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19551
BOOK REVIEWS
43
pape r s by K . Menge r
1
concerned wi th the va l id i ty o f the law of
d imin i sh ing r e tu rn s . Th i s i s : Let the product of
applying y {e.g.,
dollars) to x(e.g., acres) be E(x, y). Then for h>0
E(x, 3>
2
+ h) - E{x, y
2
) < E(x, yi+h)- E(x,
yi
)
provided y2>yi>y~y(x). Va r ious p rop er t ies of p ro
duc t ion func t ions
E such a s mono tony , supe r - add i t i v i t y , supe r
-homogene i ty , de
pendence a r e i n t roduced . Example s a r e p r e sen t ed t
o show how the se
a re r e l a t ed among themse lve s and w i th t he above l aw
and s im i l a r
p ropos i t i ons . These r e l a t i ons a r e summar i zed g r
aph ica l l y . Th i s e s say
can be rec om m end ed b y ma the m at ic ian s to the i r f r
iends in soc ia l
science as an example of how one discipl ine can be used in
another .
The conc lud ing essay by Morgens te rn i s concerned wi th the
ro le o f
exper iment and comput ing in economics . Apar t f rom d iscuss
ions o f a
more ph i lo soph ica l na tu r e t he r e a r e i nd i ca t i
ons o f some compu ta
t iona l exper iments o f immedia te in te res t to economis t s
.
O L G A T A U S S K Y
Ma thematics and plausible reasoning. By G . P lya . V o lum e I
, Induc
tion and analogy in mathematics, 16 + 280 pp . , $5.50; vo lum e
II ,
Patterns of plausible inference,
10
+ 190 pp . , $4 .50 . Pr in ce to n Un iver
s i ty Pre ss , 1954. 2 vols . , $9.00.
Th e re a r e m an y so -ca ll ed pop u la r books on m a th em
a t i c s . Some
of them turn ou t to be o f in te res t to p rofess iona l
mathemat ic ians on ly
(o r , pe rhaps , t o p ro fe s s iona l ma thema t i c i ans in
ovoas w el l ) . O ther s
a re so non- technica l as to be wel l wi th in the reach of any
educa ted
layman, and , consequent ly , the i r sub jec t ha rd ly
deserves to be ca l led
ma thema t i c s . Mos t o f t he t ime P lya manages t o s t e
e r an admi rab l e
cou r se be tw een the se tw o ex t r emes . The tw o vo lumes
unde r r ev i ew
a r e , however , no t un i form in th i s respec t ; the f i r
s t i s more the mathe
mat ic ian ' s vo lume and the second the ph i losopher ' s . S
ince th i s rev iew
is addres sed to m ath em at ic ian s , i t wil l d i scuss th e
f irs t vo lum e in m ore
de t a i l , and , i t may w e l l be cha rged , w i th more
sympa thy , t han t he
second .
The book as a whole i s o rgan ized a round the cen t ra l thes
i s tha t a
good guess i s qu i te as impor tan t as a good
proof
As in his l i t t le book
How to solve it, P l y a a d v o c a t e s t h a t t h e m a t h
e m a t i c i a n s h o u l d t h i n k
an d ta lk (a t h i s desk an d in th e c lass room ) ab ou t
the theo ry of guesses
as wel l as th e the ory of p roofs . C er ta in ly , l e t us l
ea rn pro v ing , he
1
Bem erkungen zu den Ertragsgesetzen and Weitere Bemerkungen zu
den Ertrags
gesetzen in Zei tschrif t f. Nat ionalko nom ie vol . 7 1936) pp
. 25 -26 and 388-3 97.
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44
BOOK REVIEWS
[May
says ,
b u t a l so let us learn guessing. The funct ion of the f i rs
t volume
in the theory of p laus ib le reasoning i s to p rov ide some
concre te
m a t h e m a t i c a l r a w m a t e r i a l
the second vo lume i s more in te res ted in
an abs t rac t ph i losophica l d i scuss ion of the pa t te rns
tha t the f i r s t
vo lume ind i ca t e s .
