Publications of The University of Texas Publications Committees: GENERAL: FREDERIC DUNCALF MRS. F. A. PERRY J. F. DOBIE C.H. SLOVER J. L. HENDERSON G. W. STUMBERG H.J. MULLER A. P. WINSTON OFFICIAL: E. J. MATHEWS KILLIS CAMPBELL C. F. ARROWOOD C. D. SIMMONS E. C. H. BANTEL BRYANT SMITH The University publishes bulletins four times a month, so numbered that the first two digits of the number show the year of issue and the last two the position in the yearly series. (For example, No. 3001 is the first bulletin of the year 1930.) These bulletins comprise the official publica­tions of the University, publications on humanistic and scientific subjects, and bulletins issued from time to time by various divisions of the University. The following bureaus and divisions distribute bulletins issued by them; communications concerning bulletins in these fields should be addressed to The University of Texas, Austin, Texas, care of the bureau or division issuing the bulletin: Bureau of Business Research, Bureau of Economic Geology, Bureau of Engineering Research, Interscholastic League Bureau, and Division of Extension. Communications concerning all other publications of the University should be addressed to University Publications, The University of Texas, Austin. Additional copies of this publication may be procured from the University Publications, The University of Texas, Austin, Texas THE UNIVE8SlTT OF TBXil PBUI, AUl'ftll THE UNIVERSITY OF TEXAS BULLETIN No. 3043: November 15, 1930 THE TEXAS MATHEMATICS TEACHERS' BULLETIN Volume XV, Number 1 J!IVBLISHBD BY THB VNIVBRSITY POUR TIMBS A MONTH, AND BNTBRBD Al 8BCOND·CLASS MATTBR AT THB POSTOPPICB AT AUSTIN. TBXAB. ttNDBR THB ACT OP AUGUST :H, 1912 The benefits of education and of uaef11l knowledge, generally diffuaed through a community, are eHential to the preeervation of a free govern• ment. Sam Houston Cultivated mind ia the guardian genius of Democracy, and while guid­ed and controlled by virture, the no• bleat attribute of man. It ia the only dictator that freemen acknowledge, and the only security which fr-men desire. Mirabeau B. Lam&r THE UNIVERSITY OF TEXAS BULLETIN No. 3043: November 15, 1930 THE TEXAS MATHEMATICS TEACHERS' BULLETIN Volume XV, Number 1 Edited by MARY E. DECHERD Adjunct ProfeHor of Pure Mathematic• MATHEMATICS STAFF OF THE UNIVERSITY OF TEXAS W. N. Barnes Helma L. Holmes P. M. Batchelder Goldie P. Horton H. Y. Benedict E. G. Keller J. W. Calhoun E. C. Klipple C. M. Cleveland R. G. Lubben A. E. Cooper R. L. Moore H. V. Craig M. B. Porter Mary E. Decherd J. H. Roberts E. L. Dodd W. P. Udinski H. J. Ettlinger H. S. Vandiver O. H. Hamilton C. W. Vickery Teachers of mathematics in Texas are cordially invited to use this bulletin for the expression of their views. The editor assumes no responsibility for statements of facts or opinions in the articles. TABLE OF CONTENTS Horizontal Line Graphs________________________________ H. J. Ettlinger............ 5 Pedro De Paz: Arithmetic__________________________ E. C. Casteneda.......... 8 Making Business Arithmetic Interesting.... Laura Wendt.............. 11 Vitali~ing Elements of Variation in a Solid Geometry Course________________________________________ Ruth Cottingham ........ 17 A Note on an Interesting Relation............ O. S. Hollabaugh. ....... 21 Algebra in the Geometry Course ................ Cicely Goff .................. 22 Teaching 11A Algebra__________________________________ Mary E. Decherd______ 26 The Three Famous Problems of Greek Geometry (Chap. 1)--------------------------------Della Houssels. ........... 62 Brown University Mathemat:cal Frizes for Freshmen --------------------------------------------------·-----------------------·-··----------68 Freshman Tests on High School Algebra and Geometry-------------------------------------------··------------------------------------· 70 HORIZONTAL LINE GRAPHS DR. H. J. ETTLINGER The University of Texa,s The first-class postage law which determines the number of two-cent stamps to be placed on a letter is known to every high-school student; the rule reads as follows: The postage is two cents for every ounce or fraction thereof. If we let p represent the number of cents of postage and w the number of ounces of weight, then the equation or formula which represents the first-class postage law is: If w is a plus whole number, p = 2w. If w is not a plus whole number, p = 2W, where W is the first plus whole number greater than w. A table of values would read somewhat as follows: w 1/4 1/3 5/6 1 11/2 13/4 2 2 1/8 2 1/6 2 2/5 2 4/5 3 p2 2 22 4 4 4 6 6 6 6 6 This would not be a complete table of values, but represents some weights which might be met with by an individual in his correspondence or, certainly, by a postal clerk. In high-school algebra textbooks of recent vintage, bar graphs of population, cotton production, and other statistics are given. The postage law above is an exceedingly inter­esting type of bar graph, which, in fact, bridges a gap between bar graphs and the more standard graphs on coordinate paper. The graph of this postage law consists of pieces of hori­zontal lines as follows: p 1 1 I 0 r---7 I I I I I I ii : I 1 I I ~ I I I I I I l I I I ---1 I t I .2 The University of Texas Bulletin From w ==-O up to and including w -1, p is equal to 2, which is represented by a horizontal line segment two units above the w-axis. From w ~1 up to and including w -2, p is equal to 4, which is represented by a horizontal line segment four units above the w-axis, or two steps higher than the first. The complete postage law then would be represented by the horizontal pieces of a flight of steps each two units above the one before it. Another example of this same kind of law is the funda­mental fact of biology that a living cell divides into two cells, these two each divide into two others, and so on. If we assume that this process takes place at the end of each second, and we take account of the number of cells as compared with the number of seconds, we obtain the following: the number of cells during the first second is 1, the number of cells during the next second is 2, the number of cells during the third second is 4, the number of cells during the fourth second is 8, and so on. This biological law gives a horizontal bar graph as follows: N (n1.1mber o{ cells) 10 g 2 1 0 I I r--: I I :----! I ' ' I I I ~ I I 2. 3 5 t (511c;on 5 These graphs are illustrations of the kind in which we have pieces of horizontal lines. The National Committee on Mathematical Requirements recommended in its report that high-school students be given simple exercises in count­ing squares "under" a given graph. In the case of a graph like the two illustrations given this is a simple matter, since there are a whole number of squares "under" the curve. For any other graph, such as a circle, we may ob­tain an "approximate" answer by replacing the given graph by one which consists of short pieces of horizontal lines; in each case we may determine the distance from the hori­zontal axis up to the piece by taking the ordinates at equally spaced points, following the pattern of the postage law and the biological law. PEDRO DE PAZ: ARITHMETIC C. E. CASTANEDA Latin-Atnerican Librarian, The University of Texas While in Mexico this spring it was the writer's privilege to chance upon a copy of Pedro de Paz' Arithmetic of which a brief mention was made in the previous issue of this bulletin while discussing "The Earliest Arithmetics in America." The rare little book is entitled Arte para aprender todo el menor del Arithmetica, sin Maestro. Dirigido al Doctor Don Diego de Guevara y Estrada, Chantre de la Sancta Iglesia Metropolitana de Mexico. Hecho por Pedro Paz Contador de W, dicha sancta Iglesia. Con privilegio. lmpreso en Mexico por loan Ruyz. AnOI de 1623. It is octavo size and in perfect state of preserva­tion. The owner of this precious textbook, of which only one other copy is known to exist, is Senor Gomez de Orozco, of Tizapam, Mexico. The book consists of 181 folios of text besides 7 pre­liminary leaves in which appear the license of the viceroy to print, dated March 24, 1623, and a prologue in verse and another in prose. The work comprises the four simple rules of addition, subtraction, multiplication, and division, a surprisingly full treatment of fractions, a discussion of proportion and the uses and applications of the rule of three simple and complex, and simple algebraic equations. It is truly a remarkably comprehensive arithmetic for the time of publication. A good idea of the work and of the preparation of the author for writing it as well as the reasons for its publication are given in the prologue in prose which we translate herewith. When I determined to begin this work (curious reader) it was my intention to write a book that should contain not only the whole of the art of Arithmetica Menor y Ma,yor in its entirety but such short cuts and simplified methods as I had discovered or worked out, arranged in such a manner as to improve each and every one of them as best I could, with the help of my dull understanding. (If this art should repay me for the time I have given it, which has been my whole life from the time of my childhood, it would have to be most liberal, considering the affection with which I have cherished it). When I considered, however (supposing the book finished) that certain characters were required [for its printing] which because of their frequent use could not be employed in long examples in a more technical book, I checked myself and chose the means most adaptable to bring it out as the brief work [I present now]. It seems to me that in this form I have succeeded in the aims I desired. For, what is more needed in the state than a book whereby one may learn without a teacher the necessary rules for dealing with whole or fractional ciphers, to carry on all manner of transactions, without the exception of a single one, and without omitting a thing of importance? From this I infer the need of a simple book of rules and common characters, printed at whatever the cost, for its usefulness will repay for the effort. It is specially fit for all states, classes, ages, trades, and business, all of whom lack the means to learn this without a teacher, for all those that desire it do not have the means to afford it. Those that do will find it less