REVIEWS OF
RATIONAL GEOMETRY
BY
GEORGE BRUCE HALSTED
A B. and A. M. (Princeton University); Pli. D. (Johns
Hopl<ins University); F. R. A. S.; Ex-Fellow of Prince-
ton University; twice Fellow of Johns Hopkins Univer-
sity ; Intercollegiate Prizeman ; sometime Instructor in
Post-Graduate Mathematics. Princeton University ; Mem-
ber of the American Mathematical Society ; Member of
the London Mathematical Society; Member of the Society
for the Promotion of Engineering Education ; Member of
the Mathematical Association ; President of the princj^tpn
Alumni Association of Texas; Fellow and Past-P'resident
of the Texas Academy of Science ; Professor of Mathe-
matics in Kenyon College; Vice-President of i\iz Ameri-
can Association for the Advancement of Scfjence, and
Chairman of Section A (Mathematics and Astronomy))'
Non-Resident Member of the Washington Academy of
Sciences; Member of the Society of Arts; Membte d'Hori-
neur du Comite Lobachefsky ; Miembro de la Sociedad
Cientifica "Alzate" de Mexico ; Socio Corresponsal de la
Sociedad de Geografia y Estadistica de Mexico; Alitglied
des Vereins zur Foerderung des Unterrichts in der Mathe-
matik und den Naturwissenschaften ; Mitglied der D.eut-
schen Mathematiker-Vereinigung ; Societaire Perpetual de
la Societe Mathematique de France ; Socio Perpetuo del
Circolo Matematico di Palermo.
920387
RATIONAL GEOMETRY
Rational Geometry, a Text-book for the Science of
Space, par George Bruce HalsTED. — Un vol.
in 12, VIII -I- 285 pages, 247 figures. John
Wiley & Sons, Newyork, 1904.
Les recents et si remarquables travaux de M.
Hilbert sur les fondements de la geometrie, ma-
gistralement analyses par M. Poincare dans ses
articles de la Revue des Sciences et dans son Rap-
port sur le y concours du prix Lobatschefsky
[1903], ne pouvaient manquer a bref delai
d'eveiller I'attention des geometres et d'exercer
une influence profonde et decisive sur leurs ouv-
rages. On devait certainement s'attendre a voir
publier des Traites didactiques dont les hardis et
erudits auteurs, rompant resolument avec les habi-
tudes et traditions de plus de vingt siecles, essaier-
4 HALSTED'S RATIONAL GEOMETRY
aient d'harmc3niser*l"enSe'i^ement de la geometrie
avec It/s'Jdpc^s jVji^yelifs.,. jMa'lgje.- que M. Hilbert
eut pris deja lui-meme soin d'indiquer et de
jalonner d'une maniere precise la route a suivre,
la taclie etait loin d'etre aisee. Elle devait attirer
particulierement M. George Bruce Halsted, le
savant professeur de Kenyon College, un des plus
ardents defenseurs de la geometrie generale aux
Etats Unis, bien connu par ses nombreuses publi-
cations dans les Revues "Science" et "z/lmcricau
Matlieniaticjl Moutlily" , et surtout par ses belles
traductions anglaises de Saccheri, Bolyai et Lobat-
schetsky. La "Rational geometry" de M. Hal-
sted, encouragee par M. Hilbert, marque une
epoque dans I'histoire des livres destines a I'en-
seignement. Nous alions analyser en detail les
chapitres de cet ouvrage.
Pour constituer une geometrie vraiment ration-
nelle, deux choses etaient indispensables : en
premier lieu, t^tablir une liste complete des axiomes
en s'efforcant de n't^w oublier aucun; ensuite,
supprimer totalement le rule de I'intuition qui a
HALSTED'S RATIONAL GEOMETRY '- 5
occupe jusqu'ici Line place telle en geometrie que
nous faisons dans cette science presque a chaque
instant usage de propositions intuitives sans nous
en apei'cevoir le moins du monde. Dans ce but,
les axiomes qui expriment les relations mutuelles
pouvant exister entre les etres geometriques, point,
droite, plan, espace, ont ete suivant la methode
de M. Hilbert, repartis en cinq groupes: Con-
nexion ou association, ordre, congruence, axiome
des paralleles ou d'Euclide, axiome d'Archimede
ou de continuite.
Dans le chapitre I, M. Halsted definit les etres
geometriques et expose les sept axiomes de con-
nexion. De ces axiomes decoulent naturellement
les propositions habituelles.
Deux droites distinctes ne peuvent avoir deux
points cummuns.
Deux droites distinctes ont un point commun
ou n'en ont aucun.
Deux plans distincts ont en commun une droite
ou n'ont aucun point commun.
Un plan et une droite qui n'y est pas situee ont
un point commun ou aucun.
6 HAJ.STED'S RATIONAL GEOMETRY
Par une druite et un point, ou dt-iix droites qui
ont un puint commun, on peut faire passer un
plan et un seul.
— DanP le chapitiv 11 viennent, au nombre de
quartrp, ItfS axiomes de I'ordre qui precisent I'ar-
rangeitiiBnt des points caracterise par le mot enti'e.
Ces axlomes sont completes par la definition du
segnient qui ne doit eveiller aucune idee de
mesure: Deux points A et B de la droite A defin-
issept le sejiment AB ou BA; les points de la
driiite sitiies entrc A et B sont les points du seg-
ment. De la la distinction entre les deux
I'ciyotis d'une droite separes par un point,
entre les deux regions du plan separees par
une droite. — Points interieurs et exterieurs a un
polygone. — Notons pour memoire I'axiome 4 ou
axiome de Pasch. Si A, B it C sont trois points
non colIint'Liircs ct a ////<' droite du plan nc passant
par aiiiiin d'eux, lorsque a renfermc un point du
segment A B, elle en a un autre sur B C ou sur A C.
11 est evident que si le plus petit role etait laisse a
I'intuition, on ne songerait pas a enoncer cette
HALSTED'S RATIONAL GEOMETRY 7
proposition dont on fait inconsciemment un si fre-
quent usage.
Le chapitre 111 develuppe les axiomes de con-
gruence : segments, angles, triangles, et I'auteur
y formule en ces termes precis le theoreme general
de congruence.
5/ A B C... A' B' C... sont deux figures con-
gruentes, et que P designe un point quelconque
de la premiere, on peut toujours trouver de fagon
univoque dans la deuxieme un Point P' tel que
les figures ABC... P, A'B'C" ... P' soient con-
gruentes.
Ce theoreme exprime I'existence d'une certaine
transformation unique et reversible qui nous est
familiere sous le nom de deplacement. La notion
de deplacement est done bassee sur celle de con-
gruence, ce qui est absolument logique.
Le chapitre suivant est consacre a I'axiome de
la parallele unique et aux propositions qui en sont
la consequence. La plupart sont classiques, nous
n'y insistons pas; mais il en est d'autres que nous
avons eu jusqu'ici I'habitude de considerer comme
HALSTED'S RATIONAL GEOMETRY
intuitives et qui ne le sont pas. M. Halsted les
demontie avec raison; ce sont celles-ci: Tout
segment a un point milieu; tout angle a un rayon
bissecteur.
Chapitre V — Circonterence.
Chapitre VI — Problemes de Construction.
Toutes les constructions decoulant des theoremes
bases sur les cinq groupes d'axiomes peuvent etre
graphiquement resolues par la regie et le trans-
porteur de segments (Streckeniibertrager de M.
Hilbert) et ramenees a ces deux traces fonda-
mentaux: Tracer une droite; prendre sur une
droite donnee un segment donne.
Chapitre- VII — Egalites et inegalites entre
cotes, angles etarcs.
Chapitre VIII — Calcul des Segments. En se
basant sur les axiomes des groupes I, II, IV et en
mettant systematiquement de cote Taxiome
d'Archimede dont on s'est passe dans ce qui
precede et dont on peut egalement se passer dans
ce qui suit, on arrive a creer, independamment de
toute preoccupation metrique, un calcul de seg-
HALSTED'S RATIONAL GEOMETRY ^>
merits ou les operations sont identiques a celles
des nombres. Sommes et produits de segments.
Sommes d'arcset d'angles.
Chapitre IX. — Proportions et similitudes.
Deux triangles sont dits semblables quand leurs
angles sont respectivement congruents. 11 eutfallu
dire la un mot de IVxistence de tels triangles;,
c'est une lacune bien facile a combler. La simili-
tude conduit naturellement au tbeoreme de Thales
et aux proportionnalites qui t-n decoulent.
Chapitre X — Equivalence dans le plan. La
mesure des aires planes peut etre obtenue sans
le secours de I'axiome d'Archimede parce que
deux polygones equivalents peuvent etre con-
sideres comme sommes algebriques de triangles
elementaires en meme nombre et deux a deux con-
gruents, quoique de dispositions differentes. Par
definition I'aire d'une triangle egale le demi
produit de la base par la hauteur; deux polygones
equivalents ont meme aire et reciproquement.
Theoreme de Pythagore et carres construits sur
les cotes d'un triangle. Le chapitre se termine
10 H^LSTED'S RATIONAL GEOMETRY
par Line note historique courte, mais interessante
SLir le numbre tt.
Chapitre XI — Geometrie du plan, differant
pen de notre cinquieme livre usuel.
Le chapitre XI 1 est consacre aux polyedres et
volumes. M. Halsted commence a bon droit par
le theoreme d'Euler; il appelle par dcuififion
Volume du tetraedre le tiers du produit de la base
par la hauteur, et prouve que le volume d'un
tetraedre egale la somme des volumes des tetrae-
dres en lesquels on le partage d'une fagon quel-
conque. L'auteur examine quatre methodes de
division particulieres, la division la plus generale
pent etre obtenue au moyen de ces dernieres, et
il en est de meme pour un polyedre.
Les chapitres XIII et XIV nous donnent I'etude
de la sphere, du cylindre et du cone, avec le
mesure de leurs surfaces et volumes. Pour le
volume de la sphere. Ton fait usage de I'axiome
de Cavalieri: Si deux solides compris entre
deux plans paralleles sont coupes par un plan
HALSTED'S RATIONAL GEOMETRY 11
quelconque parallele aux deux premiers suivants
des aires e;c^ales, ils ont meme volume.
Chapitre XV Spherique pure ou Geometrie a
deux dimensions sur la sphere: Ce Chapitre ne
pouvait manquer de trouver ici sa place. M.
