The Volterra Chronicles
Th e Life an d Time s
of an Extraordinar y Mathematicia n
1860-194 0
This page intentionally left blank
https://doi.org/10.1090/hmath/031
HISTORY OF MATHEMATICS • VOLUME 31
The Volterra Chronicles
Th e Life an d Time s
of a n Extraordinar y Mathematicia n
1860-194 0
Judith R. Goodstein
»AMS
The London
Mathematical
AMERICAN MATHEMATICA L SOCIET Y jOCiety
Editorial Boar d
American Mathematica l Societ y Londo
J o s e p h W . D a u b e n Ale
Peter Dure n Jerem
K a r e n P a r s h a l l , Chai r R o b i
Michael I . Rose n Stev
n Mathematica l Societ y
x D . D . Crai k
y J . Gra y
n Wilson , Chai r
e Rus s
Cover a r t : A n o n y m o u s , P o r t r a i t o f V i t o Volterra , Pisa , ca . 1 879 , T h e B u r n d y Library ,
C a m b r i d g e , M a s s a c h u s e t t s . Oi l o n canvas .
2000 Mathematics Subject
Classification.
Primar
y 01 A60 , 01 A70 .
For a d d i t i o n a l informatio n a n d u p d a t e s o n thi s book , visi t
www.ams.org/bookpages/hmath-31
Library o f C o n g r e s s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a
Goodstein, Judit h R .
The Volterr a chronicle s : th e lif e an d time s o f a n extraordinar y mathematician , 1 860-1 94 0 /
Judith R . Goodstein .
p. cm . — (Histor y o f mathematics , ISS N 0899-242 8 ; v. 31 )
Includes bibliographica l reference s an d index .
ISBN-13: 978-0-821 8-3969- 0 (alk . paper )
ISBN-10: 0-821 8-3969- 1 (alk . paper )
1. Volterra , Vito , 1 860-1 940 . 2 . Mathematicians—Italy—Biography . 3 . Mathematics —
Italy—History—19th century . 4 . Mathematics—Italy—History—20t h century . 5 . Jews —
Italy—Social conditions—1 9t h century . 6 . Jews—Italy—Socia l conditions—20t h century .
7. Italy—Politic s an d government—1 9t h century . 8 . Italy—Politic s an d government—20t h cen tury. I . Title .
QA29.V64G66 200 7
510.92—dc22 200605 75
2
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10 9 8 7 6 5 4 3 21
1
2 1 1 1 0 09 0 8 0 7
Contents
Illustrations i
x
Introduction. "Th
1
e Jewis h Mathematician "
Chapter 1 . " A New
1
Er a I s Dawning, " 1 86 0 1
Chapter 2 . "This , Abov e All , I Promise,
1
" 1 863-1 87 0
9
Chapter 3 . "Tha t Damne d Passion, " 1 874-1 87 7 2
5
Chapter 4 . "Lon g Liv e th e Republic, " 1 878-1 88 2 3
7
Chapter 5 . "Professo r b y Deed, " 1 880-1 88 3 5 1
Chapter 6 . "Ou r Professo r o f Smal l Intervals, " 1 883-1 89 3 6 1
Chapter 7 . "Th e Lif e I Live, " 1 887-1 89 5 7 1
Chapter 8 . "Demonstration s o f Thei r Resentment, " 1 893-1 90 0 8 1
Chapter 9 . "Go d Liberat e U s fro m Hi s Symbols " 9 1
Chapter 1 0 . "I t I s th e Greates t Desir e o f M y1
Life, " 1 90 0 0
3
Chapter 1 1 . "Mos t Importan t fo r1
1
Ou r Fatherland "
7
Chapter 1 2 . "Wil l The y Creat
1
e a Ne w World? " 3
3
1
Chapter 1 3 . " A Political Man " 4
7
Chapter 1 1
4 . " A Professor i n America " 6 1
1
Chapter 1 5 . "Empire s Die " 7
7
Epilogue 20 1
Appendix A . Si r Edmun d Whittaker , "Vit o Volterra , 1 860-1 940 " 20
5
Appendix B . O n the Attempt s t o Appl y Mathematic s t o the Biologica l
and Socia l Science s 24
7
Appendix C . Scienc e a t th e Presen t Momen t an d th e Ne w Italia n
Society fo r th e Progres s o f th e Science s 26 1
viii CONTENT
S
Acknowledgments 27 1
Selected Bibliograph y 27
3
Notes 28 1
Index 30
3
Illustrations
1. Ma p o f Italy , 1 87 0
2. Ma p o f Ancona , circ a 1 86 4
3. Ancon a ghetto , circ a 1 81 3
4. Sau l an d Edoard o Almagi a
5. Alfons o Almagi a an d co-workers , 1 86 8
6. Cesar e Arzela , undate d
7. Enric o Betti , undate d
8. Guid o Castelnuovo , circ a 1 89 1
9. Uliss e Dini , undate d
10. Griffit h Evans , circ a 1 91 2
11. Giovann i Battist a Guccia , undate d
12. Sofi a Kovalevskaya , undate d
13. Tulli o Levi-Civita , circ a 1 90 0
14. Antoni o Roiti , circ a 1 90 0
15. Giovann i Vailati , circ a 1 90 0
16. Vit o Volterr a an d Carl o Somigliana , 1 88 1
17. Volterr a trai n pass , circ a 1 88 4
18. Graduatio n class , Scuol a Normale , 1 88 8
19. Vi a Po , Turin , circ a 1 89 0
20. Angelic a Volterra , 1 90 0
21. Portrai t o f Edoard o Almagia , circ a 1 90 0
22. Virgini a Almagia , circ a 1 90 0
23. Vit o Volterr a an d Virgini a Almagia , 1 90 0
24. Palazz o Fiano-Almagia , circ a 1 90 0
25. Vit o an d Virgini a Volterra , circ a 1 90 1
26. Clar k Universit y meeting , 1 90 9
ILLUSTRATIONS
27. Identificatio n documen t issue d t o Volterra , 1 91 8
28. Alber t Einstei n an d Federig o Enriques , Bologna , 1 92 1
29. Postcar d inscribe d b y Volterra , undate d
AUSTRI A
1. Map of Italy, 1870.
Credit: from George Holmes, ed., The Oxford History of Italy (1997). By permission of
Oxford University Press (OUP). Click here for OUP's web site.
2. Map of Ancona, ca. 1864.
Credit: Baedeker's Central Italy, 1874 edition/courtesy of the Department of Special Collections, Charles E. Young Research Library, UCLA.
3. Th e Ancona ghetto , ca . 1 81 3 . The ghett o wa s abolishe d i n 1 860 , and th e construc tion of a new street, named for King Vittorio Emanuele, running from Piazz a Cavour to
Piazza de l Teatro, effectively opene d u p Via Lata an d Via Bagno, two of th e principa l
streets of the old ghetto.
Credit: E. Sori, "Una comunita crepuscolare: Ancona tra Otto e Novecento," Proposte
e Ricerche, n. 14.
4. Portrait by an unknown artist of Saul Almagia and his son Edoardo, ca. 1852.
Credit: Edoardo Achille Almagia.
5. Alfonso Almagia (seated, secon d from right ) and co-workers in the central office o f
the Italian National Bank (today the Bank of Italy) in Florence, Italy, 1868.
Credit: Banca d 'Italia, Archivio storico della Banca d Italia, Album del personale della
Banca Nazionale del Regno d'Italia, Ritratto di gruppo dei dirigenti e degli impiegati
della Sezione I - Divisione I della Direzione Generale, 1868.
6. Cesare Arzela (1 847-1 91 2 )
Credit: Biblioteca
delVAccademia Nazionale dei
Lincei e Corsiniana, Fondo
Volterra.
FOTOGRARA;jfcS;>
* VA N LIN T
7. Enric o Betti (1 823-1 892 )
Credit: Biblioteca delVAccademia Nazionale dei
Lincei e Corsiniana, Fondo Volterra.
8. Guid o Castelnuovo
(1865-1952)
Credit: Fonti iconografiche,
Biblioteca Matematica
Giuseppe Peano, Department
of Mathematics, University
of Turin.
9. UlisseDini (1 845-1 91 8 )
Credit: Institut Mittag-Leffler.
sM
W$$$£$':10. Griffith Evan s (1 887-1 973 )
Credit: Biblioteca
dell 'Accademia Nazionale dei
Lincei e Corsiniana, Fondo
Volterra.
'M
3T& :f™
11. Giovann i Battista Guccia (1 855-1 91 4 )
Credit: Biblioteca dell'Accademia Nazionale dei
Lincei e Corsiniana, Fondo Volterra.
12. Sofi a Kovalevskaya (1850-1891)
Credit: Institut Mittag-Leffler.
13. TullioLevi-Civita(1 873-1 941 )
Credit: Institut Mittag-Leffler.
14. Antoni o Roiti (1 843-1 921 )
Credit: Biblioteca delVAccademia
Nazionale dei Lincei e Corsiniana,
Fondo Volt erra.
15. Giovann i Vailati (1 863-1 909 )
Credit: M. Calderoni, U. Ricci, and
G. Vacca, eds. Scritti di G. Vailati
(Leipzig, JohannA. Barth, 1911).
16. Vito Volterra (left) an d Carlo
Somigliana, Pisa, 1881.
PI Credit: Pontificia Academia ScieniJgl tiarum, 1942.
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17. Vit o Volterra's Train Pass, Pisa, ca.
1884.
Credit: from a private collection.
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18. Mathematic s graduates, Scuola Normale Superiore di Pisa, 1 888 . The dedication on
the reverse reads: "Al chiarissimo prof. Vito Volterra i Normalisti laureandi in matematica
dell'anno 1 887-1 888 . Pisa, 31 Maggio 1888."
Credit: Biblioteca delVAccademia Nazionale dei Lincei e Corsiniana, Fondo Volterra.
19. Via Po, Turin, ca. 1 890 . The University Building is situated in the middle of Via Po
(the main thoroughfare, o n the left sid e of the street). The Mole Antonelliana, visible in
the upper left corner , held the title of tallest brick construction in Europe for many years.
Credit: Fonti iconografiche, Biblioteca Matematica Giuseppe Peano, Department of
Mathematics, University of Turin.
20. Angelica Volterra, 1 900 .
Credit: The Burndy Library,
Cambridge, Massachusetts.
21. Amelia Almagia Ambron's portrait of Edoardo Almagia, ca. 1900.
Credit: from a private collection.
^P^T
22. Virginia Almagia, ca. 1900.
Credit: from a private collection.
23. Vito Volterra and Virginia Almagia, 1 900 .
Credit: from a private collection.
24. The Palazzo Fiano-Almagia, a t the corner of Piazza in Lucina and the Via del Cor so
in Rome, where Volterra lived from 1 90 0 to 1 940 . The palazzo runs the length of the
block on the Corso, with another entrance on Via in Lucina, on the south side of the
building. Standing in the heart of the city's historic center, the building had the added
advantage of being around the corner from th e Chamber of Deputies and a short walk to
the Senate.
Credit: Edizioni Quasar s.r.L di Severino Tognon.
25. The professor an d his wife in his library at home, ca. 1 901 . Durin g World War II, the
janitor of the building walled over the entrance to the room, and Volterra's rare book collection, prized for its many unique items, survived the war intact.
Credit: Biblioteca delVAccademia Nazionale dei Lincei e Corsiniana, Fondo Volterra.
WmmJ]^w^w&* •%•£.
26. Leadin g physicists and mathematicians met at Clark University in 1909 to celebrate
the school's twentieth anniversary. Speakers and Clark participants included: Front row:
Robert Williams Wood (first on the left), Eliakins Hastings Moore (second from left) ,
Vito Volterra (third from left) , A. A. Michelson (fourt h fro m left) , William Edward Stor y
(second from right). Second row: Arthur Gordon Webster (second from left) , Edward
Burr Van Vleck (fourth fro m left) , Ernest Fox Nichols (third from right). Third row:
Ernest Rutherford (thir d from left) , Carl Barus (fifth fro m left , wit h beard). Fourth row:
Robert Hutchings Goddard (far left).
Credit: Clark University Archives.
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27. After establishin g Italy's Office for War Inventions in 1 91 7 ,
Volterra made many trips to France
and England to promote military
cooperation amon g the Allies. In
his application to visit these countries in May 1 91 8 , he states that the
purpose of the journey is an official
mission in connection with the War
Ministry.
Credit: from a private collection.
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28. Albert Einstein (in profile) an d Federigo Enriques, Bologna, 1921.
Credit: Federico Enriques.
29. On a postcard bearing his own picture, Volterra wrote his epitaph for Mussolini' s
Italy: "Empire s die, but Euclid's theorems keep their youth forever," ca. 1931.
Credit: The Burndy Library, Cambridge, Massachusetts.
Epilogue
Vito Volterra lived just lon g enough t o witness the wreckage of the brav e
new world int o which he had bee n bor n i n 1 860 . H e did no t liv e to se e a new
era ris e fro m th e ashe s o f th e old—th e restoratio n o f huma n freedom s an d
parliamentary democrac y i n muc h o f Wester n Europe ; th e ne w prominenc e
accorded scienc e an d technolog y i n the aftermat h o f the Manhatta n Projec t
and th e onse t o f th e Spac e Ag e an d th e Col d War ; an d th e post-wa r revi talization o f th e scienc e an d mathematic s communit y i n Italy . Nearl y al l of
his friend s an d famil y survive d th e wa r an d th e horror s o f the Naz i occupa tion o f Rom e an d norther n Italy , includin g Naples , i n 1 943 . Italian s b y an d
large had n o stomach fo r Hitler' s Fina l Solution , an d manage d t o thwart th e
Germans' effort s t o roun d u p an d depor t Jew s i n occupie d Ital y throug h a
combination o f non-cooperation , obstruction , an d outrigh t sabotage . I t i s
estimated tha t eight y to ninet y percen t o f the country' s Jew s (includin g sev eral thousan d wh o sough t safet y ther e fro m elsewher e i n occupie d Europe )
survived th e Holocaust .
A fe w o f Volterra' s mor e senio r associate s di d no t liv e lon g enoug h t o
experience th e ris e o f Fascism . Antoni o Roiti , wh o ha d bee n th e youn g
Vito's mentor , an d professor , an d a truste d confidan t o f bot h mothe r an d
son throughou t hi s life , die d i n 1 921 , th e sam e yea r a s Virginia' s father ,
Edoardo Almagia . O n Decembe r 1 8 , 1 921 , Volterr a opene d a meetin g o f
the Lincei' s clas s o f physical , mathematical , an d natura l science s wit h a n
announcement o f Roiti' s deat h and—i n accordanc e wit h Roiti' s explici t in structions tha t ther e b e n o commemoratio n o r las t honors—adjourne d th e
session a s a gestur e o f mourning .
After he r husband' s death , Virginia' s mother , Eleonora , live d ou t th e
rest o f he r lif e i n Rom e an d passe d awa y i n 1 932 . Tulli o Levi-Civita , th e
dean o f relativity theor y i n Italy , die d i n 1 941 , a year afte r Volterra . Enric o
Fermi wo n th e Nobe l Priz e i n physic s i n 1 938 , went t o Stockhol m t o collec t
it, an d wit h hi s wife , Laura , whos e famil y wa s Jewish , continue d o n t o
the Unite d States , wher e h e woul d spen d th e res t o f hi s life . I n 1 942 , h e
oversaw th e firs t controlle d nuclea r chai n reactio n i n a n atomi c pil e h e ha d
constructed unde r th e stadiu m a t th e Universit y o f Chicago , an d soo n afte r
he becam e on e o f the leader s o f the Manhatta n Projec t t o buil d th e atomi c
bomb.
Italy ha d declare d wa r o n th e Unite d State s i n Decembe r 1 941 , thre e
days afte r th e Japanes e attac k o n Pear l Harbor . Afte r th e Allie s liberate d
201
202
EPILOGUE
Sicily wit h onl y toke n oppositio n i n Augus t 1 943 , th e res t o f th e countr y
erupted i n politica l turmoil , culminatin g i n th e overthro w o f Mussolini , th e
dissolution o f the Fascis t Party , an d th e establishmen t o f a new governmen t
under Marsha l Pietr o Badoglio . I n September , th e ne w governmen t surren dered t o th e Allies ; the German s retaliate d b y occupyin g Ital y a s fa r sout h
as Naples . Th e Allie s entere d Rom e i n th e summe r o f 1 944 . Th e follow ing spring , Mussolin i an d hi s mistres s wer e captured an d sho t b y partisans .
Several week s later , th e Germa n arm y i n Ital y gav e u p th e fight, an d th e
entire peninsul a wa s liberated .
After th e war , Francesc o Severi , who m th e Fascist s ha d installe d a s a
kind o f puppe t doye n o f Italia n mathematics , foun d himsel f unde r investi gation b y th e "Reconstitutio n Commission, " whic h ha d bee n establishe d i n
1944 to restor e th e function s o f th e Lincei . Th e commission' s specifi c man date wa s t o loo k int o allegation s o f wartim e collaboratio n agains t member s
who had bee n accuse d o f taking a n activ e part i n Fascist politica l lif e or wh o
had remaine d loya l t o Mussolin i afte r h e wa s depose d i n Septembe r 1 943 .
Severi wa s subsequentl y absolve d o f an y crimina l activit y i n a repor t tha t
concluded tha t h e ^'had no t receive d fro m fascis m anythin g more " tha n wa s
his du e a s a distinguishe d scientist. * Hi s "mora l rectitude, " th e commis sion added , wa s neve r calle d int o question . Som e mathematician s remaine d
skeptical, nevertheless .
As fo r othe r Jewis h member s o f th e Volterr a circle , Federig o Enrique s
and Guid o Castelnuov o live d throug h th e Naz i occupatio n o f Rome , chang ing hidin g place s frequently . I n 1 94 1 Castelnuov o organize d a clandestin e
university fo r Jewis h student s i n Rome , operate d i n conjunctio n wit h th e
Fribourg Institu t Techniqu e Superieur , a privat e school , i n Switzerland . H e
and Enrique s taugh t ther e wit h a numbe r o f othe r professors , bot h Jewis h
and non-Jewish , unti l th e occupatio n force d th e school' s closur e i n th e fal l
of 1 943 .
After th e liberatio n o f Rome, Castelnuovo , the n sevent y nine , was calle d
out o f retiremen t t o breath e ne w lif e int o Italy' s pre-Fascis t scientifi c or ganizations. Appointe d genera l commissione r o f Italy' s Nationa l Researc h
Council an d presiden t o f its mathematics committee , h e played a major rol e
in reviving and reconstituting bot h organization s an d contribute d als o to th e
rebirth o f the Lincei , whos e presiden t h e became i n Apri l 1 945 . H e held th e
post unti l hi s deat h seve n year s later , an d wa s name d a Senato r fo r Lif e i n
Italy's Parliamen t i n 1 949 . Th e Institut e o f Mathematic s a t th e Universit y
of Rom e wa s name d fo r hi m i n 1 953 .
Volterra's first cousi n Roberto Almagia, the geographer wh o had been a n
enthusiastic proponen t o f Italian nationalis m i n the year s befor e Worl d Wa r
I, rose to become the head of the Italian school of geographers in the interwa r
years. I n 1 938 , h e becam e a gues t o f th e Vatica n librar y an d publishe d
* Francesco Sever i t o Beniamin o Segre , Octobe r 1 5 , 1 945 , Beniamin o Segr e Papers ,
private collection .
EPILOGUE
203
under th e pseudony m o f Bernard o Varenio ; afte r th e wa r h e resume d hi s
academic career at the University of Rome. H e headed the National Researc h
Council's committe e fo r geography , geology , an d oceanograph y fro m 1 94 5
until hi s death , i n Rome , i n 1 962 .
Virginia Volterr a remaine d i n Rom e afte r he r husband' s death . Whe n
the racia l law s prohibite d Italia n Jew s fro m havin g servants , sh e ha d aske d
the family' s librarian , Guald a Caput o Massimi , t o com e t o th e hous e les s
frequently, fearin g tha t th e police , wh o kep t surveillanc e o n th e palazzo ,
would cal l Mrs . Massim i dow n t o polic e headquarter s fo r questioning . I n
1943, afte r th e German s occupie d Rom e an d bega n roundin g u p th e city' s
Jews for deportation, Massim i took Virginia into her home and concealed he r
there fo r mor e tha n a month . Later , he r so n Edoardo , wh o ha d joine d th e
resistance movemen t i n Rome , cam e an d foun d a secur e plac e fo r Virgini a
with a grou p o f nun s i n Vi a Vicenza , clos e t o th e city' s mai n trai n station .
After th e war, Volterra' s fou r childre n resume d thei r professiona l career s
in th e la w an d th e sciences ; al l ar e no w deceased . (I t migh t hav e please d
Volterra t o kno w tha t a numbe r o f hi s direc t descendent s no w resid e i n th e
United States. ) Thei r ^mother live d ou t he r day s i n Rom e an d die d i n 1 96 8
at th e ag e o f ninety-three . Virgini a Volterr a neve r los t he r livel y spiri t o r
her strea k o f independence . Th e stor y goe s tha t afte r th e war , sh e refuse d
to gree t o r eve n acknowledg e individual s lik e Severi , who ha d ostracize d he r
husband afte r th e racial laws were passed. Wit h magnificen t disdai n Virgini a
would behav e a s if these forme r colleague s o f her husban d di d no t exist . Sh e
never offere d an y explanatio n fo r he r behavior . Thos e wh o remembere d th e
old day s understood .
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APPENDIX A
Sir Edmun d Whittaker , "Vit o Volterra ,
1860-1940"
"Vito Volterra " b y Si r Edmun d Whittake r i s reprinte d wit h permissio n o f
the publisher , Th e Roya l Society , fro m Obituary Notices of Fellows of the
Royal Society of London, 3 (1 941 ) , 691 -729 .
205
VITO VOLTERR A
1860-1940
wa s bor n a t Ancon a o n 3 Ma y i860 , th e onl y
child o f Abram o Volterr a an d Angelic a Almagia . Whe n h e wa s
three month s ol d th e tow n wa s besieged b y th e Italian arm y an d
the infan t ha d a narro w escap e fro m death , hi s cradl e bein g
actually destroye d b y a bomb whic h fel l near it .
When h e wa s barel y tw o year s ol d hi s fathe r died , leavin g
the mother , no w almos t penniless , t o th e car e o f he r brothe r
Alfonso Almagia , a n employe e o f th e Banc a Nazionale , wh o
took hi s siste r int o hi s hous e an d wa s lik e a fathe r t o he r child .
They live d fo r som e tim e i n Terni , the n i n Turin , an d afte r tha t
in Florence , wher e Vit o passe d th e greate r par t o f his youth an d
came t o regar d himsel f a s a Florentine .
At th e ag e of eleve n h e bega n t o stud y Bertrand' s Arithmetic
and Legendre' s Geometry, and fro m thi s tim e o n hi s inclinatio n
to mathematic s an d physic s becam e ver y pronounced . A t
thirteen, afte r readin g Jule s Verne' s scientifi c nove l Around the
Moon, he tried t o solv e the problem of determining the trajector y
of a projectil e i n th e combine d gravitationa l field of th e eart h
and moon : this i s essentiall y th e 'restricte d Proble m o f Thre e
Bodies', an d ha s bee n th e subjec t o f extensiv e memoir s b y
eminent mathematician s bot h befor e an d afte r th e youthfu l
Volterra's effort : hi s method wa s t o partitio n th e time int o shor t
intervals, i n eac h o f whic h th e forc e coul d b e regarde d a s
constant, s o tha t th e trajector y wa s obtaine d a s a successio n o f
small paraboli c arcs. Fort y year s later , i n 1 91 2 , he demonstrate d
this solutio n i n a course o f lectures give n a t th e Sorbonne .
When fourtee n h e plunge d alone , withou t a teacher , int o
Joseph Bertrand' s Calcul differentiel: h e doe s no t see m t o hav e
had acces s t o an y wor k o n th e integra l calculu s a t thi s time ,
VITO VOLTERR A
692
OBITUARY NOTICE S
and whe n he attacked various specia l problems relatin g t o centre s
of gravity , h e discovere d fo r himsel f tha t the y coul d b e solve d
by mean s o f a n operatio n (integration ) whic h wa s th e invers e
of differentiation .
His family , whos e mean s wer e slender , wishe d hi m t o tak e u p
a commercia l career ; whil e Vit o insiste d o n hi s desire t o becom e
a. man o f science . Th e struggl e betwee n vocatio n an d practica l
necessity becam e ver y acute : an d th e famil y applie d t o a distan t
cousin wh o ha d succeede d i n th e world , t o persuad e th e bo y t o
accept thei r views . Thi s man , Edoard o Almagia , wh o die d a t
the ag e o f eight y i n 1 921 , wa s on e o f th e mos t celebrate d civi l
engineers and financiers i n Italy in the latter part of the nineteenth
century : a s a contracto r o f publi c work s h e constructe d man y
important railway s an d harbour s a t home an d abroad , includin g
the harbour s o f Alexandri a an d Por t Sai d i n Egypt : th e pro prietor o f vas t estate s i n Tuscan y an d th e Marches , h e wa s
renowned fo r hi s charities: and it was in the course of excavations
at hi s palac e i n th e Cors o Umberto—onc e th e Palazz o Fiano Ottoboni—that th e discover y wa s mad e o f th e sculpture s o f th e
Ara Pad s o f th e Empero r Augustus , whic h ar e no w amon g th e
treasures o f th e Muse o dell e Term e i n Rome .
His intervie w wit h hi s youn g relativ e turne d ou t differentl y
from wha t th e famil y ha d expected . Impresse d b y th e boy' s
sincerity, determinatio n an d ability , th e olde r ma n thre w hi s
influence o n th e sid e o f science , an d turne d th e scale . Professo r
Roiti offere d a nomination a s assistant in the Physical Laborator y
of th e Universit y o f Florence , thoug h Vit o ha d no t ye t begu n
his studie s there : it wa s accepted , an d no w th e di e was cast . Th e
young aspiran t entere d th e Facult y o f Natura l Science s a t
Florence, followin g th e course s i n Mineralog y an d Geolog y a s
well a s in Mathematic s an d Physics . In 1 87 8 he proceede d t o th e
University o f Pisa, where h e attende d th e lecture s o f Dini, Bett i
and Padova : i n 1 88 0 h e wa s admitte d t o th e Scuol a Normal e
Superiore, wher e h e remaine d fo r thre e years ; an d here , whil e
still a student , h e wrot e hi s first origina l papers . Unde r th e
influence o f Din i h e ha d becom e intereste d i n th e theor y o f
VITO VOLTERR A
693
aggregates an d th e functions o f a real variable, an d he gav e som e
examples [3 ] x which showe d th e inadequacy , unde r certai n cir cumstances, of Riemann's theor y o f integration, an d adumbrate d
the development s mad e lon g afterward s b y Lebesgue .
In 1 88 2 h e graduate d Docto r o f Physic s a t Pisa , offerin g a
thesis on hydrodynamic s i n whic h certai n results , actuall y foun d
earlier b y Stokes , were rediscovered independently . Bett i at onc e
nominated hi m a s his assistant . I n 1 883 , when onl y twenty-thre e
years of age, he was promoted t o a full professorship o f Mechanics
in th e Universit y o f Pisa , whic h afte r th e deat h o f Bett i wa s
exchanged fo r th e Chair of Mathematical Physics . He now set u p
house in Pisa with his mother, who u p to that time had continue d
to liv e wit h he r brother . I n 1 88 8 h e wa s electe d a non-residen t
member o f th e Accademi a de i Lincei : i n 1 89 2 h e becam e pro fessor o f Mechanic s i n th e Universit y o f Turin , an d i n 1 90 0 h e
was called t o th e Chai r o£ Mathematical Physic s in Rome, a s the
successor o f Eugeni o Beltrami . I n July o f tha t yea r h e marrie d
Virginia Almagia , on e o f th e daughter s o f th e distinguishe d
relative wh o ha d firs t mad e i t possibl e fo r hi m t o follo w a
scientific career . Sh e had inherited intellectua l brilliance from he r
father, an d grea t beaut y fro m he r mother , an d a s th e wif e o f
Vito Volterr a too k upo n hersel f al l th e care s whic h migh t hav e
distracted he r husban d fro m hi s scientifi c work , undertakin g th e
education o f thei r childre n an d th e administratio n o f al l thei r
possessions. Si x childre n wer e bor n o f th e union , o f who m fou r
now survive . Hi s mothe r stil l live d wit h them , an d die d a t th e
age o f eight y a t th e Palazz o Almagi a i n Marc h 1 91 6 .
We mus t no w procee d t o a n accoun t o f Volterra' s scientifi c
work. Instea d o f considerin g th e individua l paper s on e b y on e
in chronologica l order , w e shal l grou p the m accordin g t o
subjects: an d shal l consider firs t thos e relatin g t o functionals .
Afunctional ma y b e introduced a s a generalization o f the ide a
of a function y o f severa l independen t variable s <f> l9 </> 2, . . . , <f> n>
say y(i>i, <^2> • • • > 9^) - k e t u s s u P P o s e d* at th e set of variables ,
<£i> ^2> • • • > 4>n>> i s modifie d fro m bein g a finit e se t to bein g a n
1
Th e numbers i n square brackets refe r t o th e bibliography a t the en d o f this notice.
44
OBITUARY NOTICE S
694
enumerably-infinite se t an d finally t o bein g a continuou s set .
To represen t this analytically , w e ca n regar d <f> x a s a functio n o f
its suffi x x: the n th e functiona l y i s a functio n o f al l th e value s
that th e functio n <f>(x) take s whe n x lie s i n som e interva l
AAtfAB. Th e functio n (f>(x) i s arbitrary, an d is , s o t o speak , th e
independent variabl e o f whic h th e functiona l y i s a function .
This definitio n ma y readil y b e extende d t o a functiona l y de pending o n severa l functions </>i(x) , ^ (^ • ' • • anc^ moreove r o n
certain parameter s t h * 2> • - * > }' bein g a function i n th e ordinar y
sense o f th e t 9$. We ma y als o introduc e function s o f severa l
variables <f>(x l9 x 2, . . .) i n plac e o f the functions <f> f(x).
The transitio n fro m ordinar y function s t o functionals corre sponds exactl y t o th e transitio n fro m th e theor y o f maxim a an d
minima o f function s o f severa l variables , t o th e calculu s o f
variations: an d indee d th e integrals , whic h i n th e calculu s o f
variations ar e t o b e mad e maxim a o r minim a b y choosin g th e
functions involve d i n them i n a certain way, constitut e a familia r
and importan t exampl e o f functionals .
Volterra seem s to have conceive d th e idea of creating a general
theory o f th e function s whic h depen d o n a continuou s se t o f
values o f anothe r function , a s earl y a s 1 883 : bu t hi s first pub lished wor k o n th e subjec t [1 7 ] di d no t appea r unti l 1 887 . Th e
rxamt functional was introduce d late r b y Hadamar d an d ha s no w
replaced Volterra' s origina l nomenclature .
The first ste p i n th e theor y mus t evidentl y b e t o exten d t o
functionals th e well-know n fundamenta l concept s o f th e theor y
of functions : th e continuit y o f a functiona l i s first defined , an d
the derivativ e an d th e differentia l hav e thei r analogues . Th e
partial derivativ e wit h respec t t o a particular variabl e passe s int o
the derivativ e o f th e functiona l y wit h respec t t o <f> a t a certai n
point o f th e interva l o f definitio n o f <f> 9 sa y x= f . T o th e tota l
differential, whic h i s a linea r for m i n th e differential s o f th e
independent'variables wit h th e partia l derivative s a s coefficients ,
there correspond s th e tota l variatio n o f th e functional , whic h i s
an integra l ove r th e variatio n o f th e independen t variable s '<(>(£)
with th e derivative s o f th e functiona l y wit h respec t t o tf> a t th e
VITO VOLTERR A
695
point f a s coefficients . B y repeate d applicatio n o f thes e opera tions, highe r differentia l coefficient s an d highe r variation s ar e
easily defined . Th e highe r differentia l coefficient s wit h respec t
to 4> a t th e point s g l9 £ 2, . . . ar e show n t o b e symmetrica l
d^J1 Vfixxv") Jd^ v f (xx y )
at thes e point s (correspondin g t o — V ~ =
' ~). I n
C/vVt/ y {jy\jj\>
some cases , however , a slightl y generalize d definitio n o f th e
variation is necessary in which, beside s the integral, ther e appear s
a finit e o r enumerably-infmit e numbe r o f term s linea r i n th e
variation o f</ > and th e derivative s o f <f> a t certai n exceptiona l
points: th e latte r i s more ofte n tha n no t th e cas e in th e Calculu s
of Variations , wher e th e exceptiona l point s ar e usually th e limit s
of integration .
In tw o othe r note s o f th e sam e year , th e concep t o f functions
of a line was introduce d [1 8] . Le t a close d curve d lin e L i n
space o f n dimension s b e specifie d b y equation s Xi = <{>i(t)
(i = 1 , 2 , , . . , 41) an d suppos e tha t t o ever y suc h lin e L ther e
corresponds a definit e valu e o f a quantit y y: the n y i s calle d a
function of the line L . Obviousl y y i s a functiona l o f th e <£'s :
but i t i s no t th e mos t genera l typ e o f functional , sinc e i t i s
invariant when the <f>'s are replaced by othe r functions 0 obtaine d
from the m b y a chang e o f paramete r t = t(u), ^(w ) — <f>i(t):
j , i n fact, depend s on the line L but not on its mode of parametric
representation. Volterr a define d th e derivativ e o f a function o f a
line wit h respec t t o th e lin e a t a certai n poin t o f it , an d the n
defined th e variation . H e nex t introduce d th e ide a o f a simple
function of a line: let Lx an d L 2 be two contour s having in commo n
an arc which is traversed in opposite directions in the two circuits,
and le t L x + L 2 denot e th e contou r obtaine d b y deletin g thi s
common arc : the n a simple function of a line is define d t o b e a
function havin g the property <f> (Lx +L 2 ) = 4> (1^) +</ > (L2). Volterr a
established severa l importan t theorem s regardin g thes e simpl e
functions.
A remarkabl e applicatio n o f simpl e function s o f a line i s con tained i n a serie s o f papers , th e firs t o f whic h appeare d i n th e
same yea r [1 9] . Le t L b e a line , F an d<E > tw o function s o f thi s
696
OBITUARY NOTICE S
line, L + A L a lin e whic h i s identica l wit h L excep t i n th e
neighbourhood o f a fixed poin t M , an d let F + A F = F( L + AL),
<3> +A<l > —<3>(L + AL) . N O W le t th e deformatio n A L tend t o zero ,
so A F an d A * ten d t o zero . I f AF/A<E > tend s t o a limi t whic h
depends onl y o n M an d i s independen t o f th e sequenc e o f
diminishing deformation s AL , the n th e function s F an d< & wer e
said b y Volterr a t o b e connected with each other in Riemanns sense:
he suggeste d tha t this correspond s t o th e relatio n betwee n tw o
complex variable s z an d w fo r whic h dw/dz i s independen t o f
the wa y i n which th e limi t i s approached, dependin g merel y o n
the valu e o f z. H e develope d th e theor y o f thes e functions ,
showing tha t i t depend s o n certai n partia l differentia l equations ,
which correspon d t o th e Cauchy-Rieman n equations .
In tw o mor e note s [21 ] he gav e a theory o f th e differentiatio n
and integratio n o f connecte d function s o f a line , definin g firs t
the 'connexion ' betwee n a n ordinar y functio n an d a function o f
a line : th e limi t d<&/dF introduce d abov e i s show n t o b e a n
ordinary function 'connected ' with both F and <!>. Iff (a n ordinary
function) an d F ( a function o f a line) ar e 'connected' an d withou t
singularities inside a closed surface cr, the n /fdF = o : this corresponds t o Cauchy' s integra l theore m i n th e theor y o f analyti c
functions. Morera' s convers e o f Cauchy' s theore m ca n als o b e
extended t o 'connected ' function s o f a line . Integratio n an d
differentiation a s introduce d i n thes e paper s ar e invers e opera tions. Th e theor y ha s som e connexion s wit h th e theor y o f
analytic function s o f tw o variables .
The whol e theor y o f thi s generalizatio n o f th e function s o f
a comple x variabl e wa s systematicall y presente d i n a consider able memoi r [25 ] i n 1 889 : an d a t th e sam e tim e i n a serie s o f
notes [26 , 27, 28 ] th e ide a o f function s of' a lin e wa s extende d
by considering , instea d o f a line, an y sub-spac e S r in a spac e S„
of an y numbe r n o f dimensions : i n particular , th e notio n o f
'conjugate functions ' (dependin g o n a n S r _! and S„_ r_x respec tively) was developed . Differentia l parameters , correspondin g t o
the v an d A of ordinary theory , wer e introduced : a s in ordinar y
697
VITO VOLTERR A
potential theory , th e vanishin g o f th e second differentia l para meter i s a necessar y conditio n fo r th e existenc e o f a conjugat e
function (th e second differential paramete r of which also vanishes).
In th e followin g yea r Volterr a showe d [30 ] tha t b y mean s o f
his functiona l calculu s th e Hamilton-Jacob i theor y o f th e in tegration o f th e differentia l equation s o f dynamic s migh t b e
extended t o genera l problems o f mathematica l physics . The ide a
was, tha t wherea s th e equation s o f dynamic s aris e fro m varia tional problem s relatin g t o simpl e integrals , th e equation s o f
physics aris e fro m variationa l problem s relatin g t o multipl e
integrals, whic h mus t b e regarded a s functionals o f the boundar y
of th e fiel d o f integration .