burdensome by the use of this Arte, being free to use the second book in which the Arte Mayor will be explained after they have mastered the present one, the second being now in preparation. This will be printed with the best characters used to this day, making everything clear and easy to learn. It will contain many rules of geometry (for the solution of which the simple operations are essen­tial) and for this reason I recommend that the present volume be mastered first, because without it it will be impos­sible to understand the other. For this reason it was highly advisable and proper to publish this first that it may, through its shortcomings, reveal to me my deficiencies and point the things in which I must exercise greater care in the next, which I promise to prepare in like manner, God willing, all for His better service and your own. Thus it is evident that de Paz intended to publish a second volume of his Arte de Arithmetica Mayor, but if he ever published it, no copy of it is known. As stated before the book is remarkably full in its treatment of not only the simple rules but of fractions and simple equa­tions. The Arithmetica of Pasamonte published in 1649 is better explained in the light of the thoroughness of the work of de Paz. It is not without an antecedent. The work of his predecessor gives an insight into the knowledge with regard to arithmetic that was prevalent in Mexico in the first half of the Seventeenth Century. MAKING BUSINESS ARITHMETIC INTERESTING LAURA WENDT Gradua,te Student in The Uni1'61'8ity of Tea:as Teachers of mathematics in high schools are frequently confronted with the problem of justifying the place of mathematics in a student's course of study. If the student is struggling along with the square on the hypotenuse or with two pipes filling a cistern while a third is emptying it, it is difficult to make him see the practical value of the subject. The teacher of business arithmetic, however, should find a ready answer to such questions. The teacher of business arithmetic can and must sell his subject to the class before he can hope to teach it. He must create and maintain interest in it and a real desire on the part of the student to master it. Much of the suc­cess of the subject depends upon the teacher. He must be vitally interested and wide awake; his personality and in­terest must be felt. He must win the confidence of the class. No text is capable of telling a teacher how to con­duct a class or of making the subject of real lively interest; the individuality and personality of the teacher must put the work over. Attitudes are contagious; the teacher's atti­tude toward the subject will very soon become the attitude of the class toward it. There are many devices which a teacher can use to create interest. The students should be asked to consider them­selves business men and women trying to arrive at certain conclusions. The business man is daily confronted with problems that he must solve. These do not come to him labeled and classified; but he must classify, arrange, and arrive at conclusions. In doing this he uses common sense, and the students, picturing themselves as business men, must apply this same common sense to their problems. As business men depend largely upon bookkeeping for the oper­ation of their business, these future business men must learn The University of Texas Bulletin bookkeeping, which is really only applied business arith­metic. A13 business men, the students have in mind the mastery of the language of business, which is the language of :figures, the most nearly universal of all languages. To manipulate this language with accuracy and speed must be their pur­pose. In the vocabulary the teacher should include such terms as Trade Discount, Interest, Net Proceeds, Assets, Liabilities, Inventory, and many others that must be thor­oughly understood. Good habits of keen observation and careful comparison must be formed at the beginning. The teacher should ask the pupils or have them ask each other as many mathe­matical facts as they can about the room in which the class meets, its length, height, width, floor space, window space, number of desks, and such questions. By careful questioning on his part, the teacher can develop in the class what may be called a mathematical method of observa­tion, the ability to observe, to count objects and measure quantities, and in this way "tie up" the subject with every­day activities. Along with the observation habit should come the habit of making comparisons. The students can compare the length and width of the room or the size of the doors and windows. These comparisons may be ex­pressed in simple terms at first, as twice as long, one-half as wide, and then in terms of decimals, percentage, or ratio, as the teacher may see fit. In this way the student begins to think in terms of arithmetic. If the student can observe and gather the data and state or construct problems of his own, a foundation is laid for real development later in the course. After these foundations have been laid, work should begin in earnest. The same plan of work should not be used every day, for it will become a routine and the stu­ dents will lose interest. Since the four fundamental opera­ tions must be mastered, these can furnish the first part of the class work a number of times. Short, snappy drills or rapid calculation should be given for the first few minutes of each recitation, with enough variations in the drills to keep them interesting. One day the forty-five combinations in addition can be used and the record kept, another day drill on multiplication can be given. Problems such as this never fail to keep the class alert : 3X4-2+5X7+3-=? These problems should be given slowly at first, with increas­ing speed and a greater number of terms later. It is fine drill for the student to give the class such a problem. He must think clearly and quickly and his progress will please him greatly. If drill in addition is needed, such a device as the follow­ ing may be used : 8 1 9 4 7 5 3 6 2 The teacher points to the numbers in rapid succession and calls for the sum of the group pointed out. This may also be used for multiplication. The drill might also be on the writing of figures. Stu­dents should be able to write at least 150 figures per minute, and drill on this is interesting. As the speed increases the figures should not lose their legibility, so drill is really necessary. Then there may be drill on calling numbers rapidly and distinctly. A number of standard tests, such as the Curtis tests, can be obtained and may be used for drill purposes. Drill and competition will keep the students alert, with friendly competition against themselves, their classmates, or even the teacher. After the four fundamentals have been studied the teacher is confronted with the problem of fractions, com­mon and decimal, with their applications. Fractions can easily be justified. The students can think of various trans­actions which involve fractions; in fact they can hardly think of one that does not. Farmers seldom sell chickens The University of Texas Bulletin that weigh an exact number of pounds, mothers do not always buy an exact number of yards of material, butchers cut meat in pieces which involve fractions. If the prob­lems are practical, the work will be more interesting. Auto trips, baseball percentage scores, and such topics usually interest the students. Interest, an application of percentage, can be made more realistic if approached properly. All students understand the matter of renting a house, and if interest is discussed as a matter of rent on money, the student can understand it. The sixty-day 6 per cent method which the banks use will prove interesting because of the time saved when using it. The study of compound interest tables, drafts, notes, in­terest on bank deposits, are all subjects of practical use. Bankers should be consulted as to forms and methods used. Students should be assigned to interview various bankers on these subjects. The bankers are glad to discuss them and give the students blank forms of notes and negotiable papers with which they should be familiar. The same method can be used for commission and brok­ erage. If the problems can be related to the local business organizations, the students can see that they are really learning something to be of use later. The students can discuss the advantages and disadvantages of a man's work­ ing on a commission basis, both to him and his employer. In the same way, stocks and bonds can be made real. Stu­ dents can secure stock certificate forms and bonds from banks or homes to study and compare. They can discuss the stock market, transfer of bonds, government bonds, and other realistic material. It is surprising how little some students know of such things as the stock exchange and how interested they become in these things if other students discuss them or report on them. There is another group of problems which are in them­ selves interesting. These are the problems of the merchant and manufacturer. Local merchants can be interviewed on such questions as trade and cash discounts received on in­ voices or allowed to customers, turn-over, figuring profits on ~ost or sales, marking of goods, seasonal variations in the demand for articles, and other questions equally as interesting. The manufacturing industries of the town can be discussed and visited. The problem of making a payroll can be made interesting if the students can see one or talk to the paymaster or clerk about it. The time-clock is another interesting device used by merchants and manufacturers, and students are interested in seeing it used. The superin­tendent of a factory can give them interesting facts about raw materials, cost of production, and distribution of vari­ous overhead costs. Then when these problems come up in the text, the students know they are really being solved in home factories. Another topic the value of which can be easily shown is that of graphs. Students are usually interested in pictures. Graphs are pictures designed to transfer readily to the vision certain facts. Before beginning the construction of graphs, the students should study all forms in general and collect typical ones in magazines or papers. It is worth mentioning that the president of the American Telephone and Telegraph Company, one