Publications of·the University of Texas Publications Committee-: FREDERICK DUNCALF . C. T. GRAY KILLIS CAMPBELL E. J. MATHEWS D. B. CASTEEL C.E.ROWE F. W. GRAFF A. E. TROMBLY The University publishes bulletins six times a month, so numbered that the first two digits of the number show the year of issue; the last two the position in the yearly series. (For example,No.1701 is the first bulletin of the year 1917.) ·· VThese comprise the official publications of the University, publications on humanistic and scientific subjects, bulletins prepared by the Bureau of Extension and by the Bureau of Government Research, and other bulletins of general educational interest. With the exception of special num-. bers, any bulletin will be sent to a citizen of Texas free on request. All communications about University publi­ cations should be addressed to University Publications, Aus­ tin. 108-1216-5807-lSh University of Texas Bulletin No. 2063: November 10, 1920 Texas Mathematics Teachers' Bulletin Volume VI, No. 1 PUBLISHED BYTHE UNIVERSITY SIX TIMES A MONTH, AND ENTERED AS SECOND-CLASS MATTER AT THE POSTOFFICE AT AUSTIN, TEXAS, UNDER THE ACT OF AUGUST 24, 1912 The benefits of education and of useful knowledge, generally diffused through a community, are essential to the preservation of a free govern­ment. . Sam Houston Cultivated mind is the guardian genius of democracy. • . • It is the only dictator that freemen .acknowl­ edge and the only security that free­men desire. Mirabeau B. Lamar University of Texas Bulletin No. 20€3: November 10, 1920 Texas Mathematics Teachers' Bulletin Volume 6, No. 1 ~dited by J. W. CALHOUN, Associate Professor of Applied Mathematics, and H.J. ETTLINGER, Adjunct Professor of Pure Mathematics This Bulletin is open to the teachers of mathematics in Texas for the expression of their views. The editors assume no responsibility for statements of facts or opinions in articles not written by them. PUBLISHBD BY THB UNIVBKSITY SIX TIMES A MONTH, AND BNTBRBD AS SBCOND·CLAIS MATTER AT THE POSTOFFICE AT AUSTIN, TBXAS. UNDER THE ACT OF AUGUST 24, 1912 CONTENTS Editorials. The Work of the National Committee on Mathematical Requirements ........H. J. Ettlinger. . . . . . 5 A Series of Papers on Funda1nental Con­cepts of Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Mathematics the Common Denominator of the Exact Sciences. . ........................ T. McN. Simpson... . 8 An Elementary Comparison between the Eucli­dean Geometry of the Plane and the Geom­etry of the Surface of a Sphere........... H. J. Ettlinger......14 Codes and Ciphers......................... Renke Lubben. . . ...18 An Account of Some Philosophical Theories of Geometry. . ..............·............... Elizabeth Hutchings. 25 . The Foundations of Geometry...............H. H. Hammer......29 Junior High School Mathematics (a reprint) .....................35 A P:roblem Inv9lving Indeterminate Equations. H. J. Ettlinger......50 The Straight Edge ......................... A. N. Onymous .....54 MATHEMATICS FACULTY OF THE UNIVERSITY OF TEXAS P. M. Batchelder H. Y. Benedict A. A. Bennett J. W. Calhoun C. M. Cleveland Albert Cooper Mary Decherd E. L. Dodd H. J. Ettlinger • Helma L. Holmes· Goldie P. Horton Jessie M. Jacobs J. N. Michie R. L. Moore Anna M. Mulliken M. B. Porter C. D. Rice THE WORK OF THE NATIONAL COMMITTEE ON MATHEMATICAL REQUIREMENTS During the past summer the National Committee on Mathematical Requirements issued a preliminary report on "Junior High School Mathematics." This is the second re­port issued by the Committee since its appointment three years ago to make a thorough study of the entire field of the teaching of mathematics. In this number of the Bul­letin the above is reprinted in full. The report contains a great deal of general information on the plan of the junior high school and the growth of the movement in the last ten years. The committee outlines a course of study in mathematics to begin with the seventh grade. The underlying principles are much the same as those laid down in the first preliminary report, to develop real power in dealing with numerical and space relation­s,hips and to plan the contents of the course so that it will serve: the student best at the time he is taking the work. The "number-pair" idea or the notion of the dependence of one number on another is the cen1ent which holds the course together and makes it · a connected and natural se­quence of topics, instead of a disjointed hodge-podge. The writer is convinced more strongly than ever that the fact that two sets of numbers are related is one that can be ap­preciated by a student at a very early stage. If our teach­ers of mathematics will saturate themselves with this prin­ciple, they will find that their classes will absorb it almost unconsciously as their studies continue. There is a real difficulty in reaping the fruits of the two reports of the National Committee in that our teachers of mathematics of the present day have received their train­ing under the old formal system and unless strenuous University of Texas Bulletin efforts are made to acquire these new ideas, the value of these reports to the schools. of this state will be negligible. The Committee is making every effort to place these ideas before the teachers of the Southwest not only by means of the printed page but also by word of mouth. The writer has been asked by Chairman Young to act as spokesman for the Committee for the State of Texas and the Southwest. Addresses have been delivered at Comanche, San Antonio and Corpus Christi at the time of the county institutes. Professor Young is eager to have a representa­tive before every large educational meeting. If such a meeting is to be held