BULLETIN OF THE UNIVERSITY OF TEXAS 1916: No. 70 DECEMBER 15 1916 The Texas Mathematics Teachers' Bulletin (Vol. 2, No. 2, December 15, 1916) Published by the University six times a month and entered a.a second-class matter at the postoffice at AUSTIN, TEXAS Publications of the University of Texas Publications Committee: W. J. BATTLE C. HARTMAN E. c. BARKER J. L. HENDERSON G. c. BUTTE A. c. JUDSON R. H. GRIFFITH J. A. LOMAX The University publishes bulletins six times a month. These comprise the official publications of the University, publica­tions on humanistic and scientific subjects, bulletins prepared by the Department of Extension and by the Bureau of 1ifonic­ipal Research, and other bulletins of general educational in­terest. With the exception of special numbers, any bulletin will be sent to a citizen of Texas free on request. All communica­tions about University publications should be addressed to the Editor of University Publications, University of Texas, Austin. A. C. BALDWIN I SONI• AUSTtfl BULLETIN OF THE UNIVERSITY OF TEXAS 1916: No. '10 December 11'S 1918 The Texas Mathematics Teachers' Bulletin (Vol. 2, No. 2, December 15, 1916) Edited by J. W. CALHOUN Adjunct Professor of Pure Mathematics and C. D. RICE Associate Professor of Applied 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. CONTENTS. An Invitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 What Great Men Say About Mathematics ............ . . . ......32 The Geometry Original. .......... . ....... J. G. Dunlap.. . . . . . 4 The Mathematics of Investment............ E. L. Dodd.. . . . . . . 9 Literal Arithmetic ....... . .............. C. D. Rice.........19 On Postulational Systems.. . ..............A. A. Bennett......22 The Straight Edge....................... A. Nonymous .....33 Published by the University six times a mo:n.th and entered as second-class matter at the postofflce at AUSTIN, TEXAS The benefits of education and of useful knowledge, generally difTused 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 acknowl­edge and the only security that free­.men desire. Mirabeau B. Lamar. AN INVITATION It is the desire of the editors of this Bulletin that it shall render to the teachers of mathematics in Texas the greatest possible amount of service. To this end they invite their readers to suggesct any topics in connection with secondary mathematics that they desire to have discussed, to mention any particular difficulties they wish to have removed. The editors do not agree to discuss any topic proposed or to remove any difficulty mentioned-they scarcely wish to place so large an order,-but they do agree to do the best they can to secure the desired discussion or to suggest a remedy for the difficulty. Any reader having any good ideas of his own or special methods that he has found helpful is invited to send a paper dealing with them, and if its character seems to warrant the publication of the whole, any part, or a synopsis, the editors will be glad to give it space. THE ORIGINAL EXERCISE IN GEOMETRY J. G. Dunlap, Principal Cleburne High School The ''original exercise'' in this paper means that group of geometric principles whose truth must be established and prob­lems to be solved as distinguished from the theorems demon­strated in the text. The use of the original exercise in impress· ing and emphasizing geometric truths is of comparatively recent date. The experienced teacher will find it an extremely fertile field for developing accurate thinking. The earliest manuscripts in geometry were, of course, very primary, and the necessity of some means of fastening in mind the fundamental principles was not so great. As the science was gradually developed from the old Euclidean scroll, the field being ex­tended, the application of the principles then known called for and brought into use the exercise. The demonstration of a theorem must in form be essentially deductive or inductive-synthetic or analytic. Each has its peculiar use, and, to some extent at least, involves the other; analysis to discover -and synthesis to demonstrate the truth or falsity of the exercise. In the solution of practical exercises and problems the ability to investigate and reason for one's self is the necessary prerequisite to success. This ability is not inherent in the pupil, but must be acquired by long and earnest hours of application to study. Happy that teacher who e-an inspire enthusiasm in his pupils and make them fond of the task of solving the tedious original. And at this point I believe the pupil gets his most lasting benefit-a doggedness of pur­pose, a determination to win. It has been my observation that most of the failures in geometry are due to a lack of tenacity. Nothing in the field of secondary mathematics is quite as good in developing tenacity of purpose as the mastery of the original -nothing quite so good in developing the ability to concentrate the mental faculties. I am convinced that, if there is a superi­ority of the German