of the largest corporations in the world, was first brought to the notice of the executives because he was asked to prepare certain data and he placed the data in graphic form. He was just out of college and it was at a time when graphs were coming into prominence. From that time on his progress was very rapid, and he is now president of this large corporation because he saw fit to place certain data before his superiors in graphic form and thereby won recognition. With this as a background the students will see the purpose of solving some of the problems. There are two more types of problems which can be made of great interest to students, insurance and taxation. The modern business world could not operate without in­surance protection, and if the teacher can develop in the class an appreciation of the real value of insurance to both the individual and the business man, the time will have been well spent. Divisions of the subject, such as rates and The University of Te:tas Bulletin risks, rates and ages, rates on wooden structures as com­pared with brick or concrete structures, and rates in the country and city, can be assigned to various members for discussion with local insurance agents. Various kinds of policies should be brought in and discussed, not with re­spect to the technicality of insurance, but in order to have them visualize a policy and better understand the problems involved. In the same manner, taxation is a problem that touches almost every family. While one may not pay a direct tax of any kind, yet every one pays an indirect tax on some commodity that he purchases. While some of the problems in taxation are old, the income tax problem is always new. Students can secure income tax blanks from a local bank or adviser, and use these as a guide. Although the law changes in regard to rates, exemptions, and such questions, the basis is the same and the problem is of vital interest. Other topics could be discussed, but the problem in each case is to relate the topic studied to the local conditions of the community. Teachers should always be searching for means to create interest in the topics, not only to the end of getting better results with the problems and in the class work, but also of increasing interest in relation to local conditions and proving to local business men and parents that he is not merely teaching arithmetic, but business prac­tice as well. VITALIZING ELEMENTS OF VARIATION IN A SOLID GEOMETRY COURSE RUTH COTTINGHAM Highland Park High School, Dallas, Texas In recent years we have heard much about the combi­nation of plane and solid geometry in a one-year course. Perhaps this is a good plan, but I shall vote against any such measure for the reason that a half-year devoted to solid geometry affords me the opportunity of presenting to a group of more or less mature students facts of general mathematical significance not included in other high-school courses. It is true that we do not need an entire half-year for the propositions within the covers of a solid geometry textbook. Time spent thus, in my opinion, is wasted. But it is my purpose here to state a few of the elements of variation that tend to make more vital both previous and subsequent knowledge of mathematics, thus broadening the horizon of thinking of the average student. The first unit of the course provides for an abundance of analytical thinking. The student has probably received a certain amount of this type of training before he reaches the senior year, but it is my belief that nothing affords more opportunity for the enrichment of his thought proc­esses than his contact with the problems of three-dimen­sional space. The magnitude and reality of his own world stagger him, and he realizes that he must find solutions to his problems, or "grope in the darkness." Generally speaking, we know that not until the primary stage of maturity has been reached does the student find himself capable of grasping the entirety of a situation. It is then that the game becomes fascinating and we find him eager to attack the most difficult problems. It is sometimes very amusing to observe his self-sufficiency. Most of us have laughed quietly at the serious attempts to unwind some time-worn mathematical complexity. Psychologically speak­ing, it would seem that the senior year of high school is The Univer8ity of Tems B'Ulletin the appointed time for intensified training in analytical thinking. After the student becomes acquainted with the realms of three-dimensional space to some extent, a peep into a world of four dimensions usually stimulates his interest and imag­ination. Just here I may say that I have found quite valuable a five or ten minute discussion on present-day scientific discov.eries and their relations to mathematics. You may think the students know nothing worthy of dis­cussion in connection with modern science, but I dare say you will find them surprisingly intelligent. Ask them a few leading questions such as : "Who is Einstein?" "Have you read anything recently of general mathematical inter­est?" If you get no response, ask the same questions to­morrow. I can't guarantee what your luck will be, but mine has been quite gratifying. I must confess, however, that the teacher who undertakes such a scheme is forced to increase continually her sources of information relating to the various fields. After these little roundtable dis­cussions concerning current happenings of the scientific world it is well to introduce topics of mathematical history, including the outstanding discoverers and discoveries of previous times. After presentation of the series of topics I ask each student to select one and make a comprehensive written report at some distant date. I usually cite a few references and urge him to find many others. A complete bibliography must be included in the report. This serves as the traditional term paper. The list of topics is exceed­ingly flexible and if the student is particularly interested in some other phase of mathematical relationships, I am ~ad to allow him to report upon a subject of his own choosing. As a whole the class takes great pride in these reports, and frequently individuals ask my permission to prepare another one. A certain boy came to me a year after his credit on the course was in the permanent record and expressed a desire to add something to his paper. He said that he had found more material on his subject. A The Texas Mathematics Teachers' Bulletin 19 number of others have come to me telling of added infor­mation they have acquired relative to their topics. Surely some lasting values result from work of this nature. Another little exercise that most of the pupils enjoy is the solution of a real problem that relates in some way to our own city or to some nearby field of interest. In this collection I have problems relating to concrete driveways, parking stations, silos, floating duck blind, White Rock Lake, concrete culverts, grain elevators, road dumps, water supply tanks, swimming pools, and numerous problems re­lating to neighboring oil fields. Besides reviewing some of the outstanding geometric principles in the light of three dimensions, we have specific possibilities in connection with various solids. The class usually enjoys furnishing its own specimens in the form of buckets, ice cream cones, tennis balls, etc., so that the teacher is relieved of this task. And, too, they are much more real than those solids the teacher may happen to have in the supply cabinet. The cone is one of my "pets." We have so much fun slicing it so as to observe the conic sec­tions. (Incidentally, we usually have to resort to imagina­tion or the specially prepared cone, for the ice cream cone is almost too brittle.) Just here a glimpse at analytic geometry can do no harm. I see no reason why the class should not learn a few elementary facts concerning the circle, the ellipse, hyperbola, and parabola. The individuals of the class probably have more of a "mind-set" for this work now than they will have next year when they launch into freshman college mathematics amidst the whirl and newness of college life. Again, there is much to be said about the pyramid and its relation to the history of archi­tecture; also the sphere, and its beautiful significance in the study of astronomy. The last unit of this "variegated" course I hesitate to call vitalizing. The reader must decide that for me. I think, however, that you will agree with me that the last unit is probably the most essential. In our study of the specific solids we, of course, stumble over the formula again. The University of Texaa Bulletin Just a little innocent play with formulas for areas and volumes is not amiss, I think. And then when we begin to substitute actual values in the formulas my seniors begin to be troubled. "Why should a perfectly good course be ruined by including 'mean' arithmetic solutions!" Never­theless, a bit of review brushes away the cobwebs from the arithmetic, and the skies are blue once more. Just in this connection I find it quite valuable to include a few lessons on approximation, significant figures, degree of accuracy, and the slide rule. Perhaps you will be ready to offer criticisms about the arrangement of this sort of course. It may not be peda­gogically correct. My contentions are simply these: (1) Not having a program of general mathematics in the schools of Texas and some other states, we need a course in the senior high school within which to incorporate cer­tain "facts and fancies" of mathematical significance. (2) My experience has been that all the mathematics courses in the senior high school except solid geometry require so much time for drills, tests, and measurements, if the work is to be effectively taught, that we do not have time for "extras." (3) Such a course certainly is in accord with certain phases of the philosophy of education, for there have been many evidences of the fact that the stu­dent's activities involved have led on to other activities which have provided in part for the continual adjustment of the individual in an ever-changing society. A NOTE ON AN INTERESTING RELATION 0. 