Lectures on the History of Mathematics

and Mathematical Education

I. Grattan-Guinness*

(received 2 February, 1978)

1. Introduction.

I was invited by the New Zealand Mathematical Society to be their

first Visiting Lecturer, and during October and November 1977 I visited

each of the six universities and lectured to the departments of mathe­

matics and to the local associations of mathematics teachers on various

aspects of the history of mathematics and mathematical education. I am

very grateful to Kevin Broughan and Graham French of the University of

Waikato for the organisation of the trip, and to the hosts at each

university for their hospitality.

This article takes the form of short essays based on the lectures,

followed by a collective bibliography. I have prefaced some essays by

explanatory comments, and concluded each one with a list of appropriate

references. I have not indulged in heavily footnoted or detailed

historical exposition, nor cited the hundreds of primary works on which

the historical judgements are based. The bibliography lists several

secondary sources from which the reader can retrieve such information.

I have omitted a general lecture on the golden section, delivered at

Dunedin: the mathematical results involved are in [13].

2. The Calculus and its Broadening into Mathematical Analysis.

This essay conflates two closely related lectures, called respectively

'What was and what should be the calculus?' and 'On early difficulties

with limits: the emergence of mathematical analysis'.

* First New Zealand Mathematical Society Visiting Lecturer.

Math. Chronicle 7(1978) 105-123.

105

The invention of the calculus is justly attributed to Newton and

Leibniz; but the customary historical opinion that their success was

based on noticing the inverse relationship between differentiation and

integration rather misses the point. The basis of their achievement was

the realisation that, given a function f of a variable x , the purpose

of the two calculi is to calculate new functions of x : for differen­

tiation, the derivative f r or some suitable alternative, and forf X

integration the indefinite integral J f(u)du or an analogue. Their

predecessors tended to find only particular points (rather than regard

that point as a new function) when doing differential calculus, and to

compute only definite integrals in the integral calculus.

Newton1 s calculus is fairly similar to a modem approach inasmuch

as his fluxion x of a variable x 'flowing' with respect to

(conventionally understood) time corresponds to a derivative, and has

a notion of limits embodied in it. However, that notion was not

explicated, so that the foundations of his scheme are shaky. The inverse

relationship takes the form that x is the fluent of x . Newton

realised that the behaviour of the fluxions depended on some underlying

variable, and sc he calculated ratios of fluxions x/y , where y

would flow uniformly. This move corresponds to the modem technique of

assigning y as independent variable.

Newton's system did not bring the calculus into prominence: this

was due to the tradition set by Leibniz.

Leibniz's calculus was developed in the 1670s, ten years after

Newton's, but was published first (in the 1680s). This system is quite

different, for limits are eschewed. For a variable x the basic idea

is of the differential dx , an infinitesimally small increment on x .

The conceptual difference between limits and differentials is fundamental

for a line x , for example, a limiting case would involve a point,

whereas the differential dx is also a line (a very short one). d is

a dimension-preserving operator on dx , whereas limits involve change

of dimension. Just as x is a variable, so is ax ; and its differentii

is (using the last example) the very very short line ddx (or d2x) ; and

so on to higher orders.

The inverse relationship came to Leibniz from realising that the

construction of the tangent involves the subtraction of ordinates while

the calculation of areas involves their summation. For summation he used

the symbol '/' , the elongated form of 's ’ in 'summa'. / , like d ,

preserves dimension; Jx is a very long line, jjx a very very long one,

and so on. The inverse relationship takes forms such as d j x = x .

The slope of the tangent is dy / dx , which for him means literally

'dy * dx' . The area under the curve is / y dx , the sum of rectangles

y high and dx wide.