After this , som e year s elapse d befor e th e wor k o n functional s
was continued . I n 1 892-1 89 4 h e publishe d a numbe r o f paper s
[35*3<S> 39 s 40 , 4I 5 42 , 44 ] o n th e Partia l Differentia l Equation s
of Mathematica l Physics , especiall y th e equatio n o f cylindrica l
waves
d2u _ d 2u d
2
dt2 dx
dy
2
u
r
For thi s equatio n h e obtaine d a n expressio n o f th e solutio n i n
terms of the initial values of u and du/dt y whic h ma y be regarde d
as th e analogu e i n tw o dimension s o f th e Riemann-Gree n
formula fo r th e propagation o f waves in on e dimension/H e als o
enquired ho w Kirchhoff' s well-know n expressio n o f Huygens '
Principle i n the wave-theor y o f light coul d b e extended t o spac e
of tw o dimensions , o r o f mor e tha n thre e dimensions : an d h e
obtained a formul a whic h give s th e valu e o f a cylindrica l wav e
function a t a poin t a t th e instan t t, i n term s o f th e disturbanc e
at point s Q o f a give n curv e a t th e instan t /.—PQ/ c an d a¥
previous instants. 2 Thi s formul a was use d late r b y Sommerfel d
in hi s work o n th e diffractio n o f X-rays .
It was afte r thi s tha t Volterr a bega n hi s celebrate d researche s
in th e theor y o f Integra l Equations . H e ha d first me t wit h a n
integral equatio n i n a pape r [ n ] o f 1 884 , dealin g wit h th e
2
Thes e result s ar e brough t int o relatio n wit h mor e recen t wor k b v Hadamard ,
Marcel Ries z an d others , in Bake r an d Copson , Huygens Principle (Oxford , 1 939) -
698
OBITUARY NOTICE S
distribution o f electri c charg e o n a segmen t o f a sphere : th e
problem, a s he showed , depend s on th e solutio n o f wha t woul d
to-day b e calle d a n integra l equatio n o f th e first kin d wit h a
symmetric nucleus , I t was , however , no t unti l 1 89 6 tha t h e
seriously too k u p wor k i n thi s field, applyin g hi s theor y o f
functionals t o wha t wa s a t tha t tim e calle d th e 'inversio n o f
definite integrals' , an d obtainin g result s [56 , 57 , 58 ] whic h wer e
greatly admired. In these papers, he regarded the integral equatio n
of th e secon d kin d wit h a variable limi t o f integratio n
<f>(y) =AY) +/JA*)
s
( x ' y) dx
(where f i s th e unknow n function) , whic h i s no w generall y
called a n integral equation of Volterrds type, as th e limitin g cas e
of a system of linear algebrai c equations ; th e nt h equation o f thi s
system contain s onl y th e first n unknow n quantities , s o th e
system ca n b e solve d b y recurrence . Fro m th e solutio n o f th e
algebraic system , h e passe d t o th e correspondin g solutio n o f th e
integral equation , obtainin g a formul a
AY) = t(y) +J?<I>(X)T(X, Y¥
X
where T(x , y) i s a function—th e resolvent nucleus, a s i t woul d
be calle d to-day—whic h ca n b e constructe d b y a simpl e proces s
from th e give n function S(x, y). Unlik e later writers—Fredholm ,
Hilbert, Schmidt—Volterr a use d th e analog y wit h th e linea r
algebraic equation s onl y heuristically , th e final result s bein g
proved independently .
In orde r t o dea l wit h integra l equation s o f th e first kin d
6{y) - 0(a) ^f ayf(x)Yi{x, y)dx
(where/is th e unknown function), h e differentiate d thi s equatio n
with respec t t o y> thu s obtainin g a n integra l equatio n o f th e
second kin d whic h coul d b e solve d b y th e metho d alread y
found. A difficult y arise s whe n H(y , y) vanishe s o r become s
infinite a t certai n points : h e discusse d certai n type s o f thes e
'singular' nuclei , an d mastere d them .
All thes e investigation s wer e afterward s extende d t o th e cas e
VIT0 VOLTERR A
699
of multiple integrals, and als o to simultaneou s system s of integra l
equations, involvin g severa l unknow n functions . I n 1 89 7 h e
showed [65 ] tha t hi s metho d i s applicabl e t o integra l equation s
with bot h limit s variable : th e rang e o f integratio n actuall y
considered wa s ay^x^y wher e — 1 A«< 1.
In a lecture [68 ] o n th e oscillation s o f liquid s unde r th e influ ence of gravitational forces (th e problem of seiches), he advocated
the applicatio n o f infinit e determinant s t o th e theor y o f integra l
equations—a metho d whic h becam e o f grea t importanc e late r i n
the wor k o f Fredholm .
In a pape r writte n o n th e occasio n o f th e centenar y o f Abel' s
birth [86] , he applie d hi s theor y of'th e inversio n o f a definit e
integral' t o a proble m o f stability . Afte r pointin g ou t tha t Abe l
was th e first t o solv e a n integra l equatio n o f th e first kin d o f
Volterra's typ e (namely , i n th e theor y o f th e tautochrone) , h e
discussed th e proble m o f th e stabilit y o f a fluid mas s rotatin g
about on e o f it s principa l axe s an d consistin g o f concentric ,
similar, and similarly situated layers: this configuration h e prove d
to b e unstable . A simple r proof , independen t o f th e theor y o f
integral equations , wa s als o given : bu t i t wa s pointe d ou t tha t
this latte r proo f i s les s general , a s regard s th e condition s t o b e
satisfied b y th e functio n representin g th e densit y o f the liquid .
Meanwhile, a wid e interes t i n integra l equation s ha d bee n
awakened b y Fredholm' s theory , whic h wa s publishe d i n
Swedish in 1 90 0 and in French (i n the Acta Mathematica) i n 1 903 .
Volterra [1 01 ] pointe d ou t th e connexio n betwee n Fredholm' s
theory an d som e problem s o f his ow n theor y o f functionals: th e
solution o f a n integra l equatio n is , indeed , a simpl e cas e o f th e
solution o f a functiona l equation . I n th e sam e paper , an d als o
in th e sevent h o f hi s Stockhol m lecture s o f th e sam e year , h e
discussed certai n transcendenta l integra l equation s originating ' in
the 'Taylo r expansion ' o f th e theor y o f functionals .
To thi s perio d o f his life belon g som e celebrate d researche s i n
the theor y o f elasticity , whic h wer e importan t no t onl y o n thei r
own accoun t bu t als o becaus e the y suggeste d muc h o f hi s sub sequent wor k i n Pure Mathematics . Perhaps th e mos t notabl e o f
70 0
OBITUARY NOTICE S
these wa s th e theor y o f wha t h e calle d distorsioni, a ter m fo r
which Lov e introduce d th e Englis h renderin g dislocations. I n
elastic solids which occupy a multiply-connected regio n o f space,
the displacement s ma y b e many-valued functions , correspondin g
to deformation s fo r whic h certai n fundamenta l result s o f th e
ordinary theor y o f elasticit y ar e untrue . A s a simpl e example ,
suppose tha t a thi n slic e o f materia l i s cu t ou t o f a n anchor-rin g
and th e ne w surface s thu s forme d ar e brough t int o contac t and ,
after havin g bee n twisted , joine d together . Ther e i s the n a n
initial stres s eve n i n th e absenc e o f al l externa l forces , an d ther e
are certai n discontinuitie s i n th e displacements , althoug h stres s
and strai n ar e continuous . Th e join t ma y i n fac t b e regarde d
either a s a sea t o f discontinuitie s i n th e displacement , o r els e a s
a barrie r ( a branch-cut ) fo r th e many-value d function s whic h
represent th e component s o f th e displacement .
Such system s had bee n introduced earlier , notabl y i n Larmor' s
attempt t o explai n electron s a s place s o f intrinsi c strai n i n th e
aether: bu t Volterr a wa s th e firs t t o develo p i n 1 905-1 90 6 [IOO ,
103] a fairl y genera l theor y o f thes e 'dislocations' . A compre hensive accoun t o f it , wit h som e improvement s b y E . Cesaro ,
was publishe d i n 1 90 7 [1 07] . H e firs t determine d th e many valued displacemen t i n a multiply-connecte d region , corre sponding t o a give n one-value d strain : b y ai d o f th e formula e
thus obtained , h e wa s abl e t o discus s th e typ e o f discontinuit y
at th e barrier : thi s h e prove d t o b e o f th e typ e o f displacemen t
of a rigid body . I n particular , h e discusse d i n som e detai l th e
possible displacement s i n a hollow cylinder , an d als o in a syste m
of thi n rods : i n th e forme r cas e h e wa s abl e t o compar e hi s
deductions wit h th e result s o f experiment . Onc e durin g th e
war, whe n h e wa s o n a missio n t o Englan d t o discus s scientifi c
questions o f commo n interes t t o the . Allies, h e returne d afte r a
tiring da y t o th e Colleg e wher e h e was a guest , an d foun d tha t
his kind host s ha d place d roun d th e wall s o f his room a numbe r
of model s o f cylinder s subjecte d t o Volterr a dislocations . H e
was deepl y touched , an d ofte n recalle d thi s incident in late r life .
His wor k in elasticit y wa s th e origi n o f hi s theor y o f integro -
VITO VOLTERR A
701
differential equations , i.e. integral equations between the unknown
function an d it s (partial ) derivatives . I n 1 90 9 [1 1 4 ] h e studie d a
particular typ e o f suc h equations , an d showe d tha t th e solutio n
of this integro-differential equatio n was equivalent t o the solution
of a simultaneou s syste m consistin g o f thre e linea r integra l
equations an d a partia l differentia l equatio n o f th e second order .
Integro-differential equation s occu r i n variou s branche s o f
mathematical physics . Thus , fo r certai n substances , th e electri c
or magneti c polarizatio n depend s no t onl y o n th e electro magnetic fiel d a t th e moment , bu t als o o n th e histor y o f th e
electromagnetic stat e o f th e matte r a t al l previou s instant s
(hysteresis). Whe n th e term s correspondin g t o thi s physica l fac t
are introduce d int o th e fundamenta l equations , thes e becom e
integro-differential equation s [1 1 5] .
A similar situatio n i s foun d i n 'hereditar y elasticity ' (a s Picar d
called it) , t o whic h tw o o f Volterra' s note s o f th e sam e yea r
are devote d [1 1 9 , 1 20] . H e assume s 'linea r heredity' , i.e . tha t
the strai n i s a linea r functiona l o f th e stress ; i n this cas e th e
fundamental equation s ar e system s o f linea r integro-differentia l
equations, an d h e showe d tha t th e strai n i n a definit e interva l o f
time can be determined, provide d tha t th e forces i n the body an d
the stress or strain on its surface ar e known for thi s time-interval .
In 1 91 0 Volterra introduce d int o th e theor y o f functiona l th e
fruitful notion s o f composition an d o f permutable functions [1 28] .
The composition of tw o function s F(x , y) an d $ (x 9 y) i s define d
to b e the formatio n o f the integra l
which i s denote d b y F#<1 > , the compositio n bein g said t o b e o f
the first kind i f th e limit s o f integratio n ar e x an d y 9 an d o f
the second kind i f th e limit s ar e constant s a an d b: thes e tw o
cases evidently correspon d t o Volterra' s an d Fredholm' s integra l
equations respectively . Tw o function s F an d < £ are sai d t o b e
permutable i f thei r compositio n i s commutative. I n th e firs t note ,
he introduce d permutabilit y o f th e firs t kind , an d i n particula r
investgitaed function s<3 > which ar e permutabl e (o f the first kind )
70 2
OBITUARY NOTICE S
with a give n functio n F . H e transforme d th e definin g equatio n
F-fc</> = </>* F into a n integro-differentia l equatio n an d b y solvin g
the latte r showe d tha t ever y functio n permutabl e (o f th e first
kind) wit h F ca n b e represente d b y a linea r aggregat e o f F an d
compositions (o f the first kind ) o f F with itself . Sinc e the opera tion o f compositio n i s evidentl y a n extensio n o f th e operatio n
of matri x multiplication , t o matrice s i n whic h th e row-numbe r
and column-numbe r tak e a continuou s sequenc e o f values , i t is ,
obvious tha t Volterra' s resul t correspond s t o th e well-know n
theorem tha t ever y matri x which is permutable with a matrix F
(whose laten t root s ar e al l simple ) mus t b e a polynomia l i n F .
The applicatio n o f Volterra' s theore m t o th e cas e whe n F i s a constant yielde d th e importan t resul t tha t th e aggregat e o f all .
functions permutabl e o f th e first kind wit h a constant i s identical
with th e aggregat e o f all functions o f (y — x).
In thre e note s o f th e sam e yea r [1 25 , 1 27 , 1 29 ] th e theor y o f
composition an d permutable function s wa s applied t o th e theor y
of integra l an d integro-differentia l equations . Conside r a n
algebraic relatio n F(z, £ ) = o betwee n tw o variables . Le t th e
variables b e replace d b y tw o functions/an d <j> an d le t al l multi plications o f z wit h itsel f o r wit h £ be replace d b y composition s
of th e correspondin g functions . W e thu s obtai n a n integral equation betwee n f an d <£ . I f i t i s possibl e t o represen t th e
solution o f F(z , £ ) = o (wher e £ is th e unknown ) b y a power series i n z, the n this representation , whe n z ha s bee n replace d
by / an d th e multiplication s b y compositions , wil l yiel d th e
solution (f> o f th e integra l equation . Thu s th e solutio n o f a n
integral equatio n ha s bee n mad e t o depen d o n th e solutio n o f
an algebrai c equation . B y a simila r proces s th e solutio n o f a n
integro-differential equatio n ma y b e deduce d fro m th e solutio n
of a differential equation .
The result s o f thes e note s wer e applie d t o th e proble m o f
hereditary elasticit y i n tw o note s [1 24 , 1 26 ] i n whic h Volterr a
solved th e fundamental integro-differentia l equation s fo r th e cas e
of a n isotropi c sphere . I n th e secon d not e h e als o solve d a quadratic integra l equation .
VITO VOLTERR A
70 3
In 1 91 1 h e too k u p th e proble m o f integro-differentia l
equations wit h constan t limits , i.e . o f th e Fredhol m typ e [1 33] .
As we hav e seen , th e solutio n o f a n integro-differentia l equatio n
can b e reduce d t o th e proble m o f findin g th e 'fundamenta l
solution' o f a certai n associate d differentia l equation . Thi s
'fundamental solution ' i s represente d b y a serie s proceedin g
according t o power s o f th e paramete r whic h occur s i n th e
integro-differential equation : whe n thi s paramete r i s zero , th e
integro-differential equatio n whic h Volterr a was studying ,
reduced t o Laplace' s partia l differentia l equatio n i n n variables ,
and hi s 'fundamenta l solution ' t o th e elementar y solutio n o f
Laplace's equation . I n orde r t o solv e th e integro-differentia l
equation, a serie s o f composition s wa s derive d fro m th e funda mental series : thi s serie s o f composition s i s alway s convergen t
in th e cas e o f composition s o f th e first kind , bu t i n th e cas e o f
integro-differential equation s o f th e Fredhol m type , th e com positions ar e o f th e secon d kin d an d th e serie s forme d o f the m
do no t alway s converge . Volterr a therefor e no w gav e anothe r
theory fo r linea r integro-differentia l equation s o f Fredhol m
type, which , however , i s applicabl e onl y whe n th e coefficient s
of th e integro-differentia l equatio n ar e permutabl e function s (o f
the secon d kind) . Th e chie f featur e o f th e ne w theor y i s tha t
the fundamenta l solutio n appear s no w a s th e solutio n o f a
linear integra l equatio n o f Fredholm' s typ e (instea d o f bein g
the solutio n o f a differentia l equation) .
In anothe r not e [1 34 ] h e determine d al l th e function s per mutable (o f th e firs t kind ) wit h a give n functio n o f orde r tw o
(i.e. suc h tha t ^>{x, x) = o an d f — j ^ = 0 ) showin g tha t the y
depend o n a n arbitrar y functio n o f on e variable : h e als o solve d
the equatio n ifj*ifj—<l> completely , o n th e assumptio n tha t</ > is
a functio n o f orde r two . I n a furthe r not e [1 35 ] h e pointe d ou t
that fo r function s o f th e specia l for m
Hx> y)
n
=
2
a iKfi(x)<f>K(y)
if K =-- 1
70 4
OBITUARY NOTICE S
the questio n o f permutabilit y i s identica l wit h th e questio n o f
commutability o f product s o f th e correspondin g matrice s (#,*) .
Thus fro m matrix-algebr a h e wa s abl e t o fin d al l function s per mutable wit h F (i.e . al l matrice s permutabl e wit h (a**)) . Th e
theory wa s applie d t o th e solutio n o f a n integral equatio n o f th e
nth degree .
Later in 191 1 he published [1 31 ] a general survey of his method
of utilizing the theory of composition and of permutable function s
in orde r t o solv e integra l an d integro-differentia l equations .
In 1 91 2 h e gav e a mor e detaile d expositio n [1 36 ] o f th e
theory o f integro-differentia l equation s wit h a variabl e uppe r
limit an d o f ellipti c type : an d (i n th e sam e memoir ) th e theor y
of'hereditary elasticity * an d o f electri c an d magneti c hysteresis .
He als o [1 39 ] extende d hi s theor y o f hereditar y elasticit y b y
considering vibrations : this le d hi m t o integro-differentia l
equations o f hyperboli c type , an d t o som e fundamenta l result s
regarding vibration s o f hereditar y type . I n man y simpl e case s
the 'hereditary ' solutio n ma y b e obtaine d fro m th e commonl y
known on e b y replacin g th e trigonometri c function s o f th e
known solution s b y certai n transcendenta l function s whic h h e
now defined .
In th e followin g yea r (1 91 3 ) h e returne d onc e mor e [1 47 ] t o
integro-differential equation s o f ellipti c typ e an d complete d th e
investigations o f 1 91 1 b y considerin g i n greate r detai l th e cas e
of a n od d numbe r o f (spatial ) dimensions . I n a lectur e t o th e
Fifth Internationa l Congres s o f Mathematician s a t Cambridg e
[149] h e deal t wit h transcendenta l integra l equations .
In th e sam e year appeare d i n boo k for m hi s lecture s a t Rom e
on integra l an d integro-differentia l equation s [1 45] , an d hi s
lectures a t th e Sorbonn e o n function s o f line s [1 46] ; i n thes e a
full accoun t i s given o f th e theor y o f functional, th e basi s bein g
always th e transitio n fro m a finit e numbe r o f variable s t o a
continuously infinit e number . Thes e work s di d muc h t o mak e
Volterra's idea s widel y known . O n th e invitatio n o f th e BerH n
Mathematical Societ y h e als o delivere d a lecture [1 52 ] outHnin g
the fundamental notion s o f the functional calculu s an d indicatin g
VITO VOLTERR A
705
applications t o th e calculu s o f variations , integra l an d integro differential equations , th e theor y o f quadrati c form s i n a n
infinite numbe r o f variables , hereditar y elasticity , mechanic s
and electro-magnetism .
In 1 91 4 he publishe d a couple o f notes [1 55 , 1 56 ] dealin g wit h
functional derivativ e equations , i.e . equation s betwee n a func tional an d it s derivative . Afte r studyin g som e simpl e type s o f
such equation s (linea r functiona l derivativ e equation s o f th e firs t
and secon d order ) h e showed tha t thes e equation s correspon d t o
the differentia l equation s o f th e ordinar y theory . H e als o dis cussed a syste m o f integro-differentia l equation s correspondin g
tc a canonica l syste m o f equation s i n dynamics , an d obtaine d
the functiona l derivativ e equatio n whic h correspond s t o th e
Hamilton-Jacobi equation .
In a substantia l memoi r produce d o n th e ev e o f th e war—i n
fact, writte n fo r th e Napie r tercentenar y i n Edinburg h i n
July 1 91 4—bu t no t publishe d unti l 1 91 6 [1 62] , h e gav e a
systematic expositio n o f hi s theor y o f compositio n o f th e first
kind, introducin g som e importan t ne w ideas . Th e mos t notabl e
of these is the 'zero't h compositiona l powe r o f a function' whic h
plays th e par t o f unit y an d is essentiall y identica l wit h Dirac' s
d-function. B y th e ai d o f thi s unit , compositiona l fraction s an d
hence negativ e compositiona l power s ar e easil y defined . Pro ceeding furthe r i n th e sam e direction , a definitio n o f th e
'logarithm b y composition ' i s obtained , an d th e theor y o f
'functions b y composition ' developed . I n particular , th e funda mental notion s o f th e Calculu s ar e extende d t o th e domai n o f
functions b y composition , suc h as : derivativ e o f a functio n b y
composition whic h i s prove d t o b e a (new ) functio n b y com position, an d th e definit e integra l b y compositio n o f a functio n
by composition .
This theory of functions b y composition was the subject ofthre e
lectures delivere d a t th e Ric e Institute , Houston , Texa s [1 74] .
Before describin g th e scientifi c wor k o f th e las t twenty-fiv e
years o f Volterra' s life , le t u s tak e u p agai n th e threa d o f hi s
personal history . I n Marc h 1 90 5 h e wa s create d a Senato r o f
706
OBITUARY NOTICE S
the Kingdo m o f Italy— a grea t honou r fo r a ma n stil l com paratively young—an d abou t thi s tim e h e was appointe d b y th e
Government a s Chairma n o f th e Polytechni c Schoo l a t Turin ,
and Royal Commissioner . Th e way was open for him t o becom e
a grea t figure i n political and administrative life : but h e preferre d
the caree r o f a pur e scientist , an d too k a n activ e par t i n publi c
affairs o n onl y tw o occasions—th e Grea t War o f 1 91 4-1 91 8 , an d
the struggl e wit h Fascism .
In July 1 91 4 h e was , accordin g t o hi s custo m a t tha t tim e o f
year, a t hi s countr y hous e a t Ariccia , whe n th e wa r brok e out .
Almost a t onc e hi s min d wa s mad e u p tha t Ital y ough t t o joi n
the Allies : an d i n concer t wit h D'Annunzio , Bissolati , Barzila i
and others , he organize d meeting s an d propagand a whic h wer e
crowned wit h succes s o n th e 24t h o f Ma y i n th e followin g
year, whe n Ital y entere d th e war . A s a Lieutenant i n th e Corp s
of Engineer s h e enliste d i n th e army , and , althoug h no w ove r
fifty-five year s o f age , joined th e Ai r Force . Fo r mor e tha n tw o
years h e live d wit h youthfu l enthusias m i n th e Italia n skies ,
perfecting a ne w typ e o f airshi p an d studyin g th e possibilit y
of mountin g gun s o n it . A t las t h e inaugurate d th e syste m o f
firing fro m a n airship , i n spit e o f th e genera l opinio n tha t th e
airship woul d b e se t on fir e o r explod e a t th e firs t shot . H e als o
published som e mathematica l work s relatin g t o aeria l warfare ,
and experimente d wit h aeroplanes . At th e end of these dangerou s
enterprises h e wa s mentione d i n dispatches , an d decorate d wit h
the Wa r Cross .
Some day s afte r th e capitulatio n o f Gorizi a h e wen t t o thi s
town whil e i t wa s stil l unde r th e fir e o f Austria n gun s i n orde r
to tes t th e Italia n instrument s fo r th e location o f enemy batterie s
by sound . A t th e beginnin g o f 1 91 7 h e establishe d i n Ital y th e
Office fo r Wa r Inventions , an d becam e its Chairman , makin g
many journey s t o Franc e an d Englan d i n orde r t o promot e
scientific an d technica l collaboratio n amon g th e Allies . H e wen t
to Toulo n an d Harwic h i n orde r t o stud y th e submarin e war ,
and i n Ma y an d Octobe r 1 91 7 too k par t i n th e Londo n dis cussions regardin g th e Internationa l Researc h Committee , t o th e
VITO VOLTERR A
70 7
executive o f which he was appointed. H e was the first t o propos e
the us e of helium a s a substitute fo r hydrogen , an d organize d it s
manufacture.
"When in 1 91 7 some political parties—especially th e Socialists—
wanted a separat e peac e fo r Italy , h e strenuousl y oppose d thei r
proposals: afte r th e disaste r o f Caporetto , h e wit h Sonnin o
helped t o creat e th e parliamentar y hhc whic h wa s resolve d t o
carry o n th e war t o ultimat e victory .
On th e conclusion o f the Armistice in 1 91 8 , Volterra returne d
to hi s purel y scientifi c studie s an d t o hi s teachin g wor k i n th e
University. Th e mos t importan t discoverie s o f hi s lif e afte r th e
war wer e i n th e fiel d o f mathematica l biology , an d o f thes e w e
must no w giv e an account .
The titl e o f hi s discours e [85] , at th e opening o f th e academi c
year followin g hi s electio n t o th e Roma n Chair , show s tha t
already i n 1 90 1 h e was intereste d i n th e biologica l application s
of mathematics . Hi s ow n researche s i n thi s field , stimulate d b y
conversations wit h th e biologis t D r Umbert o D'Ancon a o f th e
University o f Siena , bega n a t th e en d o f 1 925 : hi s firs t an d
fundamental memoi r [1 89 ] (reprinte d wit h modication s an d
additions a s [1 93] : summarize d briefl y i n Englis h [1 91 ] : an d
more full y [202] . Critica l summar y b y J. Peres , Rev. gen. Set.
pur. appL 38, 285-300, 337-341 (1 927) ) o n the subject appeared i n
the following year . Th e theor y was develope d i n severa l furthe r
papers [1 94 , 1 95 , 1 96 , 1 98 ] an d i n th e winte r 1 928-1 92 9 wa s
made th e subjec t o f a cours e o f lecture s delivere d b y Volterr a
at th e Institu t Henr i Poincar e i n Paris . Thes e lectures , togethe r
with a historica l an d bibliographica l chapte r compile d b y
D'Ancona, wer e publishe d i n 1 93 1 [21 0] .
The entitie s studie d i n thes e investigation s wer e biological
associations, i.e . system s o f anima l (o r plant ) population s o f
different species , livin g togethe r i n competitio n o r allianc e i n a
common environment : an d th e theor y i s concerne d wit h th e
effects o f interactio n o f thes e population s wit h on e anothe r an d
the environmen t a s expressed i n thei r numerica l variations .
At th e beginnin g o f these researches , Volterr a wa s unaware o f
708
OBITUARY NOTICE S
similar wor k alread y i n existence . Som e o f thi s 3 referre d onl y
to special problems. Other work, such as that of W. R . Thompso n
on parasitology , thoug h mor e genera l i n characte r an d allie d t o
Volterra's studies , differe d i n thi s respect : tha t i t wa s necessar y
to regar d generation s a s distinct an d th e number o f a populatio n
as varying discontinuousl y i n time . I t wil l b e obvious , however ,
that ther e ar e man y problem s relatin g t o species - i n whic h
generations overlap , where the number o f a population ca n mor e
appropriately b e regarded , wit h sufficien t approximation , a s a
continuously varying functio n o f th e time : an d i t i s to problem s
of thi s kin d tha t Volterra' s method s apply . Suc h problem s ha d
been studied to some extent by A. JVXotka, and some of Volterra's
simpler result s referrin g t o association s o f tw o specie s ha d bee n
anticipated b y him .
To understan d th e idea s o n whic h Volterra Y theory i s based ,
we ma y conside r first th e b y n o means trivia l cas e o f a singl e
species. Let th e numbe r o f th e populatio n a t th e tim e t be N(f) .
The simples t assumptio n o f constan t birt h an d deat h rate s lead s
immediately t o th e 'Malthusian ' equatio n
•* - e N w
giving o n integratio n th e geometri c la w o f increas e
N = N 0 ^ . (2
)
Under mor e genera l conditions , ther e wil l b e correction s t o b e
applied t o th e right-han d sid e o f (1 ) : an d i t i s o f cours e suc h
corrections whic h falsif y th e famou s prediction s o f Malthu s
regarding huma n populations .
If th e environmen t wil l suppor t onl y a limite d numbe r o f
individuals, w e mus t suppos e tha t th e 'coefficien t o f increase '
€ is n o longe r constant , bu t a decreasin g functio n o f N . I t i s
simplest t o assum e tha t thi s decreas e is hnear s o that w e have th e
Verhulst equatio n
^
3
= ( e
_ ,
N ) N (3
e.g . th e analysi s o f the Ross-Martini malari a equations .
)
VITO VOLTERR A
709
where €, X ar e constants, e being calle d i n Volterra's late r paper s
the coefficient of auto-increase o f th e population , t o distinguis h i t
from th e whol e coefficien t o f increas e e— XN. Integratin g (3) ,
we obtai n th e well-know n 'logisti c curve '
N=€/(x+-kr"j- (4
)
which i s verifie d observationall y i n man y context s i n biolog y
and economics .
Under mor e genera l condition s furthe r correction s mus t b e
allowed for ; thu s i n a complicate d cas e w e migh t hav e t o
consider a n integro-differentia l equatio n suc h as
-jt = L - X N + Ksinirf— fQ N ( r ) / ( f - T ) M N + a .
The tota l coefficien t o f increas e her e consist s o f e , the coefficien t
of auto-increas e o f th e population , togethe r wit h thre e cor rections: — XN representing th e Verhuls t effec t o f competitio n
within th e species : a periodic ter m arisin g perhap s fro m seasona l
variations o f th e environmen t an d generall y producin g force d
oscillations i n N : an d a n integra l representin g som e delaye d
effect, suc h a s th e intoxicatio n o f a close d environmen t b y th e
accumulation o f wast e products . Th e fina l constan t a indicate s
immigration a t a uniform rate .
Volterra's concern , however , i s wit h association s o f 2 , 3 , . . .
or n species , an d henc e wit h 2 , 3 , . . . o r n differentia l equation s
(or integro-differential equation s if delayed effect s ar e envisaged).
He investigate d i n considerabl e detai l th e associatio n o f tw o
species, on e o f whic h feed s o n th e other : w e ma y cal l the m
predators and prey. Fo r thi s cas e w e hav e th e Lotka-Volterr a
equations
(prey) ~^
=
fa-y^Nx
(5)
(predators) —^ = ( — e 2 + yaNi)N s
dt
By themselves , th e pre y woul d increas e an d th e predator s di e
out: henc e th e coefficient s o f auto-increas e (ci , — e 2) ar e respec 4=>
OBITUARY NOTICE S
7io
tively positiv e an d negative . Th e competitio n terms , sinc e the y
depend on encounters , ar e proportional t o N X N 2 and ar e positiv e
for th e predators , negativ e fo r th e prey . B y integratin g thes e
equations, Volterr a deduce d th e existenc e o f periodi c fluctuations, whos e perio d i s independen t o f y l9 y 2 (so , e.g. , i t woul d
not b e affecte d b y increase d protectio n o f the prey) : th e averag e
values o f N x an d N 2 d o no t depen d o n thei r initia l values , an d
are give n b y
N^ca/ya, N
2=e
1
/ y1 . (6
)
The existenc e o f periodic fluctuations i n biological association s
was alread y wel l know n fro m observation : bu t ecologist s ha d
generally considere d tha t i t wa s necessar y t o see k a n explanatio n
of th e fluctuation s i n som e externa l cause , such a s the seasons , o r
human interference . Partl y a s a resul t o f th e Lotka-Volterr a
analysis, i t i s no w generall y admitte d tha t periodi c fluctuations
in a constan t environmen t ma y unde r som e circumstance s b e
sufficiently explaine d b y th e mer e fac t o f interaction .
If th e population s considere d i n Equation s (5 ) ar e differen t
species o f fish, th e effec t o f fishin g (i.e . unifor m destructio n o f
both specie s o f fish proportionatel y t o thei r numbers ) woul d b e
to increas e e 2 a n d t o decreas e e ^ Fro m (6 ) i t follow s tha t th e
mean numbe r o f predator s (N 2) i s decreased , an d tha t o f thei r
prey (N x) increased . Similarly , a cessatio n o f fishing , suc h a s
occurs o n a larg e scal e i n tim e o f war , wil l b e t o th e relativ e
advantage o f th e predato r species . Thi s effec t ha d alread y bee n
observed b y D'Ancon a i n hi s statistica l stud y o f th e Adriati c
fisheries ove r th e perio d 1 905-1 923 . H e found , tha t i s t o say , a
temporary increas e i n th e mea n relativ e frequenc y o f th e mor e
voracious kind s o f fish , a s compare d wit h th e fis h o n whic h
they preyed , durin g th e year s 1 91 4-1 91 8 .
In th e Legons sur la Theorie mathematique de la Lutte pour la
Vie [21 0 ] a genera l theor y o f n specie s i s developed , an d th e
suggestion o f a dynamica l analog y i s introduce d b y th e dis tinction betwee n wha t h e calle d conservative and dissipative
associations. Withou t goin g int o th e detail s o f th e definition , i t
VITO VOLTERR A 7 I
I
may b e explaine d tha t th e associatio n (o f on e species ) repre sented i n equatio n (i ) i s conservative , tha t o f equatio n (3 ) i s
dissipative. Equation s (5 ) represen t a conservativ e association :
while i f w e modif y the m b y takin g accoun t o f competitio n
within eac h species , just a s (3 ) wa s obtaine d b y modificatio n o f
(i), w e obtai n th e equation s o f a-dissipativ e association , namely ,
it
<2
The effec t o f thi s chang e i s tha t th e fluctuations o f N x an d N 2
are no w damped : th^ t is , thei r amplitude s diminis h an d th e
association tend s wit h increasin g tim e t o it s equilibriu m state .
This, Volterr a showed , i s a genera l propert y o f th e association s
which h e calle d dissipative , an d give s a n obviou s poin t t o th e
mechanical analogy . H e regarde d conservativ e association s a s
representing idea l situation s no t generall y attaine d i n nature, an d
supposed tha t actua l association s ar e mor e ofte n o f th e dissipa tive type. 4
Finally, i n chapte r 4 o f hi s book , Volterr a extende d th e
theory o f tw o specie s t o th e cases , s o importan t i n man y
biological problems , wher e delaye d effect s occur . Th e tw o
differential equation s wer e no w replace d b y a pai r o f integro differential equations . He solve d the m b y a method o f successiv e
approximations, an d discusse d a t lengt h th e analog y o f thi s cas e
with hi s previou s studie s o f hereditar y phenomen a i n elasticit y
and electromagnetism .
The extensio n o f thi s stud y o f delaye d effect s t o a n associatio n
of n specie s wa s tackle d onl y i n Volterra' s las t publicatio n o n
mathematical biolog y [235] , whic h appeare d i n 1 939 . I n th e
4
G . F . Gause , The Struggle for Existence (Baltimore, 1 934) , describe d experiment s
which aime d a t reproducin g som e o f Volterra' s mathematica l model s wit h simpl e
biological models , usin g yeas t cell s an d differen t specie s o f Protozo a i n competition .
The attempt s wer e no t altogethe r successful , bu t i t ma y b e note d tha t th e periodi c
oscillations eventuall y obtaine d i n a predator-pre y experimen t wer e o f diminishin g
amplitude.
712
OBITUARY NOTICE S
intervening years , two ver y differen t development s o f the theor y
occurred, i n wha t ma y b e calle d th e 'applied ' an d 'pure ' aspect s
of th e subject .
Firstly [232] , Volterr a collaborate d wit h D'Ancon a i n sur veying th e relevan t biologica l literatur e fo r confirmatio n an d
further applications . Th e inheren t difficultie s o f thi s tas k aros e
rather fro m th e immens e variet y o f specia l condition s occurrin g
in differen t case s an d th e correc t assessmen t o f thes e condition s
in an y particula r case , tha n i n th e mathematica l analysi s o f th e
conditions onc e identified . Th e fac t tha t th e author s wer e abl e
to fin d th e appropriat e mathematica l mode l i n man y differen t
cases is a vindication o f Volterra's technique .
Secondly [222 , 223 , 224 , 225 , 226 , 227] , o n th e 'pure ' side ,
he extende d int o th e ver y cor e o f classica l dynamic s th e sug gestion o f a n analog y whic h ha s bee n notice d above . Thi s ne w
development, whic h h e describe d i n a lectur e [228 ] a t th e
University o f Geneva o n 1 7 June 1 937 , may b e explained simpl y
by referenc e agai n t o th e predator-pre y equation s (5) . Wit h a
change o f notation, thes e ca n be writte n
Ni = ( e r - ^ ) N„ N
2
=( - e
2
+ ^l)N 2.
We deduc e easil y
aN x +j8N 2 - c ^ Ni +j8e 2N2 = O ,
aNx + £N 2 ~ a>eif Njt + pe 2J N 2dt = a constant, sa y H. (8 )
Introducing wha t h e calle d th e quantities of life o f th e tw o
populations,
x =f N
±dt,
y —f N 2dt9
and regardin g thes e a s analogou s t o co-ordinate s i n dynamics ,
we hav e i n (8 ) th e resul t whic h Volterr a calle d th e conservation
of demographic energy, namely,
T + V = H,
where T = aX-\-fly =
actua l demographi c energ y
V = — ae xx + fie 2y = potentia l demographi c energy .