8. HOLLABAUGH Vickery High School This interesting numerical relation grew out of a prob­lem which I gave to my class in solid geometry. The prob­lem was to find the diagonal of the school room with the three dimensions being given. The class solved the prob­lem but suggested in a pleasant way that next time I should give one that would be a "perfect" square. I suggested that they find three numbers that would give a square. The next day they said that 3, 4, and 12 and 4, 5, and 20 would give a square. We observed that 12 and 20 were the products of these numbers. We tried other combinations and found that they would work the same way. We also noticed that the result was always one more than the prod­uct of the two numbers. We stated the truth in the fol­lowing statement : The sum of the squares of any two consecutive numbers plus the square of their product will always give a (per­fect) square and the result will be one more than the prod­uct of the two numbers. If x is one number and x +1 the next number above, x(x +1) is the product. Then (x)'+ (x+l)"+ [x(x+l)]'= x• + x• + 2x + 1 + x• + 2x• + x• = x• + 2x' +3x1 + 2x + 1 = (x2 + x +1) 2 ALGEBRA IN THE GEOMETRY COURSE CIGELY GOFF Austin High School In the study of geometry there is much need for many of the principles which are taught in algebra. It is my purpose here to consider a few of the more important of these. The equation that is found in geometry differs from the algebraic equation in that the terms represent geometric figures such as angles, lines, arcs, triangles, instead of alge­braic symbols. This enlargement of the use of the equation to express an equality of geometric concepts, just as later it must be extended to the use of trigonometric functions, baffles the student at first. Sometimes in algebra the stu­dent has failed to recognize the real meaning and the equa­Uon has simply been a problem to be solved, to obtain a value for some unknown quantity. In geometry the con­creteness of the quantities aids in clarifying the meaning of the equation. The axioms of addition, subtraction, mul­tiplication, and division must be stated each time instead of being mechanically used as they too often are in algebra, even to the point where the student fails to know what he is doing and if asked why, can give no other reason than that that is the way to do it. The axiom of substitution or, as it is so often used, the equating of two magnitudes which are equal to the same magnitude, is very important and is frequently used in geometric proofs. All of these axioms are fundamental in equation work and a successful handling of the equation demands a clear understanding and an accurate use of them. Most of the remaining axioms deal with inequalities, a subject which is omitted from many of the newer texts of secondary algebra and hence comes as a new subject when met in geometry. Some teach­ers feel that it is better to leave the theorems of inequality until the close of the course in plane geometry; certainly its postponement until late, perhaps at the close of Book One, is desirable. One form of the equation much used in geometry is the formula. Too much attention cannot be given it in algebra, for it is one of the practical phases of geometry and is used extensively in the study of the physical sciences. The for­mula is a literal equation, in which any letter may become the unknown. Great care should be taken that all formula work be done in good form, which should be the parallel form that is used in the proof of theorems ; a reason should accompany each statement that is made in the solution of the formula. Another type of equation that occurs fre­quently in problem work in geometry is the quadratic. Mr. Breslich in his Sec 1 1 3. Simplify + -\!B vs +v5 v3-v5 The University of Texas Bulletin (2) ( 1) (2) ( 1) (2) (1) (2) "­ 2y'l28 4. Simplify "­ 3\132 v2 5. Simplify -­ ~ 4_ 4_ 6. Simplify y'2. 2y'32 n • 7. Simplify v'x•nIy•n­ 8. Simplify y'a' +a•;4 9. Solve 2-z= 32 x-•-y 10. Simplify --­ X-1 -ylfl 1 11. [a21•x-1!2/ (ax) •1•--;-a-112x; x"l'a0r 12. Solve in three ways: x-• -2x-1 -3 = 0. Assignment: "Race" on exercises in §86. Hand in pp. 185-186, "Review"; p. 189 :7, 12, 14, 17, l(e), 4(d); p. 190:5(d), 6(f). APRIL 3 Class discussion on last quiz. Assignment: Rework last assignment. Hand in examination questions solved. Study §86. Work problems, p. 445. APRIL 4 Class discussion on last assignment. Assignment: Hand in pp. 444 :18, 19, 23, 24, 25; pp. 445:27; p. 433:233, 235, 227; p. 432:221; p. 430 :206; p. 428 :189; p. 426 :169; p. 423 :140. In working this assignment check each problem carefully with the outlines previously given on Ex­ponents and Radicals. APRIL 7 Quiz. Simplify (x..•; x•)" + (x•; x•-•)"-• Class discussion on last assignment. (3) Assignment: Short quiz on p. 156. Hand in Review, pp. 218-219; p. 423 :140; p. 426: 169; p. 428 :189. APRIL 8 (1) Quiz. 1 1 1. Simplify ----+----+2v'5 v5-v7 v5 + v7 2. Simplify v'x'/9 + x• (2) Discussion of quiz and of last assignment. (3) Assignment: Quadratics, pp. 220-225. Hand in pp. 222 :22-28; p. 225 :31-35. APRIL 9 a-b (1) Quiz. Simplify --­ alfl -blf' alf' + blfl (2) Class discussion on exponents and quadratics. (3) Assignment: §§101, 102. Note discussion. Hand in p. 230 :6-19. Quiz on p. 155, first column. APRIL 10 (1) Quiz. --1 8/ ­ 1. Simplify v'x-•z-v'x-1 v'z• • 2. Simplify (a••1;a1-•)•--;-a-•;a (2) Discussion on quiz and p. 230. (3) Assignment: Study pp. 227-230. Hand in p. 230 :6-26. Quiz on p. 189 and p. 155. (1) Class discussion on radical equations. APRIL 11 Consider (1) vx +9-v'x= 9 (2) -v'x + 9-v'x= 9 (3) -v'x+9+ vx=9 The University of Texas Bulletin (4) '\/X +§ + \/X-9 In each of these equations, the value found for x is 16. Does 16 satisfy these equations? Why is 16 found in (1), (2), and (3)? Can any one of these four equations be solved without solving the other three? (2) Assignment: Wentworth, p. 280. Hand in any ten examples. Quiz on the 14th on p. 189 or radical equations. Note examples solved on pp. 227, 228, 229. APRIL 14 (1) Quiz: 4x +1-3y4x +~-= 6. (2) Class discussion on radical equations. (3) Assignment: Study pp. 232-234. Hand in Review, p. 231; p. 234 :1-4. Quiz on p. 229 :7-16; p. 117 :36-44; p. 182 :21-25. APRIL 15 a.n......,___ (1) Quiz. Simplify v64D/4"". (2) Class discussion on last assignment. (3) Assignment: E. & C. Review pp. 229-230. Hand in p. 234 :5-18. APRIL 16 (1) Quiz: 1. Solve vi-vx -12 = 2 2. Solve y9x'+5-3x=l 3. Solve yx +5 + vx=3== 2'\/x. (2) Class discussion on quiz. (3) Assignment: Hand in E. & C., p. 235 :19-30. APRIL 17 (1) Quiz. Simplify ['\/72ir/3 X 9°] (21t .. )·s. ( 2) Class discussion on assignment. (3) Assignmerit: p. 235 :31-34; "Review." For quiz: Chapters VI and VII. .A>PRIL 18 a,1/"b'I' a -b (1) Quiz. Simplify ---.--­ a'/I +b'I" a'I' + b'I' (2) Class discussion : Wentworth, p. 235. (3) Assignment: Wentworth, p. 283:7, 8, 10, 13, 16-22. APRIL 22 1 (1) Quiz. Simplify (21/•-• 211•n) 1/" (2) Class discussion on W., p. 283. (3) Assignment: Hand in any 8 from p. 230 :18, 20; p.231:1,3,6;pp.234-235:10, 11, 15, 18,20,28,34. Six weeks' quiz on exponents, radicals, quadratic form. Good reviews on p. 219, p. 189. Note p. 452 :90. APRIL 23 (1) Six weeks' test : x"l"-1 x-1 1. Simplify 2. Solve y'2x-3-vx+ 1= v15x-4 3. Solve x•;a _ x-•;a = 17/4. These three form the quiz. If any one has time, credit will be given for working the following: 4. Solve 2x" + 3x-5y'2x1 +3x + 9 = -3 5. Simplify (2.... -2 X 2•) X (2-• X 2-•-•) 6. Solve 8x-• + 19x-'1" -27 = O. (2) Assignment: Hand in pp. 411-413 :2, 3, 6, 11, 16, 19, 30; p. 238 :1, 2. APRIL 24 3 (1) Quiz. Find value of ( 125 )-•;a +--­ 2 +2-l (2) Class discuBBion on last assignment. (3) Assignment: p. 238 :1-10. The University of Texas Bulletin APRIL 25 3n(3•-l) • ( 1) Quiz. Simplify (2) Class discussion on exponents and p. 238. (3) Assignment: Hand in any ten on pp. 238-240. Work five more on these pages. APRIL 28 (1) Class discussion on pp. 238-240. (2) Assignment: Quiz on E. & C., p. 155. Hand in Wentworth, p. 154:2, 3; 155:7, 8; 156:14, 15; 157 :24, 25; 158 :30, 31; 159 :37, 38; 160 :43, 44. APRIL 29 (1) Quiz. Simplify (ax)-' -(ax)" (2) Class discussion on word problems. (3) Assignment: Quiz on E. & C., p. 156. Hand in Wentworth, p. 161 :48-51; p. 162 :55, 56; p. 163 :62-68. APRIL 30 (1) Prove 4 = 1 8=8 2=2 8-2=8-2 8 -8 = 2 -2; if we divide by 2 -2. 4 = 1. Where is the fl.aw? Prove that all A are isosceles. Make LACS = L BCS Make AE=EB Draw ET .L to AB CS and TE meet at 0 Draw OF and OG .L to CB and AC, respectively, .6CGO = .6COF (Why?) Then .6AOG = .6.BOE (Why?) CG == CF (Why?) AG=BF AC = CB. Where is the flaw? Also discussion on word problems. {2) Ass!gnment : Hand in E. & C., p. 128 :1; p. 242, Review 1, 2, 3, 4, 5. Test on exponents. Study pp. 155-156. MAY 1 (1) Quiz: 1. Simplify aD+lall-1 2. Simplify 1oa+l x 1oa-1 2· x (2°-1). or 2··1 x 2•-l (2) Class discussion on logarithms. (3) Assignment: E. & C., pp. 185-186 :2-5; p. 281 :2; p. 129 :44, 47. MAY2 (1) Quiz. Solve either The denominator of a fraction is 6 more than the The hypotenuse of a right triangle is 20. The sum of the other two sides is 28. Find the lengths of the sides. (2) Class discussion on logarithms. (3) Assignment: Hand in p. 281 :3, 4, 5; p. 189, Review. numerator. What is the fraction if its value is .25? or, MAY 5 (1) Quiz. Solve either 1 or 2. 1. The length of a rectangle exceeds its width by 8 ft. If each dimension were 2 ft. more, the area would 125 sq. ft. more. Find its dimen­sions. The University of Texas Bulletin (2) (3) 2. Find the time between 1and2 o'clock when the hands of a clock point in opposite directions. Class discussion on logarithms. Assignment: E. & C., pp. 194-195. Study for quiz Wentworth's fractional equations. The following is the discussion on logarithms that was given in class on May 1, 2, 3, and 5. A logarithm is an exponent. 10· . 10° . 10· = 10· 10• = 100; 2 is the log. of 100 to the base 10. 10• = 1000; 3 is the log. of 1000 to the base 10. 10° = 1; 0 is the log. of 1 to the base 10. 1 10-• = -; -2 is the log. of .01 to the base 10. 100 1 10-• = --;-4is the log. of .0001 to the base 10. 10000 Or, the log. of 1000 is 3, etc. What do you know about the logarithms of num­bers between 10 and 100? E. g., 10 < 25 < 100 Or 10' < 10· < 10• Hence 1 < x < 2, or, 25 = 101+ The table records the decimal part of the log. Hence from the table 25 = 10'· ..... Consider 625 : 100 < 625 < 1000 10· < 10· < lo·; 2 < x < 3, . ·. 625 -lo··. From the tables then 625 = 10•··-. But (25)"= 625, (101.8879)°= 10•,.,... (note differ­ence in last digit). 