over the American child in mathematical development, it is due largely to the preponderance of the exercise in the German text. Too many pupils memorize the proof, if given in full in the te·xt, thus relying entirely on the memory and neglecting the reason. Nothing could be more harmful than this process, and yet I realize that it is one of the very things we have to fight, and one that is especially preva­lent among beginners. Nothing helps like the knowledge that one must depend entirely upon himself. The original furnishes this field of activity as, in my opinion, no other does. Smith says: "The great value of teaching originals is in developing the power to think along correct lines of logical thought; if properly handled they make the pupil think more intensely and interestedly than any other subject fitted to pupils of the same age.'' The subject matter of the original appears in so many forms that its mastery involves a many-sided view of the sub­ject. The pupil must be resourceful, and if blocked in one avenue of attack, try another. This brings confidence in him­self and the ability to do something for himself, rather than a dependence on the text. When once the pupil feels the joy of having accomplished something for himself, he has, indeed, a stimulus of no mean account. Too many pupils feel over· whelmed with the apparently impossible task and surrender. Sisson says: ''Without enthusiasm no mathematics. Geometry is a human book-not divine-therefore a very imperfect book. Geometry is the product of the human mind and not of the hand; therefore the subject concerns the intellect and i~ not mechanical.'' With this viewpoint, which is undoubtedly cor­rect, the aims, which a.re several, may be reduced to one all important one-the training of the mind to habits of correct thinking and to reasoning logically and accurately to a correct conclusion. Nothing in all the high school course is so well adapted to developing the power to express one's thoughts concisely and elegantly as the original of geometry. I daresay the student of English has found few agencies in the correct and direct expression of his thoughts more helpful. Three· fourths of its value is disciplinary. The mind should direct the hand. Mathematical reasoning is that process which step by step arrives at a definite conclusion. The original, as its name indicates, has within itself the suggestions of the line of reason­ing that lead to a solution. At this point are brought to play those powers of observation which must suggest to the beginner the tools needed and the method of attack. To the beginner this is the most difficult point-''getting started'' ; ''finding out what you want to do." These and kindred statements express the pupil's conception of the task before him. In this step-by-step process each is made up of two distinct parts-the statement of a fact that leads the mind in the proper direction toward a conclusion, and the authority for such step. When the beginner has recognized this truth, he has made at least one step in solving the exercise. One pupil who can and does master with fair accuracy the originals as the class proceeds soon becomes the leader. And in this the value of the original is recognized by the class. If this leadership is properly direct­ed by a skillful teacher, growth and development of a friendly rivalry will assert itself and the effect on class work will be beneficial. The grouping of originals immediately after the basic theorem is a step in the right directon and is productive of splendid results. It enables the student to center-fire, as it were, and holds him on the principle until it is mastered. Many pupils fail utterly because they are unable to translate English into geometric terms, thereby losing the meaning of the exer­cise. Again, many lose sight of the all-important motive for studying geometry-the training of the reasoning faculties. They take for granted certain relations because it looks that way and are unable to pick up the train of thought when the statement is challenged. This weakness can be corrected by persis.tent efforts on the part of the teacher in having the pupil attack the problem from another point. Our work in geometry is impaired and our success often disappointing because we omit the originals. The original furnishes the material for practice in the clinic and many originals with a few well­digested theorems will bring better results of a permanent nature than many theorems and few applications of them. While it is true, as Loomis says, that problems do not consti­tute a necessary part of the science of geometry, forming no part of the chain of connected truths embodied therein, yet because of their importance as applications of geometric prin­ciples, they are of the utmost educational value and should be studied in connection with the theorems upon which they depend. After all, may it not be truthfully said that a pupil's knowledge of geometry is measured by his ability to