Q A 685 B62 1896 MAIN UC-NRLF B M EMfl THE SCIENCE ABSOLUTE OF SPACE Independent of the Truth or Falsity of Euclid's Axiom XI (which can never be decided a priori). BY JOHN BOLYAI M TRANSLATED FROM THE LATIN BY DR. GEORGE BRUCE HALSTED PRESIDENT OF THE TEXAS ACADEMY OF SCIENCE FOURTH EDITION. VOLUME THREE OF THE NEOMONIC SERIES PUBLISHED AT THE XEOMON 2407 Guadalupe Street AUSTIN. TEXAS, U. S. A. 1896 TEANSLATOR'S INTRODUCTION. The immortal Elements of Euclid was al- ready in dim antiquity a classic, regarded as absolutely perfect, valid without restriction. Elementary geometry was for two thousand years as stationary, as fixed, as peculiarly Greek, as the Parthenon. On this foundation pure science rose in Archimedes, in Apollon- ius, in Pappus; struggled in Theon, in Hypa- tia; declined in Proclus; fell into the long decadence of the Dark Ages. The book that monkish Europe could no longer understand was then taught in Arabic by Saracen and Moor in the Universities of Bagdad and Cordova. To bring the light, after weary, stupid cen- turies, to western Christendom, an English- man, Adelhard of Bath, journeys, to learn Arabic, through Asia Minor, through Egypt, back to Spain. Disguised as a Mohammedan student, he got into Cordova about 1120, ob- tained a Moorish copy of Euclid's Elements, and made a translation from the Arabic into Latin. M306H52 iv TRANSLATOR'S INTRODUCTION. The first printed edition of Euclid, pub- lished in Venice in 1482, was a Latin version from the Arabic. The translation into Latin from the Greek, made by Zaniberti from a MS. of Theon's revision, was first published at Venice in 1505. Twenty-eight years later appeared the editio princeps in Greek, published at Basle in 1533 by John Hervagius, edited by Simon Grynaeus. This was for a century and three- quarters the only printed Greek text of all the books, and from it the first English transla- tion (1570) was made by "Henricus Billings- ley," afterward Sir Henry Billingsley, Lord Mayor of London in 1591. And even to-day, 1895, in the vast system of examinations carried out by the British Gov- ernment, by Oxford, and by Cambridge, no proof of a theorem in geometry will be ac- cepted which infringes Euclid's sequence of propositions. Nor is the work unworthy of this extraor- dinary immortality. Says Clifford: "This book has been for nearly twenty-two centuries the encourage- ment and guide of that scientific thought which is one thing with the progress of man from a worse to a better state. TRANSLATOR'S INTRODUCTION. v "The encouragement; for it contained a body of knowledge that was really known and could be relied on. "The guide; for the aim of every student of every subject was to bring his knowledge of that subject into a form as perfect as that which geometry had attained." But Euclid stated his assumptions w4th the most painstaking candor, and would have smiled at the suggestion that he claimed for his conclusions any other truth than perfect deduction from assumed hypotheses. In favor of the external reality or truth of those as- sumptions he said no word. Among Euclid's assumptions is one differing from the others in prolixity, whose place fluc- tuates in the manuscripts. Peyrard, on the authority of the Vatican MS., puts it among the postulates, and it is often called the parallel-postulate. Heiberg, whose edition of the text is the latest and best (Leip- zig, 1883-1888), gives it as the fifth postulate. James Williamson, who published the closest translation of Euclid we have in English, in- dicating, by the use of italics, the words not in the original, gives this assumption as elev- enth among the Common Notions. vi TRANSLATOR'S INTRODUCTION. Bolyai speaks of it as Euclid's Axiom XI. Todhunter has it as twelfth of the Axioms. Clavius (1574) gives it as Axiom 13. The Harpur Euclid separates it by forty- eight pages from the other axioms. It is not used in the first twenty-eight pro- positions of Euclid. Moreover, when at length used, it appears as the inverse of a proposition already demonstrated, the seventeenth, and is only needed to prove the inverse of another proposition already demonstrated, the twenty- seventh. Now the great Lambert expressly says that Proklus demanded a proof of this assumption because when inverted it is demonstrable. All this suggested, at Europe's renaissance, not a doubt of the necessary external reality and exact applicability of the assumption, but the possibility of deducing it from the other assumptions and the twenty-eight propositions already proved by Euclid without it. Euclid demonstrated things more axiomatic by far. He proves what every dog knows, that any two