The conceptual difficulty for Leibniz's calculus is the idea of a

differential, which obeys the law

x + dx = x , (1)

or more generally

(dx)S + (dx)^ = (dx)S , (fx + cl'x = ( f x , t > s > 0 . (2)

However, Leibniz and his followers the Bemoullis were happy to accept

such laws. The principal difficulty was the realization that any

expressions involving second or higher-order differentials were indeter­

minate, in that they depended on how the differentials of the various

variables related to each other. Euler eventually solved this problem

with his theory of differential coefficients. Let y be a function of

x , where dx has a constant differential (this corresponds to assigning

x as independent variable, and thus also to Newton's idea of uniform

flow), and their differentials be related by

dy = p dx , ddx = 0 . (3)

p is a function of x , so that

dp = q dx , ddx = 0 . (4)

107

and so on. Higher-order differentials of y are handled as follows.

From (3),

d dy = d(dy) = d[jp dx) = dp dx + p ddx = q(dx)2 . (5)

Thus the indeterminate d dy is replaced by the fully determined

q(dx)2 , and so on for higher-order differentials, p, q , . . . . are the

first, second .... differential coefficients of y with respect to x .

The differential coefficient is of equal importance with the differential.

It equals the derivative but differs conceptually in that it is computed

by algebraic means (and is always assumed to exist). The integral is

conceived as the anti-differential.

Lagrange continued the algebraic approach with his theory of

de'Hved functions. He assumed that any function f has a convergent

Taylor expansion

f(.x+h) = f^x) + hf'{x) + h2f "(x )/ 21 + .... (6)

and defined its derived functions f , f ” ,... (Lagrange introduced

these notations) as the appropriate terms in (6). Their calculation

was so algebraic a process as (allegedly) to be free of reliance on

infinitesimals, limits and the like.

Lagrange pushed the algebraic approach to extreme and so paved the

way for its end: the view was too good to be true, and various kinds

of counter-example were being found anyway. A fresh approach was

introduced by Cauchy in the late 1810s in his Paris lectures. He set

up the outlines of mathematical analysis as we know it by bringing the

calculus, the convergence of infinite series and the theory of functions

under the umbrella of limits. That is to say, he offered not merely the

definition of a limit (which had been uttered more or less vaguely

before) but a proper theory: the arithmetical theorems, the notion of

upper limit, and applications of limits in other definitions. In the

case of the calculus, he did have our derivative, for it was truly the

limiting value of the difference quotient:

f'ix) = lim [{f(x+i) - /(ar))/£] . (7)£-> 0

But he also defined the differential:

df(x) = lim [{fix+ah) - fix )) /a.} (8)a->0

where a is infinitesimal (a quantity which decreases to limit zero)

but h is finite. Setting i = oh yields from (7) and (8):

df(x) = hftx) (9)

and putting f(x) = x in (9) yields

dx = h , ( 10)

so that (9) is finally

df(x) = f ’ (x)dx . ( 11)

Thus df{x)/dx is a ratio, as with Leibniz, but its notions are defined

via limits, as with Newton. This is a curious mixture of ideas, and

affects his beautiful treatment of the integral, which is now defined as

an area with the area itself analysed as the limit of a sequence of

partition sums lf[x)&x . In the limiting case becomes '/' ,

but kx is dx and thence k from (10), so that for Cauchy the

integral can be written by the (to me incomprehensible) fftx)h .

Elsewhere in analysis Cauchy made notable innovations. The

definition of the local continuity of a function in terms of the mutual

smallness of (f(x+h) - f(x)) and h is there, and a basic sprinkling

of theorems, including the fundamental and mean value theorems of the

calculus. Convergence is clearly stated, and the development of

convergence tests inaugurated. These ideas are also applied to complex

variables, in which Cauchy was a great pioneer.

The deficiencies in Cauchy's programme (and in the works of his

contemporaries whom he influenced) are these. Firstly, he was very

competent at handling single limits, but did not notice the additional

techniques required for multiple limit problems. Secondly, he noticed

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the need for certain existence theorems, but his proofs were vitiated by

a lack of structure for the real line. Thirdly, the role of infinitesimals

is rather enigmatic. Fourthly, the work is often largely verbal, which

causes some unwanted ambiguities in reasoning.