100 < 243 < 1000 1o· < 10· < 10·; 2 < x < 3, . ·. 243 = 1o·+, and from the table we get the decimal, or 243 ­ 10··-. 7909 625 x 243 = 10.. x 10...... . . . 151875 = 10°'1810, Verify by finding log. of 151875 approximately. 151000 = 10•·1"° 152000 = 10•·1BU. Therefore, presumably, the above result is correct. Get log. 54.2, log. 542; log. 57.6, log. 576. Check by the products 54.2 X 10, 57.6 X 10. Interpolation : 27 [86500__ __ _______ ____ 9370 ] ] 100 86527_________ ___ __ ( ?) q 5 [ 86600_ __ __ _ ____ _ 9375 By principle of proportional parts: q 27 27 x 5 -=-, Q= =1.35; 5 100 100 since 10000 < 86527 <100000, 10" 9871 86527 = • Assignment: Get logarithms of 683 x 65245 1. 6794 3768 x 672 2. 538 Get log 828: 18 = 101.!!.'.::a 46 = 101· 00"' 100 < 828 < 1000 828=10·· 828 = 10··01 1 828 = 10··•1 0 • • Note difference in last digit. To find the number corresponding to a given loga­rithm. 104.9371 86257 --= ---= 102""2 325 102·""9 4249_____________ _ ,266] ] 16 3 (4252_____ _ _______( ?) q 1 [ 4265_ _______ _ _____267 The University of Texas Bulletin s Then 3: 16 = q: 1, and q --X 1 -.2, 16 . ·. 266.2 is the number required. Since 10 ..... is greater than 100 and less than 1000, 10 ..... = 266.2. 3 (4249 ________________2660]Better, 16 4252_______________ ( ?) q 10 42 65 ________________26 7 0 3 q -=-,and q=2. 16 10 Hence 2662 is the succession of digits corre­sponding to 4252 as the decimal part of the logarithm. The required number is 10•·... and hence 100 < N < 1000, :. 10'·... = 266.2. MAY 6 (1) Quiz. Simplify: 1 1 --+-­ 1-a l+a 1. a 1 --+-­ 1-a l+a 2. 2/5y3 x 4/9y5 + 6/7y2 263 x 9643 3. Find by logarithms 567 (2) Discussion of logarithms. (3) Assignment: "Reviews" on pp. 177-178, 183, 185, 189-190, 195, E. & C. Hand in any of these that you have previously failed to hand in. MAY 7 (1) Class discussion. 263 x 9643 10...... x 10 ...... -----.... 10··-==4473. 567 10··... To explain 4473: 10• < 10··-< 10' 3 (6503____,______ ____44700 ) )10 6506________________447300 q 1000 ( 6513________________448000 3 -x 1000 ~300. 10 This illustrates the fact that any number of O's can follow 447 and 448. Many such examples were worked in class by the students. Consider also the following: 653 x 42 10....., ---= --= 10-·•0•11 • 86738 10··­ Since the tables give no negative decimals, -.5001 can be written in the equivalent form -1 + .4999, or l.4999. We can subtract thus: -1+5.4381 4.9382 1.4999 To find the corresponding number : 2 (4997________________ 31600) ) 14 4999 ·----·----------31614 q 100 ( 5011______________31700 2 100 Q=-X 100=-=14+ 14 7 316.14 10··­ Now =--, 10· 10' and .31614 = 101·•.... (2) Assignment: Long division with expressions hav­ing exponents. Also E. & C., p. 252 :1, 2; p. 235 :3, 4, 5, 6, 7, 9, 10; p. 281, 7. The University of Texas Bulletin MAY 8 (1) Quiz. 6_ 6_ 1. Divide 1/ x"I"-1/y•1• by y x -y y. 2. What is the log. of (2134) •/•? (2) Class discussion. 364 Find -by logarithms. 895 102·"°11 364 009 -=--=101·' • 895 102.9018 To perform the subtraction, we write -1+3.5611 2.9518 1.6093 We find 10'·"098 = 40.673, but we want 1Ql·0008 ; hence we divide this equation by 10' and have 10""""" 40.693 --=--­ 10' 10' 10'·"'°" = .40693 (3) Assignment: Look over "Reviews" on pp. 208-210, E. & C. Hand in "Review," pp. 211-212 :2, 3, 4, 5, 6, 7, 8, 9, 11, 15. MAY 9 (1) Class discussion: The description of an equation in quadratic form is important. Note­ 1. 8x" + 64x" = 8, a quadratic in x•. -4_ 2. y x• -3y x• = 40, a quadratic in x"I'. 3. (2x-3)1 -(2x-3)-= 6, a quadratic in 2x-3. 7x1 4. -5x + 8y7x' -5x + 1 = -8 can be written as a quadratic in y7x' -6x + 1: (7x1 -5x + 1) + 8y7x"-5x + 1+7-O. 5. x'I' -13/ 6 +x-'I' ~0 can be written 6x -13x'I' + 6 = 0, a quadratic in x11•. 4_ ­ 6. 3yx-2yx ==--20 or 2x'I' -3x'I' -20 = 0, a quadratic in x'I'. 7. 6x' + 6x + \!x(x + 1) = 7, a quadratic in yx(x+l). (2) Assignment: Rework, laying book aside, problem worked out on p. 281. Hand in p. 376 :22, 25, 26, 27, 32; p. 281, 9; p. 287, 21; p. 447, 55; p. 423 :146. MAY 12 (1) Class discussion. Multiply by logs: 16.73 x 2.854 = 101·-x 10°·'""' = 10""'"" = 47,744. 4 [6785________________47700 ] ] 9 6789________________47744 q 100 [ 6794______________ _47800 q 4 400 -=-, Q=-=44.4 100 9 9 (2) Assignment. Review as many recent assignments as you can for the quiz. MAY13 Club day. MAY 14 (1) Three weeks' quiz: 1. Solve 4x-•1• -3x-'18 -27 = 0. 2. Solve \! x + 4 -yx= y_x_+_3,,,_/..,,..2. The University of Texas Bulletin (2) (1) (2) (3) (1) a: -y -4, {a:' +y' -18xy, 3. Solve x' +y' _ or x +Y _ { 40 12 4. Solve: A and B can do a piece of work together in 18 days and it takes B 15 days longer to do it than it does A. In how many days can each do it alone? or The area of a square may be doubled by in­creasing its length by 10 ft. and its breadth by 3 ft. Find the length of its side. 5. A substitute for any one of the above 4: (x11/m-11) m-11 -7-(x11/xm). Assignment to be handed in: E. & C., p. 241 :28, 29; p. 283 :11, and Review. Note carefully p. 282, §121. MAY 15 Quiz. Solve {x'-xy +y' = 13• x• +3xy +y' = 61 Class discussion on graphic solutions; graphed in class x -y = 4. Also consider the quadratic ex­pression x• -6x +5 , and the quadratic equation x• -6x +5 = 0. Let y = x' -6x +5, and note that this graph cuts the x-axis at 1 and 5, the roots of the quadratic equation. Roots real and unequal: y'36 -20 = y'16 -== 4. Assignment in Wentworth: p. 367 :23, 24, omitting decimals. Solve by algebra and by graph x +y = 7, d ~ xy = 12, { x -y = 1 an lx' +y" = 25 MAY 16 Quiz. Simplify: 1. (2/x"-1 l/x"•1y•-"/"; • 2. [ (Xm) m-1/m]•/Cm+u 3. (xm••;x11) m (n/xm••) m-11; (2) (3) (1) (2) (3) The Texas Mathematics Teachers' Bulletin 4. (a•/-+-a<•-J'>/X) <••7>/J'; 5. (a•;a••J' -+-a•-J';a•) <••J'>/J', Class discussion on quiz and on the following: 1. Get the logarithm of .00234. 234 10··­ -=--=lO"L.... 10' 10· 2. Get the logarithm of .0673. 673 101.­ -=--=loL"""° 10· 10· Assignment: Construct four quadratics to solve in four ways each, Wentworth, p. 276 :28 and 29. MAY19 Quiz on high-school algebra. 1. Find the value of 3/5 +5/2-4/7. 2. Divide 0.02683 by 0.007213 to two decimal places. 3. Simplify: 2 + yS x-1/x (a) (b) v2 4. Factor completely: (a) y•-81; (b) a'b +3a• +ab+ 3a 5. Solve: 7x/2 -11==3x/4 . {2m-7n=2, 6. Solve simultaneously: 6m + 4n = 31 7. Solve: 3x'-x-14 = 0. 8. If a and b are the sides and c the hypotenuse of a right triangle, find the side a and the area if b= 12, C= 15. Graphed y = x• -4x -5. Assignment: 1. Kinds of exponents- a. Positive integral-why? b. Negative } c. Fractional -why? d. Zero The University of Texas Bulletin 2. Operations- a. Multiplication1 b. Divisio~ How performed? c. Involution J d. Evolution Learn outline. Review thoroughly pp. 147­ 162. Note exponents, pp. 444-445. Hand in four difficult examples. Plot Y=x2 -6x-7. Get log. of .02345. MAY 20 (1) Quiz. Simplify: 1. [ ( x•) •• ( x-•) "]1/0 Roots real and unequal (graph drawn). In y = x' + 4x + 4, let x• + 4x + 4 = 0; b" -4ac = 16 -16 = 0, and the roots are real and equal (graph drawn). Then y = x• +4x +5 gives the quadratic equation x'+4x+5=0 -4 ± y'l6-20 whose roots are 2 -2 ± y'-1 ; and the roots are "imaginary" (graph drawn). (3) J\ssignment to be handed in: Any eight of p. 413: 19, 23, 24, 29, 30; p. 414 :34, 39, 41, 42, 43. Also: 1. (.00678)-'12 x (.0345) •;a 2. Graph equations used in solving x' +y' = 169, { xy =60 MAY 22 (1) Class discussion. (4a-•;a) -"1', (31/•a-•)-•1•, (16a-•; slb') -•/• Also (.0234)'/1 = (10..... ') •;a= 10..""' The actual steps in performing this calculation are: 234 10··-23.4 10···00 • -=--,or--=-­10' 10' 10· 10· and .0235 = 102.acoo and .0234 = 102...,, ( 10 ··-)'I"= 10-.­ -2 + .3692 2 or 3 I-6 +2.7394 3 l-4+ .7384 -2 +.9131 The University of Texas Bulletin (2) Assignment: E. & C., pp. 154-155:15, 16, 19, 22; p. 437 :268; p. 432 :221; p. 416 :65; p. 377 :33, 37; pp. 243, 244. ( .00671) -•(I Find the value of by logs. (. 0234) -•/• S l x• +y' = 169, ove { xy = 60, and draw the graphs. MAY23 (1) Class discussion on the last assignment. (2) Assignment: E. & C., Ch. XII; p. 293 :3, 19, 24. Hand in Review, p. 293. Hand in Review, p. 301, 17. Review Wentworth, p. 77. Solve in four ways: x• -4x -5 = 0. MAY26 (1) Class discussion on graphs and logarithms. .548 x 1.98)'1' = 1()1.,... x 10··­ ( 0077 39.6 X 2.74 101 X 10°".,. • 10·­= --= (10-•) 11' = .1 ur·­ (2) Assignment. Hand in the five hardest review problems that you can find, Wentworth, p. 404. MAY27 (1) Class discussion. Worked in class the "hard" problems called for in previous assignment. (2) Assignment: p. 424, 153; p. 281, 6; p. 240, 19; p. 189, 16; p. 169, 49, 63; p. 171, 12; p. 172, 20; p. 104, 17; p. 185, 17; p. 239, 13. Solve v'x +2 +v'x -1 -y'3x +3 = O x_,,I' -4x..,,I' -32 = 0 3x' +15x -2y'x• +5x +1= 2. MAY28 (1) Class discussion a general review of the term's work. (2) Assignment: The final examination given the seniors: 1. (a) Simplify 3y5/32 +2/3y40/9-yl/10 (b) Solve and check: 12 V5+x+V5-x=--­ \15-x 2. Plot and check by algebraic solution: x• + y•= 25, { x-y+l=O 3. Solve by factoring (a) 8x"-10x = 3 ~?+ 3~x== 134 4. Solve by two methods : x• -ex = 112 (ax -ac) 2y3-5 5. (a) Simplify \13+5 x-1 x•f•-1 (b) Simplify x'l"-1 x'I" + 1 6. (a) Simplify [ ( x••b) • x ( x•b) -·r1<••-•> a a a-b a+ b (b) Simplify b a --+-­ a-b a-b 3-• -2-· 7. (a) Simplify (b) Solve for l in t = yl/ g 8. The width of a room is %. of its length. If the width were 4 ft. more and the length 4 feet less, the room would be square. Find its di­mensions. 9. A can do a piece of work in 6 days, B in 5 days, and C in 4 days. How long will it take them all working together to do the work? 10. Find the logarithm of 0137 )'/" if 13 in the 7 column is 1367 -·-~ 42 in the 8 column is 6314 ( 42.85 42 in the 9 column is 6325 Answer any eight. Final examination for llA's. 1. Find by logarithms: 32.48 x ( .288). 