in your part of the state during the year, notify the writer or Professor Young. Mathematics teachers should study these reports and in­corporate the ideas contained in them in their teaching. Inform your fellow-teachers, in or outside of mathematics, and all other fellow-citizens, of the objects of the Commit­tee's investigations. It is a. piece of work which will be of incalculable advantage to every man and woman of this state. · H. J. ETTLINGER. A SERIES OF PAPERS ON FUNDAMENTAL CON­ CEPTS OF MATHEMATICS During the summer of 1920 a course was given at the University of Texas on the "Fundamental Concepts of Alge­bra and Geometry." The editors offer to the readers of the Bulletin a series of eight papers which were written as part of the work of the above course. These papers were writ­ten with the view of publishing them for an audience of high school mathematics teachers. Two papers are to be found in this number. Six others will appear in the other two Bulletins to be issued during the year. MATHEMATICS THE COMMON DENOMINATOR OF THE EXACT SCIENCES Perhaps every teacher of mathematics has heard the plea of the ignorant, "I don't care to study mathematics any fur­ther; I am going to specialize in physics or chemistry." One might as well say, "I shall study grammar no more; I must begin on the writing of my novel," or "I do not wish to waste time studying anatomy; I . intend to be a surgeon." Mathematics is the basis, the tool, and the language of exact science. · The great Kant said, "Science is real science only to the extent that it is mathematical," and in our O\Vn time Karl Pearson has remarked in his Grammar of Science, "The description in terms of motion, the brief for1nulae expressing the changes in time and space of geometri~al concepts, is the whole content of natural science." One hears mathematics spoken of as the "handmaiden of science" ; I am willing to go further ip. this personifica­tion of mathematics and speak of it as the historian, the judge, the interpreter, the discoverer, and the prophet of science. Science rests its general statements upon qualitative ex.. periment; for its exact formulations its experiments be­come· quantitative. This requires standards and methods of measurement. As the historian of science, mathematics is concerned, with the record_ing of these measurements, their classification, their graphical presentation-in a word, the foundation material of exact science is statistics and the study of statistics is but one branch of mathematics. To be proficient in the use of statistics one needs no great amount of mathematics beyond arithmetic, algebra, graph­ical analysis, theory of probability, method of least squares, calculus, differential equations, and real function theory. As the judge of exact science, mathematics weighs the consistency of experimental results, the relative significance to be attached to discordant results, the leanings and the Mathematics Teachers' Bulletin leadings of various hypotheses, the harmony or discord between theory and experiment. As the interpreter of science, mathematics undertakes to combine into compact formulas the wealth of experi­mental data, thus making hundreds of pages of figures tell their story in a few lines of symbolism, which symbolism is the common language of all sciences and of all tongues. Mathematics thus becomes the great economist, reducing volumes to pages, summing in sentences the determinations of years. Further, in its interpretative role mathematics undertakes to exhibit the causal relations between events. It is not satisfied to accept the fact that things happen in a certain order; it thrusts itself with all its power into the question why they so happen. Thus mathematics becomes a discoverer in science. Math­ematics is itself a method of investigation an~ so it is con­tent to take the material for investigation from any science. All that it asks is that the data be accurate, reasonably con­sistent, and related-it does not undertake to explain things outside of their relation to other things, for mathematics· is a science of relations. The tremendous discoveries of science will be found associated in no insig·nificant number of in­stances with the names of men of great mathematical at­tainments, and the discoveries have culminated in the study rather than in the laboratory. The same power which makes mathematics a discoverer makes it a prophet, for the same analysis applied to the same experimental data which serves to explain one phenomenon may·serve to show the inevitability of others not yet ob­ilerved. A few illustrations, chosen from the various sciences and briefly cited, will make clearer the important way in which mathematics underlies the exact sciences. In physics, for instance, consider Boyle's familiar law connecting the pressure and volume of a gas, pv=c. This is an empirical law expressing the result in four symbols of a very large number of independent and .carefully per­formed experiments. It admits of a very simple geomet­ University of Texas Bulletin rical picture, one branch of an equilateral hyperbola, from the graph of which pressures corresponding to different volumes of the same gas may be rapidly read. But ex­ periment soon reveals that this law expresses. an accurate relation between pressure and volume only provided the temperature remains unchanged. In order to take account of temperature the law must be modified to the form pv=rt. The geometrical picture of this is a surface, the sections _of which for constant values of t are hyperbolas. If now one associates the law of pressure and volume with the kinetic theory of gases the problem complicates itself still further and a master mind like Kelvin