1616-1703), a very able mathematician, was appointed professor at Oxford in 1649 and held the post until his death 54 years later. In Wallis' Arithmetica infinitorum (1656) Cavalieri's method of counting up lines was systema- tised and in it is found the general formula which we write as

I xndx = n+ . This was stated for n not only a positive integer 0 n+lf but also for all rational numbers except - 1. Wallis was apparently unaware of earlier efforts (e.g. by Torricelli and Fermat) to establish this result. Following Cavalieri, Wallis' approach was by measuring the lengths of parallel lines in a triangle and in the corresponding parallelogram of which the triangle is half (see Fig. 6). By considering the ratios

2

1 1

2 2 2

Fig. 6

3

0 1

3 3 3 3

lines of triangle lines of parallelogram

namely 0+1 1 1+1 2

0+1+2 1 2+2+2 2

0+1+2+3 1 =- etc. 3+3+3+3 2

it was obvious that

area of triangle 1 area of parallelogram 2

This suggests to us x dx=-. 0 2

9

0 1

1 1 la2 12a

l

o

4 0''

0

0 0 Fig. 7

Then taking squared terms he compared the parabola with the enclosing rectangle (Fig. 7).

0+1 1 1 1+1 3 6

0+1+4 1 1 4+4+4 3 12

0+1+4+9 1 1 9+9+9+9 3 18

0 + 1 +4 + 9+16 1 1 16+16+16+16+16 3 24

and so on. Thus Wallis showed that

area of parabola 1 area of enclosing rectangle 3

which suggests to us

Sx2dx = 1 o 3

27



y=Jx

x Fig. 8

Wallis proceeded to consider higher powers in the same way (for the interested reader to investigate). However, he did not stop with positive integer powers. For example, he looked at the parabola y2 = x, or rather half of it, given by y = 1 (Fig. 8).

Since the parabola y= x2 (Fig. 7) has area 1/3 of the corresponding rectangle then the parabola y2= x must have the area 2/3, it being the reflection of y= x2 in the line y = x, and this is confirmed by considering

10+/1+12+ ... +/1k 1k +k + Vk + ... +1k

which does approach 2/3 (for the reader who has a calculator to check!).

1 By writing 2/3 in the form Wallis was led to conclude

1/2+ 1 that -[ could be considered to be x2, and by this procedure he inferred that whether p is an integer or a fraction

OP+ P + 2P+ . . . + kp kP+kP+kP+ . . . +kp

1 tends to

p+l as k increases. He had in effect established

xPdx-= 1

0 p+1

It was against such a background that Newton and Leibniz independently "invented algorithmic procedures which were' universally applicable and which were essentially the same as those used at the present time" (Ref 3, page 188).

Newton and Leibniz Newton was able to show that for the curve y = axm/n the area

under the curve is given by -n ax(m + n)n. This he did not by m+n adding up infinitesimal areas but by considering the rate of change of area. He thus linked the integration process with what to him was the more fundamental process of differentia- tion. Newton was the first to make clear this connection - the Fundamental Theorem of the Calculus. Newton proceeded to apply the method to a very wide class of problems, dealing with the integration of (1 - x2)n when n was not an integer, for example.

Leibniz, although far less knowledgeable generally in mathematics than Newton, was able to arrive at equivalent results, and it was he who introduced the I dx notation. However, neither Newton nor Leibniz could offer a satisfactory explana- tion of the infinitesimal quantities used in their discoveries nor of the limit concept implied. They had developed a powerful tool but could not really explain how it worked nor prove its validity to the satisfaction of their critics! The limit concept had been alluded to by Newton (and many before him), but it was not until Cauchy, and later Weierstrass, expressed the limit and derivative clearly in algebraic, rather than geometric, terms that calculus could be put on a firm footing - and that was to take another 200 years.

In a later article the development of differential calculus will be described, and fuller discussion of Newton's and Leibniz's contributions will be made.

References

1. Eves, H. (1969) An Introduction to the History of Mathematics. 3rd edn, Holt, Rinehart and Winston.

2. Baron, M. E. (1969) The Origins of the Infinitesimal Calculus, Pergamon Press.

3. Boyer, C. B. (1959) The History of the Calculus and Its Conceptual Development, Dover.

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Historical Topics: The Development of Integral Calculus