Halsted y precise d'abord ce que devinnent a la
surface de la sphere les axiomes d 'association,
d'ordre et de congruence, il en deduit simplement
et naturellement les proprietes elementaires, trop
negligees dans I'ensignement, des triangles
spheriques.
Trois notes terminent I'ouvrage, et sont rela-
tives; Tune a theoreme de Tordre, la deuxieme
au compas, et la troisieme ci la solution des pro-
blemes.
Ainsi qu'on le voit par cette analyse, le livre de
M. Halsted constitue une innovation et une tenta-
tive de vulgarisation des plus interessantes. Pour
lui donner plus de poids aupres des etudiants a
qui il est destine, I'eminent professeur de Kenyon
College y a ajoute 700 exercices formant un choix
12 HALSTED'S RATIONAL GEOMETRY
excellent et varie. Nous souhaitons a cet ouvrage
de notre distingue ami tout le succes qu'il merite.
P. Barbarin.
President de la Societe des Sciences physiques
et naturelles de Bordeaux.
I'Enseignement JWathematique
du 15 Mars 1905.
HALSTED'S RATIONAL GEOMETRY 13
SCIENTIFIC BOOKS.
Rational Gt'omctiy. By GEORGE BRUCE Hal-
STED. New York and London, John Wiley
and Sons. 1904. Pp. viirr285.
For over two thousand years there has been
only one authoritative text-hook in geometry.
" No text-book," says tlie British Association,
"that has yet been produced is fit to succeed
Euclid in the position of authority!" There is, in
fact, little improvement to be made in Euclid's
work along the lints w'.iich he adopted, and
among the multitude of modern text-books, each
has fallen under the weight of criticism in pro-
portion to its essential deviation from that ancient
autlvjr.
This does not mean tiiat Euclid is witliout
defect, but starting from his discussion of his
famous parallel postulate, the modern develop-
ment has been in the direction of the extension
of geometrical science, with the place of that
author so definitely fixed that the system which
lie developed is called Euclidean geometry, to
distinguish it from new developments. The de-
14 HALSTED'S RATIONAL GEOMETRY
fects of Euclid arise out of a new view of rigorous
logic whose objections seem finely spun to the
average practical man, but which are based upon
sound thought. The key to this modern criticism
is the doubt which the mind casts upon the relia-
bility of the intuitions of our senses, and the
tendency to make pure reason the court of last
resort. Thus, the sense of point between points,
the perception of greater and less and many other
tacit assumptions of the geometrical diagram, are
the vitiating elements on which modern criticism
concentrates its objections.
As an evidence of the ease with which the
senses can be made to deceive, take a triangle
ABC, in which AC is slightly greater than BC.
Erect a perpendicular to AB at its middle point to
meet the bisector of the angle C in the point D,
From D draw perpendiculars to AC, BC, meeting
them respectively in the points E, F. Let the
senses admit, as they readily will in a free-hand
diagram, that E is between A and C, and F
between B and C; then fmm the equal right
triangles AED=BFD, DEC = DEC, we find
AE=BF, EC=FC, and, by adding, AC = BC,
whereas AC is in fact greater than BC.
HALSTED'S RATIONAL GEOMETRY 15
Are we to take our eyes as evidence that one
point lies between two other points, or how are
we to establish that tact? This query alone lets
in a flood of criticism on all established demon-
strations. The aim of modern rational geometry
is to pass from premise to conclusion solely by
the force of reason. Points, lines and planes are
the names of things which need not be physically
conceived. The object is to deduce the conclu-
sions which follow from certain assumed rflations
between these things, so that if the relations hold
the conclusions follow, whatever these things
may be. Space is the totality of these things;
its properties are solely logical, and varied in
character according to the assumed fundamental
relations. Those assumed relations which de-
velop space concepts that are apparently in accord
with vision constitute the modern foundations of
Euclidean space.
Mr. Halsted is the first to write an elementary
text-book which adopts the modern view, and in
this respect, his " Rational Geometry " is epoch-
making. It is based upon foundations which
have been proposed by the German mathema-
tician, Hilbert. in point of fact, the book con-
16 HALSTED'S RATIONAL GEOMETRY
tains numerous diagrams, and is not to be dis-
tinguished in this respect from ordinary text-
books, but these are simply gratuitous and not
necessary accompaniaments of tlie argument, de-
signed especially for elementary students whose
minds would be unequal to the task of reveling in
the domain of pure reason. Also, in opening the
book at random, one does not recognize any great
difference from an ordinary geometry. In other
words, those assumed relations are adopted which
lead to Euclidean geometry, in this respect the
author is appealing to the attention of elementary
schools, where no geometry other than the prac-
tical geometry of our world has a right to be
taught.
The first chapter deals with the first group of
assumptions, the assumptions of association.
Thus, the first assumption is that hco iiistinct
poUits determtne a stniio/it line. This associates
two things called points with a thing called a
straight line, and is not a definition of the straight
line. The definition of a straight line as the
shortest distance between two points involves at
once an unnamed assumption, the conception of
distance, which is a product of our physical
HALSTED'S RATIONAL GEOMETRY 17
senses, whereas the rational development of ge-
ometry seeks the assumptions which underlie and
are the foundations of our physical senses. hi
the higher court of pure reason, the testimony of
our physical senses has heen ruled out, not as
utterly incompftent, hut as not conforming to the
legal requirements of the court. However, there
is no ohjection to shortness in names, and a
straight line is contracted into a straioht, a seg-
ment of a straight line, to a sect, etc.
In the second chapter we find the second group
of assumptions, the assumptions of betweenness,
which develop this idea and the related idea of
the arrangement of points, hi the next chapter
we have a third group, the assumption of congru-
ence. This chapter covers very nearly the
ordinary ground, with respect to the congruence
of angles and triangles, and all the theory of
perpendiculars and parallels which does not
depend upon Euclid's famous postulate. This
postulate and its consequences are considered in
chapter IV.
All the school propositions of both plane and
solid geometry are eventually developed, although
there is some displacement in the order of propo-
18 HALSTED'S RATIONAL GEOMETRY
sitions, due to the method of development.
Numerous exercises are appended at tlie end of
chapters, which are numbered consecutively from
1 to 700.
Undoubtedly the enforcement upon logic of a
a blindness to all sense perceptions introduces
some difficulties which the ordinaiy cjeometries
seem to avoid, but as in the case of our concep-
tfon of a blind justice, this has its compensation
in the greater weight of her decisions. It seems
as if the present text-book (uight not to be above
the heads of the average elementary students, and
that it should serve to develop the logical power
as well as practical geometrical ideas. Doubtless,
some progressive teachers will be found who will
venture to give it a trial, and thus put it to the
tests of experience. At least the work will appear
as a wholesome contrast to many elementary
geometries which have been constructed on any
fanciful plan of plausible logic, mainly with an
eye to the chance of profit.
Arthur S. Hathaway,
rose polytechnic instisute.
ISCIHNCE, Feb. 3, 1905.']
HALSTED'S RATIONAL GEOMETRY 19
HALSTED'S RATIONAL GEOMETRY.
''liatioiia/ Geo})h'try, a Text-book for the Science
of Space. By GEORGE BRUCE HALSTED.
New York, John Wiley & Sons (London,
Chapman & Hall, Limited). 1904.
In his review of Hilbert's Foundations of Ge-
ometry, Professor Sommer expressed the hope
that the important new views, as set forth by
Hilbert, might be introduced into the teaching of
elementary geometry. This the author has en-
deavored to make possible in the book before us.
What degree of success has been attained in this
endeavor can hardly be determined in a brief re-
view but must await the judgment of experience.
Certain it is that the more elementary and funda-
mental parts of the " Foundations " are here pre-
sented, for the first time in English, in a form
available for teaching.
The author's predisposition to use new terms,
as exhibited in his former writings, has been ex-
hibited here in a marked degree. Use is made of
the terms sect for segment, straight in the mean-
20 HALSTED'S RATIONAL GEOMETRY
ing of straiglit line, betweenness instead of order,
copunjtal for concurrfnt, costraight for collinear,
inversely for conversely, assumption for axiom,
and sect calculus instead of algebra of segments.
Not the slightest ambiguity results from any of
these substitutions for the more common terms.
The use of sect for segment has some justifica-
tion in the fact that segment is used in a different
sense when taken in connection with a circle.
Sect could well be taken for a piece of a straight
line and segment reserved for the meaning usu-
ally assigned when taken in connection with a
circle.
The designation, betweenness assumptions,
which expresses more concisely the ci^ntent of
the assumptions known as axioms of order in the
translation of the "Foundations" of Hilbert, is
decidedly commendable. As motion is to be left
out of the treatment altogether, copunctal is bet-
ter than concurrent. Permitting the substitution
of straight for straight line, then costraight is pre-
ferable to collinear. hiversely should not be sub-
stituted for conversely. The meaning of the
latter given in the Standard Dictionary being
accepted in all mathematical works, it is well that
HALSTED'S RATIONAL GEOMETRY 21
it should stand. The term axiom* lias been used
in so many different ways in matliematics tliat it
seems best to abandon its use altogether in pure
mathematics. The substitution of assumption for
axiom is very acceptable indeed.
The first four chapters are devoted to statements
of the assumptions and proofs of a few important
theorems which are directly deduced from them.
The proof of one of the betweenness theorems
(§29), that every simple polygon divides the plane
into two parts is incomplete, as has been pointed
out,t yet the proof so far as it goes, viz., for the
triangle, is perfectly sound. It is so suggestive
that it could well be left as an exercise to the stu-
dent to carry out in detail. The fact that Hilbert
did not enter upon the discussion of this theorem
is no reason why our author should not have done
so. Hubert's assumption V, known as the Archi-
* " The familiar definition: An axiom is a self-evident
truth, means if it means anythins:, that the proposition
which we call an axiom has been approved by us in the
light of our experience and intuition. In this sense ma-
thematics has no axioms, for mathematics is a tormal
subject over which formal and not material implication
reigns." E. B. Wilson, BULLETIN, Vol. ii, Nov., 1904,
p. 81.
tDehn, Jahresbericht d. Deutschen Math.-Vereinigung,
November, 1904, p. 592.