is led to the notion of an absolute zero of temperature at which either pressure or volume must become zero. By the lique­ faction of rare gases physicists have already succeeded in producing temperatures closely approaching this absolute . zero, but its prediction preceded by years any very near approximation to its attainment~ If one makes even a casual study of the commercial ap­. plications of electricity, he finds that theory keeps abreast of and frequently outstrips experiment. It pays the Gen­eral Electric Company to pay Steinmetz a salary reputed to duplicate that of the president of the United States, be­cause Steinmetz is a wizard with the complex function theory. The connection between apparently so real a thing as electricity and apparently so unreal a thing as the com­plex function is one of the wonders of science . • Chemistry has as: one of its outstanding monuments the periodic table. Its construction is one of the triumphs of genius in correlation data. The gaps in the original table were no less significant than its inclusions, for if there was an underlying reason for the similarity of certain proper­ties of elements by columns, thei:e was reason to believe that there was a cause for the gaps. It was strongly sus­pected that the reason was simply that the elements be­longing in these gaps had not yet been isolated. This has Mathematics Teachers' Bulletin proved true in numbers of instances. Thus the table pre­dicted the discovery of elements. Physics and chemistry unite today in a field known as physical chemistry, the fundamental problem of which is the determination of the constitution of matter. The phys­icist has ceased to be satisfied with the molecule and the chemist is ·no longer satisfied with the atom; both are now struggling with the sub-atomic. For the physicist the goal may be an identification of electricity and matter, and an explanation of mass, hardness, and the cause of gravita­tion. For the chemist the goal may be a foundation of one rather than seventy odd fundamental entities and an ex­planation of atomic weight, of valence, and of the stability and instability of different chemical compounds. Words of Karl Pearson are significant in this connection. It is on these points of the constitution of ether and the structure of the prime atom that physical theory is at present chiefly at fault. There is plenty of opportunity for careful experiments to define more narrowly the perceptual facts we want to describe scientifically; but there is still more need for a brilliant use of the scientific imagination. There are greater conceptions yet to be formed than the law of gravitation or the evolution of species by natural selection. It is not probJems that are wanting, but the inspiration to solve them; and those who shall unravel them will stand the compeers of Newton, Laplace, and Darwin. Indeed it may safely be said that the future progress of physics .and chemistry lies probably in the perfecting of their mathematical theory, and today the process is re­tarded because of the limited number of experimental phys­icists and chemists with sufficient mathematical knowledge and skill to formulate the problem ''and to advance its solu­ .tion. And when these few do give their results they give them to a small audience. · Biology too is increasing its demands upon mathematics and is developing an almost distinct field of · bionietry­a word, by the way, perhaps ·inadequately defined in the Standard dictionary and not li~ted in the Brittanica. Dar­win. laid down certain great fundamental notions connected 12 University of Texas Bulletin with a theory of evolution. His work. is being carried on in many ways, and in our own time the mathematical theo­ries of Mendel are giving neW1 explanation to problems of genetics, and serving to account not merely for normal hereditary traits but for cases of regression, for what are sometimes called "sportS'." Karl Pearson, from whom two or three quotations have already been given, has been for many years one of the most active in developing the statis­tical study of biological problems. It is therefore interest­ing to find him saying, I allow no distinction ultimately between the physical and biological branches of science. As the latter advance, mere descriptions of sense-impressions are more and more likely to be formulae describing conceptual motions; such ·is, indeed, the confessed aim of those somewhat embryonic studies, "cellular dynamics" and "protoplasmic mechanics." Biology needs today far more trained mathematicians than it has. Last year at the University of Chicago a doc­tor of philosophy in zoology was back taking a course in calculus because he had discovered the need of it in his re­search. In physiology, we are told, the mathematical problems of nerve action are similar to those of the submarine cable. The time rate of healing of a wound is given by an ex­ponential curve; daily plotting of the patient's: progres3 against this curve will disclose the development of a path­ological condition. One of the most romantic of the instances of mathemat­ics applied to experiment~! data is in astronomy. It is an oft told tale but one forever inspiring. Tycho Brahe in the sixteenth century,.with meagre instrumental equipment but wonderful skill and persistence, accumulated a great mass of observations on the sun and planets, remarkable for their accuracy, considering the fact that Tycho died before the . · invention of the telescope. His pupil Kepler studied these observations for thirty years and finally reduced them to three "laws." Newton found these laws ready to hand and proved that they were a necessary consequence of one greater law of gravitation. Herschel