VITO VOLTERR A
713
Moreover, i f we introduc e th e functio n
O=(a^logpc + ftylogy) + l ( x j / — xy) — V,
it ca n b e show n tha t th e origina l fluctuation equation s (5 ) ar e
equivalent t o th e Lagrangia n equation s o f motio n
' dt\df)
lt\dx) dx~~
dy~
It then follows a s in classical dynamics that, as far as the variations
of Ni , N 2 ar e concerned , th e propertie s o f th e associatio n ma y
be summe d u p i n a variational principl e
t
df&dt = 0 . (9
)
to
The analog y ha s bee n describe d s o fa r fo r th e specia l associatio n
(5); bu t Volterr a worke d i t ou t generall y fo r a conservativ e
association (an d als o fo r certai n type s o f dissipativ e associations )
of n species.
Yet on e mor e analog y wa s describe d i n a lectur e [229 , 230 ]
to th e Reunio n International e de s Mathematiciens . Generalizin g
the analysi s o f 'cessatio n o f fishing' describe d abov e b y mean s o f
equations (5) , (6) , i t referre d t o th e change s i n th e equilibriu m
state of a conservative association of n species, caused by variatio n
of thei r coefficient s o f auto-increase ; an d i t too k th e for m o f
some principle s o f reciprocit y no t unlik e thos e whic h appea r i n
the theorie s o f elasticity an d electrostatics .
Biologists hav e bee n ap t t o criticiz e Volterra fo r preoccupyin g
himself s o elaboratel y wit h abstrac t mathematica l model s base d
on simplifyin g assumption s remot e fro m th e complexitie s o f
nature. Yet this , after all , is the procedur e o n which the triumph s
of physica l scienc e hav e bee n founded . I t woul d b e ras h t o sa y
whether th e analogie s wit h physica l scienc e whic h h e unearthe d
will remai n wha t the y appea r t o b e a t first , an d certainl y are ,
at least, a cleve r an d remarkabl e tour de force—or whether the y
will eventuall y b e see n as the germ s o f a profound biodynamics ,
essential t o th e theoretica l an d economi c biolog y o f th e future :
OBITUARY NOTICE S
714
what i s beyon d disput e i s tha t hi s contribution s t o pur e
mathematics wil l b e i n deman d mor e an d mor e inescapeabl y a s
mathematical biolog y develops .
While th e researche s whic h hav e las t bee n describe d wer e
Volterra's chie f interes t durin g th e late r year s o f hi s life , h e stil l
from tim e t o tim e publishe d contribution s t o pur e analysis .
In 1 92 4 appeare d th e well-know n Volterra-Pere s boo k [1 85 ]
on th e theory o f composition an d permutable functions, markin g
the completio n o f his work i n this field: an d a course o f lecture s
on th e theor y o f functional s an d o f integra l an d integro differential equations , give n i n Madri d i n 1 92 5 b y invitatio n o f
the Facult y o f Scienc e o f th e University , wer e publishe d i n
Spanish i n 1 92 7 [1 92 ] an d i n a n Englis h translatio n [204 ] i n
1930.
Still mor e significan t fro m th e moder n poin t o f vie w wa s a
work o n th e genera l theor y o f functionals writte n i n collabora tion wit h J. Peres , o f whic h th e first o f thre e projecte d volume s
appeared i n 1 93 6 [233] . Thi s first volum e contain s th e genera l
principles o f th e functiona l calculu s an d it s application s t o th e
theory o f integra l equations : th e secon d volum e was planne d t o
contain th e theorie s o f composition , o f permutabl e functions , o f
integro-differential equation s an d functional derivativ e equations ,
and o f Volterra' s generalization s o f analyti c functions : th e thir d
volume woul d dea l wit h som e subsidiar y topic s an d wit h th e
applications o f th e functiona l calculus . Accoun t wa s t o b e take n
of th e moder n theor y o f function s an d o f abstrac t spaces , an d
the complet e wor k woul d therefor e hav e bee n o f grea t im portance.
In 1 93 8 Volterra published , i n th e Bore l series of monographs ,
a work [234 ] written in conjunction wit h B. Hostinsky, concern ing researche s whose origins belong to his earlier period. In 18871888 he had written tw o note s [20 , 23] and a substantial memoi r
[16], dealin g wit h th e theor y o f substitution s o r matrices , th e
infinitesimal operation s whic h ca n be performed upo n these , an d
their application s t o th e theor y o f linea r differentia l equations .
He regarde d th e n 2 element s o f a matri x o f order n a s function s
VITO VOLTERR A
715
of a variabl e x whic h was suppose d t o b e rea l i n thes e earlie r
papers, th e extensio n t o comple x variable s bein g give n i n a
later memoi r [75] : and h e define d th e derivative an d th e integral
of a matrix wit h respec t t o x, showin g tha t thes e tw o operation s
are inverse . Th e valu e o f thes e concept s i s show n whe n w e
consider a system of linear differentia l equation s o f the first orde r
- ^ = 2 <Hl(x)y; ( 1 = 1 , 2 ,. . . , «).
ax J = I
Volterra showe d tha t th e elements o f th e 'integral ' o f th e matri x
of th e aij(x) yield s a fundamenta l syste m o f solution s o f th e
differential equations . Ther e i s n o difficult y i n extendin g th e
formulae s o a s t o includ e th e cas e of non-homogeneou s system s
of differential equations , or the case of more than one independent
variable. Theories o f total differentials o f matrices, and o f doubl e
and curvilinea r integral s wer e the n developed : an d th e trans formation o f thes e latte r int o eac h othe r le d t o th e introductio n
of differentia l parameters . Whe n th e variabl e x i s suppose d t o
be complex , th e integra l o f a matri x alon g a closed contou r ca n
be defined , an d a calculu s o f residue s developed , th e residue s
depending o n th e singularitie s o f th e element s o f th e matrix : i n
short, th e mai n idea s o f th e theor y o f function s o f a comple x
variable can be carried over into matri x theory : an d an extensio n
is thu s obtaine d o f th e well-know n result s o f Fuch s o n th e
expansion o f th e solution s o f a linear differentia l equatio n i n th e
neighbourhood o f on e o f its singula r points . Volterra the n wen t
on t o stud y matrice s whos e element s ar e one-valued an d regula r
functions o f positio n o n a Rieman n surface : thes e h e calle d
algebraic matrices , an d thei r integral s h e name d Abelian matrices:
the theor y i s a n interestin g analogu e o f th e ordinar y theor y o f
algebraic function s an d thei r integrals . Th e late r chapter s o f th e
1938 monograph contai n many development s an d appHcations o f
Volterra's origina l work , du e chiefl y t o Hostinsky : on e o f th e
most important relates to the solution of the celebrated functiona l
equation whic h Professo r S . Chapma n introduce d i n 1 92 8 (Proc.
Roy. Soc, 1 1 9 ) i n connexio n wit h a problem i n diffusion .
7i6
OBITUARY NOTICE S
Volterra's scientifi c activit y overflowe d i n man y domain s
quite outsid e hi s mor e usua l field s o f research . Th e studen t o f
topology, fo r instance , who read s Professor Lefschetz' s admirabl e
monograph o n Analysis Situs (Paris , 1 924) , find s therei n th e
photographs o f a numbe r o f ingeniou s model s constructe d b y
Volterra i n orde r t o sho w ho w tw o manifolds , define d i n ver y
different ways , ma y nevertheles s b e homeomorphou s t o eac h
other.
There remain s t o b e tol d th e melanchol y stor y o f hi s late r
years. I n 1 92 2 Fascism seize d th e rein s in Italy . Volterr a wa s on e
of th e ver y fe w wh o recognize d fro m th e beginnin g th e dange r
to freedom o f thought, an d immediately oppose d certai n change s
in th e educationa l system , whic h deprive d th e Italia n Middl e
Schools o f thei r liberty . Whe n th e opponent s o f th e Fascis t
Government i n the House of Deputies withdrew altogethe r fro m
the debates , a smal l grou p o f Senators , heade d b y Volterra ,
Benedetto Croc e an d Francesc o Ruffmi , appeared , a t grea t
personal risk , a t al l th e Senat e meeting s an d vote d steadil y i n
opposition. A t tha t tim e h e wa s Presiden t o f th e Accademi a de i
Lincei an d generall y recognize d a s th e mos t eminen t ma n o f
science in Italy .
By 1 93 0 th e parliamentar y syste m create d b y Cavou r i n th e
nineteenth centur y ha d bee n completel y abolished . Volterr a
never agai n entere d th e Senat e House . I n 1 931 , having refuse d
to tak e the oath of allegiance imposed by the Fascist Government ,
he wa s force d t o leav e th e Universit y o f Rome , wher e h e ha d
taught fo r thirt y years : an d i n 1 93 2 h e wa s compelle d t o resig n
from all ItaHa n Scientifi c Academies. 5 Fro m thi s tim e fort h h e
lived chiefl y abroad , returnin g occasionall y t o hi s country-hous e
in Ariccia . Muc h o f hi s tim e wa s spen t i n Paris , wher e h e
lectured ever y yea r a t th e Institu t Henr i Poincare : h e als o gav e
lectures i n Spain , i n Roumania , an d i n Czechoslovakia . O n al l
these journeys h e was accompanie d b y hi s wife , wh o neve r lef t
him, an d learn t typewritin g i n orde r t o cop y hi s papers fo r him :
6
H e wa s however , o n th e nomination o f Pop e Piu s XI, a member o f the Pontifica l
Academy o f Sciences , and thi s honour h e retaine d unti l hi s death .
VITO VOLTERR A
717
he wa s accustome d t o sa y tha t th e signatur e 'V . Volterra ' i n hi s
later work s represente d no t Vit o bu t Virgini a Volterra .
In th e autum n o f 1 938 , unde r Germa n influence , th e Italia n
Government promulgate d racia l laws , an d hi s tw o son s wer e
deprived o f thei r Universit y position s an d thei r civi l rights : a t
their father' s suggestion , the y lef t thei r nativ e countr y t o begi n
a ne w lif e abroad .
He ha d a remarkabl e powe r o f inspirin g affection . Whe n i n
the las t month s o f hi s lif e th e ne w law s forbad e hi m t o hav e
Italian servants , al l o f the m refuse d t o leave : a mai d wh o ha d
been wit h hi m fo r mor e tha n twent y years , an d wh o wa s force d
to leave , die d o f sorro w a week afterwards .
In Decembe r 1 93 8 h e wa s affecte d b y phlebitis : th e us e o f
his limb s wa s neve r recovered , bu t hi s intellectua l energ y wa s
unaffected, an d it was after thi s that his two last papers [235 , 236]
were.published b y th e Edinburg h Mathematica l Societ y ari d th e
Pontifical Academ y o f Science s respectively . O n th e mornin g o f
11 Octobe r 1 94 0 h e die d a t hi s hous e i n Rome . I n accordanc e
with hi s wishes , h e wa s burie d i n th e smal l cemeter y o f Ariccia ,
on a little hill , nea r th e country-hous e whic h h e love d s o muc h
and wher e h e ha d passe d th e serenes t hour s o f hi s nobl e an d
active life .
He ha d receive d countles s honours . H e wa s electe d a Foreig n
Member o f ou r Societ y i n 1 91 0 , an d ha d receive d a simila r
distinction fro m almos t ever y Nationa l Academ y an d Mathe matical Societ y i n th e world ; an d h e wa s a docto r honoris causa
of man y Universities : i n thi s country , o f Cambridge , Oxfor d
and Edinburgh . Th e photograp h whic h i s reproduce d a t th e
head o f thi s memoi r wa s take n whe n h e receive d th e honorar y
Sc.D. o f Cambridg e i n 1 900 . In Augus t 1 93 8 h e was offere d th e
honorary doctorat e o f the University o f St Andrews, and wishe d
to trave l t o Scotlan d t o receiv e it , bu t wa s forbidde n b y hi s
medical attendant . I n hi s nativ e lan d h e ha d th e Gra n Cordon e
della Corona d'ltali a an d th e Croc e d i Guerra, an d was a Senato r
of th e Kingdo m an d a Knigh t o f SS . Mauric e an d Lazarus , I n
France h e wa s a Gran d Officie r d e laLegio n d'Honneur : h e ha d
7i8
OBITUARY NOTICE S
also th e order s o f Leopol d o f Belgium , o f S . Carl o o f Monac o
and o f th e Pola r Sta r o f Sweden . O n 2 3 Augus t 1 92 1 h e receive d
from Kin g Georg e V th e dignit y o f a n honorar y K.B.E. : an d
this wa s a n immens e gratificatio n t o him , fo r h e wa s deepl y
attached t o hi s man y friend s i n thi s country .
In th e word s i n whic h hi s deat h wa s announce d t o hi s fellow members o f th e Pontifica l Academ y o f Sciences , w e believ e hi m
ex hac vita in scientiarum profectum sedulo impensa ad sapientiae
aeternitatem transisse.
E. T . WHITTAKE R
[Acknowledgment o f hel p receive d i n th e preparatio n o f thi s notic e
is gratefull y mad e to Dr Enric o Volterra, Dr A . Erdelyi an d D r I . M. H .
Etherington.]
LIST O F PUBLICATION S O F VIT O VOLTERR A
1. Su l potenzial e d i un ' elissoid e eterogene a sopr a s e stessa . Nuovo
dm., 9, 221 -22 9 (1 881 ) .
2. Alcun e osservazioni sull e funzioni punteggiat e discontinue. G. Mat.,
19, 76-8 7 (1 881 ) .
3. Su i principii de l calcol o integrale. G. Mat., 1 9 , 333-37 2 (1 882) .
4. Sopr a una legge d i reciprocity nell a distribuzion e dell e temperatur a
e dell e corrent i Galvanich e costant i i n u n corp o qualunque .
Nuouo Cim., 1 1 , 188-192 (1 882) .
5. Sopr a alcun i problem i d i idrodinamica . Nuovo Cim., 1 2 , 65-9 6
(1882).
6. Sull a apparenz e elettrochimich e ali a superfici e d i u n cilindro . Atti
Accad. Torino, 18, 1 47-1 6 8 (1 882) .
7. Sopr a alcun e condizion i caratteristich e dell e funzion i d i un a
variabile complessa. Ann. Mat. para appl. (2), l l 5 1 -5 5 (1 883) .
8. Sopr a alcun i problem i dell a teori a de l potenziale . Ann. Scu. norm.
sup. Pisa (1883). (Tes i di abilitazione. )
9. Sul T equilibri o dell e superficie flessibili ad inestendibili . Atti Accad.
Lincei, 8, 21 4-21 8 an d 244-24 6 (1 884) .
10. Sopr a un problema d i elettrostatica. Nuovo Cim., 1 6 , 49-57 (1 884) .
11. Sopr a u n problem a d i elettrostatica . R. C. Accad. Lincei (3) , 8 ,
315-318 (1 884) .
1
VITO VOLTERR A 7
9
12. Sull a deformazion e dell e superfici e flessibili e d inestendibili . R. C.
Accad. Lincei (4) , I , 274-27 8 (1 885) .
13. Integrazion e d i alcun e equazion i difFerenzial i de l second o ordine .
R. C. Accad. Lincei (4) , I , 303-30 6 (1 885) .
14. Sull e figure elettrochimich e d i A . Guebhard . Atti Accad. Torino,
18,329-336(1885).
15. Sopr a un o propriet a d i un a class e d i funzion i trascendenti . R. C.
Accad. Lincei (4) , 2 , 21 1 -21 4 (1 886) .
16. Su i fundament i dell a teori a dell e equazion i difFerenzial i lineari .
Parte prima . Mem. Soc. ital. Sci. nat. (3) , 6 9 no . 8 , 1 0 4 pp. (1 887) .
17. Sopr a l e funzion i ch e dipendon o d a altr e funzioni . JR . C . Accad.
Lincei (4) , 3 , 97-1 05 , 1 41 -1 4 6 an d 1 53-1 5 8 (1 887) .
18. Sopr a l e funzion i dipendent i d a linee . R. C. Accad. Lincei (4) , 3 ,
225-230 an d 274-28 1 (1 887) .
19. Sopr a un a estension e dell a teori a d i Rieman n sull e funzion i d i
variabili complesse . I . R. C. Accad. Lincei (4) , 3 , 281 -28 7 (1 887) .
20. Sull e equazion i difFerenzial i lineari . R. C. Accad. Lincei (4) , 3 ,
393-396 (1 887) .
21. Sopr a un a estension e dell a teori a d i Rieman n sull e funzion i d i
variabili complesse . II , III . R. C. Accad. Lincei (4) , 4 , 1 07-1 1 5
and 1 96-20 2 (1 888) .
22. Sull e funzion i analitich e polidrome . JR. . C. Accad. Lincei (4) , 4 ,
355-362 (1 888) .
23. Sull a teori a dell e equazion i difFerenzial i lineari . R. C. Circ. mat.
Palermo, 2 , 69-7 5 (1 888) .
24. Sull a integrazione d i un a sistema di equazioni difierenzial i a derivat e
parziali che si presenta nell a teoria dell e funzioni conjugate . R. C.
Circ. mat. Palermo, 3 , 260-27 2 (1 889) .
25. Su r un e generalisatio n d e l a theori e de s fonction s d'un e variabl e
imaginaire. le r Memoire . Acta Math., Stockh., 1 2 , 233-28 6
(1889).
26. Dell e variabil i compless e negl i iperspazi . R. C . Accad. Lincei (4) ,
5, 1 58-1 6 5 an d 291 -29 9 (1 889) .
27. Sull e funzioni conjugate . R. C. Accad. Lincei (4) , 5, 599-61 1 (1 889) .
28. Sull e funzion i d i iperspaz i e su i lor o parametr i difFerenziali . R. C.
Accad. Lincei (4) , 5 , 630-64 0 (1 889) .
29. Sull e equazion i difFerenzial i ch e provengon o d a question i dell e
variazioni. R. C. Accad. Lincei (4) , 6 , 43-5 4 (1 890) .
30. Sopr a un e estension e dell a teori a Jacobi-Hamilton de l calcol o dell e
variazioni. JR. . C. Accad. Lincei (4) , 6 5 1 27-1 3 8 (1 890) .
31. Sull e variabil i compless e negl i iperspazi . JR. . C . Accad. Lincei (4) , 6>
241-252 (1 890) .
72 0
OBITUARY NOTICE S
32. Sopr a l e equazion i d i Hertz * Nuovo Cim. (3) , 29 , 53-6 3 (1 891 ) .
33. Sopr a l e equazion i fundamental i dell a elettrodinamica . R. C. Accad.
Lincei (4) , 7 , 1 77-1 8 8 (1 891 ) an d Nuovo Cim. (3) , 29 , 1 47-1 5 4
(1891).
34. Enric o Betti . Nuovo Cim. (3) , 32 , 5- 7 (1 892) .
35. Sull a vibrazion i luminos e ne i mezz i isotropi . JR. . C. Accad. Lincei
(5), 1 , 1 61 -1 7 0 (1 892) .
36. Sull e ond e cilindrich e ne i mezz i isotropi . R . C . Accad. Lincei (5) , I ,
265-277 (1 892) .
37. Su l principi o d i Huygens . Nuovo Cim. (3) , 31 , 244-25 5 (1 892 ) an d
32, 59-6 5 (1 892) .
38. Su r le s vibration s lumineuse s dan s le s milieu x birefringents . Acta
Math. Stockh., 1 6 , 1 53-21 5 (1 892) .
39. Sull e vibrazion i de i corp i elastici . R. C. Accad. Lincei (5) , 2 , 389 397 (1 893) .
40. Sull e integrazioiie dell e equazion i difFerenzial i de l mot o d i u n corp o
elastico isotropo . R. C. Accad. Lincei (5) , 2 9 549-55 8 (1 893) .
41. 42 . Su l principi o d i Huygens . Nuovo Cim. (3) , 33 , 32-3 6 an d
71-77 (1 893) .
43. Eserzis i d i fisica matematica . I . Sull e funzion i potenziali . Riv. Mat.,
4, 1 -1 4 (1 894) .
44. Su r le s vibrations de s corp s elastique s isotropes . Acta Math., Stockh.,
18, 1 61 -23 2 (1 894) .
45. Sull a teori a de i moviment i d d pol o terrestre . As.tr . Nachr., 1 38 ,
33-52 (1 895) .
46. Sull a teori a de i mot i de l pol o terrestre . Atti Accad. Torino., 30 ,
301-306 (1 895) .
47. Su l mot o d i u n sistem a ne i qual e sussiston o mot i intern i stazionarii .
Atti Accad. Torino, 30 , 372-38 4 (1 895) .
48. Sopr a u n sistem a d i equazion i difFerenziali . Atti Accad. Torino, 30 ,
445-454(1895).
49. U n teorem a sull a rotazion e de i corp i e su a applicazion e a l mot o d i
un sistem a nei quale sussistono mot i interni stazionarii. Atti Accad.
Torino, 30 , 524-54 1 (1 895) .
50. Su i mot i periodic i de l pol o terrestre . Atti. Accad. Torino, 30 , 547 561 (1 895) .
51. Sull a teori a de l mot i de l pol o nell e ipotes i dell a plasticit a terrestre .
Atti Accad. Torino, 30 , 729-74 3 (1 895) 52. Sull a rotazion e d i u n corp o i n cu i esiston o sistem i ciclici . R. C.
Accad. Lincei (5) , 4 , 93~9 7 (1 895) .
53. Su l mot o d i u n sistem a ne i qual e sussiston o mot i intern i variabili .
R. C. Accad. Lincei (5) , 4 > 1 07-1 1 0 (1 895) .
VITO VOLTERR A 7 2 1
54. Sull e rotazion i permanent i stabil i d i u n sistem a i n cu i sussiston o
moti intern i stazionarii . Ann. Mat. pura. appl. (2) , 2 3 , 269-28 5
(1895).
55. Osservazion i sull a mi a not a 'Su i mot i periodic i de l pol o terrestre' .
Atti Accad. Torino, 30 , 81 7-82 0 (1 895) .
56. Sull a inversion e degl i integral i definiti . JR. . C. Accad. Lincei (5) , 5, .
177-185 (1 896) .
57. Sull a inversion e degl i integral i multipli . R. C . Accad. Lincei (5) , 5, ,
289-300 (1 896) .
58. Sull ' inversione degl i integral i definiti . Atti Accad. Torino, 3 1 , 3 1 1 323, 400-408 , 537-56 7 an d 693-70 8 (1 896) .
59. Letter a a l Presidenta Brioschi . R. C. Acc,ad. Lincei (1 896) .
60. Osservazion i sull a not a precedent e de l Prof . Lauricell a e sopr a un a
nota d i analog o argument i dell ' Ing . Almansi. Atti Accad. Torino*
3 1 , 1 01 8-1 02 1 (1 896) .
61. Lezion i d i meccanica . Prim e nozion i d i cinematica . Livorno . Giust i
98 pp . (1 896) .
62. Sull a rotazion e d i u n corp o i n cu i esiston o sistem i policiclici . Ann.
Mat. pura appl. (2) , 24 , 29-5 8 (1 896) .
63. U n teorem a sugl i integrali multipli . Atti Accad. Torino, 32 , 859-86 8
(1897).
64. Su l principio d i Dirichlet. JR.. C. Circ. Mat. Palermo, 11,83-86 (1 897) .
65. Sopr a alcun e question i d i inversion e d i integral i definiti . Ann. Mat.
pura appl (2) , 25 , 1 39-1 7 8 (1 897) .
66. Sull e scaric a elettric a ne i ga s e sopr a alcun e fenomen i d i elettrolisi .
R. C . Accad. Lincei (5) , 6 , 389-40 1 (1 897) .
67. Sull a scaric a elettric a ne i gas . Editore R . Giusti . R o m a 1 897 .
68. Su l fenomen o dell e seiches . Nuovo' Cim., 8 , 270-27 2 (1 898) .
69. Sull e funzion i poliarmoniche . Atti 1 st. Veneto (7) , 1 0 , 233-23 5
(1898).
70. Sopr a un a class e d i equazion i dinamiche . Atti Torino, 3 3 , 451 -47 5
(1898).
71. Sull a integrazione d i un a class e d i equazion i dinamiche . Atti Torino,,
3 3 , 542-55 8 (1 898) .
72. Su r l a theori e de s variation s de s latitudes . Acta Math., Stockh., 22 *
201-296 (1 898) .
73. Su r l a theori e de s variation s de s latitudes . Astr. Gesellsch., 3 3 ,
275-329 (1 898) .
74. Sull a scaric a elettric a ne i ga s e sopr a alcun e fenomen i d i elettrolisi .
Nuovo Cim. (4) , 7 , 53-5 7 (1 898) .
75. Su i fundament i dell a teori a dell e equazion i differenzial i lineari s
(Parte seconda) . Mem. Soc. Ital. Sci. Nat. (3) , 1 2 , 3-6 8 (1 899) .
72 2
OBITUARY NOTICE S
76. Sopr a un a class e d i mot i permanent i stabili . Atti Accad. Torino, 34 ,
247-255 (1 899) ;
77. Su l flusso di energia meccanica . Nuovo Cim. (4) , I0 9 337-35 9 (1 899) .
78. Su l flusso d i energi a meccanica . Atti Accad. Torino, 34 , 366-37 5
(i899).
79. Sopr a alcun i applicazion i dell a rappresentazion e analitic a dell e
funzioni de l Prof . Mittag-Leffler . Atti Accad. Torino, 34 , 492 494 (1 899) .
80. Sopr a alcun e applicazion i dell e legg i de l flusso d i energi a meccanic a
nel mot o d i corp i di e s i attraggon o coll a legg e d i Newton . Atti
Accad. Torino, 34 , 805-81 7 (1 899) .
81. Necrologi a de l Prof. Eugenio Beltrami. Annuario Univ. Roma (1 900) .
82. Betti , Brioschi , Casorati , troi s analyste s italien s e t troi s maniere s
d'envisager le s question s d'analyse . Congres intern. Math., 43-5 7
(Paris 1 900) .
83. Sugl i integral i linear i de i mot i spontane i a caratteristich e in dependent!. Atti Torino, 35 , 1 86-1 9 2 (1 900) .
84. Su r le s equation s au x derivee s partielles . Congr. intern. Math. 377 378 (Pari s 1 900) .
85. Su i tentativi di applicazione dell e matematiche all e scienze biologich e
e sociali . Discors o lett o i l 4 novembr e 1 90 1 ali a inaugurazion e
dell' anno scolastic o nell a R. Universit a d i Roma. , 2 5 pp . (1 901 ) .
86. Su r l a stratificatio n d'un e mass e fluide e n equilibre . Acta Math.,
Stockh., 27 , 1 05-1 2 4 (1 903) .
87. Su l numer o de i component i indipendent i d i u n sistema . R. C.
Accad. Lincei (5) , 1 2 , 41 7-41 9 (1 903) .
88. Commemorazion e de l Soci o Stranier o G . G . Stokes . R. C. Accad.
Lincei (1 903) .
89. Su r le s equation s differentielle s d u typ e parabolique . C. R. Acad.
Sci. Paris, 1 39 , 956-95 9 (1 904) .
90. Relazion e su r viaggi o compiut o da l Prof . V . Volterr a pe r incaric o
avato dall a Commission e nominat a pe r i l riordinament o de l
Politecnico d i Torin o (1 904) .
91. U n teorem a sull a teori a dell a elasticita . R. C. Accad. Lincei (5) , 1 4 ,
127-137 (1 905) .
92. Sui r equilibr o de l corp i elastic i pi u volt e connessi . R. C. Accad.
Lincei (5) , 1 4 , 1 93-20 2 (1 905 ) an d Nuovo Cim. (5) , 1 0 , 361 -38 5
(1905).
93. Sull e distorsion i generat e d e tagl i uniform! . R. A. Accad. Lincei (5) ,
14, 329-34 2 (1 905) .
94. Sull e distorsion i de i solid i elasticit i pi u volt e connessi . R. C. Accad.
Lincei (5) , 1 4 , 351 -35 6 (1 905) .
VITO VOLTERR A
723
95. Sull e distorsion i de i corp i elastic i simmetrici . R. C. Accad. Lincei
(5), 1 4 , 431 -43 8 (1 905) .
96. Contribut o ali o studi o dell e distorsion i de i solid i elastici . R. C .
Accad. Lincei (5) , 1 4 , 641 -65 4 (1 905) .
97. Not e o n th e applicatio n o f th e metho d o f image s t o problem s o f
vibrations. Proc. London Math. Soc. (2) , 2 , 327-33 1 (1 905) .
98. Oper e de l Prof . Alfred o Cornu . Atti Accad. Torino (1 905) .
99. Fondazion e d i u n Politecnic o nell a Citt a d i Torino . Discors o
pronunciato i n Senato , giugn o 1 906 .
100. Sull ' equilibri a de i corp i elastic i pi u volt e connessi . Nuovo Cim. (5) ,
10, 361 -38 5 (1 905 ) ; (5) , i i » 5-20 , 1 44-1 61 , 205-22 1 an d 338 347 (1 906) .
101. Su r de s fonction s qu i dependen t d'autre s fonctions . C. R. Acad.
Scu Paris, 1 42 , 400-40 9 (1 906) .
102. L'economi a matematic a e d i l nuov o manual e de l Prof . Pareto .
G. Economisti (1 906) .
103. Nuov o studi i sull e distorsion i de i solid i elastici . R. C. Accad. Lincei
(5), 1 5 , 51 9-52 5 (1 906) .
104. Su i tentativ i d i applicazion e dell e matematich e all e scienz e bio logiche e sociali . Arch. Fisiol. (1 906) .
105. Lecon s su r Integratio n de s equation s differentielle s au x derivee s
partielles, professee s a Stockhol m (fevrier-mar s1 906 ) su r
Pinvitation d e S . M . l e R o i d e Suede . Uppsala , pp . 8 2 (1 906) .
106. Le s mathematique s dan s le s science s biologique s e t sociales . Revue
du mois (1 906) .
107. Su r l'equilibr e de s corp s elastique s multiplemen t connexes . Ann.
icole norm. (3) , 24 , 401 -51 7 (1 907) .
108. Parol e pronunziat e all e fest e giubilar i d i August o Righi . Bologn a
1907.
109. I l moment o scientific o present e e l a nuov a Societ a Italian a pe r
il Progress o dell e Scienze . Atti Soc. Ital. Progr. Sc, 3-1
4
(1908).
n o . Parol e pronunziat e a l Congress o dell a Societ a Italian a pe r i l P r o gresso dell e Scienze . Atti Soc. Ital, Progr. Sc. (1 908) .
i n . L a matematich e i n Itali a nell a second a met a de l secol o X I X .
4th Math. Congr. Rome, I , 55-6 5 (1 909) .
j 12. Sull ' applicazion e de l metod o dell a imagin i all e equazion i d i tip o
iperbolico. 4th Math. Congr. Rome, 2 , 90-9 3 (1 909) .
113. Giovann i Vailat i (Necroligia) . Period. Mat. Inseg. Sec. (3) , 6 ,
289-292 (1 909) .
114. Sull e equazion i integro-differenziali . R. C . Accad. Lincei (5) , 1 8 ,
167-174 (1 909) .
72 4
OBITUARY NOTICE S
115. Sull e equazion i dell a elettrodinamica . R. C . Accad. Lincei (5) , 1 8, .
203-211 (1 909) .
116. Parol e pronunziat e a l Congress o dell a Societ a Italian a pe r i l P r o gresso dell e Scienze . Atti Soc. Ital. Progr. Sc. (1 909) .
117. Parol e de l Presid e dell a Facolt a d i Scienze . Onoranz e a l Prof .
Cremona. (1 909) .
118. Alcun e osservazioni sopra proprieta atte ad individuare un a funzione .
R. C. Accad. Lincei (5) , 1 8 , 263-26 6 (1 909) .
119. Sull e equazioni integro-differenzial i dell a teori a dell ' elasticita. R. C.
Accad. Lincei (5) , 1 8 , 296-30 1 (1 909) .
120. Equazion i integro-differenziali dell a elasticita nel caso della isotropia..
R. C. Accad. Lincei (5) , 1 8 , 577-58 6 (1 909) .
121. Commemorazion e d i Valentin o Cerrati . R o m a 1 909 .
122. Lecture s delivere d a t Clar k Universit y ; Worcester , Mass. , 1 909 .
123. Parol e pronunziat e avant i feretr o d i Stanisla o Cannizzaro . Nuovv
Cim. (5) , I9 > 387-38 9 (1 91 0) .
124. Soluzion e dell e equazion i integro-differenzial i dell ' elasticita nel caso
di un a sfer a isotropa . R. C . Accad. Lincei (5) , 1 9 , 1 07-1 1 4
(1910).
125. Question i general i sull e equazion i integral i e d integro-differenziali .
R. C. Accad. Lincei (5) , 1 9 , 1 69-1 8 0 (1 91 0) .
126. Deformazion e d i una sfer a elastica , soggett a a date tensioni , ne l cas o
ereditario. R. C. Accad. Lincei (5) , 1 9 , 239-24 3 (1 91 0) .
127. Osservazion i sull e equazion i integro-differenzial i e d integrali . R. C,
Accad. Lincei (5) , 1 9 , 361 -36 3 (1 91 0) .
128. Sopr a l e funzion i permutabili . R. C. Accad. Lincei (5) , 1 9 , 425-437 (1910).
129. Sull e equazion i permutabili . JR. . C. Accad. Lincei (1 91 0) .
130. Espacio , tiemp o i mass a segu n la s idea s modernas . An. Soc. dent.
Argent., 70 , 223-28 3 (1 91 1 ) .
131. Sopr a un a propriet a general e dell e equazion i integral i e d integro differenziali. R. C . Accad. Lincei (5) , 20 , 79-8 8 (1 91 1 ) .
132. Parol e de l President e dell a Facolta . Onoranz e a l Prof. D e Helgnero .
Roma, 1 91 1 .
133. Equazion i integro-differenzial i co n limit i costanti . R. C . Accad.
Lincei (5) , 20 , 95-9 9 (1 91 1 ) .
134. Contribut e ali o studi o dell e funzion i permutabili . R. C. Accad.
Lincei (5) , 20 , 296-30 4 (1 91 1 ) .
135. Sopr e l e funzion i permutabil i d i 2 a speci e e l e equazion i integrali .
R. C. Accad. Lincei (5) , 20 , 521 -52 7 (1 91 1 ) .
136. Su r le s equation s integro-differentielle s e t leur s applications . Acta
Math., Stockk, 35 , 295-35 6 (1 91 2) .
VITO VOLTERR A
725
137. Lemon s su r l'integratio n de s equation s differentielle s au x d e r i v e s
partielles. Paris, 1 91 2 . (Republicatio n o f 1 05. )
138. Lecture s delivere d a t th e celebratio n o f th e twentiet h anniversar y
of th e foundatio n o f Clar k University , unde r th e auspice s o f th e
department o f physics . N e w Yor k an d London , 1 91 2 .
139. Vibrazion e elastich e ne l cas o dell a eredita . JR. . C Accad. Lincei ($) y
2 1 , 3-1 2 (1 91 2) .
140. Sull e temperatur e nel T intern o dell e montagne . Nuovo Cim., 4, .
111-126 (1 91 2) ,
141. L'evolutio n de s id£es fondamentale s d u calcu l infinitesima L Revue
du mois, 1 3 , 257-274 (1 91 2) .
142. L'applicatio n d u calcu l au x phenomene s d'heredite . Revue du mois >
I3> 556-57 4 (1 91 2) .
143. Onoranz e a l Prof. Valentin o Cerrati . Roma , 1 91 2 .
144. Henr i Poincar e : L'oeuvr e mathematique . Revue du mois, 1 5 *
129-154 (1 91 3) .
145. Lemon s su r le s equation s integrale s e t le s equation s integro differentielles, profess& s a l a Facult e de s science s d e R o m e e n
1910, pubise s pa r M . Tomassett i e t F . S . Zarlatti . Collectio n
Borel, pp . 1 6 4 (1 91 3) .
146. Lefon s su r le s fonctions d e lignes , professees a l a Sorbonn e e n 1 91 2 ,
recuellies e t redig& s pa r J . Peres . Collectio n Borel , pp . 23 0
(1913).
147. Sopr a equazion i integro-differenzial i avent i i limit i costanti . R. C.
Accad. Lincei (5) , 22 , 43-4 9 (1 9*3) 148. Su i fenomen i ereditarii . R. C. Accad. Lincei (5) , 22 , 529-53 9
(19x3).
149. Sopr a equazion i d i tip o integrale : Proc. $th intern. Congr. Math., I *
403-406 (1 91 3) 150. Som e integra l equations . Bull. Amer. Math. Soc. (2) , 1 9 , 1 70-1 71 :
(1913).
151. Onoranz e a l Prof . Dott . G . B . Guccia . R. C. Circ. Mat. Palermo*
(1914).
152. Le s probleme s qu i ressorten t d u concep t d e fonction s d e lignes. .
S. B. berl. math. Ges., 1 3 , 1 30-1 5 0 (1 91 4) 153. Osservazion i su i nuclei dell e equazion i integrali . R. C. Accad. Lincei
(5), 2 3 , 266-26 9 (1 91 4) .
154. Henr i Poincare . Nouvelle Collection Scientifique. Paris , 1 91 4 .
155. Sull e equazion i all e derivat e funzionali . R. C. Accad. Lincei (5) , 23, .
393-399 (1 91 4) .
156. Equazion i integro-differenzial i e d equazion i all e derivat e funzional L
R. C. Accad. Lincei (5) , 2 3 , 551 -55 7 (i9*4) 46
726
OBITUARY NOTICE S
157. Dre i Vorlesunge n iibe r neuer e Fortschritt e de r mathematische n
Physik, gehalte n i n Septembe r 1 90 9 a n de r Clar k Universitat .