22 HALSTED'S RATIONAL GEOMETRY
medes assumption, part of the assumption of
continuity whicli our author carefully avoids using
in the development of his subject, is placed at the
end of Chapter V, in which the more useful pro-
perties of the circle are discussed. For the be-
ginner in the study of demonstrative geometry, it
has no place in the text. For teachers and former
students of Euclid who will have to overcome
many prejudices in their attempts to comprehend
the nature of the "important new views" set
forth in the " Foundations" it has great value by
way of contrast. Contrary to Sommer's state-
ment in his review of the " Foundations " (see
Bulletin, volume 6, page 290) the circle is not
defined by Hilbert in the usual way. It is defined
by Hilbert and likewise by Halsted according to
the common usage of the term circle. The defi-
nition is — if C be any point in a plane a, then the
aggregate of all points A in a, for which the sects
CA are congruent to one another, is called a circle.
The word circumference is omitted entirely, with-
out loss.
In the chapter on constructions we have a dis-
cussion of the double import of problems of con-
struction. The existence theorems as based on
HALSTED'S RATIONAL GEOMETRY 23
assumptions I — V are shown to be capable of gra-
phic representation by aid of a ruler and sect-car-
rier. In this the reader may mistakenly suppose
on fust reading that the author had made use of
assumption V, but this is not the case. While in
the graphic representation the terminology of mo-
tion is freely used, it is to be noted that the ex-
istence theorems themselves are independent of
motion and in fact underlie and explain motion.
The remarks, in §157, on the use of a figure, form
an excellent guide to the student in the use of this
important factor in mathematical study. In chap-
ter VIII we find a discussion of the algebra of
segments or a sect-calculus. The associative and
commutative principles for the addition of seg-
ments are established by means of assumptions
IIlj and III^. To define geometrically the pn)duct
of two sects a construction is employed. At the
intersection of two perpendicular lines a fixed
sect, designated by 1, is laid off on one from the
intersection, a and b are laid off in opposite senses
on the other. The circle on the free end points
of 1, a and b determines on the fourth ray a sect
c = ab. This definition is not so good as the one
given by the "Foundations," as it savors of the
24 HALSTED'S RATIONAL GEOMETRY
need of compasses for the construction of a sect
product, althouj:^h the compasses are n(jt really
necessary. It seems that it is not intended that
this method be used for the actual construction oi
the product of sects, in case that be required, the
definition being suited mainly to an elegant demon-
stration of the commutative principle for multipli-
cation of sects without the aid of Pascal's the-
orem. Were it necessary to accept the truth of
Pascal's theorem as given in the "Foundations,"
a serious stumbling block has been met, and
Professor Halsted's definition would be altogether
desirable. All that is required of Pascal's theorem
for this discussion is the special case where the
two lines are perpendicular, and with this proved,
in the simple manner as presented in this book,
using Hubert's definition of multiplication, the
commutative principle is easily proved. As the
author makes use of Pascal's theorem to establish
the associative principle, so he might as well have
used it to establish the commutatix'e principle,
thus avoiding his definition of a product.
The great importance of the chapter on sect
calculus is seen when its connection with the
theory of proportion is considered. The propor-
HALSTED'S rational geometry 25
tion a : b :: a' : b' {a, a' , b, b' used for sects),
is defined as the equivalent of tlie sect equation
ab' =a'b, following the treatment of the "Foun-
dations." The fundamental theorem of propor-
tions and tiieorems of similitude follow in a man-
ner quite simple indeed as compared with the
Euclidean treatment of the same subject. It is in
the chapter on Equivalence that the conclusions
of the preceding two chapters, taken with as-
sumptions Ij.^, II, IV, have perhaps their most
beautiful application, in the consideration of areas.
This subject has been treated without the aid of
the Archimedes assumpti(jn, as Hilbert had shown
to be possible. Polygons are said to be equiva-
lent if they can be cut into a fmite number of
triangles congruent in pairs. They are said to be
equivalent by completion if equivalent polygons
can be annexed tn each so that the resulting poly-
gons so composed are equivalent. These two
definitions are quite distinct and seem necessary
in order to treat the subject of equivalence with-
out assumption V, Three theorems (§§ 26\, 265,
266) fundamental for the treatment are quite
easily proved, but the theorem Euclid I, 39, if
two triangles equivalent by completion have equal
26 HALSTED'S RATIONAL GEOMETRY
bases then they have equal altitudes, while not
difficult of proof, requires the introduction of the
idea of area. The author points out that the
equality of polygons as to content is a construct-
ible idea with nothing new about it but a defini-
tion. It is then shown that the product of alti-
tude and base of a given triangle is independent
of the side chosen as base. The area is defined
as half this product. With the aid of the dis-
tributive law it is then shown that a division of
the triangle into two triangles by drawing a line
from a vertex to base, called a transversal parti-
tion, gives two triangles whose sum is equivalent
to the given triangle. This aids directly in the
proof of the theorem, — if any triangle is in any
way cut by straights into a certain finite number
of triangles \ then is the area of the triangle
equal to the sum of the areas of the triangles A,
This theorem in turn aids in the proof of a more
general one (§ 281), viz., if any polygon be parti-
tioned into triangles in any two different ways, the
sum of the areas A^ of the first partition is the same
as the sum of the areas A^^ of the second and
hence independent of the method of cutting the
polygon into triangles. As the author says, this
HALSTED'S RATIONAL GEOMETRY 27
is the kernel, the essence of the whole investiga-
tion. It deserves complete mastery as it facili-
tates the understanding of a corresponding theo-
rem in connection with volumes. The area of a
polygyn is defined as the sum of areas of tri-
angles ^^ into which it may be divided, whence
it follows as an easy corollary that equivalent
polygons have equal area. The proof of Euclid
I, 39 is then given with other theorems concern-
ing area.
The mensuration of the circle discussed in this
chapter, beginning with § 312, Dehn character-
izes* as an "energischen Widerspruch." It does
not so impress the present writer. The author
does not claim that the sect which he calls the
length of an arc is uniquely determined. It is
defined in terms of betweenness — not greater than-
the sum of certain sects and not less than the
chord of the arc. Even with a continuity as-
sumption it cannot be uniquely determined. But
the question as to whether the sect can be deter-
mined uniquely or not can well be left, as the
author leaves it, for the one student in ten thou-
sand who may wish to investigate tt while the
*L. c, p. 593.
28 HALSTED'S RATIONAL GEOMETRY
others are occupying their time at wiiat may be to
them a more profitable exercise. The definition
of the area of a sector (§ 323), as Dehn says, *
"Sielit im ersten Augenblicke noch sclilimmer aiis
als sie in Wirklichkeit ist." Plane area has thus
far been expressed as proportional to the product
of two sects. The author could well choose the
area of the sector as k r (length of arc) and,
taking the sector very small, the arc and length of
arc may be considered as one, in which case
k /'(length of arc) becomes the area of a triangle
with base equal to length of arc, and altitude r,
whence k = K We then have the sector area
defined in terms of betweenness, since the arc
length which is included in this definition was
thus defined. What geometry comes nearer than
this, admitting all continuity assumptions? In
any case it can be but an approximation and the
author assumes this.
The geometry of planes is next considered, in
Chapter XI, and the author passes to a considera-
tion of polyhedrons and volumes in Chapter XII.
The product of the base and altitude of a tetra-
* L. c, p. 594.
HALSTED'S RATIONAL GEOMETRY 29
hedron having been shown to be the same regard-
less of the base chosen, the tetrahedron is made
to play the same role in the consideration of vol-
umes that the triangle did in the treatment of
areas. Its volume is defined as s- the product of
base and altitude. The partitioning of the tetra-
hedron analogous to the partitioning of the tri-
angle discussed in a previous chapter is employed
to prove another "kernel" theorem, namely, if a
tetrahedron 7 is in any way cut into a finite num-
ber of tetrahedra T^ then is always the volume
of the tetrahedron T equal to the sum of the vol-
umes of all the tetrahedra T^. This is one of the
features of the text as a text. Two proofs of the
theorem are given. The second one, that given
by D. O. Schatunovsky, of Odessa, is quite
long. The beginner is liable to get hopelessly
swamped in reading it as when reading some of
the "incommensurable case" proofs of other
texts. He can well omit It. The volume of a
polyhedron is defined as the sum of the volumes
of any set of tetrahedrons into which it may be
cut. With the introduction of the prismatoid
formula and its application to finding the volumes
of polyhedrons we have reached by easy steps
30 HALSTED'S RATIONAL GEOMETRY
another climactic point in tlie text. The volumes
of any prism, cuboid and cube follow as easy co-
rollaries. Contrary to the plan followed in the
treatment of areas, the consideration of volume is
wholly separated from the consideration of equiv-
alence of polyhedra. No attempt is made to treat
the latter. If the treatment of it be an essential
to be considered in a school geometry then a very
serious difficulty has been encountered. The
writer believes this is one of a few subjects that
may well be omitted from a school geometry. The
tendency has been, in late years, too much in the
other direction. Dehn's criticism* of the proof of
Euler's theorem (§379) is just, but it serves to
point out but another minor defect of the book.
In the proof the terminology of motion is used
in the statement: "let e vanish by the approach
of B to y4," but this is not an essential method
of procedure. The demonstration may well be
begun thus — if the polyhedron have but six edges
the theorem is true. If it have more than six
edges, then polyhedra can be constructed with
fewer edges. Given a polyhedron then with an
* L. c, p. 595-
HALSTED'S RATIONAL GEOMETRY 31
edge e determined by vertices A and 'B, construct
another with edges as before excepting that those
for which 'B was one of the two determining
points before shall now \vivq.A in its stead. Then
the new polyhadron will differ from the given one,
in parts, under the exact conditions as stated in
the remainder of the pn^of. The restriction to
convex polyhedra, if essential, should be made
clear.
\x\ the discussion of pure spherics. Chapter XV,
which has to do with the spherical triangle and
polygon, we have an excellent bit of non-euclid-
ean geometry whose results are a part of three
dimensional euclidean geometry. The plane is
replaced by the sphere, the straight by the great
circle or straightest, and the planeassumptions by
a new set on association, betvv^enn^ss and con-
gruences applicable only to the sphere. The pre-
sentation is easy to comprehend and in fact much
of the plane geometry of the triangle can be read
off as pure spherics. The proof of the theorem
(§ 567) — the sum of the angles of a spherical
triangle is greater than two and less than six right
angles — assumes that a spherical triangle is always
positive. The theorem can be proved in the usual
32 HALSTED'S RATIONAL GEOMETRY
way by § 548 and polar triangles, whence it fol-
low^s as a corollary that the spherical triangle is
always positive, if it be desirable to introduce the
notion of a negative triangle. In the next and
last chapter, within the limits of three pages, the
definitions and twenty-two theorems relating to
polyhedral angles are given. All these follow so
directly from the conclusions on pure spherics that
the formal proofs are unnecessary. One of
our widely used school geometries devotes as
many pages to the definitions and a single theo-
rem. This furnishes a sample of many excel-
lencies of arrangement in the text.