solve the original? It is doubtful whether it is best to demand that the proof shall be submitted in a set form without variation. One of the chief ends to be attained is clearness of expression. However, it cannot be attained at once. Would it not be better to be master of facts than a slave to the form? As the pupil ad­vances, the polished form of proof should appear. If the de­mand for correctness of form is urged too far, the pupil loses the unfolding of the exercise in his effort to be correct in form. Young says: "Not all elegance and verbal accuracy that are to be attained later need be inflexibly required at first." Much of our trouble in teaching this part of geometry is caused by rushing the pupil over a mass of truths without time for diges­tion on his part. "Better plod at first and rush later," should be our motto. Many times we have heard the expression, ''I just know it is so,'' and kindred expressions, given as reasons for certain steps taken in demonstration. To accept such state­ments is not wholly bad, for it shows on its face that the pupil has within him the capacity for reasoning. The reason will come later, if he is properly directed. Much of our labor has been lost because we have not directed the mind of the pupil in the prQper way. Nothing, in my opinion, in the high school course calls for more patience on the part of the teacher than the direction of a slow student in attacking the original exer­cise. Again, no fixed rule of attack can be laid down, though some suggestions may be helpful. The first thing, of course, is to get a clear conception of the exercise. Skillful question­ing on the part of the teacher will give the student a start, as he calls it, and will help him discover the relation between the hypothesis and some theorem already learned. At this point the imagination is brought into use. Unless the pupil catches the spirit of geometric analysis, he will not succeed in finding the proof. Analysis is the soul of interpretation and interpre­tation is the key to the solution of the original exercise. Slaught says: "Many a high-school pupil who can play the game of hypothesis and conclusion-and likes it-has never recognized geometry as a basic fact in the tiles on the floor, in the decorations on the walls, in the arches and windows of great buildings, in fact in all the mechanical and architectural developments of this and of every age. When once he compre­hends this, when he takes his 'originals' from actual conditions about him, then the 'game' assumes a new significance to him and geometry becomes a. fact instead of a theory, a part of life inatead of a mere school creation.'' Whether or not the stu­dent is able to reach the end just mentioned depends largely on the teacher-whether or not we recognize several distinct periods in the mental development of the child and the a.da.p­tation of subject matter and methods of teaching. This means in geometry a readjustment of subject matter, a more accurate classification of basic truths and a. multiplication of originals for the application of these truths. In fact, the humanizing of the subject will do much to improve the success in teaching it -make the originals as practical as possible, thus appealing to his observation and experience for material. Plato's school of geometry was for mature men. Plato and Euclid, if living, would be astonished at our methods of teach­ing geometry and at the personnel of the average class. Yet with all the advancement of the subject, it can be made more practical without losing prestige as a subject which develops and trains the mind in accurate thinking and logical reasoning. At this point in the pupil's development the teacher must do more than ask questi-0ns. He must bear in mind that the pupil in the beginning of a subject does not receive and understand as quickly as the teacher. He must remember that the pupil does not generalize, but must learn to do so; that the subject is not developed, but is in the process of being developed. We as teachers are much given to "overshooting," as it were, the pupil-forgetting that he is not a man and therefore does not reason a.s one. After all, "the pupil must learn to do by doing," and will succeed if properly directed. (Read before the Mathematics Section ot the Texas State Teachers' Association, December 1, 1916.) THE MATHEMATICS OF INVESTMENT FROM AN ARITHMETIC VIEWPOINT EDWARD L. DODD INTRODUOTION To solve some of the problems in investment, algebra and calculus of an advanced type are necessary. But many prob­lems are susceptible of an arithmetic treatment. Algebra is commonly viewed as a higher branch of mathemat­ics than arithmetic. And in a certain sense it is. But a clear arithmtic grasp of a fundamental principle may represent higher intellectual activity than the corresponding algebraic reasoning which leads to a formula. And, in addition, it will probably be immensely more useful. Much the same relation exists between plane geometry and analytic geometry. The latter is rated as the higher branch of mathematics. But when the same subject is treated both in plane geometry and analytic geometry, the plane geometry usually calls forth the higher intellectual activity. In this presentation of some principles in the Mathematics of Investment from the arithmetic standpoint, letters will be used from time to time to stand for numbers. 