Deutsch vo n Erns t Lamia . Arch, Math. Phys., Lpz. (3) , 22 ,
97-181 (1 91 4) . Als o Leipzig , pp . 8 4 (1 91 4) 158. Th e theor y o f permutabl e functions . Princeton , 1 91 5 .
159. Henr i Poincare . Rice Inst. PamphL, 1 , 1 33-1 6 2 (1 91 5) 160. Sull e corrent i elettrich e i n un a lamin a metaUic a sott o l'azion e d i
un camp o magnetico . Nuovo Cim. (1 91 4 ) an d R. C. Accad.
Lincei (5) , 24 , 220-234 , 289-303 , 378-39C an d 533-54 3 (1 91 5) .
161. JVtetod i d i calcol o degl i element i d i tir o dell ' artiglieri a aeronautica .
R. C. Inst. Centr. Aero. Roma, 1 91 6 .
162. Teori a dell e potenz e de i logaritmi e dell e funzioni d i composizione *
Mem. Accad. Lincei (5) , 1 1 , 1 67-26 9 (1 91 6) .
163. Th e generalisatio n o f analytic functions . Rice Inst. PamphL, 4 , no. 1 ,
53-101 (1 91 7) .
164. Relazion e sull a mission e i n Inghilterr a e d i n Francia . Roma , 1 91 7 .
165. O n th e theor y o f wave s an d Green' s method . Rice Inst. PamphL,
4, no . 1 , 1 02-1 1 7 (1 91 7) .
166. Inaugurazion e dell ' Instituto Central e d i Biologia Marina i n Messina.
Venezia, 1 91 7 .
167. Pietr o Blaserna . JR . C. Senato, Roma, 1 91 8 .
168. Relazion e dell a conferenza interalleat a sull a organisazione scientifica .
R. C. Accad. Lincei (1 91 9) .
169. L'entent e scientifique . Nouv. Revue Italie (1 91 9) .
170. L e congre s d e mathematique s d e Strasbourg . Nouv. Revue Italie
(1920).
171. Su r Tenseignemen t d e la physiqu e mathematiqu e e t quelque s point s
d'analyse. Enseign. Math., 2 1 , 200-202 (1 920) .
172. Su r l'enseignemen t d e la physiqu e mathematiqu e e t quelque s point s
d'analyse (conferenc e generale) . C. R. congr. intern, math., 81 -9 7
(1920).
173. Commemorazion e d i Augusto Righi . Atti Pari. Senato Regno Roma,
1920.
174. Function s o f composition . Thre e lecture s delivere d a t th e Ric e
Institute i n th e autum n o f 1 91 9 . Rice Inst. PamphL, J 9 no . 4 ,
181-251 (1 920) .
175. Sagg i scientific! . Bologna , 1 920 .
176. Osservazion i su l metod o d i determinar e l a velocit a de i dirigibili .
Rassegna Marittima Aero. Roma, 1 920 .
j 7 7 . G . Lippman n (Necrologia) . R. C. Accad. Lincei (5) , 30 , 388-38 9
(1921).
378. A . Roit i (Necrologia) . R. C . Accad. Lincei (5) , 30 , 47 7 (1 921 ) .
VITO VOLTERR A 7 2
7
179. Th e flow o f electricity i n a magnetic field. Berkeley , pp . 7 2 (1 921 ) .
180. Funzion i d i linee , equazion i integral i e integro-differenziali . An.
Soc. dent. Argent (1 921 ) .
181. Le s equation s au x derivee s fonctionelle s e t l a theorie d e la relativite .
Enseign. Math., 2 2 , 77-7 9 (1 922) .
182. Mouvemen t d'un e fluide e n contac t ave c u n autr e e n surface s d e
discontinuity C. R. Acad. Sci. Paris, 1 77 , 569-57 1 (1 923) .
183. Su r le s fonction s permutables . Bull. Soc. Math. Fr., 52 , 548-56 8
(1923)184. D a annunci o dell a mort e de l Soci e Corrad o Segr e e n e rimpiange
perdita. R. C. Accad. Lincei (5) , 3 3 , 459-46 1 (1 924) .
185. Lemon s su r l a compositio n e t le s fonction s permutables . Paris ,
1924.
186. D a annunci o dell a mort e de l Soci o stranier o Fusakich i Omor L
JR.. C. Accad. Lincei (5) , 3 3 , 43 (1 924) .
187. Arthu r Gordo n Webster . Worcester , Mass. , 1 924 .
188. Commemorazion e de l President e F . d'Ovidio . R. C. Accad. Lincei
(1925).
189. Variazion i e fluttuazioni de l numer o d'individu i i n speci e animal i
conviventi. Mem. Accad. Lincei, 2 , 31 -1 1 3 (1 926) .
190. Loi s d e fluctuation d e l a populatio n d e plusieur s espece s coexisten t
dans l a mem e milieu . Association Frangaise Lyon, 96-9 8 (1 926) .
191. Fluctuation s i n th e abundanc e o f a specie s considere d mathe matically. Nature, n 8 5 558-56 0 (1 926) .
192. Teori a d e lo s funcionale s y d e la s ecuacione s integrate s e integro difFerenciales. Conferencia s explicada s e n l a Faculta d d e l a
Ciencias d e l a Universitad , 1 925 , redactada s po r L . Fantappie .
Madrid, pp . 20 8 (1 927) .
193. Variazion i e fluttuazioni i n speci e animal i conviventi . R. Comit.
Talass. Italiano, Memori a cxxxi , Venezia , 1 927 .
194. Sull e fluttuazioni biologiche . JR. . C . Accad. Lincei (6) , 5 , 3-1 0
(1927).
195. Legg i dell e fluttuazioni biologiche . R. C . Accad. Lincei (6) , 5 , 61 -6 7
(1927).
196. Sull a periodicit a dell e fluttuazioni biologiche . R. C. Accad. Lincei
(6), 5 , 463-47 0 (1 927) .
197. Essa i mathematiqu e su r le s fluctuations biologiques . Bull, de la Soc.
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198. Un a teori a matematic a sull a lott a pe r Tesistenza . Scientia, 4 1 ,
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199. Loi s d e fluctuations d e la population d e plusieur s espece s coexistan t
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203. I n memori a d i Traian o Lalesco . Revista Universitara Bucarest, I ,
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204. Theor y o f functional s an d o f integra l an d integro-difFerentia L
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London, 1 929 .
205. L a teori a de i funzional i applicat a a i fenomen i ereditari . Atti Congresso Bologna, I , 21 5-23 2 (1 929) .
206. Alcun e osservazion i su i fenomeni ereditarii . R. C. Accad. Lincei (6), .
9, 585-59 5 (1 929) .
207. L a theori e de s fonctionelle s applique e au x phenomene s hereditaires .
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208. Sull e fluttuazioni biologiche . R. C. Semin. mat. jis. Milano, 3 *
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209. Sull e meccanic a ereditaria . R. C. Accad. Lincei (6) , I I , 61 9-62 5
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210. Lefon s su r l a theori e mathematiqu e d e l a lutt e pou r l a vie . Paris, .
214 pp. (1 931 ) .
211. (Wit h V. D'ANCONA) L a concurrenza vitale tra le specie nelT ambiente
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212. Ricerch e matematich e sull e associazion i biologiche . G . 1 st. ItaL
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APPENDIX B
On th e Attempt s t o Appl y Mathematic s t o th e
Biological an d Socia l Science s
The Inaugural address delivered at the formal opening of the academic
year at the University of Rome in 1 901 , published in Annuario dell'Universit a
1901-1902 and reprinted in Giornal e degl i Economisti , Serie II, vol. 23,
1901. It was also printed in French in Revu e d u Moi s 1 :1 (Paris, Soudier,
1906) and in Archivi o d i fisiologia III:2 (Firenze, Gennaio, 1 906).
Anatole France , tha t brillian t philosophe r an d novelis t an d deligh t o f so
many discerning readers, recounts the following anecdote . "Som e years ago,"
he says , " I visite d th e museu m o f natura l histor y i n a grea t Europea n cit y
with on e of its curators , wh o describe d t o m e with immens e satisfactio n th e
fossil animals . H e instructe d m e ver y wel l u p unti l th e Pliocene , bu t whe n
we foun d ourselve s i n fron t o f th e firs t trace s o f huma n beings , h e turne d
away, an d t o m y question s replie d tha t thi s wa s no t hi s displa y cabinet .
I realize d m y indiscretion : Neve r as k a scientis t abou t th e secret s o f th e
universe tha t canno t b e foun d i n hi s displa y cabinet. "
Although a mind a s keen an d paradoxica l a s that o f Anatole Pranc e wa s
bound t o conclud e fro m thi s simpl e encounte r tha t scientist s ar e th e leas t
curious peopl e i n th e worl d an d unintereste d i n wha t lie s outside thei r ow n
display cabinets , w e will be careful no t t o dra w th e sam e conclusio n bu t wil l
instead conside r th e episod e a n exampl e o f th e natura l an d ofte n justifie d
reluctance o f scholar s t o discus s idea s an d mak e statement s outsid e thei r
own field .
Among scientist s th e curiosit y t o rang e fa r an d wid e i s very strong , a s is
their desir e to rummag e throug h th e displa y cabinet s o f their peer s i n orde r
to bette r gaug e th e valu e o f thei r own . Sometime s a collegia l stocktakin g
overcomes whateve r constraint s Anatol e Prance' s frien d fel t i n fron t o f a
stranger. An d suc h curiosit y i s muc h greate r i n thos e devote d t o th e stud y
of mathematic s tha n i n thos e engage d i n othe r disciplines .
The mathematicia n possesse s a wonderfu l an d valuabl e tool , fashione d
over th e centurie s b y th e cumulativ e effort s o f th e keenes t mind s tha t eve r
lived. H e has , s o t o speak , th e ke y tha t open s a passag e t o th e hidde n
mysteries o f th e universe—th e mean s t o summarize , i n a fe w symbols , a
synthesis embracin g an d connectin g th e vas t an d disparat e result s o f man y
sciences.
247
248 B . MATHEMATIC S APPLIE D T O TH E BIOLOGICA L AN D SOCIA L SCIENCE S
While he devotes his life and hi s mental power s to refining an d perfectin g
his tools, to fit the m fo r th e subtlest inquir y an d a n ever greater understand ing o f th e facts , h e i s continuall y badgere d b y a growin g wav e o f scholars ,
who be g hi m fo r hel p an d frequentl y fo r mor e tha n h e ca n give . Onl y th e
rarest o f spirit s ca n roa m th e spher e o f number s an d th e abstraction s o f
geometry an d logic , remainin g indifferen t t o an d apar t fro m al l tha t live s
and move s around the m an d laborin g purely fo r the glory of human thought .
Instead, i t i s mor e natura l t o brea k ou t o f th e circl e o f pur e mathematica l
analysis—to gathe r information , t o compar e th e result s of the various meth ods a t you r disposal , t o classif y the m i n preparation fo r thei r application , t o
direct the m towar d thei r prope r use , to improv e the mos t useful , t o reinforc e
the weakest , t o creat e method s eve n mor e powerful .
But tha t curiosit y i s most intens e i n the biologica l an d socia l sciences —
which mathematic s ha s onl y recentl y begu n t o penetrate—becaus e o f a
strong desir e t o fin d ou t whethe r th e classi c method s tha t produce d suc h
wonderful result s i n th e mechanical-physica l science s ca n b e applie d wit h
equal succes s t o thos e ne w an d unexplore d fields .
Succumbing t o th e desir e t o giv e th e impressio n tha t a mathematicia n
can entertai n thes e matters , a s compare d wit h th e classi c application s o f
mathematics, I wil l allo w mysel f t o abando n th e are a o f m y ow n studie s
in orde r t o touc h o n a field—i n a ver y limite d way—whic h i s closel y tie d
to th e majo r problem s o f th e philosoph y an d histor y o f scienc e an d i s it self extensive . Indeed , t o trac e an d compar e th e paths , ol d an d new , tha t
mathematics ha s taken a s it infiltrated th e various branches of science, gauging it s effect s an d counte r effects , examinin g i n dept h th e mutua l relation s
thus engendered , whil e offering a grand panoram a an d super b synthesi s o f a
great par t o f th e complete d wor k o f huma n though t an d providin g a guid e
to futur e progress—al l thi s woul d b e a hug e endeavo r an d quit e beyon d m y
powers.
First o f all , i t i s important t o clarif y a subtl e poin t concernin g ou r sub ject. Som e people expec t to o littl e fro m mathematics , other s to o much ; thi s
accounts fo r th e suspicion s o f the on e an d th e hyperenthusias m o f the othe r
toward it s ne w applications . I f i t i s true , a s th e sayin g goes , tha t yo u ge t
out o f somethin g onl y wha t yo u pu t int o it , an d tha t analysi s add s nothin g
essential t o th e postulate s formin g th e basi s o f mathematics , i t i s als o tru e
that mathematic s i s the roya l road t o arrivin g a t genera l laws and th e sures t
guide t o formin g ne w hypotheses—tha t is , t o refinin g an d perfectin g thos e
same postulate s tha t for m th e basi s fo r ever y singl e treatment—indeed , i t
offers th e mos t precis e wa y t o tes t them , t o brin g the m ou t o f th e real m o f
abstraction int o that o f reality. I n truth, ther e i s no better too l than th e cal culus wit h whic h t o mak e accurat e estimate s o f the long-ter m consequence s
of dat a obtaine d fro m observatio n an d experiment .
But th e histor y o f scienc e reveal s a mor e direc t contributio n o f math ematics t o th e perceptio n an d understandin g o f nature . Whe n w e us e th e
B. MATHEMATIC S APPLIE D T O TH E BIOLOGICA L AN D SOCIA L SCIENCE S 24 9
calculus t o establis h th e exac t cours e o f tw o apparentl y differen t phenom ena, we often find a n identity, or , a s we say, we find they ar e governed b y th e
same equations—i t i s a singl e ste p t o conclud e tha t th e tw o constitut e tw o
representations o f a singl e phenomenon . Thi s wa s th e procedur e Maxwel l
followed t o reac h th e conclusio n tha t electromagneti c perturbation s ar e th e
same as light—a momentou s discover y opening the wa y to the work of Hert z
which has so influenced moder n physic s and inspire d th e practical invention s
of Ferrari s an d Marconi .
None ca n predic t t o wha t vas t horizon s th e geomete r wil l b e le d o n th e
narrow an d thorn y pat h o f th e calculus . Di d Lagrang e suspect , whe n h e
conceived o f analytica l mechanics , no t onl y tha t h e was creatin g a powerfu l
tool an d sur e guid e fo r al l th e mos t difficul t problem s o f th e scienc e o f
motion an d equilibriu m bu t tha t hi s formula s woul d on e da y become—i n
the hand s o f me n o f geniu s lik e Maxwel l an d Helmholtz—s o comprehensiv e
as t o embrac e an d dominat e al l th e phenomen a o f the physica l world ?
Nevertheless, whil e suc h i s th e importanc e o f analysis , on e mus t als o
consider it s limitations. Unfortunately , professiona l mathematician s ar e separated fro m th e res t o f the worl d b y a barrier o f symbols , whic h len d a n ai r
of myster y t o thei r work—s o muc h s o tha t thos e no t priv y t o th e secret s
of algebr a an d th e calculu s sometime s labo r unde r th e delusio n tha t thei r
methods ar e o f a differen t natur e fro m thos e o f everyda y reasoning . Man y
people make a similar error with regard to the capability of machinery whos e
mechanisms ar e hidden .
Well, th e ga p i s no t a s grea t a s i t migh t first appea r betwee n th e crud e
reasoning enablin g thos e ignoran t o f th e calculu s t o determin e th e cours e
of certai n phenomen a an d th e mechanis m o f the force s governin g the m an d
the subtl e reasonin g o f the geometer , wh o derives i n the sam e cas e a precis e
result fro m a convolute d se t o f algebrai c symbol s i n a manne r ofte n aston ishing eve n t o thos e traine d i n an d accustome d t o analytica l argument . O n
the contrary , i f we examine the matte r carefully , w e see that th e subtle r pro cess is nothing mor e than th e crud e process perfected an d refined ; moreover ,
in th e min d o f th e geomete r th e crud e reasonin g precede d an d guide d th e
calculation, pointin g th e wa y t o g o an d suggestin g wha t t o attempt . I n a
sense, i t represent s th e framewor k fro m whic h th e analytica l edific e i s built .
But whe n w e see the complete d work , w e are confronte d wit h a magnificen t
monument strippe d o f all its scaffolding an d supports. Th e props that serve d
to suppor t th e dom e durin g constructio n hav e vanished , an d i t appear s t o
the astonishe d ey e o f th e beholde r a s a miracl e o f construction .
Therefore, i t i s neither wit h undu e hopes , no r ofte n dangerou s illusions ,
but als o withou t indifference , tha t w e ough t t o welcom e ne w attempt s t o
apply th e calculu s t o whateve r specie s o f phenomena .
The transitio n o f scienc e fro m wha t I wil l cal l it s pre-mathematica l er a
to th e mathematica l er a i s characterize d thus : Element s unde r stud y ar e
examined i n a quantitativ e rathe r tha n a qualitativ e way ; i n thi s transition ,
definitions tha t sugges t a n ide a o f th e elements , i n a mor e o r les s vagu e
250 B . MATHEMATIC S APPLIE D T O TH E BIOLOGICA L AN D SOCIA L SCIENCE S
picture, giv e wa y t o thos e definition s o r principle s tha t determin e them ,
offering instea d a way t o measur e them .
What importanc e i n Newtonia n mechanics , fo r example , ca n th e prim itive notio n o f force—a s expresse d b y th e definitio n "Forc e i s th e caus e o f
motion"—have whe n compare d wit h th e firs t tw o laws , whic h essentiall y
provide a wa y t o measur e it ? S o littl e tha t i n som e moder n attempt s t o
reformulate mechanics , thi s word—"force, " th e las t verba l residu e o f per sonification i n th e inanimat e world—ca n b e don e awa y wit h an d replace d
with th e component s that , combined , giv e it s magnitude .
In ligh t o f this classi c example , an d man y analogou s one s I coul d easil y
cite, w e joyfully salut e Galton' s attempt s t o measur e quantitativel y certai n
elements of the theory of evolution, suc h as heredity an d variation. 1 Wherea s
Galton ha s taken onl y the firs t ste p on this path, an d we may hav e to accep t
the criticism s o f hi s result s an d revis e muc h o f wha t h e ha s done , w e als o
must recogniz e tha t a ne w da y ha s dawne d wit h th e ris e o f the method s h e
pioneered.
However, t o translat e natura l phenomen a int o arithmetica l o r geometri cal language i s to ope n a new avenu e fo r mathematic s rathe r tha n simpl y t o
apply th e tools o f analysis. T o study th e law s of the variatio n o f measurabl e
entities, t o idealiz e thes e entities , t o stri p the m o f particula r propertie s o r
attribute som e propert y t o them , t o establis h on e o r mor e elementar y hy potheses tha t regulat e thei r simultaneou s an d comple x variation—al l thi s
marks th e momen t whe n w e lay th e foundation s o n whic h w e ca n erec t th e
entire analytica l edifice . An d i t i s then tha t w e see the brillianc e an d powe r
of th e tool s tha t mathematic s offer s i n abundanc e t o thos e wh o kno w ho w
to us e them .
Although politica l economics , fo r example , i s onl y a t th e beginnin g o f
this path , it s practitioner s hav e experimented wit h ho w simply i t represent s
—as i n a picture—the mechanism s connectin g th e element s o f the economi c
world, an d ho w algebrai c computatio n expresse s th e degre e o f chang e o f
each wit h th e others , o r accordin g t o th e condition s i n whic h the y fin d
themselves; wherea s pre-mathematica l economic s neve r achieve d th e tota l
picture becaus e i t wa s force d t o examin e eac h o f those relation s b y itself , i n
isolation fro m th e others .
To manipulate concept s s o that w e can introduc e measurement ; t o mea sure them; t o deduce laws; from thos e to work backward t o hypotheses; fro m
those t o deduce , thank s t o analysis , a scienc e o f idealize d entitie s bu t stil l
a rigorousl y logica l one ; to the n compar e thi s scienc e t o reality ; t o rejec t o r
modify, a s contradiction s ar e foun d betwee n th e result s o f calculation s an d
the rea l world , th e fundamenta l hypotheses ; an d thereb y t o arriv e a t ne w
facts an d analogies , o r t o argu e fro m th e presen t wha t th e pas t wa s an d
1
Natural Inheritance, b y Franci s Galton , Londo n 1 889 . I n a shor t bu t interestin g
article b y [Charles ] Davenpor t i n whic h th e theor y o f thes e studie s i s explaine d (Science,
N. S. , XII : 31 0) , h e observe s tha t Galto n wa s le d t o hi s researc h mainl y b y th e wor k o f
[Adolphe] Quetelet .
B. MATHEMATIC S APPLIE D T O TH E BIOLOGICA L AN D SOCIA L SCIENCE S 25 1
what th e futur e wil l be—thus d o I summarize a s briefly a s possible the birt h
and evolutio n o f a scienc e wit h a mathematica l character .
The pat h i s long an d rugge d an d strew n wit h difficulties . Conside r tha t
whereas ancien t trace s o f huma n civilizatio n attes t t o astronomica l mea surements mad e b y primitiv e people , celestia l mechanic s itsel f i s only thre e
centuries old . Wha t a wonder , therefore , ar e th e results—eve n i f limite d
compared wit h ou r ambitiou s hope s an d demands—tha t th e calculu s ha s
been abl e t o obtai n i n thos e science s tha t onl y yesterda y wer e i n th e pre mathematical perio d an d ar e stil l strugglin g t o escap e it .
But le t us see these analytica l method s in action, an d appreciat e them a s
they tackle these ne w problems. Amon g the physica l sciences , on e disciplin e
has alway s led , whil e th e other s hav e graduall y followed , imitatin g i t an d
taking i t a s example. Thi s scienc e i s mechanics, an d i t constitutes , togethe r
with geometry , i f no t th e mos t brillian t the n surel y th e mos t dependabl e
and secur e body o f knowledge i n which the huma n min d glories . No t amon g
the biologica l bu t amon g th e socia l sciences , w e ca n identif y a discipline —
pure economics—tha t i s shape d b y mechanics , ha s use d it s procedure s an d
methods, an d ha s achieve d simila r results .
Mechanics, lik e al l th e othe r physica l science s an d lik e economics , owe s
its succes s t o th e method s o f th e infinitesima l [calculus] , th e mos t subtl e
and a t th e sam e tim e mos t powerfu l analytica l too l eve r conceived . Th e
fundamentals o f infinitesima l analysi s ar e no t b e easil y o r briefl y explained ,
even i f w e stri p i t o f nonessential s an d la y bar e th e skeleto n o f thi s super b
and nobl e edific e tha t s o many geniuses , fro m Archimede s t o Newton , hav e
constructed. S o I wil l no t try . I wil l sa y onl y tha t natura l phenomen a o f
whatever kin d displa y a n obviou s complexity: Wha t i s happening no w is the
result o f al l tha t ha s occurre d i n th e past ; change s tha t tak e plac e a t on e
point i n spac e depen d upo n an d ar e relate d t o thos e tha t occu r everywher e
else.
To discover al l those hidden ties whose consequences ar e so apparent—t o
take al l this i n a t a glance, to maste r it—seem s a t first no t onl y difficul t bu t
impossible, ye t i t i s a necessar y tas k i f w e wan t t o for m a complet e ide a o f
a phenomenon . Ho w d o the method s o f the infinitesima l [calculus ] extricat e
us fro m suc h a tangle , whic h beset s u s o n al l side s an d seem s t o chok e of f
any avenu e o f escape ?
Let u s imagin e a sequenc e o f event s i n a n infinitel y shor t tim e an d a n
infinitely smal l space . I t the n become s possibl e t o distinguis h th e dominan t
changes o f variables fro m thos e tha t ar e comparativel y negligible , an d i f we
can measur e th e former , o r establis h relation s betwee n them , w e ca n the n
use thos e dat a t o wor k backwar d fro m wha t happen s a t a certai n instan t
in a particula r plac e t o wha t wil l happen , a s tim e goe s by , everywhere—o r
wherever th e elementar y law s obtain . Establishin g thos e elementar y law s
is known a s settin g dow n differentia l equations ; workin g throug h the m ste p
by ste p an d calculatin g eac h elemen t i s known a s integratin g th e equations .
The geomete r ca n perfor m th e latte r operatio n eve n i f h e ignore s (a s ofte n
252 B . MATHEMATIC S APPLIE D T O TH E BIOLOGICA L AN D SOCIA L SCIENCE S
happens) th e specifi c proble m t o whic h hi s formula s wil l b e applied ; jus t
as th e obscur e an d humbl e miner , dee p i n th e bowel s o f th e earth , enriche s
humanity wit h a wealt h o f energy , unawar e tha t th e fue l h e laboriousl y
quarries fro m th e groun d wil l powe r a factory , illuminat e ou r night s wit h a
thousand flames , o r prope l a shi p throug h distan t seas .
Thanks t o th e infinitesima l calculus , w e can , fo r instance , plo t th e mo tions o f the heavenl y bodies , expoun d th e la w o f the harpstring' s vibration ,
and calculat e th e effect s o f th e mos t powerfu l machines , an d i t i s wit h jus t
such method s tha t th e differentia l equation s o f economic s wer e formulated .
A comparison betwee n mechanic s an d economic s is easily made. Let' s imag ine wha t impression s a practitione r o f mechanic s migh t ge t fro m th e stud y
of economics. 2
The concep t o f Homo economicus, whic h ha s prompte d s o much discus sion an d provoke d suc h enormou s difficult y tha t ther e ar e stil l thos e wh o
refuse t o accep t it , comes s o naturally t o ou r mechanis t tha t h e i s surprise d
at th e suspicion s arouse d b y thi s abstract , schemati c being . H e see s i n
Homo economicus a concep t simila r t o thos e that , b y lon g habit , hav e be come familia r t o him . H e i s use d t o idealizin g surface s a s frictionless , wire s
as inextensible , solid s a s undeformable , an d t o substitutin g perfec t liquid s
and gase s for th e natura l kind . No t onl y has he made a habit o f all this, bu t
he als o know s th e advantage s o f doin g so .
If th e mechanis t look s a littl e further , h e realize s tha t i n hi s science , a s
in economics , everythin g i s reduce d t o a n interpla y o f tendencie s an d con straints, th e latte r limitin g th e former , whic h reac t b y generatin g tensions .
Sometimes equilibriu m arises , sometime s motion—s o ther e ar e static s an d
dynamics i n bot h sciences .
We have alread y mentione d th e concep t o f force i n mechanics ; i t ha s de scended fro m th e height s o f metaphysics t o th e plai n o f measurable entities .
Similarly, i n economics , w e n o longe r speak , a s Jevon s did , o f th e mathe matical expression o f unmeasurable quantities. 3 Pareto , instead o f beginning
directly fro m th e notio n o f "ophelimity, " a s h e did i n hi s Course of Political
2
Compare Principi di economia pura b y Maffe o Pantaleoni , Firenze , 1 894 , an d hi s
Scritti vari di economia, Palerm o 1 904 . Se e Mathematical Investigations in the Theory
of Value and Prices, b y Dr . Irvin g Fishe r (Trans. Connecticut Academy, IX , Jul y 1 892) .
An expositio n o f th e mechanica l mode l imagine d b y Fishe r wa s mad e b y Colone l [En rico] Baron e i n vol . VIII , Seri e 2 a o f Giomale degli Economisti. I n th e Encyklopddie
der Mathematischen Wissenschaften (Leipzig , Teubner) , [Vilfredo ] Paret o publishe d a n
interesting articl e i n 1 902 : Anwendungen der Mathematik auf Nationaloconomie, i n whic h
the fundamenta l ideas , th e differen t theorie s an d th e principa l result s o n thi s subjec t ar e
summarized an d accompanie d b y a ric h bibliography .
3
The Theory of Political Economy b y W . Stanle y Jevons , Londo n 1 888 . I t i s interest ing t o follo w th e evolutio n o f Jevons ' ideas , whic h ca n b e linke d t o thos e o f Laplac e an d
Bernouilli and , accordin g t o Pantaleoni , t o Jevons ' studie s unde r [Augustus ] D e Morgan' s
supervision (cf . Contributo alia teoria del riparto delle spese pubbliche, enclose d i n Scritti
vari di Economia politica, mentione d above) .
B. MATHEMATIC S APPLIE D T O TH E BIOLOGICA L AN D SOCIA L SCIENCE S 25 3
Economy* suggest s tha t w e begi n fro m purel y quantitativ e concepts , wit h
his indifference curves , which correspond s o well to the standar d leve l curve s
and equipotentia l surface s o f mechanics. 5
Molecular an d atomi c theorie s lead u s to understan d th e innermos t con stitution o f bodies a s discontinuous : Lame , Cauchy , an d other s wh o define d
the mathematica l theor y o f elasticity—whic h demonstrate s it s importanc e
and practica l application s daily—succeede d onl y when they passed, i n a true
stroke o f genius , fro m th e discontinuou s t o th e continuous . Now , lik e the m
and lik e Fourie r wit h th e theor y o f heat , th e economist s assum e tha t th e
quantities o f goods a t one' s disposal , whic h ar e by nature discrete , may var y
continuously.
Finally, ou r mechanis t recognize s i n th e logi c o f economi c equilibriu m
the sam e reasonin g h e uses to establis h th e principl e o f virtual work . Whe n
he i s confronte d b y th e differentia l equation s o f economics , h e immediatel y
wants to appl y to them th e proven integral method s h e knows so well.6 Thu s
we see a discipline belonging to the [socia l sciences] that, whil e preserving it s
originality, is progressively assimilating the tools of mathematics—and whic h
in th e shor t perio d fro m th e appearanc e o f work s b y Whewell , Cournot ,
Gossen an d Walra s t o th e presen t day , ha s attempte d t o appl y th e idea s of
mathematics. 7
The applicatio n o f mathematic s t o th e biologica l science s appear s t o
be i n it s initia l stages , eve n thoug h interes t i n i t grow s da y b y day . I t i s
true tha t rathe r recentl y a so-calle d "biomechanical " schoo l wa s founded ,
4
Pareto, Vilfredo . Cours d'Economie Politique Prof esse a VUniversite de Lausanne.
Lausanne, 1 896 .
5
Summary o f chapters of a new treatise on political economy by Prof. Pareto , Giornale
degli Economisti, II XI, XX. - Se e als o the Encyklopadie der Math. Wiss § § 3, 4- an d th e
appendix t o Manuale di Economia politica b y Paret o (Milano , Societ a Editric e Libraria ,
1906).
6
See Amoroso , Luigi , Sulle analogie fra Vequilibrio meccanico e Vequilibrio economico
(Modena, 1 91 0 ) - Contributo alia teoria matematica della dinamica economica. (Roma ,
1912).
7
The oldes t paper s o n politica l economic s b y thes e author s are : Whewell ,
William, "Mathematica l expositio n o f som e doctrine s o f politica l economics, " Cam bridge, Phil. Trans.,
VIII , 1 82 9 ; Cournot , Antoin e Augustin , Recherches sur les
Principes Mathematiques de la Theorie des richesses, 1 83 8 ; Gossen , Herman n Heinrich ,
Entwickelung der Gesetze des menschlichen Verkehrs und der darausfliessenden Regeln
furmenschlichen Handeln, Brauschweig , 1 854 ; Walras , Leon , Elements d'economie politique pure ou theorie de la richesse sociale, Paris , 1 874 . Leo n Walra s i s the so n of Antoin e
Auguste Walras , als o a n economis t an d th e autho r o f De la nature de la richesse et de
Vorigine de la valeur. Paris , 1 831 . I n orde r t o find th e oldes t trace s o f th e idea s an d
principles o f mathematica l econom y i t i s necessar y t o g o bac k t o Giovann i Cev a (bor n i n
1647 o r 1 648) , mathematicia n an d hydrauli c engineer . Th e titl e o f hi s economi c wor k i s
De re numeraria, Mantova , 1 71 1 . Compar e th e articl e b y Pantaleon i o n Giovann i Cev a i n
the Dictionary of Political Economy, edite d b y R.H . Ingli s Palgrave , London , 1 894 .
254 B . MATHEMATICS APPLIE D T O TH E BIOLOGICA L AN D SOCIA L SCIENCE S
but i t doe s no t see m t o exhibi t characteristic s tha t woul d indicat e th e tru e
beginning o f a mathematica l era. 8
There ar e als o som e branche s o f physiology , suc h a s physiologica l op tics an d physiologica l acoustics , t o whic h me n suc h a s Helmholt z hav e con tributed thei r largel y mathematica l training. 9 Ther e i s als o wha t migh t b e
called a physiological thermodynamics: 10 th e classic studies of the circulatio n
of th e blood—tha t is , o n th e motio n o f fluids i n th e elasti c an d contractil e
vessels; the mechanical-physiologica l studie s o n walking, runnin g an d jump ing;11 and man y other s I will no t bothe r t o mention . I n al l o f these, th e us e
of th e calculu s i s ver y advance d an d ha s produce d ver y usefu l results ; bu t
those admirabl e an d matur e investigation s see m t o belon g t o th e variou s
branches o f mathematical physic s an d mechanic s rathe r tha n constitutin g a
new field i n whic h mathematic s ha s foun d a n origina l application .
Leaving them asid e for thi s reason, w e come, of course, to those fledgling
endeavors tha t ar e tacklin g ne w problem s pertainin g t o biology . Thei r re sults hav e no t ye t reache d th e leve l o f reliabilit y o f th e investigation s jus t
mentioned; the y stil l rais e som e doubts , but , i f onl y fo r thi s reason , the y
arouse ou r curiosit y al l th e more . The y concer n th e problem s o f classifica tion an d evolution—problems , moreover , s o closel y relate d tha t i n geneti c
theories the y ten d t o depen d o n eac h other .
It i s obviou s o n onl y cursor y inspectio n tha t th e mathematica l studie s
begun i n thi s field exhibi t al l th e characteristic s o f th e first stage s o f scien tific research—tha t is , o f a perio d o f orientation ; i n th e end , w e find thes e
studies to b e dominate d b y the metho d o f mathematical analog y an d statis tical method s base d o n th e calculatio n o f probabilities an d o n th e theor y o f
errors. Moreover , th e researches of the so-called biometrica l schoo l are indistinguishable fro m th e classi c statistica l studie s typica l o f social phenomena .
The metho d o f analog y i n mathematica l physic s i s certainl y no t new .
Nowadays, w e hav e abandone d man y illusion s abou t givin g a mechanica l
8
See Roux , Vilhelm , Gesammelte Abhandlungen iiber Entwickelungsmechanik der Organismen, Leipzig , 1 895 .
9
Helmholtz, Hermann , Handbuch der phisiologischen Optik. Hamburg , 1 894 . - Die
Lehre von den Tonempfidungenals physiol. Grundlage fur die Theorie der Musik. Braunschweig, 1 877.
10
Compare Les transformations d'energie dans Uorganisme, b y Andr e Broc a (Rap ports presente s a u congre s internationa l d e Physiqu e reun i a Pari s e n 1 900 ) t . Ill , Paris ,
1900.
11
Theorie der durch Wasser oder andere inkompressibele Fliissigkeiten in elastischen
Rohren fortgepflansten Welle, b y Wilhel m Weber . (Bericht e d.k . Sachs . Ges . D . Wiss .
Math. Phys . Klass e XVIII , 1 866) . Compar e th e pape r b y E . H . Weber , Ueber die
Anwendung der Wellenlehre auf die Lehre von Kreislaufe des Blutes und insbesondere auf
die Pulslehre. Ibid., 1 850 . Mechanik der menschlichen Gehwerkzeuge. Eine anatomischphysiologische Untersuchung b y W . WEBE R an d E . WEBER , 1 836 . Thes e studie s wer e
preceded b y a lon g serie s o f works , amon g whic h th e dee p researc h b y [Giovann i Alfonso ]
Borelli i s especiall y memorabl e (De motu animalium. Roma , 1 630) . Abou t th e so-calle d
school heade d b y Borelli , se e fo r exampl e The History of Medicine, b y [Kurt ] Sprengel .
Venezia, 1 81 4 .
B. MATHEMATIC S APPLIE D T O TH E BIOLOGICA L AN D SOCIA L SCIENCE S 25 5
explanation o f th e universe . I f w e ar e n o longe r confident o f explainin g al l
physical phenomen a b y laws like that o f universal gravitation , o r b y a singl e
mechanism, w e substitut e mechanica l model s fo r thos e collapse d hopes —
models tha t ma y no t satisf y thos e wh o loo k fo r a ne w syste m o f natura l
philosophy bu t d o suffic e fo r thos e who , mor e modestly , ar e satisfie d b y
analogy, an d especiall y mathematica l analogy , tha t somewha t dissipate s th e
darkness enshroudin g s o man y phenomen a o f nature .
A mechanica l mode l o f a phenomeno n i s a simpl e devic e designe d onl y
to bea r som e sor t o f relationshi p t o th e phenomenon ; th e sol e conditio n
is tha t whe n i t operates , it s part s wil l move , o r change , accordin g t o th e
same law s tha t gover n chang e i n th e correspondin g variabl e element s o f th e
phenomenon: Thes e elements are assumed to be its fundamental parameters .