followed with the discovery of Uranus, whose position after fifty years falled to agree with its theoretical place by a slight but intolerable amount of a few minutes of arc. Adams and Leverrier formulated the difficulty mathematically, made the hypothesis of a trans-Uranian planet, and predicted its place. Challis and Airy in England failed to follow up Adam's work observa­tionally. but Galle in Berlin found Neptune within· a degree of the spot where Leverrier bade him search. We have not yet come to expect the same high degree of accuracy in the social sciences which has been reached in the natural sciences, yet the increasing use of statistical methods in psychology, in education, in economics and busi­ness administration, even indeed in certain studies of lit­erary forms and usages in poetry and prose, makes it ap­parent that President G. Stanley Hall was not overstating the~ case when he spoke of mathematics as "the ideal and norm of all careful thinking." Today the great handicap of science is the scarity of men who are mathematicians by gift and by training~ Even in the light of the wide extension of modern applications of modern developments in mathematics it is hard to say more than Roger Bacon said in his Opus Majus six centuries: ago, Mathematics is the gate and key of the sciences. . . . Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of this world. And what is worse, men who are thus ignorant are unable to perceive their own ignorance and so do not seek a remedy. Randolph.;..Macon College, T. McN. SIMPSON, JR. Ashland, Virginia. AN ELEMENTARY COMPARISON BETWEEN THE EUCLIDEAN GEOMETRY OF THE PLANE AND THE GEOMETRY OF THE SURFACE OF A SPHERE* To most students of the plane geometry course in the second year of the high school the theorems proved are very real. The book-page or the blackboard provides a. conve­nient domain in which the facts of Euclid appear to . be verified by experience. It is also true that gradually the validity of these spatial relationships are transferred to the surface of the earth by means of practical problems in mensuration. Continuing his mathematical studies, the student learns in solid geometry the usual propositions con­cerning the surface of a. sphere without, however, properly correlating the latter with the plane situation. Seldom, if ever, as he goesi on in the study of plane trigonometry does he have a definite realization that, for human beings in­habitin~ the earth, plane geometry is an ideal world, use­ful as a close approximation in some cases, inaccurate in others. It is, therefore, of considerable~ interest to make a comparisonJ between the two kinds of geometry, that of the Euclidean plane and that of the surface of the sphere, in order that both similarities1and dissimilarities may be pointed out and their practical, mutual relationship be es­tablished. We begin with the simplest form of geometrical object, the point as defined in some manner or accepted as unde­fined. It is the zero-dimensional element in our two geo­metrical realms. We shall think of our plane and of the sphere,t defined as in our high school courses, and .notice that they are both two-dimensional aggregates of points, both ·without any boundary, but the plane is infil)ite in ex­tent, while the sphere is finite. Consider now a pair of points in each. Define a line as the collection of points determined by stretching an inextensible string so as to pass through *Read before the Pentagram during the summer term, 1920. tWe shall use "sphere" in the sense of "surface of the sphere." Mathematics Teachers' Bulletin the given points with the plane and sphere perfectly smooth. It is readily recognized that the first "line" is our ordinary straight line in the plane; the second "line" is a great circle of the sphere. A comparison of the facts will also show that the "line" is infinite in length in P* and finite in length in S.* In P, the .line is definitely determined for any pair of points, but in S two points at opposite ends of a diameter determine an infinite set of such lines, e~ g. the meridians on the earth's surface through the poles. All other points on S determine a unique line. In all cases on S and P the line is the shortest distance between the two given points. If, now, we consider a pair of distinct lines., we find that · in P there are two possibilities, either the lines do not in­ tersect at· all, in which case they are parallel, or: else they intersect in one and only one point.. In S two distinct lines always intersect in two points. In both cases, two half­ lines determine an angle. As' our next object of interest let us consider three ·non­ collinear points. In P this configuration determines a tri­ angle with three sides, three vertices, and three angles.. On S three non-collinear points determine eight triangles, each with three sets of elements as above. There are certain properties of a triangle which are equally true in P and on S. For example, the sum of two sides of a triangle is greater than the third side, and their difference is less than the third sid~. That not all the relations are the same is apparent from the sum of the angles. In P this number is 180,0 on Sl this number is always greater than 180° and less than 540.0 The area of a plane triangle can be as large as you please, while on S it cannot exceed the area of the sphere. ' The various cases of congruence and symmetry are very similar. The possibilities are these: ( 1) three sides re­spectively equal, (2) two sides and an included angle re­spectively equal, or (3) one side and two adjacent angles respectively equal. In P two congruent .triangles can ·be *Let P=plane and S=sphere. University of Texas Bulletin .. made to