M or e tha n half of th e fi rs t vo lum e is occu pied by prob
lem s (a t th e
end of each chap te r ) and the i r so lu t ions (a t the back
of the book) .
T h e orga n iza t io n of thes e p rob lem s is s imi la r to
th a t of the wel l
kno wn prob lem co l lec t ion b y Po ly a and Szeg . Som et im
es th e v ery
s t a t emen t and some t imes t he so lu t i on o f one p rob l
em sugges t s t he
nex t p rob l em by work ing th rough a g raded se r ies o f p
rob lems of th i s
sor t the reader ge t s a very good ins igh t in to the i r sub
jec t . The
problems a re a lmos t a lways non- t r iv ia l . The f i r s t
p rob lem in the f i r s t
ch ap te r i s to guess the ru le accord ing to which th e
success ive te r m s of
th e fol lowing seq uen ce a re c ho sen : 11, 31 , 41 , 61 , 71
, 101, 131, .
(For the convenience of the reader of th is review the solut ion
is g iven
be low . ) The l a s t p rob l em in t he l a s t chap t e r
conce rns some compu ta
t ions involv ing the to rs iona l r ig id i ty o f a beam wi th
square c ross -
sec t ion . Many of the p rob lems inv i te the reader to guess
an answer ,
to examine a s i tua t ion , o r to con jec ture a theorem.
The prob lems , by the way , a re no t ca l led prob lems; they
a re ca l led
example s and commen t s . Rough ly speak ing , t he example s a
r e
ma thema t i ca l exe rc i s e s and t he commen t s a r e ph i
lo soph ica l d i s
cou r se abou t t hem. The qua l i t y o f t he commen t s i s
qu i t e va r i ab l e .
Som e a r e ha rd ly more t ha n w eak jokes ( t he r e is one
ab ou t t he dif-
fe rences am on g log ic ians, m ath em at ic ian s , phy s ic i
s t s , and eng ineers ) ,
w h i l e o the r s a r e pene t r a t i ng r emarks on ma thema
t i ca l me thodo logy
(e .g . , a discussion of the effect of s t rengthening the
hypothesis of a
theo rem to be p roved by i nduc t ion ) .
T h e grea t e r po r t ion of the bo dy of vo lu m e I ( i .e .
, of the m ate r ia l
d i s t in c t from th e prob lem s an d the i r so lu t ions)
is m ath em at ic a l ex
pos i t ion of unusua l ly h igh ca l iber ; on ly about 40
pages ( spr ink led
th rough the vo lume) a r e devo ted t o ou t r i gh t ph i l o
soph iz ing . A n idea
of the contents can be gained from the fol lowing l is t of the
main
ma thema t i ca l i dea s t ha t occu r i n t he va r ious chap
t e r s . (The t i t l e s o f
the chap te rs would be of l i t t l e use here ; Chapte r VI ,
fo r ins tance , i s
en t i t l ed A more general statement.) I . G o ldbac h ' s con
j ec tu r e . I I . T he
Py tha go rean t heo rem ; Eu le r ' s m e tho d of eva lu a t i
ng )w = i
n
~
2
- H I .
The Eu le r f o rmu la ,
V E + F = 2.
IV. Sums of squares ; in par t icu la r ,
the four -square theorem. V. I f a
n
> 0 (w = l , 2, 3, ) , th en
l im sup
n
((ai + a
n
+i)/a
n
)
n
e.
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*955l
BOOK REVIEWS
245
V I .
Euler 's recurs ion formula for the sum of the divisors of n. V
I I .
M a th em a t i c a l i nduc t ion . Some non - rou t ine app l
i c a t i ons , such a s t he
fol lowing on e: I f th e polyg on P is convex and con ta ined
in t he
poly gon
