REVIEWS OF RATIONAL GEOMETRY BY GEORGE BRUCE HALSTED A B. and A. M. (Princeton University); Pli. D. (Johns Hopl 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