The removal of these defects was effected mainly between 1860 and

1900 under the inspiration of Weierstrass’s Berlin teaching. He and his

followers brought in multivariate analysis (including distinctions

between various modes of uniform and non-uniform convergence) and

theories of irrational numbers. They banished infinitesimals (though a

few heretics tried to preserve them), and greatly increased the use of

symbolism. They also tidied up certain other unclarities (such as

distinguishing between lim and lub) and brought in some basic set

topology.

It is this movement which brought the calculus and classical

mathematical analysis into modemly recognisable forms. Basically, in

this field of mathematics, the Weierstrassiar.s tended to be

FOR: rigour and generality, almost as ends in themselves;

AGAINST: geometry (as being merely intuitive) and infinitesimals (for

lack of rigour);

INDIFFERENT: to applications of mathematics to empirical problems.

This rings a bell, doesn't it? I shall conclude this survey with

some educational points.

A benefit from history is to be able to criticise our heritage as

well as to honour it. All this century the Weierstrassian influence

has been profound, and with educationally dubious consequences. We

teach 'impeccable* mathematics, but we do not motivate it. We achieve

levels of rigour, but do not contrast it with lower levels of rigour

and thus show its worth. In other words, we supply answers, but raise

no questions; we provide solutions to the students, but give them no

problems. We reject the beautiful intuitions of geometry just because

on some occasions they let us down, and thus also deprive the student

of the valuable educational lesson that a piece of mathematics is good

only to a point and then needs revising. We rob mathematics, including

its foundational aspects, of its atmosphere of discovery and creation.

There are also some technical criticisms to make of normal teaching

practice. Because of the excessive worship of rigour and the extolling

of limits, we have rejected the Leibniz-Euler tradition of differentials,

one of the most successful interprises in the whole history of mathematics,

the form of calculus which set up the subject as important in mathematics,

and a beautifully fertile source of mathematical models for all kinds of

continuous phenomena. Such methods are being revived in teaching under

the inspiration of non-standard analysis, but the older methods stand as

important mathematics in their own right. Further, although the history

has been forgotten, all sorts of terminologies and notations remain:

differentials, differential coefficients, derived functions, derivatives,

differential equations; dy/dx (as a whole symbol), dy/dx (as a ratio),

and so on. It is hopelessly confusing, and made worse by the failure to

emphasise distinctions which the history can teach us: between algebraic

and analytical versions of the calculus, between univariate and multi­

variate analysis, and so on.

These educational practices are of long standing. It is high time

they stopped.

References: [l], [2], [5], [6], [10], [14], [15], [17] .

3. The History of Mathematics and Mathematical Education.

I have put this essay next, as I drew heavily in lectures on

encapsulated versions of the previous essay to exemplify general points.

Apart from in Dunedin, I lectured on this topic to audiences including

teachers, and so brought in other areas of mathematics.

Often a piece of mathematics begins as a research topic, then passes

into teaching at post-graduate level; then it possibly reaches the

111

undergraduates and might finally achieve some form for school consumption.

The route is partly short-circuited if the mathematics is motivated by

mathematical education; examples were given in the last section with

Cauchy's and Weierstrass's teaching of mathematical analysis.

I can deal only with a few aspects of a large subject, but the

history of mathematical education will make an excellent start. The

points which I made at the end of the last section can be generalised

with fair justice to many aspects of mathematical education: thef*

excessive extolling of rigour, the over-emphasis on allegedly basic

concepts, and so on. It is partly because of alleged educational needs

that such views were espoused in the first place; but the effects on

education have been most unfortunate.

When the 'New Mathematics' programmes were introduced in the late

1950s, some ingenious teaching aids and games were brought in for the

very young, and emphasis on fundamental parts of mathematics given to

the older children. However, the results for the older ones have not

matched up to expectation: they do not seem to have any greater liking

for or understanding of mathematics, and are less numerate than they

used to be.