sides of a triangle are together greater than the third. Yet after he has finished his demonstration, that straight lines making with a transversal equal alternate angles are parallel, in order to TRANSLATOR'S INTRODUCTION. vii prove the inverse, that parallels cut by a trans- versal make equal alternate angles, he brings in the unwieldy assumption thus translated by Williamson (Oxford, 1781) : "11. And if a straight line meeting two straight lines make those angles which are in- ward and upon the same side of it less than two right angles, the two straight lines being produced indefinitely will meet each other on the side where the angles are less than two right angles." As Staeckel says, "it requires a certain courage to declare such a requirement, along- side the other exceedingly simple assumptions and postulates." But was courage likely to fail the man who, asked by King Ptolemy if there were no shorter road in things geometric than through his Elements? answered, "To geometry there is no special way for kings!" In the brilliant new light given by Bolyai and Lobachevski we now see that Euclid un- derstood the crucial character of the question of parallels. There are now for us no better proofs of the depth and systematic coherence of Euclid's masterpiece than the very things which, their cause unappreciated, seemed the most notice- able blots on his work. viii TRANSLATOR'S INTRODUCTION. Sir Henry Savile, in his Praelectiones on Euclid, Oxford, 1621, p. 140, says: "In pul- cherrimo Geometriae corpora duo sunt naevi, duae labes ..." etc., and these two blemishes are the theory of parallels and the doctrine of proportion; the very points in the Elements which now arouse our wondering admiration. But down to our very nineteenth century an ever renewing stream of mathematicians tried to wash away the first of these supposed stains from the most beauteous body of Geometry. The year 1799 finds two extraordinary young men striving thus 4 ' To gild refined gold, to paint the lily, To cast a perfume o'er the violet." At the end of that year Gauss from Braun- schweig writes to Bolyai Farkas in Klausen- burg (Kolozsvar) as follows: [Abhandlungen der Koeniglichen Gesellschaft der Wissen- schaften zu Goettingen, Bd. 22, 1877.] " 1 very much regret, that I did not make use of our former proximity, to find out more about your investigations in regard to the first grounds of geometry; I should certainly thereby have spared myself much vain labor, and would have become more restful than any one, such TRANSLATOR'S INTRODUCTION. ix as I, can be, so long as on such a subject there yet remains so much to be wished for. In my own work thereon I myself have ad- vanced far (though my other wholly hetero- geneous employments leave me little time therefor) but the way, which I have hit upon, leads not so much to the goal, which one wishes, as much more to making doubtful the truth of geometry. Indeed I have come upon much, which with most no doubt would pass for a proof, but which in my eyes proves as good as nothing. For example, if one could prove, that a rec- tilineal triangle is possible, whose content may be greater, than any given surface, then I am in condition, to prove with perfect rigor all geometry. Most would indeed l,et that pass as an axiom; I not; it might well be possible, that, how far apart soever one took the three vertices of the triangle in space, yet the content was always under a given limit. I have more such theorems, but in none do I find anything satisfying." From this letter we clearly see that in 1799 Gauss was still trying to prove that Euclid's is the only non-contradictory system of geome- x TRANSLATOR'S INTRODUCTION. try, and that it is the system regnant in the external space of our physical experience. The first is false; the second can never be proven. Before another quarter of a century, Bolyai Janos, then unborn, had created another pos- sible universe; and, strangely enough, though nothing renders it impossible that the space of our physical experience may, this very year, be satisfactorily shown to belong to Bolyai Janos, yet the same is not true for Euclid. To decide our space is Bolyai's, one need only show a single rectilineal triangle whose angle-sum measures less than a straight angle. And this could be shown to exist by imperfect measurements, such as human measurements must always be. For example, if our instru- ments for angular measurement could be brought to measure an angle to within one millionth of a second, then if the lack were as great as two millionths of a second, we could make certain its existence. But to prove Euclid's system, we must show