Experience show s that suc h model s hav e bee n useful , an d stil l are , t o orien t
us i n th e newes t an d darkes t scientifi c fields, wher e w e ar e gropin g t o find
our way .
We therefor e welcome , wit h th e sam e interes t tha t marke d th e accep tance of the mechanical models of electrical induction an d the thermal cycle 1 2
by Maxwel l an d Boltzmann , th e bol d attemp t o f our celebrate d astronome r
Schiaparelli t o buil d a geometrical mode l fo r th e stud y o f organic form s an d
their evolution; 1 3 al l th e mor e so , becaus e i t woul d no t b e to o difficul t t o
transform Schiaparelli' s mode l fro m geometrica l t o mechanical , makin g i t
even mor e intuitive .
We mus t distinguis h tw o part s i n th e wor k o f th e Italia n astronome r i n
order t o bette r understan d it : On e concern s th e prope r geometrica l repre sentation o f variations i n the organic world, th e othe r relate s to a hypothesi s
that, i f no t intrinsicall y new , i s a t leas t expresse d i n a ne w form , becaus e
the autho r ha s applie d hi s model , puttin g i t immediatel y t o th e test .
Even thos e scarcel y acquainte d wit h th e mos t elementar y notion s o f
geometry kno w tha t line s ar e classified : W e have , fo r example , th e straigh t
line, the circle , an d th e curve s of the famil y o f conical sections, comprise d o f
the ellipse , th e hyperbola , an d th e parabola . Schiaparell i trie d t o establis h
a paralle l betwee n th e classificatio n o f curve s i n a famil y an d an y syste m o f
organic entitie s sharin g certai n character s an d belongin g t o th e sam e group ,
be i t order , class , o r kingdom . Th e curve s an d th e entitie s bot h follo w a la w
of correlatio n betwee n thei r parts , suc h tha t eac h depend s o n th e valu e o f
certain parameter s tha t ca n b e expresse d a s a point—s o passag e fro m on e
form t o anothe r ca n b e characterize d b y th e movemen t o f thi s point .
If w e accep t tha t th e natur e o f organi c entitie s i s describe d b y analo gous parameters , Darwin' s speciatio n hypothesi s finds a mode l i n a simila r
movement, closel y correspondin g t o th e la w o f natura l selectio n base d o n
12
Compare Antoni o Garbasso , Fisica d'oggi, filosofia di domani (Milan o 1 91 0 ) - Vorlesungen ilber Maxwells Theorie der Elektricitdt und des lichtes, b y Dr . Ludwi g Boltz mann, Leipzig , 1 891 .
13
Studio comparativo tra le forme organiche naturali e le forme geometriche pure, b y
Prof. [Giovanni ] Schiaparelli , Milano , Hoepl i 1 898 .
256 B . MATHEMATICS APPLIE D T O TH E BIOLOGICA L AN D SOCIA L SCIENCE S
the struggl e fo r existence . However , Schiaparell i recognize s i n bot h th e
inorganic an d organi c world a general la w that lead s him to modif y th e Dar winian structur e b y addin g a new hypothesis, wit h whic h h e arrive s a t wha t
he calls the principl e o f controlled evolution , o r fixed types. I n the inorgani c
kingdom, h e see s a marke d tendenc y emergin g fro m th e genera l rang e o f
phenomena towar d creatio n o f specific, well-determine d type s quit e distinc t
from on e another , thei r serie s an d classe s proceedin g b y marke d difference s
and no t b y insensible gradations ; thi s tendency i s even more apparen t i n th e
organic world . Therefore , i n hi s geometrical scheme , h e posits a discret e se ries of points correspondin g t o the forms predestine d t o be the types of those
species that , becaus e o f a comple x o f circumstance s unknow n t o us , ar e th e
only form s possible . Accordin g t o thi s ne w hypothesis , evolutio n cease s t o
be unfettered , a s in the pur e Darwinia n theory , bu t remain s boun d b y thes e
fixed points, departur e fro m whic h woul d generat e reaction s comparabl e t o
elastic forces .
These consideration s o f on e o f th e mos t crucia l problem s t o occup y th e
human min d ar e s o closel y linke d t o th e geometrica l mode l tha t w e canno t
imagine an y wa y to expres s the m withou t recours e t o th e languag e i t offers .
This alon e woul d b e enoug h t o rende r Schiaparelli' s endeavo r worth y o f th e
highest consideration , becaus e i t i s n o smal l thin g t o offe r a languag e t o a
science, especiall y whe n it s sourc e i s the clea r sprin g o f geometry . S o man y
theories hav e died , an d mos t ar e burie d i n oblivion ; still , a vestig e o f the m
remains, indicatin g tha t the y di d no t pas s useles s int o th e earth . Tha t the y
gave birt h t o a singl e ter m o f ou r languag e i s enoug h fo r u s t o declar e tha t
the remot e glea m o f their existenc e brightens , eve n today, th e grea t torc h of
knowledge. Acros s th e centuries , somethin g o f them stil l live s an d i s of use.
Schiaparelli's work , however , doe s no t resolv e matter s bu t rathe r add s
a ne w questio n t o th e man y crowdin g th e field o f biology . Eve n th e fiercest
detractors o f th e biometrica l schoo l canno t den y tha t it s ai m i s t o respon d
to th e innumerabl e question s an d problem s arisin g fro m th e outsize d con ceptions o f Lamarck, Geoffro y Saint-Hilaire , an d Darwin—startin g wit h ob servation an d measuremen t an d discussin g [th e problems ] i n ligh t o f know n
methods alread y note d o r o f ne w method s i t i s devising . Oppositio n t o
[this work ] fault s it s application s a s perhap s to o specific , an d als o som e o f
its results—thoug h no t th e mathematica l metho d itsel f o n whic h the y ar e
founded. I t i s exactl y tha t poin t whic h w e want t o emphasiz e today. 1 4
14
See: Geor g Dumcker , Die Methode der Variations-statistik (Archiv fur Entwicklungsmechanik der Organismen b y W . Roux , VIII , 1 899) . - C.B . DAVENPORT : Statistical Methods. Ne w York, 1 899 . Compar e Las Matematicas y la Biologia, b y Ange l Gallard o
(Anales de la Sociedad Scientifica Argentina, t. II). Bueno s Ayres , 1 901 . - / metodi somatometrici in Zoologia b y G . Cattane o (Riv. Di biologia generale, Aprile-Maggi o 1 901 ) .
- Se e th e article s b y Prof . CAMERAN O i n Atti dell fAce. Di Torino 1 900-0 1 an d thos e
by Prof . Andre s i n Rend. 1 st. Lomb. 1 897-901 . - J . Ludwi g publishe d extensiv e bibli ographies o n biometrica l studie s i n volume s XLII I an d XLI X o f th e Zeitschrift fur Math.
Und Physik. Th e yea r 1 90 1 marke d th e birt h o f th e journa l Biometrika wit h th e ai m o f
gathering an d spreadin g biometrica l researc h (Biometrika. A Journal for the Statistical
B. MATHEMATIC S APPLIE D T O TH E BIOLOGICA L AN D SOCIA L SCIENCE S 25 7
Better tha n anyon e else , Pearson ha s show n wh y this ne w path wa s chosen, and nobod y ha s more clearly outlined it s scope and import. 1 5 Accordin g
to Pearson , on e mus t abandon—give n th e presen t stat e o f our knowledge —
the ide a o f a mechanism o f inheritance, an d renounc e th e hop e o f obtainin g
a mathematical relatio n betwee n an y one particular paren t an d an y one par ticular descendant . Th e cause s of natural inheritanc e i n particular case s ar e
so complex tha t the y d o not admi t a n exac t treatment . On e mus t therefor e
begin b y examinin g a grea t numbe r o f case s e n mass e an d wor k one' s wa y
down littl e b y littl e t o increasingl y limite d classes ; on e shoul d neve r estab lish genera l rule s fro m singl e examples . I n othe r words , on e mus t procee d
by statistica l methods , no t b y a consideratio n o f typica l cases . Nowaday s
this ma y perhap s see m discouragin g t o th e practica l ma n o f medicine , wh o
is mor e intereste d in , fo r example , th e hereditar y patholog y o f a particula r
family tha n i n averag e probabilitie s pertainin g t o a n entir e clas s o f people .
On th e othe r hand , i t goe s to sho w tha t i n th e stud y o f heredity , a s i n tha t
of variation, w e find ourselve s confronte d b y a hug e numbe r o f smal l cause s
all actin g simultaneously , s o that i t i s impossible t o isolat e an y on e o f them .
Thus ther e i s n o wa y t o find ou r bearing s othe r tha n t o resor t t o thos e
procedures s o manifestl y usefu l i n attackin g simila r problems : tha t is , pro cedures base d o n th e calculu s o f probabilities , tha t mos t singula r an d re markable o f mathematica l branches . I f w e wer e t o analyz e an y on e o f ou r
conscious judgments , w e would b e sur e t o find, mor e o r les s concealed , th e
calculation o f a probability. Yo u migh t sa y tha t th e simples t o f men, await ing the daw n and the rising sun, in some sense owes his faith i n the coming of
day t o a n unconsciou s applicatio n o f Bernoulli' s theore m o f larg e numbers .
However, th e theor y o f probabilit y i s th e onl y par t o f mathematic s whos e
principles are not rigorously set and are still open to criticism and discussion .
What, fo r instance , i s the soli d basi s o f th e fundamenta l propositio n o f th e
theory o f errors ? Ye t everybod y believe s it , because , a s Lippman n onc e re marked t o Poincare , th e experimentalist s imagin e tha t i t i s a mathematica l
theorem whil e th e mathematician s conside r i t a n experimenta l fact .
Whatever confidenc e w e ma y hav e i n it s basis , th e theor y o f probabil ity certainl y ha s produce d an d i s producin g incontestabl e an d incalculabl e
benefits t o al l th e sciences . Eve n enumeratin g them , lik e discussin g th e
aforementioned genera l problem s an d apparen t contradictions , woul d tak e
too long . Le t u s loo k instead , withou t goin g int o particular s an d jus t a s a n
example, a t ho w th e ne w schoo l treat s on e o f th e problem s i t ha s begu n t o
examine.
Imagine a large numbe r o f individuals o f a certain species . I f their form s
cluster aroun d a n averag e type , w e wil l find, a s w e mov e awa y fro m that ,
Study of Biological Problem. Founde d b y W . R . F . Weldon , Franci s Galto n an d Kar l
Pearson. Edite d b y Kar l Pearson , Cambridg e Universit y Press) .
15
"Mathematical Contribution s t o th e Theor y o f Evolution . Il l Regression , Heredit y
and Panmixia, " b y Kar l Pearso n (Phil. Transactions of the Royal. Society of London
S.A., CLXXXIX) , London , 1 897 .
258 B . MATHEMATIC S APPLIE D T O TH E BIOLOGICA L AN D SOCIA L SCIENCE S
fewer an d fewe r individuals . Galto n represent s thi s graphically , measurin g a
particular orga n and constructing a curve that expresse s the relation betwee n
its siz e an d th e greate r o r lesse r numbe r o f correspondin g individuals . W e
then find a lin e tha t geometer s cal l th e curv e o f erro r o r frequenc y an d
statisticians cal l th e Quetele t curve . Suc h a se t o f individual s i s known a s a
monomorphic group .
However, i t ma y b e tha t when , fo r a certai n se t o f individuals , w e con struct th e curv e as we said, we do not find a frequency line . Thi s means tha t
the individual s ar e clusterin g aroun d tw o or mor e distinc t type s rathe r tha n
one, i n which cas e the curv e can b e decompose d int o tw o or mor e frequenc y
lines. Th e se t i s then know n a s dimorphi c o r polymorphic. 1 6
The breaku p o f a polymorphi c grou p int o it s element s the n become s a
purely geometrica l question , whic h Pearso n ha s partl y resolved , an d corre sponds t o th e divisio n o f a specie s int o it s varieties. 1 7 I f w e ca n follo w suc h
a divisio n throug h time , an d observ e a grou p changin g fro m monomorphi c
to polymorphi c o r vic e versa—o r also , quit e simply , i f w e ca n discove r a
tendency t o divisio n o r to recombination—w e wil l have understood i n detai l
a fundamenta l datu m o f evolution , sheddin g unexpecte d ligh t o n question s
of variation an d regression , continuit y o r discontinuit y [i n that species] .
But ther e i s more: A frequenc y curv e whil e remainin g suc h ca n assum e
different shapes , or , a s w e say , ca n chang e it s characteristi c parameters .
Recognizing th e variation s o f parameters tha t correspon d t o a group an d it s
subgroups in successive generations an d recognizing the correlations betwee n
parameters correspondin g t o th e various organs, toda y constitut e a comple x
and extensiv e chapte r i n whic h th e subtl e insight s o f Laplac e an d Bravais 1 8
on probabilitie s find importan t applications .
In thi s way , w e ca n establis h mathematica l definition s fo r fundamenta l
elements i n th e scienc e o f heredit y an d [natural ] selection ; s o tha t thes e
concepts see m to emerg e fro m th e fo g i n which they wer e shrouded an d tak e
precise shap e i n ou r minds .
Highly interestin g result s hav e alread y bee n obtaine d i n thi s field, o n
various subjects . So , fo r example , Pearso n ha s foun d tha t mora l trait s ar e
inherited a s readil y a s physica l traits. 1 9 H e ha s als o discovere d tha t civi lized race s exhibi t mor e variabilit y tha n savag e ones. 20 Davenpor t studie d
16
Compare Materials for the Study of Variation treated with special regard to Discontinuity in the Origin of Species b y Willia m Bateson , London , 1 894 .
17
"Contributions t o th e Mathematica l Theor y o f Evolution " b y Kar l Pearso n (Phil.
Trans. Roy. Soc. London (A), CLXXXV . London , 1 895) . - Th e solutio n give n b y
Pearson i s vali d onl y fo r th e divisio n o f a dimorphi c group : Prof . D e Helguer o ha s give n
an interestin g simplificatio n o f th e Pearso n metho d (Biometrika IV , 1 , 2 Jun e 1 905) .
18
"Analyse mathematiqu e su r le s probabilites de s erreurs d e situatio n d'u n point, " b y
A. Bravai s (Memoires presentes par divers savants a I'Academic Roy ale des Sciences de
VInstitut de Franc, t . IX) . Paris , 1 846 .
19
"On th e inheritanc e o f the menta l an d mora l character s i n Ma n an d it s Compariso n
with th e inheritanc e o f th e physica l Characters. " The Huxley lecture for 1 903.
20
The Chances of Death and other Studies in evolution, 2 vol., London , Arnold .
B. MATHEMATIC S APPLIE D T O TH E BIOLOGICA L AN D SOCIA L SCIENCE S 25 9
the phylogen y an d geographica l distributio n o f certai n animals, 21 Duncke r
[studied] bilatera l symmetr y i n animals, 22 D e Vrie s vegetabl e hybrid s an d
monstrosities,23 Ludwig the specifi c character s o f various vegetable species; 24
and we could cit e a great man y othe r notabl e investigations , fo r whic h I refe r
you t o specialize d bibliographies. 25
Of th e vas t collectio n o f fact s befor e us , I trie d t o emphasiz e two , th e
most prominent , i n particular: th e great advance s made by political econom ics i n recen t times , eve r sinc e th e disciplin e tha t Descarte s an d Lagrang e
would no t hav e hesitate d t o cal l analytica l economic s becam e a scientifi c
field i n it s ow n right , an d th e eve n mor e recen t beginning s o f quantitativ e
and statistica l researc h i n biology .
The ne w economic studie s find thei r mathematica l counterpar t i n infini tesimal procedures, which economists ar e already confident i n using; an d th e
new directions i n biology find thei r counterpar t i n the law s of large number s
and in the calculation o f probabilities, tools that a whole school has adapted .
With th e first o f thes e powerfu l an d wonderfu l instruments , ou r mind s ca n
penetrate deepl y int o th e mysterie s o f th e infinitel y small ; wit h th e second ,
we take th e lon g view , i n a n attemp t t o encompas s th e vas t contour s o f a n
infinitely grea t mas s o f facts .
In th e sam e wa y tha t th e microscop e an d th e telescop e hav e reveale d
to th e histologis t an d th e astronome r tw o world s that th e nake d ey e canno t
examine, thes e mathematica l method s ope n ne w an d unknow n horizon s t o
us; like those two optical devices, these two analytical instruments ar e in par t
different, i n part quit e alike. Bu t ther e is something that render s their magi c
far mor e wonderful tha n tha t o f any conceivable system of lenses: Bot h sho w
21
"Quantitative Studie s i n th e Evolutio n o f Pecten. " Proc. Amer. Acad. Arts and
Sci. Compariso n o f som e Pecten s fro m th e Eas t an d th e Wes t Coast s o f th e U.S . Reprinted fro m th e Mar k Anniversary , 1 903 , ecc.
22
"Symmetric un d Asymmetri e be i bilaterale n Thieren " i n Arch. Entw.-mech, XVII ,
533-682.
23
"Sur l a loi de disjonctio n de s hybrides." Comptes rendus de VAc. des Sci. de Paris,
26 Marz o 1 900 . - Su r l'origin e experimenta l d'un e nouvell e espec e vegetale , ibid. , 1 900 .
- L a lo i d e Mende l e t le s caractere s constant s de s hybrides , ibid. , 2 Feb . 1 903 . -Di e
Mutationslehre; Veit , Leipzi g 1 902 , 2 vol. ecc .
24
"Beitrage zu r Phytarithmetik, " Bot. CentralbL, LXXI , 1 897 . - Uebe r Varia tionskurven. Ibid. , LXXV , 1 898 . - Variationstatistisch e Problem e un d Materialen .
Biometrika, I , 1 1 -29 , 31 6-8 , ecc.
25
In recen t years , studie s relate d t o thi s matte r hav e bee n ver y numerou s an d t o
mention the m explicitl y i s outside th e scop e o f this article . Beside s th e alread y mentione d
magazine (Biometrika, Cambridge) , se e th e Journal of Genetics (edite d b y W . Bateso n
and R . C . Punnett , Cambridge) , The Eugenics Review (London ) an d others . Th e mos t
complete sourc e fo r bibliographica l searche s i s th e International Catalogue of Scientific
Literature publishe d b y the Internat. Counci l a t th e Roya l Society of London. Se e Volumes
L. (genera l Biology) , M . (Botany) , N . (Zoology) , P . (Anthropology) , Q . (Physiology) ,
and th e chapter s Variations , Evolution , Method s an d Apparat i (weight s an d measures ,
biometrics), etc .
260 B . MATHEMATIC S APPLIE D T O TH E BIOLOGICA L AN D SOCIA L SCIENCE S
us onl y wha t i s usefu l t o see ; the y serv e t o concea l th e superfluitie s tha t
would clou d ou r vision .
I could als o mention th e hopes, perhaps th e dreams , fo r th e futur e whic h
other method s offer—suc h a s energetics , fo r example ; thes e thing s hav e no t
yet bee n successfull y teste d i n th e socia l an d biologica l sciences . Bu t i t
would tak e m e fa r afield , s o I will conclud e m y tal k b y alludin g t o th e past .
If we take a quick look at th e birt h an d developmen t o f the mos t origina l
and fertil e idea s that hav e transformed an d revitalized huma n knowledge , we
immediately se e that a conspicuous numbe r o f them ar e du e to the geniu s of
Italians. Remainin g withi n th e scientifi c discipline s discusse d today , I wil l
note tha t Giovann i Ceva , i n th e seventeent h century , firs t conceive d an d
proposed th e concept s an d principle s whic h toda y infor m economics ; an d
the vestige s o f probabilit y theor y ca n b e foun d i n th e fourteent h centur y i n
a commentar y abou t Dante .
And fro m thos e far-of f epochs , acros s th e centuries , t o toda y continue s
unbroken th e successio n o f Italian s wh o hav e le d u s int o th e moder n world ,
in whic h Ital y no w play s suc h a grea t part .
APPENDIX C
Science a t th e Presen t Momen t an d th e Ne w
Italian Societ y fo r th e Progres s o f th e Science s
This talk inaugurated the first congress of the Italian Society for the
Progress of the Sciences, held in Parma on September 25th, 1 907. It was
published in the proceedings of the Society (Rome, 1 908) and in Scientia,
Year I, V. II: 4 (Bologna, 1 907).
More tha n thirt y year s hav e elapse d sinc e th e las t congres s o f Italia n
scientists wa s held , i n Palermo . Toda y thi s nobl e institutio n reawaken s
and salute s th e ne w su n shinin g befor e it. 1 Sinc e th e las t congress , th e
material an d socia l conditio n o f Ital y ha s undergon e profoun d change , an d
scientific knowledg e ha s rapidl y develope d an d mature d worldwide . Th e
scientific advance s appearin g i n thi s brie f perio d o f tim e hav e renewe d no t
just ou r dail y lives but als o the general direction o f our culture , breeding an d
strengthening a completely new, modern, origina l spirit—what I would call a
scientific consciousness—tha t dominate s ou r time s just a s othe r intellectua l
climates dominate d earlie r times . Thi s scientifi c spirit , whic h ha s b y no w
pervaded ever y leve l o f ou r society , hig h an d low , aros e no t jus t fro m th e
genius o f our finest minds , t o who m w e owe the grea t discoverie s an d ideas ,
but als o fro m th e day-to-da y activitie s o f societ y a s a whole , whic h make s
constant us e of them. Ou r healthies t an d livelies t energie s ar e today revive d
by thi s guidin g spirit ; heedin g it s call , ou r ol d associatio n rise s again .
We can be sure that th e public' s attitud e towar d scienc e is very differen t
today fro m wha t i t wa s onl y fifty year s ago . Indeed , th e genera l public , a s
never before , ha s witnesse d th e birt h an d developmen t o f ou r generation' s
discoveries in the scientifi c laboratory , thei r transfe r t o the factory floor, an d
finally thei r invasio n o f everyda y life .
Thus, a t thi s momen t i n histor y w e ar e presente d wit h vas t number s
of peopl e wh o ar e fascinate d b y th e invention s tha t hav e rapidl y produce d
so muc h comfor t an d wealt h an d s o fundamentall y influence d thei r habit s
and thei r way s o f thinking . The y ar e tryin g t o understan d th e truth s o f
science, an d t o kno w the m i n detail ; mor e important , the y expec t fro m
science materia l an d socia l progress . I t i s perhaps thi s stat e o f expectation ,
x
See V. Volterra, "Propost a d i un a associazion e italian a pe r i l progresso dell e scienze "
(1906), Saggi scientifici (Bologna : Zanichelli , 1 920 ; Zanichell i reprint , 1 990 , wit h a n in troduction b y Raffaell a Simili) , 81 -95 .
261
262 C
. SCIENC E A T PRESEN T AN D SOCIET Y FO R PROGRES S O F SCIENCE S
so characteristic o f our times, that ha s fostered th e aforementione d scientifi c
spirit.
I wil l tr y t o illustrat e m y thesi s wit h th e familia r compariso n betwee n
the developmen t o f stea m an d tha t o f electric power :
Historically, th e us e o f the forme r precede d th e us e o f the latter ; i n fac t
the practica l application s o f electricity , alon g wit h it s transmission , hav e
taken hol d i n jus t th e las t thirt y years , a s w e al l know . [James ] Wat t an d
[George] Stephenso n wer e practical-minded people , who rose by dint o f their
genius from thei r workshops to the Academy of Science and the highest level s
of industry ; the y demonstrat e that—a t leas t i n thos e heroi c time s whe n
the stea m engin e wa s born—th e workshop s wer e th e sourc e o f th e mos t
ingenious an d celebrate d inventions . Onl y the n di d science , investigatin g
the operatio n o f industria l engines , construc t tha t wonderfu l monumen t o f
thermodynamics, whos e principle s gover n al l natura l phenomena .
The contrar y i s true o f electricity .
The [electric ] batter y an d it s immediat e application s cam e straigh t ou t
of th e physic s laborator y o f th e Universit y o f Pavia . Wit h th e inductio n
principle, [Michael ] Farada y lai d th e basi s fo r al l th e application s o f elec tricity, fro m th e dynam o t o th e telephone . [Antonio ] Pacinotti' s [iron ] ring ,
Galileo Ferraris's rotatin g field [motor] , and th e discover y o f electrical wave s
are al l th e resul t o f researc h carrie d ou t i n scientifi c laboratories .
In summary , whil e th e inventio n o f the therma l engin e wa s th e startin g
point fo r muc h theoretica l research , theoretica l electrodynamic s created , o n
its own , th e man y an d wonderfu l application s o f electricity . I n thi s case , a s
in numerous others, the history of words summarizes an d mirrors the histor y
of a long , slo w evolutio n o f ideas . Fo r instance , "temperature, " originall y
a vagu e an d approximat e expressio n o f atmospheri c conditions , assumed ,
little b y little , a mor e scientifi c import , unti l thermodynamic s conceive d o f
it a s th e integratin g facto r o f a differentia l expression . B y contrast , th e
notion o f "potential"—originatin g wit h [Marqui s Pierr e Simo n de ] Laplac e
in celestia l mechanics , enriche d b y [Car l Friedrich ] Gauss , transplante d int o
electrostatics b y [George ] Green , an d introduce d int o electrodynamic s b y
[Gustav Robert ] Kirchhoff—i s i n ou r da y know n a s "voltage, " a nam e o n
the lip s of eve n th e humbles t laborer s i n the farthest , mos t isolate d reaches ,
where just on e electri c ligh t illuminate s th e nigh t i n a poverty-stricke n vil lage.
Everywhere apparent , amon g peopl e o f ever y stripe , benefitin g u s i n
every circumstanc e o f ou r existence , enlivenin g an d intensifyin g ou r lives ,
the application s o f electricit y ar e a demonstration , unparallele d i n thei r
effectiveness, o f th e powe r o f scientifi c researc h an d th e usefulnes s o f th e
most abstrac t thinking .
While ther e i s thu s a direc t connectio n betwee n th e practica l an d th e
scientific life , scienc e professionals , b y a natura l correspondence , ar e als o
drawn t o th e multitud e o f thei r fello w men ; thei r existenc e i s no t confine d
to thei r laboratorie s an d offices ; the y fee l compelle d b y their daily , intimat e
C. SCIENC E A T PRESEN T AN D SOCIET Y FO R PROGRES S O F SCIENCE S 26
3
contact wit h society , to participat e i n the lif e o f that society . Th e demeano r
of th e moder n scientis t i s thu s ver y muc h differen t fro m tha t o f th e savan t
of a fe w year s ago . I n comparin g thes e tw o distinctl y opposit e types , on e
thinks o f tw o grea t me n whos e geniu s encompasse s th e worl d o f physics :
Gauss an d Lor d Kelvin .
The forme r worke d alon e for fift y year s in modest Gottingen , neithe r ap proached no r approachable , makin g publi c onl y wha t h e fel t wa s perfected ,
while jealously guarding , o r confidin g onl y t o clos e friends , th e newes t an d
most origina l o f hi s thoughts—thos e tha t woul d late r provok e suc h excite ment an d suc h a n intellectua l revolution . Th e latter , toda y ou r greates t liv ing scientist , brough t hi s fertil e an d protea n effort s t o bea r i n bot h worlds ,
bravely tacklin g an d popularizin g th e mos t origina l an d singula r theorie s
his geniu s coul d devis e an d playin g hi s par t i n th e gran d modernizatio n o f
England, t o universa l admiration .
Nevertheless, ho w many point s o f contact ther e ar e betwee n th e tw o scientists! I f Lor d Kelvi n joine d Europ e an d Americ a wit h th e transatlanti c
telegraph, Gaus s wa s th e firs t t o imagin e a n electri c telegrap h linkin g hi s
observatory t o th e physic s laborator y o f hi s frien d [Wilhel m Eduard ] We ber. Th e crystalline , geometrica l eleganc e o f Lord Kelvin' s theor y o f image s
is alon e comparabl e t o th e harmoniou s divin e beaut y o f th e propertie s o f
numbers discovere d b y Gauss . No t th e for m o f their genius , then , bu t thei r
character—and eve n more, the dissimilarit y o f their circumstances—wa s th e
source o f the marke d differenc e betwee n them .
The intimat e connectio n o f science with practica l lif e ha s not , moreover ,
diminished it s majesti c an d solem n character— a characte r nourishe d an d
enlivened b y wha t I hav e calle d th e scientifi c spirit . Dynamos , th e giganti c
monuments o f ou r presen t era , accomplis h thei r wor k silentl y an d a t hig h
speed i n marvelous, moder n edifices—no t smokin g an d roaring , lik e the fac tories o f old, bu t brigh t an d quiet ; suc h place s recallin g i n thei r solem n an d
austere grandeu r th e monument s o f another era , th e ancien t cathedral s tha t
raise thei r wonderfu l spire s t o th e sky . Beneat h thos e air y arcade s raise d
by th e artisan s o f th e Middl e Ages , th e sou l fill s wit h a solem n emotion ,
an empath y wit h th e hope s an d dream s o f long-ag o centuries . A n emotio n
just a s profoun d adhere s t o th e sacre d place s o f ou r moder n industries ; ou r
hearts ar e overcome with pride an d satisfactio n an d a serene faith i n a secure
future.
In th e recen t movemen t o f scienc e towar d practica l applications , Ital y
may well have benefited mor e than an y other country , s o the aforementione d
scientific spirit—althoug h i t develope d late r her e tha n elsewhere—i s mak ing increasingl y rapi d an d gratifyin g progres s befor e ou r ver y eyes . Ther e
was, no t s o many year s ago , much gloo m ove r the stat e o f our economy , bu t
by virtu e bot h o f peopl e an d circumstance s i t ha s miraculousl y an d unex pectedly improved : A n abundan t sourc e o f wealt h spran g fro m industry , a
source w e had though t denie d t o ou r countr y b y Natur e itself .
264 C
. SCIENC E A T PRESEN T AN D SOCIET Y FO R PROGRES S O F SCIENCE S
When th e 1 88 4 Turi n Expositio n sa w th e appearanc e o f th e firs t elec tric transformers—whic h woul d b e compare d t o th e lever , th e fundamenta l
mechanical device—th e see d tha t woul d produc e s o much wealt h wa s sown :
The potentia l energ y containe d i n ou r mountain s an d ou r river s flowe d ont o
the plain, powering thousands of our factories an d lighting up our cities. Th e
wires w e se e spreadin g lik e a ne t abov e ou r house s an d stretchin g fa r awa y
are th e mos t eloquen t evidenc e o f ou r economi c prosperity . I n th e lonel y
Roman countryside , the y ru n paralle l t o th e magnificen t aqueducts . Li t b y
a fier y sunset , the y onc e spok e t o Lor d Kelvi n i n a languag e a s solem n a s
the majesti c ruin s o f ancien t Rome .
I hav e u p t o no w trie d t o describ e th e effect s o f recen t scientifi c devel opments o n the moder n world—an d I mentioned briefl y th e evolutio n o f th e
scientist—as bes t I could withi n th e narro w scop e allowe d me . I brought t o
light, an d onl y fleetingly , jus t on e aspec t o f th e gran d landscap e o f science .
This aspec t w e may conside r it s exterior ; th e interior , whic h i s doubtless o f
greater interest , ha s remaine d concealed .
Nevertheless, th e value of science does not li e only in its practical utility ,
nor d o it s strengt h an d it s suppor t res t solel y wit h th e publi c wh o mak e
use o f it s result s an d admir e it s producers . Th e valu e o f science , whic h
inspired i n [Henri ] Poincar e s o man y dee p thought s an d s o man y eloquen t
pages, i s reveale d i n anothe r way— a nobler , highe r way : I t i s manifes t i n
the intimat e characte r o f th e wor k itself , i n th e satisfactio n i t provides . I n
the pure , disintereste d searc h fo r th e truth , whic h i s th e suprem e goal , th e
researcher's greates t jo y i s in learning , no t i n knowing .
But i t i s no t m y tas k t o spea k t o yo u abou t thes e interio r aspect s o f
science. Tha t wil l b e mad e clea r i n th e plenar y talk s give n b y illustriou s
scientists o n th e thre e grea t branche s o f science—th e physical-chemical , bi ological, an d socia l sciences ; i n th e openin g addresse s o f th e president s o f
the variou s sections; in the progress reports o f the chapter s o f different disci plines; in origina l communication s an d discussions . I n short , onl y th e entir e
proceedings o f thi s congres s ca n presen t th e spectacl e o f al l tha t live s an d
breathes withi n th e scientifi c world ; i t wil l sho w yo u th e mysterie s w e ar e
feverishly tryin g to solve, the victories we have achieved, an d the disappoint ments w e hav e suffered ; howeve r cruel , thes e mus t no t b e ignored .
The presen t momen t offer s n o more than a chance for a fleeting overvie w
of th e variou s disciplines . To o man y fundamenta l discoverie s ar e rapidl y
accumulating an d waitin g t o b e classifie d an d organized , whil e a deep , in tense, and, I would say, almost ruthles s critique scrutinizes and dissects every
thought, ever y speculation , underminin g systemati c construction s tha t onl y
yesterday seeme d poise d t o def y th e centuries . Today , the y ar e nothin g bu t
scattered ruin s upo n whic h a fe w ar e alread y tryin g t o rebuild .
But I must no t fai l to mention what ever y careful observe r alread y know s
from persona l experience—tha t almos t al l scientifi c discipline s ar e toda y i n
great crisis , on e i n th e condition s unde r whic h the y wor k an d on e i n th e
philosophical though t tha t inform s th e work .
C. SCIENC E A T PRESEN T AN D SOCIET Y FO R PROGRES S O F SCIENCE S 26
5
The firs t o f thes e present s a singula r contrast : While , o n th e on e hand ,
the acquisitio n o f technica l skil l require s specializatio n an d th e divisio n o f
scientific labo r (s o much s o that a n entir e lifetim e i s often barel y enoug h t o
acquire th e abilitie s essentia l t o progress) , o n th e othe r hand , th e variou s
disciplines have so interpenetrated eac h other that i t is not a t al l clear how we
can move forward i n one without a deep understanding o f many others—an d
not onl y those generally thought t o be closely related bu t als o new fields no w
seen to b e relate d a s well. Collaborativ e efforts—commo n an d mor e intens e
in th e establishe d sciences , suc h a s astronomy , an d i n th e grea t school s
that aris e aroun d me n o f genius , a s i n th e mos t advance d countries—ten d
to coordinat e an d disciplin e individua l energies , bu t w e ar e stil l fa r fro m
achieving th e balanc e tha t alon e ca n generat e th e econom y o f effor t w e al l
aspire to .
But thi s is not th e onl y crisis : Th e other , o f interest t o the philosophica l
thinker, i s eve n mor e striking .
It i s a n ancien t belie f that , i n science , hypothese s ar e mean s an d no t
ends, tha t tomorro w w e ca n abando n wha t toda y w e embrace—s o ancien t
a belief , i n fact , tha t th e Gree k astronomer s foun d al l cosmi c hypothese s
acceptable, provide d the y wer e abl e t o us e the m t o calculat e th e position s
of the stars .
But th e presen t perio d differ s fro m thos e preceding , becaus e suddenl y
not just singl e hypotheses bu t als o great principles , som e o f which ha d bee n
unquestioned, universall y accepted , an d taugh t almos t a s dogma , hav e be come th e subjec t o f debate , whil e system s tha t ha d lon g seeme d burie d fo r
good hav e been unexpectedl y resurrected . Perhap s posterit y wil l regard th e
present momen t a s w e regar d th e Renaissance , whe n th e ver y foundation s
of ou r worldvie w wer e laid .
At th e cente r o f the moder n critica l movemen t tha t le d t o thi s perio d of
unrest i s undoubtedly mathematics .
Scarcely a centur y ha s passed—a s [Gosta ] Mittag-Leffle r acutel y ob serves i n th e beautifu l page s dedicate d t o th e memor y o f [Niel s Henrik ]
Abel2—since tha t grea t analys t proclaime d tha t mathematic s i s a n en d i n
itself, an d find s it s mode l i n itself . An d yet , w e ca n add , i n thi s centur y
more tha n i n an y other , mathematic s move d beyon d th e limit s o f it s intrin sic activit y an d enriche d field s fa r fro m it s own , givin g ris e t o a ne w an d
flourishing philosophy .
As Abe l reasoned , mathematic s fel l bac k upo n itself , firs t t o build , an d
then t o consolidat e th e theor y o f function s an d th e geometr y tha t woul d
provide th e foundatio n fo r recen t research . Thi s ha s led , b y a profoun d an d
assiduous stud y o f it s ow n method s an d concepts , th e analysi s o f though t
to suc h perfection, suc h sharpness , flexibility , an d power , tha t [mathematic s
now] penetrates an d affect s al l scientific an d philosophica l speculation . Thi s
2
G. Mittag-Leffler , Niels Henrik Abel (Revue du Mois, t . VI , 1 0 Jul y -1 0 August ,
1907).
266 C
. SCIENC E A T PRESEN T AN D SOCIET Y FO R PROGRES S O F SCIENCE S
is how , t o mentio n a singl e famou s example , a purel y geometrica l pape r b y
[Eugenio] Beltrami, 3 derive d fro m researc h o n non-Euclidea n geometr y b y
Gauss, [Nicola i Ivanovich] Lobatschewski , an d [Bernhard ] Riemann , becam e
of suc h universa l importanc e tha t i t cas t ne w ligh t o n th e fundamental s o f
logic itself . Th e moder n critiqu e initiate d b y mathematician s ha s success fully invade d th e physica l science s an d produce d ne w current s o f thought .