I think that these consequences were only to be expected, and that

lack of knowledge of history could have prevented those changes being

made in the first place. For the emphasis on 'basic' or 'fundamental'

mathematics is only the son of the Weierstrassian emphasis on rigour for

its own sake, and so is subject to the same criticisms as given above.

Further, some of the New Mathematics was originally created as part of

the Weierstrassian approach -- namely, the advocacy of set theory by

that rabid Weierstrassian Georg Cantor, and its eventual assertion as

fundamental to mathematics by him and others. (See the next section for

some details of these developments.) Now such advocacy was quite

legitimate for the research-level mathematical/philosophical enterprises

at hand: it is their imitation at the much lower levels of teaching

that is so misguided. Such methods teach nothing about either set theory

or the mathematics in which it is being used. The same points can be

made for the other trumpeted features of 'New' mathematics; namely, the

emphasis laid on algebraic laws, equivalence relations, groups, and so on.

They became prominent in the late 19th century as legitimate but

extremely erudite research topics, and their imitation in the schoolroom

is again educationally silly. The ignorance of these historical facts

is evident even in the name that the enthusiasts gave to their approaches.

'New' mathematics? 80 years old at least? Is Elgar 'New' music?

I do not wish this criticism to be identified with the criticism

that the New Mathematics was just a fresh lot of jargon for the teachers,

so that all they could do was to parrot it out to their pupils. New

jargon can also be worthwhile mathematics: for example, Leibniz's

differential calculus was both in the late 17th century. My criticism

is that the jargon is not worth teaching at school, whether it is 'New'

or not, and that it should largely be abandoned. I do offer this

criticism in a pretty comprehensive form - in contrast to those

enthusiasts who admit that the reforms may have gone a bit too far

(but are basically valid). To me they are basically an error3 although

some of the less heavily advocated parts are welcome. For example,

the introduction of linear algebra was overdue, and probability and

numerical mathematics could perhaps be taken further than they are.

What should we do instead? Let me make first the following

contrast. Mathematical education has long over-emphasised formal

presentations of mathematics. Now if we look at the history of the

mathematics involved, we often find that the historical record shows a

development opposite in direction to the formal one: in other words,

the formal presentation follows a sequence of topics which were (to

some extent) introduced in chronologically backwards order. So

instead of trying to imitate the formal presentation in education, we

should try to make more use of the historical record, relive some of

the older techniques, show our youngsters some old mathematics for

113

what it was - motivated intellectual material which solved certain

problems (which are stil] relevant) but only solved them to a degree

and had to be replaced by superior alternatives.

Another point needs discussion. I mentioned in the last section

that the calculus was algebraic in Euler's time and analytical since

Cauchy. Thus geometry was not relied on. The demotion of geometry

was marked also in the various developments of algebra since 1870

(although geometry did provide examples of some of those new ideas).

Now at the same time geometry was going through its most exciting

phase: the development of non-Euclidean geometry, their reconciliation

with Euclidean geometry, the improvement of Euclid's axiom system, and

so on. Sadly, the demoters of geometry have won in modem mathematical

education. Thus another major reform needed is to restore geometry to

its rightful place as the central part of mathematical education

throughout school and into university. Parts of its history will be

educationally useful too.

How much history of mathematics can we use in mathematical

education? The question depends on the branch of mathematics involved,

and on whether it is advocated for teachers or for the pupils. Some

branches (such as geometry, calculus, analysis and algebra) have an

educationally more amenable history than do others (probability and

statistics, linear algebra), though in all branches it is useful. As

for levels, quite substantial amounts could be used at university level

or its equivalent (which includes the important category of teacher

training), but would be more limited to schoolchildren. Finally, in

order not to become bogged down in historical details, we need in

education to pick out principal developments from the history without

labouring the historical nuances too much - an approach which I have

called 'history-satire'. The aim is to teach mathematics through its

history - including the difference between modem and historical

evaluations of a piece of mathematics - not to teach history of

mathematics itself.