that a triangle's angle-sum is exactly a straight angle, which nothing human can ever do. However this is anticipating, for in 1799 it seems that the mind of the elder Bolyai, Bolyai Farkas, was in precisely the same state as TRANSLATOR'S INTRODUCTION. xi that of his friend Gauss. Both were intensely trying to prove what now we know is inde- monstrable. And perhaps Bolyai got nearer than Gauss to the unattainable. In his * 4 Kurzer Grundriss eines Versuchs," etc., p. 46, we read: "Koennten jede 3 Punkte, die nicht in einer Geraden sind, in eine Sphaere fallen, so waere das Eucl. Ax. XI. bewiesen." Frischauf calls this "das anschaulichste Axiom." But in his Autobiography written in Magyar, of which my Life of Bolyai contains the first transla- tion ever made, Bolyai Farkas says: "Yet I could not become satisfied with my different treatments of the question of parallels, which was ascribable to the long discontinuance of my studies, or more probably it was due to myself that I drove this problem to the point which robbed my rest, deprived me of tran- quillity." It is wellnigh certain that Euclid tried his own calm, immortal genius, and the genius of his race for perfection, against this self-same question. If so, the benign intellectual pride of the founder of the mathematical school of the greatest of universities, Alexandria, would not let the question cloak itself in the obscuri- ties of the infinitely great or the infinitely small. He would say to himself: "Can I prove xii TRANSLATOR'S INTRODUCTION. this plain, straightforward, simple theorem: ^rhose straights which are produced indefin- itely from less than two right angles meet. [This is the form which occurs in the Greek of Eu.1.29.] Let us not underestimate the subtle power of that old Greek mind. We can produce no Venus of Milo. Euclid's own treatment of proportion is found as flawless in the chapter which StoU devotes to it in 1885 as when through Newton it first gave us our present continuous number-system. But what fortune had this genius in the fight with its self-chosen simple theorem? Was it found to be deducible from all the definitions, and the nine "Common Notions/' and the five other Postulates of the immortal Elements? Not so. But meantime Euclid went ahead without it through twenty-eight propositions, more than half his first book. But at last came the practical pinch, then as now the tri- angle's angle-sum. He gets it by his twenty-ninth theorem: "A straight falling upon two parallel straights makes the alternate angles equal." But for the proof of this he needs that re- calcitrant proposition which has how long been keeping him awake nights and waking TRANSLATOR'S INTRODUCTION. xiii him up mornings? Now at last, true man of science, he acknowledges it indemonstrable by spreading it in all its ugly length among his postulates. Since Schiaparelli has restored the astron- omical system of Eudoxus, and Hultsch has published the writings of Autolycus, we see that Euclid knew surface-spherics, was famil- iar with triangles whose angle-sum is more than a straight angle. Did he ever think to carry out for himself the beautiful system of geometry which comes from the contradiction of his indemonstrable postulate; which exists if there be straights produced indefinitely from less than two right angles yet nowhere meet- ing; which is real if the triangle's angle-sum is less than a straight angle? Of how naturally the three systems of geom- etry flow from just exactly the attempt we suppose Euclid to have made, the attempt to demonstrate his postulate fifth, we have a most romantic example in the work of the Italian priest, Saccheri, who died the twenty-fifth of October, 1733. He studied Euclid in the edi- tion of Clavius, where the fifth postulate is given as Axiom 13. Saccheri says it should not be called an axiom, but ought to be dem- onstrated. He tries this seemingly simple xiv TRANSLATOR'S INTRODUCTION. task; but his work swells to a quarto book of 101 pages. Had he not been overawed by a conviction of the absolute necessity of Euclid's system, he might have anticipated Bolyai Janos, who ninety years later not only discovered the new world of mathematics but appreciated the transcendent import of his discovery. Hitherto what was known of the Bolyais came wholly from the published works of the father Bolyai Farkas, and from a brief article by Architect Fr. Schmidt of Budapest "Aus dem Leben zweier ungarischer Mathematiker, Johann und Wolfgang Bolyai von Bolya." Grunert's Archiv, Bd. 48, 1868, p. 217. In two communications sent me in Septem- ber and October 1895, Herr Schmidt has very kindly and graciously put at