While the study of the foundations of geometry
has been, during the last century, afield of study
bearing the richest fruitage for the specialist in that
line, the results of the study have not hitherto
served the beginner in the study of demonstrative
geometry. It seems, however, the day is at iiand
when we can no longer speak thus. With the
book before us. and others that will follow, we are
about to witness, it is hoped, another of those
important events in the history of science whereby
what one day seems to be the purest science may
become the next a most important piece of applied
HALSTED'S RATIONAL GEOMETRY 33
science. Such events enable us to see with Pres-
ident Jordan * that pure science and utilitarian
science are one and the same thing.
Commendable features of the text are, a good
index, an excellent arrangement for reference,
brevity in statement, the treatment of proportion,
areas, equivalence, volumes, a good set of original
exercises, and the absence of the theory of limits
and "incommensurable case" proofs.
S. C. Davisson.
INDIANA UNIVERSITY,
January, 1905.
[From the Bulletin of the American Mathematical
Society, 2d Series, Vol. XL, No. 6, pp. 330-336.]
* Popular Science Monthly, vol. 66, no. i, p. 8i (No-
bember, 1904).
34 HALSTED'S RATIONAL GEOMETRY
Rational Geometry. By George Bruce Hal-
sted, A. B., A. M. (Princeton), Ph. D. (Johns
Hopkins). Price ^1.75. Chapman & HaH.
Although so many books on elementary geome-
try are continually appearing, no apology need he
offered for the publication of the present work.
It has nothing in common with the ordinary
text-book, except that it deals with the same sub-
ject. Prof. Halsted yields to none in his rev-
erence for the marvellous work achieved by
Euclid; nevertheless, he belongs to that school of
mathematicians which maintains that Euclid's
system is not infallible; that his theory is, in fact,
built up from an imperfect and incomplete set of
fundamental axioms to which he himself tacitly
and, perhaps even unconsciously, added. In the
opinion of Prof. Halsted and kindred thinkers it
has become necessary, for the advancement of
truth, that the system which has held sole sway
for so many centuries should give place to another
and a better one. Unlike many of the writers
who undertake the task of reforming Euclid, Prof.
Halsted shows no tendency to be content with less
HALSTED'S RATIONAL GEOMETRY 35
rigid proof: on tlie contrary, he urges the neces-
sity for the utmost rigour; and this, we venture
to think, is one of the strongest of his many
strong claims to consideration. He asserts that
the principles which form the groundwork of his
book secure both greater simplicity and increased
rigour for his demonstrations. Hilhert's "Foun-
dations of Geometry" furnish the basis for the
present treatise. Accustomed as we are to the
small number of simply worded axioms which are
met with in Euclid, it is somewhat difficult to ac-
quire readily a comprehensive grasp of the five
groups of "assumptions" considered essential by
Hilbert, and, seeing that an authority as notable
as Poincare failed to detect the redundancy of one
of Hubert's "betweenness assumptions," no
humbler mathematician need hesitate to reserve
for a time any definite expression of opinion as
to the extent to which Hilbert's "assumptions"
are deserving of being regarded as unimpeachable.
Mone, however, will dispute the care and the effort
to attain perfection which mark the drawing up,
the classification, and the enunciation of the "as-
sumptions"; none can fail to recognize how in
Prof. Halsted's hands they yield simple and de-
36 HALSTED'S RATIONAL GEOMETRY
lightful proofs of many of the propositions with
which every student of matliematics is familiar.
Four only of the five groups of "assumptions"
are used in the present work, viz., those in which
the ideas of "association," of "betweenness,"
of "congruence," and of parallelism claim atten-
tion. The Archimedean principle of continuity is
avoided in demonstrating the theory of proportion,
and in its place stands a sect calculus which fur-
nishes for geometry an analogue to the operations
of algebra as applied to real numbers. The asso-
ciative, commutative, and distributive laws which
govern algebra are shown to apply equally to the
sect calculus for geometry. The charm of many
of the author's methods of proof has been re-
ferred to: it exists in a marked degree in the sixth
chapter, where the originality displayed in the
solution of problems is specially attractive. When
Hubert's "Foundations of Geometry" appeared
there at once arose in the mind a doubt as to the
possibility — at any rate, at the present time — of
adapting the system to the needs of the immature
student; but the production of Prof. Halsted's
work shows that no cause for the doubt really
existed.
[From The Educational Times, December i, 1904.]
HALSTED'S RATIONAL GEOMETRY 37
Rational Geometry, based on Hilhert's Foun-
dations. By G. B. Halsted. New York: John
Wiley & Sons, 1904, pp. 285.
We could have wished that Mr. Halsted's
plan had included a commentary; the matter is
<set out with Euclidean severity.
Hubert's first quarrel with the traditional geom-
etry is about congruence. When is one finite
straight line AB (which Mr. Halsted calls a
"sect") to be considered congruent with another
sect XY? Euclid answers: When AB can be
moved so as to coincide with XY. But, of course,
AB must not alter in length while it is being
moved. Now, what does this mean? It means
that if A'B' is any position of AB during the
translation, then A'B' is to be congruent with AB.
But what does congruent mean? This is just
what we are trying to define. And we are arguing
in a circle. "To try to prove the congruence as-
sumptions and theorems with the help of the mo-
tion idea is false and fallacious, since the intuition
of rigid motion involves, contains, and uses the
38 HALSTED'S RATIONAL GEOMETRY
congruence idea." To define congruence of sects
and angles without motion, Hilbert resorts to a set
of assumptions. It is curious tliat lie is forced to
assume Euclid 1. 4 as far as the equality of the
base angles: he can then prove the equality of
the bases.
He is unable to prove the congruence of tri-
angles which have congruent two pairs of angles,
and a pair of sides not included (Euclic 1. 26,
Case 2). This appears to lead to a second am-
biguous case, as would happen in the surface of a
sphere.
We learn that "no assumption about parallels
is necessary for the establishment of the facts of
congruence or motion." Playfair's axiom is
adopted.
Tile chapter on "constructions" is interesting.
Apparently all figures whose existence can be de-
duced from assumptions admit of construction
with ruler and "sect-carrier," c.q. trisection of
sect is possible, and trisection of angle impossible.
Hilbert shovvs that there are constructions possible
with ruler and compass which are not possible
with ruler and sect-carrier.
Coming to area, we find the rejection of intui-
HALSTED'S RATIONAL GEOMETRY 39
tion leads as along a thorny path. For reasons
which we dimly apprehend, Mr. Halsted refuses
to associate numbers with sects (he never gives a
numerical measure of the length of a line), and
will have nothing to do with limits. (Hilbert is
more generous.) Two polygons are defined as
equivalent if they can be cut into a finite number
of triangles congruent in pairs. After proving the
equivalence of parallelograms on the same base
and between the same parallels, Hilbert is seized
with misgivings — perhaps all polygons are equiv-
alent. These doubts are resolved, and the section
ends with the demonstration that "a polygon
lying wholly within another polygon must always
be of lesser content than the latter."
A similar procedure is necessary in dealing with
the volumes of polyhedra. The area of a sector
of a circle is defined as the product of the length
of its arc by half the radius. Product is defined
satisfactorily, and Mr. Halsted lias a right to define
"area of sector" as he likes; but this definition
gives no clue to what would be meant by the area
of an ellipse, say. No general definition is given
of the area of a curved surface; but in § 453 we
are told that the lateral area of a right circular cone
40 HALSTED'S RATIONAL GEOMETRY
IS the same as that of a sector of a circle with the
slant height as radius and an arc equal in length
to the length of the cone's base. Is this a latent
definition? Again, the area of a sphere is defined
as the quotient of its volume by one-third its
radius.
The volume of a sphere (or other curved sur-
face) is virtually defined by Cavalieri's assump-
tion: "If the two sections made in two solids be-
tween two parallel planes by any parallel plane
are of equal area, then the solids are of equal
volume." A sphere is then compared in an in-
genious way with a tetrahedron.
C. Godfrey.
WINCHESTER COLLEGE,
England.
[The Mathematical Gazette, Vol. Ill, pp. 180-182.]
HALSTED'S RATIONAL GEOMETRY 41
Rational Geometry. By Prof. George Bruce
Halsted. New York: John Wiley & Sons,
publishers. London: Chapman & Hall, Limited.
The modern standpoint permits many simplifi-
cations in the development of geometrical theory,
of which our author skillfully avails himself. Of
the many notable features of this book it suffices
to mention only the treatment of Proportion,
Equivalence, Areas, Volumes, Pure Spherics, the
absence of the theory of limits, of a continuity
assumption, the presence of the ruler as a sect-
carrier displacing the compasses. This volume of
285 pages contains all that is essential to a course
in elementary geometry. The language is simple,
the logic exact, the exposition masterly, as was to
be expected from Dr. Halsted. The book seems
admirably adapted to class use. The already
great indebtedness of teachers of geometry to Dr.
Halsted has been manifoldly increased by the pub-
lication of this book, which, in the opinion of the
writer and with no intended disparagement of
others, is the most important contribution to the
42 HALSTED'S RATIONAL GEOMETRY
text-book literature of elementary geometry that
has appeared. And now that the way has been
opened may we not hope that the teachers of ge-
ometry in the secondary schools and colleges will
see to it that the present generation of pupils shall
receive the benefits rightly accruing to them
through the profound researches of the present
and last centuries on the foundations of geometry.
T. E. McKlNNEY.
MARIETTA, O.
[From the review in The American Mathematical
Monthly.]
HALSTED'S RATIONAL GEOMETRY 43
HALSTED'S GEOMETRY IN HINDUSTAN.