References: [6], [8], [ll], [16].

4. Georg Cantor's Influence on Bertrand Russell.

The names of Cantor and Russell register set theory and mathematical

logic in the minds of the reader, but I wish to draw attention also to

'influence1, for that is our key word. Personal influence, influence

through intermediaries and publications, influence under general philo­

sophical or mathematical traditions, positive influence and influence

by reaction: influence is an historiographically complicated notion.

Examples of all these forms occur in the Cantor-Russell relationship.

The two never met, although Cantor sent some eccentric letters to

Russell in 1911 which Russell unfortunately published in his

Autobiography. Russell knew nothing of set theory when an undergraduate

at Cambridge - the subject was virtually unknown there in the early

1890s - but he became acquainted with it a few years later, in 1896.

He was trying to write on the foundations of knowledge, but saw the

role for set theory only after learning of mathematical logic through

Peano at the 1900 International Congress of Philosophy. He then began

to develop his 'logicist' philosophy of mathematics: that pure mathe­

matics is part of (mathematical) logic.

Bv this time Cantor had brought his set theory to a very general

form, as a possible foundation to mathematics, and had also discovered

two paradoxes (of the greatest cardinal and the greatest ordinal).

Russell found his own paradox (of the set of non-self-belonging sets)

in 1901, deriving it from considering Cantor's power-class theorem.

The two men had differing views on the paradoxes: Cantor thought the

trouble lay in assuming the existence of 'too big' classes, but Russell

saw that his paradox-generating class was not too big in that sense,

and put the blame on 'too complicated' defining properties. His own

solution, in his and Whitehead's Prinoipia Mathematica (1910-13), was

115

based (for him) on a ’vicious circle principle' forbidding certain kinds

of defining property as legitimate.

To show in some more detail the nature of Cantor's influence on

Russell, I shall briefly describe three aspects of Russell's system:

the axiom of infinity, the axiom of choice, and irrational numbers.

Cantor had no axiom of infinity, for he had no axioms. Instead

he had, for example, the second principle of generation, which

posited the existence of an ordinal at the end of a progression of

ordinals (o> at the end of 1, 2, 3, ..., for example). Russell

disliked this approach, preferring to state axioms, but for some

years he thought that the axiom of infinity was provable by means of

mathematical induction. He finally abandoned this hope, but for a

surprising reason. In 1906 he attempted a proof using his 'substitu­

tional theory', a logical system using only propositions, their truth-

values, and constants. He became chary of assuming false propositions

as abstract objects (for what could such objects be?), and so abandoned

the substitutional theory -- and his latest attempted proof of the

axiom of infinity with it.

\

The axiom of choice was first made public in Zermelo's 1904 proof

of Cantor's well-ordering theorem. However, Russell had come across a

form of it independently, in connection with defining the multiplication

of an infinitude of cardinals. Thus he called his form of the axiom

'the multiplicative axiom'. Zermelo was happy to assume the new axiom,

but Russell was always reluctant to do so. There was a very interesting

discussion of the axiom in the 1900s concerning the places where it was

needed (many were found in Cantorian set theory), the possibility of

reproving theorems without it, and the philosophical differences between

the various forms that were investigated.

Cantor's theory of irrationals was introduced as part of

Weierstrassian analysis described in section 2. Cantor 'associated’ a

number b with a Cauchy sequence { } of rationals, and wrote

Ivm a = b . (1)n

n-*°°Russell misunderstood (1) to mean that Cantor assumed the existence of

the limit. He also (with justice) disliked the idea of ’association*

in Cantor's theory, and developed his own theory of irrationals defined

in terms of classes of segments of rationals. His own theory is valid,

but his claims for its superiority over all other forms are doubtful..

He regarded his own theory as the only means of guaranteeing the

existence theorems for irrationals; but what does he mean by 'existence'?