my disposal the results of his subsequent researches, which I will here reproduce. But meantime I have from entirely another source come most unex- pectedly into possession of original documents so extensive, so precious that I have determined to issue them in a separate volume devoted wholly to the life of the Bolyais; but these are not used in the sketch here given. Bolyai Farkas was born February 9th, 1775, at Bolya, in that part of Transylvania (Er- TRANSLATOR'S INTRODUCTION. xv dely) called Szekelyfrld. He studied first at Enyed, afterward at Klausenburg (Kolozsvar), then went with Baron Simon Kemeny to Jena and afterward to Goettingen. Here he met Gauss, then in his 19th year, and the two formed a friendship which lasted for life. The letters of Gauss to his friend were sent by Bolyai in 1855 to Professor Sartorius von Walterhausen, then working on his biography of Gauss. Everyone who met Bolyai felt that he was a profound thinker and a beautiful character. Benzenberg said in a letter written in 1801 that Bolyai was one of the most extraordinary men he had ever known. He returned home in 1"7% and in January, 1804, was made professor of mathematics in the Reformed College of Maros-Vasarhely. Here for 47 years of active teaching he had for scholars nearly all the professors and no- bility of the next generation in Erdely. Sylvester has said that mathematics is poesy. Bolyai' s first published works were dramas. His first published book on mathematics was an arithmetic: Az arithmetica eleje. 8vo. i-xvi, 1-162 pp. The copy in the library of the Reformed Col- lege is enriched with notes by Bolyai Janos. xvi TRANSLATOR'S INTRODUCTION. Next followed his chief work, to which he constantly refers in his later writings. It is in Latin, two volumes, 8vo, with title as fol- lows: TENTAMEN | JUVENTUTEM STUDIOSAM | IN ELEMENTA MATHESEOS PURAE, ELEMEN- TARIS AC | SUBLIMIORIS, METHODO INTUI- TIVA, EVIDENTIA QUE HUIC PROPRIA, IN- TRODUCENDI. | CUM APPENDICE TRIPLICI. | Auctore Pro- fessore Matheseos et Physices Chemiaeque | Publ. Ordinario. | Tomus Primus. .| Maros Vasarhelyini. 1832. | Typis Collegii Re- formatorum per JosEPHUM, et | SlMEONEM KALI de felso Vist. | At the back of the title: Imprimatur. | M. Vasarhelyini Die | 12 Octo- bris, 1829. | Paulus Horvath m. p. Abbas, Parochus et Censor | Librorum. Tomus Secundus. | Maros Vasarhelyini. 1833. | The first volume contains: Preface of two pages: Lectori salutem. A folio table: Explicatio signorum. Index rerum (I XXXII). Errata (XXXIII XXXVII). Pro. tyronibus prima vice legentibus no- tanda sequentia (XXXVIII LJI). Err ores (LIU LXVI). TRANSLATOR'S INTRODUCTION, xvii Scholion (LXVII LXXIV). Plurium errorum haud animadversorum numerus minuitur (LXXV LXXVI). Recensio per auctorem ipsum facta (LXXVII LXXVIII) . Err ores recentius detecti (LXXV- XCVIII). Now comes the body of the text (pages 1502). Then, with special paging, and a new title page, comes the immortal Appendix, here given in English. Professors Staeckel and Engel make a mis- take in their " Parallellinien " in supposing that this Appendix is referred to in the title of " Tentamen." On page 241 they quote this title, including the words ' * Cum appendice triplici," and say: "In dem dritten Anhange, der nur 28 Seiten umfasst, hat Johann Bolyai seine neue Geometrie entwickelt." It is not a third Appendix, nor is it refer- red to at all in the words ' ' Cum appendice triplici." These words, as explained in a prospectus in the Magyar language, issued by Bolyai Parkas, asking for subscribers, referred to a real triple Appendix, which appears, as it xviii TRANSLATOR'S INTRODUCTION. should, at the end of the book Tomus Secun- dus, pp. 265-322. The now world renowned Appendix by Bolyai Janos was an afterthought of the father, who prompted the son not "to occupy himself with the theory of parallels," as Staeckel says, but to translate from the Ger- man into Latin a condensation of his treatise, of which the principles were discovered and properly appreciated in 1823, and which w-as given in writing to Johann Walter von Eck- wehr in 1825. The father, without waiting for- Vol. II, inserted this Latin translation, with separate paging (1-26), as an Appendix to his Vol. I, where, counting a page for the title and a page "Explicatio signorum," it has twenty- six numbered pages, followed by two unnum- bered pages of Errata. The treatise itself, therefore, contains only twenty-four pages the most extraordinary two dozen pages in the whole history of thought! Milton received but a paltry 5 for his Paradise Lost; but it was at least plus 5. Bolyai Janos, as we learn from Vol. II, p. 384, of ef