The San Francisco call., 1906-06-06
TELLS Of WORK
IN GE0ME TRY
Book on Researches Is
Issued by Professor
Wilczynski.
Contribution Important
to Advancement
of Science.

result of Professor f. j. WilczynskPs
trip to Europe last year, when he per-
formed research work in the universi-
! ties of the Continent, is found in q pub
lication of which he is the author, just
Taubner OF Leipzig. Germany. The
work is rated by great mathematician
as a remarkable contribution to the lore
on the subject of geometry.
Professor Wilczynski is research
associate of the Carnegie Institution of
Washington, d. c., and professor of
mathematics at the University of Cali
fornia He describes, for the public. the
character of his Investigations and
work, his writing for this purpose being
issued from the office of President
Wheeler today The matter is as fol-
lows:
In the geometfical Investigations of
the last century, one of the most fun
damental distinctions has been that
between metrical and prejective geom-
etry 1t is curious fact that this
fo to gvn y not Fise
all justified by the nature of things
There are certain properties of curves,
surfaces. etc, which may be deduced
for the most general CONGREGATIONS of
their kind depending only Upon the
knowledge that certain conditions Of
continuity are fulfilled in the vicinity
called infinitesimal properties and are
naturally treated by the methods of the
differential calculus.
The curious fact to which we have
referred is that, but for rare exceptions,
these infiintesifnal properties have been
dealt with only from the metriea1 point
of view. Protective geometry. which
has made such progress in the. course
dained to consider the lnfiniteiy small
parts into which its configurati0ns may
be deeomposed. 1t has gained the pos
sfbility of making assertions about its
CONGREGATIONS as whole, only by linn
iting it's field to the consideration OF
dpnec cessary in differential geometry
Between the metrical differeutial
geometry Of Monge and and Gauss aBd nd,oa the
lot and P1ucker; there is left there
fore; the field OF protective differential
geometry whose MIXTURE partakes some
what of both. The theorems of this
Kind of geometry are concerned with
protective properties of the infnitesi-
NAL elements: As In the ordinary diff
feren tial geometry, the process of inte-
grafton may lead to statements. con-
eerning properties Of the configqration
as a whole. But, of course, such inte-
Even With this limitation, however,
which lies in the nature of things the
field of PROTECTIVE differential geometry
gy than has been done heretofore.
But. few Investigations belonging to
contributions are those of Haiphen.
Who has developed an admirable theory
of plane and space curves from this
point of view. The author has; in the
differential geometry of ruled surfaces.
In this book we shall conine ourselves
to the consideration of these simplest
viii Wi III follow. ssAe''i '"eV y s1' ia'e'
In presenting the theories of Hall
phem | have nevertheless, followed my
own methods both for the sake of uni
have have attempted 70 Indicate clearly
to other authors. The general method
of treatment and the results which are
not specifically attributed to others
The theory of ruled surfaces has been
developed by ME in series of papers,
published principally in the Tfansac-
cefs eiens Or s9e s han sS
it unnecessary to refer TO them in de-
tall the treatment here given being
the original papers. In particular. some
errors have been corrected: / hope that
no serious mistakes have been allowed
to pass over into the present work.
To finish these personal remarks, /
may add that Chapter II contains
number of Important additions to the
the1ry As G,,i5 genei5aaly preS;ented without
pleteness- The canonicA1 development
of Chapter III has also been added by
the author.
The examples collected at the end of
each chapter are of two kinds. Some
of them are mere exercises. Some,
however, are of very different ma
ture. They are essentially suggestions
for such further Investigations as APP
pear to me to be of promise and inn
portance. have also. in the body of
the book. taken the privilege of point
ing out further problems which seem
to be Of interest. Many others will
readily suggest themselves. 1t is my
sincere wish that these suggestions
may be helpful toward further de-
velopment of this fascinating subject
The instructor In an American uni
versity hnds his time fully occupied by
of science. The Carnegie Institution
of. Washington, in recognition of this
fact, makes 1t part of its policy to
give a certain, number OF men the OPP
porrtunitg n,or to devote sot all Ct their time
have had the honor and the good for
tune of finding my efforts aided and
encouraged by the support OF this mag
nlilcers Iha erssd mstltu1ion; WITHOUT this aid
| the light OF day for several years /
take this opportunity to express to the
Carnegie Institution my fullest graft
tude for its help and for the. generosity
and move, ulfettered by. unnecessary
conditions and regulations.
In these last two years have had
occasion to make use of libraries at
Gottingen, Paris, Cambridge and Rome.
For these privileges. i am indebted to
Professors Klein. Darboux Forsyth and
| lCastelnuovo. who met my wishes with
the greatest OF courtesy.
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