Mechanics—not fo r the first time—wa s th e road b y which this ne w directio n
arrived.
But a n importan t developmen t ha s com e alon g whic h ha s tende d t o
change th e ver y positio n o f thi s disciplin e withi n th e physica l sciences .
Everyone i n ou r generatio n ( I thin k I ca n sa y openly ) wa s taugh t thos e
principles tha t w e nowaday s cal l mechanistic ; an d i n fac t th e ide a tha t al l
phenomena, a t leas t thos e studie d i n physics , coul d b e reduce d t o motion s
and al l fell int o the orbi t o f classical mechanic s wa s a dogma that al l school s
used to bo w to, its origins lost i n the mist s of long-ago Cartesian philosophy .
But littl e b y littl e th e mechanisti c theorie s hav e changed ; difficultie s
have accumulated ; idea s beginnin g wit h thos e o f [William ] Rankin e an d
strenuously supported b y [Ernst ] Mach (wh o however holds a unique positio n
in the philosophy o f science), and develope d b y others, hav e made their way .
Many have fought beneat h the flag bearing that celebrate d inscription: "Wa r
Against th e Mechanica l Mythology. "
Energetics cam e alon g an d classifie d mechanic s a s just anothe r physica l
science, n o longe r th e basi s fo r al l o f them . A ne w commo n basi s wa s
established, founde d o n broade r an d mor e comprehensiv e principles .
This framewor k o f ideas , th e subjec t o f s o muc h discussio n an d debat e
among mathematician s an d natura l scientists , doe s not , however , represen t
the extrem e limi t tha t ma y b e arrive d at . Th e critiqu e develope d b y math ematical physicists , comparabl e t o ultramicroscopi c observatio n a s oppose d
to ordinar y microscopy , i s now scrutinizin g an d castin g doub t o n thes e ver y
principles.
In truth , th e moder n concept s o f th e electrica l constitutio n o f matter ,
while relate d t o atomi c an d kineti c ideas , reintroduc e principle s simila r t o
those of the ol d physical mechanics; on the other hand , the y engende r a profound revolutio n i n th e concept s o f mas s an d inerti a establishe d b y Newto n
as the basis of all natural philosophy— a revolutio n mad e even greater b y th e
new theorie s o f relativit y bein g develope d today . W e ca n wel l understan d
that th e dominan t principle s of half a century ag o scarcely surviv e the stor m
that seemingl y overwhelm s them .
This crisi s reverberate s throughou t th e natura l sciences ; an d i n th e
meantime, i n th e heaven s a s o n th e earth , a thousan d thing s ar e reveale d
that philosoph y coul d no t eve n drea m of , fro m th e effect s o f ligh t o n th e
movement o f star s t o ne w source s o f terrestria l heat .
3
Eugenio Beltrami , Saggio di interpretazione della geometria non-euclidea. Giornale
di Matematiche, vol . VI . Se e als o Opere matematiche b y Eugeni o Beltrami , v . I , p . 374 ,
Milano 1 902 .
C. SCIENC E A T PRESEN T AN D SOCIET Y FO R PROGRES S O F SCIENCE S 26
7
Moreover, whe n w e realize that littl e by little the theorie s base d o n [par ticle] emissio n see m o n the ris e again , whil e a few year s ag o the wav e theor y
was the sol e victor i n the fiel d o f phenomena propagatin g ove r distance , ou r
surprise increases—t o se e ne w an d unexpecte d speculatio n joine d b y th e
no-less-unexpected ancien t idea s that , lik e ghosts , ris e fro m grave s though t
forever closed .
Perhaps mor e tha n i n physic s itself , thi s revolutio n i n idea s show s u p i n
its sister science, chemistry, wher e new ideas of atomic structure hav e shaken
up th e ol d classica l theories ; wher e th e alchemica l drea m i s reborn , ful l o f
mystery an d promise ; wher e a ne w an d flourishing field, physica l chemistry ,
rich i n result s an d i n hopes , ha s arisen .
In th e virgi n field o f physica l chemistr y w e encounte r opposin g trends ,
and i t i s very difficult t o decide to which of them th e mos t interestin g result s
are due . I n fact , if , o n th e on e hand , th e purel y kineti c theorie s hav e pro duced th e discoverie s o f [J . D.] Van der Waals , Arrhenius , an d man y others ,
energetics, o n th e other , ha s foun d nothin g t o destro y o r chang e bu t muc h
to buil d o n profitably ; i n thi s field, it s beneficia l influenc e ha s bee n deepl y
felt.
There i s a characteristi c kin d o f reasonin g tha t I woul d no t hesitat e t o
call energetics. Powerfu l an d fertile , it s origins g o back t o [Sadi ] Carno t an d
his memorabl e cycle ; i t dominate s al l o f theoretical physica l chemistry , an d
the doctrine s i t ha s inspire d hav e ha d th e mos t importan t consequences .
But th e application s o f thi s kin d o f reasonin g hav e extende d muc h further ,
into differen t sciences : I mus t no t fai l t o mentio n tha t wha t I conside r th e
most characteristi c an d stimulatin g resul t o f mathematica l economics—th e
general demonstration tha t th e differential equatio n o f economic equilibriu m
is infinitely integrable—ca n b e base d o n it. 4 Le t u s presum e an d hop e tha t
many othe r result s ca n b e s o obtained .
This notio n naturall y lead s m e to th e subjec t o f the influenc e o f mathe matics and the innovations it has produced i n the social sciences, but I would
be oversteppin g th e boundarie s I se t fo r myself . Thes e limit s allo w m e t o
note onl y i n passin g tha t th e infan t physica l chemistry , o f whic h w e hav e
just spoken , ha s imparte d ne w dat a an d thu s a new directio n t o physiology .
With regar d t o th e biologica l sciences , I wil l mentio n onl y briefl y th e
crisis affectin g fundamenta l concept s o f life , evolution , an d heredit y tha t
has so disrupted Darwinia n doctrine . Afte r guidin g minds for hal f a century ,
Darwinism, i n the wake of recent zoologica l an d botanica l research , seem s t o
be losing , i f no t it s importance , perhap s th e predominanc e i t onc e enjoyed .
Several factors hav e combined, an d ar e combining, t o produce this transformation o f thought , an d I canno t mentio n al l o f them , bu t the y certainl y
include observation s arisin g fro m scienc e an d technology , ne w experimenta l
methods i n physiological chemistry , an d no t leas t thos e o f biometry (alway s
4
Compare V . Volterra, "L'economi a matematic a e il nuovo manual e de l Prof. Pareto '
(Giornale degli Economisti, Apri l 1 906) .
268 C
. SCIENC E A T PRESEN T AN D SOCIET Y FO R PROGRES S O F SCIENCE S
a muc h appreciate d sourc e o f dat a an d definitiv e laws) , [al l of which] heral d
a ne w er a fo r biology. 5 I t seem s t o m e tha t w e ar e seein g th e outlin e o f
methods tha t on e da y ma y hav e broa d application .
There exists , a s [Emile ] Picar d observes , a mechanic s o f heredit y tha t
contrasts with the classical one, according to which the present stat e of a system alon e determines its future. Th e concept o f function tha t ha s dominate d
mathematics i n the last centur y ha s been extended, an d this extension raise s
new questions , whic h hav e le d t o usefu l results . I t i s eas y t o forese e tha t
this migh t constitut e a hereditar y mechanic s capabl e o f representin g wit h
greater precisio n th e elastic , magnetic , an d othe r suc h phenomen a i n whic h
hysteresis i s o f grea t importance. 6 I wonde r wha t th e futur e hold s fo r thi s
mechanics, i f i t succeed s i n penetratin g th e field o f biologica l phenomena .
But i t i s no t saf e t o mak e an y kin d o f prophecy . Th e histor y o f scienc e
teaches u s tha t sometime s al l i t take s i s th e discover y o f on e trivia l fac t t o
overturn th e mos t well-founde d predictions . Extrapolatio n i n a field whos e
laws ar e uncertai n o r unknow n i s a dange r I inten d t o avoid .
But i t i s time fo r m e t o conclud e m y tal k an d summariz e m y thoughts .
I wanted t o la y before yo u two facts: th e growin g proximity betwee n th e
public and scientists , due to the predominantly scientifi c spiri t o f the presen t
age, an d th e grea t crisi s that toda y afflict s s o many branche s o f knowledge .
Correspondingly, huma n societ y ha s developed ne w needs—needs tha t ever y
civilized countr y mus t meet , les t it s intellectua l lif e languis h o r com e t o a
halt an d th e sourc e o f it s prosperit y dr y up .
The interna l crisi s threatenin g s o man y discipline s require s widesprea d
and ope n discussio n amon g scholars , i n which they ca n expres s the thought s
that occup y them , th e doubt s tha t troubl e them, th e difficultie s tha t thwar t
them, th e hope s tha t driv e the m on . Book s an d paper s d o no t an d neve r
will serve this function; th e nee d is precisely to talk abou t an d liste n to idea s
that nobod y dare s ye t t o publis h o r tha t wil l neve r b e published .
The ol d academie s ar e to o closed ; th e teachin g institution s hav e othe r
goals; th e individua l scientifi c societie s ar e too narro w t o b e usefu l fo r thes e
purposes, whic h ca n b e fulfille d onl y withi n a societ y tha t include s expert s
in al l disciplines , lik e the associatio n w e inaugurate today .
On th e othe r hand , ever y da y w e see the proliferatio n o f scientific work s
and publication s tha t engag e th e large r public , who flock to conference s an d
lectures i n eve r increasin g number s an d wit h eve r increasin g curiosity . Bu t
no boo k wil l ever sho w how a scientifi c though t i s born an d formed , ho w a n
initially vagu e idea takes shap e i n the min d o f the researcher ; n o lecture wil l
ever illuminat e it , jus t a s n o displa y i n a zoologica l museu m wil l eve r full y
represent lif e itself .
5
See V . Volterra , "Su i tentativ i d i applicazion e dell e matematich e all e scienz e bio logiche e sociali " (1 901 ) , i n Saggi scientific^ 1 -35 .
6
See V . Volterra , "L'applicazion e de l calcol o a i fenomen i d i eredita " (1 91 2) , i n Saggi
scientific^ 1 89-21 8 .
C. SCIENC E A T PRESEN T AN D SOCIET Y FO R PROGRES S O F SCIENCE S 26
9
Well, whateve r th e publi c canno t lear n fro m book s an d lecture s wil l
become clea r when the y atten d th e discussion s amon g the scientists , sinc e i t
is in spontaneou s an d livel y debate tha t th e developmen t o f ideas artificiall y
popularized b y thos e wh o kno w to o muc h appea r i n thei r trues t light .
Not thi s alon e doe s ou r countr y as k o f it s risin g institution—no t onl y
the satisfactio n o f intellectua l curiosit y bu t als o th e usefu l promotio n an d
encouragement o f fruitfu l stud y an d ne w an d vita l research . Ever y day ,
industrialists, businessmen , an d thos e i n th e profession s tur n t o science ,
continually urge d b y a growing multitude wh o hope that scienc e will provide
a solutio n t o th e new , complex , an d pressin g problem s the y fac e an d wh o
invoke a scienc e tha t wil l prov e victoriou s ove r eve r risin g difficulties .
These questions , o f such interes t t o scienc e an d technology , ca n b e effec tively pose d onl y befor e a n ope n an d libera l associatio n lik e our own , whic h
welcomes me n i n divers e fields , becaus e eve n to formulat e the m require s th e
cooperation o f differen t disciplines . I t wil l the n b e u p t o th e laboratorie s
and th e scientifi c institute s t o develo p an d resolv e them . Thi s i s wh y th e
committee ha s warml y an d enthusiasticall y promote d th e ne w Society , wh y
it ha s convene d al l o f you an d no w rejoice s i n seein g ho w man y o f you hav e
gathered her e fro m you r schools , laboratories , an d practica l occupations .
We ar e al l warme d b y a n equa l passio n fo r thi s newbor n Society , whic h
our vote s consecrat e t o a grea t an d nobl e end ; it s destin y smile s o n u s wit h
equal hope ; it s futur e seem s linke d t o th e futur e o f ou r countr y itself , a s i t
moves o n towar d it s hig h destiny .
In conclusion , m y thought s retur n t o th e compariso n I dre w a littl e
while ag o betwee n th e presen t er a an d th e Renaissance . I n tha t tim e o f
the wonderfu l restoratio n o f intellectua l life , Ital y becam e th e ver y cente r
of universa l scientifi c thought . Today , I ventur e t o wis h tha t th e destin y
reserved fo r u s no t b e a lesse r one , a s th e pur e an d authenti c Italia n sou l
rises an d take s shape , revivin g ou r though t an d restorin g t o u s ou r ancien t
country.
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Acknowledgments
Asked onc e wh y sh e wrot e abou t families , th e Italia n novelis t Natali a
Ginzburg replied , "becaus e that i s where everything starts , wher e the germ s
grow." I n th e cas e o f Vit o Volterra' s "germs, " I hav e benefite d greatl y
from th e generou s hel p offere d ove r th e year s b y famil y members , includ ing Vit o Volterr a Jr. ; Edoard o an d Nell a Volterr a an d thei r daughte r Vir ginia Volterra ; Silvi a D'Ancona , Edoard o Achill e Almagia , Luis a Almagia ,
Ginetta Montecorboli , Fiorenz a Almagia , an d Mass a Mori . I ow e a spe cial deb t o f thank s t o Liber a Levi-Civita , wh o welcome d m e int o he r hom e
in Rom e i n th e summe r o f 1 971 , an d pu t he r husband' s extensiv e collec tion o f letter s an d persona l document s a t m y disposal . Pie r Vittori o an d
Susanna Silberstei n Ceccherini , an d Francesc o an d Teres a Scaramuzz i hav e
been faithfu l companion s o n m y lon g voyag e t o tel l th e stor y o f th e golde n
age o f Italia n mathematic s throug h th e eye s o f Volterr a an d hi s circl e o f
mathematicians. Fo r permissio n t o quot e fro m th e Vit o Volterr a Collectio n
and th e Levi-Civit a Paper s a t th e Accademi a Nazional e de i Lincei , I than k
Edoardo Vesentini , pas t presiden t o f th e Lincei , an d Giovann i Conso , th e
current president ; an d Willia m Roberts , Universit y Archives , U C Berkeley ,
for permissio n t o quot e fro m th e Griffit h Evan s papers .
Thanks als o go to th e man y archivists , librarians , an d historian s i n Ital y
who patientl y answere d m y questions , an d foun d an d supplie d copie s o f
documents, locate d out-of-prin t books , an d i n som e cases , photographs . I n
Florence, Teres a Porcella , a t th e Archivi o dell o Stat o Civil e de l Comun e d i
Firenze; i n Rome , Marc o Guard o an d Enric a Schettini , a t th e Accademi a
Nazionale de i Lincei , an d Emili a Campochiaro , Cristin a Cannizzo , a t th e
Archivio Storic o de l Senat o dell a Repubblica, an d Federic o Pommier , a t th e
Biblioteca de l Senato della Repubblica; i n Pisa, Sandr a d i Majo, Rosangael a
Cingottini, Millett a Sbrilli , an d Stefan o Pieroni , a t th e Scuol a Normale ,
and Giovann a Tanti , a t th e Archivi o d i Stato ; i n Turin , Livi a Giacardi ,
Sandro Caparrini , Clar a Silvi a Roero , an d Paol a Novaria , a t th e Universit y
of Turin; i n Bologna, Fulvi o Cammaran o an d Mariangel a Mafessanti , a t th e
University o f Bologna ; an d i n Crema , Maur o D e Zan .
Giovanna Bergonz i an d Ann a Mari a Galoppin i provide d informatio n
about anarchists , women, an d higher education in late 1 9t h century Tuscany ;
Vittoria Amighett i use d he r pharmaceutica l trainin g t o diagnos e Volterra' s
heat ras h a t Pisa ; Brun o D i Porto , Riccard o D i Segni , an d Lionell a Viterb o
educated m e abou t Ba r Mitzvah s an d othe r Jewis h ceremonia l tradition s
271
272
ACKNOWLEDGMENTS
in nineteenth-centur y Italy ; Franc a Fo a Ascoli , presiden t o f th e Comunit a
Ebraica di Ancona; Barbar a Martinell i a t the Comunit a Ebraica d i Pisa, an d
the Archivi o Storic o dell a Comunit a Ebraic a d i Rom a graciousl y searche d
their record s o n m y behalf .
Closer t o home , I wis h t o than k Le v Ginzbur g fo r a mini-tutoria l o n
mathematical biology ; Judit h Nolla r an d Shad y Peyvan , an d th e entir e
interhbrary loa n offic e i n Millika n Library , a t th e Californi a Institut e o f
Technology, wh o neve r flinche d a t m y bibliographi c question s an d requests ;
Bonnie Ludt , wh o deal t wit h th e endnote s an d muc h els e wit h he r usua l
efficiency; Charlott e Erwin , fo r he r assistanc e wit h translatin g Germa n doc uments; an d al l the staff o f the Caltec h Archives, who kept the place runnin g
smoothly whil e I wa s of f doin g research . I a m als o gratefu l fo r th e hospi tality o f Je d Buchwal d an d th e Burnd y Librar y an d th e Dibne r Institut e
for th e Histor y o f Science an d Technolog y o n th e Massachusett s Institut e o f
Technology campus , wher e th e Vit o Volterr a Collectio n wa s house d befor e
it move d i n 200 6 t o th e Huntingto n Librar y i n Sa n Marino , California . I
also benefite d fro m readin g Ti m Sluckin' s unpublishe d manuscript , "Math ematical Appreciatio n o f Vit o Volterra. "
In additio n t o interviewin g member s o f th e Volterr a family , Carlott a
Scaramuzzi, a 1 99 4 Caltec h Summe r Undergraduat e Researc h Fellow , tran scribed an d translate d man y Italia n letters ; Elis a Piccio , wh o joine d th e
project i n 2002 , picke d u p th e rein s fro m Carlott a an d ha s don e a n equall y
fine jo b a s a translator , indefatigabl e Interne t fac t checker , an d jack-of-al l
trades researc h assistant .
Finally, I have ha d th e hel p of two consummate editors , Sar a Lippincot t
and Heid i Aspaturian , i n shapin g th e narrativ e backbon e o f thi s biography .
In som e ways , The Volterra Chronicles i s thei r stor y too , an d I a m grate ful no t onl y fo r thei r kee n eye s an d appreciatio n o f th e Englis h language ,
but als o for thei r collegialit y an d war m friendship . Donal d Babbit t deserve s
a roun d o f thank s als o fo r servin g a s m y mathematic s mento r an d fo r in troducing m e t o Serge i Gelfand , th e edito r a t th e America n Mathematica l
Society, wh o kep t fait h wit h th e boo k an d th e author .
Selected Bibliograph y
The startin g poin t fo r thi s biograph y i s the collectio n o f Vito Volterra' s
scientific paper s an d correspondenc e a t th e Accademi a Nazional e de i Lin cei, i n Rome , an d Giovann i Paoloni' s importan t documentar y history , Vito
Volterra e il suo tempo, publishe d o n the occasion of an international confer ence a t th e Lince i i n 1 99 0 in memor y o f th e mathematician . Famil y paper s
collected b y Luis a Almagi a an d Edoard o Achill e Almagi a provide d furthe r
details abou t th e Almagia-Volterr a family .
In Rome , th e Bess o Foundatio n house s a n importan t collectio n o f con temporary journals an d books relating to Rome around th e turn o f the twen tieth century . Th e Italia n Stat e Archives , situate d o n th e ground s o f th e
Esposizione Universal e Rom e (th e intende d sit e o f a n aborte d 1 94 2 World' s
Fair) contai n th e record s o f the Accademi a d'ltalia , th e Consigli o Nazional e
delle Ricerche , an d othe r governmen t agencies—record s tha t revea l ho w th e
Fascist stat e viewed it s scientifi c community .
Oral historie s provide d m e wit h anothe r importan t sourc e o f informa tion abou t Vit o Volterra , hi s family , an d condition s i n Ital y i n th e 1 930s .
Interviews wit h hi s persona l librarian , Guald a Massimi ; tw o o f hi s grand daughters, Dr . Silvi a D'Ancona an d Professor Virgini a Volterra; an d variou s
other famil y members , includin g hi s cousin s Edoard o A . Almagia , Ginett a
Montecorboli, an d Luis a Almagia , figure i n th e book .
Archives i n th e Unite d State s tha t contai n informatio n relevan t t o th e
life an d time s o f Volterr a includ e th e Griffit h Evan s Paper s a t th e Bancrof t
Library i n Berkeley , California , th e Genera l Educatio n Boar d files i n th e
Rockefeller Archiv e Cente r i n Nort h Tarrytown , Ne w York , th e Georg e E .
Hale Paper s a t th e Californi a Institut e o f Technology , i n Pasadena , an d th e
Vito Volterr a collectio n a t th e Huntingto n Library , Ar t Collections , an d
Botanical Gardens , i n San Marino , California . Secondar y source s in Englis h
on Volterra an d o n the histor y o f Italian mathematic s ar e scarce. I n Italian ,
there is a considerable list of books about th e development o f mathematics i n
Italy between 1 86 0 and 1 940 . Thes e books are rarely translated int o English .
The Italia n mathematicia n Giorgi o Israe l ha s writte n extensivel y o n Vit o
Volterra; hi s most recen t book , The Biology of Numbers (Birkhauser , 2002) ,
written i n collaboratio n wit h An a Milla n Gasca , explore s th e developmen t
of Volterra's idea s in the field of mathematical biology ; anothe r recen t book ,
Scienza e razza nelVItalia fascista (Societ a editric e i l Mulino, 1 998) , writte n
273
274
SELECTED BIBLIOGRAPH Y
in collaboratio n wit h Pietr o Nastasi , deal s wit h issue s o f rac e an d anti Semitism. Othe r importan t work s to appear i n recent year s include Umbert o
Bottazzini's Va J Pensiero (Societ a editrice i l Mulino, 1 994) ; Raffaella Simili ,
ed., Vito Volterra, Saggi Scientifici (Zanichelli, 1 990) ; an d La Matematica
Italiana dopo L'Unita (Marco s y Marcos , 1 998) , a n importan t collectio n
of essay s b y som e o f th e leadin g Italia n historian s o f mathematic s workin g
today, edite d b y S . D i Sieno , A . Guerraggio , an d P . Nastasi . I n general ,
these books ar e aimed a t workin g mathematicians an d specialize d historian s
of mathematics .
There i s a smal l numbe r o f book s an d article s dealin g wit h Volterra' s
circle o f mathematicians . Emm a Castelnuov o ha s writte n a n articl e "Fed erigo Enrique s e Guid o Castelnuovo " (1 997 , Bollettino U.M I.) abou t he r
father, Guid o Castelnuovo ; Giovann i Enrique s offer s memorie s o f hi s fa ther, Federig o Enriques , i n hi s boo k Via d'Azeglio 51 (Zanichelli, 1 983) ;
the mathematicia n an d historia n o f scienc e Dir k Strui k ha s touche d o n hi s
association wit h Tulli o Levi-Civit a i n Rom e i n th e 1 920 s i n a n unpublishe d
autobiography h e share d wit h me . Emili o Segr e (n o relatio n t o Beniamino )
writes abou t th e Rom e mathematician s i n hi s ow n autobiography , A Mind
Always in Motion (Universit y o f Californi a Press , 1 993) . I n 1 95 9 Dove r
republished th e firs t Englis h translatio n o f Volterra' s Theory of Functionals and of integral and Integro-Differential Equations. Th e Dove r editio n
includes a prefac e b y Griffit h C . Evan s an d th e Roya l Societ y biograph y
and bibliograph y prepare d b y Si r Edmun d Whittaker . Evan s supplemente d
Whittaker's bibliography , bu t th e biograph y i s abridged .
The standar d politica l history o f Italy durin g the perio d o f Volterra's lif e
is Deni s Mac k Smith' s Modern Italy (Universit y o f Michiga n Press , 1 997) .
The standard histor y o f Italian Jewr y i s Attilio Milano' s Storia degli ebrei in
Italia (Turin , 1 963) ; Renz o D e Felice' s accoun t o f Italia n Jew s unde r Mus solini, The Jews in Fascist Italy: a history (Enigm a Books , 2001 ) remain s
the authoritativ e wor k o n th e subject . A smal l selectio n o f othe r work s
consulted i n th e preparatio n o f thi s biograph y i s appended .
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Notes
Notes t o Introductio n
1
Albert Einstei n t o Michel e Besso , The Collected Papers of Albert Einstein, trans . A . Beck , vol . 5 (Princeton : Princeto n Universit y Press , 1 987) ,
374.
2
Einstein t o Tulli o Levi-Civita, Collected Papers, trans . A . M. Hentschel ,
vol. 8 (1 998) , 89 .
3
George Eller y Hal e t o Joh n J . Carty , Jul y 3 , 1 91 8 , Georg e Eller y Hal e
Papers, Bo x 1 0 , Archives, Californi a Institut e o f Technology .
4
G. E . Hal e t o Vit o Volterra , Jun e 30 , 1 93 2 an d V . Volterr a t o G. E .
Hale, Dec . 24 , 1 934 , Hal e Papers , Bo x 41.
5
Vito Volterra , "Troi s analyste s italiens : Betti , Brioschi , Casorati, " Bull.
Am. Math. Soc. 7 (1 900) , 60 .
6
Dirk J . Struik , "Som e Mathematicians I Have Met," Proc . AM S Specia l
Session, Join t Mathematic s Meeting , Cincinnati , Jan .1 994 , i n Bettyn e
A. Case , ed. , A Century of Mathematical Meetings (Providence : America n
Mathematical Society , 1 996) , 253.
7
Francesco G. Tricomi , "Matematic i italian i de l prim o secol o dell o stat o
unitario," Mem. Ace. Sci. Torino, cl . sci. , fis., 4t h ser . 1 (1962): 3
8
Ferdinand Gregorovius , The Ghetto and Jews of Rome, trans . M . Hada s
(New York : Schocken , 1 948) , 83-86.
9
Ibid., 90 .
10
David I . Kertzer , The Kidnapping of Edgardo Mortara (Ne w York :
Knopf, 1 997) .
11
Ercole Sori, ed., La Comunita ebraica ad Ancona: la storia, le tradizioni,
Vevoluzione sociale, i personaggi (Ancona : Comun e d i Ancona , 1 995) , 46.
12
P. Levi , Preface , i n Vivia n B . Mann , ed. , Gardens and Ghettos: The
Art of Jewish Life in Italy (Berkeley : Univ . o f Californi a Press , 1 989) , xvi.
281
282
NOTES
Notes t o Chapte r 1
1
Quoted i n Mari o Natalucci , Ancona attraverso i secoli, vol. 3 (Citt a d i
Castello: Union e Art i Grafiche , 1 960) , 252-253.
2
Ercole Sori , "Un a 'comunit a crepuscolare' : Ancon a tr a ott o e nove cento," i n Sergi o Anselm i an d Vivian a Bonazzoli , eds. , La presenza ebraica
nelle Marche secoli XIII-XX (Ancona : Universit y o f Ancona , 1 993) , 1 96 197.
3
Ibid., 1 99 .
4
Ibid., 201 .
5
F. Borioni , Le feste anconitane nel settembre delV anno 1 8^1 per la
faustissima venuta e dimora del N.S. Gregorio XVI Regnante felicemente
(Ancona, 1 841 ) , 51.
6
Edoardo Almagia to his sons, May 23, 1907, The Volterra papers, privat e
collection (V P hereafter) .
7
E. Almagi a diary , VP .
8
Mose Volterra , "A l su o affettuos o fratell o Abram o Volterr a e d egregi a
cognata Angelic a Almagia, " [Mar . 1 6] , 1 859 , VP.
9
Saul Almagi a t o Robert o Almagia , Octobe r 4 , 1 860 . I a m indebte d t o
Luisa Almagi a fo r a cop y o f thi s letter .
10
Edoardo Volterr a t o Virgini a Volterra , Ma y 24 , 1 94 7 an d Vit o Terni ,
same date , Vit o Volterr a files , Institut e Archives , Californi a Institut e o f
Technology.
Notes t o Chapte r 2
1
Angelica Volterr a t o Virgini a Camiz , Sept . 1 8 , 1 864 , i n Th e Volterr a
papers, privat e collectio n (V P hereafter) .
2
A. Volterr a t o V . Camiz , Mar . 5 , 1 865 , an d A . Volterr a t o Alfons o
Almagia, Jan . 22 , 1 865 ; A. Volterr a t o V . Camiz , Sept . 1 8 , 1 864 , VP .
3
A. Volterr a t o Alfons o Almagia , Mar . 5 , 1 865 , VP.
4
Charles Neider, ed., The Complete Travel Books of Mark Twain (Garde n
City: Doubleday , 1 966) , 1 63-1 66 .
5
A. Volterr a t o V . Camiz , Aug . 20 , 1 866 , VP .
6
Ibid, Oct , 7 , 1 866 , VP .
7
A. Volterr a t o A . Almagia , Aug . 1 5 , 1 869 , VP .
NOTES 28
8
Ibid., Aug . 1 5 , 1 869 , VP.
9
Vito Volterr a t o A . Almagia , Jul y 29 , 1 869 , VP.
10
u
3
Ibid, Aug . 1 5 , 1 869 , VP.
I b i d , Jul y 29 , 1 869 , VP.
12
Ibid, Aug . 22 , 1 869 , VP.
13
A. Volterr a t o A . Almagia , Aug . 1 5 , 1 869 , VP.
14
V. Cami z t o A . Almagia , Aug . 22 , 1 869 , VP.
15
V. Volterra to A. Almagia an d Fortunata Almagi a to A. Almagia, [Aug. ]
1869; A . Volterr a t o A . Almagi a [Sept. ] 1 869 , VP.
16
V. Volterr a t o A . Almagia , Sept . 5 , 1 869 ; V . Volterr a t o A . Volterra ,
same date , VP .
Notes t o Chapte r 3
1
Angelica Almagi a t o Edoard o Almagia , Aug . 1 6 , 1 875 , in Th e Volterr a
papers, privat e collectio n (V P hereafter) .
2
[Joseph Peres?], "Materiau x pour une biographie du mathematicien Vit o
Volterra," Archeion 2 3 (1 941 ) : 326 .
3
Vito Volterra , "L'Evolutio n de s idee s fondementale s d u calcu l infini tesimal," i n Josep h Peres , ed. , Legons sur les fonctions des lignes (Paris :
Gauthier-Villars, 1 91 3) , 8-9 .
4
Edoardo Almagi a t o Vit o Volterra , Oct . 1 876 , VP.
5
6
"Materiaux pou r un e biographie, " 328 .
Alfonso Almagi a t o E . Almagia . Nov . 1 3 , 1 876 , VP.
7
Ibid, VP .
8
E. Almagi a t o A . Almagia , Nov . 20 , 1 876 , VP .
9
A. Almagi a t o E . Almagia , Nov . 26 , 1 876 , VP.
10
The exchange is related i n Alessandro Faedo, "Discors i pronunciati nell a
cerimonia tenut a press o l a Scuol a Normal e Superior e d i Pis a i l giorn o 2 0
novembre 1 960, " i n Vito Volterra nel primo centenario della nascita (Rome :
Accademia Nazional e de i Lincei , 1 961 ) , 43.
n
E . Almagi a t o A . Almagia , Jun e 1 3 , 1 877 , VP.
12
Ibid. [Aug. ] 1 877 , VP.
13
E. Almagia , "Da l libr o d i memorie, " Mar . 1 2 , 1 867 , VP .
14
E. Almagi a t o Alfons o Almagia , [Aug. ] 1 877 , VP.
284 NOTE
15
S
A. Almagi a t o E . Almagia , Aug . 3 , 1 877 , VP .
16
Ibid., Aug . 25 , 1 87 7 an d Sept . 8 , 1 877 , VP ; Antoni o Roit i t o A . Al magia, Sept . 1 , 1 877 , i n Giovann i Paolini , ed. , Vito Volterra e il suo tempo
(Rome: Accademi a Nazional e de i Lincei , 1 990) , 4.
17
A. Almagi a t o E . Almagia , Oct . 7 , 1 877 , VP.
18
A. Volterra to E. Almagia , Oct . 7 , 1 877 , and E . Almagia to A . Almagia ,
Oct. 1 1 , 1877, VP.
19
E. Almagi a t o Vit o Volterra , Mar . 1 5 , 1 889 , VP.
Notes t o Chapte r 4
1
Vito Volterra to Angelica Volterra, Nov. 5 , 1878, in The Volterra papers,
private collectio n (V P hereafter) .
2
Ibid., Dec . 9 , 1 878 , VP.
3
A. Volterr a t o V . Volterra , Nov . 4 , 1 878 , VP.
4
Ibid., Nov . 6 , 1 878 , VP.
5
Alfonso Almagi a t o V . Volterra , Nov . 6 , 1 878 .
6
V. Volterr a t o A . Volterra , Nov . 1 5 , 1 878 , VP.
7
Ibid., Dec . 7 , 1 878 , VP.
8
Ibid., Jan . 1 0 , 1 879 , VP.
9
Ibid., Nov . 1 0 , 1 878 , VP.
10
Giovanni Paoloni , ed. , Vito Volterra e il suo tempo (Rome : Accademi a
Nazionale de i Lincei , 1 990) , 6-7 .
n
V . Volterr a t o A . Volterra , Nov . 1 0 , 1 878 , VP.
12
A. Almagi a t o V . Volterra , Jan . 1 1 , 1879, VP.
13
V. Volterr a t o A . Volterra , Jan . 1 2 , 1 879 , VP.
14
A. Almagi a t o V . Volterra , Nov . 6 , 1 878 , VP.
15
V. Volterr a t o A . Volterra , Nov . 1 2 , 1 878 , VP.
16
Paoloni, Vit o Volterra , 5-6 .
17
V. Volterr a t o A . Volterra , Nov . 8 , 1 878 , VP.
18
Ibid., Nov . 1 7 , 1 878 , VP.
19
Ibid., Feb , 1 , 1 879 , VP.
20
Ibid., Jan . 1 4 , 1 879 , VP.
21
Fedele Romani , " I miei ricord i d i Pisa, " La Lettura 8 (1 908) : 1 20-1 21 .
NOTES 28
5
22
John A . Davis, "Ital y 1 796-1 870 : Th e Risorgimento," i n George Holmes,
ed., The Oxford History of Italy (Oxford : Oxfor d Press , 1 997) , 208.
23
V. Volterr a t o A . Volterra , Nov . 9 , 1 878 , VP.
24
0scar Browning , Memories of Sixty Years at Eton, Cambridge, and
Elsewhere (London : Joh n Lane , 1 91 0) , 1 70 .
25
V. Volterr a t o A . Volterra , Nov . 21 , 1878, VP.
26
Alessandro D'Ancona , Ricordi ed affetti (Milan : Fratell i Treves , 1 901 ) ,
243.
27
V. Volterr a t o A . Volterra , Dec . 1 1 , 1878, VP.
28
Romani, " I miei ricordi, " 1 22 .
29
V. Volterr a t o A . Volterra , [Jan.] , 1 879 , VP.
30
Ibid.
31
Ibid., Jan . 1 4 , 1 879 , VP.
32
Ibid., Apr . 1 , 1 879 , VP .
33
Ibid., Mar . 8 , 1 879 , VP.
34
Translation supplie d b y Professo r Georg e Pigman , privat e communica tion.
35
Carlo Somigliana , "Vit o Volterra," Pont Acad. Sci. Acta 6 (1942): 58 .
36
Tina Tomas i an d Nell a Sistoli Paola, L a scuola normale d i Pisa da l 1 81 3
al 1 94 5 (Pisa : ET S Editrice , 1 990) , 1 00 .
Notes t o Chapte r 5
1
Carlo Somigliana , "Tulli o Levi-Civit a e Vit o Volterra, " Rendiconti del
seminario matematico e fisico di Milano, 1 7 (1 946) : 8 .
2
Ernesto Padova , "Commemorazion e d i Enric o Betti, " Atti. R.
Veneto di Scienze, Lettere ed Arti 4 (1 892-93) : 61 4 .
1 st.
3
V. Volterra , "Alcun e osservazion i sull e funzion i punteggiat e discontin ue," i n Opere matematiche, vol . 1 (Rome: Accademi a Nazional e de i Lincei ,
1954-62), 7-1 5 .
4
Enrico Giust i an d Luig i Pepe , eds. . La matematica in Italia, 1 800-1 950
(Florence: Polistampa , 2001 ) , 1 28 .
5
Vito Volterr a t o Angelic a Volterra , Nov . 1 1
papers, privat e collectio n (V P hereafter) .
6
, 1 880 , i n Th e Volterr a
V. Volterra, "Su i principi del calcolo integrale," Opere matematiche, vol .
1, 1 6-48 .
286
NOTES
7
C. Somigliana , "L'oper a scientific a d i Vit o Volterra, " ibid. , xv .
8
The Frenc h mathematicia n Henr i Leo n Lebesgu e late r showe d ho w t o
generalize Rieman n integrabilit y t o includ e thes e pathologica l functions .
(Dr. Michel e Vallisneri , privat e communication. )
9
C. Somiglian a t o V . Volterra , n.d. , 1 881 , Vito Volterr a Collection , Ac cademia Nazional e de i Lincei , Rome .
10
n
Ibid., Ma y 4 , 1 882 .
Ibid., Dec . 9 , 1 883 .
12
Ibid., Sept . 1 9 , 1885.