Tentamen" contributed for the TRANSLATOR'S INTRODUCTION. xix printing- of his eternal twenty-six pages, 104 florins 50 kreuzers. That this Appendix was finished consider- ably before the Vol. I, which it follows, is seen from the references in the text, breath- ing a just admiration for the Appendix and the genius of its author. Thus the father says, p. 452: Elegans est conceptus similiumy quern J. B. Appendicis Auctor dedit. Again, p. 489: Appendicis Auctor, rem acumine singulari aggressus, Ge- ometriam pro omni casu absolute veram posuit; quamvis e magna mole, tantum summe neces- saria, in Appendice hujus tomi exhibuerit, multis (ut tetraedri resolutione generali, plu- ribusque aliis disquisitionibus elegantibus) brevitatis studio omissis. And the volume ends as follows, p. 502: Nee operae pretium est plura referre; quum res tota exaltiori contemplationis puncto, in ima penetranti oculo, tractetur in Appendice se- quente, a quovis fideli veritatis purae alumno diagna legi. The father gives a brief resume of the re- sults of his own determined, life-long, desper- ate efforts to do that at which Saccheri, J. H. Lambert, Gauss also had failed, to establish Euclid's theory of parallels a priori. xx TRANSLATOR'S INTRODUCTION. He says, p. 490: "Tentamina idcirco quae olim feceram, breviter exponenda veniunt; ne saltern alius quis operam eandem perdat." He anticipates J. Delboeuf's * ' Prolegomenes phil- osophiques de la geometric et solution des postulats," with the full consciousness in addition that it is not the solution, that the final solution has crowned not his own intense efforts, but the genius of his son. This son's Appendix which makes all pre- ceding space only a special case, only a species under a genus, and so requiring a descriptive adjective, Euclidean, this wonderful produc- tion of pure genius, this strange Hungarian flower, was saved for the world after more than thirty-five years of oblivion, by the rare erudition of Professor Richard Baltzer of Dresden, afterward professor in the Univer- sity of Giessen. He it was who first did jus- tice publicly to the works of I^obachevski and Bolyai. Incited by Baltzer, in 1866 J. Ho li el issued a French translation of Lobachevski's Theory of Parallels, and in a note to his Preface says: "M. Richard Baltzer, dans la seconde edition de ses excellents Elements de Geometric, a, le premier, introduit ces notions exactes a la place qu'elles doivent occuper,' : Honor to TRANSLATOR'S INTRODUCTION. xxi Baltzer! But alas! father and son were al- ready in their graves! Fr. Schmidt in the article cited (1868) says: ' ' It was nearly forty years before these pro- found views were rescued from oblivion, and Dr. R. Baltzer, of Dresden, has acquired im- perishable titles to the gratitude of all friends of science as the first to draw attention to the works of Bolyai, in the second edition of his excellent Elemente der Mathematik (1866-67). Following the steps of Baltzer, Professor Hoiiel, of Bordeaux, in a brochure entitled, Essai critique sur les principes fondamentaux de la Geometric elementaire, has given ex- tracts from Bolyai's book, which will help in securing for these new ideas the justice they merit." The father refers to the son's Appendix again in a subsequent book, Urtan elemei kez- doknek [Elements of the science of space for beginners] (1850-51), pp. 48. In the College are- preserved three sets of figures for this book, two by the author and one by his grand- son, a son of Janos. The last work of Bolyai Farkas, the only one composed in German, is entitled, Kurzer Grundriss eines Versuchs I. Die Arithmetik, durch zvekmassig kons- xxii TRANSLATOR'S INTRODUCTION. truirte Begriffe, von eingebildeten und unend- lich-kleinen Grossen gereinigt, anschaulich und logisch-streng darzustellen. II. In der Geometrie, die Begriffe der ger- aden Linie, der Ebene, des Winkels allgemein, der winkellosen Formen, und der Krummen, der verschiedenen Arten der Gleichheit u. d. gl. nicht nur scharf zu bestimmen; sondern auch ihr Seyn im Raume zu beweisen: und da die Frage, ob zwey von der dritten geschnit- tene Geraden, wenn die summe der inneren Winkel nicht = 2R, sich schneiden oder nicht? neimand auf der Erde ohne ein Axiom (wie Euklid das XI) aufzustellen, beantworten wird; die davon unabhilngige Geometrie ab- zusondern; und eine auf die Ja Antwort, andere auf das Nein so zu bauen, das die Formeln der letzten, auf ein Wink auch in der ersten g^ltig seven. Nach ein lateinischen Werke von 1829, M. Vasarhely, und eben daselbst gedruckten un- grischen. Maros Vasarhely 1851. 