This word is used in various partly conflicting senses in Russell's

writings. As well as existential quantification he had the existence of

descriptive phrases (E! (12:)(^) : 'the present King of France exists'),

the existence of classes in the (Peanoesque) sense of non-emptiness

(3!a), and the existence of classes in the sense of being definable by

a propositional function (Ea) . For irrationals he seems to have meant

only the '3!' sense of existence, whereas in mathematics existence is

often of the 'E!' type.

To conclude, let me summarise their respective philosophies of

mathematics. Both accepted the law of exluded middle, and both (like

most others of their time) were poor on distinguishing object theory

and meta-theory. Russell's logicism asserted that mathematical logic

(including the theory of descriptions) is enough to produce pure

mathematics, the class of propositions of the form p q . Cantor was

a Platonist concerning mathematical objects (for example, for him

there is one and only one set theory), an idealist with regard to

definitions (for example, the 'association' of b with {a^}), and a

formalist concerning proofs (in the sense that consistency - itself

naively assessed - of construction guarantees existence).

All in all, the influence of Cantor on Russell is a complicated

matter, but one which sheds much light on the work of both men.

References: [4], [9], [10].

117

5. On the Early Histories of the Fourier and Laplace Transforms.

Although the Fourier and Laplace transforms are closely linked

mathematical theories and were also linked historically at times, their

histories are remarkably different. The Fourier transform was set up in

a short time and became adopted fairly quickly; but the Laplace transform

took a long time to achieve comparable status, even though its origins

are earlier.

Transform theory requires: a particular form to handle; the idea of

transforming something into something else; theorems connected with this

idea; a means of inverting the transform. The history shows at times

only some of these criteria being met.

The Laplace transform has its origins in various attempts by

Euler to use

as the solution form for differential equations. He started with

relations involving (1) and looked for differential equations which

they could solve. Little beyond a suitable form for the Laplace

transform is involved here, and the results are not impressive.

The next step was a 1773 paper by Lagrange on the probability of

mean errors of observations lying within given limits for given

distributions. This work involved him in expanding

in a power series. It seems marginal to Laplace transform theory, but

the paper as a whole influenced Laplace.

Laplace has two main papers in this area in the late 18th century.

A 1779 essay on series also involved integral solutions to differential

equations, in the course of which appears

/ eo:cX{x)dx CD

j a°e °^X (x)dx ( 2)

00

/ $tz)zVdz = C + I (-1)2¾ <|>r+1(z)2y"rr=o

(3)

where <{> is the rth indefinite integral of tp . Then a 1782 paper on

asymptotic theory, where Lagrange's general influence can be seen,

includes both something like the (mis-named) 'Mellin transform'

z/(s) = / :cs4>0:)c& (4)

and also

I/(s) = / e~SX$(x)dx , (5)

which would be a Laplace transform if the limits of integration were

specified. The idea of transforming ¢) to z/ is there, and also

theorems derived from (4) and (5); for example, from (5),

hry (s) = / e~sx(ex-l)r$(x)dx . (6)

But the uses made of these ideas are not what we might now expect, for

they include transforming difference formulae and evaluating some

definite integrals.

Laplace transform theory now lay nascent. But in 1807 mathematical

physics received a jolt when Fourier submitted his first major paper on

heat diffusion to the Institut de France. There was controversy about

various of his results, including his 'Fourier series' solution methods,

but Laplace seems to have liked them. In 1809 he solved a problem♦

which Fourier left unresolved in 1807: to solve the diffusion equation

v = v. (7)xx t

for an infinite range of values of x . Applying

v (2:,0) = <p(x), -« < x < “ (8)

to (7) Laplace obtained a power-series solution; but he then used some

results from his 1782 work to transform it to the integral solution

= ~= |°° (f>(x + 2z/t) e Z dz . (9)

Fourier now saw that an integral solution was needed, and by hair-

raising escapades with infinitesimals he quickly produced the 'Fourier

119

II

transform' and its inverse in forms such as

giq) = 2̂/tt fiu) cos qu du , (10)

f(x) = / 2 / tt £?(<?) c o s <72:

0(11)

Fourier's work meets all our criteria for transform theory: he has

the form, the idea of transforming, related results, and the inverse.