In a leader in "Indian Engineering" (Published at 7,
Government Place, CalcDtta), the editor, praising Hal-
sted's Elements of Geometry, had said :
The elements of old immortal Euclid have been used as
THE text-book on the subject of geometry for twenty-two
centuries in all countries of the modern world which
derive their culture and civilization from the Greek ; in-
deed so close has been the association of Euclid with
geometry, that not unnaturally the name of Euclid is
used in common parlance as synonymous with the science
of geometry. But though he has worn the crown so well
and so long, within the last century the foundations of
the science have been examined anew by Iha mighty
intellects of Lobachevsky, Bolyai, Riemann, and others —
men worthy of a seat by the side of Archimedes and New-
ton ; and the penetrative insight of men like these has
shown that the vision of Euclid was limited, that the
boundaries of the science are not where he placed them,
that the system he reared on the basis of the so-called
twelfth axiom is not one of the necessities of the human
intellect, and that it is quite possible to construct a con-
siscent system of geometry in which both the twelfth
axiom and the thirty-second proposition of the first book
of Euclid are violated. Dr. Halsted has been one of the
foremost captains in the work of popularizing the re-
searches of the investigators we have named, and has
thus materially facilitated the exploration of the new
country. We have always regretted that these beautiful
44 HALSTED'S RATIONAL GEOMETRY
researches, so stimulating and fascinating to the imagina-
tion, are not presented in a form in which they can be
readily assimilated by the beginner, and we venture to
hope that Dr. Halsted, who is so well qualified for the
task, will deal with the subject definitely in a companion
volume to the work now before us.
[What the learned editor ventured to hope has come now
to fruition, as signaled by the following review in "Indian
Engineering," Vol. XXXVll, No. 22, June 3, igc;, by
Wm, John Greenstreet, F. R. A. S., editor of the "Math-
ematical Gazette," the official organ of the British Asso-
ciation for the ImprovemenrGeometrical Teaching:]
RATIONAL GEOMETRY
Under the above name Professor G. B. Halsted
has published a volume which is sure to attract
attention from those who have followed the work
that has been accomplished by Hilbert in the
study of the foundations of geometry. The book
before us is certain to attract more than ordinary
attention, being the first essay in the introduction
of the new ideas into the teaching of ele-
mentary geometry. The author is, of course,
well known to mathematicians all over the world,
being the most doughty and intrepid advocate of
general geometry in the United States. Time
alone will show whether the present effort will
HALSTED'S RATIONAL GEOMETRY 45
command more than a sneers d'estime. One
wonders whether the American teacher will over-
come all his prejudices and set to work to master
the new ideas so ably herein set forth. To many,
no doubt, the difficulties will be repellent, and if
that be so, when the tide turns, and general
opinion is ripe for the adoption of the new ideas,
the recalcitrants will have to be "mended or
ended." The change will not be as pleasantly
made as was the case when the proposals of the
Mathematical Association were adopted by the
universities and teaching bodies in Britain, for
British opinion had long been ripe for the change.
So far as we can judge, Americans have as yet
exhibited but a mild curiosity as to the scope of
the changes in the teaching of geometry in the
old country. The book before us makes a much
more serious demand on the patience and the in-
tellect of the teacher, and one wonders whether
the cheque will be honoured until after consider-
able preliminary delay. For this volume marks a
tremendous breach with the traditions of two
thousand years. It sounds the death knell of
intuition, and at first one can hardly think of
46 HALSTED'S RATIONAL GEOMETRY
geometry without intuition. 11 y a plus de qiiar-
ante ans que jc dis de la prose sans que j'en susse
Hen! said M. Jourdain in Moleire's Bourgeois
Gentilhomme. And a little consideration will
show how often our work and our methods in
geometry have been unconsciously intuitional.
Another point which will militate in some measure
against the success of this book in so conserva-
tive a land as Britain is the predilection of Pro-
fessor Halsted to adopt novelties of nomenclature.
We do not mean but that in most cases he may
be able to advance sufficient justification for a
course which always has great drawback's, and
especially when the change concerns words which
are wrought into the warp and woof of the lan-
guage. Sometimes the change happens to be
both timely and happy. When a word has more
than one connotation it is time that it disappeared.
For this reason it is high time that "axiom"
should be relegated to the limbo of words that
have outlived their use, and we cannot object to
the ingenious substitute — assumption . So again,
the word "segment" in "segment of a line" and
"segment of a circle" is at times, and to a cer-
HALSTED'S RATIONAL GEOMETRY 47
tain order of mind, provocative of confusion.
Segment is retained for the circle, but tlie segment
•of a line is called by the author a "sect," the
instrument for the transfer of segments (streck-
eniibertrager) being a "sect-carrier." "Co-
punctal" is hideous, but then it has a great advan-
tage over "concurrent," first because the latter
involves the idea of motion, and secondly because
the word co-punctal expresses exactly what is
intended, i.e., the possession of a common point.
But we shudder at co-straight in place of co-linear.
Hilbert's Second Group of Axioms, we beg par-
don — assumptions, defined the idea expressed by
"between," and were called axioms of order.
Professor Halsted calls them "betweenness as-
sumptions," to which there is no objection.
Chapters 1-IV state the assumptions, and give a
few theorems which miy be deduced from them.
The assumptions are divided by Hilbert into five
groups: — connection, order or betweenness, par-
allels (Euclid's), congruence, continuity (Ar-
chimedes'). The order is logical enough. First
the blade, then the ear and then the full corn in
the ear. First the definition of the geometrical
entities — point, line, plane, space; then the as-
48 HALSTED'S RATIONAL GEOMETRY
sumptions which are made as to the mutual rela-
tions of the entities. The assumptions of con-
nection are seven; by tlieir means we can show-
that two co-planar straight lines, "straights,"
cannot have two points in common; they must
have one common point or none, with similar
properties of planes. Next we have the four as-
sumptions of betweenness, first treated properly
by W. Pasch. Hilbert originally gave five, buf
the fourth was shown by R. L. Moore to be in-
cluded in the others. The last of these assump-
tions will show the reader the extent to which we
are left independent of intuition. Draw a triangle
ABC. Any co-planar line which cuts AB will
also cut either BC or AC. That is now an as-
sumption! The general theorem of congruence is
as follows: — If ABC A'B'C are two con-
gruent figures and P any point in the first we can
always find a point P' in the second such that the
figures ABC P, A'B'C P' are congruent. This
brings us to the idea of displacement, which is
logically dependent on that of congruence. The
last of the four chapters contains a signal instance
of the fading glories of intuition, tor the author
proves that every straight line has a middle point
I
HALSTED'S RATIONAL GEOMETRY 49
and that every angle has a bisector! We next
have the chapter on the circle and its properties.
We may point out an e.xcellent innovation — the
word circumference disappears. If C be any
point in a plane, the aggregate of all the points A
in the plane for which tlie sects CA are congruent
to one another is a circle. At the end of this
chapter we come to Archimedes' assumption,
which has not yet been used. We must not
omit to mention that Professor Halsted missed an
opportunity of improving the proof of the theorem
that the plane is divided into two parts by a poly-
gon. The proof as given has been shown by
Dehn to hold good in the case when the polygon
is a triangle, but not otherwise. Next comes con-
structions. Whenever a construction is dependent
on theorems based on the assumptions they re-
quire for their solution, only the straight-edge and
the sect-carrier are necessary, and thus they in-
volve only the drawing of a line and the cutting
off on it a given sect. Chapter VIII is devoted
to what used to be called the algebra of segments,
but is now the "sect-calculus." Proportion and
Similitude form the subject matter of Chapter IX,
and Chapter X deals with areas. In the twelfth
GEORGE BRUCE HALSTED
HALSTED'S RATIONAL GEOMETRY 53
GEORGE BRUCE HALSTED
La Societa Americana pel progresso delle Scienze
nel suo 50° congresso tenuto a Pittsbourgh dal 28
giugno al 3 luglio 1902 ha eletto Presidente della
sezione " Matematiche ed Astronomia " il Pro-
fessore GEORGE BRUCE HalSTED. Pigliando
occasione dalla lusinghiera e meritata distinzione
ottenuta dall'egregio nostra collaboratore (1) ed
amico, d permettiamo dire qualclie parola di Lui.
II Prof. G. B. HaLsted Iia avuto dalla natura
il dono poco comune di poter accoppiare nel modo
piu simpatico grande modestia e grande bonta
d'animo ad un' erudizione estesissima, tanto da
far stare dubbioso chi lo avvicina se in lui debba
(i) Oltre all'aver fatto conoscere negli Stati Unit! la
presente Rivista con parole di simpatia egli tradusse per
due delle niu diffuse Riviste scientifiche americane.
rAmerican Mathematical Monthly e il Science, N. S., le
note scritte nel nostro periodico da Juan J. Duran Loriga
(Charles Hermite, vol. I, pag. 2) e da P. Barbarin
(Sull'utilita di studiare la Geometria non-euclidea, vol. I,
pag. 85.)
54 HALSTED'S RATIONAL GEOMETRY
piu ammirare le doti della mente o quelle del
cuore. E pero certo che chi relazione con lui e
tratto ad affezionarglisi sinceramente.
Nato il 25 novembre 1853 da una famiglia di
studios! che prese larga parte alia rivoluzione
americana, ha 11 vanto di essere diretto discen-
dente di queirAbramo Clark che fu il firmatario
della dichiarazione d' indipendenza. Comincio
giovanissimo a mostrare la sua predilezione per lo
studio del scienze esatte col distinguersi in esse e
meritarsi continui premi nelle classi successiva-
mente frequentate, benche varie ore egli dovesse
sottrarre agli studi per guadagnarsi i mezzi di con-
tinuarli. Ma le sue doti di studioso furono ancor
piu apprezzate al suo ingresso nella carriera dell'in-
segnamento e gli valsero un pubblico elogio dell'il-
lustre Prof. Sylvester ed una calorosa raccomand-
azione di questi al Prof. Borchardt quando il gio-
vane Halsted si reco in Germania a compiervi un
corso di perfezionamento. Piu tardi fu lo stesso
Sylvester a presentarlo alia Societa Matematica
di Londra. Nel 1879 ricevetti il grado di Dottore
in Filosofia neH'Universita Johns Hopkins, e dopo
aver inaugurato ed anche diretto per qualche
HALSTED'S RATIONAL GEOMETRY 55
tempo "Post Graduate Instuction" in Princeton.
La sua opera di scienziato fu opera feconda e si
esplico in campi diversi rendendone il sui nome
popolarissimo in ogni categoria di studiosi. Negli
otto anni passat all'Universita John Hopkins pub-
blico in diverse riviste scientifiche una lunga serie
di note e memorie sui quaternioni, sui determin-
anti, sulla storia delle matematiche, sull'algebra
moderna: pubblico inoltre un 'opera didattica, di-
venuta classica, la Geometria metrica (1), ove,
con principi e metodi nuovi e esposto quanto si
riferisce alia misura delle lunghezze, delle aree,
dei volumi, degli angoli. Quest'opera fu tanto
bene accolta no solo in America, ma anche in
Inghilterra, da valergli I'onore di essere quasi
integralmente riportata da Wm. Thomson nella 9^
edizionedeir "Encyclopaedia Britannica" alia voce
"Mensuration," e da far scrivere al venerando
Sim. Newcomb: "Halsted e autore del tratto sulla
misura che e il migliore e il piu complete che io
conosca."