coincide by translation in P or by reflection in a line and translation. On S, likewise, two congruent tri­angles can be superposed by translation or sliding' on S; two symmetric triangles, by reflection in the surface of the sphere, that is, by turning "inside out," followed by transla­tion. There is, however, one case presenting a striking difference. On S, if two triangles have three angles of the one equal to the three angles of the other, the triangles are either symmetric or congruent ; in P they are merely sim­ilar. Corresponding .,~o these several cases of congruence, we have the methods of constructing a triangle for these cases. The co;nstructions on both P and S are very similar except for the last case. Other constructions are exactly similar in p· and on S, e. g., to draw the perpendicular bisector of a line segment or to draw the bisector of an angle, to erect a perpendicular to a given line at a given point on· or off the line. Having pointed out the similarities and dissimilarities between the geometry of p and that of s, it is possible to bring them yet closer together and to establish their mutual practical relationships by considering a map of Son P. We refer now to a representation similar to ordinary geographic ma~s in which points on the sphere correspond unambigu­ously to points in the plane, that is, if\ Q is. a point on the sphere and Q' its representative on the map, then to Q on S there corresponds only Q' in P and conversely. A central or gnomoni~ projection with the center of the sphere as· the center of projection upon a tangent plane at a point, T, will yield a map in which lines on S go over into lines in P. Hence a spherical triangle on S corresponds to a plane tri­angle in P. For a very small portion of S, near T, the approximation is very good and forms our justification for the use of plane trigonometry to solve practical problems on the earth's surface not involving large distances. We, therefore, substitute in this case for the real situation on S an ideal one which. is so close to it that we cannot measure the difference. Just as we approximate to the' spherical surface near T Mathematics Teachers' Bulletin by a portion of the surface of the tangent plane at T, so we may be replacing the real three dimensional world about us by an approximating Euclidean space such that the error of approximation cannot be detected. This has been sug­gested very recently by the late French mathematician, Poincare, who constructed a world differing from ours but to which our world of experience, regarded as an approxi­mation, approached. Recently renewed impetus has been given to this thought by Einstein's space-time theory which represe~ts our world as four dimensional in three directions of space plus one of time, to which the Euclidean. universe regarded as independent of time approximates. H. J. ETTLINGER. CODES AND CIPHERS At various times since man first learned to write-during wars, in times of political controversy, or on diplomatic · missions-he has felt the need of conveying messages in­telligible only to himself and to the receiver. The number and variety of ways in which this has been accomplished and the ingenuity displayed are amazing. By use of even the simplest rules, messages can be composed that are baf­fling to the unitiated; and by combining the more complex systems one may evolve a combination . puzzling even to an expert.. The subject of deciphering and reading codes should be difficult enough for a year's college course; and those who completed such a course would no doubt find ~ifficulty in solving all but the easiest codes. However, there are ex­perts who can solve ordinary ciphers without trouble; Edgar\ Allen Poe is said to have been able to decode any that were submitted to him. Codes and ciphers are modes of conveying information by writing, secretly, according to a certain ·confusing ar­rangement of letters or words, or the use ot symbols, let­ters, or words· for other letters or words ; ancj so con­structed that the message-can be understood readily only by those. "."ho are in possession1 of the code or device ac­ . cording to which the words or letters are arranged. There are systems of conveying secret messages otherwise than by changes in the order of words and letters, which ·can not properly be called codes or ciphers. For instance, a ·foreign language may .be employed. The British used the Greek language with great success during the Sepoy re­bellion in India in 1857. The ancient Spartans used leather thongs for conveying secret messages. The sender and the recei':er each had a stick of the same size. A narrow thong was wrapped tightly about the stick so that no part of the stick was left uncovered and so that there was nowhere more than one thickness of leather. The characters of the letter to be sent were written upon this surface. When ' 19 Mathematics. Teachers' Bulletin the thong was unwrapped the characters or letters were broken up and were unintelligible to anyone who did not wrap them upon a stick like that of the sender and in a· similar manner. This device was called a scyiale. An­other device, called the grille, has been used in.more modern times. In this case the sender and the receiver have each an identical card, perfo·rated at certain intervals with rec-. tangular holes or grilles. An apparently harmless message is written on paper of the same size as the grilled card; the words that can be read through the holes when the card is placed over the paper constitute the actual secret message. For instance, we might receive the following message that would arouse no suspicion. if it fell into the wrong hands: My dear X: The lines I now send you are forwarded by