THE ANALYTICAL SOCIETY:
MATHEMATICS AT CAMBRIDGE UNIVERSITY
’IN THE EARLY NINETEENTH CENTURY
by
Philip Charles Enros
■ - i . ' ■ ^ ,
a Institute for the History and Philosophy
of Science and Technology
A thesis submitted in conformity with the requirements
for the Degree of Doctor of Philosophy in the
University of Toronto
© Philip Charles Enros 1979
This work is licensed under a Creative Commons Attribution 4.0 International License. %
UNIVERSITY OF TORONTO
SCHOOL OF .GRADUATE STUDIES
PROGRAM OF THE FINAL ORAL EXAMINATION
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
OF
PHILIP CHARLES ENROS
10:00 a.m., Friday, October 5, 1979
Room 111, 63 St. George Street
THE ANALYTICAL SOCIETY: MATHEMATICS
AT CAMBRIDGE UNIVERSITY IN THE EARLY NINETEENTH CENTURY
Committee in Charge:
Professor C.R. Morey, Chairman Professor E. Barbeau Professor M.-Crowe, External Appraiser Professor S. Eisen Professor'R.J. Helmstadter Professor T.H. Levere, /Supervisor Professor I. Winchester
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract
The thesis is a study of the Analytical Society (1812-
1813) of Cambridge University. Its purpose i£ to present a
detailed history of the Society, of which little has
previously been known, in or/3er to obtain an insight into
the reasons for the transition in Cambridge mathematics in
the early nineteenth century. A large part of. the content^of
the thesis is based'on research-in extensive manuscript
sources, especially various Charles Babbage and John Herschel
collections.
Two chap'ters in the thesis are devoted to theNjjackground
to the Analytical Society. The curriculum of the University
of Cambridge and the prominent position of mathematics there
are examined. And the widespread lament about the decline of
the mathematical sciences in England (1790-1815) is discussed
and shown to have two "Sonnected features: a debate over
analytical and synthetical mathematics, and a new view, for
England, about the relationship" of mathematics and society.
The ’lament along with, the Cambridge curriculum helped^o
provoke both the founding of the Analytical Society and the
later changes in Cambridge mathematics.
The Analytical Society is dealt with in a long chapter.
It was a short-lived association of a small but remarkable
group of students at Cambridge University. The Society was
a manifestation of a larger movement towards Continental
analytical methods in-British mathematics. The Analytical
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Society did not attempt to reform Cambridge mathematical
studies as is often assumed. However, the Society is -an
important illustration of the ways in which various features J s ~ \ of Continental mathematics were being imported into England.
>S\ Two other chapters are given over to an examination of
the influence (mostly indirect) of the Analytical Society.
It promoted and encouraged the mathematical work and vision
of some of its members, in particular, the mathematical
concerns -of Charles Babbage, John Herschel and Edward
BromheadJ f rom 1814 to 1822. Also, several of its former
members initiated an informal movement that led to a reform
of Cambridge mathematical studies (1813-1820s). This transition
to analytics took place through the structure of the studies
of Cambridge. Thus not only did some members, of the Analytical
Society do creative mathematics after the Society's
dissolution, but many former members were also involved in
changing Cambridge' mathematics. All of these activities
including the formation of the Analytical Society were
expressions of the members' image of mathematics. 55 The main theme of the thesis is the importance, for
understanding early'nineteenth-century Cambridge mathematics,
of an intellectual and social' framework which was composed
of three key elements: ideas concerning the nature of
mathematics (analytics versus ^synthetics) } ideas about the
purpose of a university (a liberal education), and a set of
expectations concerning mathematics and science best described
as professionalism. The thesis contends that the presence of
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. this particular framework in/-early nineteenth-century England
explains the activities of the Analytical Society and the
revival of Cambridge mathematics. This thesis thus provides
a valuable insight- into the stste and nature of mathematics
at Cambridge in the early nineteenth century.
J
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Biographical Sketch <•
Philip Charles Enros was born in Chicoutimi, Quebec,
August 30th 1950. He received primary and'secondary
education in Montreal. He gained a B.Sc. (summa cpm laude")
in mathematics from Loyola College, Montreal, in 19 71. From
that date he has been a graduate stpdent (M.A. 1973) in' .the
Institute for the History and Philosophy of Science and
Tethnology, University of Toronto.
Mr. Enr|os^ s graduate studies have been supported by a ■'
Bell Canada Centennial Fellowship (1971-73), a Canada Council
Doctoral Fellowship (1973-76), and an Ontarid Graduate
Scholarship (1976-77). He was a Lecturer in-1977-78 in the
Department of History, University of New Brunswick., and in ... O 1978-79 in^the Institute for History and Philosophy.of Science
and Technology, University of Toronto. In 1978-7.9 he also
organized the production of a "Biobibliography of Ontario
Scientists, 1914-1939".
Mr. Enros is Secretary-Treasurer of the Canadian Society
for History and Philosophy of Mathematics, and a member of
the History of Science Society'and of the Canadian'Society
for History and Philosophy of Science. He has delivered a
number of papers^to various conferences: the latest was to
the third Workshop on the Social Histor-y of Mathematics,
Berlin, July 1979; he y £ s" also been invited to' speak to the
Davis Center Seminar on the History of the. professions,
Princeton University, in December.
Mr. Enros married Pragnya Thakkar- in 19 75. Their son)
Madhava, was born in 1976.*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Courses.
M^jor Field: History of Mathematics .
s&x 1005 History of Mathematics (A-> K.O. May S&T 1011. History of Physical Science (B+) J. MacLachlan S&T 1021 Intellectual Context of 19c. Science (A-) ■ T.H. Levere S&T 20t>0x Philosophy and Science in the 17-18c. (A-) T. Goudge S&T 2192x Philosophy of Science (A) B.C. van Fraassen t-
First Minor: History of Technology
S&T 1013 History of Technology (A) B. Sinclair S&T 2013 History of Technology (A^) B. Sinclair \
Second Minor: . '
S&T 1012 History of Biological Science (A> M.P. Winsor
Specialist Examination's
History of England m the Early Nineteenth Century (R.J Helmstadter), Mathematics (K.O. May) . and Science (T.H. Levere) in the Early Nineteenth Century.
Comprehensive Examinations
History of the Physical Sciences (T.H. Levere), History of the Biological Sciences (M.P. Winsor), History of Technology (B-. Sinclair), Philosophy of Science (B.C.. van Fraassen). •'* ' ‘
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Publications
"Person Index for Delainbre's Rapport Hjstorigue of 1810", Historia Mathematics 3^ (1976) 321-324.|
Review of J. Delambre's Rapport Histonque, Historia Mathematica 3 (1976) 342-344. ! •••------; * ' i Review of J.M. Dubbey's The Mathematical Work of Charles Babbage, American Scientist 6^(S^ (1978) 639.
Review ofJJ.M. Dubbey's The Mathematical WorTc• of Charles- Babbage,' to appear in Historia Mathematica 19 79.
"Commentary" on R.A. Jarrell's paper "Courses in the History of .Canadian Science and Technology: Their Purpose and Content", to appear in the Proceedings of the Conference on the Study of the History of Canadian. Science and Technology, Kingston, 1978.
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REVISED AUGUST 1973
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements
Thi^_ thesis is the result of much -more than my efforts
alone. My thanks are due first and foremost to many of the
members - both staff and students - of the Institute for the
History and Philosophy of Science and Technology of the
University of Toronto in the pasl; eight years. They provided
a very stimulating environment whic.h greatly influenced the
development of my views and ideas. I am indebted to the late
Ken May, my supervisor for almost all the period of my
graduate studies, and to. Trevor Levere,'my present supervisor
and a special influence on me for years before. I am very
grateful to Charles Jones for his oareful'reading and helpful
criticism of the thesis.' I am also thankful for Ed Barbeau's
comments on this work, especially on the mathematical
sections. <. / " ,
Many individuals and institutions have been kind enough
to permit me ^o examine their collections; without their help
this thesis would not have been possible. I wish to thank
Mrs. Eileen Shorland and Sir Benjamin Denis Gonville Bromhead
for permission to consillt various John Herschel and Edward
Bromhead manuscript collections. I 3m also thankful to the
staff of the libraries of the University of Toronto, the
National Maritime Museum, the Royal Society of London, the
British Library, the libraries^f Trinity College and of
St. John's College, Cambridge, the Science Periodicals
Library of Cambridge, the Cambridge University Library,
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the Humanities Research Center of the University of Texas > / and the Museum of the History of Science at Oxford'. , . , ’
I have been very fortunate in having my studies
financially supported by Bell Canada, the Canada Council, and
the Ontario Graduate Fellowship Program. I am pleased to
acknowledge'the fine typing of'this thesis by Carol Lucier '
and Lynn Sobolov.
On the more spiritual side, I am very indebted to
Connie Gardner for being herself. And, finally, my struggle',
with this thesis was given a new. and very pleasant dimension
by my wife, Pragnya, and son, Madhava.
(
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page
I. Introduction 1
II. Mathematics and'the Curriculum 'of the
University of Cambridge .in the Early
Nineteenth Century ■ 17
.. < III. The Decline of the Mathematical Sciences in
England and Their State at Cambridge
(1790-1815) 46
IV. The Analytical Society (1812-1813) 103
V. The Mathematical Concerns of Charles Babbage
and John Herschel (1814-1822) 167
VI. The Introduction of Analytics at Cambridge
University (1813-1820s) 213
VII. Conclusion 256 *
Bibliography 265.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I.; ' introduction
The first decades of the nineteeth century witnessed
a great change in English mathematics. It was a period of
■revival, marked by an abandoning of the mathematics of
Newton for that of Leibniz and of the Qop'feLnent. "By 1830",
as Morris Kline has written, "the English were able to. join in
the work of. the Continentals."^ The University of Cambridge
was ah 'important center of and force'behind this revival. '
The highlight of the changes in Cambridge mathematics has
generally been considers® to be the Analytical Society
(1812-1813). Indeed, historians of mathematics have
regarded the Society as both the herald and agent of
England’s rallying fronr its protracted slump in mathematics
during the eighteenth century.
The Analytical Society is usually mentioned in most
histories of mathematics, even though very little is 2 known about the group. There has also been very little
research done on the renewal of English mathematics in the
early nineteenth century. Much of what has-been written
has limited itself to evaluating the Society and the
renewal solely from the viewpoint'of the development of
mathematical knowledge. ' Thus accounts of these events have
centered on the switch from the Newtonian to the rival ; \
1. Kline (1972) 623.
2. See, for example, Ball (1889), Becher '(1971), Cajori (1919) , Dubbey (1978)’, Kline (1972) and Koppelman (1971/72). -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Continental notation and methods'. Since the Analytical
Society espoused the latter,- it has been viewed by
historians as a successful and influential reformer of
English mathematics. This interpretation, however, appears
very tenuous due to both the lack of research on the Society
and its context, and tt» the insufficiencies, at least in
this case, of an explanation based solely on mathematical
knowledge. The tenuity of the usual view of the Analytical
Society is nicely illustrated by the opinion of Charles
Babbage, a major figure in the Society. He saw the
Analytical Society not as being a success but rather a
miscarriage.
Such was the origin of the Analytical Society, which though it did not realise the splendid and perhaps visionary expectations of its youthful projectors has yet left some records which may redeem its existence from oblivion; and in the productions of its various members the future historian of the abstract sciences may perchance discern some gleams of genius which shall at least excuse if not justify its lofty pretensions? he may perhaps discover that there were other causes which prevented * its extension than any deficientcy in the perseverance, enthusiasm, or intellect of its promoters, and that had this monument of youthful ambition been constructed on a more congenial soil it might have contributed to the promotion of the mathematical sciences in a'/degree hot 'totally inadequate to the hopes of those who fotmed it.^- .
There would seem to be, therefore, some value for a
better understanding of early nineteenth -century Cambridge
1. Written by Babbage in 1817. Buxton ms.13, p.26.
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mathematics in an attentive reconsideration of the history
of the Analytical Society
The goal of this thesis is to examine in detail/-the
history of the Analytical Society in order to gain an
understanding of^tf£ie'NS|ociety, its concerns, activities and
influence. A consequent goal is to relate .the Society to'
the transition to Continental mathematics in order to
obtain an explanation for this transition, especially at
Cambridge University.. Thus the thesis aims to provide a
framework from which the events of the history of the
"^halytical Society may be explained_as well as tfce concerns
and activities of early nineteenth-century Cambridge • * mathematicians.
In attempting to answer the above questions much
research has been done with published primary material and
extensive manuscript sources. . A description of all of
these sources may be obtained from the footnotes and the
bibliography. Out of this wealth of material has emerged
a picture of early nineteenth-century Cambridge mathematics
which consists -of three main elements: ideas concerning
the nature of mathematics, ideas about the purpose of a
university, and a set of expectations concerning
mathematics and science which is best described as
professionalism. These three elements appear to have
been the prominent factors in the transition in England.
With these and with the very interesting patterns of
alignment of individuals of that period with respect to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4
them, it is possible to.|construct an explanation of the
record of ''the Analytical Society ana of the revival of
English mathematics. •
The first key element was a debate on the relative-
' merits of analytic.-and synthetic mathematics. These two
styles of mathematics were sharply distinguished in the
early nineteenth century. The difference had its roots in
the Greek contrast between analysis and ^synthesis, based
on reverse methods of reasoning. .The distinction acquired
a new le^el^of meaning in the sixteenth century with the
emergence of the "analytic art", or algebra. It sought
to resolve mathematical problems by reducing them to
equations.'*' The analytic art was extended in the
seventeenth and eighteenth centuries to encompass infinite
quantities and processes. Hence analysis came to designate
such areas of mathematics as algebra and differential
calculus. In the second half of the eighteenth century
there was a movement, due especially to L. Euler and J.L.
Lagrange, to regard the characteristic of analytics as the
formal manipulation of equations, or expressions. . Lagrange,
for example, sought to base the differential calculus on
1. The study of relations between finite quantities by means of equations involved a new idea of number. On the broadening of the number, concept at this time see Charles V. Jones1' The' Concept' of Qg£ an a Number, Ph.D. dissertation, University of Toronto, 1978.
V
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the formal expansion of functions in power series. 1 in the
time period cohered in this thesis, then, analytics implied
an algebraic or formal, operational approach to the subject-
matter. . • ...
Synthetic mathematics^' on the other hand, was all that / - was not algebraic: • geometry, for example. With the
restrictions, of the second half of the eighteenth century,
synthetics; also came to include all that was non-strictly
analytic. Fluxions, for instance, were not properly
analytic because they involved the idea of motion which was
a non-algebraic concept and therefore, for many, not
analytic. The Analytical Society held firmly to this new
rigorous view of analytics. The mathematical work of its
members, therefore, may be seen as a continuation of a
formalistic tendency in parts of French mathematics,
extended to a study of the "language of symbols", or the
structure of abstract analysis.
Another important facet of the debate over analytics
and synthetics was a prevalent difference in emphasis as
to their respective values. Probably due to the successes
of the analytic art, analytics was highly regarded for its
power of discovery. It was the best example of the way in
__— ...... — ■ s 1. Kline (1972) 431-432. See also pages 100-101 of I. Grattan-Guinness "The emergence of -mathematical analysis and its foundational progress 1780-1880" pp.94-148 in his (ed.)' From the' Calculus to' Set Theory,' 1S30-1910 (1979). , ' i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which reasoning was to be used. In contrast, synthetic
mathematics, such as' geometry, was prized for the clarity
and rigor of its explanations. Many mathematicians had
misgivings about the vagueness and imprecision then /.
associated with analysis. Analytic ^and synthetic
ma'thematics were thus alternative modes of doing mathematics ; in\the early nineteenth century: one promoted discovery, X the' other rigor, each possessed a distinctive style and was
valued for very different reasons. The views on the nature
of mathematics held by the members of the Analytical Society
bring into sharp relief these various aspects of the early
nineteenth-century conception of analytics and synthetics.
And the whale debate became, prominent at that time in
England because there was a widespread lament about the
inferiority of English mathematics.
The'•second prominent element of the setting for the
Analytical Society was the idea of a higher education.
The accepted ideal of a university training in early
nineteenth —century England was that of a liberal education.
This meant the molding of the character of a young man
into that of a gentleman. Such an education stressed the
transmission of the culture of the nation, or of man, to
the individual. It opposed any narrow education, that is,
any education devoted solely to specialized training for
a later career. The university, then, was to embody the
ideal of a liberal education through being the guardian of'
accepted knowledge or culture and iy transmitting this knowledge
to the young men in its care. The task
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of a university was not seen as directly including the
production of knowledge. The goal of a liberal education,
as the purpose of a university, changed very little, if
at all, in nineteenth-century England despite cries for \ ■ *- "useful" education. The content, however, did become more
comprehensive in that it included a wider range of subjects.
The ideal of a liberal education is important in
understanding the Analytical Society because it was composed
of Cambridge University undergraduates and recent graduates.
They were dissatisfied with the content and the system of t *
Cambridge studies - both of which were justified by the
ideal of a .liberal education. Such dismissal of "usefulness"
in education leads naturally to the third key factor in the
transition to Continental mathematics.
Professionalism is the third element in the framework
for considering early nineteenth-century Cambridge
mathematics. The Industrial Revolution has been regarded
as the emancipator of the. professional man."'' Although S little research has been done on the professionalization
of science in England, it is clear that professionalization
.took place there very slowly and reluctantly - the x hesitation being largely voiced as a belief in individualism.
Yet it is also the case that an increasing number of British
natural philosophers in the early nineteenth century w e r e .
1. Harold Perkin' The' Origins' of M o d e m English Society 1780-1880 (1969) 254. \
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. concerned with making science a profession.^"
The social element of professionalism acts as a
dynamical component in the framework established in this
thesis. The expectation that one could be a mathematician
or scientist, that mathematics and science should be a
9 '■ profession like other branches of knowledge or activity, "N seems to have acted as a motivation* for many individuals to
reform English mathematics in the early nineteenth century.
The element of professionalism is best understood when its
relation to the other two elements is kept in mind. A * liberal education was clearly a non-professional education,
because the training which a profession required was far
too specialized for the objectives of; a liberal education.
On the other hand mathematics, being a discipline concerned
with developing mathematics, was professional - the business
of mathematicians. Analytic mathematics, therefore, became t
firmly linked with professionalism in early nineteenth-
century British thought primarily because of the reputation
of analytics for discovery and hence advancing mathematics.
Synthetic mathematics, by contrast, became linked to a
liberal education. For a main goal of a liberal education
was the developing and strengthening of the reasoning
powers of the mind, and synthetics - especially geometry, ' ( 1, See, for example, J.B. Morrell’s articles "Individualism and the Structure of British Science in 1830"' Historical Studies' in the' Physical' Sciences' 3_ (1971) 183-204, and "London Institutions and Lyell’s Career: 1820-41" British Journal for the' History' of Science 9 (1976) 132-146.
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9
as Well as other non-analytic methods - had been
traditionally esteemed for just this value. Since analytics
was also contrasted with synthetics on this issue, and
since synthetics seemed eminently, suitable for the purposes
of a university, a further (.though indirect) connection was
established between analytics and professionalism. In an
educational institution such as Cambridge the commitment
to professionalism would have to be greater than that of
a liberal education if the transition to analytics was to
take place.
The three ^elements, analytics versus synthetics, a
liberal education, and professionalism, set the context
within which the events surrounding the Analytical Society
and the renewal of Cambridge mathematics are to be t understood. These events reveal a certain pattern of
alignment in the actors with respect to the three elements.
Most university reformers, if not all, were concerned with
making mathematics a profession, were promoters of
analytics, and did not reveal much enthusiasm for the
traditional ideal of a liberal education. Conservatives
were not interested in making mathematics a profession, if
not actually against such measures, -were supporters of
synthetics, and defended the ideal'of a liberal education
as the purpose of higher education. To the extent of the
generality of these patterns one could say that these three
elements determined the activities of the individuals
involved. It is within the framework established by these
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three elements that this thesis will seek to provide an
explanation not only of the Analytical Society but of the* ■
transition in Cambridge mathematics.
The Analytical Society was'the product of three J circumstances of early nmeteenth-century Cambridge. In
\jenferal, the situation at Cambridge, and the widespread
feeling of English mathematical inferiority combined with
new expectations of- students at Cambridge to produce the
Analytical Society and the Cambridge adoption of analytics.
The details of this process will be set forth in the
following chapters of the thesis.
Chapter IX reviews the situation at the University of
Cambridge in the early nineteenth century. It focuses on
the structure and content of Cambridge studies especially
with regards to the position of mathematics there. The
training available at Cambridge was distinctive for its >
emphasis on mathematics. Moreover, Cambridge, as an
institution, incorporated a number of incentives to pursue
mathematics at a rather advanced level. Chief among these
was the mode of acquiring honors through the Senate House
Examination which was almost completely mathematical in
content. Yet Cambridge was not thought of as a place for
training mathematicians: the ideal of a liberal education
was the underlying principle of a Cambridge education. *
This tension between the ideal of a liberal education and,
in practice, a specialized training in mathematics was
manifested in numerous complaints about Cambridge in the
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early nineteenth century. On the one hand many critics of f Cambridge called for a broadening of the content of studies'
to include subjects other than mathematics; for example,
more classics or the natural sciences. Many wished to
break the dominance of mathematics in the Senate House.
On the other hand, from a mathematical perspective, there
was room for complaint about the style or level of
mathematics being taught'since it was mainly synthetic
mathematics, as dictated by the tenets of a liberal
education. This was the position taken by the members of
the Analytical Society and by other reformers of Cambridge
mathematics. There was a situation at Cambridge where
mathematics was highly valued and fundamentally related' to
the character of its curriculum, and this situation was to
play an important role in the; reform.
The second important set of influences in the history
of the Analytical Society involved the outcry in the early
nineteenth century over the state of mathematics in England.
Chapter III examines this complaint which had two chief
characteristics: one an argument over the style of
mathematics, the other a resolution about the.1 relationship
of mathematics and society. These again relate to the
three fundamental factors outlined above.
The first feature of the lament was that many saw
the decline of the mathematical sciences in England to be
a result of the emphasis there on synthetic mathematics.
Reflecting developments in mathematics on the Continent,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ana particularly ifl France, numerous British mathematicians
pushed for the adoption of analytics in their own country
because of its ’ability to produce new mathematics. They
met with opposition from those who preferred synthetics
for a variety of reasons, such as an objection to / analytics as meaningless manipulations or adherence to the
Newtonian tradition in mathematics of fluxions and
synthetical methods. This opposition, howevfer, was largely
passive. It tended to fade into the background when
challenged by the efforts of supporters of analytics
^especially when .these efforts were concerted.
Cambridge had been the university of Newton and
mathematics was central in its curriculum, so it was seen
as the epitome of English mathematics. Its stress on
synthetics represented the state of mathematics in the
rest of the country. Some mathematicians at Cambridge,
Robert Woodhouse for example, attempted to introduce
analytics. Barriers to these efforts were created not
only by the state of Cambridge mathematics and the inertia
of many there, but also by the acceptance of a
liberal education and hence an apparent restriction to
synthetics.
The second characteristic of the outcry was the view
that a further cause of the decline was the lack of public
institutional encouragement for the mathematical sciences.
If mathematics was to prosper in England, supporters of
Continental mathematics argued, the government, the Royal
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Society and the universities had to promote mathematics -
by which they meant analytics. This aspect of the' lament
was probably prompted by the emergence in France at this
time of institutional or professional opportunities for
scientists. Thus professionalism and analytics were woven
together in still another relationship in the contentions
of the critics of the state of British mathematical science.
This association was maintained by many of those who first
introduced and adopted analytics in England. The University
of Cambridge, due to its stress on mathematics and its
institutional role, came under much criticism from these
persons not only for its teaching of synthetics: but also for
its system of studies. Thus the lament was both a
reflection of the changing circumstances of mathematics in
England as well as a tool in the efforts to transform that '
mathematics. i The two sets of circumstances described above provided
the setting for the early nineteenth-century change in -
Cambridge mathematics. Only individual actors were required to draw out the tendencies af the circumstances. This process occurred at Cambridge among the students,
many of whom brought into their university lives the
popular issues of the time. The best example of the
unfolding, by students With new expectations, of the
situation at Cambridge and of the cry about the inferiority
of English mathematics was the Analytical Society. Its
history is set forth in detail in chapter IV. The
o
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14
Analytical Society was a mathematical society: it existed “
to promote.analytics. Its members were not satisfied with "
the study of synthetics at Cambridge, with the level of
the mathematics taught there, nor with the Cambridge system
associated with those studies. But contrary to the uaual
historical view, despite this dissatisfaction with Cambridge
the activities of the Society were focused on the production
of analytical mathematics and not on the reform of Cambridge.
The Analytical Society, as Babbage’s words quoted
above' indicate, did not prosper at Cambridge. • It suffered
from the system of Cambridge studj.es which, particularly
through the Senate House Examination, emphasized synthetic
mathematics and demanded for success much time and attention
S, to its study. The existence of the Society was also
jeopardized by the lack of careers in mathematics in England
which would have served as incentives to the Society's
members to work in mathematics after graduation. Yet even
with its difficulties and short existence, the Analytical
Society seems to have had an influencef t ' on its members which
became visible, after the Society's dissolution, in the
mathematics of some of its former members as well as in the
■ later efforts of some to reform Cambridge studies.
A few of the members of the Analytical Society
- continued their efforts in mathematics even after the
Society's dissolution. Chapter V looks at their work and
S? t.' • concerns in-mathematics. It outlines the role of some of
the elements of the framework., which this thesis establishes,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15
in the members' production of new mathematics. A close
link exists between this mathematics and their ideas about
mathematics, or vision of analysis. The technical
mathematical work of such former members as Charles Babbage
John Herschel and Edward Bromhead shows that it was aimed
at revealing the foundations of pure analytiqs- in order to
advance the development of mathematics. This goal was
based on the belief that mathematics had grown to be as
powerful as it was because of analytics; a belief which J was becoming prominently embedded in the mathematical i
climate of England at that period. Thus it appears that /
their technical mathematics depended on and reinforced
their vision of analysis.
The_ vision itself was a reflection of some of their
other concerns, especially with professionalism. While
these former members were producing mathematics, they were
also attempting to make careers in.mathematics. Their lack
of success in this area served to confirm the idea that ^"\
mathematics had to be treated as a professic^i if it was to
prosper in England. Only in th^s way woul lysis, and
especially “pure .analysis, be assured of support. Their
work in pu ^ is and their vision of analysis were a
basic part of their attempts to make mathematics in England
a profession. Thus their mathematics was also an image of
the elements-of -the framework of early nineteenth-century
Cambridge mathematics.
Chapter VI reviews the introduction of analytics at \
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Cambridge. The same elements which led to the formation of
the Analytical Society and were involved in its mathematics
were also active in the transition of Cambridge mathematics.
Sparked by former members of the Society, many younger
members of Cambridge reformed the mathematical studies there, C
These "reforms" were carried out through the structure of
the University, for example, through the textbooks and the
Senate House Examination . Analytics conforming to the
Analytical Society's vision of analysis was adopted. The
reformers also sought to infuse into the Cambridge studies
a. wider range of mathematical topics and a deeper study of
them - a very non-liberal direction in education.
The changes of the 1820s in the mathematical studies
at Cambridge met with much criticism. There were some who
objected to.the particular views of analytics advocated.
Others rejected analytics and stressed the value of
synthetics within a liberal education. And many also pointed
out that the link between analytics and professionalism
conflicted with the Object of a university. Thus both the
forces for and against the changes in Cambridge mathematics
reflect the basic factors we have identified. We hope that
the approach of this thesis through the social and
intellectual framework of analytics versus synthetics,
liberal education, and professionalism, will provide insight ■) into the nature of Cambridge mathematics in the early
nineteenth century.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. II. Mathematics and the Curriculum of the University
of Cambridge in the Early Nineteenth Century
. The University of Cambridge played an important role
in the history of mathematics in England. This fact was
due not simply to the many outstanding mathematicians
associated with the University - it was the University
of Newton - but also to the tradition of mathematical
study there. The aim of this chapter is to examine the V* system of Cambridge studies and the role of mathematics
in that system in the early nineteenth century, and
thereby to establish the Cambridge context of the
Analytical Society.
Cambridge, as an early nineteenth-century institution,
blended indwell with many other aspects of pre-industrial
— f ' England. It was also a university with many distinctive
characteristics, particularly in its system of studies.
A brief contrast with Oxford, the only other English
university before 1828, and with the Scottish-universities
is useful in appreciating these features.
Oxford, like Cambridge, had been in a depressed
state in the eighteenth century. It had begun to revive
its studies in 1800, when examinations for the :B.A., A B.C.L. and M.A. degrees were established. The subjects
for the B.A. were Grammar, Rhetoric, Logic, Moral
Philosophy and the elements of Mathematics and Physics.
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Candidates were to be examined orally in all or some of
these s u b j e c t s b u t the tests were always to include
classics.'*' An honours examination was also set up but
few persons attempted it. In 1807- separate Honours
Schools in Literis Humanioribus and in Mathematics and
Physics were founded with the former consisting of the
Greek and Latin languages, Rhetoric, Moral Philosophy !
> 2 s and Logic. As for the latter school, apparently there
were few persons at Oxford capable of acting as tutors
in these subjects at first Although by 1816 some students ^
had "acquired a profound knowledge of the higher geometry,
of the Principia of Sir Issac Newton, and of the four
branches of natural philosophy.11^ However, with this
separation, mathematics and physics were no longer
obligatory subjects; professional lectures in all the
sciences declined in numbers, and the classics and logic 4 gained a new monopoly on learning. In the 1820s it was
said
That the mathematical sciences are in the lowest' possible state in Oxford may Ije assumed as an indisputable fact. They had rather gone backwards than forwards for the last 20 years. 5
1. Clarke (1959) 98.
2. Ibid. 99.
3. Anon, "A Review of 'Wainewright's Literary and Scien- t tific Pursuits of Cambridge"’ British Review 1_ (1816) / 357-375. p.365. Incidentally, this is a lovely ex- *• pression of the ideal of a liberal education.
4. Ward (1965) 15-16. See also Powell (1832) 30-38.
5. Ward (1965) 57.
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The Scottish universities were very different from
their southern counterparts. They continued, in the very
early nineteenth century, to enjoy their eighteenth-
century reputation for medicine, philosophy and science.
In further contrast to the English universities, they were
cheaper to attend than Oxford or Cambridge, wer.e more
popularly based and offered mostly professional degrees
with the professors both lecturing and giving tutorial
instruction. And in the subjects of study,- philosophy
played a predominant role with "an unusually large amount
of attention" given '"to the first principles and metat
physical ground of the disciplines. But instruction in
the Scottish universities was of a low standard; because
of their broad base they tended in some measure to do 2 "the work of secondary schools." A Royal Commission on
Scottish universities was established in 1826, after much
debate on the decline of Scottish universities,: and
following their report the English example of a classical
basis for higher education increasingly replaced the older
Scottish tradition.
What distinguished a Cambridge Oniversity education
from that available at Oxford and at the Scottish
universities was its-emphasis on mathematical studies and
its examination system. The mathematical curriculum was
of a relatively high standard at least for honours,
1. Davie (1961) 13.
2. Saunders (1950) 358.
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while the Senate House Examination (later to evolve in
part into the Mathematical- Tripos) was a rigorous final
examination for the B.A. and was mostly mathematical in
content. These facets of the curriculum will now be re
viewed in some detail.
A young man would enter one of the seventeen colleges
of Cambridge University in the early nineteenth century at
about the age of-eighteen. Generally he would be admitted t before the end of an Easter Term and, after residing the
greater part of each of the ten following terms and ful
filling certain requirements, he would receive a bachelor
of arts degree. There were three terms in an academic
year, Michaelmas Term (October 10 to December 16), Lent
Term (January 13 to the Friday after Midlent Sunday) end
Easter Term (eleventh day after Easter to the Friday
after the first Tuesday in ’.July) . Except for regular
attendance at chapel and, perhaps, at college lectures,
the undergraduate could and did sp£nd most of his time
as he pleased. As the great majority of the under
graduates did not study for honours and as the standard '
for a B.A. was quite low, many passed their time in
various idle pursuits. However, incentives such as
prizes, scholarships and fellowships also existed and
encouraged many to be "hard reading men".
One of many such persons was G.B. Airy, later
1. Wall (1798) 8, 37, 41, 61, 67, 82.
' •' I \
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Royal astronomer of England. He studied at Cambridge
from 1819 to 1823, graduating as Senior Wrangler, that
is, the first in the ranking of the hohours classes.
His daily routine began with chapel at 7 a.m., then
breakfast and attendance at college lectures from 9 to
11 a.m. After this he put his lecture notes in order,
wrote a piece of Latin prose and then usually read math
ematics. At 2 p.m. he went for a four or five mile
coui^rj? walk or perhaps rowing;, returning to dine in
the college hall at 4 p.m. After dinner he lounged until
evening chapel at 5:30 p.m. and returning about 6 p.m.
had tea. He then read quietly, usually a classical sub
ject, until 11 p.m., when he went to bed.1
The college lectures were given by the college tutors
and concentrated on mathematics and classics. The order
in which various topics were studied seems to have
varied from college to college but the subjects themselves
appear to have been the same in the colleges throughout 2 the early nineteenth century. As a specimen of these
1. Airy (1896) 26.
2. Compare such accounts as Airy (1896) , Schneider (1957), Wright (1827), Wainewright (1815) , Academicus, "A fetter on the 'Course of Studies at Cambridge and Senate-House Exam"1, Monthly Magazine 11 (1801) 115-118, 292-294. pp.115-116, and the ms. notebooks for 1809-10 of Thomas Pierce Williams of St. John'is College i.in the University Library, Cambridge.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. lectures consider those given in Trinity College, the
largest of the colleges of Cambridge, for the period
1815 to 1818.1 A freshman entering Trinity in 1815
would have had in the Michaelmas Term a mathematics
lecture at 9 a.m. on Euclid followed at 10 a.m. by
an hour1s lecture on a Greek tragedy. In the second term
these subjects were replaced by the first part of
algebra, according to Wood's text, and the 21st Book of
Livy. In the final term of his first year the subjects
were plane trigonometry and the 8th Muse of Herodotus.
It should be mentioned that the lectures were not usually
formal but were conducted on the lines of the tutors
questioning the students on assigned readings. In the
first term of his second year the Junior Soph attended
a mathematics lecture on statics and dynamics and a
classics one on the 7th Book of Thucydides. The 2nd,
3rd and 4th parts of Wood's Algebra, spherical trig
onometry and divinity in the form of St. Luke, Paley's
Moral Philosophy and Evidences of Christianity and Locke's
On the Human Understanding took up the Lent Term, with
the final term devoted exclusively to mathematics: conic
sections, popular and plane astronomy and the first three
1. The following description of Trinity's lectures is taken from Wright (1827).
with permission of the copyright owner. Further reproduction prohibited without permission. 23
sections of Newton's Principia.1 In this second year
each student also had to compose two declamations,
generally on some historical subject, and to defend them
against some opponent in'the chapel. As John Wright wrote n2 "It is usually considered a bore. The Senior Soph
would find no classics nor diivinity lectures to distract
him from mathematical studies in his third year. The first
term was spent on the remaining parts of Book I of the
Principia, the second term on fluxions, fluents and hydro
statics, and the Easter Term on optics, physical astronomy
and "a general recapitulation of the studies of the whole
three years in the working of problems." Finally, in the
tenth term after commencing studies, there were no lectures i but rather frequent examinations on subjects previously
studied in order to prepare the student, now called a
questionist, for the Senate House Examination, which
usually began on the first Monday of the Lent Term.
To encourage study of the college lecture topics,
an annual examination was given by Trinity College for
the first and second years and extended to the third in
1818. However only a few colleges had any faf£5r"bf
examination for their students. Moreover the standard
of the lectures was 'low because they had to hJe within the
reach of all the students of each year. Stills, the
1. The term 1 popular1 seems to have denoted elementary principles and notions, and the use of various' astronomical instruments.
2. Wright (1827) 1 199. -
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subjects of the lectures, if studied, must have been
sufficient to have enabled a student to obtain his degree
easily. To gain a degree in 1800, "two books of Euclid's
Geometry, Simple and Quadratic equations, and the early
parts of Paley's Moral Philosophy, were deemed amply suf
ficient."''' The better students quickly out-stripped-, the
content of the college lectures through extra reading
guided by-private tutors in an effort to gain high honours
in the Senate House Examination. Indeed, Cambridge Univer
sity's examination system, which was its pride and glory,
was very competitive. While a very little knowledge
might suffice for a degree, there was no maximum for one
who aspired to be placed high in the honours list.
Consider, for.instance, the extra mathematical
studies of John Wright who attended Trinity College from
1815 to 1819 and who, but for an accident, would have 2 been very high i n .the honours list. Wright came up to
Trinity with very little knowledge of mathematics, only
Ludlam's Elements and Walkinghame's Tutor's Assistant.
In the Easter Term he engaged a private tutor and towards
the end of his first yedr he began to read more widely
than ne^d4d for the college lectures. This he continued
rn his second year , consulting various mathematical texts,
buying. otha|fe..aTid-. practicing on as many problems as he could
~ ~ ' 4 1. Prymej_/(1870) 92. Throughout the nineteenth century there were to be many calls for broadening and raising the level of an ordinary degree.
2.. See Wright (1827).
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find.
For Problems or Deductions my resources were the Diaries, Leybourne's Mathematical Repository, Dodson's Repository, and some others, and all the Examination Papers X could lay my hands on. With " J these last constantly on the table as a conductor, I traversed the_regions of knowledge, collecting at every step something useful, .and writing them I out, generally in better form I conceived, into ^ . a "College MS."1
During the long (summer) vacation of 1817 Wright prepared
in advance for the following term's lectures ±>y working
at Newton's Principia and at various texts on fluxions
and also read parts of French works on mechanics, by
Francoeur and Poisson, and even struggled with Monge's
Gdomgtrie Analytique, Lagrange's Mgcanique Analytique,
and Laplace's Traitg de Mgcanique Cdleste.
X soon found, however, that the three latter works were at that time much too abstruse ;for my comprehension. I proceeded, indeed, as far as page the seventh of the Mecanique Celeste with some difficulty, but there came to a dead stop, for want of a previous knowledge of the doctrine of Partial Differentials, which had not yet found its way into any work on the subject of Fluxions, in the English language.'. . ^
During the first term of his third year, Wright, finding
. himself so far ahead of the college lectures, began
skipping them quite often. By the end of the term he
had studied Book 2 and a "considerable part" of Book 3
of the Principia, had worked on fluxions and had pursued
such superior French works as would lead to "those (,ne
plus ultras of Mathematical Science; the Mecanique
'1. Ibid. 1 206-08.
2. Ibid. 2 2-3
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 •v Analytique, and Mecanique Celeste."''' For the remainder
of his studies at Cambridge, Wright worked mainly by
himself, studying primarily Newton, Lagrange and
Laplace and consulting many English and, especially,
French mathematical texts.
Thus, while college lectures provided enough
guidance for those who might want to gain honours, they
were not at all adequate for obtaining high honours.
For this, private tuition and study of .more advanced
mathematical topics were necessary. The Cambridge
system with its few requirements provided the time for
this study. Also noteworthy in Wright's example is the
fact that serious students of mathematics at Cambridge
were being attracted to the study of French mathematics
by the prestige of Lagrange and Laplace.
The University, compared to its colleges, actually
did very little teaching and the little it did had no
bearing on the B.A. degree. Its.influence lay in the
area of requirements for a degree, these being, before
the 1822 introduction of the elementary Previous Exam-
ination, primarily the Disputations and the Senate House'
Examination. An undergraduate might be called upon
anytime from the Lent Term of his third year to the end
of the following Michaelmas Term to take part in the
Disputations. These j^ere debates.in Latin between
1. Ibid. 2 24-25 “ ■ -
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undergraduates of the same year, and their function, in
the early nineteenth century, was to establish a pre
liminary classification of the undergraduates for the
Senate House Examination. Once called upon, the Respondent,
ag the student was denoted, had to submit three theses
to a Moderator, an official examiner. Usually the first
thesis was on Newton's Principia, the seaond on some
other mathematical or natural philosophical writer and the--
third on some point of moral philosophy: for example,
The ninth section of Newton's first book is true. The aberration of the fixed stars dis covered by Bradley is accounted for by him on just principles. A future '-state is not dis coverable by the light of nature.1
The theses were communicated to three other students,
the Opponents, chosen by the Moderator. A public Dis
putation lasting about two hours was held three weeks
after the Respondent had been called by the Moderator.
The Opponents were supposed to bring arguments in the form
of syllogisms against each of the Respondent's theses.
During his act the Respondent would read a brief treatise
on one of his theses, usually on the third, following'>
which the first Opponent would offer five objections to
the Respondent's first thesis, three to his second and
one to his third. The Respondent and Opponent would
1. Academicus, "A Letter to.the Editor on the 'Course of Studies at Cambridge and Senate-House Exam'", Monthly Magazine 11 (1801) 115-118, 292-294. p.117. Charles Babbage kept his act (Disputation))in Feb-, ruary 1813 on the Second part of Wood's Algebra, the appendix-to Woodhouse's Trigonometry, and Dugald- Stewart on Dreams. Letter from Whittaker to Bromhead, Feb.. 16 1813; Br.ms.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28
discuss each argument until stopped by the Moderator.
Similarly the second Opponent would offer three, one and
"one arguments respectively, and the third Opponent one
argument against each thesis. It was the practice that
only the better students', as reported to the Moderators
by the College tutors, would serve as Respondents and
\first Opponents and probably only they appeared more
than two or three times in the Disputations.1
The Disputations, with the growth in importance of
the Senate House Examination, became increasingly subordi
nate to it. In 1819 William Whewell felt that the system
of Disputations did not, "at leas'd immediately, produce
any effect on a man's place in the tripos, and is there
fore considerably less attended to than used to be the
case, and in most years is not very interesting after 2 the five or six best men...." By 1830 the Respondent
and Opponents began prearranging their arguments, and in
1839 the Disputations wereu discontinued.
The Senate House Examination was by far the most
important test in qualifying for a Bachelor of Arts
degree. Almost all the undergraduates eligible for a
B.A. had to pass this examination although, as seen above, a it was much more rigorous for those aspiring to high
honours. The examination appears to have evolved from
a pgrior (about 1725) statutory, examination.1 By the
. 1. Ibid. 117 and Schneider (1957) 30, 32.