'rhis can hardly be called algebra, although a beginner in algebra may think of algebra as the mathematics of letters and of arithmetic as the mathematics of numbers-a very infelicitous conception. The use of letters for numbers is mere short-hand. Algebra really begins when we substitute for real thinking some rule such as transposing with change of sign. In many texts on Investment and Life Insurance there are ''verbal explanati-0ns. '' These are usually attempts to give arithmetic color to what has been proven algebraically. This paper is an attempt to make clear by an informal and largely arithmetic treatment a few fundamental principles in the Mathematics of Investment. THE MATHEMATICS OF INVESTMENT The interest being 4%, a deposit of $100 is treated as follows: Original deposit, principal, or capital. ...............$100.00 For 4%, the multiplier is . ........................... .04 Interest for first year. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.00 This is added to the principal. . . . . . . . . . . . . . . . . . . . . . . . 100.00 Amount at end of first year. . . . . . . . . . . . . . . . . . . . . . . . . . 104.00 Multiply again by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .04 Interest for second year. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16 This is added to the amount. . . . . . . . . . . . . . . . . . . . . . . . . . 104.00 Amount at end of second year. . . . . . . . . . . . . . . . . . . . . . . . 108.16 The computation might have been arranged thus: Principal ............................ . ............$100.00 As multiplier, use 1 + interest rate. . . . . . . . . . . . . . . . . . 1.04 Amount at end of one year. . . . . . . . . . . . . . . . . . . . . . . . . . 104.00 As multiplier use again. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.04 416 104 108.16 It is easy to see that the two methods above must give the same result, for multiplying any number by .04 and then ad years is l.04Xl.04=1.0816. The compound interest on one dollar for 2 years is .0816 dollars or 8.16 cents. In passing, it may be noticed that the simple interest is only 8 centi on the dollar; whereas the compound interest, as just shown, is 8.16 cents on the dollar. Now divide $100 by .0816. This gives $1225.50 as the capital that must be invested at 4% compound interest to yie]d an in­come of $100 payable every two years forever. If simple interest at 4% were used, this would be .08 dollars­on one dollar in two years. We would have divided $100 by .08 and obtained $1250 as the capital that should be invested. At compound interest, however, as has just been shown, the capital needed is only $1225.50. This result may be checked as follows: Capital or prineipal ........................$1225.50 Add interest at 4%. . . . . . . . . . . . . . . . . . . . . . . . 49.02 Amount at end of first year...... ........... 1274.52 Again add interest at 4%................... 50.98 Amount at end of second year. . . . . . . . . . . . . . . 1325.50 Deduct the specified income. . . . . . . . . . . . . . . . . 100.00 Capital as at beginning. . . . . . . . . . . . . . . . . . . . 1225.50 It is thus evident that a principal of $1225.50 win at 4% compound interest yield an income of $100 payable at the end of each period of nvo years, the prncipal remaining "unim­paired'' or undiminished. The algebraic verification or check of the general Rule is also interesting, but will be left to the reader. DKFINITION The capitalized cost of a structure or an article is the first cost plus the capital which would provide for an indefinite nwm­ber of renewals. It has just been shown how this capital can be computed. By addin" to this the original cost, the capitalized cost can be found. In an enterprise in which machinery or perishable equipment is involved, the percentage of profit must be based upon the capitalized cost and not upon the original CO$t. If a man buys an automobile for $800, runs it as a jitney for four years clearing $100 each year over running expenses, and if the automobile then breaks down, being worth as junk only $50, the man has not made 100/ 800 or 12%% on his investment each year; for his original capital of $800 has been almost com­pletely absorbed or destroyed. It did not take long to compute l.04Xl.04 or (1.04)' as used in a foregoing illustration. But to compute by actual multi­plication (1.04)" would take considerable time. For 1.04 would be the multiplicand and there would be forty-nine multiples, each being 1.04. 