La grande operosita di Halsted non venne certo
(i) Metrical Geometry, An Elementary Treatise on
Mensuration. Boston: Ginn and Co.
56 HALSTED'S RATIONAL GEOMETRY
meno; e stanno a fame fede i numerosi lavori che
portano il suo nome. Parecchi di essi hanno per
scopo la volgarizzazione della scienza e sono disse-
minati nel Monist, nella Educational Review, nella
Popular Science Monthly, nell 'American Mathe-
matical Monthly, ecc.
Fra le sue opere didattiche, (1) ed oltre a quella
precedentemente ricordata, merita speciale atten-
zione la "Elementary Syntetic Geometry (New
York, J. Wiley and Sons), nella quale sono logi-
camente riuniti e rigorosamente esposti i punti
principali della Geometria sintetica. Essa e una
della piu rimarchevoli nella letteratura didattica
americana e ne fa fede quanto di essa scrisse una
delle riviste piu autorevoli ed imparziali: (2)
"Per piu di 2000 anni la Geometria ebbe per fon-
damento esclusivo la congruenza dei triangoli: si
presenta ora un libro che giunge ai risultati stessi
senza fare uso alcuno dei triangoli congruenti e
(i) Altre di tali opere sono ad esempio: Mensuration—
(Ginn & Co,, Boston and London); Elements ot Geome-
try. (J. Wiley and Sons, New York), Projective Geom-
etry. (Ibidi.
(2) Bullet, of the New York Mathematical Society, t.
Ill, N.o I, pag. 8-14.
HALSTED'S RATIONAL GEOMETRY 57
con tale semplicita che, ad esempio, tutti i casi
ordinari della coiT^ruenza dei triangoli sono dimo-
strati in otto righe."
11 nome di Halsted e indissolubilmente collegato
alia volgarizzazione della Geometria non-euclidea.
(4) Da quando comincio ad appassionarsi agli
(4) Hanno per scopo la volgarizzazione della Metage-
ometria le sue pubblicazioni :
Gauss and the Non-Euclidean Geometry,— Science, N.S.,
t. XIV, pag. 705-717 (1890); The Appreciation of
Non-Euclidean Geometry,— ibid., pag. 462-465.
Lambert's Non-Euclidean Geometry,— Bui. of the New-
York Math. Society, t. Ill, pag. 78-80, 11894).
Non-Euclidean Geometry ; Historical and Expository,—
American Math. Monthly, t. I, H, III, (1894, 95, 96).
The Non-Euclidean Geometry inevitable,— The Monist,
t. IV, Chicago, 1894.
Some salient points in the History of Non-Euciidean Ge-
ometry and Hyper-Spaces,— Math. Papers read at the
Internat. Math. Congress,— Chicago, 1893.
Nicolai 1, Lobatchefsky,— Address prononced at the com-
memorative meeting of the Imperial University of
Kasan, October 22, 1893, by A. Vassilief (trad, dal
russo, con prefazione). Austin, 1894.
Darwinism and Non-Euclidean Geometry,— Boll, di Ka-
san, (2), t. VI, pag. 22-25, (1896).
The Introduction to Lobatchefsky's new elements of
Geometry,— Texas-Academy, 1897,
Scientific Books, Urkunden zur Geschichte,— Science, N.
S. t. IX, pag. 813-817, (1889).
Report on Progress in Non-Euclidean Geometry,— Proc.
of the Amer. Ass. f. adv. of Sc. t. XLVIII, pag.53-68
C1899),
58 HALSTED'S RATIONAL GEOMETRY
studi filosofici commincio ad interessarsi a questa
Geometria per divenirne in breve non solo cultore
ma apostolo entusiasta guadagnandosi il vanto di
darne la prima bibliografia, (1) cosi importante
da essere subito tradotta e ristampata in Russia.
Da pochi anni aveva avuto principio in Europa
quel periodo scientifico nel quale 1' attenzione del
geometri era stata richiamata sulle ricerche relative
ai fondamenti della Geometria e su quella Geome-
tria per la quale Sylvester proponeva la denomi-
nazione di iiltra-enclidea. J. Hoiiel in Francia e
I'immortale Beltrami in Italia erano quasi soli a
segnalare I'alta importanza dei lavori di Lo-
batchefsky, di Bolyai, ed a fare intravedere
come lo studio delle basi della scienza dovesse
assorgere alia piu alta importanza filosofica, poten-
Non-Euclidean Geometry,— Am. Math. Month., t. VII,
pag. 123-133, (1900).
Non-Euclidean Geometry for Teachers,— Popular As-
tronomy, 1900.
Supplementary Report on Non-Euclidean Geometry,—
Science, N. S. t. XIV, pag. 705-717, U901).
The Teaching of Geometry,— Educational Review, New
York, Dec. 1902, pag. 456-470.
(i) Bibliography of Hyperspace and Non-Euclidean
Geometry.— Amer. Jour, of Math. vol. I, pag. 261-266 e
384-385, (1878); vol. II, pag. 65-70, (1879).
HALSTED'S RATIONAL GEOMETRY 59
do forse diventare I'unico capace di darci le chiavi
delle origini e della formazione delle conoscenze
umane. Che meraviglia dunque che nella giovane
America nessuno si fosse ancor messo alia testa di
coloro che ambivano di essere ammessi nella scir
ola che aveva mostrato che quella Geometria che
per piu di duemila anni era stata ritenuta Tunica
possibile non poteva reggere ad una seria discuss-
ione dei suoi postulati e che altri sistemi di Ge-
ometria, egualmente rigorosi, erano possibili?
Ivi pure piu di uno aveva cominciato a discu-
tere le due proposizioni di Legendre, la cui dimo-
strazione implica I'assioma d'Archimede, ed aveva
mostrato che cosa poteva diventare questa Geom-
etria, privata di tale postulato, e quella di Euclide,
privata del suo Xl° assioma. Tutto ciu pero res-
tava nell'esclusivo dominio dei dotti, anzi di quei
pochi che erano iniziati ai nuovi studi. Halsted
si assunse I'incarico di porre alia portata di tutti i
nuovi studi, traducendo le opere del russo Lobat-
chewsky, (1) dell'ungherese Bolyai, (2) dell'-
(i) N. Lobatchefsky, — Geometrical Researches on the
Theory of Parallels— (trad, dal russo, con prefazione e
appendice),— Tokyo Sugakubutsurigiku Kawai Kiji, t.
V, pag. 6-50, (1894).
60 HALSTED'S RATIONAL GEOMETRY
italiano Saccheri: (3) fu il suo entusiasmo che
trascino molti nella via da quel sommi segnata, e
ben presto una bella schiera di nomi eletti venne
ad arrichire la falange dei cultori delle nuove idee.
Postisi al corrente dei lavori dei geometri non-
euclidei, pienamente iniziati alia tradizione filoso-
fica, dominati da spirito critico di raro vigore, con-
tribuirono in breve anch'essi a porre in evidt-nza
gli errori e i controsensi filosofici dei metageometri
ed a debellare le obiezioni ingiuste e spesso igno-
ranti indirizzate dai filosofi alia metageometria. Si
schierarono anch'essi fra coloro che vollero restau-
rare e correggere le teorie criticiste mostrandosi
Id. — The Non-Euclidean Geometry, — Geometrical Re-
searches on the Theory of Parallels, (trad, dal russo),
Austin, 1894.
Id. New principles of Geometry, with a complete
Theory of Parallels,— (trad, dal russo), Austin, J897.
(2) J. Bolyai,— The Science absolute of Space, indepen-
dent of the truth, etc. (trad, dal latino), — Austin, 1894,
e riportato anche in Tokyo Sugaku , t. V, pag. 94-135,
1894.
(3) Euclides ab omni naevo vindicatus, sive conatus
geometricus quo stabiliuntur prima ipsa universae Geo-
metriae principia, — Auctore Hieronymo Saccherio, Socie-
tate Jesu, in Ticinensi Universitate Matheseos Professore
— Mediolani. 1733.
HALSTED'S RATIONAL GEOMETRY 61
discepoli e continuatori di Kant, sintetizzando ogni
anteiiore ricerca nella Teoria dei Gruppi che per-
mise a Sophus Lie di ridurre gli assiomi della Ge-
ometria alia loro logica essenza.
E se questo contributo di gratitudine chela sci-
enza deve ad Halsted sia giusto valga a confer-
maiio il giudizio che di lui da I'illustre Prof. A.
Vassilief dell'Universita di Kasan in lettera indi-
zzatami in questi ultimi giorni:
"Nella stoiia della difiusione delle idee della
Geometria non-euclidea il nome di Halsted sara
sempre menzionato con grande stima. E state
lui a dare la prima bibliografiia delle opere sulla
Geometria non-euclidea; e stato lui ad offrire il
suo eminente appoggio all'opera del Comitate
Lobatchefsky fondato a Kasan nel 1893 alio
scopo di celebrare la memoria del grande geom-
etra russo; e stato lui a dare la traduzione in-
glese di varie opere di Lobatchefsky, ed e stato
ancor lui a far conoscere al pubblico scientifico
anglo-sassone, in una serie d'articoli sempre in-
teressanti, tutte le novita letterarie della Geom-
etria non-euclidea, Questo ardore instancabile
col quale il distinto professore si occupa
62 HALSTED'S RATIONAL GEOMETRY
"di tutto cio che si lega alia Geometria non-eu-
clidea deriva daH'interesse filosofico e gnoseo-
logico che essa offre per lui. Egli ha molto lu-
cidamente esposto questo interesse nel suo arti-
"colo "Darwinism and Non-Euclidean Geometry"
scritto a mia preghiera durante il suo soggiorno
a Kasan e del quale conservero sempre la piu
cara memoria. 11 lungo viaggio dal Texas alle
rive del Volga, fatto col solo intento di onorare
"la memoria di Lobatchefsky e anch'esso prova
dell'amore, — e posso anche dire del fanatismo,
" — del Prof. Halsted per questo ramo della Sci-
enza geometrica. Ma senza fanatismo non si
"puo fare nulla di grande, e son sicuro che la
letteratura scientifica americana ricevera hen
presto da parte di Halsted una storia completa
"della Geometrica non-euclidea, che noi non pos-
"sediamo ancora. Sara il degno coronamento dei
"suoi sforzi per propagare le idee di Lobatchefsky
"e di Bolyai nella letteratura anglo-americana."