2. Todhunter (187 6) 2 35.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. middle of the eighteenth century it had become a re
quirement for nearly all undergraduates. For the remainder % ■ of that century and for the first half of the nineteenth
the examination was constantly being refined: its
duration increasing from 2 1/2 to 8 days, its mode of
examination becoming more and more written rather than
oral, its problem papers'' being extended to include *
students other than-just those of the h i g ^ honours classes,
and, finally, the range and difficulty of its mathematical
questions being greatly augmented.1 While these extensions
in the Senate House Examination had the overall effect
of raising the level of examining of the whole student body
to a new, more thorough plane, they also were to cause ,
concern about the Resulting curbing and diluting of 2 the examination of the better students.
An example of this test was the operation of the
Senate House Examination in 1819. The undergraduates
taking the examination had previously been arranged into
eight classes by the‘results of the Disputations and
it was according to this classification that they were,
tested in the Senate House. As was usual at that time,
there were six public examiners: the senior and junior
■moderators "of the present year, who. were nominated by the
Proctors, those of the previous year, and those of the
1. For the details of these changes see Ball (1889).
2. Great Britain (1852).
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year preceding the two last or else two examiners appointed
by the Senate. There does not appear to have been any
pattern in appointing the moderators/ except that they
had to be Masters of Arts and, at least in the early
nineteenth century, to have been placed very high in tile
honours lists of recent Senate House Examinations.^
The moderators were important university officials,
conducting the Disputations and the Senate House Exam-i
ination as well as setting the questions for the latter y
\ and arranging the final hounours list.
The k^iamjnati^n lasted at least six hours each day
for five day's£-'The first three days were employed entirely
by mathematics with the fourth spent on logic, moral
philosophy, the evidences of Christianity and such i/topics.
On the fifth day a re-classification of the undergraduates
appeared and the rest of the day was spent examining the
students, expecially those of tl^ higher classes, to
determine more finely their proper ordy c of merit. For
example, on the first day of the 1819 examination, the
first and second classes were given a problem paper by
the junior moderator, George Peacock, from 8 to 9 a.m.
After a half-hour break they were given bookwork, that
is, problems or theorems read out of some text, until
11 a.mi by the senior moderator,; Richard Gwatkin. After
lunch (which many could not eat) they were given more
1. The Masters of Arts degree was not very rigorous in 'its requirements and was obtained three .jyears after ■ the B.A.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. >> 31 bookwork by Peacock from 1 to 3 p.m. After another
half-hour break they would probably have been available
to any examiner for oral examination until 5 p.m.
Then from 6 to 9 p.m. they took tea with Gwatkin in the
Combinatioh Room of St. John's College. Afterwards they
were given a problem paper to do until io p.m., when
the examination was finished for the day. While these
two classes were thus employed, the otfier six were also
busy at various times with written and oral work.^
After the examination all the students would be
listed according to their excellence into four classes
of descending order; the first three were the honours
classes and were called Wranglers, Senior Optimes and / Junior Optimes, respectively, while the last and largest
class was the poll men, or o_i uoXXo l , who took a B.A.
degree without honours. In 1819, of 179 men who obtained
a B.A., there were 19 Wranglers, 23 Senior Optimes and
17 Junior Optimes.
Soon after the Senate House Examination, the top
Wranglers, in another opportunity to show their skills,
would compete for the Smith's Prizes. This was an exam
ination in higher mathematics with two prizes of twenty-
five pounds each and was conducted by the Lucasian Profes
sor of Mathematics, the Lowndean Professor of Astronomy
1. Wright (18273) 2 62-93. v
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32
and Geometry, and the Plumian Professor of Astronomy and
Experimental Philosophy. Usually the first and second
Wranglers were also the first and second Smith's Prize-
The Senate House Examination must have caused much
anxiety among the students by its form, duration and even
the season.it was held in - in 1810 W.H. Maule had to
keep his ink bottle " in his bosom that he might
'notbe impeded in his writing" in the unheated Senate- House -
and ■ there are many accounts of the examination- having
severely affected a student's health.1 Much advice,
mostly hints on the subjects of the examination, must
also have circulated; some of it in a jesting vein:
I have a few words of advice for you respecting your conduct in the Senate house. - Keep all tight below, that nature get not the better of yo-u - Get drunk at both the Moderator's Rooms, but yet not so bad as to roll about or overstep the modesty of nature - When you get a problem you cannot do, grin in the examiner's face & tell him you know a trick worth 'two of that - For "my dear boy", this is a wicked world we ■ live in and we have always need to keep in mind Shakespeare's apothegem "Come what come may 2 Time & the ham goes through the blackest day
The examination could be very important for the A
future prospects of the abler students. A high rank
in the Senate House would probably lead to a valuable
College Fellowship - a share in the revenue of the
1. Leathley (1872) 131.
2. Letter from J. Herschel to-.'J. Whittaker, Jan.10 1814; St.J.ms.
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College: '
They eat, and drink, and sleep, What then? They eat, and drink, and sleep again.1 / This was an important consideration in early nineteenth--
century England where there were few professional positions
for a Cambridge graduate except the traditional ones of law
and the church. A fellowship would make the graduate
financially independent and might lead to better positions
in the colleges or the University, to livings in the gift
of the colleges, or to high offices in the Church of
England. Many a Cambridge graduate owed his success to
his alma mater.
Mathematics was, as has been outlined above, firrtily
embedded Lin the structure of Cambridge studies. It also
had a dominant position' in that structure through its
place in the topics of the college lectures and especially
through the emphasis and influence of the 'Senate House
Examination. Mathematics,, therefore, was regarded as
an important instrument in the education of the young
men at Cambridge. But Cambridge, as noted at the
beginning of this chapter, was better fitted in the early
nineteenth century for pre-industrial times than for the
turbulence of the period. 'With the great forces for change
at that time, the meaning of mathematics in the Cambridge
system could not avoid being called into question because
1. Quoted in Gradus ad Cantabrigiam (1824) 48.
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of the prominence of mathematics there.
In the first decades of the nineteenth century the
University of Cambridge, like many other British institu
tions, at that time, was confronted by a spirit of reform.
...just as the University in the eighteenth century reflected the dislike of that age to violent change, so in the nineteenth century it responded to the prevailing sentiment that institutions, however venerable, had duties to the present as well as obligations to the past.1
The previous century had in general been -a period of
decay for the University despite the fact that a number
of brilliant individuals were then associated with it.
Outdated statutes, the increasing expenses of education,
the neglect of teaching, a decline in the number of
students, and patronage-inspired politics had all helped ' i> 2 to contribute -tc( the stagnation of the University.
Many groups had a very low'opinion of a university
education. For example, Manchester manufacturers com
plained of its expense, its encouragement to dissipation
and especially its leading to alienation from their own
norms.^ Attempts at reform in the 1770s had led no
where, and the suppression of liberal sentiments in the
1790s, as a reaction to the events of the French Rev
olution, did not aid in relieving Cambridge's debased
1. Winstanley (194 0) 157.
2. Roach (1959) 234-235. See also Winstanley (1922) and (1935) for evaluations of eighteenth-century Cambridge.
3. A. Thackray "Natural Knowledge in Cultural Context: The Manchester Model" American Historical Review 79 (1974) 672-709. p. 690. ~
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 I
reputation. Yet the numbers of students coming up to
Cambridge steadily increased in the late eighteenth
century and rose steeply after the period of the French
Wars. This increase in student numbers was to strain
the Cambridge system and thereby to put additional
pressure on the need for change there.
With the end of the Napoleonic Wars reform movements
at Cambridge became^vigorous. The goals of these move
ments, not just at this period but for most of the nineteenth '
century, could be briefly characterized as the extension
of education in all of its facets. ~ Besides attacking the
exclusiveness of admissions to the University and %the many r rights, or privileges, of various groups, this extension
particularly involved the curriculum. Especially from
about 1815 there was substantial concern and activity
in reforming studies. Examinations for the degree of
Bachelor of Laws were instituted in 1816, and examinations
and a course^of lectures on the principles of medicine for
the degree of Bachelor of Medicine in 1819.^ At the same
time there were increasing demands that university professors
lecture on their subjects, a function which had long been
neglected.
Most undergraduates at Cambridge enrolled for the
1. Winstanley (1940) 160, 167.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V
36
Bachelor of Arts degree., And here, in particular, there
was much criticism of the course of studies and many
attempts to make it more comprehensive. Henry Brougham,
in 1825, complained of the inadequacy of the existing
university instruction:
The excellence of few individuals in each University, in classical and mathematical attainments, cannot be cited as any real exception to these remarks. The number of these proficients is extremely small, compared with that of the whole students; and there is really no medium between almost entire idleness, and such skill in making Greek and Latin verses as would astonish a first-rate German commentator, and such readiness in solving difficult problems as would surpass the belief — certainly far exceed the power of Sir Isaac Newton, were he again to visit the banks of the Granta. But the true test of a good and efficient system of instruction, is, first of all, its teaching the whole body of those whom it embraces, and making each advance according to the measure of his faculties; and, next to that,its imparting knowledge which . may remain with the students in after life.
Attempts to change Cambridge studies, which were
dominated by mathematics, met with much resistance.
A syndicate was appointed in December 1818 to consider
whether, undergraduates should "be examined, previously
to their degrees, in theological and classical knowledge, 2 as well as in mathematics, metaphysics and ethics." Its
1. H. Brougham "Review of 'The Proposals for founding an University in London considered'" Edinburgh Review 42 (1825) 346-367. pp.351-352.
2. Winstanley (1940) 66.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 i j favourable report was lost in the Caput of the University.
And, when in March 1822 a scheme for a previous examination
(that is,-previous to the Senate House Examination which '
was the final, -and most important, test for ,the B.A.) was
finally approved, it merely included one of the Gospels
or Acts of the Apostles in Greek, Paley’s Evidences of
Christianity, and one each of a Greek and Latin classic.^
The content of this examination was made so elementary
as not to divert students too greatly from their math-
ematical studies. 2 /
Similar results befell efforts to broaden the scope
of the Senate House Examination. Christopher Wordsworth’s
plan to have all the bachelor of arts' students take an
examination in classics and theology after the Senate
House Examination was rejected by the Senate in May 1821.
This vote, as D.A. Winstanley has written, was in.part
due to the colleges’ wish to retain control over the
student's instruction, their inability to offer a wide
variety of subjects, and also to the tradition of mathematical
study.^ And, when a year later a classical honours
examination was approved-— the Classical Tripos of May
1822 - the examination was a voluntary one and could
1. Gunning (1828) 97-98.
2. --Winstanley (1940) 167.
3. Ibid. 67-68.
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only be taken by those who had already achieved honours,
through mathematics, in the Senate House Examination.
The reaction by members of the-University to these attempts
to extend the course of study was stated in a review of
the debate over Cambridge studies in 1822:
It has even been openly attempted to introduce classics into the senate-house!!! Visions of the ghost of Sexths Empiricus adversus Mathematicos, and the efforts of defensive wit, levelled at ther’imputed empiricism of the measure, have haunted and employed the light corps of the exclusive mathematicians;; while their weightier reasoners have brought all their, private artillery to .bear on the frivolity of the proposed reform, and on the danger of risking the . enjoyment of a posi£ive good for contingent advantages.
Throughout the firsi half of the nineteenth century
Cambridge reform movements were to meet with little
success. Owing to the powers and interests of the colleges
and the conservative views of many in the University,
few major changes of any sort actually occurred before
1850. In the system of studies, a separate non-honours
degree examination formally came into being only in
1858, although it may be regarded as having existed in 2 practice since 1828. Mathematics continued to enjoy
its privileged position in the intellectual life of
1. Anon "Review of 'Thoughts on the Present System of Academical Education in the University of Cambridge'" Monthly Review 97 (1822) 306-315. p.307.
2. Ball (1889) 212.
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the University especially in the acquiring of honours.
It was not until midncentury that the Classical Tripos
became independent of gaining mathematical honours and
that other modes of acquiring honours, such as the Natural
Sciences and Moral Sciences Triposes, were introduced.'1'
The Mathematical Tripos, as the Senate House Examination ' \ came to be called after the introduction of the Classical
Tripos, increased in the first half of the nineteenth
century in length, rigor, and in the scope of the topics 2 it covered. Despite a few minor reforms and ameliorations,
major change at Cambridge only came about with Parliament
appointing a Cambridge University Commission in 1850.
An important factor in the turmoil at Cambridge
was the activity and expectations of students. The
first third of the nineteenth century, at both Cambridge
and Oxford, was a period which witnessed the arrival,
of "the independent student and the notion of a separate 3 student estate." As a consequence of new social values,
according to Sheldon Rothblatt, students "were coming to 4 the universities in a questioning mood". Many developments
1. Ibid. 211-213.
2.- Ibid. 211-215.
3. Rothblatt (1974) 303. See also Rothblatt (1976).
4. Ibid. (1974) 300.
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at the University at that time confirm this new student
character.
The Oxbridge clubs, the debating societies, the intellectual and sporting associations, the expeditions, the strenuous exercises, the magazine essays and poems, the animated social life and convivial ethic, all point in the same direction: toward a generation of young adults seeking distinctions, pursuing recognition, looking for' public reputations, and introducing into their university lives many of the social and intellectual ideas of their time, a time that was marked by disturbance on a national scale. 1
This movement was undoubtedly a cause of much of the
ferment at Cambridge.
Both the students', activities as well as the
attitude of the University authorities towards them are
well illustrated by two events, at early nineteenth-century
Cambridge. . In the first, the founding in 1811 of an
auxiliary branch of the British and Foreign Bible
Society, the initiative of the undergraduates was frowned
upon, especially by the heads of the colleges. Isaac
Milner, President of Queen's College and Lucasian Professor,
believed that
...if undergraduates were permitted to organize themselves for the purpose of diffusing a knowledge of the Bible, it would not be long before they .were banding together to spread subversive political ideas; and that therefore it was of the utmost importance to impress upon them that they had not come to the University to teach their . elders and betters.2
1. Ibid. 301.
2. Winstanley (1949) 21.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This view of the discipline of /the students is also
visible in the 1817 suppression of the .Cambridge Union
Society, an undergraduate debating club, by the Vice-
Chancellor of the University, James /Wood. William / Whewell, President of the Union at that time, described v v, the event in a'letter to H.J. Rose the next day and gave
the following vivid account of his interview with Wood.
"We are told you have an objection to our debates - Want to know how "far it goes - literary subjects?" "No sir - they are against the statutes - all meetings at regular times for the purpose of debate are -.hum - haw - hum - irregular. - and you have only bhree years - you have other things to do. You take too much upon you - your knowledge, your reading, your minds are not proper for &c .." "I. am afraid we are not to be allowed,-to consider the -reasons - we must submit to the authority" A move at the word authority. "But the case must have been exaggerated - two or three-'hours a*. • week" "Sir I have had a letter from a person who j once belonged to the society and who says that his prospects have been ruined and that the , prospects of several of his friends have been ruined by the time and attention he has bestowed on the Society" "Very unfortunate - but it j.s impossible this can be common" "Sir it is 1 against the statutes - you must disperse"
The‘frustration of their expectations by the structure -
of Cambridge University was to cause much disenchantment .
among the students. Student discontent became significant
when some of the student's, .upon graduation, obtained
posts at Cambridge and attempted reforms. /______/ • 1. Letter from Whewell to H.J. Rose, Mar.25 1817; ■’ W.ms.T.C. These two examples also involved other . considerations such as Milner's political position and the reaction to social unrest.
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The proper sphere of the undergraduate, from the
viewpoint of the University authorities, was study. And
‘not simply any study was expected, but only that of the
subjects taught, at Cambridge and especially those topics
which would be of use for gaining honours in the Senate
House. _ The emphasis on mathematics at Cambridge became
the fo'cus of much :student dissatisfaction. Many called
for a broadening of the course of studies to include
more non-mathematical topics, but with little success
as noted above. Some other students were unhappy with the
'•studies and the mathematics of Cambridge for other
reasons. They objected to the style of the mathematics
required at fcartfbridge. Charles Babbage, for instance,
"acquired -a distaste for the routine studies" of Cambridge .
after finding that' the tutors could not help him in hi
particular mathematical studies and that, m o r e o v e r
■ J • they tried to discourage him from these studies by saying
that they "would not be asked in the Senate House."’''
It was within this context that the Analytical Society
was formed at Cambridge with the goal of promoting
analytics, a style of mathematics different, from that
used for teaching and examining at Cambridge.
The extensive mathematical education which one
could receive at Cambridge was wasted from the point of
1. Babbage (1864) 27.
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•view of the advancement of the field, for very few
continued their mathematical’ studies after graduation. '
There was no incentive at Cambridge, beyond the Senate
House or personal interest, to pursue higher mathematics
or to teach it. And few, if any, occupations in England
in the early nineteenth century'required a training in
mathematics or for that matter, a Cambridge degree.
Furthermore the advancement of knowledge was not a
part of the- purpose of a university: it existed to educate
•gentlemen. This last position, the ideology of a liberal
education; was the framework within which the stress on
mathematics at Cambridge was given meaning.
The ideal of a Cambridge- training throughout the
early nineteenth century was that of the pervasive, though
amorphous, liberal education.'. It was best defined as
the education of a gentleman. This involved the
cultivation of all of the faculties- intellectual, moral
and social-of an individual for his own sake. The I - means and content of such an education could be and
were interpreted in many ways, but during the early
nineteenth century it generally meant a classical or
mathematical training and certainly a non-professional
one. At Cambridge, perhaps because of the great regard
for Newton, the emphasis was on mathematics as a training
for the reasoning powers of the mind. Further, in accord
1. For some studies of the meaning of a liberal education in the eighteenth and nineteenth centuries, see Rothblatt '(1976) and McPherson^ (1959).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with the tenets of a liberal education, Cambridge educated
the student not simply through its curriculum but es-'
pecially by the environment it provided. Charles Babbage,
for 'instance, valued his stay at Cambridge because of
the availability of many books'and for "the still more
valuable opportunities it affords of acquiring friends."'"
Bather thiui the gaining of any expert knowledge, for a
. Cambridge education was not very rigorous, it was the
status of a Cambridge degree along with the gentlemanly
connections, or "friends", one could form there, that
undoubtedly helped in later life.
Aside from the University's social function, it
also had the potential'of playing an important intellec
tual role through its curriculum. Mathematics had such
an overwhelming position within that curriculum that ifs
study often appeared to be incompatible with the principles
of a liberal education. Yet the emphasis on mathematics
at Cambridge was justified by its essential role in
training the jnind. Doubts,,-however, arose not on this
position but on whether there was too great a stress on
mathematics or whether just any type of mathematics
was useful in training the mind. The mathematics of the
Cambridge reformers was to incite criticism of*itself from ■
this latter standpoint. The mathematics of Cambridge before
the transition was allied to the ideal of a liberal education.
1. Letter from Babbage to Herschel, Aug.10 1814; H.ms.R.S.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Any change in the mathematics, therefore, would require the abandoning
of a liberal education or at least a modified understanding of it. Mathematics enjoyed a privileged position within
the structure of Cambridge studies. For this it came
under much criticism in the reform-minded ear ly., nineteenth
century. Of particular significance in this criticism
were the efforts of many students with new expectations.
Mathematics at Cambridge found'- its meaning in education,
in the ideal of a liberal education. The content of
Cambridge mathematics was typical of English mathematics
at that time, as will be seen in the next chapter. With
the widespread lament about the decline . of mathematical
scienge in England in the early nineteenth century much
criticism was directed at Cambridge because of its repute
for mathematics. The traditional and institutionalized
position of Cambridge mathematics as well as its links
with a liberal education made any curriculum reform
difficult. But this position also provided a means for •
change. All that was required was control .over the
teaching and examining posts. Yet some motivation was
also needed. In a university which prided itself on
being the University of Newton, criticism of English
mathematics, or, synonymously at that time, Newtonian
mathematics, would undoubtedly play a large part in any
motivation of this sort. It is this theme of contemporary
opinion of the state of mathematics in England and its
relation to Cambridge which will next be reviewed.
)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill. The Decline of the Mathematical Sciences in England
and Their State at Cambridge (1790-1815)
The situation at Cambridge was an important part of
the context from which the Analytical Society and the
transition in Cambridge mathematics emerged. The other
significant factor in this context was the widespread
lament about the'"decline of the mathematical sciences'
in England in *the early nineteenth century. Historians
of mathematics and, in particular,, of developments in
mathematics in England, have agreed in viewing eighteenth-
and early nineteenth-century British mathematics as in a
state of stagnation, if not decline.""
And, with equal unanimity, their explanation of this
situation has centered on the mathematical influence of
Isaac Newton. Consider, for example, Morris Kline's view:
England's poor performance in view of its"great activity in the.seventeenth century may be surprising, but" the explanation is readily found.. The English mathematicians had not only isolated themselves personally from the Continentals as a consequence of the controversy between Newton and Leibniz, but also suffered by following the geometrical methods of Newton. The English settled down to study Newton instead of nature. Even in their analytical work they used Newton's notation for fluxions and fluents and refused to read anything written in the notation of Leibniz.' Moreover,' at Oxford and-Cambridge, no Jew or Dissenter from the Church of England could even be a student. By 1815 mathematics in England was at its last gasp and astronomy nearly so.l
1. See, for example, Kline (1972), Boyer (1959), Ball (1889), Koppelman (1971/72),and Dubbey (1963), (1964), and (1978).
2. Kline (1972) 622. For the last sentence see Herschel (1857) 577.
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Most historians of mathematics have assigned the British
bias for the fluxional notation and for geometrical
methods a s .the cause of the mathematical slump in
England. However, this explanation is surely somewhat
confined for it fails, for exampfe,' to1 explain the lack
of mathematical investigation in Britain along geometrical lines or using fluxional notation.f Indeed, Kline's references, in the above quotation, to isolation and to
higher education confirm the view that any explanation
of the decline must focus on social and cultural factors o and not wholly on mathematical reasons. Whatever were
the causes of the decline, it is clear that fluxions
and geometrical methods were more symptoms than causes.
By the very late eighteenth century many persons in Britain
who were concerned with the .mathematical sciences began
looking to the Continent and especially to France for
advanced knowledge. It is these persons who lamented
the decline in England. And contemporary opinion usually
appealed to a much wider range of causes than historians
have since done.
Unfortunately, little research has been done on the
alleged slump in British mathematics, probably because it
is commonly held that nothing happened.^ This chapter
1. See, for example, Ball (1889) 98 where, speaking of the English mathematical school of the latter half of the eighteenth century, he says, "Its history, therefore leads nowhere, and hence it is not necessary to discuss it at any great length."
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will review the issue of a decline during the period
1790 to 1815 through an examination of the chief math^
ematical contributors of the period, their works, and their * views on the state of the mathematical sciences if? England.
An important source exists1 for securing these views.
The early nineteenth.century was, in the words of John
0. Hayden, "the heyday of periodical reviewing" in England.'*'
And it is these British reviews and magazines that have
provided the chief source in this chapter for contemporary 2 comment on the state of the mathematical sciences.
1. Hayden (1968) 1.
2. The following is a list of the periodicals I have examined and the years for which they were checked: Quarterly Review (1809-1832), Westminster Review (1824-1830), Critical Review (1802-1813), Edinburgh Review (1802-1830), Gentleman's Magazine (1800-1820),. Monthly Review (1790-1828), Athenaeum (1819-1833), British Critic (1793-1832), Universal Magazine (1799-1815), Scotts Magazine/Edinburgh Magazine (1809-1820), British Review (1811-1825), Anti-Jacobin' Review (1798-1821), Blackwood's Edinburgh Magazine (1817-1820), Monthly Magazine (1796-1820), New Monthly Magazine (1814-1620) , European Magazine (1860-1820), London Magazine (1820-1829), Literary Panorama (1806- 1819), Eclectic Review (1805-1B31). Fortunately most of these reviews did not share in the Edinburgh Magazine's opinion: "We hold Mathematics'-to be a bore in political and literary reviews." Anon "Review of the Westminster Review No.7 July 1825" Edinburgh Magazine 96 (1826) 846.
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xxx.l The Decline of the Mathematical Sciences in England
(1790-1815)
Thomas Simpson, one of the very few illustrious English
mathematicians of the eighteenth century, noted, a few years
before his death in 1761, that foreign mathematicians were
pushing "their researches farther, in many particulars, than
Sir Isaac Newton and his Followers here, have done."1 He
claimed that this advance was due to this "diligent culti
vation of the modern analysis". This claim came as a
result of his defence of his use of the analytic method.
Simpson was convinced that the analytic method was essential
to advanced mathematical investigation even though it might
lack, compared to geometry, in neatness and rigor. For
Simpson, and throughout the period from 1790 to 1815,
'analytic' indicated a use of analysis, and in a mathematical 2 context, a use of modern analysis. The term 'analysis! C' referred to the Greek concept of analysis, and thus was an
"art of reasoning" whereby one proceeded "from the thing
sought as taken for granted, through its consequences, to
something that is really granted or known; in which sense
it is the reverse of synthesis or composition, in which we
lay that down first which was the last step of the anaiy-
1. P.J. Wallis "Simpson, TJgpmas" Dictionary of Scientific Biography 12 (1975) 444.
2. Compare the entries for' these terms in the various editions of the Encyclopaedia Britannica. (1797, 1817), Ree's New Cyclopaedia (1820), Charles Hutton's A Mathematical and Philosophical Dictionary (1795-17 96) and Peter Barlow's A New Mathematical and Philosophical Dictionary (1814).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 .
sis,...."1 Mathematical analysis was divided into ancient,
or geometrical, and modern. ^Geometrical analysis was
simply the use of analysis in geometry, whereas modern
analysis was an extension of the Greek concept and indicated 2 the method of solving problems by reducing them to equations.
As an example of the difference between . the analytic
and geometric approach to a problem consider the following
example taken from Daijiel Cresswell's An Elementary Treatise
on the Geometrical and Algebraical Investigation of Maxima
and Minima, being the substance of A Course of Lectures
Delivered Conformably to the Will of Lady Sadler (second
edition 1817, Cambridge). Early in the book Cresswell
gave a geometric proof of the theorem:
The greatest parallelogram which can be inscribed in a given triangle, so as to have the vertical angle of the triangle for one of its angles, is that which is formed by drawing two straight lines from the bisection of the base, each parallel to a side of the triangle.3
He proved this theorem in the following way. Let ABC be the
given.triangle. Bisect its base BC. From this point (K)
draw KL parallel to AC and KM. parallel to AB. Let D
be any other point in BC and DH and DE be drawn parallel to
AB and AC respectively. Then parallelogram AK is greater
1. Hutton (1796) 106.
2. J. Hintikka s U. Remes The Method of Analysis. Its Geometrical Origin and its General Significance. Boston Studies in the Philosophy of Science. 25 (1975) 106. F. Viete "Introduction to the AnalyticaT-Art” pp. 313-353 in Jacob Klein.1 s Greek Mathematical Thought and the Origin of Algebra (1968).
3. pp.17-18.
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Cresswell, by use of a previous result, then drew the
straight line FDG through D so that it was bisected at D.
The triangles BKL and KCM were equal, again by a previous
result. As-LK equals MC, and LK equals AM (Euclid, Book 1,
prop. 34), AM equals MC. And as the parallelogram AK
is double the triangle MKC (Euclid, Book 1, prop. 41), it
is equal to the sum of the triangles MKC, LBK; and is half
of the triangle ABC. Similarly the parallelogram AD is
half of the triangle AFG. By a previous theorem that
Cresswell had proved, the triangle ABC was greater than
AFG and therefore the parallelogram AK was greater than the
parallelogram AD.
Later in his book Cresswell approached the same
problem analytically, as an example of the algebraic method:
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To find the greatest parallelogram which can be inscribed in a given triangle, .so as to have the vertical angle of the triangle for i>ne of its angles.b
He took AFG to be the given triangle. Then the parallelogram
AD inscribed in it, with the vertical angle A for one of
its angles, was required to be the maximum.
A
Gr
From Euclid (Book 6 , prop. 23) it was Jcnown that equiangular
parallelograms had to one another the same ratio as the
rectangles contained by the. sides about equal angles in
each. Thus the parallelogram AD would be greatest when
the rectangle EA x AH, or AE x ED, was greatest. Letting
AF be denoted by a, FG by b, GA by c, and FD by y, then-
(Euclid, Book 4, prop. 6)
FG : GA :: FD : DE
or b : c :: y : DE
Thus DE = ^ . y .
Similarly DH = || x DG = ^ ' (b-y)‘
So 5^" x 5 " ■i-s to be
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It will be greatest when y.(b-y) is, thus by the calculus
b. y ' - 2y.y'=o b y = 2
Therefore the end result of this analytical manipulation
is that D must bisect FG if the parallelogram AD is to
be the maximum in the triangle AFG.
Aside from the difference in the geometrical and
algebraical language in the above example, it is interesting
to note that Cresswell dealt with this problem, geometrically,
using only results from the first book of Euclid, thereby
stressing the simplicity of the geometrical approach.
On the other hand, the geometrical "theorem" was strictly
synthetic; one knew the result that needed proving. Whereas
the algebraic' "example" was analytic; one had to find the
greatest parallelogram. The result y = ^ was no proof
for Cresswell. To be a proof it would have to be shown
that this value of y rendered the parallelogram a maximum.
Hence this illustration from Cresswell shows up well both
the stylistic and pedagogic differences between analytics
and synthetics: the use of synthetics for' proof and
analytics for discovery',, the 'blind1 manipulation of
symbols in analytics and the step-by-step deductive
reasoning in synthetics, and the rigor of synthetics com
pared to the unconvincing operations of analytics.
Modern analysis consisted of such branches of mathe
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matics as algebra, infinite series and fluxions. It ■ ■ l brought with itself such new and Often' unacceptable
, topics as complex numbers. Opposed to it was not simply
synthesis ,\but rather methods based on synthetic geometry
after the Euclidean paradigm. .Probably due to its great
success in the seventeenth and, especially, eighteenth
centuries, the analyfcic^method acquired a mechanical,
yet somewhat mystical reverence for its power,.for through
it an order was prescribed, following which, the mind,
. independent of all else, could easily attain the unknown.
Charles Hutton's view of modern analysis in 17 96 provides
an example of this reverence.
The modern analysis is a general instrument by , which the finest inventions and the greatest improvements have been made, in mathematics and philosophy, for near two centuries past. It furnishes the most perfect examples of the Amanner in which the art of reasoning shouldnbe .employed; it gives to the mind a wonderful skill for discovering things unknown, by means of a , small number that are given; and by employing, short and easy symbols for expressing ideas, it presents to the understanding things which otherwise would seem to lie above its sphere. By this means geometrical demonstrations may be greatly abridged: a long train of arguments, in which the mind cannot, without the greatest effort of attention, discover the connection of ideas, is converted into visible symbols; and the various operations which they require, are simply effected by the combination of those %> symbols.- And, what is still more extraordinarye by this artifice, a great number of truths are often expressed in one line only: instead of which, by following the ordinary way of explanation and demonstration, the same truths would occupy whole pages or volumes. And thus, by the bare contemplation of one line of calculation, we may understand in a short time whole sciences, which
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otherwise could hardly be comprehended in several years.1
Despite a widespread respect for analysis among British
mathematicians, it was Continental mathematicians, and
in particular Euler and Lagrange, who developed analytical
methods and replaced geometric arguments with analytic 2 ones. Of particular importance was Lagrange's attempt
to base the calculus on algebra. Because of this, the
English fluxional calculus, which was formulated in
terms of motion, though a branch of analysis, came to
be regarded as non-analytical. And with the great
advances in mathematical science on the Continent-in
the late eighteenth century, non-analytic methods came
to be identified with 'firitish mathematical inferiority.
Thus analytics and synthetics were alternative
styles of mathematics in the early nineteenth century.
They were distinguished by differences in methods, rigor
add uses. In tertns of the content of the mathematics,
analytics implied a purely algebraic approach, as
illustrated in some of the works of Lagrange. The want
of British work in analytical mathematics was to be seen
by those who wished to revive British mathematics as a
cause for the decline of that✓ mathematics. \
1. Hutton (1796) 106.
2. Kline (1972)' 614.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Many of the early critics, during the period
1790-1815, of the state of British mathematics were
Scottish, Perhaps this was due to the tradition of
learning there and the ties between Scotland and the
Continent. John Robison (1739-1805)/professor of natural
philosophy at Edinburgh University, had deplored, in his
influential Encyclopaedia Britannica .(3rd ed. 1797)
article "Physics", the decline of the taste for math
ematical sciences in Britain.^- He wrote that "there
has not appeared in Britain half a dozen, treatises worth
consulting for these last forty years*.,/ and was greatly
mortified that his countrymen had to look to foreign
'writers for developments in the Newtonian philosophy.
John Leslie (1766-1832), at about the same time as
Robison, also hoped to arouse his contemporaries by 2 remarking on the state of learning on the Continent.
His exhortations, in general, were allied with a certain
view of the nature of mathematics. Leslie, who had
studied under Robison, was named professor of mathematics
at Edinburgh in 1805. and its professor of natural philosophy
in 1819, succeeding John P l a y f h ^ in both positions.
Leslie's views have been regarded by Richard Olson as
1.. This passage also appeared in the fifth^edition of ~ the encyclopaedia (1817). I have not seen the fourth edition which was, however, mostly.a reprint of the third.
2. J. Leslie "Review of F. Callet's Tables Portatives de Logarithmes, &c." Monthly Review 21 (1796) 570-574. pp.570-571.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. .57
as typical of the Scottish Common Sense interpretation
of the foundations of mathematics.1 This philosophy
had its greatest impact at the University of Edinburgh
and together with the mode of education there tended
to favour geometric over analytic methods, a tendency' 2 which continued until almost 1840. As Olson stated
Something much stronger than.a mere passive cultural inertia maintained geometry at the center of Scottish education after the useful ness of applied algebra .was recognized. -If we consider the Scots1 pedagogical emphasis on geometrical studies in conjunction with the dissatisfaction wi:th analysis which arose out of their epistemological considerations, it is hardly surprizing that geometry maintained its supreme position in Scottish mathematics well into the nineteenth century.3
Leslie did camend analysts far their skill, activity
and achievements,‘and even used modern analysis him
self, yet had fundamental reservations about its uses,
or, more properly, its abuses. For analysts, he felt,
tended to be overly enthusiastic with their method to
the neglect of external observation and so to be prone
to loose and artificial reasoning and consequently to 4 error and defect. It is not surprising then that Leslie
1. Olson (1971) 38. See’also Olson (1975).
2. Olson (1975) 252.
3. Olson (1971) 44.
4. See, for example; Leslie':s reviews of "Laplace's 'On the Motions of Light &c.'" Edinburgh Review 15 (1810) 422-426, and "Delambre1s~De 1 *Arrthmetique des'Grecs" Edinburgh Review 18 (1811) 185-213.
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stressed geometry in education for its "tendency to
invigorate the whole of the intellectual powers, and to
lay a sure and solid foundation on which to erect future
superstructures".’'' Thus while Leslie might lament the
decline of British mathematics and view the neglect of
modern analysis as one of its elements, as may be seen
in his 1835 history of the mathematical and physical
sciences in the eighteenth century, he was not willing
to introduce analytic methods intp ordinary university
education nor to use them in his work without confining
hesitancy,' as illustrated by his rejection of negative 2 .and complex numbers. In his reaction to modern analysis,
or to put it another way, to the means, as many at that
time saw it, by which Continental mathematics had become
superior, Leslie's ambivalence may be regarded as a
good example of the difficulty which any reform of British
mathematics would face*
John Playfair (1748-1819), unlike Leslie, whole- 3 heartedly endorsed analytical mathematics. Playfair had
a long association with the University of•Edinburgh,
having been for a time a student there, then the professor
of mathematics from 1785 to 1805, and finally professor
of natural philosophy from 1805 until his death. He
1. Ibid. (Review of Delambre).
'2. Leslie (1842) 576-579. This is another edition of ’ his 1835 work.
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is perhaps best remembered today for his exposition
of James Hutton's ideas in the Illustrations of the
Huttonian Theory of the Earth (1802), or for his sub
stitute wording for Euclid's parallel axiom. But in
his day he- was also known for his propagation of
continental mathematics. He'diffused analytics in
Britain through some of his university lectures, some
of his mathematical papers, and especially through his
numerous reviews in the newly-founded (1802) , whiggish,
aggressive and very popular Edinburgh Review.^
According to Playfair's viewsanalytical methods
and discovery of truth were closely allied. Algebra
was a language "invented expressly for the purpose of
assisting the mind in the management of thought: this
is its'primary destination; and the business'uf commun
icating knowledge, which is principal with respect to
other languages, with respect to it, is secondary and 2 accidental." Playfair was very willing bo defend
analytic mathematics from geometers on the basis of its
1. For an acount of Playfair^feee J.B. Morrell "Professors Robison and Playfair, and the Theopholjia Gallica: Natural Philosophy, Religion and Politics in Edinburgh, 1789-1815" Notes & Rec. Roy. Soc. London 26 (1971) 43-63.
2.-. J. Playfair "Review.of BuSe's 'Memoire sur les ' Quarititds Imaqinaires'" Edinburgh Review 12 (1808) 306-318. p.306.
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power of -discovery. •
Whatever has served for the discovery of truth, has a character too sacred to be rashly thrown aside, or to be sacrificed to the fastidious taste of those who make truth welcome only when it wears a particular dress, and appears arrayed in the costume o f ,antiquity.1
While synthesis might very well convey truth, he felt
it could not impart methods of investigating truth nor
develop the powers of invention. This was left to
analysis, and i'n all "the\most general and difficult
speculations of the pure_Jnathematics, and in all the
most important branches of the mixt, it is the latter
[algebraic analysis] only that can be employed to ad- 2 vantage." In contrast to Leslie, then, Playfair stressed
the importance of analytical mathematics in education,
for only through it could the mind be led to an understanding
of the methods of investigation. Too much emphasis on
synthesis - geometry - would restrain the natural ex
pansion of the studentts mind and so lead to disgust
and to "the extinction of the ardour that might have
enabled him to attempt investigation himself, and to
acquire both the power and the taste of discovery." 3
1. Ibid. 317.
2. J. Playfair "Review of Bishop Horsley's edition of the Elements, &c." Edinburgh Review 4 (1804) 257-272. p.270.
3. J. Playfair Review of LaPlace's Traite de Michanique Celeste" Edinburqh Review 11 (1808) 249-284. pp.283-284. ■ „
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Playfair's great concern with research and analytic
mathematics could not fail to play an important role
in his comments on the decline of the mathematical
sciences. As early as about 1782 he;.reve'aled his views
on this subject in a letter concerning his meeting with
the Astronomer Royal, Nevil Maskelyne (1732-1811),
on a trip to London.
He is an excellent observer, and a good mathematician. He is much attached to the study of geometry, and I am not sure that he is very deeply versed in the late discoveries of the foreign algebraists. Indeed, this, seems to be somewhat the case with all the English mathematicians; they despise their brethren on the Continent, and think that everything in science must be. for ever confined to the country that produced Sir Isaac Newton. Dr. Maskelyne, however, is more than almost any of them superior to this prejudice.1
Some years later he expanded his ideas on the inferiority
of British mathematics. He saw that inferiority wit
nessed by the fact that in the last half-century no
British author could be found among those who had con
tributed to the great improvements in mathematics, which
he saw as being analytical trigonometry, the methods of
partial differences and of variations, and developments 2 in methods of integration. He acknowledged a widespread
diffusion of mathematical knowleftge in England yet
also pointed out the great neglect of its higher ^branches,
which could only be found in Continental writings.
1. J. Playfair The Works of John Playfair 1_ (1822) lxxviii.
2. See page 60, footnote 3, pp.250-252,- 279-284.
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Hence
... a man may be perfectly acquainted with every thing on mathematical learning that has been written in this country, and may yet find himself stopped at the very first page of the works of Euler or D'Alembert. He will be stopped, not from the difference of the fluxionary notation, (a difficulty easily overcome), nor from the obscurity of these authors, who are both very clear writers, especially the first of them, but from want of knowing the principles and the methods which they take for granted as known to every mathematical reader.1
The attachment to synthetical methods had often
been seen as the cause of this inferiority, and Playfair 2 himself had on occasion held such a view. But Playfair
now moved fron the form of the mathematics to its social
• , . underpinning. The true cause, he argued, lay in the state
of certain public institutions, namely, the English
universities and the.Royal Society. At Oxford all
mathematics, but the elements of geometry, were neglected.^
At Cambridge, although there whs a high level of mathematical
learning, the mode of acquiring it, which was synthetic,
led naturally in Playfair1 s'thought to' a loathing of
mathematics. And the complaint directed to the Royal.
Society was that it did not offer "sufficient encourage- 4 ment for mathematical learning." -This last point takes
on a deeper meaning when it is noted that in 1783-84
1. Ibid. 281
2. See, for example, page 60* footnote 2, p. 261
3. His comments on Oxford led Edward Copleston to make a number of replies, based on the nature of a liberal ed- .ucation, which in turn led to a counter-reply in the Edinburgh Review 16 (1810) 158-187.
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a large number of mathematicians, including Bishop
Horsley, Nevil Maskelyne, Charles Hutton, Samuel Vince,a
Baron Maseres and James Glenie, seceded from the Royal
Society, at that time under the presidency of Joseph
Banks:, charging that it neglected mathematics for the
natural sciences.1 A few years later-, in 1810, Playfair
elaborated on what he saw as "sufficient encouragement"
when he praised the Paris Royal Academy of Sciences for
its promotion of the mathematical sciences by "small
pensions and great honours, bestowed on a few men for
devoting themselves exclusively to works of invention 2 and discovery." He viewed the English inadequacy in
the mathematical sciences to be a result of: the English
public's self-defeating "mercantile prejudices" which
were always prepared to demand an immediate justification
for science in terms of use. Playfair, therefore, had
moved beyond viewing the stagnation of British mathematics
as simply due to an addiction to geometrical methods.
With his concern for the advancement of the field, he
saw a greater cause in the want of institutionalized
Seq C.R. Weld History of the Royal Society (1848) 261 and Taylor (1966) TS-T9T
m ] of the World as translated by John Pond" Edinburqh Review 15 (1810) 396-417. p.398. =------
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encouragement for' the advancement of the mathematical sciences!.1-
There were also other Scots writing for the Edinburgh
Review who shared Playfair's concern over the decline of the
mathematical sciences in England. Henry Brougham (1778-1868),
a close friend of Playfair and later a famous lawyer and
Whig politician, also had a high regard for analysis.1
Another instance was the editor of the review, Francis
Jeffrey (1773-1850). He saw the chief cause of the
* y "singular decay of mathematical science in England" as the
great progress of knowledge, which consequently did not
permit a man of "liberal,curiosity" to "go beyond the 2 first elements of mathematical learning." All of these
writers did much to publicize the opinion that British
mathematical science was backward. And there were other
^Scottish critics not connected with the University of
Edinburgh nor with the Edinburgh Review who deplored
the state of British mathematics. One example is that
^of the little-known mathematician William Spence (1777-
1815). In the preface to his An Essay on the Theory of the
various Orders of Logarithmic Transcendents; &c. (1809) ,
1. H. Brougham (probably) "Review of Wallace’s "A New Method of Expressing the Coefficients of the development of the Algebraic F o rmula'Edinburgh Review 1 (1803) 506-510.p. 510. In later life he co-autdjpred (with E.J. Routh) the Analytical V ie y / of Sir Isaac Newton's Principia (1855). :------—
-2. F. Jeffrey "Review pf D. Stewart's Philosophical Essays" Edinburgh Review 17 (1810) 167-211. pp.168-169.