4_ \1525 2yl0 7y48- 4yl5­ 2. (a) Simplify --X---+---­3y27 5ylf 15y2i (b) Solve by factoring: 4x'I" -3x'JS -27 = O 3. Plot and solve algebraically : x• +y'= 25,} XY=12 Do the solutions check? v2-3 4. Simplify (a) 2v5-a a•/• -b'I" a-b (b) a'JI +b'l8 a'JS-b'JI I 2-·a·v~ 5. (a) Simplify (27) 'l"yl/4 (b) Solve and check: vx +15-vx+8=2vx 6. A can do a piece of work in 8 days, B in 10 days. A and B together with the help of C can do the work in 3 days. How long will it take C alone to do the work? 7. A rectangular grass plot 12 yds. long and 9 yds. wide has a path around it. The area of the path is 2/ 3 of the area of the plot. Find the width of the path. 1 b 8. (a) Simplify a-b a a"-b2 b ab+ b2 ab+ a• (b) Solve 7x"-5x +8y7x2 -5x +1 =-8 9. (a) Simplify a-•+ b' (b) Solve in 'three ways: 2x"-3x-5= 0 10. (a) (b) Can you do this a fourth way? Solve yx-6 +yx = 3/yx-6 Simplify 3y200 -6yl/ 50 +3yl/ 15 +5v'60 11. Solve (a) --= --­x-1 m-n -'­ (b) vx-vx==2 Pledge. THE THREE FAMOUS PROBLEMS OF GREEK GEOMETRY• DELLA HOUSSELS Austin High School CHAPTER I HISTORY The earliest history of mathematics shows that the Baby­lonians and Egyptians each possessed a highly-developed system of mathematics as early as 2000 B.C. It is not known whether the two systems sprang from a common source or were developed independently. It is known that about 2200 B.C. the Babylonians had a wide knowledge of the subject and, in some respects, had developed it to a higher degree than had the contemporary Egyptians. Babylonian tablets are now in the British Museum, the Prussian State Museum at Berlin, the Ottoman Museum at Constantinople, the University of Strasbourg, the Uni­versity of Pennsylvania, and the Palais de Cinquantenaier of Brussels. One of these tablets, of about 2200 B.C. reveals the fact that the Babylonians, even at that early date, knew how to find the area of a rectangle, of a right triangle, and of a trapezoid. An Akkadian tablet (about 2000 B.C.) gives a method for finding the diagonal of a rectangle which suggests that the Babylonians may have known the Pytha­gorean theorem long before Pythagoras was born. Various problems on these tablets show that 3 was then taken as the value of 'Tr. The same value of 7r was used at a later date (560 B.c.) by the Hebrews (I Kings, 7 :23, II Chron­icles, 4 :2). The earliest known record of Egyptian mathematics is a papyrus called the Golenishchev Papyrus. It was written about 1850 B.c. and is now in the Museum of Fine Arts at *We are publishing Mrs. Houssels' thesis because of its general interest. The remainder of the thesis will be given in subsequentissues of the Bulletin. Moscow. It contains formulae for finding the volume of a hemisphere and the volume of a frustrum of a regular square pyramid. A second papyrus, called the Rhind Papyrus, is in the British Museum. It was written about 1650 B.c. by the Egyptian priest Ahmes, and bears the interesting title "Directions for Knowing All Dark Things." It contains problems for finding the area of a circle and the relation between the area of a circle and that of the circumscribed square; finding the area of the part of a triangle cut off by a line parallel to one of the sides; problems dealing with the volumes of cylinders and parallelepipeds; and a dis­cussion of the angles formed between the base and the faces of a pyramid. Both of these papyri are thought to be copies of earlier documents of about 2000 B.c. Geometry was first introduced into Greece by Thales of Miletus (640-550 B.C.), one of the "Seven Wise Men of Greece." He studied for many years in Egypt and later founded the Ionic school of philosophy in Greece. He added many important theorems to those already known to the Egyptians. His pupil Pythagoras (569-500 B.C.) also studied in Egypt and founded a school at Croton, Italy, which advanced geometry to the status of a liberal science. Euclid (about 330-275 B.c.), a Greek geometer, taught at Alexandria. Little is known of his life except that he wrote a geometry textbook called "The Elements," which is one of the most famous books ever written. It is a compila­tion of the works of earlier writers, assembled and arranged in so logical an order that little change was made in it for hundreds of years. This book has formed the basis for all later textbooks on the subject. Archimedes (287-212 B.C.), the greatest mathematician the world has known, except Newton, lived at Syracuse in Sicily. The following books written by him survive: Equiponderance of Planes; Quadrature of the Parabola; The Sphere and Cylinder,· Measurement of the Circle; Spirals; Conoids and Spheroids; The Sand-Counter; and Floating Bodies. He used both circumscribed and inscribed The University of Texas Bulletin regular polygons and proved that the circle is the limit which each perimeter approaches as the number of sides is indefinitely increased. He proved that the area of a circle equals 1!'r2; that 1!'r2: (2r)' : : 11 :14 very nearly; that 31/7> 1!'>3 10/ 71; and reduced the problem of the quadrature of the circle to the construction of 1!'. The later Greek geometers were interested in what were known as "The Three Famous Problems." 1. The duplication of the cube, i.e., finding the edge of a cube which shall have twice the volume of a given cube. 2. The trisection of any angle. 3. The quadrature of the circle, i.e., the construction of a straight line segment equal in length to a given circle or, the construction of a rectilineal figure equal in area to a given circle. The solution of these problems with the instruments then in use-the straight edge and the compass-has baffled all who have attempted them. However, persistent efforts to find solutions to them have led to the discovery of many new theorems and have developed geometry into a wonder­ful and beautiful science. The origin of these problems is not accurately known. According to a legend, Athens was afflicted with a pesti­lence and sent to the oracle of Apollo on the Island of Delos for a remedy. Apollo ordered, through his oracle, that the size of his altar at Athens be doubled. The altar was in the form of a cube, and the Athenians immediately built a new cubical altar with each edge twice the length of the edge of the old one. As this had no effect on the pestilence, they tried placing a new altar of the same shape and size by the side of the old one. This also proved of no avail, so another delegation was sent to Delos. There they were told that the new altar must be an exact model of the old one, but of twice the volume. The Athenians then turned the problem over to the geometers, who found themselves un­able to solve it. On account of this legend, the problem of the duplication of the cube has come to be known as the "Delian Problem." Another legend concerning the same problem states that the Egyptian king Menos rejected the plans for a tomb for his son, Glaucus, because it was not large enough and ordered it to be doubled in size but the same form retained. As the tomb was in the shape of a cube, the same problem arose. Many geometers worked on this problem; Hippo­crates of Chios (about 430 B.C.) reduced it to one of plane geometry by proving that it is equivalent to finding two mean proportionals between one line and another twice as long. In the proportion a :x = x :y = y :2a, x• = ay, and y" == 2ax, therefore x' = a2y", and x' = 2a"x or x• = 2a". '!'his problem, however, is impossible of solution with straight­edge and compasses, like the original one. Archytas of Tarentum (428-347 B.C.) found a curve by using half cylinders which enabled him to find two mean proportionals between two straight lines, but his construc­tion was rejected as being mechanical rather than geomet­rical. The trisection of the right angle was known to the Pytha­goreans probably as early as 500 B.c., but the general prob­lem of trisectipg any angle baffled all attempts. Only in recent years has it been shown that it is impossible of solu­tion with straightedge and compasses. Hippias of Elis (about 420 B.C.) discovered a transcen­dental curve by means of which he could divide any angle into any number of equal parts. He also invented a machine for its construction, but Plato refused to allow such methods on the ground that they would widen the limitations of geometry and thus destroy its value. This opinion has been supported by all later geometers. Pappus (about 200 B.C.), a teacher at Alexandria, offered several solutions of this problem by using non-Euclidean methods, but they were rejected as being outside the limits of geometry. Nicomedes (about 180 B.C.) invented a curve, the con­choid, and a machine for describing it, by means of which a mechanical solution of the trisection problem was accom­ plished. Diocles, a contemporary of Nicomedes, also invented a curve, the cissoid, which furnished other mechanical solu­ tions. These furnished no help in finding a solution to the problem by means of the instruments of plane geometry alone. Of these three problems, the quadrature of the circle has received the greatest amount of attention. Anaxagoras of Chazomen ( 500-428 B.c.) is said to have been the first to attempt to construct a square equivalent to a circle. While in prison he studied this problem and wrote a treatise on it. Antiphon of Athens (480-411 B.C.) used the process of exhaustion in attempting to square the circle. His reason­ing, that a straight line segment when sufficiently small coincides with an arc, was defective and was condemned by geometers. Bryson of Heraclea, a contemporary of Antiphon, used circumscribed as well as inscribed regular polygons and made the false statement that the circle was the arithmeti­cal mean between the perimeters of the circumscribed and the inscribed regular polygons of the same number of sides. Ahmes found the area of a circle by squaring eight-ninths of the diameter, or by using 3.1604 as the ratio of the circle to its diameter. Hippocrates (about 470 B.c.) wrote the first textbook on geometry and with Antiphon was the first to use inscribed regular polygons to find the area of a circle. Ptolemy (87-165 A.D.) used 3E.. or 3.1416+ as the value 120 of 7r and about 500 A.D. a Hindu named Aryabhata, using the same method Archimedes had used, obtained 3.1416+ as the value of 11". Methuis of Holland (1527-1607 A.D.) gave the fraction 355/133 as the value of 'Ir, which correct to six decimal places. Ludolph von Ceulen of Leyden (1527-1616) computed the value of 1l' to thirty-five decimal places. 