1.04Xl.04Xl.04Xl.04X ... etc., involving forty-nine multiplications. Certain monetary tables have been constructed giving the amount of one dollar for any number of years up to 100 years at the usual rates of interest. For unusual rates of interest interpolation can often be used. But unless the consecutive rates given differ from each other by only a small fraction of a per cent, the interpolation must use second or third differences; the interpolation method used in logarithms will not be adequate. Logarithms may be used directly. The logar­ithms should be given to at least six decimals. Seven or eight place logarithms are frequently necessary. The computation side of investment problems is Yery im­portant. But it is not the purpose of this paper to dwell upon this matter. In this paper just a few rules have been given, applying to simple cases. The reader is invited to ask himself these ques­tions: How would the last "R1tle" giv1m be altered if a nom­inal rate of interest was giv·en, payable semi-annually or quar­terly? How would the Rule be altered if the income was payable monthly? Before attempting to answer these questions the reader should note (1) that 2% for a half-year is not ~quivalent to 4% for a year; (2) That an income of $100 at the end of each month is not equivalent to an income of $1200 at the end of the year. In practice income very frequently takes the form of a rent payable monthly, and nominal rates are commonly used. SUMMARX The amount S of P dollars at the rate i for n years is S P(1+i)n If n is a whole number, this can be proven. If n is any other real number, it is true by definition. The relation between the npminal rate j payable m times a year and the effoctive rate i is 1+i==(1+j/m)m (3) By means of (1) and (3), Scan be expressed in terms of P, j, n and m. The compound interest on one dollar for a specified period of time is defoned to be the amount of one dollar for that time diminished by one dollar. To find the capital needed to produce a specified income pay­able at the end of ea.eh period of a specified number of years, forever, divide the specified income in dollars by the number representing the compound interest on one dollar for ihe speci­fied number of years. This can be used to obtain the capitalized cost of a structure or article, viz. the original cost plus the present value of an indefinite number of renewals. In the case of perishable equipment the rate of profit must be computed on the basis of the capitalized cost, not the original cost. (To BE CONTINUED.) LITERAL ARITHMETIC In this and the following bulletins we will briefly indicate some simple applications that may be made by the use of the literal notation in grammar grade work before the high school. To the teacher in the high school these applications will seem very elementary and in many cases trivial, but if he would help the grammar grade teacher introduce these applications in the study of arithmetic he would find his classes in the beginning of algebra having fewer difficulties in the first few weeks. In this bulletin we will briefly take up the use of letters in the study of problems in interest. In all interest problems there are four quantities to be con­sidered: (1) The principle p. (2) The rate per oont per annum r. ( 3) The time t given in years. ( 4) The interest i. The product of pXr is the interest for one year, and the product pXrXt is the interest for t years. Hence we have the formula (1) pr t==i. Dividing both sides of (1) respectively by pr, p t, r t, we have i (2) t=­ pr i (3) r= pt i (4) p=­ rt Formula (1), as we have seen, is an expression of the defini­tion of interest by the use of the four letters i p r and t. It is therefore easily remembered and can be written down, at any time, when needed. From (1) the pupil can soon learn to find formulas (2), (3), ( 4). These four formulas show that when any three of the four numbers i, p, r, t, are given the fourth can be determined. Thus By (1) the interest i may be found when p, r, t are given. By (2) the time t may be found when i, p, r are given. By (3) the rate r may be found when i, p, t are given. By (4) the principal p may be found when i, r, t are given. It would be well for the teacher to give (1) frequently to the class and have formulas (2) (3) (4) derived from it and state the reason for the process. A few examples will be added. Example. Given p=$1200, r=5l/2% and t=2 yrs., 5 mo., 10 da. to find i. By definition we know the time t must be expressed in years. Hence t-2 yrs., 5 mo., 10 da.