the kindness of the bearer, who is my "friend. Is not the message yet delivered to my brother? Be quick about it, ~or I have all along trusted that you would act with discretion and dispatch. Yours truly, Y. Upon placing our grilled card over the message we per­ceive through the gr~lles the warning: The bearer is not to . be trusted. In actual practice, of .course, these words would not be italicized as they are in this example; they would be revealed through the grille. Vanishing ink is· also often used where secrecy is imperative. Benjamin Franklin, while upon diplomatic journeys to France during our revo­lution used this device. He would write a letter of general or personal interest, and on the margins and the blank spaces at the· foot of the page he would put the messages in secret ink. These methods give secret messages, but are not, accord­ ing to Ball, in Mathematical Recreations and Essays, what are properly called cryptographs or ciphers. He defines cryp­ tograph as "a manner of writing in which the letters or symbols are employed as used in their natural sense, but University of Texas Bulletin are so arranged that the communication is intelligible only to those who possess the key." A simple way would be by writing every word backwards, thus : The road is dangerous. Eht daor si suoregnad. Or, every third letter after a punctuation mark might be counted. The Century Magazine gives an example for ar­ranging another code of this type. Take the message : Tig Cfpwywe it a sqnhriok mbiddntl. This might suggest to us a message in Sanskrit or Hot­tentot. But, write out the second word Cfpwye; allow the first letter to remain as it is. Under the next, f, .write the letter in the word, p, put the letters o, n; go backward three letters in the next case w, v, u, four in the next, and one more each time following. It is: a good plan to ;arrange these in little triangles as shown below; have a different triangle for each word. Notice that the top row is the above message. c T~~ _'i_~· ~ . h~· e On the hypotenuse or sloping side of each triangle is one word of the deciphered message, which, very modestly on the part of the Century, reads : "The Century is a ·splendid magazine.'! • 1 g f d m . This is a simple, effective method, and would suggest solu­tion by the method of frequency, explained below, but could not be solved by that method. Take the rectangle below : ·e 0 h c s e 1 3 y n m e s t 5 6 0 n g p n t 0 0 . 1 . 1 r m 8 · 2 7 4 The first eight letters of the message are in the first and sixth columns; the second eight in the second and fifth col­umns; and the third eight in the other two columns. The two columns of figures give the key. Picking out first the letters in the first and sixth columns in the order shown in the key, and then doing likewise in the second and fifth, and the third and fourth, we get the following message: "Enemy troops in sight. Come on." We might send a connected series of letters specifying that the letters were to be counted at intervals of three; the first time, say, every first, fourth, seventh, etc. letter; then ev~ry second, fifth, etc. ; then every third, sixth, ninth, etc. Thus, dco ood n ma oeytt becomes deciphered, "Do not. come today." One might construct a sentence similar 1n appearance, but actually different in construction, by specifying that the seventh, then the third from the seventh, and the fourth letter from this one, and then the seventh again were to be counted; and that all inteverning letters were to be "dummy" or nonsignificant. Thus, "come now" might be written abcfgrcovomurmvyglutefeneatofignorwe Such a message would have the disadvantage of undue longness. The foregoing systems have relied upon a certain arrange­gent, peculiar in each case, to hide the meaning. Another type of methods, called ciphers by Ball, rely upon the sub­stitution of characters, figures, or other letters for the let­ University of Texas Bulletin ters composing the actual message. We might use the let­ters of a foreign language, Greek, for instance; or we might invent characters for the purpose, as was done by the pirates in Poe's The Gold Bug. Instead of a certain letter, we might use the letter immediately following it; for a use b, and for b use c. Thus, "The cannon are ready" would be written, "Uif dboopo bsf sfbez." Julius and Augustus Caesar are said to have used ciphers of this type. The disadvantage of these systems of representing each letter by one certain symbol is that these systems are read­ily deciphered by applying the principle of frequency. It has been shown that the letter e occurs most often in the English language. Other letters in the descending order of frequency are said to be: t, a, o, i ; n ; r, s, h; d, l ; c, w, u, m; f, y, g, p, b; v, k; x, q, j, z. Those between each of two successive semicolons denote those of approximately equal frequency. So by looking over a message of the type indicated and by finding out which symbol occurs the oft­enest we can hazard a guess that this represents e. If we find a short word of three letters with the symbol for e at the end we can guess this to be the word the; if this works out we have two more letters, t and h. Of course, this is largely a matter of trial and error or guess work; but it is likely to lead to a solution. The readiness with which this method of deciphering works shows the need of a more complicated kind of code. One way of doing this is the following: Represent a . by 11, 37, or 63 b by 12, 38, or 64 .. .. . .. . . . .. . ... . ..... z by 36, 62, or 88 Notice that there are 26 letters in the alphabet and that 11+26=37 37+26=63 12+26=38 38+26=64 etc. Then let 89 or 90 represent the end of a word. Thus1 there will be