His methods became quickly prominent, especially because the young

Cauchy discovered them independently and used them very widely.

But the Laplace transform slumbered; and this is most strange,

for not only were transform methods of solving differential equations

now of conscious interest but also Laplace was publishing three editions

of his Th6orie analytique des probabilit£s between 1812 and 1820. Yet

the connections between harmonic analysis and mathematical probability

were not noticed, even though Laplace handled generating functions

(including for continuous random variables) and formed a sort of

inverse Laplace transform, Cauchy used complex variable Fourier

transforms (which are similar to characteristic functions) and Poisson

knew about convolution forms in solutions to partial differential

equations.

Instead only the odd tremor occured in the direction of Laplace

transform theory. Abel wrote a manuscript (published 1839) on the

transformand some of its basic properties, and Murphy found an inverse

formula for special cases in 1833. Riemann considered

gis) = hix)x dilog x), s complex (12)

in 1859, and used Fourier analysis to get the inverse

(13)

yet even this result did not stimulate much interest, although P.iemann

produced it in his widely-read paper on number theory containing the

Riemann conjecture. Much of the mathematics of this time which looks

like Laplace transform theory (especially Boole's A treatise on differential

equations) actually continues the tradition of Euler of finding solutions

of the form of (1) to differential equations. But Poincar6 used the

Laplace transform in 1885, integrated 'along a conveniently chosen line1,

in analysing differential equations. Heaviside introduced his operational

calculus from 1895 on; and then at last things began to move, although

still slowly.

In order to validate Heaviside's rules, it was noticed that they

would be related to contour integrals of the form of (13), although it

took quite a bit of work (culminating in Bromwich, 1916: Mellin's own

results are partly concerned with this) to get the contour right. Then

the engineers saw the value of the transform for quickly solving

(difficult) differential equations: Carson's work of the 1920s on

electrical circuit theory was particularly influential. The systematisa-

tion of the theory was due to authors such as Doetsch (1930s), Carslaw

and Jaeger (1941), Gardner and Barnes (1942), and (on the 'pure' side)

Widder (1946) -- over 200 years after Euler started to wonder if

e X{x)dx could be a good solution form for differential equations.

References: [3], [7], [12].

REFERENCES

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derivative in the Leibnizian calculus, Arch. hist, exact, sci.,

14(1974), 1-90

2. C.B. Boyer, The history of the calculus and its conceptual

development (1939, 1949, 1959, New York (with this title)).

3. H.F.K.L. Burkhardt, Entwicklung nach oscillierenden Funktionen...,

Jber. Dtsch. Math.-Ver., 10, pt. 2 (1908).

121

J.W. Dauben, Georg Cantor: his mathematics and philosophy of the

infinite, (1978, Cambridge, Mass.).

P. Dugac, Elements d ’analyse de Karl Weierstrass, Arch. hist,

exact sci., 10(1973), 41-176.

I. Grattan-Guinness, The development of the foundations of

mathematical analysis from Euler to Riemann, (1970, Cambridge,

Mass.).

I. Grattan-Guinness and J.R. Ravetz, Joseph Fourier 1768-1830...,

(1972, Cambridge, Mass.).

I. Grattan-Guinness, Not from nowhere. History and philosophy

behind mathematical education, Int. j. math. educ. sci. techn.,

4(1973), 321-353.

I. Grattan-Guinness, Bear Russell - dear Jourdain . . . , (1977,

London).

I. Grattan-Guinness, (ed.), From the calculus to set theory,

1630-1910: an introductory history, (1978, London).

I. Grattan-Guinness, The history of mathematics and mathematical

education, Australian mathematics teacher, 33(1977) 117-126,

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