Ed e appunto cio che anch'io mi auguro nel
porgere all'egregio Professore il piu fervido
augur io e il piu affettuoso saluto.
Prof. C. Alasia.
TEMPIO (SARDEGNA),
Marzo, 1903.
HALSTED'S RATIONAL GEOMETRY 63
GEORGE BRUCE HALSTED
The Italian Biography, by Professor Cristoforo
Alasia De Quesada, translated by Miss Mar-
garet A. Gaffney, of Whitman, Massachusetts.
The American Association for the Advancement
of Science, at its 50th meeting, held in Pittsburg
from June 28 to July 3, 1902, elected as president
of the section for Mathematics and Astronomy,
Professor GEORGE BRUCE HALSTED. This flat-
tering and deserved honor conferred upon our
distinguished collaborator (1) and friend gives us
an opportunity to say a few words about him.
Nature has bestowed upon Professor Halsted
the rare gift of being able to unite in the most at-
(i) Besides having, with sympathetic words, made
this magazine known in the United States, he has trans-
lated for two of the American scientific journals of
widest circulation, the American Mathematical Monthly
and Science, N. S., the articles written in our periodical
by Juan J. Duran-Loriga (Charles Hermite, vol. I,
pag. 2) and by P. Barbarin (SulT utilita di studiare la
Geometria non-euclidea, vol. 1, pag. 85).
64 HALSTED'S RATIONAL GEOMETRY
tractive manner great modesty and great kindness
of disposition to very deep and extensive learning,
so much so as to make all who approach him
doubt whether to admire the more the gifts of his
mind or of his heart, it is certain that his asso-
ciates come to feel for him the deepest attach-
ment.
Born November 25, 1853, of a family of schol-
ars that took an important part in the American
Revolution, Professor Halsted can claim direct
descent from Abram Clark, a signer of the
Declaration of Independence. He began when
very young to show his predilection for the study
of the exact sciences, distinguishing himself in
these, and continually v\inning honors in his
successive classes, although he several times
withdrew from his studies to secure the means of
continuing them. But his gifts as a scholar were
even more appreciated when lie began teaching,
and young Halsted won a public eulogy from the
eminent Prof. Sylvester, and a warm recommen-
dation from him to Prof. Borchardt when he went
to Germany to take a finishing course. Later,
Prof. Sylvester also introduced him to the London
Mathematical Society. In 1879 he received the
HALSTED'S RATIONAL GEOMETRY 6S
degree of Doctor of Philosophy from Johns Hop-
kins University. Shortly after he organized and
for some time directed the " Post-Graduate hi-
struction " at Princeton.
His work as a scientist was fertile, illuminating
diverse subjects, thereby making his name pop-
ular among all classes of students. \n the years
passed at Johns Hopkins University he published
in different scientific reviews a long series of notes
and memoirs on quaternions, on determinants, on
the history of mathematics, on modern algebra.
He published also his Metrical Geometry [Boston,
Ginn & Co.], a text book now become a classic.
In this by new principles and methods he ex-
pounds what pertains to the measurement of
lengths, areas, volumes, and angles. This work
was so well received, not only in America, but
in England, that it had the honor of being almost
entirely reproduced by Wm. Thomson in the 9th
edition of the Encyclopaedia Brittanica under the
title, "Mensuration." It caused the venerable
Simon Newcomb to write of Dr. Halsted : "He
is the author of a treatise on Mensuration which
is the most thorough and scientific with which I
am acquainted."
66 HALSTED'S RATIONAL GEOMETRY
Prof. Halsted's great activity has never less-
ened. The numerous works that bear his name
are evidence of this. Many of these have for
their aim the popularization of science, and are
scattered through the Monist, the Educational
Review, Popular Science Monthly, etc.
Among his text-books (2) besides that already
mentioned, his Elementary Synthetic Geometry
[New York. J. Wiley & Sons], deserves special
attention. In this the principal points of Syn-
thetic Geometry are brought together logically,
and rigorously demonstrated. This is one of the
most notable books in American didactic literature,
as the following from an impartial and authorita-
tive review [Bulletin of the New York Mathe-
matical Society] testifies : "For more than 2000
years geometry has been founded upon, and built
up by means of, congruent triangles. At last
comes a book reaching all the preceding results
without making any use of congruent triangles;
and so simply that, for example, all ordinary
(21 Others of these are for example: Elements of Ge-
ometrv, (J. Wiley & Sons, New York). Projective Ge-
ometry, (Ibid.)
HALSTED'S RATIONAL GEOMETRY 67
cases of congruence of triangles are demonstrated
together in eigiit lines."
The name of Halsted is indissolubly connected
with the popularization of non-Euclidean geome-
try (4). From the time when he fust devoted
(4) The following publications of his bear upon the
popularization of Metageometry :
Gauss and the Non-Euclidean Geometry,— Science, N.S.,
t. XIV, pag. 705-717 (1890); The Appreciation of
Non-Euclidean Geometry, -ibid., pag. 462-465.
Lambert's Non-Euclidean Geometry,— Bui. of the New-
York Math, Society, t. Ill, pag. 78-80, ( 1894).
Non-Euclidean Geometry ; Historical and Expository,—
American Math. Monthly, t. 1, H, 111, (1894, 95, 96).
The Non-Eu:lidean Geometry inevitable,— The Monist,
t. IV, Chicago, 1894.
Some salient points in the History of Non-Euclidean Ge-
ometry and Hyper-Spaces,— Math. Papers read at the
Internat. Math. Congress,— Chicago, 1893.
Nicolai 1. Lobatchefsky— Address pronounced at the com-
memorative meeting of the Imperial University of
Kasan, October 22, 1893, by A. Vassiliev (translated
from the Russian, with a preface). 1894.
Darwinism and Non-Euclidean Geometry,— Boll, di Ka-
san, (2), t. VI, pag. 25-29, (1896).
The Introduction to Lobatchefsky's new elements of
Geometry,— Texas-Academy, 1897,
Scientific Books, Urkunden zur Geschichte,— Science, N.
S. t. IX, pag. 813-817, (1889).
Report on Progress in Non-Euclidean Geometry,— Proc.
of the Amer. Ass. f. adv. of Sc. t. XLVllI, pag.53-68
(1899).
68 HALSTED'S RATIONAL GEOMETRY
himself to philosophical studies he has been inter-
terested in this geometry, becoming not only its
student but also its most enthusiastic apostle, and
winning the distinction of giving its first bibliog-
raphy, (1) which was of so much importance as
to be at once translated and reprinted in Russia.
A few years before had begun in Europe tliat
scientific period in which the attention of geome-
ters was directed to researches relating to the
foundations of geometry, and to that geometry
for which Sylvester proposed the name of iiltra-
Eiididean. J. Hoiiel in France and the immortal
Beltrami in Italy were almost alone in empha-
sizing the high importance of the labors of Lo-
bachevski and of Bolyai, and in pointing out that
the study of the foundations of science ought to
Non-Euclidean Geometry,— Am. Math. Month., t. Vll,
pag. 123-133, (1900).
Non-Euclidean Geometry for Teachers,— Popular As-
tronomy, 1900.
Supplementary Report on Non-Euclidean Geometry,—
Science, N. S. t. XIV, pag. 705-717, U9oO-
The Teaching of Geometry,— Educational Review, New
York, Dec. 1902, pag.456-470.
(i) Bibliography of Hyperspace and Non-Euclidean
Geometry,— Amer. Jour, of Math. vol. 1, pag. 261-266 and
384-385, (1878,; vol. 11, pag. 65-70, (1879).
r
HALSTED'S RATIONAL GEOMETRY 69
rise to the highest philosophical importance, being
perhaps the only thing capable of furnishing the
key to the origin and formation of the human
consciousness. What marvel, then, that in young
America no one should yet have put himself at
the head of those who aspired to be attached to
the school that had shown that the geometry
which for more than 2000 years had been regarded
as the only possible one could not resist a serious
investigation of its postulates, and that other sys-
tems of geometry just as rigorous were possible?
However, more than one had there begun to
discuss the two propositions of Legendre, the dem-
onstrations of which involve the postulate of Ar-
chimedes, and had shown what this geometry
would be without this postulate, and the geometry
of Euclid without the XI axiom. All this, to be
sure, remained in the exclusive possession of the
learned, or rather, of the few who had been
initiated in the new studies. Professor Halsted
undertook the work of placing the new studies
within reach of all, translating the works of the Rus-
sian Lobachevski (1), the Hungarian Bolyai (2),
the Italian Saccheri(3). It was his enthusiasm that
(i) N. Lobachevski,— Geometrical Researches on the
70 HALSTED'S RATIONAL GEOMETRY
drew many into the way marked out by these
heights, and very soon a distinguished band of
eminent names enriched the company of the culti-
vators of the new ideas. Familiarized with the
labors of the non-Euclidean geometers, fully im-
bued with philosophic tradition, dominated by
critical spirit of rare vigor, they also contributed to
make evident the errors and philosophical contra-
dictions of paradoxers, and to overcome the unjust
and very often ignorant objections of the philoso-
phers to metageometry. They also ranged them-
selves among those who wished to restore and
Theory of Parallels— (translated from the original with
preface and appendix),— Tokvo Sugakubutsurigiku Ka-
wai Kiji, t. V, pag. 6-50. (1894).
Ibid. — 4th ed. Austin 1894.
Ibid.— Introduction to New Elements of Geometry,
with a complete Theory of Parallels,— (translated
from the Russian). Austin, 1897.
(2) J. Bolyai,— The Science absolute of Space, inde-
pendent of the truth or falsity of Euclid's Axiom, —
(translated from the Latin).— Austin, 1894, and repro-
duced also in Tokyo Sugaku , t, V. pag. 94-134,(1894).
(3) nuclides ab omnI naevo vindicatus, sive conatus
geometricus quo stabiliuntur prima ipsa universae Geo-
metriae principia, — Auctore Hieronymo Saccherio, Socie-
tate Jesu, in Ticinensi Universitate Matheseos Professore
— Mediolani. 1733.