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Spence criticized the British style of hindering mathematical
analysis with geometrical and mechanical applications.
He believed that students had to study general methods
and operations and' not be taught analysis by means of
its applications. On the Continent mathematical analysis
was studied as an independent subject; the result, he
felt, was the superiority of foreign mathematics.'*'
In Scotland, evidently, there was a widespread concern
among those interested in the mathematical sciences about
their relative stagnation. Statements about the decline
reveal the differences between analytics and synthetics,
as well as the relation of these differences to the
advancement of mathematics in England. However, the respon
sibility for the backwardness of English mathematics was
not seen as simply a result of those differences. The
social underpinnings of mathematics were appealed to. In
.England, as will now be examined-,- there was a similar anxiety/
as in Scotland about the state of the mathematical
sciences. But, in contrast, there were many individuals
opposed to any change in the style of mathematics, or at
least with strong reservations about the nature of any
reform.
John Toplis (1774/75-1857) graduated from Cambridge
as eleventh wrangler in 1801. For much of the'period
1. Spence (1819) xiii-xiv.
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1790-1815 he was head master of the Nottingliailf'Free Grammar
School (1806-1819), and for much of his life he was rector
of South Walsham, Norfolk (1824-1857). On October 13,'
1804 he sent a letter to the Philosophical Magazine on the
decline of the mathematical sciences.1 He complained:
We seem, as a nation, for this hast half century, _ to be sunk into a great degree of supineness with respect to the sciences, regardless of our former fame. The generality of the papers in the Philosophical Transactions are no longer of that importance they were formerly. We have long ceased to study those sciences int/which we took the-lead and excelled, and are content to follow, at a very humble distance, the steps of the philosophers of the continent, in those which they have - in a manner discovered and made plain by their glorious exertions.2
Toplis wondered if the cause of this neglect was due to a
contentment with the glory gained in past achievements.
And he saw the reasons for this decline to be the lack
of patronage for science in England (as compared to the
Continent), the overemphasis on classics in education,
the current fashion of studying such less noble subjects
as natural history and chemistry to the neglect of the
mathematical sciences, and the obstinacy with which
English mathematicians clung to geometrical methods. In
support of the last reason, while he allowed that geometry
1. J. Toplis "On the Decline of Mathematical Studies, and the Sciences dependent upon them" Philosophical Magazine 20 (1805) 25-31.
2. Ibid. 26.
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was most proper for training the reasoning faculties,
he displayed his contempt for those who persisted in
using the geometrical method; for
...it is confined in its application, feeble, tedious, and almost impracticable in its powers of discovery in natural philosophy. But what is called analysis possesses boundless and almost supernatural powers in its application to science; and the discoveries made by it in natural philosophy are of so surprising a nature, that to pretend to despise it, and obstinately to grovel amongst a few properties of surfaces and solid bodies, denotes a very narrow and prejudiced mind.-j
Ten years later Toplis was to publish his translation
of part of LaBlace's M^canique celeste still in the hope of
promoting his favourite science and the work of the
- 2 Continental analysts.
Few English mathematicians were as enthusiastic about
analysis or'about the Continental mode of analysis as
Toplis or many of the Scots were, 'william Wales (cl734-
1798), mathematical master of Christ's Hospital, was
convinced of the power of modern analytics but also of
1. Ibid. 28-29.'
2. J. Toplis A Treatise upon Analytical Mechanics,- Being the first Eook of the Mechanrque Celeste of P.S. Laplace (18141 Nottingham. More space m this booE was given to explanatory footnotes than to the'translation itself, indicating, as one reviewer noted (see page 76 , footnote 2) the backwardness of British mathematics.
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its "inelegance and obscurity". And he.did not believe
that algebra could replace geometry in synthetical processes
because of this obscurity and clumsiness.^ An even
stronger criticism' of analytics came from an anonymous
reviewer of Laplace m 1804. 2 He considered that the use
of pure analysis involved an exclusion of reality from the
consideration of reality. This absence would eventually
lead to paralogism and absurdity. snce he warned against
a fascination with Laplace's splendid example:
Let there be as little deviation as possible from the geometrical method: for, since motion includes the conception of lines with their various qualities of magnitudes and position, we thus keep the subject of discussion closely in view: or, to conclude in the words of a celebrated writer, - 'Let the accomplished mathematician push forward our knowledge by the employment of the symbolical analysis; but let him be followed as closely as possible by the geometer, that we may nob be robbed of ideas, and that the student may have light to direct his steps.'3 • The opinion that the reasoning faculties of the mind
were best enhanced by geometry and geometrical methods seems
to have been .widely held among English mathematicians
as was the view that'analytic methods were but affectation.
William Saint (fl.1811) , third mathematical '.assistant at
the Royal Military Academy, Woolwich, wrote of the dis-
1. W. Wales "Review of J... Williamson1 s The ^Elements of Euclid vol. 2" Monthly Review. 3 (17901 253-258, ancT” " Reviewof T. Newton ‘ s A Short~Treatise on the Conic Sections" Monthly Review 16 (1795) 389-391.
2. Anon "Review of Laplace's A Treatise on Celestial Mechanics Vol. 3" Critical Review 1 T1804) 531-511.
3. Ibid. 540-541.
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satisfaction the mind received from the operations of
algebra.1 And a reviewer in the British Review noted. 2 the repulsive form and obscurity of analytics. Others
expressed their disaffection with Continental analytic
methods by supporting the Newtonian tradition of fluxions.
William Dickson (fl.1800) "decidedly" preferred the
fluxionary theory (based0 on the concept of motion) as well
as its notation.3 John Hellins (died 1827), vicar of
Potterspury, also preferred "the Newtonian idea of the r generation of mathematical quantitites by motion, to
Leibnitz's conceit of an apposition of an infinite'sjumber 4 of infinitely small parts". And'Olinthus Gregory (1774-
1841), a prominent Dissenter, a founder of London University
and a teacher of mathematics at the Royal Military Academy,
Woolwich (1803-1838)-, continued to offer the calculus in .
its fluxional form in various editions of Hutton's Course
1. W. Saint "John Frensham" Gentleman's Magazine 81 (1811) 11-15- pp.13-14.
2. Anon "Review of Bridge's Six Lectures on the Elements of Plane Trigonometry" BritisE~Review 1 (T5lT) 105-112. pp. 106-108.
3. W. Dickson "A Translation o% Carnot's Reflections on ... Calculus" Philosophical Magazine 8 (1800} 222-240, 335-352, 9 (1801) 39-56. pp.222-223.
.4. J. Hellins "Review of Agnesi'^s Analytical Institutions (Colson's translation)" British Critic i 3 (1804) 143-156, 2£ (1804) 563-660, 25 (1804) lTT-147. p.654.
- ’ \
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of Mathematics despite attempts to replace it by the dif
ferential calculus.^-
The period 1790-1815 witnessed the French Revolution
and the Napoleonic Wars, and so it is not surprising to ,
find strong emotions such as a dislike or fear of anything
French associated -with the defence of British mathematics.
Thus Dickson wrote of the partiality of French mathematicians '2 and of their neglect of British colleagues. Hellins 5
attacked "the arrogant claims to superiority in mathematics
and philosophy, lately made by the Infidels and Atheists
in this island, as well as on the continent".3 The 'Tory
Quarterly Review, set up as rival to the Whig Edinburgh
Review in 1809, criticized the French endeavotTr to rob
Newton of the honour of being the inventor of the fluxional
calculus, "a principle which they have uniformly pursued -" 4 with regard to English men of science." And, as one last
example of the distrust of anything French, the Eclectic
:------=-----1 ----- ■ ■' ' r~\ 1. [T.T. Wilkinson] "English Matheraatical^jiiterature" Westminster Review 55 (1851) 70-83. Py?8 . Gregory had also complained of the neglect or mathematics by English natural philosophers in his Treatise on Mechanics (1806)' v. _ ^ / 2. See page 69 , footnote 3. p.40. j
3. J. Hellins "Review of Horsley's Elemental Treatises" British Critic 21 (1803) 272-284. p.272.
4. G. D ’Oyly or J. Ireland "Review of Dealtry's Fluxions" Quarterly Review 5_ (1811) 340-352. pp.340-341.
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Review noted
r ' ... that the writings of some eminent French mathematicians abound in infidel principles. Our elder men of science, we hope, are for the most part of too sober a cast to be injured by these priniples; but we tremble for the fate of the young....1
As a final instance of the English preference for
geometrical methods despite an acknowledgement of the in
feriority of British mathematics, consider the views of
the famous English natural philosopher, Thomas Young
(177^-1829). In the summer of 1798, while at Cambridge,
he had noted the inferiority of British mathematics.
In July he had written
I am ashamed to find how much the foreign math ematicians for these forty years have surpassed the English in the higher branches of the sciences. Euler, Bernouilli and d 1Alembert -have given solutions of problems which have scarcely occurred to us in this country.2-
Yet instead of embracing continental methods he endorsed
geometrical methods and attacked analytics.
... the moderns have Very frequently neglected the more essential, for frivolous and'superficial advantages. To say nothing of the needless incumbrances of new methods of variations, of combinatorial analyses, and of many other similar innovations, the strong inclination which has been shown, especially on the continent, to prefer the algebraical to the geometrical form of representation, is a sufficient proof, that instead of endeavouring to strengthen and enlighten the reasoning faculties, by accustoming them to such a
1. Anon "Review of cBonnycastle1 s Trigonometry" Eclectic Review £ (1808) 53-59. p.59
2. A . Wood Thomas Young , Natural Philosopher 1773-—1829 (1954) pp.65-66. See also G. Peacock Life of Thomas Young (1855) p.127.
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consecutive train of argument as can be fully con ceived by the mind, and represented with all its links by the recollection, they have only been desirous of sparing themselves as much as possible the pains of thought and labour, by a kind of mechanical abridgment, which at least'jonly serves the office of a book of tables in facilitating jcomputatians,. but which very often fails even of this end, and is, at tlje same time, the 1 most circuitous and the least intelligible.
And so, in common with other defenders of British
mathematics, he criticized the confusion, absurdities,
suspension of judgement and other defects which analytics
led to. Young did not stop with a criticism of modern
analysis though. In answer to Playfair's contention
that English public institutions were the cause of the
decline of British mathematics, he replied that the
principal object of the universities was not the advancement
of knowledge but its diffusion. Young neglected the issue
of governmental encouragement of science. However an idea
of his views on this issue is obtainable. For Young's" bio
grapher, George Peacock., noted that some of Young's actions
were based on the principle that science should be independent-
. 3 of the patronage of the government. One might reasonably
enquire, that, what Young saw as the reason for the decline
1. - T. Young "An Essay on Cycloidal Curves &c." British Magazine 1_ (1800) . I have consulted the reprint in his Lectures on Natural Philosoohv 2 (1807). see p.555. — 1------— _
2. T. Young "Review of Mgmoires ... de la Societe d'Arcueil vols. 1 & 2" Quarterly Review 3 (HfloT 4?2-4Sl. '
3. See page 71, footnote 2, Peacock (1855) p.476.
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of the mathematical sciences. And the only answer
Young offered was that there are "occasional fluctuations
in the scientific pursuits" of the individuals-in Great
Britain.1
' Clearly there was a resistance among many English
mathematicians to analytic methods in spite of their
recognition of the inferiority of British mathematical
science. This resistance involved an attachment and
preference for synthetic mathematics with a feeling
that analytics were wanting in rigor. The other aspect
of the lament, the social, support of mathematics, was
largely ignored by the supporters of synthetics. But,
as in the case of Thomas Young, it appears that their
position would have been one of individualism, that is-
that matheipatics should remain independent of institutional
patronage. The resistance also undoubtedly reflected
a certain amount of pride in past and present achievements,
nationalism and opposition to change.
■In spite -of the resistance to analytics and to'a
new relationship between mathematics and society, by
about 1815 many British mathematicians were
1. T. Young "Review of Laplace's Theorie de 1'Action Capillaire" Quarterly Review I (1809! 107-ll2. p. 108.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 familiarizing themselves with the' Continental mathematics.
It appears that the great advances of.. the -Fr.ench mathe- » • >> maticians were beginning to outweigh any criticism of
analytics, especially for those concerned with'developing
mathematics. In a short time the analytical movement was
to gain much momentum even though the wish for public
encouragement' for mathematics was not then to be realized.
J^mes .Ivory (1765-1842) showed, in his nearly mathe
matical work in the 1790s his understanding and adoption
of Continental mathematics.^- Another Scot, William
Wallace (1768-1843), had begun -to study French mathematics 2 about 1793. Both Wallace, in 1803, and Ivory, in 1804,
joined Thomas Leybourn (1770-1840) on 'the teaching staff
of the Royal Military'College, which at that time was at-
Great Marlow. Leybourn was editor of the very respectable
periodical The Mathematical Repository to which Wallace
and Ivory contributed. In the volume for 1809 (which
implies the matter is date 1806), Ivory.and Wallace both
used for the first time the Continental differential
notationj-instead of the British fluxional notation.
Ivory went on to be one of the foremost mathematicians
in England, specializing in the application of analysis
1. J. Ivory "A New Series for the Rectification of the Ellipsis, &c. read Nov. 7, 1796" Transactions. Royal Society of Edinburgh. £ (1798) 177-190.
2. Anon "Wm. Wallace" Monthly Notices. Royal Astronomical Society. £ (1845) 31-41. p.34. For some other Scots who were studying Continental mathematics at this time see Proceedings. Royal Society of Edinburgh 7_ (1871-72) 544, 9 footnote. ’
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Wallace wrote the article ''Fluxions" for the fourth
edition of the Encyclopaedia Britannica (1816) ,, It was
a vast improvement over the previously existing article.
In it he wrote:
We are sorry, however, to observe, that there is no work in the English language that exhibits a complete view of the theory of fluxions, with all the improvements that have been made upon it - to the present time. We cannot at present acquire any tolerable acquaintance with the subject, without consulting the writings of the foreign mathematicians.!
Wallace continued his diffusion of Continental mathematics.
by writing the article "Fluxions” (1815) for the Edinburgh
Encyclopedia. The article developed the-calculus along
Continental lines arid used the differential notation.
Wallace later succeeded John Leslie as professor of
mathematics at the University of Edinburgh (1819-1838).
William Spence, whose views on the study of mathematics
were mentioned above, also published work using Continental
methods before his early death in 1815. Thomas Knight
(fl.1809) was another who showed his mastery of foreign
mathematics. Bartholomew Lloyd (1772-1837) had in 1796
"meditated a revolution" in the mathematical courses of 2 Dublin University. After becoming professor of mathematics
there, he began to reform its mathematical studies by
1. W. Wallace "Fluxions" Ency. Brit. 8_ (1817) 697-778. p.700.
2. Anon "Bartholomew Lloyd" Gentleman's Magazine 9 (1796) 209.
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introducing Continental analytical methods. Another
Dublin University professor, John Brinkley (1763-1835),
the professor of astronomy, also "contributed materially
to the progress of the study of the Continental Mathematics
in the United Kingdom" at this time.'*' So there is no
doubt that by about 1815 Continental mathematics was being k increasingly adopted by British mathematicians.
Intriguingly the cries of decline of English mathe
matics and mathematical science did not abate, although
they now no longer appealed to the neglect of analytics
as a cause. A Monthly reviewer of Toplis's translation. .
of Laplace "painfully" admitted the lack of improvement
in the mathematical sciences, in particular in their 2 analytical branches, in England.. But he chose to point
to the little encouragement that publications in higher
mathematics found in Britain rather than to any intrinsic
mathematical reason. And he called upon the numerous
associations for the encouragement of the arts and sciences
in London to promote.mathematics.
At the same time, October 1815, Thomas Thomson
(1773-1852) published in his journal Annals of Philosophy
1. Forbes (1852) 864, footnote. Brinkley first used the ' differential notation in an article'in the Transactions. Royal Irish Academy. 1_3 (1818) 53-61. Read April 1817.
2. Anon "Review of Tcplis's A Treatise &c." Monthly Review 78 (1815) 211-213.
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his own views on the causes of the.inferiority of British
mathematics.^ He too disregarded the issue of analytical
methods, and discounted Playfair's contention about the
mode of education at Cambridge. Rather, Thomson viewed
the exclusive confinement to classics in education and,
especially, the absence of government support for mathe
maticians and for publishing mathematical works as the
chief causes.2
In extension of Thomson's remarks, a contributor
to the Annals wrote in February 1816 that after comparing
English and French mathematical publications he found
it
... impossible to deny that the mathematical sciences in France have been carried to an extent never before known, while in England they have remained in a state of almost total stagnation for nearly half a century.3
This writer also omitted any discussion of the type of
mathematics to be pursued. His "principal preventing^
causes to our progress in mathematics" - financial impos
sibility of publishing higher mathematics treatises, the
lack of stimuli such as prizes and pensions, and the neglect '
of ±he mathematical sciences by the Royal Society - were
1. T. Thomson "Review of Wainewright's Literary and Scien tific Pursuits &c." Annals of Philosophy (1815) 294-304.
2. , Ibid. 299-300.
3. .'B.' (perhaps Peter Barlow) "Observations on the Pre sent State of the Mathematical Sciences in Great Britain" Annals of Philosophy 1_ (1816) 89— 98. p.93.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. manifestations of the want of protection and encouragement
in this area. This deficiency revealed itself, he felt,
in the few alumni of Cambridge who pursued mathematical
researches, although the real defect there lay in the
superficial stimulus.to learning.1 Even in 1809, an
anonymous critic, in the Eclectic Review, had accented
social causes and neglected mathematical ones, in declaring
the chief causes for British mathematical inferiority
as being an undervaluation in England of the profession
of mathematics and the superficial mode of learning
mathematics by cramming for examinations, as was the 2 practice in the principal English educational institutions.
Thus the complaints about the state of English mathematics
and mathematical science persisted, now maintained by
a complaint of the lack of public encouragement. This
lament was a reflection of the feeling that in order for
English mathematics to prosper it had to be treated as a
profession. And these cries and feelings were to grow
in the 1820s into a general feeling that English science
had declined, and to give rise to a view of the scientist
as a professional.
Hence, there was a very widespread recognition in
Britain in the early nineteenth century of the inferiority
of British mathematics to that of the Continent. However
1. Ibid. 96.
•2. Anon "Review of Spence's An Essay &c.“ Eclectic Review 5 (1809) 1091-1103. pp.1098-1100.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 , this sitijation may have originated, the outcry^-h the
period 1790 to 1815 seems to have evolved from an aware
ness of the progress of Continental, and especially French,
mathematics. And so the revivers urged the adoption
of analytics. But the stress on analytics was but one
element of the revival scheme. They also criticized the
lack of public encouragement for research in the mathe
matical sciences; in short, they wished mathematics to
be treated as a profession. Those who most decried
British inferiority were enthusiasts for analytics and
anxious for active mathematical research. Those
who defended geometrical methods were content to stress
the advantages of geometry in training the reasoning
powers of the mind and in ensuring that truth was attained
in a clear and rigorous manner. By about 1815 the analytical
movement appears to have gained the upper hand in the de
bate over analytics, as is illustrated by'the replacement
of the Newtonian fluxional calculus with the Continental
differential calculus.- A revolution had occurred in British
mathematics.
III.2. The State of the Mathematical Sciences at Cambridge
(1790-1815)
% The last chapter examined the importance of mathematics
in the curriculum at the University, of Cambridge. Because
of its fame for mathematics, the University attracted
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 some criticism from those concerned with the state of
English'mathematics both for the type of mathematics
taught there and for the.way in which it was taught. This
section will consider1 the mathematics studied at Cambridge
and the attitudes there towards analytics. It is quite
clear that synthetics was tightly bound to the. ideal
of a liberal education. Synthetics allowed a justification
of the important role of mathematics at Cambridge; and a
gentleman's knowledge of mathematical science could be
attained through synthetics. In particular the work and
opinions of Robert Woodhouse, one of the earliest and,
probably, most influential propagators of Continental
mathematics in Britain, will be reviewed:.inithis
section.
Robert Woodhouse was born in Norwich on Aprilx28,
1773.^ He attended Caius College, Cambridge, graduating
as senior wrangler and first Smith's Prizeman in 1795.
He became a fellow of Caius in l'W8-'SJrd-r~until his
death on .December 23, 1827, occupied various, college
and university positions. Woodhouse became ax^ellow of
£he Royal Society in 1802. He was the only candidate
for the post of Lucasian Professor in 1820 - apparently
1. For biographical details consult “the articles on Woodhouse in the Dictionary of Scientific Biography, Dictionary of National Biography, Penny Cyclopaedia, Alumni CantaEriqiensis and JohnVenn1s Biographical History of Gonville and Caius College. Vol.2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81
no one else thought they had a chance against him - /but
held this chair only until 1822 when he succeeded Samuel
Vince as Plumian Professor. After 1824 the.Plumian
Professorship included superintending the new Cambridge
Observatory. He married, as so many>)other Cambridge
fellows did in their later life, in 1823 and had one
child, Robert.
As mentioned above, Woodhouse is important in this
study because of his promotion, through his books., of
Continental mathematics. Yet there is another important
way in which he diffused his ideas, which has been neglected
by historians. Between the years 1798 and 1812 Woodhouse
was th'e chief reviewer of mathematical works for the
popular Monthly Review.1 in this period he contributed
at least one review to each of the forty-three volumes
that appeared, for a total of 303 reviews or abstracts.
Not all of these reviews concerned the mathematical
sciences, but most did, and these provide an important
source for ascertaining his views on mathematics.
In a review of Samuel Vince's A Complete System of
Astronomy (1797) i Woodhouse bitterly portrayed the state
of British mathematics by pointing' outS that the name of
Newton " r \
.... is pronounced by us with a kind of rapturous enthusiasm; and in thinking of him we indulge the
1. Nangle (1955).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 feelings, and exultati.on of national pride; yet in France has been made the most just estimate of ~ 'his merit, and the noblest monument has been erected to his memory. The geometricians of the continent have done more to perpetuate his fame, than the pen of Pemberton, or the chissel of Roubilliac. - The rational and calm appreciation * of genius, by men of science, is of more weight than the high-sounding panegyric of those who know that much has been done, yet have no distinct notion of what hare been done.l
Woodhouse was concerned that this adherence to Newton had
resulted in a blind acceptande of^-the principles and
notation of the fluxional calculus. Woodhouse's
attention, as illustrated both'in his criticism of others'
work and in his own publications, was focused throughout
his life oh two points: a high regard for analytics
and a concern with the principles, or foundations, of a
subject. And it is these two points which are manifested
in his criticism and development of the calculus. His very
first review for the Monthly Bfrvjew in 17 98 attacked the
use of motion in the establishment of fluxions even thotigh
"it may appear a species of mathematical heresy, and a 2 want of proper zeal for the honour of our countrymen".
And Woodhouse called for a logical and rigorous explanation
of the principles of fluxions; a call which he himself
1. Woodhouse "Review of Vince's Astronomy vol. 1" Monthly Review 27 (1798)'121-131. p. 1257"
2. Woodhouse "Review of Hutton's Dictionary"'Monthly Review 25 (1798) 184-201, 364-383. p.194.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 would respond to in 1803. In many reviews before 1803,
though, he rejected basing fluxions on motion and even
objected to the fluxional notation.1 Here again he
felt exposed to the charge of "antipathy against' every
thing of English invention" yet he believed that
An English mathematician, if would judge im partially, must not suffer himself to be deluded , by the facility which habit has given hin^fcof ./ ' writing and understanding the fluxionary notation; he must divest himself of ,national prejudice; and he must not imagine that he basely resigns Newton's claim to the invention of fluxions, because he quits its notation' as incommodious.2'
If a concern over principles was Woodhouse's inspir
ation for rejecting fluxions, it was through analytics
that he was to put the calculus on a firm foundation.
Woodhouse's work on the calculus will be discussed a
little further on, but first his views on the "analytic
art" will be examined. Woodhouse saw the merits of
analytics lying in its power for discovery, its.abridge
ment of time and labour and in. its ability to‘express tf
general results.1 But he^also saw' its def^c^s^ and why
there was much dispute over analytics.
■■ ' . 1'. See, for example "Review of: Lacroix's Traitfe du Calcul Differential sfcf" oMonthly Review (1800731 493-505, 32 4 *5-49J ' ■ 2. Ibid. .32 ^3-4.9-S-: - is’ "' 3. Compare such ji{q®ks of .Wdodhouse' as "Review of Playfair's Geometry*!’ MgnOT$fy Reyiew 26 (1798) .154-165, "Review of Condll'lefc' s bflnquhtfe of Calculation" Monthly. Review 30 (1800) 506-51-2,% and "On- th^,Independence of the analyticalnirid geometrical-.Me-thods of Investigation &c'." Phil. Traris.692 (1802)" 85t-125.. « .. '
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 The operations of algebra are mechanical; various and intricate combinations of quantitites are produced; and many authors, not attentive to the circumstances under which they were.obtained, have given either obscure, imperfect, or perverse explanations of the principles and methods of algebra. Certain properties have been assigned to quantities as inherent and essential, which depend solely on an arbitrary notation. The plain and obvious meanings of certain formulas have been neglected, to seek for latent truths or fanciful analogies. Hence, in many treatises, the science is obscure, perplexed, and mysterious.1
’ Woodhouse felt that these defects and misuses of
analytics could be cleared away, and then the superiority
of the analytic method ovei-xthe geometric- would be clear,
especially in the realm of abstruse and intricate research.
Correspondingly, Woodhouse regarded the use of geometric
methods in research as .akin to amateurishness. For ex
ample, in a review in 1801 he praised Vinceis -work on
physical astronomy because of the few English works on
the subject, yet pointed out the necessary tediousness
and intricacy in using’ the synthetic, or geometric, 2 method. And this criticism was made even more strongly
in reviews of Abraham Robertsofi and John Hellins.
. In.the construction of his articles, we imagine that Mr. Robertson wished to demonstrate every thing more Geometrico, since otherwise that which is diffused over five pages might have been comprised in two. We are not averse to the con
1. Ibid. (Review of Condillac) 506.
2. Woodhouse "Review of Vince's Astronomy vol. 2" .Monthly Review 35 (1801) 72-8TI p.81.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sumption of ink and paper, when perspicuity and distinctness are to be the result: but, as Mr. R. was evidently writing- to mathematicians of toler able growth and manhood, who must, for the compre hension of the latter part of his paper, be well acquainted with the ordinary fluxionary processes, to such undoubtedly he would have bejn more intell igible if he had been more succinct.
Finally, in spite of his support for analytics,
Woodhouse was anxious to avoid the extremes of those who
would rather ''operate" than .know, "who look more to the 2 truth of result than to justness of'inference''. Such
persons neglected evidence and rigour in demonstration.
This view probably led to his reputation at Cambridge as
one who disliked "ultra-analysts" .3 Nevertheless, Woodhouse'
was in all his works a zealous promoter of the analytic
method, and/ hence, of Continental mathematics.
In 1803, Woodhouse1s The Principles of Analytical
Calculation was published by the Cambridge University
Press. This work not only expressed in a single argument his various criticisns of the foundations of the calculus and his views of the calculus and his views of analytics, so often previously stated in the Monthly Review and in his papers in the Philosophical Transactions but was the first British work to introduce Continental approaches
1. Woodhouse "Review of Robertson's Equinoxes' Phil. Trans■ (1807) " "Mi______(1807) 6-16, p.12. See also his reviews of Hellins in the Monthly Review 40 (1803) 418-419 and 67 (1812) 259-261. .
2. Woodhouse "Review of LaCroix's Traite des Differences et des Series &c." Monthly Review 36 (1802") 4ab -tui. p.500.
3. Todhunter (1876) 2 29-30.
with permission of the copyright owner. Further reproduction prohibited without permission. to the calculus and to use the differential notation. The Principles was not so much a polemic as'
an appeal to the reader to accept Woodhouse1s foundation
on the basis of what he felt was-natural, commodious,
concise, perspicuous, and a natural and logical order of
ideas.
In the Principles Woodhouse1s primary -concern was to
establish a rigorous, deductive foundation for the fluxional
or differential calculus.3- Woodhouse viewed past attempts to form a basis for analytical calculation"_5--T e ^ o u r modern calculus as ,not absolutely erroneous, but as en-
compassing methods which were neither natural nor commodious. 2 * Using Berkeley's argument of "shifting the hypothesis" and
his own views orr^the signification of algebraic expressions
. and on the meaning of the equality sign," = ", he rejected. 3
the theory of.limits, the fluxionai calculus and Lagrange's
basis for the calculus. He gave up the method of fluxions
because it was based on motion, a concept not accurately •
understood, and because the usual development of this^
form of the calculus was, he thought, revolting to common
sense.3 And Lagrange's basis was set aside because of its
"tedious and unnecessary prolixity" and because of various
1. For a discussion of certain mathematical aspects of this work, see’ Dubbey (1964);.
2. Woodhouse The Principles of Analytical Calculation (1803) xvii") 218.
3. Ibid. iii-iv, 211-212.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 objections, to the way in which Lagrange expanded certain
functions. ■*"
Woodhouse1s own foundation was developed along the
lines of Lagrange's but in a different and more rigorous
order.
Instead of labouring to deduce from metaphysical principles, the properties of algebraic formulas, I think it more agreeable to the natural and logical order of ideas, to consider .the rules for multi plication, involution ^.evolution; the forms for \ (x+i)m , a x+i, 'l' lx+r), f(x+i), &c. as a series \*f regular deductions; and the steps by which we a9pend to expressions, more and more general, meisely as so many successive improvements in the! language of Analysis. Between the differential calculus and the rule for multiplication, the interval is not immense. It is that compendious * ^ method of addition, which is the low basis of the most towering speculations, the humble origin of the sublime Geometry.
On this basis Woodhouse developed his method of analytical
calculation by employing the analytical art, which he felt
was strict and certain when care was taken about the deri
vation and manipulation of the ^algebraic quantities.3 •V c Woodhouse rejected geometrical methods since the connection
between algebraic expressions and geometry was merely V ' an accident' of history and was of no benefit to calculation.
Finally,- Woodhouse also replaced the fluxional with the
differential notation. He regarded notation as the
main advantage of the language of analysis, and in this
1. Ibid’, xviii-xxv.
2. Ibid■ xxv. See also p.212.
3. Ibid. 53.
(T
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 / context criticized the fluxional notation for its awkward-
ness and for not being easily capable, of extension. The
differential notation was more convenient, easily distinguished,
less ambiguous, useful as a symbol of operation, arid also
important for discovery:
.... but the notation formed by it, has other advantages besides that of conciseness: for, not only is the deductive process rendered more easy and precise by an ingenious system of signs, but even invention is thereby considerably assisted.1
Woodhouse's Principles must have been very influential
in introducing the language and methods of Continental
mathematics to those who were interested. Charles Babbage 2 learned from it the differential notation. Many articles
in Barlow's A New Mathematical and Philosophical Dictionary
(1814) and in Rees' The New Cyclopaedia (1802-1820) cite
the Principles, especially for information on foreign
mathematics.^ But it also prompted^njany mathematicians
to defend the fluxional calculus. William Hales (1747-1831),
a former professor of.Trinity College, Dublin, responded
in hj_s Analysis Fluxionum (1800) bo Woodhouse's early
reviews in the Monthly Review by arguing for the method of
1. Ibid. 23.
2. Babbage (1864) 26. As a foundational work, it appears to have had a great influence on Augustus DeMorgan, ~ see Dubbey (1964) 80. I
3. See, for example, the article "Function" in Rees and in Barlow. )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 fluxions; that is, the principle.of motion, geometrical
methods and fluxional notation. Similarly a reviewer
in the British Critic in 1804 thought that the Principles
did little to change his conviction of the "truth of the 'A 1 Even many years after-
wards, in 1811, a critic for the Eclectic Review supported-
the method of fluxions by refuting almost point "by point 2 Woodhohse's objections. There were also many who were
zealous to defend the fluxional notation. For example,
the Anti-Jacobiri Review commented:
... Mr. Woodhouse's quitting the fluxionary notation of Sir Isaac Newton for the differential onq of Leibnitz, who, though a man of eminent . and diversified talents, was certainly a plagiarist in,matters of science, strikes us as a ridiculpus piece of affectation. The two calculi differ only, in name and in notation, which in fluxions, i9 equal, at least in simplicity, to that of differentials, and unquestionably superior to it in point of conciseness. As this is the case, and as the Royal.Society of London took a great deal of pains to have Sir Isaac's claim to the invention investigated and established, we trust the- principle mathematicians in this island will never think of abandoning the notation of the’inventor for the other.3
The Principles of Analytical Calculation was not
Woodhouse' s’only effort at diffusing Continental mathe-
1. Anon "Review of Woodhouse's The Principles of Analytical Calculation'1 British Critic 23_ (1804) 74-81. The British Critic aimed at "upholding the tenets of- the Established Church and the Tory politics of the ruling governemnt". Among its contributors were many Oxford iyld Cambridge fellows, including Abraham Robertson and Samuel Vince. See Hayden (1968) 44-45.
2. Anon "Review of Dealtry's The Principles of Fluxions" Eclectic Review 1_ (1811) 390-400.
3. Anon "Review of Woodhouse'-s 'On the Integration &c. ' Phil.Trans.(1804)" Anti-Jacobin Review 23 (1806) 254-256. pT7561------
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 matics. In 1809 his A Treatise on Plane and Spherical
Trigonometry appeared. Woodhouse noted, in this work,
that trigonometry had progressed beyond merely dealing
with triangles, for in its analytical form it provided
many "convenient forms and modes of expression" for the
general language of analysis.1 And since these ex
pressions', or formulae, were "not entirely without their
use, nor invented and shewn as mere specimens of analytical
dexterity", Woodhouse presented his Treatise in the Continental 2 analytical form. The.Treatise was the first attempt in
Britain to develop trigonometry analytically, and it met
with both praise and censure. A critic for the Edinburgh
Review, perhaps Playfair, in a very favourable review,
praised Woodhouse for hi's treatment of the subject and
for his present and past work in turning the-attention of
British mathematicians to the Continent.'1 Whereas the
Quarterly Review, though mildly favourable,, was "not much
in love with the language" which Woodhouse utilized, and
1. Woodhouse A Treatise on Plane and Spherical Trigonometry (3rd ed. 1819) irii, 41, 102.
2. Ibid. 116-117.
3. Playfair (?) "Review of Woodhouse's A Treatise &c." Edinburgh Review 17 (1810) 122-135.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 criticized various facets of the work, including that
Woodhouse, like many Continental mathematicians,
... aims rather to dazzle than to convince; that' he has struggled with intricacies, till he has lost all love for .simplicity, and in pursuit of novelty, sometimes wandered info obscurity.1
However, this work appears to have been even'much more
influential than the earlier Principles. In 1834,
George Peacock wrote that Woodhouse’s trigonometry
... more than any other work contributed to revolutionize the mathematical studies of this country. It was a work, independently of .its singularly opportune appearance, of great merit, and such as is not likely, not withstanding the crowd of similar publications in the present day, to be speedily superseded in the business of education. 2
A year after his trigonometry, Woodhouse\published
A Treatise on Isoperimetrical Problems and the Calculus of
Variations. Once again he aimed at introducing English
readers to an important branch of mathematics that! had been
developed on the Continent. And in this work he again
pointed to the•fruitfulness of the differential—notation
and to the importance of analytical methods.
There is cinother point towards which I am not unwilling to draw the attention of the reader; and that is, the method of demonstration by geometrical figures. -In the first solution . of Isoperimetrical’problems, the-Bernoullis use
1. Anon "Review of Woodhouse1s A Treatise &c." Quarterly Review 4 (1810) 392-402. p.395.
2. Peacock (1834) 295-296.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 diagrams_ and their properties. Euler, in his early essays, does the same; then, as he improves the calculus he gets rid of constructions. . . . "» A similar history will belong to every other method of calculation, that has been advanced to 'any degree of perfection.1
As with his other writings, this treatise was also
criticized for its promotion of foreign mathematics. The U Eclectic Review objected to the foreign notation not only
• because it used letters to denote both an operation.and a '
''Vj, quantity but also because its adoption would lead to an ^ 2 extinguishing of the memory of Newton. Similarly the '
British Critic saw the "deficiency" of British mathematics
simply as the result of a want of application on the part 1 of British mathematicians, and defended their methc^ds
' and notation. — "
Accustomed as we have been, to admire the clearness and satisfaction of geometrical precision, we confess, that./we have yet to learn ip what the superiority of the foreign analytical calculus above our own consists; still less can we comprehend why, having so long trodden the analytical paths of mathematical enquiries, which our forefathers so successfully traversed before'us, with mile-stones of good plain English A's and B's we are to go over the same ground again, attended with the more formidable apparatus, but not more goodly show of 9 ' s and <5 1 s for our c directors; and that too/,merely because the French mathematicians have adopted them; that we shall give up our Newtonian x ’s for the more confusing dx's of a Leibnitz. Not considering, perhaps, that while we adopt the
1. Woodhouse A Treatise on Isoperimetrical Problems &c. (1810) vi-viTT
2. Anon "Review of Woodhouse's Isoperimetrical Problems" Eclectic Review 1_ (1811) 584-595. p,592. 1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. notation we tacitly allow the superiority of the mode, if not priority of claim to the analytical invention, by. giving up our fluxional theory, for the adoption of a foreign differential method.1
This reviewer was characterized by Thomas Wilson
Cfl.1811) as an "anonymous blockhead", .as one veiled in
obscurity spinning cobweb critiques on works far above 2 his comprehension. Wilson was alarmed at the great
decline of the mathematical sciences in Britain and saw
the main causes for this situation as a lack of interest
in these subjects' and, at the same time', a lack of encourage
ment. So there were persons willing arid able to see merit
in Woodhouse's work. In 1832 John Herschel praised Woodhouse1s
efforts to diffuse higher mathematics'iri Britain and saw
the pre-eminent mdrit of the Isoperimetrical Problems as
"that of appearing just at the right moment, when the- want
of any work explanatory of.what'is merely technical in 3 that calculus was beicoming urgent.
Woodhouse continued his promotion of analytical
methods in his An Elementary Treatise on Physical Astronomy
(1818). In this work he once again argued for the superiority
of the analytic over geometric method, as illustrated by
its use in physical astronomy. i______( 1. Anon "Review of Woodhouse's A Treatise &c." British Critic 32 (1811) 344-346. pp.344-345. ' —
2. Thomas Wilson "Observations on Woodhouse1s Work &c." Monthly Magazine 32 (1811) 322-324.
3. Herschel (1832) 543.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. .Take the methods as we now find them, and the superiority of the Analytical above the, Geometrical method, for efficiency, or for the obtaining the results, is indisputable. One of the results not to be obtained by the. latter is the one just mentioned in the text, namely, the retardation of Saturn's mean motion:., a second is the progression of the Lunar apogee:' a third the acceleration of the Moon's mean motion: a fourth the invariability of the mean motions of the planets. If the Geometrical method had been adhered to, Newton's system would have ^ been deprived of more than half its supports.
It is difficult to judge the extent of the influence
of Woodhouse's works in promoting analytics in the United
Kingdom. Because they were elementary they probably
had their greatest impact among students, and, in particular,
at” the University of Cambridge. To his enemies, such as
Abraham Robertson (1751-1826), and to defenders of
geometrical methods, of Newton's fluxions, of British r ■mathematics in general, Woodlouse was a mere compiler,
from French works, an often obscure and confusing author
who displayed "such a partiality for foreigners, and so
much disrespect to the great inventor of Fluxions, as
could not be expected from an Englishman, and particularly
■■ 2 frcm any Member of the University of Cambridge."/ While^-
to a few colleagues, at least a few student^, and to
later British mathematicians, he was among the first to
1. Woodhouse An Elementary Treatise on Physical Astronomy (n.d.) footnote, pp.lix-lx.
2. [Abraham Robertson] "On the Rectification of the Hyper bola" Gentleman's Magazine 85 (1815) 18-22. p.18.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission 95
realize the inferiority of English mathematics and,
through his texts, to "propagate forward to other
jninds the rising impulse of his own".^
'v Few persons at the University of Cambridge appear
to have shared in Woodhouse1 s zeal for Continental
mathematics. While there are few contemporary comments
on the state of*the mathematical sciences thdre, there
are some later sources that speak of a lively opposition
to Woodhouse's work. For instance, George Peacock in
.1834 reported vigorous attacks on Woodhouse's trigonometry
when it was published. ,
It was opposed and stigmatized by many of the_ ..older members, as tending to produce a 'dangerous innovation in the existing course of academical studies, and to subvert the prevalent taste for the geometrical form of conducting investigations and of exhibiting results which had been adopted by Newton in the greatest of his works, and which it became us, therefore, from a regard to the national honour and our own, to maintain unaltered. It was contended, . also, that the primary object of academical \ education, namely the severe cultivation and '•discipline of the mind, was more effectually attained'' by geometrical than by analytical studies, in which the objects of our reasoning are less logical and complete.2
There appears to have been very little mathematical
research activity at Cambridge at this time. Indeed,
between 1800 and 1815, of the eighteen moderators of the
Senate House Examination only five ever published any
1. Herschel (1832) 543
2. Peacock (1834) 296.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96
mathematical text, and one of these was Robert Woodhouse.
And among students, it seems that even senior wranglers
for most of our period went no further in their studies
than geometrical methods and the fluxional calculus.1
Another way of assessing the state of mathematics
at Cambridge is to examine the texts used by its students.
Throughout most of the period 1790 to 1815 the main text
books used were a series by Samuel Vince and James Wood
that appeared between 1790 and 1799. The six books of
the series covered algebra, fluxions and the four mathe
matical sciences of mechanics, hydrostatics, optics and
astronomy. The series appears to have been very-popular,
for each volume went through many editions, although
the content of each volume did not change significantly
at any time during the period 1790-1815.
James Wood (1760-1839) had graduated from St. John's
College in 1782 as senior wrangler and first Smith's
Prizeman, was a fellow of St. John'.s from 1782 to 1815,
a tutor there f r o m -1789 to 1814, and its Master from 1815
until his death. ^ J ie contributed to the series the volumes
on mechanics, optics and algebra. In both the raeahanics
and optics, principles were developed geometrically.