three double-symbols for each letter. Another good method is by using a key word like "Power­ Mathematics Teachers' Bulletin launch"; arrange the letters horizontally and vertically as shown below; and then repeat the alphabet as often as pos­sible in the checkerboard fashion indicated. Here ea.ch letter will be represented by four different sym­bols. a can be represented by oo, eu, ae, or nn; and b by ow, en, ar, or nc; that is, each letter can be represented by any of the four combinations of two letters each which are in the key word in the same horizontal and vertical column respectively, as the letter to be represented. Another system injolving a key word on ·the vertical scale, and the alphabet can be made even more difficult to decipher. Take the key word "prudentia" and arrange as shown: Here the first word of a message would be in the first hori­ zontal column; the second in the second column, that start­ ing with r; the next in the thi:rd column; and so on. Thus, University of Texas Bulletin "iwt-vevdp-·bum-vla-gsvtw" would become when de­ciphered, "The enemy has six corps." The advantage of these two systems is that the key word can be changed every time, if necessary. It is not the purpose of this paper to give an exhaustive treatise upon the subject of ciphers. Also, unfortunately, no rules, except for the simplest types, can be given for the deciphering of codes. It would require thorough study of every type of code available, years of practice, and most of all, a native talent, to do this readily. Those who are inter­ested in getting further information upon the subject, how­ever, would do well to look up .W. W. Rouse Ball, Mathemat­ical Recreations and Essays, published by Macmillan and Company, New York. The larger part of the material of this essay is taken from this source. Articles may also be found in the following periodicals : The World's Work, June, 1918. Everybody's Magazine, June, 1918. The Literary Digest, February 9, 1918 The Survey, May 17, 1919. RENKE LUBBEN. AN ACCOUNT OF SOME PHILOSOPHICAL THEORIES OF GEOMET·RY.* The purpose of Professor Russell's book is the setting forth of the main points relating to the foundations of geometry. The material is organized according to the fol­ lowing plan : Chapter I, preliminary to the logical analysis of geometry, · contains a .brief account of the beginnings and development of various systems of geometry grouped under the heading "non-Euclidean," a term meaning simply "not according to Euclid's postulates and axioms." Chapter II, preparing the ground for a constructive theory of geometry, consists of a critical analysis of previous · phi­ losophical views, the author endeavoring (1) to show such views to be partly true and partly false, and (2) to estab­ lish the truth of his own theory. Chapter III, the beginning of real constructive work, rather than mere criticism, deals: with (1) Projective Geometry, shown to be wholly a priori, · meaning simply that certain knowledge is necessary for experience. (2) Metrical Geometry, the axioms of which fall into two classes : (a) Those axioms belonging both to Euclidean and non­Euclidean and de.clared to be a priori. ( b) Those axioms distinguishing Euclidean from non­Euclidean and declared to be entirely empirical; or, in other words, having their origin jn experience itself. Chapter IV, following up the assumptions of Chapter III, shows the necessity for experience of some form of externality, as derived from sensation or intuition, and sums up the "special difficulties of space"-as "all that is possible in an essay on the Foundations of Geometry."1 *A review of Chapter II of Russell's Foundations of Geometry. 1 Russell: Foundations of Geometry, p. 201. University of Texas Bulletin We have reviewed the volume: "Foundations of Geom­etry," as a whole, in order to get a perspective on our im­mediate topic: A Critical Account of Some Previous Phi­losophical Theories of Geometry. ;\.. discussion of the phiJ losophy of the subject presupposes the tracing of the math­ematical development of the theory of geometrical axioms from the first discussions of a geometry other than Euclid's to the present time. Geometry, as a theory of knowledge, has been a part of various discussions in the past, and has colored the viewpoint of many writers. But in a criticism of various theories, we shall go only as far back as the views: of Kant, directed against Newton's belief in absolute space, which is still the basis of the present laws of motion as formulated, and against the Leibnitz theory of space. We shall take Kant's doctrine from the purely logical side: (1) since geometry is demonstrably certain ,"apo­deitic"), space is a priori-that which is presupposed, for experience, to be possible-and subjective; (2) since space is a priori and subjective, geometry must be apodeictic. Professor Russell accepts with Kant "necessity for any pos:­sible experience as the test of the a priori," but does not enter into a discussion, in this chapter, of a relationship between the a priori. and subjective, since he applies only · the logical test. There is criticized, also, Kant's distinc­tion between analytic and synthetic judgment, on the ground that these are merely different aspects of any judgments. He believes that Kant's defense of his theories of space really prove some form of externality, yet not necessarily Euclidean space. A discussion of the theories of philosophers who followed Kant shqws that little was ·added to the theory of geometry. There was one philosopher, Herbart, whose views on geom­etry are not important, according to Russell, but whose psychological theory of space and whos:e belief in classifying space with other forms of