13
Ibid., Oct . 4 , 1 885 .
14
Angelica Volterr a t o V . Volterra , Ma y 8 , 1 881 , VR
15
Alfonso Almagi a t o V . Volterra , Ja n 1 7 , 1 880 , VP.
16
Ibid., Jun e 1 2 , 1 880 , VP.
17
Ibid, Jun e 2 , 1 881 , VP.
18
Ibid., Jun e 8 , 1 881 , VP.
19
V. Volterr a t o Angelic a Volterra , Apri l 29 , 1 881 , VP.
20
V. Volterra , "Su l potenziale d i un'ellissoid e eterogene a sopr a s e stessa, "
in Opere matematiche, vol . 1 , 1 -6 .
21
Alfonso Almagi a t o V . Volterra , Jun e 8 , 1 881 , VP.
22
Angelica Volterr a t o V . Volterra , Jun e 9 , 1 881 , VR
23
Edoardo Almagi a t o Angelic a Volterra , Aug . 1 8 , 1 881 , VP.
24
V. Volterra , "Enric o Betti, " Opere matematiche, vol . 1 , 600.
25
Enrico Bett i to V . Volterra, Oc t 31 , 1882, in Giovann i Paoloni , ed. , Vito
Volterra e il suo tempo (Rome : Accademi a Nazional e de i Lincei , 1 990) , 7-8 .
26
V. Volterr a t o E . Betti , n.d. , bu t Nov .1 882 , i n Paoloni , ed . Vito
Volterra, 8.
27
Alfonso Almagi a t o V . Volterra , Dec . 1 8 , 1 882 , VP.
28
Ibid., Mar . 5 , 1 883 , VP.
29
Ibid., Mar . 7 , 1 883 , VP.
30
Antonio Roit i t o V . Volterra , n.d. , bu t Mar . 1 7 , 1 883 , i n Paoloni , ed. ,
Vito Volterra, 9 .
31
Alfonso Almagi a t o V . Volterra , Mar . 7 , 1 883 , VP.
32
Corrado Pado a t o Enric o Stracciati , Apri l 1 883 , VP.
287
NOTES
Notes t o Chapte r 6
1
H. Stuar t Hughes , Prisoners of Hope: The Silver Age of Italian Jews,
1924-1974 (Cambridge : Harvar d Univ . Press , 1 983) , 1 8 .
2
Leonardo Sciascia , "I I quarantotto," i n Gli zii di Sicilia (Turin : Einaudi ,
1958), 161 .
3
Angelica Volterra to Vito Volterra, Mar. 1 , 1885, in The Volterra papers,
private collectio n (V P hereafter) .
4
L.S.D., Trip to the Sunny South in March 1 885 (Birkenhead , U.K. : E .
Griffith & Son, 1 885) , 48-9.
5
Alfonso Volterr a t o V . Volterra , wit h postscrip t b y A . Almagia , Jul y 5 ,
1885, VP .
6
Epicarmo Corbino , Racconto di una vita (Naples : Edizion i scientifich e
Italiane, 1 972) , 28.
7
Alfonso Volterr a t o V . Volterra , Jul y 1 4 , 1 885 , VP.
8
Angelica Volterr a t o V . Volterra , Nov . 4 , 1 884 , VP .
9
Ibid., Mar . 30 , 1 884 , VP.
10
n
Ibid., Nov . 9 , 1 884 , VP .
Alfonso Almagi a t o V . Volterra , Mar . 1 1 , 1880, VP .
12
Angelica Volterr a t o V . Volterra , Jun e 1 1 , 1 884 , wit h postscrip t b y A .
Almagia, VP .
13
Ibid., Dec . 1 8 , 1 884 , VP .
14
Ibid., Apr . 25 , 1 887 , VP.
15
Ibid., Jul y 6 , 1 887 , VP.
16
Eugenio Beltrami to V. Volterra, Jul y 1 9 , 1883, Vito Volterra Collection ,
Accademia Nazional e de i Lincei , Rome .
17
Enrico Bett i t o V . Volterra , Sept . 1 5 , 1 887 , i n Giovann i Paoloni , ed. ,
Vito Volterra e il suo tempo (Rome : Accademi a Nazional e de i Lincei, 1 990) ,
15.
18
V. Volterra t o E . Betti, undate d draft , bu t probabl y Sept . 1 887 , in Vit o
Volterra Collection .
19
V. Volterra, Theory of Functional, ed . Luig i Fantappie, trans . M . Long
(London: Blacki e & Son, 1 930) , 22.
20
.
Opere matematiche, vol . 1
Lincei, 1 954-62) , 294 .
(Rome : Accademi a Nazional e de i
21
Jacques Hadamard , The Psychology of Invention in the Mathematical
Field (Princeton : Princeto n Univ. Press , 1 945 ; Dover reprint, 1 954) , 129-30.
22
Carlo Somiglian a t o V . Volterra, Ma y 4 , 1 882 , Vito Volterr a Collection .
288
NOTES
Notes t o Chapte r 7
1
Giovanni Battist a Gucci a t o Vit o Volterra , Dec . 4 , 1 887 , i n Giovann i
Paoloni, ed. , Vito Volterra e il suo tempo (Rome : Accademi a Nazional e de i
Lincei, 1 990) , 1 6 .
2
Dirk J . Struik , A Concise History of Mathematics, vol . 2 (Ne w York :
Dover Publications , 1 948) , 276 .
3
V. Volterra to Enrico Betti, Aug. 1 , 1888, Betti Archive, Scuola Normal e
Superiore d i Pisa .
4
Ann Hibne r Koblitz , A Convergence of Lives: Sofia Kovalevskaia: Scientist, Writer, Revolutionary (Basel : Birkhauser , 1 983) , 1 36 .
5
Ibid., 1 86 .
6
Roger Cooke , The Mathematics of Sonya Kovalevskaya (Ne w York :
Springer-Verlag, 1 984) , 1 1 0 .
7
V. Volterr a t o E . Betti , Aug . 21 , 1888, Betti Archive .
8
Sophie Kowalevski , "Su r l e problem e d e l a rotatio n d'u n corp s solid e
autour d'u n poin t fixe," Acta Mathematica 1 2 (1 889) : 1 71 -232 .
9
10
n
Struik, A Concise History, 237 .
V. Volterr a t o E . Betti , Aug . 21 , 1888, Betti Archive .
Ibid., Aug . 1 , 1888.
12
Salvatore Pincherle , "Ricerch e sopr a un a class e important e d i funzion i
monodrome," Giornale di Matematiche 1 8 (1 880) : 92-1 36 .
13
Eugenio Beltram i t o Jule s Hoiiel , Feb . 1 0 , 1 881 , in L . Boi , L . Giacardi ,
and R . Tazzioli , eds. , La decouverte de la geometrie non euclidienne sur
la pseudosphere: Les lettres d'Eugenio Beltrami a Jules Hoiiel (Paris : A .
Blanchard, 1 998) , 1 92 .
14
Carlo Somiglian a t o V . Volterra, Jan . 12, 1 890 , Vito Volterra Collection ,
Accademia Nazional e de i Lincei , Rome .
15
Cooke, The Mathematics of Sonya Kovalevskaya, 1 73 .
16
Gabriel Lame , Legons sur la theorie mathematique de Velasticite des
corps solides (Paris : Bachelier , 1 852) .
17
Sofia Kovalevskaya , "Ube r di e Brechun g de s Lichte s i n crystallinische n
Mitteln," Acta Mathematica 6 (1 885) : 249-304 .
18
19
Cooke, The Mathematics of Sonya Kovalevskaya, 1 73 .
V. Volterra , "Su r le s vibration s lumineuse s dan s le s milieu x birefrin gents," Acta Mathematica 1 6 (1 891 ) : 1 53-21 5 .
NOTES 28
9
20
Roger Cooke , "Sony a Kovalevskaya's Place in Nineteenth-Century Math ematics," i n Lind a Keen , ed. , The Legacy of Sonya Kovalevskaya 6 4 (Prov idence: America n Mathematica l Society , 1 987) , 36 .
21
Paoloni, ed. , Vito Volterra e il suo tempo, 75 .
22
Constance Reid , Hilbert (Ne w York : Springer-Verlag , 1 970) , 47.
23
Struik, A Concise History, 271 .
24
David E . Rowe , "'Jewis h Mathematics ' a t Gottinge n i n the Er a o f Feli x
Klein," Isis 7 7 (1 986) : 432 .
25
V. Volterr a t o Angelic a Volterra , Jul y 25 , 1 891 , in Th e Volterr a papers ,
private collectio n (V P hereafter) .
26
Ibid., Jul y 1 6 and 25 , 1 891 , VP.
27
Ibid., Jul y 1 6 , Jul y 6 , Jul y 2 5 and Jul y 27 , 1 891 , VP.
28
Ibid., Jul y 27 , 1 891 , VP.
Notes t o Chapte r 8
x
Vito Volterra , "Enric o Betti, " II Nuovo Cimento 3 2 (1 893) : 5-7 .
2
Federigo Enrique s to Guid o Castelnuovo , Oct . 3 , 1893, in U. Bottazzini ,
A. Conte, P. Gario, eds., Riposte armonie (Turin : Bollat i Boringhieri, 1 996) ,
28.
3
Vito Volterra to Ulisse Dini, n.d. i n Giovanni Paoloni , ed. , Vito Volterra
e il suo tempo (Rome : Accademi a Nazional e de i Lincei , 1 990) , 23.
4
Antonio Roit i to V. Volterra, telegram , n.d . bu t probabl y ca . Jul y 1 893 ,
Vito Volterr a Collection , Accademi a Nazional e de i Lincei , Rome .
5
Riccardo Felic i to V . Volterra, Aug . 9 , 1 893 , Paoloni, ed. , Vito Volterra
e il suo tempo, 24 .
6
A. Roit i t o V . Volterra , Sept . 21 , 1893, ibid., 26 .
7
Enrico D'Ovidi o t o V . Volterra , Sept . 23 , 1 893 , ibid., 26 .
8
V. Volterr a t o E . D'Ovidio , Sept . 30 , 1 893 , Vito Volterr a Collection .
9
E. D'Ovidi o t o V . Volterra , Oct . 5 , 1 893 , ibid .
10
V. Volterr a t o Luig i Cremona , Oct .1 3 , 1 893 , i n Paoloni , ed. , Vito
Volterra e il suo tempo, 27 .
n
L . Cremon a t o V . Volterra , Oct . 1 7 , 1 893 , ibid., 28 .
12
V. Volterr a t o Andre a Naccari , Oct . 22 , 1 893 , ibid., 28 .
13
V. Volterr a t o A . Roiti , Oct . 24 , 1 893 , Vito Volterr a Collection .
290
NOTES
14
Quoted i n Angel o d'Orsi , "U n profilo culturale, " i n Valerio Castronovo ,
Torino (Bari : Laterza , 1 987) , 523.
15
Primo Levi , i n Vivia n B . Mann , ed. , Gardens and Ghettos: The Art of
Jewish Life in Italy (Berkeley : Univ . o f Californi a Press , 1 989) , xv .
16
Quoted i n Susa n Zuccotti , The Italians and the Holocaust (Ne w York :
Basic Books , 1 987) , 1 9 .
17
Roberto Maiocchi , "I I ruolo dell e scienz e nell o svilupp o industrial e ital iano," i n Scienze e tecnica nella cultura e nella societa dal rinascimento a
oggi, Storia d'ltalia , vol . 3 (Turin , Giuli o Einaudi , 1 980) , 877 .
18
Adrian Lyttelton , "Politic s an d Society , 1 870-1 91 5, " i n Georg e Holmes ,
ed., The Oxford History of Italy (Oxford : Oxfor d Universit y Press , 1 997) ,
251.
19
Livia Giacardi, "Corrad o Segre maestro a Torino. L a nascita della scuola
italiana d i geometria algebrica, " i n Annali di storia delle Universita italiane
5 (2001 ) : 1 62-63 .
20
BeppoLevi, "L a personalidad d e Vito Volterra," Publ . Inst . Mat . Univ .
Nac. Litora l 3 (1 941 ) : 25 .
21
Quoted i n Huber t C . Kennedy , Peano (Boston : D . Reidel , 1 980) , 54 .
Notes t o Chapte r 9
1
Costantino Botto, quoted in Hubert C . Kennedy, Peano: Life and Works
of Giuseppe Peano (Dordrecht : D . Reidel , 1 980) , 101 .
2
Vito Volterra , "Replic a a d un a not a de l Prof . Peano, " Opere matematiche, vol . 2 (Rome : Accademi a Nazional e de i Lincei , 1 954-62), 21 3-21 5 .
3
V. Volterr a t o Luig i Bianchi , Ma y 1 8 , 1 895 , quoted i n Angel o Guerrag gio, "L e memori e d i Volterr a e Pean o su l moviment o de i poli, " Archive for
History of Exact Sciences 3 1 (Nov . 1 984) : 1 1 2 .
4
Giuseppe Peano, "Sopr a lo spostamento del polo sulla terra," Atti Accad.
sci. Torino 3 0 (May , 1 895) : 51 5 .
5
Guerraggio, "L e memori e d i Volterr a e Peano," 1 1 2-1 1 3 .
6
Quoted i n Herman n Brunner , "1 896-1 996 : On e Hundre d Year s o f Volterra Integra l Equation s o f the Firs t Kind, " Applied Numerical Mathematics
24 (1 997) : 83 .
7
V. Volterra , "Su i mot i periodic i de l pol o terrestre, " i n Opere matematiche, vol . 2 , 1 41 -1 51 .
NOTES 29 1
8
Guido Castelnuov o an d Carl o Somigliana , "Vit o Volterra e la su a oper a
scientifica," i n Atti della Accademia nazionale dei Lincei, Oct .1 7 , 1 94 6
(Rome, 1 947) , 1 4-1 5 .
9
G. Pean o t o V . Volterra , Ma y 30 , 1 895 , i n Giovann i Paoloni , ed. , Vito
Volterra e il suo tempo (Rome : Accademi a Nazional e de i Lincei , 1 990) ,
33-34.
10
V. Volterr a t o G . Peano , Jun e 2 , 1 895 , ibid., 34 .
n
V . Volterr a t o C . Segre , Jun e 2 , 1 895 , Vit o Volterr a Collection , Ac cademia Nazional e de i Lincei , Rome .
12
Guerraggio, "L e memorie d i Volterr a e Peano," 1 1 3-1 1 4 .
13
C. Segr e t o V . Volterra , Ma y 30 , 1 895 , Vito Volterr a Collection .
14
Kennedy, Peano, 39 .
15
Ibid, 59 .
16
V. Volterra , Opere matematiche, vol . 2 , 1 70-1 72 ; als o quote d i n Kenne dy, Peano, 60 .
17
Kennedy, Peano, 59 .
18
Ibid., 60 .
19
V. Volterra, "Replic a ad una nota del Prof. Peano, " Opere matematiche,
vol. 2 , 213-215.
20
V. Volterr a t o Tulli o Levi-Civita , Jan .1 5 , 1 896 , Tulli o Levi-Civit a
Collection, Accademi a Nazional e de i Lincei , Rome .
21
V. Volterra , "Su r l a theori e de s variation s de s latitudes, " i n Opere
matematiche, vol . 2 , 452-573.
22
V. Volterr a t o C . Somigliana , Apri l 1 9 , 1 89 6 an d C . Somiglian a t o V .
Volterra, Apri l 22 , 1 896 , Vito Volterr a Collection .
23
Prancesco Tricomi , La mia vita di matematico attraverso la cronistoria
dei miei lavori (Padova : CEDAM , 1 967) , 1 8 .
24
V. Volterra , "Sull a inversion e degl i integral i definiti, " Opere matematiche, vol . 2 , 21 6 .
25
V. Volterr a t o T . Levi-Civita , Feb . 29 , 1 896 , Tulli o Levi-Civit a Collec tion.
26
W,F. Osgood , "Th e Internationa l Congres s o f Mathematician s a t Zu rich," Bull Am. Math. Soc. 4 (1 898) : 47 .
27
V. Volterr a t o Giovann i Vailati , Aug . 25 , 1897, Giovanni Vailat i Collec tion, Universit y o f Milan .
28
Ibid., Aug . 1 4 , 1 898 , Vito Volterr a Collection .
292
NOTES
29
Ibid., Jul y 24 , 1 898 , Giovanni Vailat i Collection , Universit y o f Milan. A
copy o f the lette r supplie d throug h th e courtes y o f Dr . Maur o D e Zan .
Notes t o Chapte r 1 0
1
Oscar Wilde , The Importance of Being Earnest (Ne w York, Dove r Pub lications, 1 990) , 1 2 .
2
Angelica Volterr a t o Vit o Volterra , Nov .1 0 , 1 899 , i n Th e Volterr a
papers, privat e collectio n (V P hereafter) .
3
Ibid., Nov . 1 9 , 1 899 , VP.
4
Ibid, Nov . 1 0 , 1 899 , VP .
5
Ibid.
6
Antonio Roit i t o V . Volterra , Dec . 1 1 , 1899, VP.
7
A. Volterr a t o V . Volterra , Dec . 1 9 , 1 899 , VP.
8
Guido Castelnuov o t o V . Volterra , Mar . 2 , 1 900 , Vit o Volterr a Collection, Accademi a Nazional e de i Lincei , Rome .
9
10
A. Roit i t o V . Volterra , Mar . 8 , 1 900 , Vito Volterr a Collection .
V. Volterr a t o Valentin o Cerruti , Mar . 1 1 , 1900, VP.
n
V . Cerrut i t o V. Volterra , Mar . 1 2 , 1 900 , in Giovann i Paoloni , ed. , Vito
Volterra e il suo tempo (Rome : Accademi a Nazional e de i Lincei , 1 990) ,
41-42.
12
V. Volterr a t o A . Roiti , Mar . 1 4 , 1 900 , Vito Volterr a Collection .
13
A. Roit i t o V . Volterra , Mar . 1 5 , 1 900 , Vito Volterr a Collection .
14
Ibid., Mar . 22 , 1 900 .
15
V. Volterr a t o A . Roiti , Mar . 25 , 1 900 , Vit o Volterr a Collection .
16
V. Volterr a t o A . Volterra , Mar . 27 , 1 900 , VP .
17
A. Volterr a t o V . Volterra , Ma y 1 0 , 1 900 , VP .
18
V. Volterr a t o A . Volterra , Ma y 1 0 , 1 900 , VP .
19
Ibid, Ma y 1 2 , 1 900 , VP.
20
A. Roit i t o V . Volterra , Ma y 6 , 1 900 , Vito Volterr a Collection .
21
V. Volterr a t o A . Volterra , Ma y 1 3 , 1 900 , VP .
22
G. Castelnuov o t o V . Volterra , Ma y 1 7 , 1 900 , Vito Volterr a Collection .
23
V. Volterr a t o A . Volterra , Ma y 21 , 1900, VP .
24
A. Roit i t o V . Volterra , Ma y 24 , 1 900 , Vito Volterr a Collection .
25
Edoardo Almagi a t o Virgini a Almagia , Jun e 2 , 1 900 , VP .
NOTES 29
3
26
A. Roit i t o V . Volterra , Jun e 1 900 , undated , bu t ca . Jun e 1 9 , Vit o
Volterra Collection .
27
G. Castelnuov o t o V . Volterra , Jun e 1 7 , 1 900 , Vito Volterr a Collection .
28
G. Castelnuov o t o V . Volterra , Jun e 1 9 , 1 900 , Vito Volterr a Collection .
29
V. Volterra, "Appunt i di Vito per cattedra Roma, " undated , bu t writte n
after Jun e 28 , 1 900 , VP.
30
Ibid.
31
V. Volterr a t o Virgini a Almagia , Jun e 28 , 1 900 , V R
32
Corrado Segr e t o V . Volterra , Jun e 1 9 , 1 900 , Vito Volterr a Collection .
33
V. Volterr a t o V . Almagia , Jul y 1 , 1 900 , VP.
34
Interview o f Ginett a Montecorboli , b y Carlott a Scaramuzzi , 1 995 , p. 4 ,
Institute Archives , Californi a Institut e o f Technology .
35
V. Volterr a t o A . Volterra , Jul y 1 2 , 1 900 , VP.
36
Pietro Blasern a t o V . Volterra , Sept . 8 , 1 900 , Vito Volterr a Collection .
Notes t o Chapte r 1 1
1
Virginia an d Vit o Volterr a t o Angelic a Volterra , Jul y 28 , 1 900 , Th e
Volterra papers , privat e collectio n (V P hereafter) .
2
A. Volterr a t o Vit o Volterra , Jul y 26 , 1 900 , VP .
3
Virginia Volterr a t o Edoard o Almagia , Jul y 31 , 1900, VP.
4
Virginia Volterr a t o A . Volterra , Jul y 28 , 1 900 , VP.
5
Ibid., Aug . 6 , 1 900 , VP.
6
V. Volterra , "Betti , Brioschi , Casorati : Tr e analist i e tr e mod i d i con siderare l e question i d'analisi, " i n Saggi scientifici (1 920 ; reprint , wit h a n
introduction b y Raffaell a Simili , Bologna : Zanichelli , 1 990) , 37-38.
7
Virginia Volterr a t o A . Volterra , Aug . 6 , 1 900 , VP.
8
Corrado Segr e t o V . Volterra , Sept . 1 1 , 1900, Vit o Volterr a Collection ,
Accademia Nazional e de i Lincei , Rome .
9
David Hilbert , "Mathematica l Problems, " trans . Dr . Mar y Winto n
Newson, Bull. Am. Math. Soc. 8 (1 902) : 437-45 , 478-89.
10
V. Volterra, Theory of Functionate, Luig i Fantappie, ed. , trans. M . Long
(London: Blacki e & Son, 1 930) , vii-viii .
11
Quoted i n Benjami n H . Yandell , The Honors Class: Hilbert 9s Problems
and Their Solvers (Natick , MA : A . K . Peters , 2002) , 424.
294 NOTE
S
12
C. A . Scott , "Th e Internationa l Congres s o f Mathematician s i n Paris, "
Bull. Am. Math. Soc. 7 (1 901 ) : 76 .
13
Vito Volterr a t o Virgini a Volterra , Sept . 25 , 1 900 , VP.
14
Emilio Segre , A Mind Always in Motion (Berkele y an d Lo s Angeles :
Univ. o f Californi a Press , 1 993) , 52-3.
15
Julius Weingarte n t o V . Volterra , Nov . 1 1 , 1 900 , Vit o Volterr a Collec tion, Accademi a Nazional e de i Lincei , Rome .
16
V. Volterr a t o J . Weingarten , Nov . 30 , 1 900 , ibid .
17
J. Weingarte n t o V . Volterra , Dec . 4 , 1 900 , ibid .
18
V. Volterra , "U n teorema sull a teori a dell a elasticita, " Rend. R. Accad.
Lincei 1 4 (1 905) : 1 27-37 .
19
V. Volterr a t o J . Weingarten , Feb . 27 , 1 905 , Vito Volterr a Collection .
20
V. Volterra , "Henr i Poincare, " i n The Book of the Opening of the Rice
Institute, vol . 3 (Houston , Texas : Th e D e Vinn e Press , 1 91 2) , 912-13.
21
V. Volterra , "Sull e equazion i integro-differenziali, " Rend. R.
Lincei 1 8 (1 909) : 1 67-74 .
22
Accad.
V. Volterr a t o Giovann i Vailati , Dec . 7 , 1 900 , Vito Volterr a Collection .
23
Interview of Luisa Almagia by Carlotta Scaramuzzi , 1 996 , p. 5 , Institute
Archives, Californi a Institut e o f Technology .
24
Personal communication fro m Edoardo' s daughter Dr . Virgini a Volterra ,
Dec. 30 , 1 996 .
25
V. Volterr a t o G . Vailati , Jul y 1 , 1 901 , Giovann i Vailat i Collection ,
University o f Milan .
26
G. Vailat i t o V . Volterra , Jul y 3 , 1 901 , Vito Volterr a Collection .
27
V. Volterr a t o Virgini a Volterra , Aug . 2 , 1 901 , VP.
28
Ibid., Aug. 8, 1901.
29
Ibid., Aug. 14, 1901.
30
Ibid., Aug. 15, 1901.
31
Ibid., Aug. 17, 1901.
32
Ibid., Aug. 15, 1902.
33
Edoardo Volterr a t o Virgini a Volterra , undated , bu t summe r 1 902 , VP.
34
V. Volterr a t o Virgini a Volterra , Aug . 1 4 , 1 902 , VP.
35
Ibid., Aug . 20 , 1 902 .
36
Quintino Sell a to Rosa Sella, Nov. 1 7 , 1879, Vito Volterra files, Institut e
Archives, Californi a Institut e o f Technology .
37
V. Volterr a t o Alfons o Sella , Aug . 29 , 1 902 , ibid .
NOTES 29
38
5
V. Volterr a t o Virgini a Volterra , Sept . 6 , 1 902 , VP.
Notes t o Chapte r 1 2
1
Vito Volterra to Virginia Volterra, Oct . 5 , 1902, in The Volterra papers ,
private collectio n (V P hereafter) .
2
Vito Volterr a t o Nunzi o Nasi , Feb . 1 7 , 1902, draft, i n Giovann i Paoloni ,
ed., Vito Volterra e il suo tempo (Rome : Accademi a Nazional e de i Lincei ,
1990), 43-44 .
3
V. Volterr a t o Tulli o Levi-Civita , Mar .1 3 , 1 905, Tulli o Levi-Civit a
Collection, Accademi a Nazional e de i Lincei , Rome .
4
V. Volterr a t o Virgini a Volterra , Aug . 9 , 1 903 , VP.
5
V. Volterr a t o Virgini a Volterra , Augus t 1 8 , 1 903 , VP.
6
V. Volterr a t o Virgini a Volterra , Aug . 1 3 , 1 903 , VP.
7
V. Volterr a t o Virgini a Volterra , Augus t 1 8 , 1 903 , VP.
8
Ibid.
9
V. Volterr a t o Virgini a Volterra , Augus t 21 , 1903, VP.
10
V. Volterr a t o Virgini a Volterra , Feb . 1 3 , 1 904 , VP .
n
Hermann Weyl , "Davi d Hilbert : 1 862-1 943, " Obituar y Notice s o f Fel lows o f th e Roya l Societ y o f Londo n 4 (1 944) , 547 .
12
V. Volterr a t o Virgini a Volterra , Feb . 1 9 , 1 904 , VP .
13
V. Volterr a t o Virgini a Volterra , Feb . 21 , 1904, VP .
14
V. Volterr a t o Virgini a Volterra , Feb . 20 , 1 904 , VP .
15
V. Volterr a t o Virgini a Volterra , Feb . 25 , 1 904 , VP .
16
V. Volterr a t o Virgini a Volterra , Feb . 29 , 1 904 , VP .
17
V. Volterr a t o Virgini a Volterra , Feb . 1 8 , 1 904 , VP .
18
Stanislao Cannizzar o t o V . Volterra , Jun e 1 0 , 1 904 , Vit o Volterr a files ,
Institute Archives , Californi a Institut e o f Technology .
19
Atti Parlamentari , Legislatur a XXII , 1 s t session , 1 904-906 , Jun e 1 9 ,
1906.
20
21
V. Volterr a t o Angelic a Almagia , Aug . 20 , 1 904 , VP .
Vito an d Virgini a Volterr a t o th e Almagi a family , Aug . 28 , 1 904 , VP.
22
"L'Illustrazione Italiana, " XXXII , 1 905 , 243.
23
Guido Castelnuovo , "Vit o Volterra, " Opere matematiche, vol . 1 , ix-x .
296
NOTES
Notes t o Chapte r 1 3
x
Cesare Arzela to Vito Volterra, Jul y 1 5 , 1905, Veronica Gavagna , "Dall a
teoria dell e funzioni all'analis i funzionale : i l carteggio Arzela-Volterra, " Bollettino di storia delle scienze matematiche 1 4 (1 994) , 70 .
2
Guido Castelnuov o t o Vit o Volterra , Jul y 29 , 1 905 , Vit o Volterr a Col lection, Accademi a Nazional e de i Lincei , Rome .
3
Vito Volterr a t o Virgini a Volterra , Jul y 1 8 , 1 905 , The Volterr a Papers ,
private collectio n (hereafter , VP) .
4
V. Volterr a t o Virgini a Volterra , Jul y 1 9 , 1 905 , VP.
5
V. Volterr a t o Virgini a Volterra , Jul y 26 , 1 905 , VP.
6
V. Volterr a t o Virgini a Volterra , Aug . 4 , 1 905 , VP.
7
V. Volterr a t o Virgini a Volterra , Aug . 7 , 1 905 , VP.
8
V. Volterr a t o Virgini a Volterra , Aug . 8 , 1 905 , VP.
9
Edoardo Almagi a t o Virgini a Volterra , Aug . 1 5 , 1 905 , VP.
10
U
V. Volterr a t o Virgini a Volterra , Sept . 3 , 1 905 , VP.
V . Volterr a t o Virgini a Volterra , Aug . 29 , 1 905 , VP.
12
V. Volterr a t o Virgini a Volterra , Sept . 25 , 1 90 5 and Sept . 26 , 1 90 5 an d
Sept. 28 , 1 905 , VP.
13
V. Volterr a t o Virgini a Volterra , Sept . 28 , 1 905 , VP.
14
Vito Volterra , "Legon s sur l'integratio n de s equations differentielle s au x
derivees partielles," Opere matematiche, vol . 3 (Rome: Accademi a Nazional e
dei Lincei , 1 954-62 ) 64 .
15
V. Volterr a t o Virgini a Volterra , Feb . 1 5 , 1 906 , VP.
16
V. Volterr a t o Virgini a Volterra , Feb . 20 , 1 906 , VP.
17
V. Volterr a t o Virgini a Volterra , Feb . 1 4 , 1 90 6 an d Mar . 9 , 1 906 , VP.
18
Edoardo Almagi a t o Vit o Volterra , Feb . 23 , 1 906 , VP.
19
V. Volterr a t o Virgini a Volterra , Mar . 5 , 1 906 , VP.
20
V. Volterr a t o Virgini a Volterra , Mar . 1 0 , 1 906 , VP.
21
V. Volterra, "Propost a d i una assoziation e italian a pe r i l progresso dell e
scienze," Opere matematiche, vol . 3 , 1 48 .
22
Volterra, "Proposta, " 1 50 .
23
Ibid., 1 49 .
24
Volterra, "Proposta, " 1 51 .
NOTES 29
7
25
Giovanni B . Gucci a t o V . Volterra , Jun e 30 , 1 906 , i n Giovann i Paoloni ,
ed., Vito Volterra e il suo tempo (Rome : Accademi a Nazional e de i Lincei ,
1990), Fig . II . 3 .
26
G6sta Mittag-Leffle r t o Vit o Volterra , Aug .1 7 , 1 906 , Vit o Volterr a
Collection.
27
V. Volterr a t o Virgini a Volterra , Sept . 1 7 , 1 906 , VP.
28
V. Volterr a t o Virgini a Volterra , Sept . 21 , 1906, VP.
29
V. Volterr a t o Virgini a Volterra , Sept . 23 , 1 906 , VP .
30
E. Almagi a t o Ettor e Marchiafava , undate d bu t en d o f Oct . 1 906 , VP.
31
V. Volterr a t o Virgini a Volterra , Sept . 24 , 1 907 , VP.
32
Quoted i n Giusepp e Giuliani , / / Nuovo Cimento: novanVanni di fisica
in Italia, 1 855-1 944 (Pavia : L a Goliardic a Pavese , 1 996) , 29.
33
Orso Mari o Corbin o t o Tulli o Levi-Civita , Sept . 1 2 , 1 909 , Tulli o Levi Civita Papers , Accademi a Nazional e de i Lincei , Rome .
34
Ibid.
35
Arthur Gordo n Webste r t o Vit o Volterra , Jul y 1 2 , 1 909 , Vit o Volterr a
Collection.
36
G. Castelnuov o t o V . Volterra , Jul y 1 9 , 1 909 , Vito Volterr a Collection .
Notes t o Chapte r 1 4
x
Quoted i n J. J . O'Conno r an d E. F. Robertson, "Maxin e Bocher," [Inter net], St . Andrews , MacTuto r Histor y o f Mathematics , Augus t 200 5 [cite d 4
Aug. 2006] , available from: h t t p : //www-history. mcs. st-andrews. ac. uk/
Biographies/Bocher.html.
2
Maxime Bocher , An Introduction to the Study of Integral Equations
(New York : Hafne r Publishing , 1 971 ) , 24 .
3
Vito Volterra to Griffith Evans , Aug. 1 3 , 1913, Griffith C . Evans Papers,
Carton 1 3 , Bancroft Library , U C Berkeley .
4
Vito Volterr a t o Virgini a Volterra , Augus t 1 9 , 1 90 9 (enclosure) , i n Th e
Volterra papers , privat e collectio n (hereafter , VP) .
5
Vito Volterr a t o V . Volterra , Aug . 1 9 , 1 909 , V R
6
V. Volterr a t o Virgini a Volterra , Aug . 20 , 1 909 , VP .
7
V. Volterr a t o Virgini a Volterra , Aug . 23 , 1 909 , VP .
8
V. Volterr a t o Virgini a Volterra , ibid. , VP .
9
V. Volterr a t o Virgini a Volterra , Aug . 21 , 1909, VP .
298 NOTE
S
10
V. Volterr a t o Virgini a Volterra , Aug . 29 , 1 909 , VP.
11
V. Volterr a t o Virgini a Volterra , Aug . 31 , 1909, VP.
12
V. Volterr a t o Virgini a Volterra , Sept . 3 , 1 909 , VP.
13
V. Volterr a t o Virgini a Volterra , Sept . 3 , 1 909 , VP.
14
V. Volterra , Lectures delivered at the Celebration of the Twentieth Anniversary of the Foundation of Clark University (Worcester : Clar k Univer sity, 1 91 2) , 1 -2 .
15
V. Volterr a t o Virgini a Volterra , Sept . 7 , 1 909 , VP.
16
V. Volterr a t o Virgini a Volterra , Sept . 9 , 1 909 , VP.
17
V. Volterra , "Lectures, " 41 .
18
V. Volterr a t o Virgini a Volterra , Sept . 1 0 , 1 91 2 , VP.
19
V. Volterr a t o Virgini a Volterra , Sept . 1 3 , 1909, VP.
20
V. Volterr a t o Virgini a Volterra , Feb . 1 , 1 91 2 , VP.
21
V. Volterr a t o Virgini a Volterra , Sep t 1 9 , 1 909 , VP.
22
Federigo Enrique s t o Vit o Volterra , Dec . 1 7 , 1 909 , Vit o Volterr a Col lection, Accademi a Nazional e de i Lincei , Rome .
23
V. Volterr a t o Virgini a Volterra , Jun e 1 8 , 1 91 0 , VP.
24
V. Volterr a t o Virgini a Volterra , Jun e 26 , 1 91 0 , VP.
25
V. Volterr a t o Virgini a Volterra , Jul y 1 , 1 91 0 , VP.
26
V. Volterr a t o Virgini a Volterra , Jul y 8 , 1 91 0 , VP.
27
V. Volterr a t o Virgini a Volterra , Jul y 1 4 and Jul y 1 8 , 1 91 0 , VP .
28
V. Volterr a t o Virgini a Volterra , Aug . 3 , 1 91 0 , VP.
29
Griffith Evan s to Julian [surnam e unknown], Griffith Evan s Papers, Jul y
29, 1 968 , Carto n 1 , UC Berkeley .
30
E. Ro y Weintraub , How Economics Became a Mathematical Science
(Durham: Duk e Universit y Press , 2002) , 42-43 .
31
V. Volterr a t o Virgini a Volterra , Jan . 1 8 , and Jan . 1 9 , 1 91 2 , VP .
32
V. Volterr a t o Virgini a Volterra , Feb . 1 1 , 1912, VP.
33
V. Volterr a t o Virgini a Volterra , Feb . 1 8 , 1 91 2 , VP.
34
Edoardo Almagi a t o Vit o an d Virgini a Volterra , Feb . 20 , 1 91 2 , VP .
35
V. Volterr a t o Virgini a Volterra , Jan . 26 , 1 91 2 , VP.
36
V. Volterr a t o Virgini a Volterra , Feb . 21 , 1912, VP.
37
[V. Volterra, Legons sur les fonctions de lignes (Paris : Gauthier-Villars ,
1913)]
NOTES 29
9
38
Quoted i n Deni s Mac k Smith , Moder n Italy : A Politica l Histor y (An n
Arbor: Universit y o f Michiga n Press , 1 997) , 225.
39
V. Volterr a t o Virgini a Volterra , Jan . 30 , an d Feb . 1 , 1 91 2 , VP.
40
V. Volterr a t o Virgini a Volterra , Feb . 6 , VP .
41
V. Volterr a t o Virgini a Volterra , Feb . 7 , 1 91 2 , VP.
42
Orso M . Corbin o t o Vit o Volterra , Feb . 24 , 1 91 2 , i n Giovann i Paoloni ,
ed., Vito Volterra e il suo tempo (Rome : Accademi a Nazional e de i Lincei ,
1990), 63.
43
V. Volterr a t o Virgini a Volterra , Feb . 1 3 , 1912, VP.
44
V. Volterr a t o Virgini a Volterra , Oct . 1 7 , 1 91 2 , VP .
Notes t o Chapte r 1 5
Portions of chapter 1 5 appeared i n different for m i n Journal of the History of
Ideas.
x
Gaetano Artur o Crocco , i n "Vit o Volterra ne l I centenario dell a nascita :
1860-1960," Accademi a Nazional e de i Lince i 5 1 (1 961 ) , 26 .