3.14,159,­ 11" ­ The Texas Mathematics Teachers' Bulletin 265,358,979,323,846,264,338,327,950,288.1 This result was considered so important in Germany that there '"' became known as the "Ludolphian number." The value of '"' was computed by Vega to 140 decimal places; by Richter to 509 places; and by Shanks to 707 deci­mal places. Lambert proved in 1760 that '"' is an irrational number and in 1882 Lindemann proved that it is a transcendental number, from which it follows that the quadrature of the circle is impossible with straightedge and compasses. ?Rupert, W.W., Famous Geometrical TkeoremB and ProulemB, p. 98. BROWN UNIVERSITY MATHEMATICAL PRIZES FOR FRESHMEN Out of gratitude and respect to his Alma Mater, an alum­nus of Brown University has established a fund known as the Brown University Mathematical Prize Fund, from the inter­est of which prizes are awarded annually by the staff of the Department of Pure Mathematics on the basis of competitive examinations.-The University of Texas Catalogue. The donor provides for four prizes, one of which is of­fered to students of the calculus, while the other three are given for excellence in high-school mathematics. These prizes are offered to the regular freshmen making the best grades on a special voluntary examination to be held on the afternoon of the second Saturday in October. The examination will cover the minimum entrance requirements in mathematics, namely, elementary algebra and plane geom­ etry.-The University of Texas Catalogue. The Brown Prize Examination on high-school algebra and geometry for the session of 1930-31 was held October U, at 2 P.M., in room 172, Main Building. The questions were as follows : 1. The base ·of an isosceles triangle is 6 inches and the diameter of the circumscribed circle is 10 inches. Find the length of the sides of the triangle. 2. Given A and Bon the same side of the line CD. Draw from A and B two lines meeting C D at the same point and making equal angles with C D. 3. Given a positive and =F 1, prove a+ 1/a > 2 1 4. Find x given (3') • = -­ 3-..... Pledge. The winners of the first, second, and third prizes were, respectively, Milo Weaver, of Kirbyville, Texas, awarded $13.50; Francis A. Hale, of Chester, West Virginia, awarded $9.00; and George T. Moore, Jr., of Coleman, Texas, award­ed $4.50. The second and third prize papers were of almost equal value, Mr. Hale's paper being superior in form and arrange­ment of proof. Sixty-four students came to take the exam­ ination, and forty-seven of them submitted papers. entire list of grades is appended. The 1 -----------------------------------------------------85 1 ---------------------------------------------------------­77 1 ------------------------------------------------------------75 2 ------------------------------------------------------------­65 1 -------------------------------------------------------------­60 9 -------------·-----------------------------------------------­50 1 --------------------··--------------------------------------45 5 -------------------------------------------------------------­40 4 -------------------------­·-------------------------------35 13 -------------------------------------------------------------­25 3 -------------------------------------------------------------­20 6 ---------------------------..-----------------------------­0 The number attempting the examination is more than double the number in 1928, while the grades on which the awards were made are the highest in a number of years. THE UNIVERSITY OF TEXAS FRESHMAN TESTS ON HIGH-SCHOOL .ALGEBRA AND GEOMETRY For the last two years tests on high-school algebra and geometry have been given to the freshman classes in Math­ematics 301 and Mathematics 302. The papers have been graded by the various instructors, and the results compiled and sent to the high schools concerned. In October of this year tests were again given with the results as follows : Number of schools_____________ 361 Number of students_______________ 910 Number of students passing _____________ 261 Number of students failing __________ 649 Per cent of students passing_ _ 28.68 By comparison with the results for the two preceding years, we see that the percentage of those making 60 or more on this year's examination is less than either 1928 or 1929. However, there are several encouraging consid­erations. Had 50 been the passing grade 43.8 per cent of the students would have passed. Moreover, in several schools where only a small per cent of the students made 60 or more, the average for the entire group from the school was about 55. This average was the result of the large number of grades between 80 and 100. The list of those who made 90 or more is appended. Amarillo: Bellville: Alex Mood, 95. Viola May Dittert, 96. Austin: Cameron: Mary Lucy Dodson, 100. Frances Sharpe, 90. Margaret Ezelle, 97. Carrizo Springs: Dayton Rutledge, 97. Milton Stern, 90. Frances Eaves, 96. Chester, W. Va.: Lillian Ammann, 95. Francis Hale, 94. Elizabeth Correll, 95. Corpus Christi: Virginia Penick, 95. Hal Rachal, 90. Willie Mae Todner, 92. Corsicana: Margaret Knippa, 90. Helen Elizabeth Blackburn, 90. Jane Pearce, 90. Elizabeth Kerr, 90. Elizabeth Willie, 90. Dallas, Forrest Avenue: Laredo: Joshua Kahn, 100. Norman S. Davis, 94. Detroit, Michigan: Mission: Milton Singer, 100. Harold Ross, 94. East Orange, N. J.: Orange:Robert C. Remby, 94. Mildred Clough, 90. El Paso: Pecos: Dan P. St.ewart, 95. Mildred Ruhlen, 90. Fort Worth, Central: San Antonio, Alamo Heights: John H. DurJ1ton, 99. Robert LaPrelle, 90. John H. Durston, 92. San Antonio, Main Avenue: Fredericksburg: Rosalie Robinson, 95. Gifford Whit.e, 90. Smithville: Galveston: Thelma Insall, 90. Fred M. Arend, 97. St.ephenville: Gonzalea: Clytie Savage, 95. John Romberg, 95. Washington, D. G.: Houston, San Jacinto: Mary E. Anderson, 100. Frank Merrill, 100. Yoakum: Harry V. Baker, Jr., 93. Elizabeth Wimberley, 95. As is to be expected, the students in Mathematics 302, that is, the students who have had trigonometry in addi­tion to algebra and geometry, average more than 10 points above the students in Mathematics 301, who entered the University with credits in only algebra and geometry. Special attention is called to the records made by Beau­mont, Dallas, Ball High School in Galveston, El Paso, and Graham. The record of schools out of the state is about 32 per cent passing. It is interesting to note that Manila (P.I.), Cardenas (Cuba), McKinley (Honolulu), and Mex­ico City have each one student who took the tests, and that in each instance the student passed. The following figures may also be of interest: There are 138 students making 50-59. There are 105 students making 60-69. There are 62 students making 70-79. There are 54 students making 80-89. There are 40 students making 90-100. MARY E. DECHERD, For the Department of Pure Mathematics. Following are four of the tests used. 1. 1/2 +3/ 7-2/5 =? 2. Reduce 2/9 to a decimal (3 places). 3. Factor completely: (a) x' -16y"; (b) 3x -y-3xz +yz. 4. Solve the equations a (a) -x +d = bx+ a. c (b) 2x"-3x +1=0. 5. The width of a room is 3/4 of its length. If the width were 2 feet more and the length 2 feet less, the room would be square. Find its dimensions. 6. In the triangle ABC, DE is parallel to AC, an~ AD == 8, DB=6, BE=3. Fmd BC. 8 a a --+-­ a-b a+b 7. Simplify b a a-b a+b v5 + v3 8. Rationalize the denominator of v5-v3 1. Find the value of 3/ 5 +5/2 -4/7. 2. Divide 0.02683 by 0.007213 to two decimal places. 2 +\IS x-1/x 3. Simplify: (b) v2 x +1 4. Factor completely : (a) y'-81; (b) a'b+3a•+ab+3a. 5. Find two numbers whose sum is 11 and whose difference is 29. 6. Find altitude, dropped on side 5, in the 6 whose sides are 3, 4, 5. 7. Solve: (a) 2x2 -x-6=-0; (b) 3/x + 2~1/x-2. 8. In 6 ABC draw DE so that AD/DC= BE/ EC with DC/AD=3/7. 1. Evaluate (2% + 5/ 6) (2/7) -1 • 2. Find y7 to two places of decimals. 3. Factor completely: (a) 81y8-16x'; (b) a•-b' + 2a + 2b. v12+ v2 4. Simplify: y27 a• -ab +b' a• -b" 5. Simplify: X --­a -b a•+ b' 6. Solve for x and check value or values found: 4x-2 5 3x (a) +-x = -+5; 5 8 4 (b) 2x'-7x+5=0. 7. Find the radius of a circle whose area is 78.54 sq. in. 8. Given a triangle of sides 5, 12, 13 in length. Find the length of the altitude upon the side of length 13 of this triangle. 1. 2/3 + 3/5 -5/7 =? 2. Reduce 12/ 17 to a decimal ( 3 places). 3. Factor completely : (a) 16x'-81y"; (b) 2a+2b+a2-b2 • vs+ v12 x'-1 4. Simplify: (a) ----; (b) 2y'27 x-1/ x The University of Texas Bulletin Total Paooed Failed Abilene ------------------------6 1 5 Albany ----------------------2 0 2 Alief ---------------------------1 0 1 Alliance Acad., Cam­ bridge Springs, Pa. 1 1 0 Alvin ----------------------------2 0 2 Alice ----------------------------1 0 1 Alvarado --------------------1 1 0 A. & M. Consolidated 1 0 1 Amarillo---------------------6 1 5 Anadarko -------------------1 0 1 Anson --------------------------6 0 6 Thos. Arnold______________ 1 1 0 Athens -----------------------4 1 3 Austin _ ________________________ 136 55 81 Bainbridge (Ga.)________ 1 0 1 Baird --------------------------1 0 1 Bars ton (Kansas City) ------------------------1 0 1 Bartlett ----------------------3 0 3 Bastrop ------------------------4 0 4 Bay City_____________________ 1 0 1 Beaumont ------------------7 5 2 Beeville ------------------------7 0 7 Bellevue ----------------------1 1 0 Bellville ----------------------6 2 4 Belton --------------------------1 0 1 Benhie --------------·----------1 0 1 Big Spring__________________ 1 O 1 Bishop --------------------------1 0 1 Brady --------------------------1 0 1 Brenham --------------------1 0 1 Bryan --------------------------1 0 1 Bolton --------------------------1 0 1 Britton, S. Dak. ____ ______ 1 0 1 Total Pa..ed Failed Bruceville Eddy__________ 1 0 1 Burkburnett ---------------­ 1 0 1 C. E. Byrd (Shreve­port) -------------------­-­Caldwell --------------------­ 3 4 0 0 3 4 Calvert ------­----------------­ 1 1 0 Carrizo Springs______ ___ _ 1 1 0 Castle Hts. Mil. Acad. 1 1 0 Celina ----------------------­ 1 0 1 Chester (W. Va.)______ Cheyenne (Okla.) ____ _ 1 1 1 0 0 1 Cheyenne (ColoradoSprings, Colo.)________ 1 1 0 Chico-­------------------------­Chilton -----------------------­ 1 1 0 1 1 0 Cisco --------------------------­ 3 1 2 Clarksville -----------------­ 3 0 3 Cleburne ---------------------­ 2 0 2 Clifton Col. Acad.______ 1 0 1 Clovis ( N. Mex.)________ 1 0 1 Coleman ---------------------­ 6 1 5 Colombia (S. C.)