=2 4/ 9 yrs. By formula (1) we have i pr t~'!;l200X.05l/2X2 4/ 9 =$161.33% Example. What principal in 2 years time will produce $30 interest, the rate being 5%? By formula ( 4) we have i $30 p=-=-­r t 2X.05 = $300 It is sometimes convenient to use a fifth formula derived as follows: a=p+i P+P rt a=p(l+r t) or a (5) p=­1+vt The a which is defined to be equal to the sum of the interest and principal is called the amoimt. Example. What principle will amount to $356 in two years and eight months, the rate being 7%? Here we have a p=-­ 1+rt 356 1+273x .01 =$300 It is thus seen that by remembering the definition of interest that all rules may be written down at once for any case that may arise. The wa~te of energy in classifying interest problems into so many cases as found in most arithmetics is saved here and the student has to learn the definition of interest only and not spend his time learning a rule to fit each case. It will be well for the teacher to propose at random problem~ for the different members of the class to indicate and then find the solution. Do not let the class expect that any two successive problems will fall under the same formula. Thus propose a series something as follows: (1) Given p=$900, r=.06, t=3%; find i (2) Given r=.06, p-$500, i=$15; find t (3) Given a=$912, t=4, r=.031/2 ; find p (4) Given p=$1089, i-$200.376, t=4 yrs., 7 mo , 6 da.; find r. In this way the pupil will learn to know interest in a way that would be impossible in studying separate cases by the old methods. ON POSTULATIONAL SYSTEMS 1. Introduction. In most of the modern fields of investiga­tion in mathematics, the philosopher can hope to finu little to arouse his interest or challenge his criticism. Undoubtedly there has never been a time in which the current problems of this science have been so unintelligible to the metaphysician and to the man of affairs. But in the marvelous growth and specializa­tion in mathematics during recent years, the relations of mathe­matics to logic and to the wider branches of philosophy have not been wholly neglected. The ancient eraving for a propositional system that shall be at once simple, consistent and universal, and which, nurtured through generations of the finest of Greek thought, found at last an adequate expression in Euclid's superb masterpiece, this same scientific and esthetic longing has re­cently, also, been making insistent demands for critical investi­gation and appreciation of what has been called the Foundations of Mathemati~. The processes of counting the construction of weights and mea­sures, of interest tables and calendars, the staking of fields, and the mapping of the starry heavens, the building of houses and ships, the erection of monuments and fortifi.cations, in fact mo.st of the occupations of the artisan and the trader in­volve at some stage, specialized notions of number and of space relations. Arithmetic, geometry and even trigonometry, arose as inevitable practfoal sciences. There was probably a fair amount of collected material forming a rudimentary mathe­matical theory even as early as 2000 B. C., among the most advanced peoples of that era. Looking back to such a time, one can well believe that it was the substance and not the form of their computational science that interested the learned men of that day. Many times since then mathematical rigor has been forgotten in the excitement of suggestive discovery, and the foundation completely obscured by the splendor of the growing superstructure, and even today this situation obtains for many if not most investigators in mathematics. The ''foundations of mathematics'' as a theory instead of being historically essential to the growth of mathematical science, has been an esthetic luxury necessary indeed for an adequate appreciation and rigorous treatment of large fields of mathe­matics but slighted during many periods of rapid, if insecure, extension. Not so much as a partial understanding of the role and the significance of anxiom'.'l is essential to the grasp of geometric facts, nor even to the discovery and formulation of many intricate proofs. Thus one is not surprised to learn that the very comprehension of the problems rela.ting to the axiomatic bases wa'.'l a rare achievement of Greek thought, and since the Alexandrian era largely lost to the world until recent times. 2. The Nature of a Mathematical System. As early ack­nowledged by the Greeks, a formal mathematical proof must be pure deduction, purged of all accidental and extraneous features such as intuitive or inductive argument'.'l. The very existence of a proof implies not only definitions, but certain propositions regarded as established and certain logical processes admitted as valid. In similar manner no definition can be permitted which does not relate objects regarded as already known. It is clear that if any deductive science is to be developed certain logically basal operations, primitive objects, and elementary propositions must be accepted as initial and not requiring just­fication by the science about to be dealt with. Thus the mathe­matician presupposes acceptance of logical laws and of certain terms and propositions. For rea'.'lons of convenience principles of logic a.re only rarely explicitly investigated in connection with a mathematical system but the same remark does not hold for the terms and propositions employed. While for each indi­vidual treatment there must exist undefined objects and un­proved theorems it is in no respect essential fhat a preassigned element or