2
V. Volterr a t o Gasto n Darboux , Sept . 7 , 1 91 4 , Giovann i Paoloni , ed. ,
Vito Volterra e il suo tempo (Rome : Accademi a Nazional e de i Lincei, 1 990) ,
94.
3
Quoted i n Deni s Mac k Smith , Modern Italy: A Political History (An n
Arbor: Universit y o f Michiga n Press , 1 997) , 264 .
4
Vito Volterr a t o Virginia Volterra , Ma y 5 , 1 91 5 , in The Volterr a papers ,
private collectio n (hereafter , VP) .
5
Carlo Somiglian a to V. Volterra, Ma y 1 9 , 1915, Vito Volterra Collection ,
Accademia Nazional e de i Lincei , Rome .
6
V. Volterr a t o Virgini a Volterra , Jan . 25 , 1 91 6 , VP .
7
V. Volterr a t o Virgini a Volterra , Apri l 1 , 1 91 6 , VP .
8
V. Volterr a t o Virgini a Volterra , Apri l 1 2 , 1 91 6 , VP .
9
V. Volterr a t o Virgini a Volterra , Ma y 31 , 1916, VP .
10
V. Volterr a t o Virgini a Volterra , Jun e 2 , 1 91 6 , VP .
U
G.A. Crocco , "Vit o Volterra, " 27 .
12
Paolo Morrone , "Report, " Feb . 1 2 , 1 920 , cartell a XVII , fasc . 3 , Vit o
Volterra Collection , Rome ; V. Volterra t o Virgini a Volterra , Sept . 1 6 , 1916,
VP.
13
V. Volterr a t o Virgini a Volterra , Jul y 21 , 1915, an d Feb . 21 , 1916, VP.
300 NOTE
S
14
Emilio Lussu , Sardinian Brigade, trans , b y Marion Rawso n (Ne w York :
Grove Press , 1 970) , 3.
15
V. Volterr a t o Virgini a Volterra , Jun e 9 , 1 91 6 , VP.
16
V. Volterr a t o Virgini a Volterra , Jun e 1 2 , 1 91 6 , VP.
17
V. Volterr a t o Virgini a Volterra , Nov . 22 , 1 91 6 , VP.
18
V. Volterr a t o Virgini a Volterra , Nov . 1 5 , 1 91 6 , VP .
19
V. Volterr a t o P . Morrone , Jan . 27 , 1 91 7 , Vit o Volterr a Collection ,
Accademia Nazional e de i Lincei , Rome .
20
V. Volterr a t o Virgini a Volterra , Apri l 1 5 , 1 91 7 , VP.
21
S.L.G. Kno x t o Georg e Eller y Hale , Oct . 1 , 1 91 8 , Georg e Eller y Hal e
Papers, Bo x 26 , Institute Archives , Californi a Institut e o f Technology .
22
Knox t o Hale , ibid .
23
Judith R . Goodstein , Millikan's School: A History of the California
Institute of Technology (Ne w York : W.W . Norton , 1 991 ) , 84.
24
Judith R . Goodstein , " A Conversation wit h Franc o Rasetti, " i n Physics
in Perspective 3 (2001 ) , 294 .
25
Emilio Segre , Enrico Fermi, Physicist (Chicago : Universit y o f Chicag o
Press, 1 970) , 30.
26
Laura Fermi, Atoms in the Family: My Life with Enrico Fermi (Chicago :
University o f Chicag o Press , 1 954) , 30 .
27
V. Volterr a t o Virgini a Volterra , Jul y 1 4 , 1 921 , VP.
28
V. Volterr a t o Virgini a Volterra , Jan . 1 8 , 1 922 , VP.
29
V. Volterr a t o Virgini a Volterra , Oct . 26 , 1 922 , VP.
30
V. Volterr a t o G . Pestalozzi , Apri l 25 , 1 923 , Vit o Volterr a Collection ,
Accademia Nazional e de i Lincei , Rome .
31
"Verbale dell e riunion e tenut a i l 2 5 maggio 1 92 3 in un a sal a de l Senat o
da u n grupp o d i parlamentar i universitar i contrar i ali a riform a Gentile, "
Vito Volterr a Collection , Rome .
32
Giorgio Abett i t o Georg e Eller y Hale , Nov . 1 , 1 922 , Georg e Eller y Hal e
Papers, Bo x 1 , Institute Archives , Californi a Institut e o f Technology .
33
J.R. Goodstein , "Conversatio n wit h Rasetti, " p . 297 .
34
L. Fermi , Atoms in the Family, 42 .
35
Vito Volterra, "Variazion i e fluttuazioni de l numero d'individu i i n specie
animali conviventi, " i n Opere matematiche, V , 1 .
36
Ibid.
NOTES 30 1
37
Sharon E . Kingsland , Modeling Nature: Episodes in the History of Population Ecology (Universit y o f Chicag o Press : Chicago , 1 995) , 1 09 .
38
V. Volterra , "Th e Genera l Equation s o f Biologica l Strif e i n th e Cas e of
Historical Actions , i n Opere matematiche, V , 496 .
39
V. Volterra , "Discors o presidenzial e de l 1 925, " i n Opere matematiche,
IV, 529-531 .
40
Dirk Ja n Struik , "Viaggi o i n Italia, " Lettera Matematica Pristem 1
(June, 1 996) , 25.
2
41
Carlo Somiglian a t o V . Volterra, Feb . 7 , 1 926 , Vito Volterra Collection ,
Lincei.
42
V. Volterr a t o Luig i Errera , i n Giovann i Paoloni , ed. , Vito Volterra.
43
V. Volterra , Legons sur la theorie mathematique de la lutte pour la vie,
ed. Marce l Brelo t (Paris : Gauthier-Villar s e t Cie. , 1 931 ) .
44
TullioLevi-CivitatoV. Volterra , Feb. 27 , 1929, in Paoloni, Vito Volterra
176.
45
The document , date d Decembe r 1 9 , 1 928 , i s reproduce d i n G . Paoloni ,
Vito Volterra, Fig. VII . 6 .
46
Quoted i n Judit h R . Goodstein , "Th e Ris e an d Fal l o f Vit o Volterra' s
World," Journal of the History of Ideas 4 5 (1 984) : 61 4 .
47
Vito Volterr a t o Griffit h Evans , Jan . 7 , 1 932 , Box 1 , Evans Papers , U C
Berkeley.
48
David M . Smith , Modern Italy: A Political History, 361 .
49
Guido Corbellini , "Pe r i l centenario dell a nascit a d i Vit o Volterra , Sen ato dell a Repubblica , Sedut a de l 1 2 Maggio 1 960. "
50
Quoted i n "Guald a Massimi, " intervie w b y Carlott a Scaramuzzi , Nov .
25, 1 994 , p . 3 , Institute Archives , Californi a Institut e o f Technology .
51
Renzo d e Felice , The Jews in Fascist Italy: A History, wit h a prefac e
by Michae l A . Leeden , trans . Rober t L . Mille r (Ne w York , Enigma , 2001 ) ,
264.
52
G. Paoloni , Vito Volterra, Dec. 1 9 , 1 938 , Figure 1 0.1 .
53
Quoted i n Andr e Weil , The Apprenticeship of a Mathematician, trans .
Jennifer Gag e (Basel : Birkhause r Verlag , 1 992) .
54
Virginia Volterr a t o [nam e unknown ] Volterra , Ma y 20 , 1 940 , VP.
55
G. Castelnuovo , "Vit o Volterra, " Opere matematiche, I , xiii .
This page intentionally left blank
Index
Abel, Niel s Henrik , 1 2 8
Abelian integrals , 73-7 4
Abetti, Giorgio , 1 8 9
Abraham, Max , 2 , 1 5 9
absolute differentia l calculus , 8 1
Academie de s Science s d e France , 1 4 4
Academy o f Ital y (Accademi a d'ltalia) ,
3, 6 1
Accademia Nazional e de i Lincei , 3 , 61,
66, 68-69 , 1 38 , 1 83 , 1 89 , 1 9 4
Acta Mathematica , 53 , 69, 9 7
algebraic topology , 5
Almagia, Alfons o (uncle) , 1 4 , 1 8 , 42, 59,
64-65
in Florence , 20 , 2 3
in Terni , 1 9
Volterra's educatio n and , 2 8
Almagia, Edoard o (cousi n t o Angelic a
Almagia Volterra) , 1 5-1 6 , 25 , 27,
30-31, 56 , 1 5 3
death of , 1 9 1
living/working conditions , 31 -3 2
Volterra's educatio n and , 28-30 , 3 2
Volterra's engagemen t and , 1 03-4 ,
111, 1 1 3-1 4
Volterra's healt h and , 1 5 7
Almagia, Eleonor a (wif e t o Edoard o
Almagia), 1 0 8
Almagia, Este r Supino , 2 4
Almagia, Robert o (cousin) , 1 5 , 1 27-28 ,
173-74, 202- 3
Almagia, Sau l (uncl e t o Angelic a
Volterra), 1 4-1 6 , 1 7
Almagia, Virginia . Se e Volterra ,
Virginia Almagi a (wife )
Almagia, Vit o (fathe r o f Angelic a
Volterra), 1 2-1 4 , 1 5
Alpinism, 1 3 7
Ancona, 1 1 -1 4
Annali d i Matematic a pur a e applicata ,
5
anti-clericalism, 5 0
Anti-Semitism, 3 , 7 , 98 , 1 98-9 9
fascist regim e and , 1 98-20 0
papal governmen t and , 8-9 , 1 1
Risorgimento and , 9
Argentina, 1 69-7 1
Arzela, Cesare , 29-30 , 36 , 37-38 , 81,
147
Astronomische Nachrichten , 9 2
Autour d e l a lun e (Verne) , 2 6
Baccelli, Guido , 1 1 1 -1 2 , 1 1 4
Badoglio, Pietro , 20 2
Bakunin, Mikhail , 4 7
Basman, Fortunat a (mothe r o f Angelic a
Volterra), 1 4 , 1 9 , 6 7
Beltrami, Eugenio , 60 , 67 , 74 , 83 , 96,
108
Berlin, 1 29-30 , 1 41 -4 2
Bertini, Eugenio , 8 3
Bertrand, Josep h L . F. , 26 , 2 7
Betti, Enrico , 5-6 , 35 , 51 , 60, 66 , 81,
93, 1 1 7
Accademia de i Lince i mathematic s
prize and , 6 8
Volterra's caree r and , 40 , 56-5 9
Bianchi, Luigi , 68 , 8 6
biological sciences , mathematic s and ,
191-92, 1 9 5
Blaserna, Pietro , 1 09 , 1 2 0
Bocher, Maxime : A n Introductio n t o
the Stud y o f Integra l Equations ,
161
Borel, Emile , 1 54 , 1 8 2
Brandes, Georg , 1 5 2
Brioschi, Francesco , 5-6 , 97 , 1 1 7
British Associatio n fo r th e
Advancement o f Science , 1 4 4
304
"Carriers scientifiques " (Volterra) , 1 9 5
Cairoli, Benedetto , 4 6
"Calculus o f variation s an d th e logisti c
curve" (Volterra) , 1 9 2
Cannizzaro, Stanislao , 1 09 , 1 38 , 1 42 ,
154
Cantor, Georg , 7 2
Cantor, Moritz , 1 1 7-1 9
Capanna Regin a Margherita , 1 34-3 7
Casorati, Felice , 5 , 1 1 7
Castelnuovo, Enrico : I Moncalvo , 1 6 9
Castelnuovo, Guido , 1 42 , 1 47 , 1 59 ,
199-200, 20 2
University o f Rome , 6
Volterra's engagemen t and , 1 1 2-1 3 ,
114-15
Volterra's mov e t o Rome , and , 1 08- 9
Catania, 61 -6 4
Cavour, Coun t Camillo , 4 , 45 , 6 5
Cerruti, Valentino , 84 , 1 09-1 0 , 1 3 8
Chandler, Set h Carlo , 9 3
Chemical Histor y o f a Candl e
(Faraday), 2 6
Circolo Matematic o d i Palermo , 7 1
Clark University , 1 6 1
Concordia Society , 1 5 2
Consiglio Nazional e dell e Ricerch e
(National Researc h Council) , 1 8 6
Cooke, Roger , 7 3
Corbino, Epicarmo , 6 3
Corbino, Ors o Mario , 1 59-60 , 1 61 -62 ,
186-87
Cossa, Alfonso , 8 3
Courant, Richard , 1 1 9
Cremona, Luigi , 5-6 , 83-84 , 85-8 6
Crocco, Gaetan o Arturo , 1 77 , 1 80-8 1
Croce, Benedetto , manifest o of , 1 9 3
Curbastro, Gregori o Ricci , 9 9
D'Ancona, Alessandro , 47 , 5 0
D'Ancona, Umberto , 1 91 -9 2
D'Annunzio, Gabriele , 1 73 , 1 78-7 9
Darboux, Gaston , 71 -7 2
Darwin, George , 1 2 8
Davenport, Charles , 1 2 7
D'Azeglio, Massimo , 6 5
Decree o n Publi c Safety , 1 9 1
"deductive metho d a s a n instrumen t o f
research, The " (Vailati) , 1 0 0
Dei, Nardi , 4 0
Dickstein, Samuel , 1 0 0
differentiation an d integration , 27-28 ,
53
INDEX
Dini, Ulisse , 39-40 , 48-49 , 51 -52 , 82 ,
85-86, 1 0 8
Fondamenti pe r l a teoric a dell e funzion i
di variabil i reali , 52 , 5 3
Dirichlet, J . P . G . Lejeune , 57 , 6 9
D'Ovidio, Enrico , 81 -83 , 84-85 , 94 , 1 45 ,
183
earth's axis , wobblin g of , 91 -9 2
economics, mathematic s and , 1 26-2 7
Einstein, Albert , 1 -2 , 6 , 1 50 , 1 59 , 1 9 1
elasticity, 75 , 1 22-24 , 1 65 , 1 66 , 1 9 9
England, 1 27-2 8
Enriques, Federigo , 1 -2 , 6 , 81 -82 , 1 68 ,
202
Euler, Leonhard , 73 , 91-92, 9 4
Evans, Griffith , 6 , 1 61 , 171 , 1 7 3
Mathematical Introductio n t o
Economics, 1 2 7
Failla, Gioacchino , 1 84-8 5
Fantappie, Luigi , 1 9 9
Faraday, Michael : Chemica l Histor y o f
a Candle , 2 6
Fascism, 3-4 , 1 89 . Se e als o Mussolini ,
Benito
Felici, Riccardo , 40 , 83 , 86, 1 5 8
Fergola, Emanuele , 1 4 5
Fermi, Enrico , 4 , 1 86 , 1 90 , 1 95 , 20 1
Ferrari, Ettore , 8 9
Ferraris, Galileo , 8 8
Filippi, Filipp o de , 1 9 7
Finzi, Cesare , 34 , 39-40 , 50 , 8 6
Florence, 1 8 , 20-23 , 36-38, 46 , 56 , 6 7
Jews in , 1 5
Volterra in , 20-23 , 26 , 36-38 , 5 6
Foa, Pio , 1 34-3 5
Fondamenti pe r l a teoric a dell e funzion i
di variabil i real i (Dini) , 51 , 53
"Foundations fo r a Genera l Theor y o f
Functions o f a Comple x Variable "
(Riemann), 5
Franz, Ferdinand , 1 7 7
Fredholm, Erik , 1 3 0
Fubini, Guido , 6 1
functional analysis , 6 9
"Functions o f lines , integra l an d
integro-differential equations "
(Volterra), 1 6 8
Gallo, Niccolo , 1 1 5 , 1 1 6
Gait on, Francis , 1 2 6
Garbasso, Antonio , 1 5 9
Garibaldi, Giuseppe , 4 , 4 5
305
INDEX
Gazzetta d i Catani a (newspaper) , 6 3
Gentile, Giovanni , 1 88-89 , 1 9 3
geometrical calculus , 9 2
Gerbaldi, Francesco , 1 0 0
Giacardi, Livia , 8 9
Giolitti, Giovanni , 1 38 , 1 45 , 1 7 3
Giornale d i Matematiche , 7 4
Gottingen, 76-77 , 1 38-4 0
Gregorovius, Ferdinand , 7- 8
Gregory XVI , 1 4
Grimm, Jacob , 7 7
Grimm, Wilhelm , 7 7
Guccia, Giovann i Battista , 71 , 155-56,
185
Hadamard, Jacques : "Th e Psycholog y
of Inventio n i n th e Mathematica l
Field," 69-7 0
Hale, Georg e Ellery , 3-4 , 1 8 4
Hall, Granvill e Stanley , 1 6 1
Hermite, Charles , 5
Hertz, Heinrich , 9 2
Hicks, W . M. , 5 7
higher education , centra l governmen t
and, 8 1
Hilbert, David , 1 38-3 9
"Mathematische Probleme, " 1 1 9
Methoden de r mathematische n
Physik, 1 1 9
Histoire d ' un e bouche e d e pai n (Mace) ,
25
Hoepli, Ulrico , 1 3 8
Hooke's law , 1 2 3
Hughes, H . Stuart , 6 1
Hurwitz, Adolf , 1 00 , 1 1 7
"image charge " method , 5 7
I Moncalv o (Castelnuovo) , 1 6 9
Innocents Abroa d (Twain) , 20-2 1
integration an d differentiation , 27-28 ,
92
International Congres s o f
Mathematicians, 9 9
International Congres s o f Philosophy ,
168
International Researc h Council , 2 , 1 8 5
Introduction t o th e Stud y o f Integra l
Equations, A n (Bocher) , 1 6 1
Istituto d i Stud i Superior i Pratic i e d i
Perfezionamento (Florence) , 3 5
Istituto Tecnic o Galile o Galilei , 26-2 7
Istituto Tecnic o Gemmellaro , 6 3
Italian Physica l Societ y (Societ a
Italiana d i Fisica) , 1 -2 , 1 58-5 9
Italian Socialis t Party , 8 8
Italian Societ y fo r th e Progres s o f th e
Sciences (Societ a Italian a pe r i l
Progresso dell e Scienze ; SIPS) ,
154-56, 1 6 7
Italian Societ y o f Natura l Sciences ,
154-56
Italy, 1 4 . Se e als o Mussolini , Benit o
politics an d society , 44-4 7
Risorgimento, 4- 6
Jews
in Ancona , 1 1 -1 4
Italian mathematic s and , 6- 7
papal rul e and , 8-9 , 1 5
pre-Italian unification , 7
Roman ghett o and , 7- 8
Kirchhoff, Gustav , 5 7
Klein, Felix , 72 , 77 , 1 00 , 1 38-4 0
Knox, S.L.G. , 1 8 4
Konigsberger, Leo , 7 2
Kovalevskaya, Sofia , 72-7 5
Kronecker, Leopold , 5 , 7 9
Lagrange, Joseph-Louis , 73 , 8 5
Lame, Gabriel , 7 5
Latin America , 1 69-7 1
Lavagna scholarships , 34 , 58-5 9
Legns su r 1 'integratio n de s equation s
differentielles au x derive s partielle s
(Volterra), 1 5 3
Legendre, A . M. , 2 6
Leo XII , 1 3
Levi, Beppo , 8 9
Levi, Enzo , 8 7
Levi, Primo , 9
Levi-Civita, Tullio , 1 -2 , 6 , 81 , 97,
98-99, 1 59 , 20 1
promotion t o ful l professor , 1 3 4
"Sulla costituzion e dell e radiazion i
elettriche," 1 5 9
World Wa r I and , 1 8 2
literacy, 9 , 2 6
Lotka, Alfre d J. , 1 9 2
Lovett, Edga r Odell , 1 72-7 3
loyalty oath , 3-4 , 1 97-9 8
Lussu, Emilio : Sardinia n Brigade ,
181-82
Mace, Jean : Histoir e d ' un e bouche e d e
pain, 2 5
306
Maiocchi, Roberto , 8 8
Mandelbrojt, Szolem , 6
Manhattan Project , 20 1
Manifesto o f Italia n Racis m (Manifest o
per l a difess a dell a razza) , 3 , 1 9 8
"Manifesto o f th e Fascis t Intellectuals, "
193
Manual o f Politica l Econom y (Pareto) ,
127
Marchiafava, Ettore , 1 37 , 1 48 , 1 57-58 ,
187-88
Marconi, Guglielmo , 1 9 4
Marey, Etienne-Jules , 9 3
Margherita, Queen , 4 5
Mathematical Introductio n t o
Economics (Evans) , 1 2 7
mathematical physics , 1 5 1
Mathematical Societ y o f France , 7 2
mathematicians, profile s of , 6- 7
mathematics i n Italy , 1 -2 , 6 , 81,
199-200, 20 2
Jews and , 6- 7
"Mathematische Probleme " (Hilbert) ,
119
Matteotti, Giacomo , murde r of , 1 89-9 0
Mattirolo, Luigi , 89-9 0
Mazzini, Giuseppe , 44-45 , 64-6 5
Methoden de r mathematische n Physi k
(Hilbert), 1 1 9
Michelson, Alber t A. , 1 61 , 1 6 6
Milan Polytechni c Institute , 6
Mind Alway s i n Motion , A (Segre) ,
121-24
Ministry o f Publi c Instruction , 57 , 60 ,
81
Minkowski, Hermann , 1 38-39 , 1 6 0
"Space an d Time, " 2
Mittag-Leffler, Gosta , 53 , 72-74 , 1 3 0
Moore, E . H. , 1 6 6
Mortar a, Edgardo , 8
Mosso, Angelo , 1 33-3 4
Mossotti, Ottaviano , 5 1
Mussolini, Benito , 3 , 1 86-8 7
Mussolini Prize , 1 96-9 7
Naccari, Andrea , 83 , 86, 9 4
Nasi, Nunzio , 1 3 4
Nathan, Ernesto , 1 1 1
National Academ y o f Sciences , U.S. , 1 7 1
nationalistic imperialism , 1 73-7 4
National Researc h Council , Italia n
(Consiglio Nazional e dell e
Ricerche), 3
INDEX
National Researc h Council , U.S. ,
183-86
Nernst, Walther , 1 3 9
North America , 1 63-6 6
Nuovo Cimento , 2 , 56 , 57 , 82 , 121 ,
158-59
"On function s tha t depen d o n othe r
functions" (Volterra) , 6 9
"On th e attempt s t o appl y mathematic s
to th e biologica l an d socia l
sciences" (Volterra) , 1 25-2 7
"On th e motio n o f a syste m i n whic h
there ar e variabl e interna l
movements" (Volterra) , 96-9 7
"On th e periodi c motion s o f th e earth' s
pole" (Volterra) , 9 3
"On th e potentia l o f a heterogeneou s
ellipsoid o n itself " (Volterra) , 5 6
"On th e principle s o f th e integra l
calculus" (Volterra) , 5 3
"On th e stratificatio n o f a flui d mas s i n
equilibrium" (Volterra) , 1 2 8
"On th e theor y o f th e movement s o f th e
earth's pole " (Volterra) , 9 2
Oscar II , 1 5 0
Osgood, Willia m Fogg , 1 0 0
Padelletti, Dino , 6 0
Padoa, Corrado , 6 0
Padova, Ernesto , 51 , 57, 6 0
Painleve, Paul , 1 8 2
Paoli, Nell a Sistoli , 5 0
Paolis, Riccard o De , 6 8
Pareto, Vilfredo , 1 2 6
Manual o f Politica l Economy , 1 2 7
Paris, 6 , 71 -72 , 75-76 , 1 1 7-21 , 1 71 -72 ,
195, 1 9 8
Academy o f Sciences , 1 53-5 4
Passanante, Giovanni , 4 6
pathological functions , 5 3
Peano, Giuseppe , 81 , 91, 1 00 , 1 20 , 1 6 8
Formulario Mathematico , 9 1
Volterra and , 91 -9 8
Pearson, Karl , 1 26-2 7
Peres, Joseph , 26 , 1 71 , 1 7 3
Picard, Emile , 75-7 6
Pincherle, Salvatore , 68 , 74 , 8 1
Pisa, 36-4 4
opera, 43-4 4
politics an d society , 46-4 7
town-gown relations , 47-4 8
University of , 39 , 42-44 , 48-49 , 6 0
INDEX
Volterra in , 48-49 , 58-5 9
Pius IX , 4 6
Pius X , 8
Pizzetti, Paolo , 1 2 5
Poincare, Henri , 1 00 , 1 1 7-1 8
Prandtl, Ludwig , 1 3 9
Princip, Gavrilo , 1 7 7
"Psychology o f Inventio n i n th e
Mathematical Field , The "
(Hadamard), 69-7 0
Radicale, I I (journal) , 4 7
Rasetti, Franco , 1 8 6
Razaboni, Cesare , 6 0
"Reconstitution Commission, " 20 2
relativity theory , 1 - 2
Rendiconti de l Circol o Matematic o d i
Palermo, 7 1
Rendiconti deH'Aceademi a de i Lincei ,
69, 9 6
Ricci-Curbastro, Gregorio , 1 2 5
Riedler, Alois , 1 41 , 143
Riemann, Geor g Friedric h Bernhard , 5 ,
53, 6 9
"Foundations fo r a Genera l Theor y o f
Functions o f a Comple x Variable, "
5
Righi, Augusto , 1 45 , 1 5 9
Risorgimento, 4- 6
Rivista d i Matematica , 93 , 9 9
Roiti, Antonio , 33-35 , 37 , 40, 42 , 58,
83, 20 1
University o f Turi n and , 8 1
Volterra's educatio n and , 33-3 6
Volterra's engagemen t and , 1 06-7 ,
109, 1 1 2
Romani, Fedele , 44-45 , 47-4 8
Rome, 1 21 -22 , 1 24-2 5
Jewish ghett o in , 7- 8
University of , 2- 3
Royal Academ y o f Science s (Turin) ,
92-93
Royal Turi n Polytechnic , 1 43-4 4
Runge, Carl , 75 , 79 , 1 3 9
Rutherford, Ernest , 1 61 , 166
Salvemini, Gaetano , 1 2 5
Samuelson, Paul , 1 07 n
Sardinian Brigad e (Lussu) , 1 81 -8 2
Scandinavia, 1 29-30 , 1 50-5 2
Schiaparelli, Giovanni , 92 , 1 2 7
Schwarz, Hermann , 72 , 7 7
307
"Science a t th e Presen t Momen t an d
the Ne w Italia n Societ y fo r th e
Progress o f th e Sciences "
(Volterra), 1 5 8
Scientia (journal) , 2 , 6
Scott, Charlott e Angas , 1 2 0
Scuola Normal e Superior e d i Pisa , 6 ,
36, 5 0
Scuola Tecnic a Dant e Alighieri , 2 6
Second Internationa l Congres s o f
Mathematicians, 1 1 7
Segre, Corrado , 61 , 81, 95, 1 1 9
Segre, Emili o
A Min d Alway s i n Motion , 1 21 -2 4
Volterra's teachin g and , 1 21 -2 2
Segre, Olga , 1 1 5
Sella, Alfonso , 1 34-3 5
Sella, Quintino , 1 30-31 , 1 3 7
Severi, Francesco , 1 95 , 20 2
Siacci, Francesco , 60 , 8 2
Sicily, 61 -6 4
Simon, Herman n T. , 1 3 9
Societa Democratic a Internazionale , 4 7
Societa Italian a d i Scienz e Natural i
(The Italia n Societ y o f Natura l
Sciences), 1 5 4
Societa Italian a pe r i l Progress o dell e
Scienze (Italia n Societ y fo r th e
Advancement o f th e Sciences) ,
154-56
Somigliana, Carlo , 49 , 54-55 , 81 , 93-94,
97-98
Stendahl, 1 1
Stokes, Georg e Gabriel , 5 7
Struik, Dirk , 6
student activism , 89-9 0
"Sulla costituzion e dell e radiazion i
elettriche" (Levi-Civita) , 1 5 9
Supino, Ester , 2 4
Tacchini, Pietro , 8 3
Terni, Peppino , 2 1
Terni, tow n of , 1 9
Terni, Vito , 1 7
Theorie genera l de s fonctionelle s
(Volterra), 1 9 7
Third Internationa l Congres s o f
Mathematicians, 1 4 4
Thomson, J . J. , 1 2 8
Thomson, Willia m (Lor d Kelvin) , 5 7
Tomasi, Tina , 5 0
Tricomi, Francesco , 6-7 , 9 8
Turati, Filippo , 8 8
308
Turin, 82-9 0
Turin, Universit y of , 87-9 0
student activis m at , 89-9 0
Volterra's appointmen t to , 82-9 0
Turin Polytechnic , 1 56-5 8
Twain, Mark : Innocent s Abroad , 20-2 1
Ufficio Invenzion i (Offic e o f Inventions) ,
183
Umberto I , 45-4 6
Unione Matematic a Italiana , 1 85-8 6
United States , 1 63-6 6
Universita Israelitica , 1 3
U.S. Nationa l Academ y o f Sciences , 1 7 1
Vailati, Giovanni , 99-1 01 , 1 25-26 , 1 3 8
"The deductiv e metho d a s a n
instrument o f research, " 1 0 0
Valenti, Giulio , 1 2 5
"Variations an d fluctuation s i n th e
number o f individual s i n coexistin g
animal species " (Volterra) , 1 91 -9 2
Verne, Jules : Autou r d e l a lune , 2 6
Victor Emmanue l II , 4 , 1 6-1 7
death of , 44-4 5
Victor Emmanue l III , 1 7 9
Villino Volterra , II , 1 50 , 1 5 2
Volta, Alessandro , 1 90-9 1
Volterra, Abram o Sabat o (father) , 1 6
Volterra, Angelic a Almagi a (mother) ,
14-16, 38-39 , 61 -62 , 66-67 , 78-7 9
Abramo Sabat o Volterr a and , 1 6
death of , 1 79-8 0
in Florence , 2 0
on money , 65-6 6
in Terni , 1 9
Virginia and , 1 2 9
Volterra's engagemen t and , 1 03-4 ,
108
Volterra, Edoard o (son) , 1 7 , 1 2 5
Volterra, Enric o (son) , 1 47 , 1 49 , 1 6 7
Volterra, Luis a (daughter) , 1 91 -9 2
Volterra, Virgini a Almagi a (wife) , 1 5 ,
21, 1 03-4 , 1 1 1 -1 2 , 1 1 3 , 1 20-21 ,
152, 20 3
Angelica and , 1 2 9
marriage, 1 1 5-1 6
Volterra, [Samue l Giuseppe ] Vito , 1 -2 ,
16-17, 1 9 7
on academia , 3 8
on America , 1 66-6 7
analysis and , 6 9
appearance, 49 , 1 0 6
INDEX
on Argentina , 1 69-7 1
Army Corp s o f Engineer s and , 1 7 7
in Berlin , 1 29-30 , 1 41 -4 2
biological science s and , 1 91 -92 , 1 9 5
birth, 1 6-1 7
at Capann a Regin a Margherita ,
134-37
in Catania , 61 -6 4
childhood, 1 9-2 0
children and , 1 25 , 1 49-50 , 1 53 , 1 5 6
Croce manifest o and , 1 93-9 4
death, 20 0
education, early , 21 -23 , 25-2 6
elasticity and , 75 , 1 22-24 , 1 65 , 1 66 ,
199
engagement, 1 03-1 5
in England , 1 27-2 8
family life , 41 -42 , 1 33-34 , 1 40-41 ,
148-49, 1 56-5 7
fascist governmen t and , 1 91 , 1 96-9 8
in Florence , 20-23 , 36-38 , 5 6
functional analysi s and , 6 9
in Gottingen , 76-79 , 1 38-4 0
"il professore, " Universit y o f Pisa ,
66-69
II Villin o Volterr a and , 1 5 0
integral equation s and , 98-9 9
Istituto Tecnic o Galile o Galile i and ,
26-27
in Lati n America , 1 69-7 1
Levi-Civita, Tulli o and , 98-99 , 1 3 4
marriage, 1 1 5-1 6
on mathematica l physics , 1 5 1
medical problem s of , 1 4 7
middle- an d high-schoo l education ,
25-30
military service , 2 , 53-54 , 1 79 , 1 81 -8 5
modern mathematic s and , 1 - 3
in Nort h America , 1 63-6 6
oath o f allegianc e and , 3-4 , 1 97-9 8
on oper a i n Pisa , 5 6
in Paris , 6 , 71 -72 , 75-76 , 1 1 7-21 ,
153-54, 1 71 -72 , 1 95 , 1 9 8
Peano, Giusepp e and , 91 -9 8
personality of , 22 , 49 , 52 , 65-6 7
philosophy o f life , 1 8 0
physics degree , 5 6
in Pisa , 36-44 , 48-49 , 58-5 9
on Pis a opera , 43-4 4
at Princeton , 1 7 5
prominence of , 7 9
Rendiconti dell'Accademi a de i Lince i
and, 9 6
309
in Rome , 1 21 -22 , 1 24-2 5
Royal Turi n Polytechni c and , 1 43-4 4
in Scandinavia , 1 29-30 , 1 50-5 2
Scuola Normal e Superior e d i Pis a
and, 50 , 51 -53 , 55-5 6
Second Internationa l Congres s o f
Mathematicians and , 1 1 7-1 9
as Senator , 1 43-4 5
in Sicily , 61 -6 4
social justice and , 1 40-4 1
Societa Italian a d i Scienz e Natural i
(The Italia n Societ y o f Natura l
Sciences) and , 1 5 4
Sofia Kovalevskay a and , 74-7 5
in Terni , 1 9
theory o f liqui d jet s and , 1 8 8
Third Internationa l Congres s o f
Mathematicians and , 1 4 4
transatlantic diary , 1 6 2
in Turin , 82-90 , 87-9 0
Turin appointmen t and , 82-8 7
University o f Rom e and , 1 08-1 5 ,
121-24
University o f Turi n and , 82-9 0
Vailati, Giovann i and , 99-1 0 1
World Wa r I and , 2 , 1 79 , 1 81 -8 5
Volterra, [Samue l Giuseppe ] Vito ,
honors an d award s
Academie de s Science s d e Prance ,
corresponding member , 1 4 4
Accademia de i Lincei , administrator ,
138
Accademia de i Lincei , member , 6 6
Accademia de i Lincei , president , 3 ,
189, 1 9 4
Accademia de i Lince i mathematic s
prize, 61 , 68-6 9
British Associatio n fo r th e
Advancement o f Science , honorar y
degree, 1 4 4
British Roya l Society , foreig n
member, 1 7 1
Circolo Matematic o d i Palermo ,
member, 7 1
Civil Orde r o f Savoy , 1 5 0
International Burea u o f Weight s an d
Measures, president , 1 8 7
International Researc h Council , vic e
president, 1 8 5
Italian Physica l Society , president ,
158-59
Italian Societ y o f Natura l Society ,
president, 1 54-5 6
Italian Societ y o f Science s medal , 6 6
Mathematical Societ y o f France ,
member, 7 2
National Researc h Council , Italian ,
president, 1 8 9
Royal Turi n Polytechnic , Roya l
Commissioner, 1 4 4
Senator, 1 43-44 , 1 56 , 1 87 , 1 9 1
U.S. Nationa l Academ y o f Sciences ,
foreign associate , 1 7 1
Vatican Academ y o f Sciences ,
member, 1 9 7
War Cross , 1 8 1
Volterra, [Samue l Giuseppe ] Vito ,
writings o f
"Cahiers scientifiques, " 1 9 5
"Calculus o f variation s an d th e
logistic curve, " 1 9 2
dissertation, 5 7
"Functions o f lines , integra l an d
integro-differential equations, " 1 6 8
Legns su r 1 'integratio n de s equation s
differentielles au x derive s partielles ,
153
"On function s tha t depen d o n othe r
functions," 6 9
"On th e attempt s t o appl y
mathematics t o th e biologica l an d
social sciences, " 1 25-2 7
"On th e Motio n o f a syste m i n whic h
there ar e variabl e interna l
movements," 96-9 7
"On th e periodi c motion s o f th e
earth's pole, " 9 3
"On th e potentia l o f a heterogeneou s
ellipsoid o n itself, " 5 6
"On th e principle s o f th e integra l
calculus," 5 3
"On th e stratificatio n o f a flui d mas s
in equilibrium, " 1 2 8
"On th e theor y o f th e movement s o f
the earth' s pole, " 9 2
"Replica a d un a not a de l Prof .
Peano," 9 7
"Science a t th e Presen t Momen t an d
the Ne w Italia n Societ y fo r th e
Progress o f th e Sciences, " 1 5 8
"Sur quelque s progre s recent s d e l a
physique mathematique, " 1 6 5
Theorie genera l de s fonctionelles , 1 9 7
"Variations an d fluctuation s i n th e
number o f individual s incoexistin g
animal species, " 1 91 -9 2
310
Walras, Leon , 1 2 6
Weber, Heinrich , 7 2
Weber, Wilhelm , 7 7
Webster, Arthu r Gordon , 1 60 , 1 6 1
Weierstrass, Karl , 5 , 72 , 73-7 4
Weingarten, Julius , 1 22-2 3
Whewell, William , 1 2 6
World Wa r I , 2- 3
Italian politic s and , 2 , 1 78-7 9
outbreak of , 1 77-7 8
Volterra and , 1 79 , 1 82-8 3
Zariski, Oscar , 6-1 3