1. See an exchange of letters between Augustus DeMorgari and Sir Frederick Pollock in Ball (1889) 111-114.
with permission of the copyright owner. Further reproduction prohibited without permission. 97
His algebra had only a very short section at the end on
the application of algebra to geometry which, according
to Carl Boyer, "presents the subject about as it was in
the days of L'Hospital.
Samuel Vince .(1749-1821) had also been senior wrangler
and first Smith's Prizeman in 1775, having been an under
graduate at Caius College. He was Plumian Professor from
1796 to 1821, and also held a number of church livings ....
throughout- his life. His work was highly regarded by •'
his contemporaries for its content, but not for its 2 elegance. His astronomy consisted of elementary princi
ples and contained no physical .astronomy. Similarly his
text on hydrostatics attempted, as far as possible,
to develop its principles without the use of fluxions.
If fluxions were needed for a concept, such as the motion
of bodies in resisting mediums, then the.reader was
referred to his A Treatise on Fluxions. This treatise,
as other contemporary English works, on fluxions, contained
more examples and applications than exposition of theory.
And in -this work was found the only trace'of his opinion
of Continental mathematics - besides the style of the
Carl Boyer History 'of Analytic Geometry (1956) 256.
See, for example, Wordsworth (1877) 77, Pryme (1870) 99 and Hilken (1967) 45.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98
works themselves. Vince criticized the differential
notation:
Foreign mathematicians denote the fluxion of x by dx', which is liable to two objections: first, it is not so simple as x, and becomes s t i l l more com plex for the higher orders of fluxions; secondly, dx is a notation which also signifies the product of d multiplied by x. Every notation should . • have but one meaning.^
Besides these volumes for the series on the principles
of mathematics and natural philosophy,’Vince also wrote
A Treatise on Plane and Spherical Trigonometry, which
appeared in 1800. Unlike Woodhouse's later work,'it
was developed along geometrical lines, neglected
Continental work on the subject and continued the old
usage of the radius in its formulae.
That Vince's and Wood's books were used extensively
at Cambridge, or at least indicated the level to which
top students would aim, is confirmed not only by the Jo number of editions of their works but also by the . manuscript notebooks (1809-1810) of Thomas Pierce Williams,
who graduated as a wrangler in 1812 and was a fellow of 2 St. John'is from 1813 until 1816. His notes on algebra
correspond in subject matter to various parts of,Wood's
algebra. The notes on hydrostatics cover much the same
material as in Vince with a slightly different order and some \ -
1. S. Vince A Treatise on V.luxions (5th ed. 1818) 2, footnote. 3 2. These notebooks are preserved rn the University Library at Cambridge.
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different headings. Williams's notebooks on fluxions
follow Vince's text point by point and almost Word for
word, except - as is the case in all his notebooks -
that there are many examples in his notes not found in
Vince. The notes on astronomy contain only elementary
principles and nothing on physical astronomy. And his
notes on Nevrton'is Principia are entirely on Book One,
"On the Motion of Bodies in Unresisting Mediums".
A further indication of the general adherence
to geometrical methods at Cambridge is shown through
Daniel Cresswell's (1776-1844) An Elementary T r e a t i s e ..
on the Geometrical and Algebraical Investigation of
Maxima and Minima &c. (1812). Thi^-work, probably
written in response to Woodhouse's Treatise on Isoperimet-
rical Problems (1811)) compared the relative advantages .
of geometry and algebra. The first part of the book
presented and proved geometrically many theorems on maxima
a&d minima.. The second part briefly presented the theory
of derivatives as expounded by Lagrange and demonstrated
how maxima and minima were thus found! It then solved
a few of the theorems of the first part by this method.
Cresswell argued that while "algebra" might be more
advantageous in the investigation of mathematical truth,
yet as far as the discipline of the mind was concerned
it greatly lagged behind geometry. The chief difficulty
in "algebra"
... is seldom more than the mere translation of the conditions of" the question, into a language, the peculiarity of which is, that
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100 it is so concise as to exhibit several propositions in a small compass". !£his having once been effected, and it is seldom an arduous task to perform, the attention is then withdrawn from the things signified, and confined to the signs: and from performing the mere operations of Algebra, it will scarcely be contended that any improvement of the reasoning faculties is to be derived.1
^ Whereas in geometry the "faculties of judging, recollecting ' 2 and inventing are continually exercised." The emphasis
on the advantages of geometrical methods in education
was a tenet held by many, whether trained or not at
Cambridge. Yet within an educational institution such
as^Cambridge, espousing the ideal of a liberal education, ** 'n it must have carried additional weight. Thus the scant
•f research in mathematics, the- adherence to synthetic
mathematics and the purpose of mathematics within a liberal
education all blended together harmoniously at the
University of Cambridge. Their concurrence provided
a great obstacle to attempts to update mathematics there.
The state of mathematics at Cambridge in the
period 1790 to 1815 was similar to its1 state in the rest
of the country. Cambridge and British mathematics had
not kept pace with Continental developments. The
equilibrium which existed between a liberal education
1. Cresswell An Elementary Treatise on the Geometrioal .<■ and Algebraical Investigation of Maxima and Minima &c. C2nd ed. 1817) 12.
• ,2. Ibid. 13.
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and the state of Cambridge mathematics was confror?
by a new factor, an outcry over the inferiority of
British mathematics. Many of these deploring the
■'’situation regarded Cambridge as-the center of British'
mathematics and saw in its mathematics the proof of
British stagnation.'*' And many within Cambridge were
also recognizing this. Robert Woodhouse is one example:
the following published request from a member of .Cambridge-;
(a student,it appears)
What elementary wo: ______^ ;ed by a person who wishes to become acquainted with • what is usually termed "the modern analysis"? That one who resides in a Mathematical University should put this question may appear strange; but it is well known by many, who, like myself, have devoted a considerable portion of time to the study of mathematics according to the system adopted in this university, — that so little attention is paid to the modern I language of science, that the most admired works'of the foreign- Mathematicians are a dead letter even to many of those, who are sufficiently familiar with the works of Newton and the ablest English philosophers.2
Woodhouse attempted to introduce Continental mathematics
into England through his texts. Any effort to (reform1
Cambridge mathematics, however, was hindered by a^
conservatism expressed through a criticism of analytics,
either as a mathematical instrument or from the viewpoint
of a liberal education, and perhaps more directly through
1. See, for example, Playfair or.Brougham. "Review of Dealtry's Fluxions" Edinburgh Review 27 (1816) 87-98.
2. A.H.Z. "Inquiry concerning the means of studying the Modern Analysis." Nicholson's Journal 32 (1812) 17-18.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. an adherence to the language and principles of Newton's
formulation of the calculus. Woodhouse's efforts, focused
on promoting analytic .-mathematics, did not overcome these
barriers. It would appear that only those who were
also concerned with the other aspect of the lament -
establishing a new relationship between mathematics and
society - as Woodhouse was not, would actually bring about
a change in Cambridge mathematics.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. IV. The Analytical Society (1812-1813)
• O r Robert Woodhouse was not the only person at Cambridge
in the early nineteenth century interested in promoting
analytics. Some students manifested, through the formation
of the Analytical Society, the confluence of the structure
of Cambridge studies, the position of mathematics there,
the lament over the state of English mathematics, and the
expectations of students at that time. This chapter will
present the history of the Society in some detail: the
motivations of its members, the goals of the Society and
its fortunes at Cambridge.
On a Thursday, the seventh of May 1812, seven students-
of the University of Cambridge and one recent graduate
gathered in the graduate's rooms in Caius College and
decided to form an association to be called the Analytical 1 . 0 Society. The following Monday, at its first meeting,
several other persons joined the Society, tjpl'es and
regulations were adopted, a president chosen, a room engaged
for the Society and arrangements made for the formation of
its library of works in the mathematical and physical 2 sciences. Of the eight founders of the Analytical Society,
three were students of Trinity College - Charles Babbage,
George Peacock and Michael Slegg. The remaining four
1. Buxton ms.13, pp.24-25.
2. Ibid. 25.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. students, Richard Gwatkin, John Herschel, John Whittaker
and Henry Wilkinson, were from the other (besides Trinity)
large college of Cambridge, St. John's. And their .host was
Edward Bromhead, at that time a scholar of Caius College.'*'
It is not known who were the several new members at the
first meeting, but they may have included William Mill,
Joseph Jordan, Edward Ryan and Thomas Robinson, all students
at Trinity College and all mentioned in various sources as 2 members of the Society. The immediate cause of the
founding of the Society was a conversation between Slegg
and Babbage earlier that May, in which Slegg had mentioned
the current controversy at Cambridge over whether the Bible
was to be distributed alone or with comments. The debate
had'become acute- then with the recent founding of the
Cambridge Auxiliary of the British and Foreign Bible
1. Ibid. 24-25. There are three main sources for the history of the Analytical Society: Babbage’s autobiography, Passages from The Life of a_ Philosopher (1864) , Buxton ms.-;13 The History of the Origin and Progress of the Calculus of Functions during the years 1809 1810 1817 (1817] written by Babbage, and correspondence between various members of the Society. X have relied primarily on the two latter sources in writing this chapter, using Babbage’s autobiography very warily as it contains many inaccuracies. For example, Alexander D'Arblay, reported as a founder in the autobiography, could not have been present at the formation meeting since he only arrived in England from France in August 1812.
2. Ryan and Robinson are mentioned in Babbage’s autobiography, Babbage (.1864) 29; for Jordan see a letter from Whittaker to Bromhead, Mar. 20 1813, Br. ms.; and for Mill consult a letter from Babbage to Herschel, ca. Jan. 12 1814, H.ms.R.S.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 5 Society. 1 The * conversation inspired Babbage to suggest
instituting a society, like the Bible Society, for
distributing Silvestre Lacroix’s Traite' elamentaire de .
Calcul diffe"rentiel et de Calcul integral (1802) in order .
to help spread the "true faith" of analytics. And he drew 2 up a series of resolutions which such a society might adopt.
Slegg felt that the scheme was "too good to be lost" and'- ^
told his acquaintance, Bromhead, of it. Bromhead was so
enthusiastic about the scheme that "he invited those of his
acquaintance who were most attached to mathematical subjects
to meet at his r o o m s . T h u s did the Analytical Society
come into being.
There is a manuscript in the library of St. John's
College, Cambridge, entitled "Plan of a New Society." It
1. The Cambridge" Auxiliary of the British and Foreign Bible Society was established on December 12 1811. Its formation intensified the already existing debate on whether the Society (founded 1804) should publish the Bible without any commentary (as it did) or with • the prayer book (as the High Church group wished). Many pamphlets on the subject were written in 1812 by members of Cambridge University. See Ford K. Brown Fathers of the Victorians (1961)
2. Buxton ms.13, p.24. In his autobiography Babbage says he drew his inspiration from a poster and that his sketch of a society for distributing Lacroix's work proposed "...that we should have periodical^ meetings for the propagation of d ’s; and consigne^to perdition all who supported the heresy of dot«J^' I t maintained that the work of Lacroix was so perfect that any comment was unnecessary.” Babbag^ (1864) 28. As mentioned in footnote 1, p. 104,1 tend to. trust the account given in Buxton ms.13, although Babbage's witticism may have originated on this occasion.
3. Buxton ms.13, p.24.
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is bound into a book once belonging to Charles Babbage.
The "Plan" consists of five -resolutions relating to the
Analytical Society. This may be the document drawn up by
Babbage, mentioned previously, or it may be an early record
of the Analytical Society. In any case, as it agrees in
some of its resolutions with the early history of the
Society, it probably accurately reflects the Society’s goals.
According to the "Plan", the Analytical Society (as is also
indicated by its name) was principally interested in the
advancement of Analysis, for its members conceived "that
the Physical Sciences keep pace with the progress of ■
Analysis". And as "the extension of Analytical science
depends upon the increased comprehensiveness of its
^notation", the Society regarded "geometry, & geometrical
demonstration, as contrary to its ultimate objects" and
admitted no papers in which the fluxional notation was
employed. Clearly, then, the Analytical Society shared in
the views of those who were deploring the stagnation, or
decline, of the mathematical sciences in England.
In order to better promote its views, the "Plan” called
for the Society to rent a room to help increase
communication between its members, to start a library to
provide proper sources for study, and to subscribe to
Leyboum's Mathematical Repository and to Nicholson’s
Journal. The Society was also to "receive mathematical
1. Bound with the Memoirs of the Analytical Society (1813).
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manuscripts, & memoirs either containing original matter,
or putting any department of Analysis into a more
convenient form". Members of the Society were to assist
each other in their .'mathematical pursuits if requested.
And the Society was to meet the first Monday of each month
to hear memoirs of'a general mathematical" nature and to
transact its business. All these arrangements seem
suitable for a self-help society, which is” not surprising
given that the members of the Analytical Society were almost
all undergraduates. However, the resolutions of the Society
were not really appropriate simply to satisfy students’
requirements at the Cambridge of that time; they manifested
a much broader interest than university studies. The tone
of the Society was very strongly set towards research in
mathematics.
What-led .these students, most of whom were unacquainted
with one another, to form a society, especially a
mathematical society to promote analytics?1 Biographical
details of the dozen persons mentioned above show that all,
except Peacock, had been admitted to their colleges as
pensioners, the usual procedure for all but the poor, very
rich or aristocratic. The members, therefore, possessed a
1. Babbage wrote in 1817 that "...previous to our first meeting no two of the members (with the exception of M r Bromhead whose acquaintance were very numerous) were known to each other otherwise than by reputation ...." Buxton ms.13, pp.25-26
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certain degree of financial independence."'" Five persons
had attended schools such as Charterhouse, three had been '
privately tutored, and Bromhead had spent two years at
Glasgow University; there are no details for the pre-
Cambridge education of the remaining three students. More
interestingly*, all of the twelve, except Babbage and Ryan,
had obtained,scholarships, at their colleges, which was
usually the first step towards a fellowship. ''(They were
thus a group of very able and serious student^. Another
indication of their great ability is that seven of them
are to be found in the Dictionary o f "National Biography.
But all this information, while it may preclude certain
possible motivations, still does not identify the actual
ones.
Sheldon Rothblatt’s work on Oxbridge student sub
cultures sheds some light on the context in which the
Analytical Society was founded. He argues that the late
Georgian period marked the emergence of the independent
student and the notion of a separate student estate, and 2 that a new kind of student society arose. These new
societies were more permanent and serious than earlier
ones, more closely identified with the university and
For these and other details consult Alumni Cantabriqiensis. Peacock had entered as a sizar, a category reserved for the poor and which entailed some menial duties and often a lower social status.
Rothblatt (1974) 303.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. colleges and their purposes, and were composed of
undergraduates from various colleges rather than fellows
of single colleges.'*' The Analytical Society clearly
shared in many of these characteristics. It seems that it
would have ^een very natural at that period for students
to organize a society such as the Analytical and to have it
reflect some of the concerns of the England,of their time.
But it is still appropriate to wonder why ttiey should have
formed a mathematical society. Fortunately, further
pertinent information is available/from three members of
the Society, two of whom - John Herschel and Charles
Babbage -.were the Society^s^ mainstays.
John Herschel (1792-1871) was the only child of the
astronomer William Herschel. In his ninth year he was sent
to a private school run by a friend of his father's, Dr. 2 Gretton. The mainly classical education he acquired there .
was complemented by a tutor's instructions in the elements
of the natural sciences, m o d e m languages, literature and
music, and mathematics. The tutor was a Scottish
mathematician, Alexander Rogers, who seems to have kindled
in Herschel an interest in mathematics.*5 Roger’s letters
to Herschel of late 1808 and early 1809 show that Herschel was interested in Continental mathematics and had a copy
1. Ibid. 252-255.
2. Buttmann (19 70) 9.
3. Ibid. 9-10.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of the Mecanigue Celeste and wished to obtain works by
Lacroix.1 In October of 1809 Herschel went to Cambridge
and matriculated at St, John’s College. Despite brilliant
success as ^an undergraduate in which he "gained all the
first prizes without exception" and which culminated in the
Senior Wranglership and the first Smith's Prize in January
1813, Herschel was not altogether happy with his studies at 2 Cambridge. At the root of his dissatisfaction lay the
Cambridge curriculum, as shown in a letter to his father
shortly before the Senate House Examination.
The impatience with which I look forward to the termination of this childish course of ' -■ study is inconcievable. I see lying in my rooms books which I long to read, and which I dare not open, without which I can advance no farther.3
Long before, in a letter of March 5 1810, Rogers had
cautioned Herschel, in a tone that.sounded of personal
.^experience, not to let his zeal for mathematics get the
better of him:
...and it is unquestionably laudable that you should cultivate such studies, both- as an inexhaustible fund of rational entertainment, and as the means of extending the boundaries of science. But to him, who, unwarranted Tby circumstances, has inconsiderately attached himself to such [------]; they only prove an ignis fatuus bewildering his steps in the
• 1. Letters dated Nov. 5 1808 and Jan. 2 1809; H.ms.R.S. ■i® 2. For his successes see Herschel (1879) 120.
3. Letter dated Dec. 1 1812; H.ms.T.
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beginning of life and diverting him from the beaten paths which would have been his securest guides; for when he endeavours to render his mathematical attainments the means of bettering^ his condition, he finds them exceedingly ill qualified for such a task.
Rogers had no need to worry, for Herschel managed both to
win all the honours of the University and to pursue his
mathematical studies. The nature of these studies, even
before the founding of the Analytical Society, was
analytical as is illustrated by two short papers he 2 anonymously published in Nicholson’s Journal. ■ The first
appeared in February 1812 and was signed, "A Lover of the
Modem Analysis”.^ The article developed formulae for
certain trigonometrical functions and -showed the values
of the two series
1 lil 1 , &c 1.i 2 -a ” 7 2 2 -a 3 7% -a 4 7 2-a
3 , 7______, 11 , &c 5 5 + 5 5 ' 2 2 (1 -a) (2 —a) (3 -a) (4 -a) (5 -a) (6 -a)
to be identical and to be represented by
■Ti______1 . . " ^ 2 fa X sin. ttja 2a
f~ The second paper, dated March 23, 1812, appeared in
1. H.ms.R.S.
2. Buttmann (1970) 13. See also Herschel’s (unpublished) "Spherical Trigonometry Analytically worked" of Dec. 25 1811; H.ms.T.
3. Herschel "Analytical Formulae for the Tangent, .• Cotangent,' &c. In a letter from a Correspondent.’"', "Nicholson's Journal 31 (1812) 133-136.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 the May issue of the' Journal* and was signed "Analyticus".^
This paper developed from trigonometrical formulae
numerical expressions for it and for the square roots of
certain integers, and various expressions involving series
of trigonometrical functions. For example,
tt = 3 43 - 3.3.6.6.9.9.12.12.&c. ~ T ~ 2.4.5.7.8.10.11.13.&c.
43 = 2 . 2.4.8.10.14.16.20.22.&c. 3 .3.9 .9 .I'S'.l'S .ilTSr.'&bT
cos .A-lcos .2A+lcosT3)C-lcos.4A+&c. , J 3 . 4 2.cos.A = e 2
The second paper also referred’to Woodhouse's Trigonometry,
probably indicating Herschel's source of inspiration. This
influence is'further suggested in that in^both papers
Herschel's initial trigonometric formulae, from which he
derives his results, are to be found in the appendix to 2 Woodhouse's work. So, John Herschel, before joining the
Analytical Society, was not only conversant with Continental
mathematics but also proficient in them and, probably in Vj part because of this ability,'was dissatisfied with the
studies at Cambridge.
’sl. [Herschel] "Trigonometrical Formulae for Sin,es and Cosines. In a Letter from a Correspondent." Nicholson's Journal 32 (1812) 13-16. Herschel does refer in this paper to the correspondence of some of his results to those of Euler'and of Wallis.
2. R. Woodhouse A Treatise on Plane and Spherical Trigonometry T3rd ed., 1819] 250.
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Charles Babbage (1791-1871) was the son of Benjamin
Babbage, a partner in a very successful London banking
firm.1 He was educated at various schools and by private
tutors but appears to have been largely self-instructed in
mathematics. By the time he matriculated at Trinity College,
in October 1810, he had examined Ditton’s Fluxions (1706),
Agnesi's Analytical Institutiohs .(1801) , Woodhouse’s
Principles of Analytical Calculation (1803) , from which he
learned the notation of Leibnitz, Lagrange's Theorie des
Fonctions Analytiques (1797) and William Spence's An Essay
on the Various Orders of Logarithmic Transcendents (1809),
which had greatly.added to his mathematical knowledge, 2 especially in the idea and symbolism of -functions. So,
like John Herschel, Babbage was well acquainted with
Continental mathematics before entering Cambridge. And
also like Herschel, he had engaged in some original
research. For example, he had discovered "by a method
something like induction” the theorem
1. There is much confusion over certain of Babbage's dates - in particular (so far as I have noticed) his birth, death and marriage. I have tended to *follow Moseley (1964) who has, apparently, consulted family records. She gives Babbage's birth date as December 26, 1791 while most other sources give the same day in 1792. The latest research in the sources confirms the 1791 date; Dubbey (1978) 4-5. Babbage himself seems to have thought that he was b o m on December 26, 1792 (see Moseley (1964) 29, 267 and also B.ms.B.L. Nov. 1859) which may explain the confusion in biographical sources.
2. Babbage (1864) 26, and Buxton ms.13, p.61.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 = ln - in 2n + m.m-1 3n - &c for positive inteqers 1 * 1.2
m and n with m greater than n, only to find it later in
Euler. "*■
Babbage's pre-Cambridge mathematical work is also
significant from the -point of view of his later work on the
calculus of functions. For towards the end of 1809, having
- been inspired by a proposition in Pappus, he drew a figure,
like the following, of a hyperbola with inscribed circles
between it and its assymptotes.
And he asked
What is the ratio of the area of the curve to the sum p f the areas of all the circles? and conversely if we suppose that area to be given or to follow any law: What will be the nature of the c u r v e ? 2 -
He was especially interested in the'converse question, but
was unable to solve it. However, Babbage believed in 1817
that he had previously "employed fx=y to denote the equation
1. Letter from Babbage to Herschel, pmk July 10 1812;. B..ms. r .S. Herschel too had "discovered" this theorem towards the end of June 1812. Professor B^rbgau has pointed out to me that this theorem says A x = 0.
2. Buxton ms.13, p.5. Pappus' proposition was proposition 18 of Book 4 of his Mathematical Collection which has circles tangentially inscribed in a semicircle.
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of the curve to be found", and although he was unsure
wtether he had "at that time used a symbol to denote operation.
or whether I conceived y to be a function of x it appears
that I attempted the solution of problems which depended on
very difficult functional equations."1
On his w a y to Cambridge in October 1810 Babbage had
bought Lacroix's''three volume' Traite du Calcul differentiel
et du Calcul integral (1797-1800) and began to study it at 2 Cambridge. Babbage, once again like Herschel, soon became
dissatisfied with Cambridge studies. For he had felt sure
that his difficulties in mathematics would be removed at
Cambridge. Instead he found that not only were his
lecturers ignorant in the subjects of his interest, but they ) also advised him to pay no attention to Jthose topics since
they would not be asked in the Senate House Examination.1
Unlike Herschel, however, Babbage reacted by ignoring as
\nuch as possible the college and university system of
studies and its rewards, and pursued his own interests, in
particular by studying the works of Continental 4 > mathematicians. George Pryme recalledcthe Babbage of
1. Buxton ms.. 13, p.6. Babbage must have mentioned his problem to others, for Herschel attempted its solution. Letter from Herschel to Babbage, "1812 or 1813”; " H.ms.R.S. Babbage wrote this account of his early work in 1817.
2. Buxton ms.13, p.7.
3. ■ Babbage (1864) 26-27.
4. Ibid. 27.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 these years when he reminisced about the Trinity College
annual examination of (June?) 1811:
Among those whom I examined was Charles Babbage, who had the reputation, even in his first year, of being an excellent mathematician. On the occasion of his first examination in the-lecture- books he gave up a small roll of MS. as if in answer to my paper. I found it to contain some clear demonstrations and able remarks on a subject connected with one of my questions on the Binomial Theorem, but not properly an answer to it. I told him the next day that on this account I could not give any marks-'for it. He answered that he did not wish to be Classed, but only to show the examiners that he was not wanting in knowledge of the subject. From a similar fancy he would .not compete for Mathematical Honours on taking his degree, though X believe if he would have . done so he could easily have been Senior Wrangler.
Some information on his mathematical work while he
was at Cambridge and before the establishment of the
Analytical Society is found in his manuscript The History
of the Origin and Progress of the Calculus of Functions
during the years 1809 1810 ...... 1817 (1817).2 In this
account, which is of course selective in subject matter,
Babbage relates that about the period August and September
1811 his enquiries referred to "functional equations of
one variable and of higher orders than the first", and that
his "success seems to have been very limited".3 For
example, various problems on curves led him to attempt the
1. Pryme (187ffJ•91-92.
2. Buxton ms.13,-preserved in the History of Science Museum at Oxford.
3. Ibid. 13.
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solution of certain functional equations. One was the
problem
APQ is a curve take any point P draw the ordinate PN, take the abscissa AM=PN draw the ordinate QM and take AS=QM draw RS and so on ad infinitum. Required the nature of the curve so that PN may have to RS a given ratio that is, that the alternate ordinates may be . in geometrical progression,. n m S Babbage reduced this problem to the functional equation
rf(x) = f[f{f(x)}] where r is the-given ratio, which he solved for r = 1 with the particular solution f(x) x_ 1-x
He also attempted to solve equations such as
f(x,y) = a n ^i f(x) X f2(x) = f3(x) ,3
However, by October of that year Babbage had mostly laid 4 aside this subject. On April 7, 1812, for some unknown
reason, he transferred to another college at Cambridge,
1. Ibid. 12-14.
2- Babbage had let -f(x) = x and from this found a = fl . a+x Nr He .noted in 1817 (Buxton ms.13) that he had only been able-to solve the problem for r = 1 and with the particular solution f(X) = -x . • 1_x 3. Ibid. 16—21.
4. Ibid. 22-23.
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Peterhouse.^ And the following month saw the formation of
the Analytical Society.
So Babbage, like Herschel, was familiar with and
competent in Continental mathematics and dissatisfied with
the studies at^Cambridge. Moreover, Babbage later felt
that the founding of the Analytical Society had contributed
more than anything else to the success of.his subsequent 2 enquiries. For he felt it had brought "together those who
were engag^d^ln similar pursuits” and had therefore acted
as a constant stimulus and aid in the enquiries of its
members.^
The final member of the Society for whom there is
some relevant information is Alexander D'Arblay (1794-1837).
mother was the famous English authoress Fanny Burney,
who had married Alexandre D ’Arblay, a refugee from the
Many secondary sources, including Moseley's otherwise excellent biography, believe Babbage to have migrated because of "his conviction that he would be beaten in the Tripos examination by his friends John Herschel and George Peacock, and preferred to be first at •Peterhouse rather than third at Trinity,". Moseley (1964) 45. There is much evidence to refute this view: Babbage does not appear to have known Herschel until the founding of the Analytical Society; Babbage did not appear at all pn the honours list of 1814, while there are two senior optimes and one junior optime from Peterhouse; and, most convincingly, Babbage matriculated at Cambridge a year later than both Herschel -hnd Peacock and consequently would have been (and was) examined in the Senate House a year later than they. This slander probably arose among his detractors later in his life, or after his death.
2. Buxton m s .'13, p.23.
3. Ibid. 26. /
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French Revolution. Alexander had received his education
mainly in France and was to enter the Ecole Polytechnique
when his parents, fearful of the military conscription.
decided that he should be sent to England."*' He and his
mother arrived in England in August 1812 where he attended
for some time hi^^ousin's school at Greenwich. Through
family influence he was elected to a Tancred scholarship
and matriculated dt-Caius College in October 1813. If
Alexander became a member of the Analytical Society, as
Babbage relates in his autobiography, then he must have
done so at this time. He had not been at Cambridge for
long before he had gained a reputation for his "mathematical
talents and knowledge" and had become a friend of Robert 2 Woodhouse. However, Alexander, having been educated in
Continental mathematics, was disgusted with the style of
mathematics studied at Cambridge, and this distaste
threatened his chances for honours. Indeed his cousin
wished that Woodhouse would use his influence over
Alexander to persuade
...dear Alex to study in the Cambridge way, that is to say, to learn to solve his problems & to give their proofs by geometry instead of ,algebra or the analytical method, which is the French way & also the best; & Alex knows that.
1. Frances D'Arblay Diary and Letters of Madame D'Arblay (1778-1840) (1904-05) vol.6, pp.65-66.
2. Letter from Charlotte Barrett to Mme d'Arblay Jan. 1
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But unfortunately, when his examination comes next year, he will be expected to bring geometrical proofs instead of analytical, &, if he had not attended to them,’he may lose . the prize which he must else' infallibly obtain.
Despite his aversion, Alexander managed to graduate in
January 1818 as tenth wrangler, much to his own and h i s '
mother's relief and surprize, and*to be elected that same 2 year a fellow of Christ's College.
Not all members of the Society, however, were so well
acquainted with the higher branches, of mathematics or with
Continental mathematics as Herschel'; Babbage and D'Arblay.
When George Peacock had entered Trinity College in 1809,
"his mathematical reading had not extended much beyond • j the first year's subjects then studied at Cambridge",
which, as we have seen, was a very meagre amount.3 And
Richard Gwatkin was more than "a little alarmed" when asked
by the Dean of Hereford in 1814 to explain a difficulty in the 4 Memoirs of the Analytical Society. Yet the members of the
Society must have shared at least an interest and
enthusiasm in analytics, if not an ample knowledge of it.
1. Ibid. See also a letter from Mrs. Barrett to Mme d'Arblay 1815, in the British Library.
2. Hemlow (1958) 407.
3. Herschel (1859) 536.
4. A letter from Gwatkin to Whittaker, July 17 1814; St.J.ms. See also a letter from William Whewell to. Herschel,. Nov. 1 1818, where he remarks that Gwatkin "has been reading a good deal of good mathematics"; Todhunter (1876) 2_ 31.
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Thus the founders and early members of the Analytical
Society were, it appears, mainly motivated to form such
a society because of a mutual interest in mathematics and
especially in analysis, which was probably reinforced by
the emphasis on mathematics at Cambridge, coupled with a
dissatisfaction' with the content and system of Cambridge
studies. Their particular interests in analytics may have
been induced by such factors as the many and great advances
made in mathematics and mathematical science by persons on
the Continent, the cries of British stagnation^ in
mathematical science, or even the emphasis on geometry at
Cambridge:
Students at our universities, fettered by no prejudices, entangled by no habits, and excited by the ardour and emulation of youth, had heard of the existence of masses of knowledge, from which they were debarred by the mere accident of position. There required no more. The prestige which magnifies what is unknown, and the attraction inherent in what is forbidden, coincided in their impulse. The books were procured and read, and produced their natural effects.1
The Society apparently was not founded to change or reform
the mathematical studies at Cambridge, much less the style
of mathematics pursued in England, but rather to further ■
analysis.
As noted above, the .Analytical Society first- met on
May 11, 1812. At this meetab John Herschel was elected
President, perhaps because o^nis^publications or V
1. Herschel (1832) 545.
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J| scholarly reputation.1 The Society continued to meet until
■early in June when, because of the summer vacation, most of 2 its members left Cambridge. Bromhead, who had also been
studying Continental mathematics before the formation of
the Society, read’to the Society its first memoir, on 3 notation. Herschel read two memoirs, one being on some 4 properties of the conic sections. The other was most
probably his "Remarks on the Theory■of Analytical
Dsvelopements " which examined some of the basic principles
of that theory.5 Babbage also presented two papers to
the Society. The first, "Solutions of Problems requiring
the application of Mixed Differences", like some of his
earlier mathematical enquiries, consisted of two problems
about curves which required the solution of functional
1. Buxton ms.13, p.25.
2. Ibid. 27, 34.
3. Bromhead shared Babbage's interest in functions and had been led through his study of Arbogast to the idea of second and higher orders of functions^’ Buxton ms. 13, pp.30-31.
4. Buxton ms.13, p.37. The memoir on the properties of the conic sections was later to be published, with some additions, as "On a remarkable Application of Cotes's Theorem" Philosophical Transactions. Royal Society. 103 (1813) 8-26. For comment on"this work see the following. The memoir is preserved in H.ms.T. _ and is dated May 7, June 15 1812.
5. H.ms.T.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 3 X equations. For example, the first problem: ?
Required the nature of a curve ApP such that taki-ftg any point P and drawing an ordinate PN and normal PG, if the triangle ^ PNG be placed in such a situation that NG may become an ordinate and NP coinciding with . the abscissa, the line PG coming into the situation pg may be perpendicular to the curve.
Babbage set PN = y , AN = x
pn = y ', An = x ’,
and so NG = y d^ = yp, which ="'y' by the problem's dx
conditions, and
ng = y'dy1 = y ’p', which = y . 3 x '
By subtracting the last equation from thp first he obtained
y'p' - yp = y - y* = -(y’ - y) , a yp = -ay,
a (yp + y) = 0 t \ Integration of- this -yielded
yp + y = a where a is a constant.
y d£ = a-y, dx .
dJ& = -dy + ady, a~y
1. These-papers are referred to in Buxton ms.13, pp.32-34. Both of Babbage's papers are to be found in an uncatalogued collection of Babbage's manuscripts (hereafter referred to as B.ms.C.) preserved in the Science Periodicals Library at -Cambridge. This collection includes a number of various solved mathematical problems which were probably presented by Babbage to the Society'by being left on a table, as the "Plan" indicates. The. solution of such "interesting" problems was probably a general activity of the Society.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. \ ' ' 124
x = c - y - al.a-y, where c is a constant and 1. stands
for logarithm, and x+y-c _ , .-a 1 e — la-yj .
Babbage noted that this was only a particular solution. He
then derived an expression forthe general solution by
multiplying his original equations, yp =-y' and y'p' = y, 2 to obtain A (y p) = 0, which indicated -that A (yp+y) = 0 ^ * ... 2 was to "be integrated, on the hypothesis of y p being
constant hence yp + y = f(y 2 p) where f(y2 p) signifies any 2 function of y p." The other problem of this paper was
similar to this one.
His second paper was entitled "Memoir on the Summation
1.' Professor Barbeau has noted the mathematical difficulties in Babbage’s move from y ’p ’ - yp - (y’-y) to A yp= - A y and in "integrating" this last expression. Babbage offers no justification for these manipulations. Babbage’s solution does satisfy the problem's conditions. • For the problem reduces to showing that if f(u) = f(x)f‘(x) then f(x) = f(u)f’(u) where an=u and f is the function describing the curve. .Assuming ex+y c = (a-y) a we obtain x = -y+c-a In (a-y)'
dx = -1 + a _ y 3y (a-y) (a-y)
Thus f ’(x) = dy = a-y = a-f(x) . __ (1) d£ y ■ f(x)
Since f(u) = f(x) f ’(x)
f(u) = f(x) . a-f (xj = a - f(x) . "TTxT
So f (x) = a - f (u) = a - f(u) . f(u)
= f ’(u) f (u) . . .by (1) Craig Fraser of this Institute developed this proof.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 of certain Series of sines cosines See". This work was
__^developed along the same lines as Herschel'^ works in
Nicholson's 'Journal. So he manipulated certain trigonometric
formulae and, having derived a series from them 'by taking^ ;
logarithms and then derivatives, found very curious, results
by substitutions for some variables in the series. For \
example, taking the following sequence of trigonometric
identities
cot.A cot. A =. sin..'(A - A^) sin.A sm.Aj
cot. A sin. fA^ - sin.A^ sin.A2
cot.A - cot.A'n+1 sm..(.An - An+j.) sin.A sin.A ~7 n n+1
and adding these equations, Babbage obtained . cot.A — cot.A , J = n+1
sin. (A - A.) sin. (A. - A_) sin. (A T A s m . A sm.A.1 + sin.A, r-i- s m-- . A1 _- + s c ------sin.A n sJ m . An+1 ,, 1 1 2 n n+1
Then by letting Ar = 2n z and noting that thus
• i. . i • / n . n+1. • . n . s m . (A - A .) = sin. (2 - 2 )z = -sin.2 z = -sin.A , n n+1 n
he found
1. This should be - (cot.A - cot.An+1) = etc. Babbage has omitted a sign in each of his identities. .
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COt.2n+"''2 - COt.Z = ]Ny +••••!•• + &C 1 sin.2z • sin.2 z sin.2n+1z
Babbage multiplied this equation by dz and integrated to
get
1 .Z.sin.2n+^z - Z.sin.z = 2n+l
1 Z .tan.z + 1_ Z.tan.2z + &c' ' 1' Z.tan.2nz 7 22 2n+1
or his equation (11)
1 11 -1 1 1 , In+T 2 7 2 ,7.2. ' J sin.2 z) = ^ tan.z ^ tan.2z ^ tan.2 z | ic | tan.2nzj . sin. z
Later in the same paper Babbage found
1. The notation of the right side of this equation signifies taking a term within braces to the- power of the product of all preceding powers', as indicated above each brace; so that in this case /■' ± 1 ^ . 1__ 2 2 ^ 2^ 2n+^ (tan.z) (tan.2z) (tan.22z) (tan.2n z)
Herschel soon noted that the constant of integration had not been correctly determined,■and by the end of June 1812 both .he and Babbage found that equation 11 was properly
1 1 1 1 2 2 .2 2 .. 2sin.2z______y= tan.2z ^ tan.22z ^ &c ^ tan.2n z | .
. _n+l . 2n (2s m .2 z)
Buxton ms.13, p.34; B.ms.C;; and a letter from Herschel to Babbage, pmk July 1 1812, H.ms.R.S.
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1 'I 1 2" 7 , 2" . • f f ' f X-l (_ x+l { x2+l x4+l ■ 2n+l n+1 (x -1)
2n - for X -1 to qet ' J - T — ~ 2 ^ ~ x +1
1 1 1 1 1 7 7 7 .7 7 \ V tail'. 0 ( tan.2 0 ( t‘an'.2 0 ( &c f tan.2 0(, J [~=r- i-rr— \ \-=r J =■ ■ ■ x-l- ' ■
_n+l 7n+l (x + l K
which, by comparisoifwi^h. equation 11 led to the result
Tn+T t .. , ( , 2 . , ( ,_4. , f r_ /- ..2 ( x^l_ ( X -l f x -l ( 5c f x -1 1 (sin.2 z) TT 1 x+l 1 k ^ l 1 ^ 1 { \ x2n+1 j ' 2n+l ( 4—1) sih.z
These examples from Babbage's mathematics illustrate
-some of the problems and methods he was working on. They
also show the very operational, manipulative,character of
his work. The work of other members of the Analytical
Society shared in being purely analytical, as will be seen
later in this chapter.
With the formation of the.Analytical Society Babbage
and Herschel had become acquainted and were soon having
"frequent conversations" about mathematical topics. These
discussions often advanced their researches. On one
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■ occasion, at that time, Babbage
..mentioned the various orders of functions of one variable from which we were led to consider those of more than one variable - Of these Mr Herschel remarked that there must exist a species of partial function; thus if there is a function of two variables the second function may be taken relative' either to the first variable o r .to the second, in a manner somewhat similar to a 'partial differentials, to -this 1 immediately replied that there would also exist two different species of second and higher functions which would thus arise, the second function might be taken first relative to one variable and then relative to the other; or the second function might be taken relative to both at once, which I named the second simultaneous function. This was the origin of partial and of simultaneous functions and it may be observed that it arose from that intercourse between those who were * engaged in the same pursuits, which was so much promoted by the establishment of the Analytical Society.1
Simultaneous functions later came to play an important
part in Babbage's work on the calculus of functions. 2
Similarly, at that same period, there was much concern
with notation. Herschel, for example, suggested the
notation f ^ for the inverse of the function f, and in
particular its use for the inverse trigonometric
functions.^
«
1. Buxton ms.13, pp.27-28
2. See for example his "An Essay towards the calculus of functions. Part IX”. ' Philosophical Transactions. Royal Society. 106 (1816) 179-256.
3. Buxton ms.13, pp.2'8-29. Herschel's use of this notation was anticipated by Heinrich Burmann; see Florian Cajori's A History of Mathematical Notations vol.2 (1929) pp.176 ,270.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129
Babbage and Herschel continued to discuss their work
throughout the summer of 1812 by^-letter. There are several
references in these letters to the annoying burden of
"Cambridge cram" and Herschel could be at times quite
bitter about Cambridge.1 Still Babbage and Herschel saw
analytics as a temptation (admittedly an attraction to the
"cause of reason and truth") from their university studies 2 and seem not to have considered changing those studies.
This correspondence is further evidence that despite much
dissatisfaction with the Cambridge system, the Analytical
Society's function was not to reform that system but to
1. Now my dearly beloved I have written myself sober, so let me give you 1 3/4 words of advice - Read.- Write. - Imprison yourself - Speak to no one - Cram. - Repeat Do all this and you will then be fitted for a fellow of a College. to spend .(or rather lose) an unhappy existence in solitary idleness -. To die forgotten and unlamented, to leave your fame to the keeping of your Bedmaker who may perhaps when drunk retail an anecdote of some trivial nature, of you, and to be succeeded by some one to whom you were in life an obstacle, in death are a cypher - These are thy trophies Cambridge. - Sweet, protecting Alma, whose nourishment is from the vitals of thy sons* --- *What do you say to the idea of Alma Mater A overlaying the minds of her children, (as an old sow, rendered careless by some excessive and beastly indulgence of ht»r gluttony, overlays her pigs, -) crushing every human, every liberal and social feeling. Letter from Herschel to John Whitaker, August 1812, pmk Sept. 10 1812; St.J.ms.
2. The five letters of that summer are in H.ms.R.S. Babbage's are dated, or postmarked, iJune 20, July 10 and "Dec.22 1812 ?" but undoubtedly dates 'from September 22 or October 22 1812; and Herschel's, Jtily 1 and August. f'
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help promote mathematics, and especially analytics. r* The mathematics in this correspondence is in the same
vein as much of their earlier work in that it deals with
•manipulating series and substitution to obtain certain
results. And there are indications in these letters of
concern about the validity of their operations. For
instance, Babbage, in his first letter, asked Herschel’s
opinion concerning the following operation. Given the
expression & + ^ + ^ 2 + &c + ^
let x <1 and n be infinite, and then integrate this
expression to get ^ + ^ 2 + ^ 3 + &c ^ ^
2 3
Babbage1s query was whether this result was "always and
necessarily the same" as if the finite expression was first
integrated and then n was made infinite and x less than one.