proposition be undefined for all possible discussions. One geometer may regard "'point," "line" al!ld "order" as un­defined, and another perhaps, define all of these in terms of ''transformation,'' and ''planar element.'' Even the term ''un­defined," may be misleading since every axiom or unproved proposition is a partial implicit definition of these objects, which were indeed initially undefined. Bulletin of the University of Texas These commonplaces of a deductive science were understood even in the Middle Ages as is shown by the following quaint quotation from Thomas Aquinas. (Summa Theologia 1, 1. questio 1, art. 1-8.) "But there are two kinds of sciences. There are those which proceed from the principles known by the natural light of the mind, as arithmetic and geometry. There are others which pro­ceed from principles made known by the light of a superior science; as perspective proceeds from principles made known through geometry, and music from principles made known through arithmetic. One science may be .said to be worthier than another by reason of its certitude or the dignity of its matter. . . It should be said that . . other sciences do not prove their first principles but argue from them in order to prove other matters. One should bear in mind that in the philosophic sciences the lower science neither prov~ its own first principles nor disputes with )tim who de­nies them, but leaves that to a higher science. But the science which is highest among them, that is, metaphysics, does dispute with him who denies its principles, if the adversary will concede anything; if he concede nothing, it cannot thus argue with him but can only overthrow. his arguments.'' 3. Euclid's Pwrallel-Postulate. The modern revival of in­terest in the logical substructure of analysis has been an out­come almost exclusively of the discussions and discoveries cen­tering in the so-called "parallel-postulate" of Euclid. Except for the phenomenal excitement awakened by the non-Euclidean geometries born in the last century, it is at least improbable that the modern mathematical philosopher could rival the logical in­dependence which recent scholarship discovers in the remains of the Greeks. Euclid who lived about 300 B. C. was associated with the foundation of what may properly be called the first university, and which was situated in Alexandria. Today we appreciate Euclid's problem of reducing the science of geometry as then known to a postulational basis, and we un­derstand also his objectors. Not until after the twenty-eighth proposition has been proved does Euclid require the famous proposition which has been rendered into English as follows: ''l£ two lines are cut by a third and the sum of the interior angles on the same side of the cutting line is less than two right angles, the lines will meet on that side when sufficiently pro­duced." This proposition has given rise to much discussion. Its very order among the axioms has been changed by various commentators. Its length and apparent complexity is in strik­ing distinction to the form of most of Euclid's axioms, nor does its independence appear so obvious. It may be regarded as a converse of a previous proposition and yields the converse of yet another. Then, too, it has appeared as being more prob­ably capable of proof by means of the remaining postulates than any of the others. 4. Origin of Non-Euclidean Geometry. Saccheri, an Italian monk developed in 1733 a body of geometric theorems in which the above axiom is denied. It must be admitted that he finally concludes the entire system to be contrary to common sense and therefore worthless, but in view of the apparent attitude of the authorities of the time and the anecdotes concerning Galileo, the sincerity of his ruthless criticism might be sus­pected. In 1766 Lambert maintained that the parallel-postulate requires proof, and suggested some characteristics of the geom­etry resulting from its denial. Legendre (1752-1833) tried to prove the above proposition. He continued, of course, to regard a line as infinite in length, and proved independently of the parallel-postulate, that the sum of the angles of a triangle is at most equal to two right angles, and that if a single triangle existlil in which the sum of the angles is exactly equal to two right angles, then this is the sum for every triangle. The exis~ence of the one triangle that would complete the proof of the theorem that the sum of the angles of a triangle is equal to two right angles and hence the proof of the equivalent theorem, viz. the "parallel-postulate," he could not establish. Such was the status of the problem until after 1830, when Euclid was justified in a spectacular manner. Approximately simultaneously, Lobachevsky, a Russian, J. Bolyai, an Hun­garian, and Gauss, a German, showed the necessity of the post­ulate for ordinary geometry, by exhibiting a geometric science, different from that of Euclid, but obtained by denying this Bulletin of th1~ University of Tex