series influenced Riemann by en­couraging him to attempt to explain the nature of space. Riemann regarded space as a particular kind of manifold, entirely from the quantitative side, neglecting unduly the qualitative_ adjectives of space. Russell thinks that Rie­ Mathematics Teachers' Bulletin mann's philosophy is not well grounded, that his definition of manifold is obscure. Riemann might be further criticized in that his "definition of measurement applies only to space." Professor Russell thinks that Riemann's theory of space as a manifold is "mathematically invaluable," "but philosophically misleading." Helmholtz's view that geometry depends on physics is rejected, Russell believing that physics depends on geom.. etry. But he admits, however, in geometry, a reference to matter, not the empirical matter of physics-, but a more abstract matter which appears in space and governs spatial relations. Erdmann accepted the conclusions of Riemann and Helm­holtz. Hence, criticism of the empiricism of Riemann and Helmholtz was reinforced by a criticism of Erdmann who regarded the axioms, adjectives of space, as "necessarily successive steps in classifying space as a species of mani­fold."1 Yet his deduction was shown to have involved false assumptions. Erdmann considers that geometry alone is incapable of deciding on the truth or falsity of the axiom of congruence which he regards as empirical, and as pos­sibly false in the infinitesimal but proved by mechanics. Russell criticises Erdmann's tests of the a priori as being varied and inaccurate, hence rendering his final conclusions unsound. Lotze's theories are set forth under two heads, a discus­sion concerning: (1) actual space, and (2) philosophical theories of space. But his criticisms of Helmholtz are based wholly on mathematical mistakes arising from insufficient knowledge. More recent French writers have given nothing new to foundations of Geometry. They have contributed much on the question of number and continuous quantity. From the mathematical side, there are Calinon and Poincare; from the philosophical, Renouvier and Delbref; intermediate, there is.Lechalar. The final discussion of the chapter under consideration is on the question of absolute magnitude. 1/bUl., p. xi. From the point of view of logic, Russell finds no obstacle to non-Euclidean spaces. Hence, he concludes that all homogeneous spaces are a priori, and that differentiation between them is the field of experience. On the other hand, spaces which are not homogeneous throughout, those with­out a space-constant, are logically unsound, impossible to know, and hence condemned a priori. To summarize: Kant believ~d that the axioms and pos­tulates, foundations of geometry, were a priori synthetic , judgments necessary to a consistent line of reason. This view brought forth the arguments: of Herbart, Riemann, and others. Riemann held the axioms of Geometry to be "successive determinations of class-conceptions, ending with Euclidean space," with no regard, however, to qualitative adjectives of space. Helmholtz believed that geometry pre­supposed physics. He granted a certain dependence on matter but not the empirical matter of physics. Erdmann accepted the conclusion of Helmholtz and Riemann. Lotze believed that any theories non-Euclidean were explainable physically, which virtually considered Euclid empirical. In conclusion, Russell's opinion may be stated: "All homo­geneous spaces are a priori possible, and the decision be­tween them is · empirical."1 ELIZABE'TH HUTCHINGS. 1Jbid p. XI. THE FOUNDATIONS OF GEOMETRY* The foundation of geometry consists of choosing a set of undefined terms and unproved propositions from which we can deduce all future propositions by the methods of formal logic. Euclid attempted this problem over two thousand years ago, but in some instances he used assump­tions in his proofS' which had not been mentioned previously. Our paper will not be concerned with the psychological or philosophical questions concerning this subject but we will pay particular attention to the logical viewpoint. Let us designate points by A, B, C . . ., lines by a, b, c . . ., and planes by ex:, {3, y • • • Our undefined terms will consist of points, lines and planes. We will call points the elements of linear geometry; points anq lines the elements of plane geometry; points, lines, and planes the elements of solid geometry. In order to proceed further in our discussion it will be necessary to give a few definitions,: A point A of a straight line a divides it into two parts; each of these parts is called a half line issuing from A. Given in a plane ex: two half-lines h and k issuing from a point 0, and belonging to two different straight lines, such a pair of half lines h and k is called an angle and is denoted by (h, k). A segment consists of the two points A and B of.the line a and all points between A and B. The points A and B are called the extremities of the segment, all intervening points are called interior points, and all other points are called exterior points of the segment. The term interval is some­times used in the same sense as segment. A system of seg­ments AB, BC, CD . . . KL is called a broken line joining A with.Land it is called the broken line ABC . . . KL. If the first and last points of a broken line coincide, we have a polygon. *A review of Hilbert's Foundations of Gez1• Uz X1-Z1 Hence we may see that the solutions depend not on the numbers, a, b, c, but on their differences, ( c-b), (b-a), (c-a). . For n=3, it may be readily ascertained from equations (1)-(5) that we have twelve unknown numbers in five equations. We may assign Xu y1 z1 to' have any arbitrary positive integral values such that x1