Herschel replied that he believed that the two integrated
expressions could not differ except in their constants of
integration, which he thought differed in many cases.^
This example shows- some of-the problems of rigor., even
f for supporters of analytics, associated with the state of
analysis in the early nineteenth century. These problems
made credible the views on analytics of those, who supported
synthetics. ■■ ■
Babbage was concerned with this question because of
1. July 1 1812; H.ms.R.S.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131
theorems he had obtained by a "method of expanding functions
in horizontal lines and summing the columns vertically",
which he later was to call EHSV. In this method Babbage
expanded the function
4> (x, xn ) = Ag + AjX + A 2x2 + &c Anxn
and then for x substituted x2 , x3 , xn . to obtain
(x2 , x2n) = A q + A^x2 + A 2x4 + &c Anx2n
(j, (x3 , x 3n) = A q + A^x3 + A 2x^ + &c Anx3n
d> (x11, xnn) = A + A.x11 + A,x2n + &c A xnn . Y 0 1 2 n
He then summed all the above expressions vertically: r (j> (X, X n ) ■•+ 4> (x2,*x2n) + &c +
nA„ + xn+1-x A, + x2(n+1)-x2 A„ + &c + xntn+1) -xn A . Q — z — i— i *--- =5------2 n X~1 x2-l x - l
Making x'
From this equation Babbage derived various "pretty theorems".
For instance, for A =1, A.=l, A = 1 , and so on, and 1 1.2
(1-x) i (ex-l) + (ex -1)+ sci = x • + 1 .x2 + 1 _ x3 +&c. L J 1.1 1.2 1+x 1.2.3 1+x+x2
1. Letter from Babbage to Herschel, July 10 1812; H.ms.R.S.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. And the right side ultimately equals when x = 1
1 + 1' + 1 + See, ? : -1.22 1.2.32
which equals -1 + )exdx where, Babbage stated, after x
integrating, x = 1. Thus Babbage derived his theorem (7)
j exd x .= (1-x) | (eA-l)+(e^ -1)+ Sccc r when x = 1. ...(7)
But Babbage's concern with the validity of certain
operations really arose with his further manipulations of
his equation (1). Here he multiplied equation (1) by dx x
and integrated to get
j dx ( 4ix-A0) + j dx ( 4>x2-A0) + See =
f A 2 A 2 1 3 A 3 ‘i 1 C - log j (1-x) (l-xz) ^ T . (l-xJ) 3 . See j .
Once'again he derived theorems by ssubstitutlnn for the An and
2 3 Now the left side equals e_ + e + e + Sc, and for 1 ~ T ~ 3
x = 0, C is found to be 1 + 1 + 1 + sc. Therefore ■ 1 . T 3 1 1 1 1 2 3 1 2 3 4 -|ex-l + ex -1 + fex - 1 + SccJ = ^ 1-x ^ 1-x2 ^ 1-x3 ^ See ^
with x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 Babbage obtained I x x2 V I'' I I i - 1 e -1 + e -1 + &c\ 1 2 3 4 1-e I 1 2 ' 1 ( f 2 f 1 (1-x) X e = ll |1+x i1+x+x | See j , wjaich when he let x "approach indefinitely near to unity", / ^ex-l + ex -1 + &cj (1-x)? -1- X e 4i— m 3 J4 See. when x = 1 . Babbage;s results inspired Herschel to derive others by a similar process,'1' In particular, Herschel found t~ n r 17273 (l + n(n-l) + ( n ( n - l ) (rf'-2) (n-3) + n (n-1) (n-2)'\+ Sc I e , 1 2 V 271 1.2.3 J J n ex where the right side is d e when x = 0. He obtained dx 1+2+3 +4 +&c=2e for n = 2 and similar 1 1 172 1.2.3 results for other values of n. In addition to his correspondence with Babbage and his preparation for the approaching Senat^_HpUse 1. Letter from Herschel to Babbage, Aug. 1812; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 Examination, Herschel prepared, in the summer of 1812, a t paper for the Philosophi'c'ar Transactions Of the Royal Society. Entitled "On a remarkable Applicatipn of Cotes's- Theorem", it appears to have been based on at least one of the n^moirs he had read to the Analytical Society.3 -The paper is dated "Slough, Oct.6 , 1812", the same day Herschel left Slough for Cambridge, and was submitted to the Royal 2 Society through his father. In this work, by considering conic sections Herschel derived various equations to which he applied Cotes's theorem to obtain numerous theorems.3 For example, he derived the equation R2 _ a2 ^x—e2) 1-e . (cos. (j>) 2 where R is the distance between au point on the curve and the center of'the conic section, a is the semi-transverse axis, e the eccentricity and and a. In spite of the geometrical origin of his enquiries, Herschel was concerned only with the equation itself and not with whether it was true for all conic sections. 1. See footnote 4, p-122and a letter from Herschel to Babbage, pmk July 1, 1812; H.ms.R.S. A very small fraction of the results in this paper appeared in 1814 under Herschel's name as "New Properties of the Conic Sections" in Leybourn's Mathematical Repository 3_: (1814) 58-59, although this last work was probably submitted previous to the Phil. Trans. paper. 2. Herschel (1879) 119. The paper was published in Phil. Trans. 103 (1813) 8-26. 3. For an exposition of Cotes's theorem see Barlow (1814) . . ;v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 He transformed the above equation into R2 = a2 (1-e2 ) (T+12)2...... by letting £l-2i 1. COS . i n - 1 ^ ^-COS.(t- apply Cotes"s theorem. By letting 0 = 4^, 4^+ 2_rr , , 4^ +' 2 (n-l) tr , denoting the n n resulting values of R by R^, R2, ... , Rr , and applying Cotes's theorem, Herschel derived n R, ... R = an . (1+ X2)n (l-e2)? 2 n ------£l-2 Xn.cos.n 4^+ X2n^2 . jl-2 Xn.cos.n( ii+^) + X211 He then deduced several theorems from this expression by taking particular values of a =0 and n is odd, R, ... R - a11. (1-X 2 )n \L I n ------»rr--- 1- x2n Herschel’s paper was extremely analytical; he even denoted it by the infinite series 4(1-1_+1-1_+ &c) . Indeed, 3 5 7 Herschel had quickly laid aside any reference to the conic 1. By a. direct application of Cotes.'s theorem, (1-2 X cos k + X2) (1-2 X cos ( if . 3-2 n ) + X2 ) . . . x n (1-2 xcos ( d^+2 (n-lk ) + X2) = (1-2 Xncosn 4^ + X2n) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 sections and was not interested in geometrically 'interpreting his theorems. These theorems, however simple their algebraic expressions, it must immediately be seen, become for the most part complicated and unintelligible when geometrically enunciated. They are indeed - (if we may in any case be allowed to consider a curve as unidentified with its equation) properties rather of the equations of the conic sections, than of the curves themselves, - of a limited number of disjointed points determined according to a certain law, rather than a series of consecutive ones composing a line.-*- Thus Hershel's work,,as that of Babbage quoted' above, was very analytical in character. It consisted of formal manipulations of equations with little concern for the meaning of the individual- operations, but only with their universality and .power. . The Analytical Society resumed its meetings with the return of its members to Cambridge in October, 1812. Edward Bromhead had left Cambridge for London that summer to study law at the Inner Temple. John Brass (B.A. 6th wrangler, 1811):, after some inquiries, had decided not to become a member, and Joseph Jordan had left the Society 2 by March 20, 1813. But two others had joined the Society: Babbage1s pre-Cambridge friend John Higman, who soon became its secretary, and Frederick Maule, whose elder brother 1. J. Herschel "On a remarkable ..." Phil. Trans.103 (1813) 8-26. p.26. 2. Concerning Brass see his letter to Herschel, Nov.2 1812; H.ms.R.S. And a letter from Bromhead to Higman, Jan. 23 1813; B.ms.B.L. And for Jordan see a letter from Whittaker to Bromhead, Mar. 20 1813; Br.ms. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 William had been private tutor to Edward Ryan, and who was to die the following year.'1' And Edward Bromhead’s brother, 2 Charles, may also have become a member. Besides D'Arblay, no one else is known to have later joined the Analytical Society. Once again Babbage and Herschel dominated the meetings. Herschel presented a paper on trigonometric functions, "On trigonometrical functions of different orders", which espoused the value of the functional notation and applied these views to various trigonometrical 3 4 functions. Babbage gave four memoirs'. Three of them • were prepared that summer and were quite lengthy.5 The' first, "Memoir On the Properties of Certain Functions", consisted of seven sections, although the last six were 1. .For evidence of membership for Higman see' a letter of his to Edward Bromhead,. Han. 17 1813; Br.ms. For some interesting biographical information on Higman see a draft of a letter from Babbage to Mrs. Dugald Stewart, April 1821; B.ms.B.L. And for Maule see Babbage (1864) 29, Leathley (1872) 233-234 and a letter from Whittaker to Bromhead, Mar. 20 1813; Br.ms. 2. There is no direct evidence for Charles Bromhead’s membership even, though there is evidence that he was friendly with Herschel, Peacock and other members. Some hint of hi's possible membership might be read into ' a letter from Whittaker to Herschel, Mar. 29(18131 H.ms.R.S. 3. H.ms.T. Buxton ms.13, pp.38-39. 4.. Buxton ms. 13, pp.38-39. 5. All three memoirs still exist in B.ms.c. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 but transformations and applications of equations derived in the first section. Babbage began this section by noting that Of the various improvements in notation which have contributed to the advancement of Analysis, none seem to have been of such essential service as the happy idea of difining [sic ] the result of every operation-which can be performed on quantity, by the general name of function.1 And he continued to applaud the idea of a function and its symbolization not simply for the generalization, simplicity, and perspicuity which it brought to its subject, but because^through these merits it amply illustrated the advantages of analysis. Babbage took the functional equation ip x.'$ x J= Xx and noted that its generality was not diminished by substituting t>fx for Xx. Then he put for x, fx, f2x, ... , fnx to get the n equations - . ‘ 7 tfx. W 2X. t ^X = >i>f3X &CC &CC ^fnx. t>fnx = and multiplied these together to get his equation (b): ^n+ljc = >ffx. W 2x. W 3x __ 'lJ f n x . ...(b) $fx Next, he assumed \ . ' 1. Babbage seems ;to have obtained this (definition of a function from William Spence's' Ah Essay.. .of Logarithmic Transcendents. See footnote 2, p.113 and Spence (1819) 73. For the changing conception of a function at this time see Youschkevitch (1976) . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 f tec f tex [. te .1 *fx ' 1 ■ 1 ' 1 ■ n n n r ? n * 1 or y = 1 ofx f 6f x ( SC f or X f 1 / 1 (( \|ifx)n 1 )( i|jfn—1x)n 1 ( ^fnx)n J This became, after multiplication of both sides by 1 n+1 . (, 0f , .n+l x) , n' 1 1 .1 11 , i _n+l n n , n n n n .. . y ( Babbage then transformed i|jx. ((sc = 4>fx by putting ( i|ix)n ^ for \(ix without, a? he put it, diminishing the generality of the equation, and he obtained <()X = <(>fx . <|lX ( i)ix)n 1_ n+1 By employing this equation, he showed y( to be equal to ^ 1 I / I II I n n n n _ n n n f 4>fx f tec f <)>f x- f sc f tejc] =(= ( ijifx.y , and he [ [ ijjfx X " ~ | tfnx [ obtaine’d his equation (e)‘ 1 _1_ I I Y n n % n-1 n n 7 n n y = (jlfx • ( frfx f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ■ The remainder of section one gave four applications of equation (b) for specific values of and by equation (b) 2n+l 2n+2 2 > 4 4 8 2n 2n+^ 1+ x ■+ X Z = (l-x^+ X*) (1-X + X ) &c (1-x + X ) . (l+^CT T T + T - X ^------ ^-The second section made similar substitutions in equation (e). For example, with the same substitutions as above and with n = 2 ,' Babbage derived 1 1 1 , o - T ,A 0 , ? Z 2 2 0n ,,n+l + 1 ? n ^ x4+.x2+ l X 4- X + 1 f &C fx^ 4- X 4- 1 1 T x4- x2+ 1 1 ” |x2n+1- x2^ 1 ) i x 2n+2+ x2n+\ j 2“ Section three transformed many of the applications of the first section' into. expressions .... containing sines/ cosines and tangents by letting.xt x = 2cos 8 ; as did section four for the second section. And the remaining sections converted the various finite products of the first four sections into series by the familiar process of taking logarithms and differentiating. Babbage's second memoir had no title, but related to his method of expanding horizontally and summing vertically, and he noted, as he had in his letters to Herschel that summer, concern over the validity of the order of certain operations. The memoir reproduces many of the results in that letter of July 10, 1812, and is very full of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 errors."^ The third memoir also had no title but is concerned with the values of series of the form A^(cosB )nx + Aj(cos20 )nx2 + &c. .2 and especially . A^. .... x +. A.,...... x_ + &c (sin 6 )n - (sin2 0 )n o Babbage started with the function S (A^x1*) , substituted xv and xv-"*- for x, and added these two equations to obtain 4 where v = cos 0 J-l sin0 .- By repeated substitutions of xv and xv-^ and addition of the resulting equations, Babbage derived his theorem (a): \ 4>.(xvn)+ n (J>(xvn 2) + n.n-1 ij)(xvn 4)+ &c+ n <(>(xv n-2)+ > |a ^x ''' (cosi 6 ) S1 This he then.transformed into theorem (c) where z— log x, 'he = S^B.x1! and B. = A. (cosiB )n . . 2 1 1 ) 1 1 log v Similarly he obtained a theorem (d) by repeated substitutions of xv and xv-2, and subtraction of the resulting equations, and analogous theorems (e,f) for 't’x = S\A. 1|. By combining these equations and. 1. For an indication of the mathematical character of this * memoir see the previous discussion (pp.131-33) of the July 10th letter Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 by successive integrations and by various other transformations, Babbage derived a great number of results. For instance, by an application of (c) and (e) he obtained his equatioh (1)' ‘ (cos0)n (cos 23 )n (cos 39)1 (im.the process of which he took l-l+l-l &c = 1/2) . This equation became, by successive integration, equation (3) (1°? X) 2k + C2 (log x) 2k~2 + &c + C2k = -S xA+- x"1 (-1)1 , 1.2...2k 1.2...2k—2 i2k(cos i0)n and equation (4) (log x)2k+1 + c2 (log x)2k_1 + &c + °2k(log x) = 1.2...2k+l 1.2...2k-l 1 -S x 1 - x~x (-1)1 , i2k+1(cos'i0)n the c's being constants of integration. Equations (3) and (4) then had certain of their variables replaced by particular values or were transformed.in various ways to dbtain numerous results. Such were . , q.costi , ,Q.cos 2n - -„,cos 3h 1 = (cos0) x «(cos 20) x (cos 30) x &c, and the values of the series tan0 + tan 30 + tan ’5'0 + &c, pj+r 32k+i pm — and of the product sinn ' sin 2n . 2k+l _2k+l (cos 0) x (cos 20 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 Babbage's final paper was titled "Remarks on Interpolations1 These works of Babbage show that some of the members of the Analytical Society were producing mathematics. And his works underline the analytical character ^)f that mathematics which was strikingly different from other British mathematics. Perhaps encouraged by all these mathematical works and certainly prompted by a wish to better promote themselves and their views, the members of the Analytical Society decided in November 1812 to publish a volume of their 2 memoirs. It was hoped that several members would contribute papers1, but only Babbage, Herschel and probably 3 Maule offered to prepare an essay for the volume. Maule soon became very sick'and died the following August, so the volume was completely written by Babbage and Herschel, ( they being, in Herschel's words, "the ringleader^, if not the only actors in this literary assault upon the peace 4 and quietness of the world." After some approaches to London printers, it was decided to have the Cambridge 1. Buxton ms.13, facing p.39 and see also a letter from Babbage to Herschel marked "Dec. 22 1812?" but' which probably is Sept. 22 or Oct, 22 1812; H.ms.R.S. 2. Buxton ms.13, p.39. 3. For Maule's proposal see a letter from Herschel to Babbage, Feb.8 1813; H.ms.R.S. and a letter from Maule to Babbage, Jan.16 1813; B.ms.B.L. 4. Letter from Herschel to Babbage, Jan.12 1814; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 144 ' University Press print the work.1 Some of the difficulties the university press had with the volume are interesting in that they indicate how different the Society’s work was from the usual Cambridge texts. For there were many problems with the availability of type to print the expressions used. Thus Babbage wrote to Herschel concerning Herschel*s memoir I think they have composed about 12 [pages] but can not print them for want of a particular kind of small numerals which are daily expected.2 Whittaker told Edward Bromhead Awful Brackets for the' expressions requisite - Smith the university printer, had not any large enough, nor plenty - forced to send to town for more.3 ■And again, a month later, Whittaker wrote Smith the university printer has great difficulty in printing the stuff Babbage has written, he says he never put together such crabbed stuff in his life.4 Not everyone familiar with the Society, however, thought that the Society had been wise in deciding to publish its memoirs. William Henry Maule, elder brother of Frederick, friend of many of the Society's members, senior wrangler in 1810, and critic of the state of English 1. Letter from Maule to Babbage, Jan.6 1813; B.ms.B-.L.' 2. Letter of May 25 1813; H.ms.R.Sj 3. Letter of Feb.16 1813; Br.ms. 4. Letter of Mar.20 1813; Br.ms. - • Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 mathematics, wrote in February 1813 to his former pupil Edward Ryan If I had been at Cambridge, I should have ventured to suggest to those members of the Analytical Society with whom I am acquainted that they should have sent their memoirs, or some of them, to ■ Leyboume, instead of publishing them independently. By that mode of publication they would have obtained a wider circulation for their discoveries than by that which they have adopted, at a much smaller, or rather at no expense; This view was shared by Edward Bromhead, who thought that the Society's memoirs should have been published in the Society's name in the' Philosophical Transactions, following the example of the Society for Promoting Animal Chemistry. In spite of such feelings, the Analytical Society perservered in its views and the' Memoirs of the Analytical Society, for the year 1813 appeared in late November 1813, a year after the decision to publish.3 1. Leathley (1872) 241. For Maule1s views on English mathematics see his favourable reviews of Woodhouse’s Trigonometry and Isoperimetrical Problems in the Monthly Review 65 (1811) 36-39 and 39-45. Maule felt that the chief two causes for the lack of British progress in mathematical science were the way in which mathematics was studied in the universities and the adherence to the fluxional notation. 2. Letter from Bromhead'to Babbage, [ca. late Nov., Dec. 1813]; B.ms.B.L. On the Animal Chemistry group see N.Gt.Coley "The Animal Chemistry Club: Assistant Society to the Royal Society" Notes and Records. Royal Society. 22_ (1967) 173-185. 3. Letter from Herschel, Nov.19 1813, and a letter from Babbage, Nov.30 1813, to Edward Bromhead; Br.ms. Babbage, according to his autobiography, suggested that the most appropriate title for the Memoirs would be "The Principles of pure D-ism in opposition to the Dot-age of the.University". Babbage (1864) 29. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 There were three papers and a preface in the Memoirs all published anonymously. Several members of the Society * had felt, in early February 1813, that the authors’ names should not be affixed to the memoirs, but their reasons for this are unknown.^ In any case, Babbage and Herschel agreed; Herschel thinking that anonymity would have.the advantage of saving appearances in allowing each of them "to give a greater number of Memoifs than we otherwise- could."2 Babbage wrote the first memoir, "On Continued Products", and had it ready for the press early in 1813. His quickness is not so surprizing since this memoir was mostly composed of parts of two papers he had delivered to the Analytical Society in the fall of 1812: the first four sections of "On the Properties of Certain Functions" and parts (with errors corrected) of the untitled paper relating to the method of expanding horizontally and summing vertically. By February 16 Babbage was correcting the press, but the printing went on very slowly, with his 3 memoir not being completely printed until May. The 1. Letter from Babbage to Herschel, Feb.19 1813; H.ms.R.S 2. Letter from Herschel to Babbage, Mar.2 1813; H.ms.R.S. 3. Letter from Whittaker to Bromhead, Feb.16^813; Br.ms. Letters from Babbage to Herschel, Feb.19 and May 1 1813; H.ms.R.S. Buxton ms.13, p.42. The manuscript of this memoir is preserved in. B.ms.C. -/ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 slowness of the printing allowed Herschel to incorporate into his memoirs the results «of his. later research, which led Babbage to write iiy-fche summer of 1813 I suspect from the slow progress of the printing that my paper will appear too elementary and simple when placed by the side of yours.1 The two remaining memoirs were written by Herschel. He began his first memoir, "On Trigonometrical Series; Particularly those whose Terms are multiplied by the' Tangents, Co-tangents., Secants, Sc. of Quantities in •Arithmetic Progression; together with some singular Trains formations.", on February 7 1813 -at his home in Slough, 2 having left Cambridge in January after his graduation. The printing of'this memoir was not completed until about August.^ The Memoir was concerned, as was the third paper Babbage read '’’to the Analytical Society in the fall of 1812, with finding general methods of summing series whose terms were divided by sines, cosines and other trigonometric functions.- It may have been an extension of his own earlier paper on trigonometric functions, for it shared with that former paper a discussion of the notation of 1. Letter from Babbage to Herschel, June 30 [must be July 30] 1813; H.ms.R.S. 2. Letter from Herschel to Babbage, Feb.8 1813; H.ms.R.S. 3.' Letter from Herschel to Babbage, Aug.20 1813; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 148 functions.''' In any case, the Memoir derived various trigonometrical expressions and results. For example ■A tan. (4i-3') tan. (2i 0 '-'it ) _ ^ = cos. i—1) -sin, e II^ 3 .i ) V '4' 1 ..if (4 9 el- tan.J— I (2i-l) 1 ' °6 -it |cos . (4i+l) e+sin. ej tan. (4i-l) 5 Vt 4T-1 e Five notes, occupying a few pages more than the memoir itself, followed. Most of them contained miscellaneous results which Herschel undoubtedly felt were too good to be lost; and as late as June 27 1813, he was adding new results to the notes.3 Herschel’s attitude towards analytics, like Woodhouse's and probably most early nineteenth-century English analysts, was a very formal and manipulative one. This is illustrated in Note III where in dealing with fz (x) for z functional or imaginary, Herschel looked upon the equation as the definition of the operation: ”... the only meaning we can assign to fz (x) is, that it/is_ that function of z and x which is here connected to it by the 4 sign of equality." However , Herschel was also concerned 1. Buxton ms.13, pp.38-39. See also a letter from Herschel to Babbage, Feb.8 1813; H.ms.R.S. 2. P denotes the product 3. Letter from Herschel to Babbage, June 27 1813; H.ms.R.S. Many of the results in this letter were put into Note v, whose first paxt' had been found the previous March. Letter from Herschel to Babbage, Mar.2 1813; H.ms.R.S. 2 — 4. Memoirs (1813) 52. f (x) denotes functional composition. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 with the legitimacy of his results, and this seems, for him, to have depended on skill rather than regulation. The operation by which these equations have been derived from (B) and (C), is of such a nature, as to leave the mind unsatisfied,'and hesitating as to its legitimacy. Such cases are of frequent occurrence in the theory of exponentials; and it must be confessed, that the management of them, so as to avoid drawing conclusions manifestly absurd, is among the most delicate and at the same time interesting points in the whole theory. We seem as it were treading on the very verge o f ' Analysis, on the line which determines truth from falsehood., and feel ourselves placed in the situation of .one who fears to pursue to the utmost, the deductions of his reason, through suspicion of . some latent error, or mistrust of his own powers. Hence, once again, the reverence for analytics as an almost ’unreasonable’ tool for discovering the unknown was a part of Herschel’s view of analytics. The_individual mathematician was swept along by its power. Originally Babbage and Herschel had each planned to contribute one paper to the' Memoirs ■ But when no other members of the Society presented papers, Herschel decided in May to write a second memoir, feeling that the Society 0 would "look rather foolish, without at least a third”. And, in part prompted by his wish to have something besides trigonometrical transformations in the volume and because of his mathematical research since leaving Cambridge, Herschel decided to write on equations of differences and 1. Tbid. 64. 2. Letter from Herschel to Babbage, May 4 1813; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 their use in solving functional equations.^ The third Memoir was entitled "On Equations of Differences and their Application to the Determination of Functions from Given Conditions”. It was the longest of the memoirs, almost equalling the other two in length, and was divided into three parts. While Herschel had previously worked on finite differences, it was only about the end of May or early in June 1813, after working on Laplace’s Mdcanique Cdleste and being inspired by it, that he had sufficient material on finite differences to begin another 2 memoir. He was to work on the memoir the rest of that summer.• The first part gave a general theory of equations of differences of the first degree in one variable. The second integrated certain equations of differences. Both these parts were completed before the third which dealt with functional equations and their solution by means of finite differences. Herschel appears to have begun the researches which led to the last section sometime in July; for on July 25 he wrote to Babbage I hasten to communicate to you the results of some researches I have been making in the theory of determining functions from given conditions, in order to avail myself of your knowledge of the subject in pointing out how far my mode of proceeding exceeds in generality 1. Letter from Herschel to Babbage, Feb.8 1813; H.ms.R.S. 2. Letter from Hershel to Babbage, "Sept. 1813” [actually after May 25 or very early June 1813]; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 what has hitherto been done, or whether it exceeds it at all.l At that time Herschel was very excited about his general method of solving functional equations of the first order and any number of variables; a feeling which Babbage shared: „ I have not for a long time received so much pleasure as idle perusal of your letter gave me. Your solution of the functional equation 1fF Despite the initial successes, more was not forthcoming, and soon afterwards both Babbage and Herschel felt that the theory of functions was "unlikely to derive much farther assistance from the method of differences".3 By August 20 Herschel was'almost finished the third part, with the first two parts in the hands of the compositor. And by - 4 October 13 the "third Memoir was nearly completely pnntfed. Herschel was so proud of tbememoir that he ordered fifty 1. Letter from Herschel to Babbage, July 25 1813; H.ms.R.S. 2. Letter from Babbage to Herschel, "June 30 1813" [must be July 30] ; H.ms.R.S. 3. Buxton ms.13, p.47. " 4. Letter from Herschel to Babbage, Oct.13 1813; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. y 152 0 extra copies of it for himself. This third memoir reveals the increasing concern of Babbage and Herschel with developing what was to be a new branch of mathematics, the calculus of functions. . The preface to the Memoirs appears to have been Babbage’s idea, and he certainly did most of the work in writing it.1 The preface was intended, in Babbage’s words, 2 as "a brief outline of the history of pure analysis.” While it was mainly Babbage’s effort, many others read and commented on it. William Maule, Ryan, Higman and Herschel > all read it separately at various tlmes in 1813, and the Analytical Society met on Wednesday, May 26 to read the preface and may have gathered once again for this same 3 purpose just before the start of term in October 1813. Babbage began work on the preface soon after he had ■m finished his memoir (about January 1813) and continued to revise it until it was printed in late October and early November 1813.4 It was the last part of the Memoirs to be 1. Letter from Herschel ’to Babbage, Mar.2 1813; H.ms.R.S. 2. Buxton ms.13, p.40. 3. Letter from Babbage to Herschel, May 25 1813, and a letter from Herschel to Babbage, Oct.13 1813; H.ms.R.S. 4. Letter from Whittaker Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 153 printed. The preface began, as Babbage had wished, "metaphysically".^ It asserted that the role of mathematical analysis was "to examine the varied relations of necessary • truth, and to trace through its successive developements, 2 the simple principle t p its ultimate result”. The advantage of analysis in dealing with long and intricate trains of reason lay mainly in the accuracy, simplicity and conciseness of its language, ail of which aided the mind, and in the essence of analysis itself - the separation of 'v— ^ ' 3 the subject into its components;. These causes of the superiority Of "Analytical Science" were well illustrated • by its history, and most of the preface was devoted to a history of "pure analytics". Such areas as the resolution of equations, the differential calculus, differential equations, the methods of finite differences and of variations, functions, and number theory were outlined. A very short account of the Analytical Society, at Herschel's 4 urging, was also included. 1. letter from Babbage to Herschel, "before May 1, 1813" [but is in response to Herschel's June 27 1813 > [fetter].; H.ms.R.S. " 2. Memoirs (1813) i 3. Ibid. i-ii. This was much the same sentiment as expressed in Babbage’s 1812 paper "On the Properties of Certain Functions". See p. 137. 4. Letter from Herschel to Babbage, June 27 1813; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 154 ... that it consists of a few individuals) perhaps too sanguine in their hopes of promoting ■their favourite science, and of adding at least some trifling aid to that spirit of enquiry, which seems lately to have awakened in the minds of our countrymen, and which will no longer suffer them to receive discoveries in science at second hand., or to be thrown behind in that career, whose first impulse they so eminently partook. The time perhaps is not far distant, when such an attempt will be regarded in an honourable light, whatever may be its success.1 The preface concluded by noting that while the "golden age of mathematical literature” was undoubtedly past, there was justification for high expectations for the future. And it appealed for a digest which would reduce "into 2 reasonable compass the whole essential part of analysis”. Thus the preface stressed the power of analytics. Its chief concern was hot with applying analysis but with pure analytics. To stress analytics was very different from the usual British defence of synthetics'. To concentrate on pure analytics was to make a virtue of what was usually accepted as the flaws of analytics. Clearly the zealousness of reformers was involved here. Two hundred and fifty copies of the Memoirs were 3 printed. Of these, the Analytical Society had ordered one hundred copies, with Herschel and Babbage responsible 1. Memoirs (1813) xxi. 2. Ibid. xxi-xxii. 3. See a copy of the printer’s bill bound in a copy of the Memoirs once belonging to Babbage; St.J.ms., Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 155 for the remainder.'*' The total cost had come to ^132.9.0^ much more than the Society had expected. A subscription from the membership had been voted .at the May 26th meeting, the Society had given up its room due to the costs of printing, and finally the Society's share of the cost had 2 been equally divided and paid for by its members. Apparently the sales of the volume were very slow - not until 1816 did Babbage first meet someone who had bought 3 the Memoirs - and no profit was made by the Society. • Indeed, Herschel wrote many years later that With respect to the proceeds, I am afraid it will be better not to stir the matter, and regard them, as £ 0:0:0 - lest if any Enquiries be made of the booksellers they should bring in a bill for commission & warehouse-room, a thing little to be desired The reaction to the Memoirs at Cambridge seems to 1. Letter from Herschel'to Babbage, Jan.12 1814; H.ms.R.S. 2. Whittaker says the subscription was £3, Babbage, who did not attend the meeting, says^ 5. Letter from Whittaker to Herschel, June 1813, and a letter from Babbage to Herschel, "after May 20 before July 5" [early June ]1813; H.ms.RIS. Soon after, June 5, 100 letters of subscription were printed. Some of these are preserved in B.ms.B.L. (Add.Ms. 37203 f.80) That 100 copies were printed suggests that the Society had high hopes for its future. 3. Letter from Babbage to Bromhead, Feb.16 1816; Br.ms. 4. "Letter from Herschel to Whittaker, Feb.23 1822; St.J.ms. ■This letter indicates that ten members shared in the cost. These certainly included Babbage, Herschel, Whittaker, Peacock and Mill. There seems to have been only about ten persons in the Society by the end of 1813. Letter from Babbage to Bromhead, Nov.30 1813; Br.ms. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 156- have been one of general bewilderment. Babbage wrote Of course much nonsense is talked about them here; but Ikhave not heard criticism yet venture beyond the "Second line of the first Memoir: of which men ask "is it to_ be found in Jemmy Wood" and if not they divide by x and are lost in the cloulds of -i|) 's which follow.1 However, some, at least, approved of the work. The Master of Jesus College bought the work "because he was glad to . 2 see that kind of spirit among the young men." And Bland, one of the moderators of 'the 1814 Senate House Examination, who had privately tutored 'Herschel in mathematics for two years, asked, ironically, during the Examination a few 3 questions taken from the' Memoirs. Outside of Cambridge, the Memoirs appear to have attracted little notice. No reviews of the work appeared in any of the periodicals, x 4 much to Babbage's anguish. Herschel had sent a copy of 1. Letter to Bromhead, Nov.30 1813; Br.ms. 2. Letter from Whittaker to Bromhead, Jan.26 1814; Br.ms. 3. Ibid. A letter from Whittaker to Herschel; Jan.28 1814, H.ms.R.S., and a letter from Slegg to Babbage, Feb.4 1814; B.ms.B.L. As only problem papers were printed before 1827, and all questions from books were proposed viva /voce, Bland’s questions from the Memoirs have not been recorded. There are some similarities between ■the Memoirs and the printed Cambridge problems for 1814. ( Bland's question 16 of the Tuesday Evening paper: to find the sum of tan'.A tan.2A +' tan.3A - &c,'which 1 1 3 appears in Note V, p.63 of the' Memoirs. Question 8 of the Monday Morning paper is the same as Babbage had solved in 1812; see p . 123. F°r Bland’s tuition see a letter from Herschel to Bland, Jan. 1831; H.ms.R.S. 4. Letter from Babbage to Herschel, Aug.l 1814; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 157 the' Memoirs to Playfair whose only response was that he "had been highly gratified by the perusal of the several Essays" and especially by Herschel’s.^ If the work was noticed by others, the general response was undoubtedly the / common complaint against analytics, that it was meaningless manipulation of symbols. As William Whewell wrote in the British Critic many years afterwards In this publication 1 Memoirs] , the extraordinary complexity and symmetry of the symbolical combinations sorely puzzled the yet undisciplined compositors of that day, and led unmathematical readers to the conviction that the wfjole was a wanton combination of signs', left to find a meaning for themselves, . ...2 While the Memoirs may be regarded as the zenith of the Analytical Society’s activity, it was also, as Herschel 3 remarked, "an expiring effort". Although the Society had faced opposition from without and some minor internal dissentions since its founding, the chief cause of its dissolution at the end of 1813 appears to have been that it was a society composed-of students yet never really 1. Letter from James Grahame to Herschel, Feb.26 1814; H.ms.R.S. X 2. [W. Whewell ] "Transactions of the Cambridge Philosophical Society. Science of the English Universities” British Critic 9^ (4th series, 1831)- 71-90. p.85. Edward Bromhead felt that the Memoirs were "much more profound than I any way expected, they are too profound to do us any good, & not one mathematician in 10 can understand them." Letter to Babbage, tea. late Nov. or Dec. 1813 ]; B.ms.B.L. 3. Letter from Herschel to Bromhead, Nov.19 1813; Br.ms. 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 158 related to student interests.^ Thus Bromhead wrote to Babbage in December U813 If the Society fails, it will fail from having taken too imposing\an attitude. It ought to have been more [ common place], & more for the Capacity of Undergraduates. It was wrong to publish the Memoirs as was d The Analytical Society had been founded nearly two years before by a group of individuals sharing a common interest, and-who soon became a close circle of friends. Despite an active beginning and.an extensive reputation (it was rumoured among the English mathematical community that Woodhouse was at the head of the Society) the Society had failed to attract new members)^ The reasons for this failure were undoubtedly the same as those responsible for the inactivity of most of the Society's members: their primary concern with the university examinations on which the Society's pursuits had no bearing and their failure to develop, or in many cases even maintain, their mathematical interests after graduation. Five members had graduated in January 1813, four of 1. For external opposition see Babbage (1864) 29, a letter from J. Grahame to Herschel, June 1812; H.ms.R.S„, and a letter from Herschel to Bromhead, Nov.19 1813; Br.ms. For internal dissention see, for example, a. letter from F. Maule to Babbage, Jan.16 1813; B.ms.B.L. 2.- [Dec. 1813]; B.ms.B.L. 3. Letter from Herschel to Babbage, Aug.20 1813; H.mslR.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 159 whom were wranglers. Herschel, who had graduated as senior wrangler, left Cambridge in January but continued to do research in mathematics. He became a fellow of St. John’s in March, and an F.R.S. in May, spent the Fall at Cambridge where with Babbage £.nd Ryan he became very interested in , chemistry, and then went to London in January 1814 in order to study law.'O Peacock was second wrangler, contending with Fearon Fallows for that position rather than with / Herschel, whose position was' secure. 2 Both he and Mill, who was sixth wrangler that year, tutored privately after their degrees and obtained the two vacant fellowships at Trinity the following year. Peacock continued his interest in mathematics but Mill went on to other interests:, as first Principal of Bishop's College, Calcutta (1820-38), Regius Professor of Hebrew at Cambridge (1848-53) and Canon of Ely (1848-M).3 Thomas Robinson, 13th wrangler and second classical medallist, also went on to a career in the Church in India and at home, being chaplain to Bishop Heber, .Archdeacon of Madras (1828-35) , Professor of Arabic at 1. Herschel (1879) 120, Buttmann (1970) 13, letter from Herschel to Babbage, Oct.13 1813; H.ms.R.S. .And a letter from Herschel to Whittaker, Jan.10 1814; St.J.ms. 2. Pryme (1870) 167; for an interesting anecdote see Winstanley (1940) 152,. and for an account of part of the Smith's Prize Examination (Herschel first prize, Peacock second) see Mary Milner Life Of Isaac Milner (1842) 524-525. 3. See the D.N.B_. article. He did maintain some mathematical interest as is shown by his translation of Bridge's Algebra into Arabic while in Calcutta; see Gentleman's Magazine (18541)-205. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 160 Cambridge (1837-54) and Master of the Temple (1845-69).^ He published many religious works, but nothing on mathematics. The fifth member, Michael Slegg, did not appear on the honours c list and went to London to study law, although there are some indications that he also continued his study'of 2 mathematics. At the next graduation, January 1814, five more members of the Society completed their degrees. Gwatkin and Wilkinson were first and second wranglers respectively. Both became fellows that year, Gwatkin remained at Cambridge and gave private tuition, and Wilkinson went to London to study law. Whittaker graduated thirteenth wrangler, also became.a fellow that year and also gave private tuitions, but then pursued interests m theology and a career in the Church. The remaining two members, Babbage and Ryan, did not appear on the honours list.3 They 4 became brothers-in-law that summer by marrying sisters. 1. See the ^-N.B. article. 2. For example see a' letter from Bromhead to Whittakeyr, Feh.ll 1814; St.J.ms. 3. Babbage graduated as Captain of the Gulph, that is at the top of those candidates for honours who failed to make the honours list but were allowed a degree. Letter from Herschel to Babbage, Jan.26 1814; H.ms.R.S. Whittaker wrote that Babbage, who is in the gulph, but would without doubt have been plucked if he had not been classed, worked one of them [ a few questions were asked from the' Memoirs ], & report says it .was the only thing he did. Letter from Whittaker to Herschel, Jan.28 1814'; H.ms.R.S. 4. Moseley (1964) 52. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 Ryan then studied law, became Chief Justice of Bengal (1833), Privy Councillor (1843-65), and first Commissioner of the Civil Service.'*' Babbage continued his mathematical research while beginning what turned out to be a life-long search for a suitable position. Of the remaining members of the Society, Edward Bromhead (B.A. 1812) had left Cambridge in - the summer of 1812 to study law in London and from his country seat continued for many years his interests in mathematics. His brother Charles (B.A. 1816) became a fellow of Trinity (1818) and also for a time studied mathematics. Higman was third wrangler in 1816, took private pupils and became a fellow of Trinity in 1818. His only publication was A Syllabus Of the differential and integral calculus (1826). And, finally, D'Arblay graduated as tenth wrangler in 1818 and became a fellow of Christ's College that same year. After pursuing his mathematical and scientific interests for some years, he settled into a career in the Church and followed theological interests. To have obtained such high positions on the honours lists, as most of the members of the Analytical Society did, meant that considerable effort had been spent in preparing for the examinations, an effort which decreased participation in the Society. After graduation most members left Cambridge to take up positions which had no 1. See the D.N.B., article. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 162 mathematical or scientific connection, and most of those who remained shifted their interests. There were few meetings of the Society in 1813, and there are no records of any papers being read that year.1 With the graduation of 1814, few members remained at Cambridge'and the Analytical Society ceased to exist, even .in name. The Analytical Society had not only served to promote analytics but had inspired and encouraged the strong analytical views of its members. While as an organization it quickly faded away, its spirit, or rather the broad influences active in the formation of the Society as well as the motives of its members, flourished. Some of its members maintained a close working association. One of these was John Herschel who, as has been seen, was very active in the Analytical Society from its start. Even after leaving Cambridge in January 1813, Herschel had continued to be anxious about the Society and wrote to Babbage I trumpet its fame as much as I can, when X meet with any one who can at all enter into the subject. It is true, my opportunities are not very frequent, but what there are are precious. For heaven's sake keep up a solemn serious appearance about it at Cambridge, and be for some time very cautious whom you admit. We really now must begin to be somewhat m earnest, and avoid "everything common or unclean". What say you to determining an annual subscription, that each member may know what he has to pay. For instance three guineas (or more or less) per An:' or else a. donation ofjj jE 50 at first entrance in lieu- of it. - Do you 1. Except for a paper Herschel seems to have communicated to the Society very early in 1813. Memoirs (1813) 87, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. think one meeting in a term enough? I think no member should be elected without a written notice from himself that he wishes it, and a written certificate from' one' member that he is a fit person. Perhaps it would be as well (as soon as it can be done) to contrive to have another secretary resident in London, - at all.events to transfer the "seat of empire" thither as soon as possible. For X repeat it again and again, we must not be a "Cambridge" Analytical Society.1 Besides his contributions to the Memoirs, he had prepared a pKjpar to be read to the Analytical Society, - had advert is eX^the Memoirs in various newspapers and journals, had shown Babbage's memoir to Ivory, Wallace and Leybourn whp^declared that they never saw its equal in typography", a W had had fifty copies of his second memoir printed h he intended to use "as may seem most conducive to the \ \ 2 good ofVthe Sdciety." When the Society did fail despite all of his'vefforts, Herschel was very despondent. His views on what was wrong with English mathematics echoed the frequently expressed mournings of others: While I admire that powerful enthusiasm which, from the midst of the dry details of law can 1. Feb.8 1813; H.ms.R.S. Herschel may certainly serve as a model, although a somewhat anomalistic one, for Rothblatt's "independent student" of late Georgian Oxbridge. See p39 , footnote 3 . 2. Letters from Herschel to Babbage, Aug.20, Oct.13 1813; H.ms.R.S. I have found only two advertisements, one appeared in Leybourn' s' Repository' 4 (1819) 39, the second was in' The' Times for Thursday, Sept.23 1813, p.2 and read: "In the press-and speedily will be published, the 1 vol. of The MEMOIRS of the ANALYTICAL SOCIETY for the year 1813" Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. \ 1 6 4 draw forth your ideas in such speculations as ' your letter exhibits, I confess I am by no means, so sanguine, although not less sincerely desirous of contributing to .the introduction of a better taste in analytics than at present prevails.. - The ill success of a first undertaking (the Anal. Soc.) although it-has not in the least damped my ardour in this respect, has yet a good deal sobered it. The fire,of enthusiasm spreads only where it meets with inflammable matter to receive & cherish it - and how few, how very few are those who are disposed to enter heart & soul into a task of such gigantic labour, and such diminutive reward. Of that' few again, how small a proportion have the time, or the peculiar turn of mind so necessary to realize their plans. It is in vain to dissemble There is little or no taste for these things afloat - The math ? are not here as on the continent considered as a branch of elegant literature. They lead to no public distinctions, and afford no prospect, of pecuniary reward The publication of a Math,, work, particularly if it goes one step beyond the comprehension of Elementary readers is a dead weight & a loss to its author. - 1 7 Nevertheless, Herschel did persevere both in his mathematical research and also in attempts to promote analytics,. In close touch with Babbage, he was to pursue his interests in analytics for several years. And together with Peacock and Babbage, Herschel was to extend his efforts in the following few years to an active reform of Cambridge mathematics, an endeavour of which the Analytical Society seems never to have dreamed. The Analytical Society was a result of the conjunction of a number of circumstances of early nineteeth-century England.- Its existence and activities revealed the 1. Letter from Herschel to Bromhead, Nov.19 1813; Br.ms. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. influences of Cambridge with its studies and of the widespread belief in the inferiority of British mathematics. The Analytical Society was a mathematical society. It concentrated on studying and promoting research in analytics and in this way participating in the revival of British mathematics in the same fashion as Woodhouse, Ivory, Wallace, and others were doing. The Society did not exist to reform Cambridge studies, even though dissatisfaction with those studies had been a factor in its formation. The mathematics which some of its members produced reflected their emphasis on pure analytics. This emphasis permitted the obvious criticism and consequent neglect by others of the Society’s mathematics as meaningless manipulations of symbols. The ' Analytical Society had little direct impact on British or Cambridge mathematics. But its existence, although brief, illustrated the working at Cambridge of certain influences which were ultimately to lead to the reform of the mathematical studies there. Some of the Society’s members were to' spark the reform movement at Cambridge a few years after the dissolution of the Society. The Analytical Society, therefore, was the precursor of the Cambridge mathematical revival. The existence of the Society also served as an aid in the mathematical research of some of its members and, must also have helped to reinforce their strong views on analytics. A few of these persons were to continue to do much original research in mathematics. The work of two of them, which is the subject of the following Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 166 chapter, also displayed the effects of those same forces which had led to the formation of the Analytical Society. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V. The Mathematical Concerns of Charles Babbage and John Herschel 1814-1822 The Analytical Society is usually remembered - incorrectly - for its reform of Cambridge mathematics. Some of its former members were indeed key figures in the reform. But a greater part of their efforts was directed to research in mathematics rather than to reviving Cambridge studies.1 The dissolution of the Analytical Society at the end of 1813 had affected neither the association \ior the ardor for mathematics of many of its members. In particular, Charles Babbage and John Herschel maintained a close friendship and a considerable exchange of their mathematical investigations. This mathematics is.. ^ significant to the historian in its own right, for the history of the development of mathematical knowledge. It is also important because of its intimate relationship to some of Babbage's and Herschel's other activities in the period 1814—1822. Their mathematical work at that time provides a deeper insight into their views on analytics, y and in turn these views are connected with their efforts to promote mathematics in England. The present chapter « will outline Babbage's and Herschel*s mathematical concerns 1. Their mathematics has been largely neglected by historians. For an exception see the book-length preliminary study of Babbage's mathematics, Dubbey (1978) .. . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 168 from 1814 to 1822 outside of the Cambridge revival * movement in order to better understand their intentions, to illustrate their views on analytics, and to show that their primary concern in these efforts was to make mathematics in England a profession. While Babbage and Herschel spent most of the fall of 1813 at Cambridge performing chemical experiments, they also continued to do some work in'mathematics. Late that fall, H^schel made a breakthrough in functional equations N by giving a Solution of the second order functional 2 1 equation <(> x = x . He had been led to the solution by 2 Babbage's first paper for the Analytical Society in 1812. In addition, at about the same time, Herschel also found 2 another solution of x = x and a method t o r obtaining particular solutions of with work, on generating functions, logarithmic transcendents and first degree differential equations were published in the fall of 1814 as "Consideration of various Points of Analysis^ Dated January 29, 1814, the paper ’.presented various results in the areas named above in the language of -the calculus of generating 1. Buxton ms.13, pp.63-68. 2. Babbage's paper was "Solutions requiring the application of Mixed Differences". Ibid. 64. 3. Philosophical Transactions. Royal Society. 1Q4 - (1814) 440-468. Read May 19." Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 functions. Like Babbage in his enthusiasm for the concept of a'function, Herschel stressed the potential of the calculus of generating functions for the "speculative philosopher" to view the arrangement of analysis as a whole.1 He began by laying down a number of rules governing his use of the functional notation and the method- of separating, symbols of operation from those of quantity. Then he derived a sequence of equations the last of which was applied, using results from Spence's Logarithmic Transcendents, to the summation of certain classes of series. His results on functional equations followed, as did a concise method for deriving known f ' theorems about differential equations of the first degreeSx Herschel seems to have done little else in mathematics during most of the remainder of 1814, except for eight problems with solutions for Leybourn's Mathematical ^ 2 'Repository. In January he had gone to London against the wishes of his parents to enter Lincoln's Inn to study law, or, as Bromhead remarked, to "eat his Way to the Bar".3' Apparently Herschel had no interest in entering the Church; the profession of law was the least distasteful alternative. Yet it was his interests in science and particularly in 1. Ibid. 440-441. 2. Questions 362-369 in the Mathematical Repository 4 (1819) 61-71. 3. Buttmann (1970) 15. And a letter from.Herschel to Babbage, Sept.21 1814; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. chemistry that were strengthened during his stay in London, especially by attendance at the meetings of the Royal Society and by his acquaintance there with W.H. Wollaston and J. South.1 Babbage, who had sat for the Senate House Examination in January of 1814, remained at Cambridge until early \ y June doing some chemistry and maintaining his interest in functional equations but doing little research in this area. However,•about mid-June, Babbage's enthusiasm was re-awakened, probably by a second look at a letter of May 2 12 from W.H. Maule. In his letter Maule obtained the first general solution of i|j n (y) = y which he gave as : ip y = ‘ (j, ^ (-l)n ()> (y) ) .3 This result led Babbage to the substitution the following month on the subject of functional equations and.-made a number of discoveries.3 These results, along with others obtained in the fall of 1814 and very early » _ 1. Buttmann (1970) 16-17. 2. Buxton ms.13, pp.76-77. 3. Ibid. 71-72. And the letter from Maule to Babbage, May 12 1814; B.ms.BJL. | • 4. Buxton ms. 13, pp.93-|9 8. 5. He told Herschel about these discoveries in a long letter of Sept.22: letter from Babbage to Herschel "after Aug.3; after Aug.10"; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1815, were pu.t together about February, 1815, as "An, essay towards the calculus of functions" and published later that year.'1' Babbage's aim in the "essay" was to' present an outline of his new calculus concerning the determination of the form of an- unknown function from given conditions. Thus he was led to solving various. classes of functional equations. Babbage stated his belief that "the solution'of functional equations must be% sought by methods peculiarly their own". This was undoubtedly due to his and Herschel’s continuing view of the inadequacies of using finite differences to solve functional equations, as well as to their belief in the potential of the calculus of functions as a powerful 2 branch of analysis. Herschel felt that,Babbage's discoveries were laying "the foundation of a calculus totally new, and immensely powerful.3 Babbage used various direct methods of solution and, above all, symmetrical functions. Among other results Babbage presented a method for solving any functional equation of the first order, that is f £x , i|ix, i|iax, . . . i|n>x| where ~ 1. Buxton ms. 13, pp. 109-110. The ."essay" appeared in the Philosophical Transactions. Royal Society. 105 (1815) 389-423. Read June 15 1815. 2. Philosophical Transactions. Royal Society.- 105 (1815) 389-423. p.395. As for the futility of employing finite differences refer to p. 151 above and to a letter from Herschel to Babbage, Oct.25 1814; H.ms.R.S. and to Buxton ms.13, p.67. 3. Letter from Herschel to Babbage,' Oct-25 1814; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 172 ii, is the unknown function and a , ... ,v are known I functions. He also gave various methods for .solving functional equations of the second and higher orders. Throughout the essay Babbage was very careful to give many illustrations of his results, probably to shield his work from the usual criticism of analytics as being useless or imaginary refinement. Very few were as enthusiastic as Herschel was about Babbage's mathematics. That Babbage was not deceiving himself in his apprehension is clearly shown in a review of the ''essay". The reviewer cautioned Babbage not to be led into "some attempt calculated to produce that kind of artificial and unmeaning solution" which "although it might enable the operator to exhibit a solution to the eye" would not allow any one to "form any mental conception, or submit to any known mode of computation'.1.'1' The author of this review was probably Peter Barlow (1776 - 1862), a teacher of mathematics at the Royal 2 Military Academy at Woolwich. It is of interest to note Barlow's vision of mathematics because of its contrast to that of Babbage and of Herschel. 'Like most persons in Great Britain interested in the mathematical sciences, he ) 1. Anon "A Review of Babbagd.’s ’An essay &c.’ Phil. Trans. ■ (1815) " Monthly Review 80 (1816) 81-13715.82. 2. Barlow had started writing for the Monthly Review in April 1814; Nangle (1955) 6. See also a letter from Babbage to Herschel, June 9 1816; H.ms.R.S. * Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 173 acknowledged their stagnation there. And like those who were f&s^ncerned about the situation he saw the English neglect of analytics as well as the absence of Englishmen working in the mathematical sciences as the chief causes of the decline.^ In the same way, he stressed the importance of m o d e m analysis for discSv^ry_w]aile also 2 accepting the claims of geometry in training the mind. /"N Yet Barlow did not share in Babbage's and Herschel's enthusiasm for what he regarded -as the "utmost limits” of analysis. ) That analysis possesses iiBnense advantages in a great variety of intricate problems, it is impossible to deny: but. that it has been pushed far beyond its natural limits is also not less certain. The French character, whether in politics or science, seems /calculated to carry every thing to excess; and]it is thus that analysis has been applied by them to a variety of problems which might have been much better resolved b y other means .3 This sentiment was in accord with those expressed by many defenders of British mathematics,’ as was his sympathy for 1. See,' for example, h'is article "Increments, Methods of" in his A New Mathematical and Philosophical Dictionary (1814); or hii "Review of J.R. Delarstre's Encyclop^dle de 1 'Ingdnieur" Monthly Review 75 (1814) 488-491. p .489; or his "Mechanics" Encyclopaedia Metropolitana 3^ (1848) 1-160. p.5. 2. See his "Analysis" A New Mathematical and Philosophical Dictionary (1814) , and hii "A review of Cresswell 's An Elementary Treatise ... Maxima and Minima" Monthly Review 75 (1814) 202-206. \ • 3. P. Barlow "Review of Delambre's Astronomie" Monthly Review 76 (1815) 519-531. p.522. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / y 174 the fluxional calculus.3- Of primary■importance to Barlow 2 were the "useful branches of mathematics". Barlow's position may thus be seen as midway between'zealous preservers o'f synthetic methods and of the fluxional calculus and equally fervent promoters of Continental mathematics. And, as a consequence, Babbage's and Herschel's attitude toward mathematics, as well as that of the early Cambridge revival movement (as will be seen in the following chapter), may be justly appreciated in the British context for its radicalism. Meanwhile, Babbage had married Georgiana Whitmore apparently in late July 1814 after an engagement of two years.3 He now lightly viewed his mathematical research as philanthropy which he could no longer afford; it was time to take a serious look at his prospects. Babbage does not seem to have considered the law, and dismissed the Church for he saw little opportunity for his own advancement. He thought of obtaining some "situation connected with the mines" where he could use his knowledge of chemistry, or of employment with the Nautical 1. See his review of Cresswell mentioned on p.173, footnote 2. 2. Anon (probably Barlow)' "Review of J. Adams' The Elements of the Ellipse" Monthly Review 90 (1819) 98-100. p.99. 3. Moseley gives June as the month Babbage married (Moseley (1964) 52), but it appears from letters to Herschel of August 1 and 10 that he married in late July; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 175 Almanac.'1' But neither of these ideas were realized and Babbage continued his work on functions , moving to London 2 in November. There, early in 1815, he was lecturing on- astronomy to audiences of the Royal Institution.^ His correspondence for the remainder of 1815 is silent on the topic of an occu m, but there is little doubt that Babbage must have been attentive to potential positions. By the time Babbage's first paper for the Philosophical Transactions was before the Royal Society in the Spring' of 1815, he had already made much progress in his continuing work in the calculus of functions. ... I have bestowed some attention on functional equations involving two or more variables, and I have met with considerable success: I am in possession of methods which give the general solution of equations of all orders, and even of those which contain symmetrical functions. I have also discovered a new and direct method of treating functional equations of the first order, and of any number of variables , and this new method I have applied to t h e solution of differential Indeed, so rapid were his advances in the calculus of functions in 1815 that he exclaimed to Bromhead in August that his first essay contained scarcely a fifth of jl. Letters from Babbage to Herschel, Aug.l, 10, Sept.22 1814; H.ms.R.S. 2. Letters from Herschel to Babbage, Oct.25, Nov.; H.ms.R.S. 3. Lett'er from Herschel to pabbage, Feb.16 1815; H.mSvR.S. Letter from Whittaker to Bromhead, May 25- 1815; Br.ms. 4. C. Babbage "An essay &c." Phil. Trans. 105 (1815) 389-423. p.423. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 176 his writings."'' In the same letter he outlined his work, most of which was published the following year in the Philosophical Transactions .as "An essay towards the 2 calculus of functions. Part II". Although this paper only appeared in November of 1816 it was basically completed by the previous November and subpiitted to the Royal Society through N.H. Nollaston veryearly in 1816 3 (probably January). y "Part II" extended the methods of his former paper to solving functional equations of any order with more than one variable and presented new methods (by elimination) for solving first order functional equations and also methods for solving differential functional equations. Many of these results may be found in his correspondence with Herschel and Bromhead in 1815. Babbage1s work profitied greatly from discussions with Herschel, Bromhead and Maule. For example, Bromhead was the first to state that an inverse function admits of many values, which result was independently derived by 4 Herschel and hinted at by both Babbage and Maule. 1. Letter from Babbage to Bromhead, Aug.6 1815; Br.ms. 2. Philosophical Transactions. Royal Society. 106 (1816) 179-256“ Read March 14 1816. 3. Letters from Herschel to BaSbage, Nov.6 1815 ,■ (pmk.) Feb.7 1816; H.ms.R.S. 4. Buxton ms.13, pp.119-121. The inverse of a function f was denoted by f ^ so that if x=fy then f ^"x = y. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 177 Again, the notation used was really the product of discussions between Herschel, Babbage and Bromhead. As Babbage later wrote. In the numerous conversations which occured it ■ is almost impossible to apportion the precise share which is due to each for any suggested improvement, the same idea must frequently occur to different minds placed in nearly similar situations: thus it happens with respect to the notation of functions of more than one variable | and of higherborders than the first i t .is the joint work of Herschel and Mr Bromhead and myself. -1- Still, the mai^i lines of development, although benefiting from the contributions of others, were apparently due to Babbage. -. ■ ' Among the many results in the very long "Part II" Babbage considered his results on recurring functions, his use of the method of elimination - "a new and beautiful \ branch", his proof of the impossibility of 2 1 12 i p ' (x,y) = ij, ' (x,y) , and his use of the substitution - 1 • 2 <(> f (4> x, y) , as among the most important. It is also interesting to note the continuing role of differential equations in. guiding his efforts* Just as in his first essay, where Babbage had regarded the origin of the "determination of functions" as being in attempts to ft solve differential equations and had relied on this theory Ibid. 110. Ibid. 105-202. Babbage developed a new system of notation to express relations between functions of two or more variables. Thus, in this system, 2 1 12 i|i ' (x,y) = iji ' (x,y) meant ( i|> (x,y) ,y) = (x ,<(j (x,y)) , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 178 Bo regulate his work (see, for example p.408 of the "essay"), so also in "Part II" Babbage looked to the analogy with differential equations for .aid. For instance. 2 1 12 Babbage concluded that tjj ' (x,y) = i|j ' (x,y) was impossible in the same way "as some differential equations of three variables are known to be".^ Finally of interest in this paper are Babbage's continuing attempts to justify his work. The reader is reminded at the Beginning and at the end of "Part II” of the importance of this calculus not simply for "the recesses of this abstract science" but for "every branch of natural philosophy, where the object is to discover by calculation from the refeults of experiment, the laws which regulate the action of the ultimate particles of bodies". This will be accomplished if only "the labours of future enquirers give to it that perfection, which other methods 2 of investigation have attained". T£pse remarks must indicate, once again, Babbage's unease with the typical sort of reception which his work was receiving: emphasis on its .inutility, its little prospect of being brought to perfection, and its needless abstraction.^ Correspondingly, 1. Ibid. 138. 2. "An essay on the calculus of functions. Part II" .Phil. Trans. 106 (1816) 179-256. pp.179-180. For an account of the modem status of functional equations and their applications, see Acze 3. Anon (probably Barlow) "Review _ s 'An essay ... Part II' Phil. Trans. (1816)" Monthly Review 83 (1817) 54-57. p.55. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 179 his remarks underline the incompatibility of his work with , what was usually accepted in England as creditable mathematics. For, while Babbage undoubtedly believed in the importance of mathematics for science, his real interest was not iJith applications nor with mixed mathematics but with pure mathematics, ^id with. the structure of analysis. ... the doctrine of functions is of so general a nature, that it is applicable to every part of mathematical enquiry, and seems'eminently . qualified to reduce into one regular and uniform system the diversified methods and scattered artifices of the modem analysis;’ from, its comprehensive nature, it is fitted for the systematic arrangement of the science, and ■ from the new and singular relations which it expresses, it is admirably adapted for farther improvements and discoveries.1 Babbage was, as he remarked on so many occasions, "function mad". He saw his new calculus as "applied to everything to which the differential one is applicable it is infinitely more powerful and has besides other 2 treasures peculiarly its own". This concern developed at this time (very early 1816) into a plan to explain "the 3 whole system on the principle of identity". As he stated his vision a few years later, In divesting Analysis of all relation to number we [Babbage and Bromhead] both agree and probably 1. "An essay on-the calculus of functions. Part II" Phil. Trans. 106 (1816) 179-256. p.256. 2. Letter from Babbage to Bromhead, Aug.6 1815; Br.ms. 3. Buxton ms.13, p.202. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 180' our definitions would be nearly the same. I should consider every equation as an abridged representation of operations which if actually performed at length would cause all the terms mutually to destroy each other - In this view all series considered as infinite must be rejected from pure analysis, and the theorem of Taylor as well as all other modes of expansion would be treated of as consisting of a certain number of terms together with a remainder; this would preserve the principle of identity and ^ although in some cases -it might be tedious it would exclude a frequent cause of error in the application of Analysis to number.1 r Clearly Babbage was concerned with purifying analysis, with founding it not on number but on various principles. His public attempts at justification reveal his awareness of just, how different his views were from the prevalent mathematical ideology in England. Herschel, remarking in August 1814 on Babbage’s efforts to find a position, showed his own uneasiness with his attempts to study law. I am happy to see however that you seriously . intend to act about being useful in this world. To one who like ourselves has existed more in theory than in practise - who has' made the beautiful & the abstract his cynosurje in mockery of base utility the resolution mbst ever be difficult to form, & more so to follow, i can only wish you may have energy & determination enough to go through with it. For my drfSh part I feel that I never shall.2 _ - '■ In the fall of 1814 he returned to St. John'*'.^ College^ Cambridge, in order to concentrate on ■hi3pstudi/i'6,'of law, - . . wv 1. Ibid. 217. + 1 '^ --- ** 2. Letter from Herschel to Ba&ba^*, Au?f.*? -1§14; H.ms.R.S. ^ • '6. "• " . \ ’r ■•‘v - V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 but his preoccupation' with science continued, especially in chemistry and mineralogy.'*' In March of 1815, he even applied, without success, for the chair of chemistry at 2 Canbridge . Herschel was offered the office of assistant tutor at St. John's in May but declined it at this time because-of his determination to continue in law.3 But an illness that summer led him in the fall to abandon his A * "professional studies", apparently for health reasons, -4 and to go to Cambridge to take private pupils. His presence at Cambridge was to lead, as will be seen in the next chapter, to his involvement in attempts to reform ■ Cambridge studies. Although Herschel had done little in mathematics in 1814 and in most of 1815, his period of recuperation at Brighton in September and October of 1815 led him to resume this interest. The result was a paper for the Royal Society, dated November 17 1815, entitled "On the develcpenent of exponential functions; together with { 1. Letter from Herschel to BaJjbage, Mar.23 1815; H.ms.R.S. Herschel was attending the mineralogical lectures of E.D. Clarke. 2. Ibid. The chair was made vacant by S. Tennant’s death early in 1815. 3. Letter from Herschel to William Herschel, n.d.Jc. May 1815] ; H.ms.T. 4. Letters from Herschel to Babbage, Sept.24, Nov.6 1815; H.ms.R.S. Buttmann ((1970) 17-18) states that Herschel took up the post at St. John’s but I have seen no evidence confirming this. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. rv . . 1 8 2 several new theorems relating to finite differences” .^ This paper, using the method of separating the symbols of operation from those of quantity, gave various general formulae for expanding (developing) very general exponential functions giving, at the same time, ways of calculating the actual coefficients. The paper also illustrates by its banishment of the ideas of infinite r and finite from analysis the coincidence of Herschel's 2 view of analytics with Babbage's, as described above. The reaction of the Monthly Review to Herschel's paper was consistent with its earlier criticism: it pointed out the needless complexity and the uselessness of much of French mathematics and warned English mathematicians, and in particular Herschel, against emulating this aspect of French work. ... it will be o f .the highest importance to embrace only such subjects as will admit of useful application; and to bear in mind that it is not the intricacy of formulae, but the simplicity of them, which constitutes the beauty -of analysis. - Mr. Herschel has in two or three instances \ manifested considerable analytical talents, which we should be very loth to undervalue: but we fear that he is too fond of that sort of parade to which we have alluded, and which we , 1. Phil. Trans. Royal Society. 106 (1816) 25-45. Read Dec.14 1815. Parts'of this paper are to be found in the manuscript "Miscellaneous researches" in H.ms.T. Of special interest is the section "The Developement of.certain functions of frequent occurce in the theory of finite differences / 2. Phil. Trans■ Royal Society. 106 (1816) 25-45 pp?i5-26. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 183 should be glad to see him correct.'*' The Philosophical’Transactions for 1816 also included, besides the papers by'Herschel and Babbage, one by Edward Bromhead. Although Bromhead had ceased to be active in the Analytical Society with his move to London in the summer of 1812 in order to study law, he had kept in close touch with both^Babbage and Herschel and with certain friends and (his brother Charles at Cambridge. Due to poor health he w js to spend most of his time“arouncKh^s home ~ s in Thurlby (near Lincoln). Yet he continued his interest and studies in mathematics, as may be seen from some of the remarks made on Babbagd’s mathematics above and from 2 Bromhead's manuscript "On the Indices of Functions". About' May of 1815 Btomhead planned a book on the method of fluents, of which he seems have finished three of { ■ 3 '• the nine chapters. Although the book was never published. undoubtedly gets a glimpse of his ideas in his paper for the Royal Society which he prepared about January 1816. In *0n the fluents of irrational functions" Bromhead "attempted to generalize and systematize our 1. Anon (probably Barlow) "Review of Herschel's 'On the DevelopeiiEnt &c. ' Phil. Trans. (1816)" Monthly Review 81 (1816) 393-395. p.393. 2. This manuscript is quoted in Buxton ms.13, pp.115-119. 3. Letter from Whittaker to Bromhead, Nov.22 1815; Br.ms. Letter from Bromhead to Babbage, pmk. Nov.24 1815; B.ms.B.L. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 184 knowledge on this subject" by showing how certain generalized forms of fluents could be rationalized.^ — \ / Once again, the separation between this type of ;s\^matheinatics - the mathematics of Babbage, Herschel, and the Analytical Society - and the prevalent English view - is shown by the response to his work. "Herschel was highly pleased with it.' Ivory has also sent him a complimentary Note . . . ." and Peacock wrote of it as "a very original and ingenious paper”.2 But the Monthly Review)commented "... with all the advantages derived from Mrl Bromhead’s notes and his numerous examples, we are still doubtful whether we exactly understand what it •* 3 is that he intends to perform; ...." The reviewer, looking back over the mathematics papers in the Philosophical Transactions for that year (1816), could only regret the direction which these papers were taking.4 Bromhead remarked Have you seen the Review of our Papers in the \ Monthly Review. The ignorant scoundrel mangles ybu; in a manner, which shews he never read a 1.' Phil. Trans■ Royal Society. 106. (1816) 335-354. p.335. ' ^ 2. Letter from C.F. Bromhead to Whittaker, Feb.4 1817; St.J.ms. S. F. Lacroix An Elementary Treatise on the Differential and Integral Calculus (trans. T5F16) pp.670-'STi: ' 3. Anon (probably Barlow) "Review of Bromhead1s ’On the' Fluents &c.’ Phil. Trans. (1816)" Monthly Review 83 (1817) 58.- ^ .4. Ibid. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 185 0 tenth part of your Memoir & did not understand, at least ^in Spirit, any of it...... The wretch further objects to any improvement in Notation, . which shewing his total ignorance, I was most . happy to fipd him dismiss my Memoir in five lines r- _ as wholly incomprehensible. Can this be Barlow?-*- Babbage had had difficulties in getting his "Part II" published in the Philosophical' Transactions. He responded^ if . by censuring the Royal Society as "a damned ignorant set in everything which relates to mathematics" and was very upset by the attempts to "excuse this - ignorance by the cant about the object of their institution being the 2 promotion of natural knowledge". *— However he still became 'a Fellow of the Royal Society that year (1816). An invitation from W.T. Brande about January of 1816 for f assistance in the mathematical part of the journal of the V Royal Institution led Babbage to dnntribute a paper on Stewart's theorems for the first volume and three other short papers pubflnshed in 1817.^ Characteristically, he approached1 Stewart's geometrical propositions using the principles of analysis, ... thus endeavouring to prove that analysis is equally adapted for the demonstration of propositions which may be"known, and for the discovery of those which are unknown, even in a 'of inquiry which has hitherto been treated 1. Letter from Bromhead to Babbage, Nov.26 1817; B.ms.B.L. 2. Letter from Babbage to Herschel, Feb.21 1816; H.ms.R.S. See also a letter from Babbage to Bromhead, Feb.16 1816; Br.ms. 3. Letters from Babbage to Herschel, Feb.21, July 20 1816; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 186 by methods purely geometrical.'*' The other papers were two short works on the calculus of \ ' 2 functions and one related to a chess move. About March of 1816 Babbage once again tried to find a suitable J \ position by applying for the professorship of mathematics at the' East India College in Haileybury but was unsuccessful, he felt, because of his lack of "interest" (patronage)As Babbage wrote to Herschel, Happiness may I am convinced be obtained by an Analyst/but how he is to obtain that sine qua non of /i^ii's _world .(money to wit) I have not yet discovered. 4 "v Despite these disappointments Babbage continued his work in mathematics, particularly in the calculus of functions.'; In October of 1815^he~had been planning a third part (to his "An essay trwards the calculus of functions" which would have included the maxima and minima of functions and the application of the method of 1. C. Babbage "Demonstration of some of Dr. Matthew Stewart1s General Theorems; &c." Journal of Science and the Arts. _1 (1816) 6-24. p.7. 2. "Solution of Some Problems by means of the Calculus of Functions” The Journal of Science and the Arts 2 (1817) 371-379. "Note Respecting Elimination11 lbi3- _3 (1817) 355-357. "An Account of Euler's Method of Solving a Problem relating to the_Knight's Move at Chess" Ibid. (I have not seen this last paper). 3. Copy of letter to East India Company, Mar.11 1816; B.ms.B.L. and Babbage (1864) 473. Oi^ Babbage's disappointment see his pbem "Sir Alphabet Function" partially quoted in Moseley (1964) 60-61. .Letter from Babbage to Herschel, July 20 1816; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 187 variations.3- Babbage felt that the three parts would provide an outline of- a calculus which at some future period would rival the integral calculus.' Accordingly, while he continued his. work in 1816 - 1817 on certain aspects of this calculus - such as extending his method of elimination to obtain general solutions, or solving functional equations containing definite integrals - he began to think of publishing all of his writings in book 2 form together with a history of the field. Apparently Babbage only prepared a sketch of his "great work on functions".3 But he did write the history, during 1817 from about February to September, which still exists as' The History of the Origin and Progress of the Calculus of Functions during the years 1809 1810 ...... 1817.^ This trejiti^Se is a very valuable source for understanding the w_p±k o f •' Babbage and his friends.3* ( ■ Some of the directions which Babbage’s'work was taking in late 1816 and very early 1817 in the calculus of functions, as well as his continuing reliance on the methods of several branches of analysis (_§..g* the integral 1. Letter from Babbage tovHefschel, Oct. 28 I8I 5; H.ms.R.S. 2. Letter from Babbage to Hetschel, July 20 1816; H.ms-R.S. 3. Letter from Babbage to Br d, Nov.11 1816; Br.ms. 4. Buxton ms.13 in the ’Histo Science Museum, Oxford. 5. X have made much use of it in this thesis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. calculus) for guidance in the "cultivation and improvement" of. that calculus, are shown in his "Observations on the Analogy which subsists between the Calculus of Functions and other branches Qf Analysis", dated March 5 1817."*" Suffering at this time from "a new fi\: of the mania • Analytica", Babbage illustrated the uses of analogy for discovery by examples from his work on the calculus of functions: for example, his extension of the method of elimination to obtain general solutions, and his metho for making a functional equation, especially of the fi order, symmetrical in order to obtain a general solution. Herschel, who had been at Cambridge since October w of 1815, found that he was not happy with "cramming pupils, which is a bore & does one no credit but very 2 . ! much the contrary’!. He had done little m mathematics' while ^t Cambridge except for the translation of part of Lacroix's Traite elementaire. Yet he had sketched out "a complete course of the essential part of the pure analytics" and was at work -by the end of 1816 on a book on algebra "upon a very different plan from any algebra ' 3 that has -yet appeared". Some hint of Herschel's views 1. Phil. Trans. Royal/Society. 107 (1817) 197-216. Read ' April 17 2. Letter fjj0)(fHerschel to Babbage, July 14 1816; H.ms.R.S. 3. And a letter from Herschel to. Babbage, Dec.24 1819; H.ms.R.S. What may be some fragments of this booljs are to be found among some uncatalogued matHematical manuscripts in H.ms.T. (W0245). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 189 on algebra is gained from his comment at the end of Spence's "Outlines of a Theory of Algebraical Equations &c. ". ... [ the- general object of this 'paper ] will be accomplished should it be thought to offer a satisfactory link of connexion between the ordinary algebra and the profounder theories of the differential calculus, - subjects which are too commonly, at least in this country, regarded as essentially disjoined, and dependent on different principles.! For some reason, perhaps connected with his own anxiety about a career, Herschel reluctantly decided in October to leave Cambridge to take up his father’s 2 astronomical observations. At his family's home in 7 Slough Herschel worked on astronomy and on chemistry and mathematics. Although he had never been quite so keen about abstract mathematics as Babbage, he too became "analysis mad" in early 1817. The immediate cause seems to have been his obtaining in December 1816 some of William Spence’s mathematical manuscripts for evaluation for possible publication. I want to see you [Babbage] particularly just now for I have at last.got all my analytical mania returned glowing hot from its perihelion. Spence’s papers have set me mad...... I the day before'Yesterday struck upon an unfinished Essay full of the most beautiful / properties of strange transcendents of the form v Spence (1819) ^ 9 5 \ 2. Letter from Herschel to Babbage, Oct.10 1816; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 190 analogous to the • c ts general; properties of log— transcend— . I devoured the Essay with avidity - the field it opens is immense. - I mean to recommend its publication and everything else on the same subject I can find in the strongest terms.1 Herschel was very busy with mathematics in 1817. He continued work on, his algebra, on finite differences, on exponential functions, on editing Spence's manuscripts, and in June/ following a request from David Brewster, on 2 two articles for the Edinburgh Encyclopedia. A result of all of this mathematical activity was a paper in November 1817 for the Royal Society, "On Circulating functions, and bn the integration of a class of equations of finite differences into which they enter as coefficients".3 As in his earlier papers, Herschel dealt with his 'topic - "series in which the same relation between a certain number of successive terms recurs periodically" - in a very generalized way. Indeed, \ analysis meant a general view and uniform treatment of the 1. Letter from Herschel to Babbage, Jan.30 1817; H.ms.R.S. ■2. Letter from Herschel to Babbage, Apr.3 1817; H.ms.R.S. Letter from Herschel to Whittaker, June 13 1817; St. t John's College Library. The two articles were "Isoperimetrical Problems" Edin. Encv. 12 (1830) 320-328 (sent to Brewster Aug.8 1817) and-"Mathematics" Edin. Ency. 13 (1830) 359-383. (sent to Brewster about July 1818). 3. Phil. Trans. Royal Society. 108 (1818) 144-168. Read Feb.19 1818. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. \ 191 subject-matter.'1' Thus he devised the "analytical artifice" of circulating functions^ to include these series in an equation of finite differences which he showed how to integrate, producing the general terms of the series.- While Herschel felt his theory was "exquisitely pretty" and affording a "remarkable simplicity" and "great neatness" in its application, the Monthly Review only commented ... we believe it to be impossible within any moderate limits to render his processes • intelligible to our readers. We have some doubt, indeed, whether the memoir itself would be sufficient for this purpose.2 Herschel, like Babbage, identified modern mathematics with algebraic methods and shared in his view of the importance of developing and purifying analysis. In his .article "Mathematics^ for the Edinburgh Encyclopedia, which he submitted about July of 1818, Herschel distinguished three great periods in the history of mathematics. The first period was that of the almost exclusive use of geometrical methods. The second was a period of transition which saw the risd of algebra, .although "symbolic analysis" had not "yet attained sufficient maturity to take the whole burden of 1. Ibid. •144, 146, 147. .2. Ibid. 166. Letter from Herschel to Babbage, Nov.24 • TSTT; H.ms.R.S. Anon (probably Barlow) "Review of Herschel's 'On Circulating &c.' Phil. Trans. (1818) " Monthly Review 87 (1818) 63. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 192 investigation on itself." Gradually analysis revealed its power and elegance until in the recent third period it had become the basis of, mathematics. ’ The last and greatest revolution of mathematical '■'science was rapidly approaching, when symbolic language, found adequate to every purpose, became the universal medium of mathematical inquiry, and when those extraneous notions which, during the foregoing period, had insinuated themselves into its principles/ were purged away.-*- The task of developing analysis was a difficult one for exactly the-same reasons that made analysis valuable: its abstractness and its generality. And hence the reliance by Babbage, Herschel and Bromhead on analogy with "^ther, better developed, branches of analysis. As Babbage wrote in November 1815, In truth this Calculus [of functions] in some respects resembles metaphysics it is infinitely general obscure abstract and absurd - 'I' 2x is like Moses which led the children of Israel into the Wilderness, Ah Herschel why did we follow the too tender too seducing .? 2 And again, two years la.^er, in September 1817 " ^ I have formerly had occasion to remark the very delicate mature of the reasoning which has been employed on this subject [calculus of functions] and the communications of Mr Bromhead prove that his enquiries have been equally obstructed by the doubt and hesitation which accompanies it: this though) undoubtedly it- in part arises from the novelty of the subject must chiefly be asCjribe'd to its great generality; quantity and 1. J. lHerschel "Mathematics" Editu^Ency. JL3- (1830) 3^9-383. pp.360-361. f - ' y . j 2. Letter from Babbage to/Herschel/ Nov.13 1815; H.ms.R.S. ,-v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 193 its relations as to magnitude being almost entirely discarded-we go in quest of abstract form, nor is it surprizing that we should miss our way amongst the laxity and latitude of conditions which bind together innumerable species.1 Babbage and Herschel's heightened zeal for analysis in 1817 led to renewed proposals for reviving the Analytical Society or for establishing a "Royal Mathematical Institute", for having it centered in London, and for 2 issuing a second volume of the' Memoirs. But nothing was realized of these schemes to professionalize. Babbage, 9 for example, felt that another volume of the Memoirs 3 would be too expensive. Bromhead too had been "a little touched" at this time. He had a reputation as a skillful analyst, as attested by Whewell, By the bye - we had Bromhead here a little while back who was as usual absolutely overflowing with theories - more particularly mathematical - The rapidity & extent of his generalizations is absolutely overpowering - .... 4 Like Herschel, Bromhead admired Babbage's calculus as "the 1. Buxton ms.13,p.249. 2. Letter from Herschel to Babbage, Jan.30 1817; H.ms.R.S. Letters from Babbage to Bromhead, Mar.14, Dec.15 1817;' Br.ms. Letter from Bromhead to Babbage, Dec.20 1817; B . ms .A. L. A ffl ^ 3. LettJBr from Babbage to Bromhead, Dec.15 1817; Br.ms. 4. Letter from Whewell to Herschel, Mar.6 1817; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ■ 194 1 most dangerous & Protean thing I ever met with'!. Perhjaps ✓ inspired by Babbage's calculus of functions, he had prepared a paper on a new calculus analogous to the. differential which he called a calculus of factors and factorials. 2 But it never appeared in the Philosophical Transactions as intended probably because of Hers'chel' s criticism. Your paper has confirmed me in the idea I entertained some years ago, when I broke off a . train of/ investigation "on the Calculus of Products"' (bearing the same -analogy to differences that the factorial does,to the differential Calculus) from a conviction that no results differentfrom those of the latter calculi could be derived from it. In fact the expression "analogous to" should throughout your paper be replaced by "identical with"3 Bromhead continued to work in mathematics in the next few years: on notation, on the solution of an algebraic equation of n dimsneiotisCCaid on the idea suggested in the Memoirs of the Analytical Society of a .digest of analytic formulae to which Babbage, Herschel, Whewell, Peacock and Whittaker were to ^mtribute. The . only mathematical item which he published was the article 1. Letter from Bromhead to Babbage, Nov.21 1816; B.ms.B.L. 2. Letters from Bromhead to Babbage, Nov.21 1816, Mar.20 1817-; B.ms.B.L. 3. Letter from Herschel to Bromhead, Apr.4 1817; Br.ms. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 195 "Differential Calculi^" for the Supplement of the fifth edition of the Encyclopaedia Britannica.^ Written in the year following May 1818 it sought to present "a systematic view of the calculus in its latest form", that i s , founded on a "pure analytical basis” with "some farther modification of the principles, and some generalization of the formulae.”2 Bromhead, in true analytical fashion| saw Newton's limiting^ratios as a practical rule for finding differentials, and fluxions as a beautiful illustration of differentials by comparing (them to velocities. Bromhead1s analytical theory depended on Taylor's formula 3 and the work of Arbogast, Lagrange and Woodhouse. After 1819 Bromhead appears to- have ^aone very little work in mathematics. He was ill and found that the countryside led to stupor. He became very involved.in locaY affairs, especially in various Lincoln institutions.^ There, some years later, he'was to come into contact with two of the most remarkable British mathematicians of the nineteenth ■1. "Differential Calculus" Encyclopaedia Britannica (5th ed., 1817), Supplement (1816-18247- 3 pp.568-572. 2. Ibid. p.568. ' . 3. Ibid. p.572.^ ^ ■\ 4. Francis Hill/Georgia Lincoln (1966)' p.277. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 196 century, George Green and George Boole.'1' He died, as second baronet, on March 14 1855. After much activity in mathematics in 1817, Herschel did considerably less mathematics afterwards. Perhaps this was due in part to his feeling that very few persons / 2 cared—about abstract mathematics. About June 1818 he was performing experiments on polarization and his chief interests now became light and chemistry.^ Following a request from Brewster for paper! for his new journal, the' Edinburgh Philosophical Journal, Herschel completed an old work of his and submitted it in January 1819. This paper, "On the Application of a new mode of Analysis to the Theory and summation of certain extensive classes ~ , ' " c of Series", presented a new proof of the fundamental __- theorem of his 1816 paper for the Royal Society, and 4 used it to find the sums of certain classes of series. 1. Bromhead was a subscriber to Green's Essay (1828) A. which he introduced to Whewell. Todhunter (1876) 3. David Phillips "George Green: His Academic Career” pp.63-89 in George Green Miller Snienton (Nottingham Castle, 19767"! Bromhead was president of the Lincoln Mechanics Institute (founded 1833) with George Boole teaching there and his father, John Boole, honorary curator. Francis Hill Victorian Lincoln (1974) pp.147-148. 2. Letter from Herschel to Babbage, Mar.10 l^lf^^.ms.R.S. 3. Todhunter 2_ (1876) 24. Letter from Herschel to Whewell, Aug.19 1818; W.ms.T.C. j 4. Edin. Phil. J. 2 (1820) 23-33. I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. \ 197 / In that same year, 1819, four problems with solutions by him appeared in Leybourn 'sC^lathematical Repository. ^ His last paper in .pure mathematics was for the newly founded Cambridge Philosophical Society and was dated 2 February 5 .1820. In it he resumed his former attempts j to resolve functional equations by reducing them to ) finite differences. In this case, the functional equations were of the first order and more than one variable where the later variables depended on the first. The functional equations were reduced to equations of * £ni1finite differences by the use of an. interpolation theorem of Lagrange. Sometime in 1820 or 1821 Herschel's "views of life" 3 changed. . Even though he nwas awarded the Royal Society's Copley Medal for his mathematical papers in the Philosophical Transactions, he no longer had "the keen relish for abstract mathematical studies", preferring the 4 physical to the mathematical sciences. Whereas he had Questions 406-409, Mathematical Repository 4 (1819) 232-159. \ ^ 2. "Ontthe Reduction of certain Classes of Functional Equations to Equations of Finite Differences” Transactions. Camb. Phil. Soc. 1_ (1822) 77-87. 3. Letter from Herpchel to Babbage, Dec.2 1821; H.ms.R.S. 4. Ibid. See also a letter from Herschel to Whewell, Aug. 17 1826; W.ms.T.C. Humphry Davy, in presenting the medal in 1821, censured "vague metaphysical abstractions" in mathematics and, ironically, praised Herschel for not following that path and, instead, applying his "formulae'"-. H. Davy The Collected Works of Sir Humphry, --DavyDavy 7_ (1840) 18-19. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. previously been interested in obtaining ^professorship at Cambridge, he now had no wish to either reside regularly at Cambridge or to lecture.'*' A larger world and a more varied scene are necessary for my happiness, and as far as mere science is concerned, X had rather pass my days among those who are advancing eagerly and.irapidly S , and running a race with ardour, than in goading / up the, hill the sluggish paces of any established ( institution under the Sun. V Herschel was entering a larger world. In Januaty^ of- 1820 he, with Babbage, helped to found the Astronomical V 3 Society in London. And about December of 1821 he became a member of the Council of the Royal Society. These early successes were to be followed by a long and illustrious career in English science, ending only with \ his death on May 11 1871. After much work in mathematics in 1817, -Babbage, like Herschel, began 1818 at a rather low ebb. Deeply concerned with obtaining a position, and very ambitious, Babbage renounced mathematics. Mathematics is unprofitable and moreover is not thought anything of - the fame of an English mathematician is not worth much - farewell therefore to x's and y ’s.- j \ I 1. Letter from Herschel to Whewell, Aug.17 1826; W.ms.T.C. And letters from Herschel to Babbage, Mar.10 1818, Apr. 4 1820; H.ms.R.S. 2. Letter from Herschel to Babbage, Dec.2 1821; H.ms.R.S. 3. G.J. Whitrow "Some prominent personalities and events in the early history of the Royal Astronomical Society" Q-O^-R. Astronomical Soc- 11 (1970) 89-104 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -*0 199 I now devote much time to Chemistry and mineralogy and other reputed'usefull things and have re-established my laboratory and set up a cabinet.^ However Babbage could not for long remain separated from mathematics, and by April of 1818 was very busily investigating and discovering porisms by means of fmictions. These researches were published five years later as "On the Application of Analysis to the Discovery of Local . 2 Theorems and Porisms". In presenting the various theorems and porisms Babbage stressed the power of the calculus | of functions. For analysis, "the language of symbols", | should not be neglected, he felt, even if it seemed i | abstruse or isolated, because of "the latent affinity i | between departments of mathematics, usually regarded as • 3 the most opposite." Other papers by Babbage in the late 1810s included a very short "Demonstration of a Theorem relating to Prime Numbers", which was submitted to Brewster by Herschel in 4 January 1819, following Brewster's request for papers. s ^ It was followed that year by Babbage's major paper "On / • some new Methods of investigating the Sums of several 1. Letter from Babbage to Bromhead, Feb.27 1818; Br.ms. 2. Transactions. Roy. Soc. Edinburqh. 9.(1823) 337-352.- Read May 1 1820. 3. Ibid. 337-338. 4. Edinburqh Philosophical Journal 1^ (1819) 46-49. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 200 Classes of infinite Series".'