HALSTED'S RATIONAL GEOMETRY 71
correct the critique theories, showing themselves
disciples and continuers of Kant, making synthesis
of every anterior research into the theory of groups
which enabled Sophus Lie to reduce the axioms
of geometry to their logical essence.
How deserved is the gratitude which science
owes to Halsted is shown by an appreciation of
him by the illustrious Prof. A. Vassilief of the
University of Kasan in a letter received from him
a few days ago:
" in the history of the diffusion of the ideas of
non-Euclidean geometry, the name of Halsted will
always be mentioned with great respect. He gave
the first bibliography of the works on non-Euclid-
ean geometry. He gave his eminent support
to the work of the Lobachevski committee estab-
lished at Kasan in 1893 for the purpose of honor-
ing the memory of the great Russian geometer.
He has translated into English various works of
Lobachevski, and has also in a series of articles
always interesting made the Anglo-Saxon scientific
world acquainted with the latest literature of non-
Euclidean geometry. The indefatigable zeal with
which the distinguished professor has occupied
himself with all that is related to non-Euclidean
72 HALSTED'S RATIONAL GEOMETRY
geometry is derived from tine philosophical and
gnoseological interest it has for him. He has
most lucidly set forth this interest in his article,
'Darwinism and Non-Euclidean Geometry,' writ-
ten at my request during his stay at Kasan, of
which I shall always retain the pleasantest mem-
ory. The long journey from Texas to the Volga,
made for the sole purpose of honoring the memory
of J^ob ^ir^^^'^kL 's also proof of the love — per-
haps one might say enthusiastic devotion — Prof.
Halsted has for this branch of geometrical science.
But without such devotion nothing great can be
done. It is assured that American scientific liter-
ature will soon receive from Prof. Halsted a com-
plete history of Non-Euclidean geometry, which
we do not yet possess. it will be a fitting culmi-
nation of his labors to propagate in Anglo-Ameri-
can literature the ideas of Lobachevski and of
Bolyai."
And it is precisely this which 1 also presage in
presenting to the noble professor the warmest
well-wishing and the most affectionate salutation.
Prof. C. Alasia.
tempio (sardinia),
March, 1903.
HALSTED'S RATIONAL GEOMETRY 73
COMMENTS OF MATHEMATICIANS
Sehr geehrter Herr College!
Der Internationale Mathematikercongress zu
Heidelberg, dem ich in voriger Woche beiwohnte,
hat mich bisher verhindert, Ihren fiir Ihr schones
Text-book on Rational Geometry den Dank auszu-
sprechen, der auch ohne ihr liebenswiirdiges
Schreiben vom 1. 8. sehr bald erfolgt ware.
Ic'ii habe ihr Buch mit dem grossten Interesse
gelesen und mich gefreut, dass wir nun endlich
eine Elementargeometrie besitzen in der die Pro-
portionslehre ohne das Archimedische Postulat
entwickelt ist.
Ich bin sicher, das ihr Buch ein Vorbild sein
wird fiir viele andre, die je nach den ortlichen Be-
diirfnissen in andern Landern verfasst werden
werden.
Darf ich mir eine Bemerkung erlauben, so ist
es die, dass nicht recht ersichtlich ist, wozu Sie
74 HALSTED'S RATIONAL GEOMETRY
das Archimedische Postulat iiberhaupt anfiihren,
da Sie es weder in dtzT Lehre vom In halt der Poly-
gone und Polyeder noch zum Beweise der As-
sumption VI 1, p. 259, beniitzen. Aber vielleicht
sind mir die Stellen, an denen es gebraucht wird,
entgangen.
Ihre Lehre vom Volumen der Polyeder hat mich
um so mehr interessiert, als ich selber friiher
einen ahnlichen Versuch gemacht habe.
Ich war aber doch nicht so ganz davon befrie-
digt, dass der Begriff des Volumens ganzlich von
dem der ^quivalenz abgelGst wurde, mochte das
auch ohne das Archimedische Postulat nicht
moeglich sein. Ich erkenne aber die Berechtigung
Ihrer Auffassung vollkommen an.
Dass die Assumption VI 1 sich aus dem Ar-
chimedische Postulat beweisen lasst, ist Ihnen
gewiss nicht entgangen, vielleicht aber bemerken
Sie nicht, dass die Assumption VI 2 sich ganz
einfach ohne Beni^itzung des Archimedische Pos-
tulat auf VI 1 zuriickfiihren lasst.
Indem ich Ihnen nochmals meinen besten Dank
fiir den Genuss ausspreche, den Sie mir durch
die Ueberreichung Ihres Buch verschafft haben,
HALSTED'S RATIONAL GEOMETRY 75
zelchne ich mit dem Ausdrucke def Hochschat-
zung als Ihr ergebenster
F. SCHUR.
FREUDENSTADT,
1 8. 8. 04.
[From Professor Friedriech Schur of Karlsruhe, one of
two greatest living authorities on elementary geometry.]
My Dear Dr. Halsted:
1 have just received your letter and a day or so
since the copy of the Rational Geometry. It is
an excellent piece of work and will do much good,
I am sure.
It is certainly a very necessary thing to have
the scientific truths of geometry put into such
perfect shape and so available for the understand-
ing of students. This, it seems to me, is the true
popularizing of mathematics.
Your old teacher Sylvester would rejoice, I
know, in the work you have been doing.
I thank you ever so much for the copy of your
book and also in behalf of mathematical teaching
in this country. Yours most truly,
W. H. Echols.
Professor of Mathematics in the
University of Virginia.
UNIVERSITY OF VIRGINIA,
August 5, 1904.
76 halsted's rational geometry
My Dear Professor Halsted:
Your Rational Geometry is a beautiful piece of
work which in my opinion is destined to have a
marked influence on the teaching of elementary
geometry.
1 think every teacher of geometry should make
a careful study of this book.
Yours very truly,
P. A. Lambert.
Professor of Mathematics in Lehigh University.
BETHLEHEM, PA.,
Aug. i6, 1904.
Messrs. John Wiley & Sons:
Dear Sirs — Halsted 's Rational Geometry con-
stitutes a new departure, and its production is
eminently characteristic of its author. His aim
marks an epoch in the teaching of the subject in
this country. All teachers would be greatly prof-
ited by its perusal. If the influx of new ideas in
geometry is to produce an early effect in the coun-
try as a whole, it will have to do it through
reaching the teachers. For such a use the Ra-
tional Geometry is emniently well adapted.
If such works as the Rational Geometry and
HALSTED'S RATIONAL GEOMETRY 77
Professor Halsted's several productions on the
non-Euclidean geometry receive reading by our
teachers of geometry, the educational effect will
most likely be far greater than if these works had
a limited actual use in our schools. Here's hoping
that the youth of our country will get the new
ideas and ideals through the medium of their
teachers. Yours very truly,
JOS. V. Collins.
State normal school,
Stevens Point, wis.,
Oct. 21, 1994.
My Dear Dr. Halsted:
1 have looked over your Rational Geometry
with great interest. The book should be read by
every teacher. Very sincerely,
H. E. HAWKES.
Professor of Mathematics, Yale University.
20CARMEL ST., NEW HAVEN.
Professor Halsted's Rational Geometry is a
book that every teacher and student of mathe-
matics should possess. It combines clearness and
simplicity with rigor, in which last quality Euclid,
and still more some of his modern rivals, are sadly
deficient. It takes into account and utilizes the
78 HALSTED'S RATIONAL GEOIVIETRY
results of all the centuries of inv^estigation since
Euclid; in fact a book like this is unthinkable
without Lobachevsky, Bolyai or Hilbert. No
tacit assumptions as in Euclid, nostraight-line-the-
shortest-distance "axiom" as in most of our mod-
ern text-books, no doubtful, erroneous and irrele-
vant statements concerning non-Euclidean geome-
tries as in some other text-books, but a reliable,
complete and rigorous system of geometry such
as could have been written only after the modern
investigations on the foundation of geometry had
been concluded in their essential features. It is
the first book of its kind in our country and in
any country (Italy perhaps excepted), and this
fact alone makes good its claim to the attention of
teachers and students of mathematics.
JOHN ElESLAND.
Instructor, U. S. Naval Academy, Annapolis, Md.
My Dear Professor Halsted:
For simplicity of form and rigor of logic your
book is a veritable model.
Yours very truly, ARNOLD Emch.
Professor of Mathematics, Univ. of Colorado.
Nov. 21, 1904.
halsted's rational geometry 79
My Dear Professor Halsted:
Your Rational Geometry came from the pub-
lisliers some days ago. 1 have read consecutively
the first ten chapters.
Its simple, rigorous logic, its accurate, concise,
terse English mark your book as a masterpiece of
geometrical exposition.
It appeals to me as simpler and easier than the
usual text. Boys and girls who are ready for
demonstrative geometry should have no difficulty
with it.
You have done them and the teachers and the
cause of mathematics in this country a great
service.
1 am delighted with the book. If my studies
entitled my opinion to any weight in such matters
(and they do not) I should say that your book is
the most important contribution to the text-book
literature of elementary geometry since Euclid.
Most truly yours,
Thos. E. McKinney.
Professor of Mathematics in Marietta College.
Secretary of the Association of Ohio Teachers
of Mathematics and Science.
MARIETTA, OHIO,
Aug. 4, 1904-
80 HALSTED'S RATIONAL GEOMETRY
From the Preface written by Poincare for the
American edition of his "Science and Hypoth-
esis."
Je SLiis tres reconnaissant a IW. Halsted qui a
bien voulu, dans une traduction claire et fidele,
presenter mon livre aux lecteurs americains. On
sait que ce savant a deja pris la peine de traduire
beaucoup d'ouvrages europeens et a ainsi puis-
samment contribue a faire m.ieux connaitre au no-
veau continent la pensee de I'ancien. . . .
D'ailleurs M. Halsted donne regulierement
chaque annee une revue des travaux relatifs a la
geometrie non-euclidienne, et il a autour de lui un
public qui s'interesse a son oeuvre.
II a initie ce public aux idees de M. Hiibert et il
a meme ecrit un traite elementaire de Rational
Geometry, fonde sur les principes du celebre sa-
vant allemand.
Introduire ce principe dans I'enseignement, c'est
bien pour le coup rompre les ponts avec I'intuition
sensible, et c'est la je I'avoue, une hardiesse qui
me parait presque une temerite.
Le public americain est done beaucoup mieux
prepare qu'on ne le pense a rechercher I'origine
de la notion d'espace.
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