8vo. pp. 88. In this book he says, referring to his son's Appendix: "Some copies of the work pub- lished here were sent at that time to Vienna, to Berlin, to Goettingen. . . . From Goet- tingen the giant of mathematics, who from TRANSLATOR'S INTRODUCTION, xxiii pinnacle embraces in the same view the and the abysses, wrote that he was sur- r ised to see accomplished what he had be- gun, only to leave it behind in his papers." This refers to 1832. The only other record that Gauss ever mentioned the book is a letter from Gerling, written October 31st, 1851, to Wolfgang Boylai, on receipt of a copy of "Kurzer Grundriss." Gerling, a scholar of Gauss, had been from 1817 Professor of As- tronomy at Marburg. He writes : 4 ' I do not mention my earlier occupation with the theory of parallels, for already in the year 1810-1812 with Gauss, as earlier 1809 with J. F. Pfaff I had learned to perceive how all previous at- tempts to prove the Euclidean axiom had mis- carried. I had then also obtained preliminary knowledge of your works, and so, when I first [1820] had to print something of my view thereon, I wrote it exactly as it yet stands to read on page 187 of the latest edition. "We had about this time [1819] here a law professor, Schweikart, who was formerly in Charkov, and had attained to similar ideas, since without help of the Euclidean axiom he developed in its beginnings a geometry which he called Astralgeometry. What he commun- icated to me thereon I sent to Gauss, who xxiv TRANSLATOR'S INTRODUCTION. then informed me how much farther alre had been attained on this way, and later expressed himself about the great acquisitio which is offered to the few expert judges ii, the Appendix to your book." The ' ' latest edition ' ' mentioned appeared in 1851, and the passage referred to is: "This proof [of the parallel-axiom] has been sought in manifold ways by acute mathematicians, but yet until now not found with complete sufficiency. So long as it fails, the theorem, as all founded on it, remains a hypothesis, whose validity for our life indeed is suffici- ently proven by experience, whose general, necessary exactness, however, .could be doubted without absurdity." Alas! that this feeble utterance should have seemed sufficient for more than thirty years to the associate of Gauss and Schweikart, the latter certainly one of the independent discov- erers of the non-Euclidean geometry. But then, since neither of these sufficiently real- ized the transcendent importance of the mat- ter to publish any of their thoughts on the subject, a more adequate conception of the issues at stake could scarcely be expected of the scholar and colleague. How different with Bolyai Janos and Lobachevski, who claimed TRANSLATOR'S INTRODUCTION. xxv at once, unflinchingly, that their discovery marked an epoch in human thought so momen- tous as to be unsurpassed by anything re- corded in the history of philosophy or of science, demonstrating as had never been proven before the supremacy of pure reason at the very moment of overthrowing what had forever seemed its surest possession, the axioms of geometry. On the 9th of March, 1832, Bolyai Farkas was made corresponding member in the math- ematics section of the Magyar Academy. As professor he exercised a powerful in- fluence in his country. In his private life he was a type of true originality. He wore roomy black Hungarian pants, a white flannel jacket, high boots, and a broad hat like an old-time planter's. The smoke-stained wall of his antique domicile was adorned by pictures of his friend Gauss, of Schiller, and of Shakespeare, whom he loved to call the child of nature. His violin was his constant solace. He died November 20th, 1856. It was his wish that his grave should bear no mark. The mother of Bolyai Janos, nee Arkosi Benka Zsuzsanna, was beautiful, fascinating, xxvi TRANSLATOR'S INTRODUCTION. of extraordinary mental capacity, but always nervous. Janos, a lively, spirited boy, was taught mathematics by his father. His progress was marvelous. He required no explanation of theorems propounded, and made his own dem- onstrations for them, always wishing his father to go on. "Like a demon, he always pushed me on to tell him more." At 12, having passed the six classes of the Latin school, he entered the philosophic-cur- riculum, which he passed in two years with great distinction. When about 13, his father, prevented from meeting his classes, sent his son in his stead. The students said they liked the lectures of the son better than those of the father. He already played exceedingly well on the violin. In his fifteenth year he went to Vienna to K. K. Ingenieur-Akademie. In August, 1823, he was appointed "sous- lieutenant" and sent to