*' The basic methods of this work go back at least to the period of the Analytical 2 Society. Babbage had resumed his interest in summing infinite series in 1817, especially in conjunction with similar work by Herschel.^ But it was a year later, about November of 1818, that Babbage resolved the difficulties surrounding his method of EHSV. It is these processes, with some use of functional equations, and with the usual \ formalistic approach to infinite series, which are presented in "On some new Methods &c.”.| Another paper was "An Examination of some Questions t/onnected with Games of Chance".^ As usual. Babbage-examined these questions as an "application of some very abstract, propositions of analysis to a subject of constant; occurrence".’’ Basically his results wer^an extension <3 of Herschel1 s notion of circulating functions. His final paper of this time was "Observations on the Notation employed in the Calculus of Functions".® The work 1. Phil. Trans. Royal Society. 109 (1819) 249-282. Dated March 25 1819. Read April 1 1819. 2. See pp.131-33, 147-49above. 3. Letter from Babbage to Bromhead, Aug.24 1817; Br.ms. 4. Transactions ■ Roy. Soc. Edinburgh. 9_ (1823) 153-177. Read Mar.21 1820. 5. Ibid. 177.’ 6. Transactions. Camb. Phil. Soc. 1 (1822) 63-76. Dated Feb.26 1820. Read May 1 1820. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 201 stressed the important influence of notation in mathematical reasoning with examples from the calculus of functions. Meanwhile Babbage continued to seek .a position connected with mathematics. In late 1818 he attempted but did not get a seat on the Board of Longitude.^ Still, he hoped, as he put it, "for a share of the loaves and 2 - fishes". And, besides, if useful things could not support their disciples then, he felt, "each hungry analyst” could be allowed to do whatever he liked.^ About August of 1819 Babbage applied for the Professorship of Mathematics at Edinburgh but did not succeed because, as 4 he complained, he was not Scottish. In January 1820 he helped to establish the Astronomical Society in London.5 This resulted in his losing Sir Joseph Banks1 patronage for 'a seat on the Board of Longitude that same year.5 An/d, the following year, he' found he had no chance for /the Lucasiak Professorship at Cambridge.7 Letter from Babbage to Herschel, Dec.l 1818; H.ms.R.S. Ibid. Letter from Babbage to Herschel, Feb.19 1819; H.ms.R.S. Babbage (1864) 474. A.S. "Obituary. C. Babbage" Monthly Notices. Roy. Astron. Soc. 32_ (1872) 101-109. p.101. Babbage (1864) 474. Letter from Herschel to Babbage, Dec.2 1821; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 202 Despite these disappointments Babbage continued his work in mathematics. His final mathematical effort in the period 1814 - 1822 was towards his Essays o n .the Philosophy of Analysis. Only two of the essays were published, the rest exist at varied levels of development in manuscript form in the British library.'1' Babbage had been collecting ' x 2 I materials for these essays since 1816. The essays \ represent the best enunciation of Babbage’s views on analytics as well as the best indication of the importance^ of the philosophy of discovery in Babbage's thought. For ^ Babbage felt that ... the highest object a reasonable being could pursue was to endeavour ‘to discover those laws of mind by which man's intellect passes from theknown to the discovery of the unknown. This'^quest undoubtedly was a major part of Babbage's ■motivation in pursuing analytics which was seen by many as the most perfect example of reasoning from the 4 known to the unknown. In particular, three of the , essays - Induction, Generalization and Analogy - along 1. B.msiB.L. ms.37202.j See Dubbey's discussion of this work"" (Dubbey (1978) ^93-130) a good deal of which, however, I find fault with. 2. Draft of a letter from Babbage to Brewster, June 20 1821; B.ms.B.L. Note that this important letter has been incorrectly filed among Babbage’s other letters as June 20 1824. 3. Babbage (1864) 486. And again* on pp.428-429. 4. See pp. 54-55 of this dissertation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 203 with notes at the end-of the manuscript illustrate his views oiiNT.nvention and indicate some of his mathematical and philosophical sources for these views.'*' Besides these three essays, the Philosophy' Of Analysis in finished form would have included eight others: On I Notation, On the Influence of Signs in Analytical Reasoning, General Notions respecting Analysis, Of Artifices, Of Problems requiring new methods containing many inquiries of interest respecting games, Of the Law of Continuity, Des Rapprochements, and Of the value of a 2 hint book. The latter three seem never to have been written although materials for them were collected.3 The . fourth and fifth essays are sketchy and not of particular interest here. The first two essays were the ones published. Babbage had been asked by Brewster in November 1818, . to contribute the articles "Notation" and "Porism" to the 1. Of particular influence on Babbage was the work of Dugald Stewart. Draft of a letter from Babbage to Stewart, Aug. c.25 1819; B.ms.B.L. See also various references at the end of the Essays and p. , footnote , of this dissertation. One of Babbage's sons, born in 1819, was christened Edward (after Bromhead) Stewart (after D. Stewart). Letter from Babbage to Bromhead, Jan.29 1820; Br.ms. 2. Draft of a letter from Babbage to Brewster, June 20 1821; B.ms.B.L. (see footnote 1 p. 27) And see the Essays themselves. N 3.. Ibid. i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 204 Edinburgh Encyclopedia.1 The article "Notation” is basically the same as "On Notation". 2 It sets down certain general maxims for notation and illustrates these points. The other published essay, "On the Influence of Signs in Mathematical Reasoning", together with "General Notions respecting Analysis" offer a clear survey of Babbacp's view of analytics..^ Echoing the sentiment of the "Preface" of the Memoirs of the Analytical Sociel^,.,^, Babbage felt that analysis at that time consisted of a \ confused and intricate collection of notations, methods, \ unfinished theories, particular contrivances and partial views — "a mass of materials of a very heterogeneous 4 nature". Analysis, the language of signs, had to be put in order. This task was complicated by the great generality of the language of signs which meant that the language could only be approached through its applications, such as geometry- or' arithmetic or the differential 1. Letter from Brewster to Babbage, Nov.22 1818; B.ms.B.L. The articles were submitted by Babbage by 1822, letter from Brewster to Babbage, Feb.25 1822; B.ms.B.L. 2. Draft of a letter from Babbage to Brewster, June 20 1821; B.ms.B.L. And compare the manuscript of "On Notation" in B.ms.C. to the article. "On Notation" is not included among the' Essays in B.ms.B.L. 3.. "On the Influence &c." appeared in Transactions. Camb. Phil. Soc. 2_ (1827) 325-377. Read Dec.16 1821. 4. Ibid. 326. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission 205 calculus. By extending the methods and definitions of these applications, or branches, the principles on which analysis rested could be obtained. 2 For Babbage the basis of pure analysis,was the principle of identity, that is that each stage of the reasoning was reducible to a pure identity (at least when the operations on each side of the equation were'carried out) . For example, infinite series were excluded from pure analysis because they were not analytical expressions 4 for they could not be rendered identical. However infinite series were not banished from mathematics. They were a part of the application of analysis to number.6 , In this way Babbage dought a firm and accurate foundation for analysis which would meet such objections as Berkeley’s.6 Analysis, purified and standardized, could then be applied with complete confidence - any difficulties arising due only to the subject-matter to which it was applied. The three stages of application were: translating the question into the. language of analysis, 1. "General Notions respecting Analysis" B.ms.B.L., pp.41-42. 2. Ibid. 43. 3. Ibid. 4 . Ibid. 47. 5. Ibid. -47-50. 6. Ibid. 43-52. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 206 carrying out the necessary operations on the analytical expression, and translating the results-of the analytical process into ordinary language.'*' The success of mathematical reasoning, its certainty, arose from the sub ject-ms(tter (whose foundations rested on definition) and the mode in which trains of thought were handled - 2 analysis being particularly powerful in this last regard. Babbage's work in the calculus of functions was therefore both an attempt to develop a new and powerful branch of analysis and at the same time a means of comprehending the principles of pure analysis and, ultimately, of better understanding the mind’s inventive faculty. In 1821 Babbage inquired about having the Essays published in Brewster's journal only to be turned down f a variety of reasons: therefwas a great backlog of papers, Babbage's propose^ essays c o u M only be published by the journal over a five year period which was (too long, and especially because the subscribers to the journal were general readers who could not be expected to follow 3 Babbage’s work. However Brewster did suggest that the Essays be published as parts of the Supplement of the 1. "On the Influence &c." Transactions. Camb. Phil. Soc. 2_ (1827) 325-377. p.346. 2 . Ibid. See the whole paper and the summary on p.377. 3. Letter from Brewster to Babbage, July 3 1821; B.ms.B.L Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 207 Edinburgh' Encyclopedia. ^ But Babbage seems to have looked elsewhere for publication. Peacock was very anxious to obtain them for the' Transactions of the^ Cambridge Philosophical Society, so Babbage^ent the Essays 2 to him. "On the Influence of Signq^-STc." was' read before the Society and published in its-^Trahsactions. None of the other essays appeared in the' Transactions, mainly owing, it seems, to Babbage's desijl^pthat they be published together or as a continued series; a wish which, Peacock felt, the 3 Council of the Society could not grant. Significantly, while returning the Essays to Babbage, Peacock noted the great interest in Babbage’s Difference Engine: ... all the' world is talking of it as the wonder . of the day, when tables are calculated, equations solved & theorems invented by steam-4 The Essays basically mark the end of "Babbage1s efforts in maCHeifiatlcs. ' Increasingly his attention was diverted from V * his firstj love, mathematics,- to a preoccupation with his calculating engines. Although he'^S^s eventually to obtain the Ludasian Professorship at Cambridge (182 8-1839), he was 1. Ibid. 2. Letters from Peacock to Babbaqp, July 15, Nov.7 1821; B.msB.'L. 3. Letter from Peacock to Babbage,. May 7 1822; B.ms.B.L. 4. Ibid. ,. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. never to return to anything like the level of his < mathematical activity that marked the period 1814 - 1822. 1 • % As Whewell noted in 1829, Babbage’s "anxiety about the success and fame of his machine is quite • devouring and 2 unhappy." After a career which contraste,d very sharply with that of his good friend John Herschel, and which can only be described as tragic, Charles Babbage died on. October 18 1871. By 1822, then, neither Babbage nor Herschel was continuing his mathematical work. .What had happened to eliminate their early enthusiasm for mathematics? To a certain extent changing interests were' a factor, but the main causes were undoubtedly their inability to find positions which could have 'Sostered their mathematical interests and the hostile reception their mathematics was given. That there were but very few career opportunities in mathematics in England - and those usually not available on mathematical merit alone - is not surprizing.-for that time and was due to a variety of circumstances. Also, the general distaste for their mathematics was quite naturally expressed as a preference for synthetics or for useful ...... 1. Babbage was proposed (unknown to himself) for the chair by his friends arid won with 8 votes over W. Maddy (2 votes) and J. Hind (1 vote). See "Lucasian Professorship. - Voting Sheets. Mar;6 1828" in Cambridge University Archives. • ’ 2. Letter from Whewell to Jones, Feb.4 1829; Todhunter (1876)' 2_ p.97. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 209 mathematics. What is noteworthy is that Babbage and Herschel could see themselves^ap^mathematicians; that is, that their ’useless’ mathema/tics was mathematics worthy of a mathematician and that fojj/ms of public support for such a discipline should exist. | These 'expectations thus X reflected what were seen by many as the causes of the decline of the mathematical sciences in England: synthetic mathematics, the lack of govermnenbiland public support, the large costs of publication, and they apathy of Royal Society. In short, their expectations disclosed their wish to have mathematics treated as other professions were. \ Professionalism has been-jLdeiftified by six features: • J a commitment to a calling, a formalized organization, the' existence of full-time occupationsy-a system of educatio: a service orientation, and autonomy.1 The first tw< features are clearly displayed in the history of the Analytical Society and in the activities of some of its former members as detailed in this chapter. And there was at least a desire for full-time occupations. The fourth feature partially existed in practice at Cambridge and was to be the object of much effort by the reformers, as will be examined in the next chapter. Service orientation existed in the sense of production of mathematics for 1. ■* Wilbert E. Moore ' The Professions: Roles and Rules (1970) . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 210 mathematicians or other groups. This feature was not prominent in the professionalization of mathematics, or of science, on account of the subject-matter. Finally, autonomy was expressed through the stress on pure analytics and the attempt to separate it from other mathematics. Mathematics was riot a .profession (in the modern sense) in early nineteenth-century England and did not at th'at time become so.'*' But it is significant that some persons, in particular reformers such as Babbage and Herschel, expected it to be a profession. This self-awareness or desire to professionalize appears to have been an important and common motivating element in .the background of the activities of the Analytical Society and of the reformers of Cambridge mathematics. ’Babbage's and Herschel's mathematics reflected their concern with professionalism. The role of professionalism in their choice of supporting analytical mathematics is quite clear. Analytics was professional mathematics and synthetics was amateurish precisely because analytics was believed to be the path to discovery, the way of research, in mathematics. This, of course, was the business of a ematician. Their selection of technical'mathematical problems to solve, the ways in which they were solved, as 1. For one historian of science’s view on the senses of professionalism in the early nineteenth century see Cannon (1978) chapter 5, especially p.150. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. well as their efforts to develop certain branches of mathematics - the calculus of functions for example - depended on and also served to reinforce their vision of pure analytics, as this chapter has illustrated. And the stress on pure analytics was a further refining of the link between analytics and professionalism. For besides the value of analytics for doing research, the emphasis by Babbage and Herschel on purifying analytics and standardizing that knowledge was an attempt to separate their knowledge-.-from that of other practitioners and users of mathematics. It was an attempt to define what was really mathematics, an important aspect of professionalization. Thus the mathematical concerns of Babbage and of Herschel during 1814 - 1822 should be judged in the light of their intent to make mathematics a profession. rWithin this view their efforts to organize to find positions, their criticisms and their critics, as well as their enthusiasm for analytics and the work they did in mathematics may all be seen as related aspects of this intention. The Analytical Society had arisen from circumstances in which professionalism was a significant aspect. Some former members of the Society continued their research in mathematics after the Society's failure and also attempted to pursue careers in mathematics. Their inability to find such careers must have confirmed their view that if Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 212 mathematics - that is analytical mathematics - was to prosper in England it would have to be treated as a profession. Their mathematics was distinctive and also manifested the influence of their concern with professionalism. Babbage’s and Herschel's mathematics was later to exercise some influence on British mathematics • but it did not have much immediate impact.^ Due to the neglect and criticism of'their work and their lack of success in finding positions connected with mathematics, they were to abandon the subject by about 1822, However, at the same time that Babbage and Herschel were producing their mathematics and searching for positions, they were also endeavouring to reform the Cambridge mathematical studies. It was within this effort, as the next chapter will examine, that they introduced their vision of analytics. And once again the element of professionalism appears to have been a key motivating factor. /* \ -- \\ i : ' ' / 1. For its influence see Koppelman (1971/72) and Dubbey (1977). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VI. The Introduction of Analytics at Cambridge University (1813-1820s) Historians of mathematics have been content in their writings about the renewal of Cambridge mathematics in the early nineteenth century to describe it as the t . displacement of the Newtonian tradition by the Continental. i While not incorrect, it is not a full description of the renewal; neither is it sufficient for an explanation of the change in Cambridge mathematics. The adoption of analytics at Cambridge involved much more than simply .a ■6 switch in traditions. Once again, the social and intellec tual context of the switch' p r o v i d e s insight into what produced that change. The chief elements of this context I were professionalism, liberal education and the debate i • X over analytics and synthetics - the same characteristics important for understanding the Analytical Society or the i mathematics which the Society's members produced. . This chapter will examine the introduction of analytics at Cambridge within the framework of the elements just men tioned. In particular, the structure and ideology of Cambridge University were important factors in determining the mode and rate of the adoption, as well as the style and content of Cambridge mathematics. As noted in chapter II, the University of Cambridge r f I ! ' - . " i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 214 was the object of much agitation for reform in the early nineteenth century. In particular, many were dissatisfied with the system of Cambridge studies. There is no doubt that a mood of imperfection or, as Herschel wrote, "a sense of our deficiencies" prevailed at Cambridge, especially among the ycruth.''' Reflecting this general mood at Cambridge there were many besides the members of the Analytical Society who, like Richard Whitcombe, were unhappy with the system of studies, or, like John Ashbrid^e, were J 2 motivated to study Continental mathematics. Yet i/'t was from among former members of the Analytical Society that the stimulus for a Cambridge mathematical revival emerged. The Analytical Society had been a product of a number of the elements which defined the state of early nineteenth- century Cambridge mathematics. The individual members of. the Society had acted within this framework. Though it had had but a very short existence, the Society did survive long enough to establish friendships built on common concerns, to promote analytics, and to foster the develop ment of a distinctive view of analytics - as shown in the 1. Herschel (1832) 542. 2 2. For Ashbridge see Gentleman1s Magazine (1820 ) 635. On Whitcombe see a letter from him to Whewell, June 3 [1817] ,- W.ms.T.C. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. previous chapter by the mathematical concerns of Charles Babbage and John Herschel. In the light of their pro- j fessional attitude towards mathematics, it was hardly j i surprizing that those members of the Analytical Society I who maintained a connection with Cambridge, along with others of similar persuasion, would.become involved in attempts at reform and would especially seek to promote their view of mathematics. Indeed, the element of profes sionalism, as reflected in (the presence of non-mathematical causes in the lament over the— state of English mathematics, appears to have been a factor dommon to all of the reformers at Cambridge. It was, therefore, 'probably the main cause of the reformers' activities.^For not all of those who promoted analytics were reformers: Robert Woodhouse is an example. Through his. writings he argued for and helped to diffuse analytics, yet he was not actively involved in the reform of Cambridge studies. The absence of this aspect in Woodhouse is also manifested in his state ments about the inferiority of English mathematics: these never went beyond individual efforts or the differences between analytics and synthetics. Thus, prompted by their belief that if mathematics was to prosper in England it had to be treated as a profession, former members of the Analytical Society attempted^tOy^eform the system of Cambridge mathematics. They were to do this through the structure of Cambridge studies. / Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The first way in which the mathematical concerns of the Analytical Society were diffused at Cambridge was through the usual wranglers' practice of private tuition, made pos sible by the meagre college teaching. George Peacock, for in stance, had a number of pupils in 1813 who were taught the calculus "in a manner purely analytical" and mechanics, not on Wood's system but on "a system of his own".'*' Peacock continued to promote his views when appointed assistant tutor and college lecturer at Trinity in 1815. This was a second way in which the reformers utilized the structure of Cambridge studies. John Herschel, who had already acted as an examiner at St. John's in December of 1813 and of 1814, went up to Cambridge in the fall of 1815 with the intention of taking private pupils, determined to instill into them "the principles of the true functional faith and practice." This design had '"the full and enthusiastic support of Babbage. What a glorious opportunity you have of spreading the true faith young converts are the only ones to hope from. I consider JFWH as the Apostle of Analysis as a missionary to untutored savages who have never heard of the Glorious truth that _ dxdy d^z I!! I hope you will purge away the diagrams dydx which like cobwebs have obstructed their progress in the paths of truth - 3 1. Letter from C. Bromhead to E. Bromhead, Nov. 12 1813; Br.ms. 2. Letter from Herschel to Babbage, Sept. 24 1815; H.ms.R.S. On Herschel's difficult paper at St. John's in 1813 ("Nobody could do anything.") see a letter from C. Bromhead to E. Bromhead, Dec. 29 1813; Br.ms. 3. Letter from Babbage-to Herschel; [Nov:l3] 1815;' H.ms.R.S.’ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 217 Similarly, some few years later, Gwatkin and Whittaker were cramming their pupils with "d's".^ The study of French mathematics was becoming quite extensive at- Cambridge, and this stirred the British Review to comment in May 1816 that ... we have of late seen, or fancied we have seen, in some individuals who are actively engaged in promoting the mathematical sciences in that cele brated University, a strange and unnatural desire to make every thing that is Newtonian give place to any thing that is foreign.^ The increasing popularity at Cambridge of French mathematics, and especially of Lacroix's works, probably helped to suggest to Babbage, Herschel and Peacock-in j December of 1815 that they could further promote Continental mathematics by translating Lacroix's Traitd gldmentaire de Calcul diffdrentiel et de Calcul integral.3 The need for and use of mathematics textbooks at Cambridge was a third avenue for the introduction of analytics to Cambridge students by reformers. By January 6 1816 it was decided that the first third of the Traitd (Differential Calculus) would be translated by Babbage, witdi Peacock and Herschel 1. Letter from Babbage to Herschel, May 17 1817; H.ms.R.S. 2. Anon "Review of Dealtry's Fluxions and 'Fluxions' Edin. Ency." British Review 7 (1816) 421-437. p.435. 3. On the popularity of Lacroix see a letter from C. Bromhead to E. Bromhead, Nov.12 1813; Br.ms. See a letter from Babbage to Herschel, Dec.28 1815; H.ms.R.S. And Babbage (1864) 39. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 218 dividing the remainder (Integral Calculus).1 Soon after wards arrangements were made with J. Deighton and Sons of 2 Cambridge to publish 1000 copies of the translation. Printing began in February, but was delayed as Babbage did not complete his part until June.1 The translation was printed by around September, _Herschel's appendix 4 by about October and the notes by early December. Al though the work had originally been iiitfended to appear in time for the beginning of term in October, it only appeared 5 ’I " in December, probably on the 13th. ■ It was an immediate success, two hundred copies being sold at Cambridge within a month.® About July of 1814 Herschel had begun work on the "Appendix" of the translation.7 As many important subjects I 1. Letter from Herschel to Babbage, Jan.6 1816; H.ms.R.S. 2. Letter from Herschel to Babbage, Feb.7 1816; H.ms.R.S. 3. Letter from Herschel to Babbage, [Feb.26 1816]; H.ms.R.S. - And a letter from Babbage to Herschel, [pre-July 10] 1816; H.ms.R.S. ; 4. Letter from Herschel to Whittaker, Sept.2 1816; St.J.ms. Letter from Herschel to Babbage, Oct.10 1816; H.ms.R.S. Letter from Peacock to Herschel, Dec. 3 1816; H.ms.R.S. 5. Letter from,Peacock to Herschel, Dec.3 1816; H.ms.R.S. 6 .xvLetpen_.ff6ml^Afeock, to Babbage, Jan.^t'Sf^, 3?] 1817; B.ms.B.L. 7. Letter from H^^Rfie'l to Babbage, July 14 1816; H.ms.R.S. \ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I I 219 ! I ! * w e r e f e l t to be either missing or imperfectly dealt''With in Lacroix's "Appendix", Herschel decided to prepare a ! ! j new ode. The 115 page work dealt with the calculus of ! differences, a topic in which.he had been very interested. | Herschel's plan was to have it supplant the old standard I i work, William Emerson's The Method of Increments (1763), ; particularly among Cambridge students. Through the ..mew [ work he hoped that they would get "a tinge of the true i ' ’ I faith" and together with other English readers "be let j into.a few secrets which have hitherto been contraband in j this country" , such as the calculus of generating functions., i the method of separating symbols of operation from those of j quantity, equations of differences and of mixed differences, j and functional equations reducible to equations of finite ' 2 differences. Herschel was "very well satisfied" with the "Appendix" and it appears to have become in the following 3 : years a standard English work on finite differences. Besides.the "Appendix", the translation of Lacroix also contained a 131 page section of sixteen "Notes"; the first twelve were written by Peacock and the remaining : four by Herschel. Peacock's notes were "principally designed to enable the Student to make use of the principle 1. Ibid. See also Lacroix (1816) iii. 2. Ibid. 3. Letter from Herschel to Babbage, Oct.10 1816; H.ms'.R.S. See D. Lardner An Elementary Treatise on the Differential and Integral Calculus (1825) vii-viii, and T.G. Hall "Calculus of Finite Differences" Encyclopaedia Metropolitana (1830) 2 227-304. p.304. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 220 of Lagrange."''' For Lacroix, had used the method of limits to establish the principles .of the calculus inste&d of Lagrange's method .(of developing functions in series) which the translators felt was the^ijiore correct and natural ~ method".2 In "Note (B) " ^Peacodjkr' showed , how the differential calculus was established by Lagpange' s method. And he < / contrasted this method with the unsatisfactory ones of limits and of infinitesimals While both of the latter methods had some advantages these were outweighed by numerous objections. The chief of these, "that wh-ich we consider as . insuperable", was tfre, tendency of these methods "to separate the -principles andvdepartments of the 4 Differential Calcul’usv£rom those of common Algebra." rrliis^tjas a reflection of the attitude, as seen in the last chapter, of many of the members of the Analytical- Society' towards Analysis. Peacock then went on to compare the Fluxional Calculus with the Differential.^ He rejected the former calculus as clear.ly inferior. 1. Lacroix (1816) iv. 2. Ibid. iii. Lacroix, in his larger treatise on the . calculus, Traite du Calcul diffferentiel et du Calcul integral 3 v.ols. (1797-1800), had used Lagrange's approach to the calculus. 3. Lacroix (1816) 611-614. 4. Ibid. 612. 5. Ibid. 614-620. / Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The consideration of motion, which is essential to the method of fluxions, is foreign to the spirit o^ pure Analysis; and the analogy by which the name >hnd properties of a fluxion are transferred to a modification of the difference of a function, is strained and unnatural-. The different orders *. of fluxions also are involved in considerable ob scurity, and we are utterly unable to comprehend^ the connection' which they respectively bear to tKeir primitive function. In the brevity of^its demonstrations, and in " the facility of its applications, it is unquestionably inferior to all the other methods; and the mixture of mechanical.and geometrical considerations upon which it is founded, are little calculated to assist us in investigating the properties of func tions which are always algebraical in their form, and generally in their nature also. Clearly Peacock was promoting the same vision of analytics ■ ^ as Babbage and Herschel held. Peacock also criticized i the fluxional notation as very often complicated, unsym- metrical, awkward and often incapable of representing various theorems _ 2 % JChe success of the translation of Lacroix led Peacock to proclaim, that "the fluxionists are now nearly talked down" and that "in a very few years, the dottites will be driven from the field entirely."3' However, the victory . of their-* viewpoint was not quite so complete as Peacock i _ ,, .i imagined. On the one hand there were those who, while V willing to adopt the differential calculus in preference to fluxions, were not convinced that the proper foundation 1. Ibid. 618. 2. Ibid. 618-620. 3. Letters from Peacock, to Babbage, Dec.10 1816, Jan. [2?,3?] 1817; B.ms.B.L^ r (. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. \ 222 for the calculus was Lagrange's. William-Wallace, for example, in his articles "Function" and especially "Fluxions", for the Edinburgh Encyclopedia, felt that Lagrange had "underrated the value of the theory^of~ limits". Wallace based his exposition of the calculus on ^that theory. This view was not compatible with that jheld by the former members of the Analytical Society, fierschel, for instance, in thanking Wallace for^a^popy of "Fluxions", praised the -"elegant manner in which the doctrine of limits is laid down" in it but could not accept limits as a proper foundation for the calculus.1 And, on the other hand, the flux'ionists were hardly "talked down". The British Review was very willing to criticize the differential calculus and to reject the differential notation while supporting Newton as the 2 ; real and sole inventor of the calculus. The Monthly Review did not share in the"~t^nslators1 "devotedness for French mathematical^ and especially in their "tendency to undervalue the Fluxional Analysis".3 And a former senior wrangler and fellow of Trinity, Daniel Mitford Peacock (1767/68-1840), wrote a long pamphlet, A Comparative c. 1. Letter from Herschel to Wallace, Sept.23 1815; H.ms.R.S. Herschel made reference to the difficulties it involved in a discussion of imaginary functions. 2. Anon "Review of Dealtry's Fluxions and 'Fluxions' Edin. Ency-" British Review 1_ (1816) 421-437. 3. Anon (probably P. Barlow). "Review of Lacroix's An Elementary Treatise &c." Monthly Review 87 (1818) 179-185. pp.180-183. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. View of The Principles o^ t h e Fluxional and Differential. Calculus (1819, 86pp.), in defence of fluxions. He attempted to refute George Peacock's objections to fluxions by showing the necessity of the concept of motion for the differential calculus and by finding fault with the ex- ' \ position of fluxions in "Note (B)". D.M. Peacock regarded B the inclination to see the calculus as "merely a branch of pure Algebra" as leaving the calculus ope^ to insur mountable objections, particularly when trying to relate the calculus to its various applications.'*' He criticized Lagran^e^s method as merely an assumption and favoured 2 the fluxional notation. Finally, while D.M. Peacock felt that the fluxional calculus deserved the "countenance and support” of the University solely on account of its superior fundamental principles, he could not end his ^pamphlet without adding a few remarks o n ■"what is really useful, and what is not" in academical education. And so he contrasted the available treatises on fluxions with the Lacroix. They were preferable for they directed "the attention to things themselves rather than to mere algebraic functions" and so illustrated "the usj and application of the calculus" more than Lacroix. They also were 1. Peacock (1819) 41-60. 2. Ibid. 54-56, 61-69. 3. Ibid. 85. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 224 ... more in the habit of deriving the solution of particular problems immediately from first principles, and so leading the student by easy and progressive steps to proceed for himself in all similar cases, instead of giving general theorems in the first instance, a method which may suit the experienced analyst, but is ill- adapted to the raw student. D.M. Peacock, like many others'at Cambridge, had no concern with pure mathematics or with its advance. This subject did not befit academical education which was to be "Strictly 2 confined to subjects of real utility." His closing remarks, therefore, revealed an essential aspect of the' conflict ov-^ :s at Cambridge; one which involved questions hbo.ut-~th£~4?urpose of a.iUniversity. Babbage, Herschel and Peacock had little but scorn for D.M. Peacock's pamphlet. They seem to have decided that there would be no point in answering it. As Babbage wroteto Bromhead Have you heard of the last dying struggle of the old school the advocates of dotage Peacock ... has written a pamphlet against Lacroix-and his translators containing sundry incidental knocks at Euler D'Alembert and Lagrange -hoping to ex plode the new system and eject it from the lec tures, this is too tough .a job for his powers ‘ . and he will only break his teeth ig the attempt - The book is its own antidote ... s Yet, while the pamphlet did not appear to have much 1. Ibid. 69-70. 2. Ibid. 85. 3. Letter from Babbage to Bromhead, Dec.l 1819; Br.ms. See also a letter from Herschel to Whewell, Dec.l 1E19; W.ms.T.C. And a letter from Peacock to Babbage, Nov.23 1819; B.ms.B.L. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 225 influence, Babbage, Herschel and Peacock seem not to have realized the grea The translation of Lacroix's Traite elfementaire was but one response by Herschel, Babbage and Peacock to Cambridge mathematics. Unlike D.M. Peacock„ they were concerned with pure mathematics, with its progress, and with "professional" mathematics being studied at Cambridge. Herschel, for instance, like William Spence, saw the cause of the backwardness of English mathematics as reflected in the "style and character" of English elementary treatises,. especially in those on fluxions.1. Aside from neglecting the achievements: of Continental mathematics, these treatises did not develop any abstract theory or focus on the analytical methods themselves' but instead concentrated on all sortp of useless particular applications 2 without any uniformity of method or pervading principle. Herschel had hoped that i ... a state of science was at length arrived when this cloud of consecrated puerilities might be dispersed, and that the attention of the elementary reader, no longer distracted by an impertinent detail of^trivial applications might be allowed to concentrate itself upon the real 1. J. Herschelr. "Review of Dealtry's The Principles of Fluxions" (1816, 19pp.) Unpublished; H.ms.T. p.2. On Spence see chapter III, of my dissertation, pp. 64-65. 2. Ibid. 4-6, 9. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. difficulties and essential principles of a vast and complicated system.1 The effect of these treatises on students, wrote Herschel, was to squander away the force and vigor of their minds, to quash their spirit of inquiry >and to destroy their 2 relish for mathematical speculation.- And so the usual treatises served to "obstruct and choke instead of ren dering accessible, the approaches to mathematical knowledge".^ Thus one aspect of any mathematical revival in England would have to be, in the reformers': view, a revision of the usual course of study with a concentration on pure mathematics and with an eye on keeping pace with the general advance ment of thq field. The purpose of Cambridge mathematical studies, for the reformers, was therefore quite different from that held by supporters of a liberal education. The reformers :wished to make students well-versed in modern mathematics, the others wished to educate a gentleman. In this spirit, Herschel and Peacock, especial- ly , felt that a replacement for the whole elementary course at Cambridge was "essential to a system of radical 4 A reform". Herschel's planned. "Algebra" was to be part of 1. Ibid. 1. 2. Ibid. 13. 3. Ibid. 18. 4. Letter from Herschel to Babbage, July 14 1816; H.ms.R.S. Letters from Peacock to Herschel, [Nov.14 1816], Mar.17 1817; H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 227 this mathematical reform as was-a work which Peacock was } 1 to write on the application of algebra to geometry. But the only work which they were to produce within this aim, besides the translation of the Lacroix, was A Collection of Examples (1820). Suggested by Babbage at the beginning of December 1816 and fervently approved of by Peacock' - he announced it as "ready for publication in the course of a few months" in the "Advertisement" to the Lacroix - the 2 Examples was planned as a sequel to the Lacroxx. It was to contain additional problems and examples to various parts of the Lacroix in the hope of attracting the Cambridge student. And so it would promote both the 3 translation and, of course. Continental mathematics. However, Babbage was not very enthusiastic about actually 4 composing the work. He had had doubts about the value of Herschel's and Peacock's attempts to reform Cambridge studies. The laudable designs and exertions which you communicate gave me much pleasure I have no doubt of the ultimate success of the true faith 1. Ibid. And a letter from Herschel 'to Babbage, Dec. 24 1816; H.ms.R.S. 2. Lacroix (1816) iv. Letter from Peacock to Herschel, Dec. 3 1816; H.ms.R.S. See also Babbage (1864) 39-40 and ' . Peacock (1820) 1 iii. 3. Ibid. 4. Letter from Herschel to Babbage, Jan.30 1817; H.ms.R.S. *» Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 228 but I have many as to the question whether its propagation will derive any profit from its establishment.1 And now, in early 1817, he felt that the Examples was a nuisance "when one has some half dozen original papers 2 dying to be written". With Herschel's encouragement Babbage did collect some examples on differential equations in 1817.5 But Peacock later felt that this was^done rather slovenly and needed much rearrangement and alteration. , 4 Peacock fi/nally did most of this work over completely. In the end Babbage contributed the collection of integrals to volume one and the 42 page ;Examples of the Solutions of Functional Equations of volume two.5 Herschel had almost finished his part of the Examples by October of 1817. . Yet due to the delays caused by the printer and by Peacock's ill health, other engagements and indolence, Herschel was to add "pretty things" to his part .in the following three years. It appeared as A Collection of Examples of the Applications of the Calculus of Finite Differences in volume two and was 17 6 pages long. 1. Letter from Babbage to Herschel, July 20 1816; H.ms.R.S. 2. Letter from Babbage to Herschel, Feb.27 1817; H.ms.R.S. 3. Letter from Babbage to Herschel, Nov.11 1817; H.ms.R.S. 4. Letters from Peacock to Herschel, Apr.l 1818, Aug.8, Dec.5 1819; H.ms.R.S. ^5. Letter from Peacock to Babbage, Nov.7 1820; B.ms.B.L. / Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 229 Peacock did not begin work on his part until about r June of 1817. Printing began that December and went on l very slowly for the next two years when/Peacock's part on / the differential and integral n-almilng (Jvol.l, 514pp.)' 1 C J was finished. It then took another year until the two- volume A collection of Examples of the Applications of the Differential and Integral Calculus, of the Calculus of Finite Differences and of Functions was published on 2 October 26 1820. One thousand copies were printed and the work sold for an expensive 30 .shillings.3 The total cost of publication came to about ^400 and while Peacock hoped the sale would bring in j£l000, and the work was selling well, it appears that the authors did not make 4 much money on it. It must have been a popular work at Cambridge because many years later (in 1840), D.F^_ Gregory sought to bring out a second edition, but he was unable to and published instead his own collection of examples.5 1. Letter from Peacock to Herschel, Dec.5 1819; H.ms.R.S.' 2 . Letter from Peacock to Babbage, Nov.7 1820; B.ms.B.L. 3. Ibid. 4. Letter from Peacock to Herschel, Nov.16 18201820 4 H.ms.R.S. Letter from Peacock to Herschel, Feb.18 1822;l B.ms.B.L. And a letter from Peacock to Babbage, Apr.l 3;827; B.ms.B.L. ' 5. Letters from D.F. Gregory to Babbage, June 6 , 16 18,40; B.ms.B.L. Gregory's work was Examples of the processes of the Differential and Integral CaTculus (1841). See pp.iii-iv. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 230 Babbage, as noted above (p.227 ), was not very interest ed in participating in attempts to reform Cambridge studies. Herschel was concerned; he even wished that there would exist some incentive for the graduate to continue his studies in mathematics. / v ' 0 that men in the''~2 or 3 years after their Bachelor's degree had some inducement to read upon a broad and manly plan in the Mathematical way as they have in the Classical & Theological departments - we should soon see a change for the better in the state of Cambridge Mathematics. But Herschel disliked "cramming pupils, which is a bore 2 & does one no credit but very much the contrary." After leaving Cambridge in October 1816 he was hardly involved in the Cambridge scene, though he did maintain his hope that the University would be active in advancing 3 science. In the following years both Babbage and Herschel were to transfer their reform-mindedness from Cambridge and mathematics to England and science. It was the zealous and indefatigable Peacock who was to push reform,at Cambridge. Peacock was in the midst of many projects of reform in late 1816. He was behind the proposal to appoint a syndicate to erect an Observatory, which was eventually built in 1822-1823. He was involved in schemes to change the second year examinations at Trinity and to -replace 1. Letter from Herschel to Whittaker, Jan.25 1817; St.J.ms. 2. Letter from Herschel to Babbage, July 14 1816; H.ms.R.S. 3. Letter from Herschel to Brewster, Dec.17 1819; H.ms.R.S. See also a letter from Herschel to Buckland, Aug.16 1832; St.J.ms. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 231 astronomy there with the calculus and the theory of .. t equations. And through his teaching post^a-t Trinity he was propagating "the true faith effectually".3 But most important of all was his appointmen\ as a moderator of the' 1817 Senate House Examination. He planned a number of reforms both in the way in which the examination was conducted and in its. content. The control of the very important Senate House Examination was the fourth feature of the structure of Cambridge studies which was used by the reformers to introduce analytics. Peacock’s objections to the system of examination were based on his view that the Senate^House sacrificed an understanding and knowledge of mathematics to cramming.3 So he found fault with the system of marking and particularly 4 with the viva voce examination.- His views on the content of the examination stressed "good" mathematics, especially the differential calculus. Peacock, for a time, managed to get the other moderator, John White, to agree to many of his ideas for reform. Of the two Examiners, Miles 1. Letter from Peacock to Babbage, Dec.10 1816; B.ms.B.L. 2. Ibid. 3. Letter from Peacock to Herschel, Mar.4 1817; H.ms.R.S. 4. Ibid. And a letter from Peacock to Herschel, i)ec.3 1816r H.ms.R.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. * 232 Bland opposed Peacock's views from the start, which Peacock had expected. Peacock asked Herschel to write to the other Examiner, Fearon Fallows, to "recommend reform & to urge him to accede to it".1 Herschel wrote to Fallows concerning , ... the lamentable defects which have too long been suffered to exist in many parts of our system of Mathematical study, and which can only be rectified by a strong bias in the course of examinations, in the opposite direction - there is in fact no other means by which the studies of men can be directed but by modelling their examinations accordingly, for they always have and ever will continue to read with direct reference to that ultimate object 2 He continued by stating the same objections as Peacock had, and ended by hoping that these blemishes would be removed or at least lessened. - . Peacock's efforts at reform in early 1817, however, failed. Due to the influence of the "older members" of \ the University,.White and Fallows, who had at first been favourable to Peacock's schemes, ended up opposing them. All that Peacock could do was confined to his own papers , where he used the differential calculus.3 And even this change caused quite a stir. Whittaker wrote to Bromhead that Peacock had "made himself very unpopular, by his 1. Ibid. 2. Letter from Herschel to Fallows, Dec.8 1816; St.J.ms. 3. Cambridge Problems (1836) 338-358. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / 233 quixotism & want of discretion". Whewell, in his account for Herschel of Peacock's' papers, said He has stripped his analysis of its applications & turned it naked among them - Of course all the prudery of the university is up and shocked at the indecency of the spectacle - The- cry is "not enough philosophy". 2 And Peacock himself, disappointed, described the reaction to his endeavour: The examination was much as usual White & Fallows are entirely of the old school & the influence of their examination was so great as completely to overpower my examination: the introduction of d's into the papers excited much remark: Wood, Vince, Lax & Milner were very angry & threatened to protest against analytic s, French mathematics S. I believe that I may consider myself as entirely to the success of the Johnians in the examination for my escape from some public proceeding against me. 3 r Despite this setback Peacock was determined bo continue his attempts at reform. A few months after the Senate House Examination Peacock went out of his way, on the occasion of D'Arblay's disputation, to display French mathematics. D'Arblay wrote to his mother ...Peacock who to my great surprize officiated for that day & that day' only, tho' the week belonged to Mtf White (no doubt that he went to M£ White & told him that he wished to have my day) made me several questions calculated to bring into play all my french mathematics - LaGranqe, - & LaCroix ^ 1. Feb.3 1817; Br.ms. 2. Letter from Whewell to Herschel, Mar.6 1817; H.ms.R.S. Quoted in Todhunter’ (1876) 2 16. 3. Letter from Peacock to Herschel, Mar.4 1817; H.ms.R.S. 4. Letter from D'Arblay to Mme. D'Arblay, Mar.7 1817; British Library. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 234 Peacock, wishing, that Herschel could be at Cambridge to help silence his opponents, pledged his resoluteness to reform. I assure you, my dear Herschel, that I shall never cease to exert myself to the utmost in the course of reform & that I shall never decline any office which may increase my power to effect it: .... I shall pursue a course even more decided than hitherto, .-. .. X have considerable influence as a lecturer & will not neglect it .... I have no [doubt] respecting the ultimate success of these plans, but the period in which they may be effected, may be abridged most materially by our personal exertions.... 