________ 1 1 0 Columbus ------------------­ 1 1 0 Comanche ----------------­ 2 0 2 Cooper ---------­-------------­Corpus Christi____________ 1 9 0 1 1 8 Corsicana -------------------­ 8 3 5 Cripple Creek (Colo.)Crockett --------­-----------­ 1 2 0 0 1 2 Cuero -------------------------­ 2 0 2 Daingerfield ---------------­Bryan St. (Dallas)____ 2 2 0 1 2 1 Forrest Ave. (Dallas) Highland Park (Dal­las) ----------·--------------­ 10 7 6 4 4 3 The Texas Mathematics Teachers' Bulletin 7o Total Pa...,d Failed North Dallas(Dallas) Oak Cliff (Dallas)____ Sunset (Dallas)______ W o o d r o w Wilson (Dallas) -------------­West Dallas (Dallas) Decatur (Ill.)______ Decatur --------------­De La Salle College (Manila, P. I.) ______ Del Rio___________________ Denison ------------------Denton -----------------­ Detroit CentraL___ Detroit Northern________ Deweyville ---------------Dormont (Pittsburg, Pa.)-----------­ Eagle Lake_____________ Eagle Pass______________ Eastland -------­East Orange (N.J.) __ E. T. S. T. C.___________ Edinburg ------­Edna ----------­ El Campo_________ Elgin -----------------Eldorado ---------­ El Paso__________ Ennis ------------ Falfurrias -------- Farmersville _______ Florence -----------------Floresville -----------Floydada --------------Follett -----------Foraker (Okla.>-------­ Foreman (Ark.) _______ Ft. Collins (Colo.) ____ Ft. Stockton_______________ Central (Ft. Worth) North Side (Ft. Worth) -------------------­Polytechnic (Ft. Worth) ------------------­Franklin --------------------­Frederick (Okla.) ------ Fredericksburg ---------­Freeport ----------------­Gainesville -----------------­ Ball High (Galves­ton) ---------------------­ George School, Pa._____ Giddings -----------------Gilmer --------------------­Gonzales ---------------------­ 2 5 2 6 1 1 1 1 2 2 1 3 2 1 1 1 3 1 2 2 1 2 4 1 1 7 3 1 2 1 3 2 1 1 1 1 5 15 1 4 2 1 4 3 1 20 1 1 3 5 2 2 O 3 0 0 0 1 1 0 0 1 1 0 1 0 2 0 2 1 0 1 0 1 0 3 1 0 0 0 0 0 1 1 0 0 1 5 0 0 2 0 1 2 0 9 1 0 1 2 O 3 2 3 1 1 1 0 1 2 1 2 1 1 0 1 1 1 0 1 1 1 4 0 1 4 2 1 2 1 3 2 0 0 1 1 4 10 1 4 0 1 3 1 1 11 0 1 2 3 Total Pa11ed Failed Goldthwaite --------------1 0 1 Goose Creek_____________ 2 1 1 Gorman -----------2 1 1 Grand Prairie_________ 1 0 1 Greenville------------1 0 1 Groesbeck -------1 0 1 3 3 Graham -------------6 Granger -------4 1 3 0 2 Groom ------------------2 Groveton -------------1 0 1 Hamilton -----------------1 0 1 Harlandale ------------2 1 1 Harlingen ----------2 0 2 Harrison, Ark._________ 1 1 0 Hearne -------------1 0 1 Hempstead -------------2 1 1 Henderson ------------3 2 1 Henrietta -------------1 1 0 Hereford ------------------4 0 4 High Bridge, N.J._ 1 0 1 Hillsboro -------------2 0 2 Hollis -----------1 1 0 Hondo--------------1 0 1 Honey Grove____________ 1 0 1 Hot Springs, Ark______ 1 1 0 Central (Houston)___ 2 0 2 Heights (Houston)___ 1 1 0 Jeff Davis (Houston) 3 2 1 Reagan (Houston)_ 1 1 0 Sam Houston (Hous­ ton) ----------------9 4 5 San Jacinto (Hous­ ton) __ • 19 5 14 Hubbard -----------------1 0 t Hull-Daisetta 1 0 1 Humble------------------1 0 1 Huntsville -------------1 0 1 Hutto -----------------4 1 3 Jacksboro (1904) ____ 1 0 1 Jackson, Miss._______ 1 0 1 Andrew Jackson (Jack­ sonville, Fla.)_____ 1 1 0 Jacksonville -----------1 1 0 Jasper, Mo.__________ 1 0 1 Jellico ----------------------1 0 1 Johnstown, Pa.__________ 1 0 1 Jonestown, Miss.______ 1 0 1 Joplin, Mo.______________ 1 0 1 Junction-------------------2 0 2 Katy ---------------------------1 0 1 Kaufman ------------------1 0 1 Kenedy ----------------------1 1 0 Killeen -----------------------1 0 1 Kirby Woodville, Tex. 1 1 0 Kirbyville --------------------4 0 4 Kountze -------------------1 0 1 The University of Texas Bulletin Total Puoed Failed Lake Forest Acad.___ _ 1 1 0 Lamesa -----------------------­1 1 0 Lampasas -----------------­3 0 8 Lancaster ------------------­1 0 1 La Progresiva (Car­ denas, Cuba> ----------1 1 0 Laredo ------------------------6 3 3 Lawrence (Ala.) ____ __ __ 1 0 1 League City________________ 1 1 0 Liberty ------------------------3 0 3 Llano --------------------------4 2 2 Lockhart --------------------4 1 3 Longview --------------------1 1 0 Lowell (San Francis­co) --------------------------1 0 1 Lufkin -----------------------3 0 3 Luling -----------------------1 0 1 Lyford _----------------------3 1 2 Mansfield (La.) ----------2 0 2 Manuel 0 j i n o g s (Mex.) --------------------1 0 1 Marlin ------------------------6 1 5 Marshall --------------·----3 1 2 Masonic Home____________ 1 1 0 Mathis ------------------------1 0 1 McAllen ----------------------3 0 3 McCallie ( Chattanoo­ga, Tenn.) ---------------­1 1 o McGregor -----------------­2 O 2 McKinley (Washing­ton, D.C. >----------------1 1 0 McKinley (Honolulu) 1 1 0 McKinney --------------------3 0 3 Memphis --------------------1 0 1 Mercedes --------------------4 1 3 ~er~dian --------------------3 1 2 ex1a ---·----------------------2 0 2Mexico City (Ameri­can) ·-----------------------1 1 0 Miami (Fla.)--------------1 1 0 0 2 SitE~~~~-:::::::::::::::::::::: f 0 1 0 1 Mills (N.M.)______________ 1 0 1Mineral Wells 3 1 2 Mission ________ __:::::::::::::: 3 2 1 Montezuma ---------------1 0 1Montgomery (Ala.)__ 1 1 0 Monthalia --------------------1 0 1Monticello (Miss.) ___ ___ 1 0 1 Moody ------------------------1 0 1 Mt. Pleasant ------·--------2 0 2 Munday ----------------------1 1 0 Mupel --------------------------1 0 1 •No paper. Total Pa11ed Failed Muskogee (Okla.)______ 1 0 1 Nacogdoches --------------1 0 1 Nogales (Ariz.) ----------1 0 1 Navasota -------------------1 0 1 Nevada ------------------------1 0 1 New Braunfels__________ 1 0 1 New Philadelphia_____ 1 0 1 Nicholas Senn (Chi­ cago) ----------------------1 0 1 Nixon --------------------------2 0 2 N. M. Mil. Inst.---------1 1 0 Oak Park (Ill.)__________ 2 0 2 Odessa ------------------------1 0 1 Central (Okla. City) 1 0 1 Orange ------------------------3 1 2 Orchard ----------------------1 0 1 Our Lady of the Lake 1 0 1 Overton ----------------------1 0 1 Palacios ----------------------1 0 1 Palestine -------------------4 2 2 Panhandle ------------------2 0 2 Paris ----------------------------2 0 2 Payne --------------------------1 0 1 Peace Inst. (Raleigh, N. C.) ______ __ _______ __ ___ __ 1 1 0 Pearsall ---------------------3 1 2 Pecos --------------------------4 1 3 Petrolia ----------------------1 0 1 Pharr --------------------------1 0 1 Philadelphia (Miss.) 1 0 1 Pilot Point_______ ___________ 1 0 1 Pinckneyville (Ill.) ___ 1 0 1 Pineville --------------------1 0 (*) Plainview ------------------1 0 1 Port Arthur________________ 4 0 4 Port Lavaca________________ 3 1 2 Post ( 55) --------------------1 0 1 Poteet --------------------------1 0 1 Quanah ------------------------2 0 2 Ralls ----------------------------1 0 1 Rankin ----------------------3 0 3 Raymondville ------------2 0 2 Richmond -------------------1 0 1 Rising Star -----------------1 1 0 Roaring Springs________ 1 0 1 Robstown --------------------1 0 1 Rockdale ---------------------1 1 0 Rock Island (Ill.)______ 8 1 2 Rocksprings --------------1 0 1 Rockwall -------------------1 0 1 Rogers --------------------1 1 0 Roosevelt (Des Moines, Ia.>------------1 0 1 Rosenberg ------------------1 1 0 Total Paaaed Failed Total Passed Failed Roswell (N. Mex.)____ Round Rock_______________ 1 1 0 0 1 1 Runge ---------------------­Rush Springs (Okla.)San Angelo_________________ 4 1 1 1 0 1 3 1 0 Alamo Heights (San Antonio) ------­---------Brackenridge (S a n 4 1 3 Antonio) -----------­---­Main Avenue (San 6 1 5 Antonio) ----------­----­San Augustine___________ San Benito_______________ 26 1 2 7 0 0 19 1 2 San Gabriel_____________ 1 0 1 San Germain, P. R.___ 1 1 0 Polytech Hi San Jacinto ----------------­ 1 0 1 S. M. B. A. CampMarlamont ______ 1 0 1 San Marcos________________ San Saba_____________ Saratoga ______ 2 1 2 0 0 2 2 1 0 Schreiner ---­-------­ 1 0 1 Schreiner Inst.____________ 4 0 4 Scoville (N.Y.C.) ___ 1 1 0 Seguin -----­------­------­Lutheran Acad. (Se­ 2 1 1 guin)Seward -----------------­Park (N.Y. 1 0 1 C.) --------­---1 Seymour --------­---­1 Sherman ------------­1 0 0 0 1 1 1 Shine -------------­ 1 0 1 Shiner --------------------­ 2 0 2 Shreiner ---------­ 1 0 1 Smithville --------­ 3 1 2 Somerville ----------­ 1 0 1 Sour Lake --------------­ 2 1 1 S. W. T. S. T. C. Prep. ----------------­S. P. N. S. (Wis.)_ 1 i 0 1 1 0 Stamford ----------­--------­ 2 0 2 State H o m e High (Corsicana) -----­ 2 1 1 Stephenville ----­------­St. Benedicts (Atchi­ 1 1 0 son, Kans.) -------------­St. Chas. (Mo.)__________ 1 1 0 0 1 1 St. Josephs (Victo­ria) -----------­--­-----­-1 St. Louis University High -----------­----------­1 St. Louis C o 11 e g e (Honolulu) --------­--­1 1 0 0 0 1 1 St. Mary's Academy (Austin) --------------­St. Thomas College (Houston) ------------­St. Xaviers (Cincin­nati, Ohio>------------­W. C. Stripling High Sulphur Springs________ 3 1 1 1 1 0 0 0 0 0 3 1 1 1 1 Central (Syracuse, N. Y.) ------------------­Taft ---------------------­ 1 2 0 1 1 1 Taylor ----------------Teague --------------------­Temple --------­---------­Terrill Prep. (Dallas) 9 1 2 3 0 0 0 0 9 1 2 3 Thorndale -----------­ 2 1 1 Thornton ----­----------Tivy (Kerrville)________ 2 1 2 0 0 1 T. M. I. (San An­ tonio) -------­Topeka (Kans.)_______ 4 1 0 0 4 1 Troy ----------------­-1 Troup -----------­1 Tulia -------------­3 0 0 0 1 1 3 Tyler --------­---------­Urbana (Ill.) --­---------Urseline A c a d em y 2 1 1 0 1 1 (Dallas) ------­--Uvalde ----­------------­ 1 1 1 0 0 1 Valley Mills______ 1 0 1 Vernon ------------­----­ 2 0 2 Villia Maria, Bryan (59) ----------------­Waco --------­-----------­---­ 1 12 0 2 1 10 Ward Belmont (Nashville, Tenn.) 1 0 1 Western (Washing­ton, D. C.) _________ 1 1 0 Weimar -------­-----­Wellington ___________ 11 0 0 1 1 Weslaco (58) ----­--------­Wesley College(Greenville) _________ 1 1 0 1 1 0 Wesleyan Inst. (SanAntonio) ____________ West _______ _ ____________ West Columbia_ __________ West Dallas.._______ _____ 1 3 1 1 0 0 0 0 1 3 1 1 Westmoorland College 1 0 1 W. T. S. T. C. High_ 1 0 1 Wharton ----­--­--­--­----­ 3 0 3 Whitesboro (N. Y.)__Wichita Falls______________ 1 12 0 3 1 9 Wilkinson (Ind.)._______ 1 1 0 The University of Texas Bulletin Total Paned Failed Total Pa11ed Failed Wills Point___________ 1 1 0 Woodstock Communi- Winnsboro (56) ________ 1 0 1 ty (Ill.)____________________ 1 0 1 Wolvin (Texas Yazoo City (Miss.) ____ 1 0 1 City) ---------------------Woodville ------------------­ 1 1 0 1 1 0 Yoakum ---------------------Yoe (Cameron) __________ 6 7 1 3 5 4 Yorktown-------------------­ 3 1 2