Temesvar, where he was to present himself on the 2nd of Sep- tember. From Temesvar, on November 3rd, 1823, Janos wrote to his father a letter in Magyar, of which a French translation was sent me by Professor Koncz Jozsef on February 14th, TRANSLATOR'S INTRODUCTION, xxvii 1895. This will be given in full in my life of Bolyai; but here an extract will suffice: "My Dear and Good Father: "I have so much to write about my new inventions that it is impossible for the mo- ment to enter into great details, so I write you only on one-fourth of a sheet. I await your answer to my letter of two sheets; and perhaps I w^ould not have written you before receiving it, if 1 had not wished to address to you the letter I am writing to the Baroness, which letter I pray you to send her. * * First of all I reply to you in regard to the binominal. *## # * #### "Now to something else, so far as space permits. I intend to write, as soon as I have put it into order, and when possible to pub- lish, a work on parallels. "At this moment it is not yet finished, but the way which I have followed promises me with certainty the attainment of the goal, if it in general is attainable. It is not yet attained, but I have discovered such magnifi- cent things that I am myself astonished at them. "It would be damage eternal if they were xxviii TRANSLATOR'S INTRODUCTION. lost. When you see them, my father, you yourself will acknowledge it. Now I can not say more, only so much: that from nothing I have created another wholly new world. All that I have hitherto sent you compares to this only as a house of cards to a castle. "P. S. I dare to judge absolutely and with conviction of these works of my spirit before you, my father; I do not fear from you any false interpretation (that certainly I would not merit), which signifies that, in certain regards, I consider you as a second self." From the Bolyai MSS., now the property of the College at Maros-Vasarhely, Fr. Schmidt has extracted the following statement by Janos : "First in the year 1823 have I pierced through the problem in its essence, though also afterwards completions yet were added. ' * I communicated in the year 1825 to my former teacher, Herr Johann Walter von Eck- wehr (later k. k. General) [in the Austrian Army], a written treatise, which is still in his hands. 4 ' On the prompting of my father I trans- lated my treatise into the Latin language, and TRANSLATOR'S INTRODUCTION, xxix it appeared as Appendix to the Tentamen, 1832." The profound mathematical ability of Bol- yai Janos showed itself physically not only in his handling of the violin, where he was a master, but also of arms, where he was unap- proachable. It was this skill, combined with his haughty temper, which caused his being retired as Cap- tain on June 16th, 1833, though it saved him from the fate of a kindred spirit, the lamented Galois, killed in a duel when only 19. Bolyai, when in garrison with cavalry officers, was provoked by thirteen of them and accepted all their challenges on condition that he be per- mitted after each duel to play a bit on his violin. He came out victor from his thirteen duels, leaving his thirteen adversaries on the square. He projected a universal language for speech as we have it for music and for mathe- matics. He left parts of a book entitled: Principia doctrinae novae quantitatum imaginariarum perfectae uniceque satisfacientis, aliaeque dis- quisitiones analyticae et analytico - geome- tricae cardinales gravissimaeque; auctore TRANSLATOR'S INTRODUCTION. Johan. Bolyai de eadem, C. R. austriaco cas- trensium captaneo pensionato. Vindobonae vel Maros Vasarhelyini, 1853. Bolyai Farkas was a student at Goettingen from 1796 to 1799. In 1799 he returned to Kolozsvar, where Bolyai Janos was born December 18th, 1802. He died January 27th, 1860, four years after his father. In 1894 a monumental stone was erected on his long-neglected grave in Maros-Vasarhely by the Hungarian Mathematico-Physical So- ciety. APPENDIX. SCIENTIAM SPATII absolute veram exhibens: veritate aut falsitate Axiomatis XI Euclidei (a priori haud unquam decidendd] in- dependentem: adjecta ad casum fal- sitatis, quadratura circuli geometrica. Vuctore JOHANNE BOLYAI de eadem, Geometrarum in Exercitu Caesareo Regio Austriaco Castrensium Capitaneo. EXPLANATION OF SIGNS. The straight AB means the aggregate of all points situated in the same straight line with A and B. The sect AB means that piece of the straight AB between the points A and B. ^ The ray AB means that half of the straight AB which com- mences at the point A and contains the point B. The plane ABC means the aggregate of all points situated in