1 In the following years Peacock gained a reputation at Cambridge as a liberal reformer. He continued to spread the spirit of analysis at Trinity and at the 1819 Senate House Examination where he once again was 2 moderator. ■ This time he had the cooperation of the other moderator, Richard Gwatkin (a former member of the ^Anj&ytical Society).., and of Fallows. Peacock "had as much analytics in his paper as ever but he took upon . himself to be scandalized (not without reason) at the ignorance and superficial knowledge of applications of mathematics which he found ....”3 Peacock planned 1. Letter from Peacock to Herschel, Mar.17_1817; H.ms.R.S. 2. Letters from Peacock to Herschel, Mar.7 1818, Jan.13 1819; H.ms.R.S. 3. Letter from Whewell to Rose, Mar.17 1819; W.ms.T.C. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 235 various textbooks in mathematics, such as a mechanics, an analytical and physical optics, and a differential and integral calculus, but none of these ever appeared. J Besides the Cambridge Observatory he played a prominent role in the establishing of the Cambridge Philosophical 2 ^ Society m 1819. In the following years Peacocks,continued to do some work in mathematics; he is always remembered for his pioneering Algebra (1830). He was deeply involved in teaching and in various Cambridge schemes of reform. Peacock also extended his activity to English science, and to the Church' of England after becoming Dean of Ely in 1839. He died on November 8 .1858. The mathematical, work, the accent on analytic^’and the efforts towards reform of Babbage, Herschel and Peacock acted as a catalyst on the early nineteenth- century Cambridge scene. Their work and their views attracted others at Cambridge, who collectively formed a loose, Cambridge mathematical revival movement. It was ''this movement which carried on in the spirit of 1. Letters from Peacock to Herschel, Mar.17 1817, Aug^8 1819, May 14 1821; H.ms.R.S. 2. Herschel's draft of an obituary of Peacock for the Royal , ^Society (Herschel (1859)); National Maritime Museum, Herschel Archive. Many people felt that the Analytical Society had inspired the Cambridge Philosophical Society. See, for example, Bromhead1s opinion in his letter to Babbage, Mar.7 1821; B.ms.B.L.- Or C. Lyell's view in his "Scientific Institutions [a review of the transactions of various societies]" Quarterly Review 34 (1826) 153-179. p.169. ' Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Babbage, Herschel and Peacock. Working"through the structure of Cambridge, the movement resulted in the adoption of analytics at Cambridge University in the • 1820s. Perhaps, the best representative of this movement was William Whewell. Although Whewell came up to Cambridge in the fall of 1812, he does not appear to have ubeen a member of the Analytical Society. The first reference to him by any member of the Analytical Society dates from July of 1814.''' It seems that it was only after his graduation in January 1816 as second wrangler and second Smith's prizeman that Whewell resolved to study 2 Continental mathematics. He was very interested in Babbage's work on the calculus of functions; indeed, Babbage was driven into a new "fit of the mania Analytica" from talking with Whewell and Peacock in January 1817.'" Whewell .was; apparently working on integrals at about 4 this time. And he was filled with zeal for the mathematics of Herschel, Babbage and Bromhead, thinking that in England at that time 1. Letter from Gwatkin to Whittaker, July 17 1814; St.J.ms. 2. Letter from Herschel to Babbage, pmk.Feb.7 1816; H.ms.R.S. 3. Letter from ^ 4. Letter from Bromhead to Whittaker, Feb.20 1817; St.J.ms. ^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 237 ' ...there are■the materials for a new era of English mathematics perhaps of mathematics tljemselves - I hope to see all science reduced Tinder the dominion of mathematics and all mathematics resolved into the eternal relation ^ . of symbols that is the inflexible laws :of thought - ... Whewell also had a low opinion of Cambridge mathematics. .1. • In a sketch of a proposed Cambridge.mathematical periodical, . he saw it as made up of two parts: the first, full of Cambridge cram; The second must be everything that will be useless •*£0 a Cambridge man - i.e. good Mathematics - Generalizations - Extensions of Processes - Illustrations of the principles and spirit of methods - Analyses of the metaphysical principles of mathematics pure & mixed - Analogies illustrated - Extracts from good books little known - account of foreign & new mathematical books - in short any thing worth reading - In March 1817 Whewell, in apparent agreement with Peacock's view of the necessity of such a work, began to translate^Lacroix’s work on the application of algebra to geometry.3 This task had been proposed by his friend 4 H.J. Rose. Despite Whewell's completion of most of his partoof the translation, the work was never published, probably because Rose did not finish his share.^ 1. Letter from Whewell.to Bromhead, May 4 1817; Br.ms. 2. Entry for June.3 1817 in Whewell’s diary (R.18.93); W.ms.T.C. 3. This was part of S.F. Lacroix's Traite elementaire de trigonometrie rectiligne et spherique et d 'application de 1' algebre a la qeometrie. (1798).. 4. Letter from Whewell to Herschel, Mar.6 1817; H.ms.R.S. Quoted in Todhunter (1876) 2 16. 5. Letters from Whewell to Rose, Apr.11, 15, May 21, June 26, -Aug.30 1817; and a letter from Herschel to Whewell, June 18 1817; W.ms.T.C. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 238 Since his graduation Whewell had been working as a private tutor in Cambridge, a task which he did not enjoy. This burden was somewhat lifted when he became a fellow of Trinity College—in October 1817. A year later he became assistant/ tutor arffl mathematical lecturer; at Trinity.'*' Whewell saw this post as an'opportunity to reform the mathematics of the University. I have it now in my power to further this laudable object by the situation I have taken of assistant Tutor (i.e. Mathematical Lecturer) here. Whatever may be the disadvantages of the office this is one 'of its advantages. I shall have a permanent and official interest in getting the men forwards - I shall have an opportunity of directing their reading - and I shall write books (good ones of course) and be able to put them in circulation - By using such powers wisely but discreetly much may be done.2 With the additional time which his fellowship afforded, Whewell had begun "dabbling in some of the creeks of the ocean of analysis" and in particular began a-"good book" on mechanics to replace Wood's.3 His An Elementary Treatise on Mechanics.appeared in the fall •J of 1819, although it was basically finished by November.. 4 of 1818. Whewell had'two objects in mind while 1. Todhunter (1876) 1 11, 2 27-28. 2. Letter from Whewell to Herschel, Nov. 1 1818; H.ms.R.S Quoted in Todhunter (1876) 2 30. 3. Letter from Whewell to Bromhead, Oct. 9 1817; Br.ms. Letter from Peacock to Herschel, Apr. 1 1818; H.ms.R.S 4. Letter from Whewell to Herschel, Nov. 1 1818; H .ms . R. S Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 239 preparing this book. The first was to firmly establish and to logically arrange the principles of the subject, a task neglected, he felt, by both French and English authors.'*' The other aim was that it would serve as a lecture book for second year students. Thus it was mostly a classification of problems with some general;principles. It was to be "readable without a very perfect or extensive knowledge of pure mathematics" and was to illustrate the application of the differential calculus to the subject. In short, it was to be a book which would enable those who wished, to proceed' ■ to 'the higher branches of mechanics. 2 Babbage, Herschel and Peacock all thought that the work would benefit Cambridge.^ With the same goals Whewell published four years later a work intended as a second volume of his mechanics, his Treatise on Dynamics (1823). There were, of course, many others who were part of the mathematical revival movement. As Whewell noted in 1831, < ... there has been at Cambridge a succession of mathematical students, who have rejoiced ;to dis port themselves in'the wild and wondrous region of analytical generalities and symbolical involutions, sometimes to the perplexity and dismay of an older race of reasoners, accustomed to more palpable 1. Ibid. Letter from Whewell to Bromhead, Apr.13 1819; Br.ms. And his An Elementary Treatise on Mechanics (1819) iii-vi. . 2. Letter from Whewell to Herschel, Nov.l 1818; H.ms.R.S. Letter from Whewell to Bromhead, Apr.13 1819; Br.ms. 3. Letter from Babbage to Bromhead, Dec.l 1819; Br.ms. Letter from Herschel to Whewell, Dec.l 1819; W.ms.T.C. Letter from Peacock to Herschel, Aug.8 1819; H.ms.R.S., Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 240 objects of thought and narrower rules of combination.'*' The 1820s saw the old standard textbooks entirely supplanted by new analytical ones or by translations of French texts. Among these were Robert Woodhouse1s Treatise on Astronomy (.1821-1823) ; Henry Coddington's An Elementary Treatise on Optics (1823) based in part on Whewell's lectures and intended to be "suited to the present'state of Mathematical 2 knowledge"; George Biddell Airy's Mathematical Tracts on Physical Astronomy (1826) to meet the "entire neglect of the analytical mode of treating Physical Astronomy ... in our Mathematical System";^ Henry Parr Hamilton's The Principles of Analytical Geometry (1826) designed to illustrate "the importance.of Analytical Geometry, as a 4 Method of Investigation"; Ralph Blakelock's translations of J.L. Boucharlat's to Elementary Treatise on the dif ferential and integral calculus (1828) and of L.B. Francoeur's A Complete Course of Pure Mathematics (1829- 1830); Henry Moseley's Treatise on Hydrostatics and Hydrodynamics (1830); and a large number of short and 1. IW. WhewellJ "Science in Enqlish Universities" British Critic 9 (1831) 71-90. p.85. 2. to Elementary Treatise on Optics (1823) iii. 3. Mathematical Tracts on Physical Astronomy (1826) iii-iv. 4. The Principles of Analytical Geometry (1826) iii. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 241 long treatises on the differential and integral calculus which mostly were based, following the views of Herschel, Babbage and Peacock, on Lagrange's method.'1' And numerous works were a-lso published by persons outside of Cambridge. For instance, Dionysius Lardner,. a graduate in 1817 of Trinity College, Dublin, published a great number of analytical t: in the mathematical sciences. A devotee of analytics r wrote such treatises as A System of Algebraic Geometry (1823), An Elementary Treatise on the Differential and Integral Calculus (1825), and An Analytic Treatise on Plane and Spherical Trigonometry - dedicated to Babbage - (2nd ed., 1828) in order to contribute "to the great work of improvement" in progress at the University 2 of Cambridge. Of equal importance, if not greater, for the adoption of analytics at Cambridge was the influence of the Moderators on the Senate House Examination. Peacock, aware of the significance of the examination in the Cambridge system of honours, had used his position as moderator in 1817 and again in 1819 to alter the content of the examination in accordance with his views. His aim was to influence the course of study of the undergraduates.; The emphasis on 1. See, for example, Arthur Browne's A Short View of the First Principles of the Differential Calculus (1824), Thomas Jephson's The Fluxional Calculus (1826-1830) , and Charles Myers' An Elementary Treatise on -the Differential Calculus (1827). 2. An Elementary Treatise on the Differential &c. (1825) v-vi. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 242 analytics was continued by Whewell and by Henry Wilkinson, a former member of the Analytical Society, when they were moderators in 1820. And this stress was maintained by the moderators throughout the 1820s, most of whom were also tutors in their colleges and, many, authors of such treatises as those mentioned above.''' he contents of all of these analytical treatises quickly found their way into \ the Senate House. 2 . \ The reform of the Senate House'wks not limited, as shown by Peacock's original aims, to/simply the style of mathematics. Aside from changes in /the mode of examination, with analytics came a much wider range and deeper study of mathematical subjects than before 18^JK'^__By''l832 Herschel could happily report on the role of the moderators in the revival of Cambridge mathematics: They were carried away with the stream, in short, or replaced by successors full of their newly- acquired powers. The modern analysis was adopted in its largest extent, and at'this moment we be lieve that there exists not throughout Europe a centre from which a richer and purer light of mathematical instruction emanates through a com-^ munity, than one, at least, of our universities. The 1820s was a period of transition in Cambridge mathematics. Augustus DeMorgan, enrolled at Trinity .from 1. For lists of the moderators see Historical Register (1917). 2. "Report of the Board of Mathematical Studies, 1849" Great Britain (1852) 452-456. p.454. 3. "Answers from William Hopkins" Great Britain (1852) 461-471. p.463. See also p.113. 4. Herschel (1832) 545. [ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 243 1823 to 1827, .felt that he had been a student "at Cambridge in the interval between two systems", the geometrical and the analytical.'*' And many others at the time, even out side of Cambridge, were aware of the great changes occurr ing there. Both Charles Lyell and Baden Powell, for instance, noted with satisfaction the introduction of 2 analytics and the progress of Cambridge studies. A different evaluation, however, was given to the changes in mathematics in the 1820s by those attached to the Newtonian methods or otherwise opposed-to analytics. Samuel Butler, head-master of Shrewsbury, deplored in 1822 the fact that Cambridge had "deserted the track of 3 geometry, and forsaken the path our mighty master trod." And J.H. Monk, tutor at Trinity and Regius .Professor of Greek, while disagreeing with Butler's view that the labors of Newton were neglected at Cambridge, conceded that "within the last six or seven years, too much stress has 4 been sometimes laid upon the French analytics". The exchange between Butler and Monk, however, had not arisen because of opinions about the type of mathematics studied 1. DeMorgan (1882) 306. v 2. C. Lyell "State of the Universities - a review of five works on education" Quarterly Review 36 (1827) 216-^68. B. Powell "Progress 85f Mathematics - a review of eleven mathematics texts" The London Review 1 (1829) 467-4^6. See also Powell (1834) 367-368. 3. Eubulus [S. Butler] Thoughts on the Present System of Academic Education in the University of Cambridge (1822) . 4. Philograntus [J.H. Monk] A Letter to the Right Reverend John, Lord Bishop of Bristol, &c. (1822). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 244 at Cambridge but rather on account of the growing dis content in and out of Cambridge with the course of studies there. In particular, there was a great deal of agitation for a more comprehensive course of studies; one which would be less exclusively concentrated on mathematics. One result of this criticism was the approval of the Classical Tripos in 1822. Another was a more rigorous justification of the role^csf mathematics in University education. Here, the ideology of Cambridge - a liberal education - was to gain renewed prominence, and be used in restricting the extent of the analytical reforms and in defining the type of mathematics to be taught at Cambridge. Mathematics had been valued at Cambridge, within the ideal of a liberal education, for its training of the reasoning powers of the mind, an important aspect of the purpose of intellectual education in the University. As seen in chapter three, geometric methods were generally acknowledged as superior to analytics in strengthening the reasoning faculty. Not everyone agreed, especially those promoting analytics. John Brinkley, for instance, felt that analysis had been so improved that it could be used with "unerring certainty" to deduce conclusions.'*' However; along with the increasing adoption of analytics at Cambridge in the 1820s and with the criticism of the 1. J. Brinkley "Review of R. Woodhouse's An Elementary Treatise on Astronomy (1818)" Quarterly Review 22 (1819) 129-149. pp.132-133. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 245 curriculum came a renewed emphasis on the value of geometric methods in education.' A graduate of Cambridge wrote in 1825 for the London Magazine: But I must inform your common readers, that- geometry is not the fashion, at present, in mathematics; ...... our college pursuits, or the mathematics on which we pride our selves, are not founded in geometry but on alge bra: .... The whole is a system of conjuration, if I may use such a word for want of a better. Not only is there no one step that can be called reasoning, but the man who works this engine, does not even know, from one minute bo another, what he is doing; nor does he see one inch beyond the unmeaning symbol which he substitutes or trans poses, multiplies or divides, squares or cubes. Likewise for Arthur Browne, fellow of St. John's, it was geometry that made the study of mathematics so valuahle for mental discipline and hence an important part of 2 the Cambridge studies. Geometry, he felt, accustomed the mind to reason.and to think, to arrange, combine and clearly express ideas, and to form comprehensive views Browne feared that ... unless some effectual barrier be raised against the introduction of French mathematics, our University, which has long been, and still continues to be, the seat of sound learning and religious education, will, in time, become a mere school ^ of useless subtleties, and Analytical refinements. 1. Cantabrigiensis "The Regrets of a Cantab" London Magazine 3. (1825) 437-456. pp.455-456. 2. A. Browne "Preface" A Short View of the First Principles of the Differential Calculus (1824) i-xxii. 3. Ibid. vii-xiv. 4. Ibid. xvi-xvii. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 246 Connected with the stress on geometry in education was a view of the purpose of a university which rejected _ its role as a discoverer of knowledge. Thus the London Magazine saw the triumph of analytics over geometry as "one more proof how strongly the tide of opinion at Cambridge sets in towards the belief, that men are congregated in those Boeotian flats for the promotion of science, .rather than of education.And Browne felt that any superiority which analytics had over geometry was valuable only to those who made mathematics or science a profession. But the object of the University, he thought, was not to expand science but to diffuse religious knowledge and to supply men qualified for offices in the Church and in 2 the State. There were some in England.who, like the members of the Analytical Society and the mathematical revival movement, felt that professional mathematics were a part of university studies and that research was part of the university's purpose. Baden Powell, for instance, saw the valuing of the mathematical sciences " solely as instruments of education" as one of the causes of their 3 decline in England. And he felt that the progress of' . 4 knowledge was. an important part of the university system. 1. Anon "The Cambridge University" London Magazine 4 (18261 289-314. p.303. ------2. See page 245, footnote 2, pp.xiv-xv, xviii-xx. 3. B. Powell "Progress of Mathematics - a review of eleven mathematics texts" The London Review 1 (1829) 467-486. pp. 479-481. 4." Powell (1832) 44-45. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24-7 This notion of advanced studies as part of the business of a university does not seem to have prospered very well at Cambridge. It was the ideal of a liberal education which flourished there. The ideal was even found among former members of the.revival movement. Many of those who had belonged to the "true faith" became revisionists, that is, they altered their ideas - and especially those concerning analytics - to fit the circumstances at Cambridge. For example, G.B. Airy later called for a move away from /'pure mathematics and felt that geometrical methods and elementary studies were best for Cambridge^education.1 And H.P. Hamilton in-.the fourth edition (1838) of his An Analytical System of Conic Sections stated that ... experience has since taught him, [H.P.H.] that this method [as in the previous editions] of treating Conic Sections, although sanctioned -by the practice of distinguished Continental Writers, is too scientific, if he may be allowed the expression, for 'elementary instruction.2 But it is William Whewell, once again, who serves as the best example of a revisionist. As seen earlier in this chapter, Whewell had been dedicated to reform and to the "true faith" of analytics. However,, even at this period he had shown some indications of dissent and conservatism. He had disagreed with Peacock's stress on pure analysis Lin the Senate House Examination 1. Airy (1896) 274-280. 2.. An Analytical System of Conic Sections (4th ed. 1838) iii. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a c ^ of 1817, thinking that analysis should have been intro- < 1 duced through its applications, a form suited to the taste of the University.'*' And although Whewell was very interested in analysis, he seems never to have come close to any of 2 _ the field's frontiers. Whewell's deviation is clearly shown first in his texts, An Elementary Treatise on Mechanics (.1819) and the Treatise on Dynamics (1823) . As previously ■ discussed, Whewell's object in the Mechanics was to pre pare an elementary treatise which would establish the topic's principles and include many problems. Now, while Herschel thought that this work was superior to anything else at Cambridge and so had to do much good, he of?ly \ regarded it as a temporary "stepping stone to some more finished & systematic.elementary work ... in the transition state of Cambridge reading".3 His reservation was that it made too great a concession to the cramming system of Cambridge. It would have been much better, he-said, if it had "conformed a little more to the taste of the age 4 & a little less to that of the University." Similarly Whewell's Dynamics, while written in the language of. analysis, was a very elementary work, full of «> 1. Letter from.Whewell to Herschel, Mar.6 1817; H.ms.R.S. ^ Quoted in Todhunter (1876) 2 16. 2. See, for example, a letter from Whewell to Rose, Apr.15? 1817; W.ms.T.C. And letters'to Bromhead, Oct.9 1817, Apr. 13 1819; Br.ms. -- 1 3. Letter from Herschel to Whewell, Dec.l 1819; W.ms.T.C. See also Herschel (1832) 545-546. 4.. Ibid. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. * 249 problejns to show the application of the general formulae. Published just.after the establishment of the Classical Tripos, Whewell acknowledged in this work the need to r 'give a knowledge of the branches of mathematical science \ to "a greater number of persons than we can expect to form into profound analysts."1 This need required a series of introductory treatises like his own which combined the advantages of analysis and of geometry. 2 It is interest ing to note the reaction of the Westminster Review,1 because like Herschel it too was interested in the advancement > y of knowledge. Apologizing for complaining of the style of the book at a time "when we are so lamentably back ward in the staple of abstract and applied mathematics", the reviewer felt compelled to note the deficiencies of 3 WhewellVs books as works of science. They were inadequate “• . as "a compact and'luminous development of the theories of Mechanics". And while they were valuable as collections of problems, the reviewer thought that the texts suffered from being intended for the peculiarities of the .Cambridge system rather'than as a "scientific deduction" of the . “ 4 general principles of mechanics. Whewell's response to 1. Treatise on'Dynamics (1823) vi. 2. Ibid. ^ 3. Anon "Review of Whewell's Treatise on Dynamics" West minster Review 2_ (1824) 311-324. p.311. 4. Ibid. 323. ¥ ( Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. .( . ■ 250 this' charge would probably have been the same as that of the Brutish Critic.^ The Critic acknowledged the inferior ity of the old English school but at the same time felt that the modern English student had imbibed too much of 2 1 the spirit and taste of the Continent. French analjjtics had a number of advantages but also many flaws, especially for the learner. So the Critic was glad to see works such as Whewell1s which kept away from extremes of analytics and diffuseness of style and yet did both "embrace the modern improvements of the French school, and retain the solid qualifications of an English character.^ Whewell's enthusiasm for analytics deteriorated as cj his views on the purpose of a university became clearer. •In a defence of English Universities in 1831 against an accusation that they neglected, "modern knowledge and improvements", Whewell argued that the function of a 4 university was to educate, not to discover. Undoubtedly 1. Anon "Review of Bland's Hydrostatics and Whewell's Mechanics and Dynamics" British Critic 23 (1825) 163-174. t 2. Ibid. 166 1 3. Ibid. - 4. [W. Whewell] "Science in English Universities" British Critic 9 (1831) 71-90. " ^ -- v. t .. > ft Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. this view along with changes in the Senate House Examination, which was increasingly adapting its first few days to a more elementary level for the lower classes, led Whewell to renounce even the limited form of the "true faith" of combining analytics and geometry. The fourth edition of his Mechanics in 1833 omitted the general analytical processes of the previous editions. He did this because of the Cambridge system, but also justified it in terms of his developing theory of knowledge with its views on the role of mathematics in training the reason.^ The evolving relation among these factors may be seen in 2 Whewell's various works on education and philosophy. Suffice it to note here that Whewell went from revisionism to renouncement to denouncement. For in his last mathematical book. The Doctrine of Limits . (1838), Whewell attacked what had been practically the central tenet of the "true faith" of the Analytical Society and of the Cambridge.revival movement: he criticized the acceptance of Lagrange's attempt to base the calculus on algebra. The temporary favour which the project found in the eyes of some mathematicians, arose, as I conceive, from the. persuasion that mathematical 1. An Elementary Treatise on Mechanics (4th ed.1833) viii-ix. 2. Thoughts on the Study of Mathematics as part of a Liberal Education (1835)., "Remarks on Mathematical Reasoning and on the Logic of Induction" The Mechanical Euclid (1837) 143-182, On'.the Principles of English University Educ ation (1837), The Philosophy of the Inductive Sciences (1840), and Of a Liberal Education in General (1845). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 252- truths are exhibited in their most genuine shape when they are made to depend upon definitions alone; an opinion of which I hope I have made the falsity apparent.1 Despite the opinions of such eminent persons as Whewell, jthe tendency at Cambridge in the first half of the nineteenth century was for the course of studies to become more and^more symbolic and analytical. However, it appears/that general opinion^ on mathematical studies was slowly aligning itself along the pattern .of Whewell's thoughts. By 1850, in Whewell's view, the "mischievous tendency" of analytics had been successfully checked.3 This break, or reversal, had occurred in two ways. First, geometry and elementary mathematics were stressed for their men-talr^training and the mathematical sciences were made subservient to'the goals of intellectual dis cipline, for which it was required that ... all unnecessary exuberance of an analytical calculation be repressed, and that among methods of connecting assumed principles with their remote consequences the more lucid should be preferred to the more powerful, and the subject matter pressed on the attention, and not suffered to become overlaid and lost in symbolic detail.^ 1. The Doctrine of Limits (1838) xii. See his criticism of Dugald Stewart's (and Babbage's) views of. definition in mathematics in his "Remarks &c." in the Mechanical Euclid, cited in footnote 128. 2. Great Britain (1852) 451. 3. Ibid. 500. 4. Ibid. 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 253 * And secondly, in agreement with Whewell1s view that only certain "permanent studies" were educationally valuable, many subjects of study were either limited or completely excluded. So those mathematical sciences without clear fundamental concepts, or involving "questions about which Mathematicians are not sufficiently agreed”, or leading to long analytical processes, were rejected.'*' Thus Cambridge studies were officially committed to a disavowal of professionalism and its attendant advancement of knowledge; that is, it was committed to an affirmation of a liberal education. Such was the situation at Cambridge, that this philosophy of a liberal education became so very pervasive that even those who attempted to defend analytics did so only within the framework of its competence for training the student's mind - without, apparently, any ■ consideration for the promotion of mathematics or the 2 training of mathematicians. Hence, the ideology of Cambridge had been used to determine both the style and the content of Cambridge mathematics. Thus it was that only part of the predisposition of the Analytical Society anc^ of the mathematical revival movement was to be realized at Cambridge. The mood of 1. Ibid. 115. ■2. See, for example, R.L. Ellis in Great Britain (1852) 444-448. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 254 these groups had been formed in a context in which the decline of the mathematical sciences was prominent. The result was a reform group which stressed the necessity of students learning current, advanced mathematics - analytics. Through private tuition, college lectures, textbooks and the Senate House Examination - in short, through the structure of Cambridge studies - the reformers were quickly successful in promoting the study of analytics. While analytics, as mathematical knowledge, could be swiftly introduced, it was much more difficult to effect changes which would alter the traditional view of the University, especially in a period where there was much conservatism at Cambridge. The element of professionalism which was implicitly attached to analytics and which had been an important motivating factor among the reformers, did not take root at Cambridge. It quickly died out, rejected by the circumstances of Cambridge, frustrated by the ideal of a liberal education. Indeed such were those circumstances that there was a reaction at Cambridge which was to limit analytics and to reaffirm the importance of synthetics within the framework of the ideology of Cambridge. While the analytical movement could not fulfill itself at Cambridge, it was at least able to modify some of its circumstances. The movement's legacy was a revived course of mathematical studies at Cambridge which, if it did not instruct most of its students in advanced mathematics or mathematical science, it at Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 255 least gave a very good preparation to those so inplined. C- / Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VII. ' Conclusion This thesis was an attempt to obtain insight into the nature of Cambridge mathematics in the early nineteenth century by focusing on the Analytical Society. In particular, it was. concerned with identifying those factors which gave rise to the Society and to -the reform of Cambridge mathematics. There were three main elements in understanding these events: arguments favoring and opposing analytics and synthetics, the ideal of a liberal education, and professionalism. These three elements and their interrelations were manifested in the early nineteenth century in two significant sets of circumstances. The first was the structure of Cambridge studies and the position of mathematics there. The second was a widespread lament about the inferiority of British mathematics and mathematical science. Both reformers and conservatives acted within the framework^established by these elements and circumstances. Indeed, two opposing positions arose: the liberal position emphasized analytics, professionalism and the study of research mathematics; the conservative one stressed synthetics, amateurism,»and a liberal education as the purpose of a University. These positions, along with the three elements, were used in the thesis to construct an explanation of 'the Analytical Society and of the movement to introduce Continental mathematics to Cambridge in the early nineteenth century. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 257 The Analytical Society (1812— 1813) was a short-lived association of a small but remarkable group of students at Cambridge University. In many ways it was typical, of a new breed of Oxbridge student societies in the early nineteenth -century. But the Society was distinct from other groups in its aim, which was a reflection, of concerns about the inferiority of British mathematics. Prompted by a familiarity or a proficiency or simply an enthusiasm for Continental mathematics, as well as by a widespread lament about the decline of English mathematical science and by a dissatisfaction with the system and content of Cambridge studies, the members of the Analytical Society resolved to contribute to the revival of English mathematical science by studying and advancing analytics. They pursued this goal through such usual features of a society as meetings, where papers were read,^md by a publication, the Memoirs of the Analytical SocietyV for the year 1813. The Memoirs was a collection of papers in advanced mathematics and not a translation:or popularization of any Continental work. This underlines the point that the Analytical Society saw itself as a mathematical society participating in the revival of English mathematics by attempting to produce analytical --mathematics. The concerns of the Analytical Society with the state of British mathematics were characteristic of those of many British mathematicians in the early nineteenth century. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 258 At that time there was a .widespread recognition of the stagnation of British, mathematics when compared to that of the Continent. This recognition involved both differences in the styles of British and Continental mathematics as well as differences in the relationship of mathematics to aspects of British and French society. The result was .an attempt on the part of several individuals, especially those concerned with research, to revive British mathematics by introducing French analytical mathematics. The main obstacles to the diffusion of Continental mathematics were that few persons were interested in advanced mathematics and that among those who did care there was much criticism of analytics especially focusing on its flaws as a tool of reason. Many mathematicians, and particularly those in teaching positions, favoured the use of geometrical methods. Despite the opposition it seems that the accomplishments of French mathematical science were too great a counterbalance. By about 1815 analytical mathematics seems to have dominated the work of English practicing mathematicians, as is best illustrated by the replacement of the fluxional calculus by the differential. The Analytical Society then was but one manifestation of a larger movement in British mathematics. It is an important illustration of a way in which Continental analytical mathematics was being imported Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 259 and diffused in England at that time.1 The Analytical Society’s efforts had little direct impact. The Society barely survived two years. It failed because its views did not harmonize with the content of Cambridge studies which dominated through the influence of the important Senate House Examination. Upon graduation many of its members left Cambridge to pursue careers. Most of them soon devoted themselves to interests other than mathematics. For while mathematics was an interesting and perhaps even important pursuit, it was not a career in early nineteenth-century England. Despite the Society’s failure, it was indirectly influential in two respects. It acted as a catalyst for the mathematical work of a number of its members. And, secondly, several of its members were, a few years after the withering of the Analytical Society, to initiate a mathematical revival movement at Cambridge. The tendencies of the mathematics of the Analytical Society were much more radical, when compared to the general conception of mathematics in Britain, than those of many other revivers. For even such a zealous promoter of Continental mathematics as Robert Woodhouse, the work 1. This thesis has concentrated on the influence of the feelings about the state of British mathematics on the formation and activities of the Analytical' Society. Much more research remains to be done on the mathematical currents on the Continent, the meaning of analytics, and on comparing the ways in which British mathematicians were adopting certain of these currents. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 260 of the Analytical Society members entered too much into the spirit (which Woodhause disliked) of the "ultra-analysts". This inclination in the work of its members was best shown in the series of mathematical papers by Charles Babbage, John Herschel and Edward Bromhead, particularly in the ten- year period following the dissolution of the ^Society. Their mathematics was very analytical and hence very abstract and general. This character made it open to the usual attacks on analytics. Peter Barlow, a reviver of British mathematics, found their emphasis to be too extreme and called for meaningful, useful mathematics. While much of their mathematics could be criticized for its often seeming to be meaningless manipulations, this appearance was an expression of their particular concerns in mathematics. Abstractness and generality were precisely why Babbage, Herschel and Bromhead valued analytics. Their work had developed within a vision of mathematics which embraced pure mathematics. It sought new and more powerful branches of analysis and at the same time was an effort to comprehend the general structure of pure analysis. This vision of pure analytics was an image of their desire to see mathematics in England professionalized. The associating of mathematics with its public support had been characteristic of the lament about the state of British y— mathematics, again probably because of . the example of thJ) French. Analytics,.almost by definition, was professional If analytics*was to prosper in England then Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 261 mathematics would have to be treated as a profession there. The experiences of Babbage and.of Herschel after graduation in searching for a career only served to accentuate the connection between their analytidal mathematics and the need for professionalization. Furthermore, their particular vision of mathematics, as expressed in their efforts to purify and standardize analysis, was a reflection • of their expectations that mathematics could be a profession in England. Thus at least the shape of their^athematics was influenced by the same framework as had given rise to the Analytical Society. Mathematics, however, did not become a profession in early nineteenth-century England. Babbage, Herschel and Bromhead soon abandoned mathematics for other, more promising interests. Yet the mathematics that they created app'ears to have been very influential, both in its approach and in its techniques, for various later developments in English mathematics such as Boole's logic and Peacock's algebra.'1' Circumstances at Cambridge had in part prompted and 1. The influence of Babbage and Herschel's mathematics and of their professional attitude still remains to be explored. The themes of support for science and of the decline of English science of the late 1820s surrounding efforts to reform the Royal Society (in which Babbage and Herschel were greatly involved) seem to have evolved - from and still reflected earlier precursors concerning mathematical science. And much more study remains to be done of the mathematical techniques of Babbage and of Herschel and to the nature of their interactions with later British mathematics. For examples of this see Koppelman (1971/72) and Dubbey (1977). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. had also permitted the appearance of the Analytical Society. It was truly a "Cambridge" Analytical .Society. The University was renowned in early nineteenth-century Britain for its emphasis on mathematics. This fame attracted the criticism of many revivers: Cambridge mathematics, which was synthetical, was proof of British stagnation, for how many mathematicians had Cambridge produced? Yet the Analytical Society was not occupied with reforming Cambridge mathematics. Soon after its expiration, however, several of.its members including Babbage, Herschel and especially George Peacock attempted to reform Cambridge mathematical studies and to mold them to their view of analytics. That they attempted such a task, whereas other revivers such as Robert Woodhouse did not, suggests that they were aroused into promoting their views in the Cambridge system of studies by their strong association of analytics and professionalism. Through the structure of Cambridge studies, that is, through private tuition, college lectures, textbooks such as the translation of Lacroix's Traitd ^lementaire (1816) and the compilation of A Collection of Examples (1820), and especially by Peacock's efforts to 1 change the mathematics in the Senate House Examination (1817, 1819), they were able to initiate a movement to change Cambridge mathematics from a geometrical to an- analytical style. Many other recent graduates were stimulated to form a loose, mathematical, revival movement at Cambridge. Much with permission of the copyright owner. Further reproduction prohibited without permission. 263 ,■ of the mathematical emphasis of this movement paralleled the stress on analytics of the Analytical Society as may be seen in the choice of foundations in various calculus texts. The movement managed in a short time in the 1820s to make Cambridge mathematics analytical. This transition , occurred mainly by the writing of various mathematical textbooks and through the influence of the Moderators on the Senate House Examination: in short, by using the Cambridge system of studies to change the content of those studies. ' While'analytical mathematics was fairly easily adopted at Cambridge - opposition to the change -did not prove to be very effective•- its implications of professionalism faltered. With the increasing amount of criticism of the Cambridge curriculum in the first half of the nineteenth century, the ideal of a liberal education as the purpose of Cambridge was reaffirmed. Not only was non-professional education stressed, but analytics at Cambridge was limited % and there was a renewed emphasis on an alternative style of mathematics, synthetics, as can be seen in the works of William Whewell. In spite of this apparent step backwards, the analytical movement had succeeded in introducing at Cambridge -a wider\range and a deeper study of mathematical topics. It h^d brought about a modernization , p ' of the mathematical curriculum which appears to have laid the basis for the vigorous school of British mathematical physics. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 264 The Analytical Society Was a significant episode in the history of early nineteenth-century English mathematics .because it transcended the circumstances in which it found itself by its analytical mathematics and by its role in reviving the state of Cambridge mathematics. This thesis has presented an account of those events. But the history r of the Society is also significant to the historian of >• mathemat/ics because it reveals the intellectual and social framework which produced the Analytical Society. This framework, as the thesis has illustrated, is useful not only for understanding the Society and its activities, but also for providing an explanation of the state and nature of mathematics in early nineteenth-century Cambridge. The reform movement with its goal of the study of advanced, .pure mathematics had been tempered by the purpose of a t / ' university. Mathematics was important at Can±rridge,..but '• its role there reflected the relationship of mathematics to English society at that time: mathematics was used', to educate gentlemen, not to tram mathematicians.^ - . w ‘pc Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 265 Bibliography Listed below are those sources which were most directly relevant to my dissertation. More specific sources may be found' in the footnotes. An important example of the latter, which should be pointed out here, are the book reviews to be found in the large periodical literature (see chapter III, page 48/ footnote 2). Manuscript Sources Abbreviation Description H.ms.R.S. Collection of Sir John F,w. Herschel manuscripts, mostly correspondence, irT~^h^ library of the Royal Society, London. H.ms.T. Collection of Sir John F.W. Herschel manuscripts in the Humanities Research Center, University of Texas, Austin. B.ms.B.L. Collection of Charles Babbage manuscripts, mostly correspondence,x£ri the British Library, London^/ "B.ms.C. ColJ^ction of Charles Babbage manuscripts in the Science Periodicals Library, Cambridge University, Cambridge-^ Buxton ms. Collection of Charles Babbage manuscripts Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and an unpublished' biography of him by Harry Wilmot Buxton in the Museum of the History of Science, University of Oxford, Oxford. Br.ms. Private collection of Sir Edward Ffrench Bromhead correspondence in the possession ' of his descendent Sir Benjamin Denis Gonville Bromhead, Thurlby Hall, Thurlby, Lincolnshire. W.ms.T.C. Collection of William Whewell manuscripts in the library of Trinity College, Cambridge University, Cambridge. St.J.ms. Collection of' Sir John F.W. Herschel, Sir" r Edward F.. Bromhead, John Whittaker, and Charles Babbage manuscripts in the library of St. John's College, Cambridge University, v Cambridge. Printed Sources Aczel, J. Lectures on Functional Equations and Their Applications (1966). Airy, Wilfrid (ed.) Autobiography of Sir George Biddell Airy (1896). Alumni Cantabriqiensis (1922). Analytical Society Memoirs of the Analytical Society, for the year 1813. Archibald, R.C. "Notes on Some Minor English Mathematical Serials" Mathematical Gazette 14 (1929) 379-400. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 267 Babbage, Charles Passages from the Life of a Philosopher (1864). Ball, W.W.R. The Origin and History of the Mathematical Tripos j(1880) . A History of the Study of Mathematics at Cambridge (1889). Barlow, P. . A New Mathematical and Philosophical Dictionary (1814). Becher, H. William Whewell and Cambridge Mathematics Ph.D. dissertation. University of Missouri, Columbia (1971). Boyer, C.B. • The History of the Calculus and its Conceptual Development (1959). Buttmann, G. The Shadow of the Telescope:. A Biography of John Herschel (1970). Cajori, F. A History of the Conceptions of Limits and Fluxions in Great Britain from Newton to Woodhouse (1919). Cambridge Problems: being a collection of the printed questions proposed to the candidates for the degree of bachelor of arts, at the General Examinations, from the year 1801 to the year 1820 inclusively (1836). Cannon, W.F. 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Further reproduction prohibited without permission. 269 a word-for-word publication of his Ph.D. thesis with chapter 3 being taken from a chapter in his M.Sc. thesis. Dubbey1s 1977 article is simply taken from chapter 5.- Foote, G.A. "The Place of Science in the British Reform Movement 1830-1850" Isis 42 (1951) 192-208. Forbes, J.D. "Dissertation Sixth: Exhibiting a " general view of the progress of mathematical and physical science principally from 1775 to 1850" Encyclopaedia Britannica 8th ed. 1 (1852) 795-996. Great Britain. Cambridge University Commission. Report of Her Majesty's Commissioners appointed to inquire, into The State, Discipline, Studies and Revenues of the University and Colleges of Cambridge': together with the Evidence and an Appendix (1852) Parliamentary Papers, Command Number 1559. Gunning, H. (ed.) The Ceremonies observed in the Senate House of the University of Cambridge (1828). Hayden, J.O. The Romantic Reviewers,,1802-1824 (1968). Hemlow, J. 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Further reproduction prohibited without permission. 272 Powell, B. The Present State and Future Prospects of Mathematical and Physical Studies in .the University of Oxford (1832) . — Historical View of the Progress of the Physical and Mathematical Sciences from the Earliest'Ages to the Present Times (1834) . - Pryme, G. Autobiographical Recollections (1870). Roach, J.P.C. ^The Age of Newton and Bentley, 1660-1800'5’\A History of the County of % Cambridge and the Isle of Ely (1959) 210- 2 35. Vol.3 of The Victoria History of the Counties of England. "The Age of Reforms, 1800-82" A History of the County of Cambridge arid the Isle of Ely (1959) 235-265. Vol.3 of The Victoria History of the Counties of England. Rothblatt, S. The Revolution of the Dons (1968). . "The Student Sub-culture'' and the Examination System in Early^L9th Century Oxbridge" in L. Stone (ed.) The University in Society 1 (1974) 2477303. Tradition and Change in English Liberal Education (1976). 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Youschkevitch, A.P. "The Concept of Function up to the Middle of the 19th Century" Archive for History of Exact Sciences 16 (1976) 37-85. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.