the same plane as the three points (not in a straight) A, B, C. The hemi-plane ABC means that half of the plane ABC which starts from the straight AB and contains the point C. ABC means the smaller of the pieces into which the plane ABC is parted by the rays BA, BC, or the non-reflex angle of which the sides are the rays BA, BC. ABCD (the point D being situated within /ABC, and the straights BA, CD not intersecting) means the portion of / ABC comprised between ray BA, sect BC, raj' CD; while BACD designates the portion of the plane ABC comprised between the straights AB and CD. J_ is the sign of perpendicularity. || is the sign of parallelism. / means angle. rt. / is right angle. st. / is straight angle. ^ is the sign of congruence, indicating tnat two magni- tudes are superposable. AB-CD means /CAB=/ACD. x=a means x converges toward the limit a. A is triangle. Qr means the [circumference of the] circle of radius r. area Qr means the area of the surface of the circle of radius r. THE SCIENCE ABSOLUTE OF SPACE. the ray AM is not cut by the ray [3] BN, situated in the same plane, but is cut by every ray BP comprised in the angle ABN, we will call ray BN parallel to ray AM; this is designated by BN II AM. It is evident that there is one such ray BN , and only one, pass- ing through any point B (taken out- side of the straight AM), and that the sum of the angles BAM, ABN exceed a st. Z ; for in moving BC around B until BAM+ABC=st. /, somewhere ray BC first does not cut ray AM, and it is then BCNAM. It is clear that BN II EM, wherever the point E be taken on the straight AM (supposing in all such cases AM>AE). If while the point C goes away to infinity on ray AM, always CD=CB, we will have con- stantly CDB=(CBD< NBC); butNBC=0; and so also ADB=0. FIG. 1. can not SCIENCE ABSOLUTE OF SPACE. FIG. 2. 2. If BN II AM, we will have also CN II AM. For take D anywhere in MACN. If C is on ray BN, ray BD cuts ray AM, since BN II AM, and so also ray CD cuts ray AM. But if C is on ray BP, take BQ II CD; BQ falls within the Z ABN (1), and cuts ray AM; and so also ray CD cuts ray AM. Therefore every ray CD (in ACN) cuts, in p each case, the ray AM, without CN itself cutting ray AM. Therefore always CN II AM. 3. (Fig. 2.) If BR and CS and each II AM, and C is not on the ray BR, then ray BR and ray CS do not intersect. For if ray BR and ray CS had a common point D, then ( 2) DR and DS would be each II AM, and ray DS ( 1) would fall on ray DR, and C on the ray BR (contrary to the hypothesis). 4. If MAN>MAB, we will have for every point B of ray AB, a point p C of ray AM, such that BCM=NAM. For (by 1) is granted N BDM>NAM, and so that FIG. 3. MDP-MAN, and B falls in SCIENCE ABSOLUTE OP SPACE. 7 NADP. If therefore NAM is carried along AM until ray AN arrives on ray DP, ray AN will somewhere have necessarily passed through B, and some BCM=NAM. 5. If BN II AM, there is on the straight w iN AM a point F such that FM^BN. For by 1 is granted BCM>CBN; and if CE-CB, and so EC^BC; evidently BEMBEM and . Therefore in revolving the hemi-plane BCD around BC until it begins to leave the ray AM, the hemi-plane BCD at last will fall upon the hemi- plane BCN. For the same reason this same will fall upon hemi-plane BCP. FIG. 7. Therefore BN falls in BCP. More- over, if BR II CP; then (because also AM II CP) by like reasoning, BR falls in BAM, and also (since BRIICP> in BCP. Therefore the straight BR, being common to the two planes MAB, PCB, of course is the straight BN, and hence BN II CP.* If. therefore CP II AM, and B exterior to the plane CAM; then the intersection BN of the planes BAM, BCP is II as well to AM as to CP. 8. If BN II and === CP (or more briefly BN II =*CP>, and AM (inNBCP) bisects J_ the sect BC; then BN II AM. For if ray BN cut ray AM, also ray CP would cut ray AM at the same point (because MABN= c MACP) , and this would be common to the rays BN, CP themselves, al- * The third case being put before the other two, these can be demonstrated together with more brevity and elegance, like ease 2 of 10. [Author's note.] io SCIENCE ABSOLUTE OF SPACE. though BN II CP. But every ray BQ (in CBN) cuts ray CP; and so ray BQ cuts also ray AM. Consequently BN II AN. . 9. If BN II AM, and MAPlMAB, and the Z, which NBD makes with NBA (on that side of MABN, where MAP is) is (st.Z-Z RFM=ZFGN), since RSllFM. But as RSl'l GN, also ZFRS is not <-ZFGN; and so ZFRS and ZRFM+ZFRS=ZGFM+Z 14 SCIENCE ABSOLUTE OF SPACE. FGN=st.Z. Therefore also ZDCP+ZCDQ =st.Z. II. If R falls without the sect FG; then ZNGR=ZMFR, and let MFGN^NGHL= LHKO, and so on, until FK>FR or begins to be >FR. Then KO II HL II FM (7). If K falls on R, then KO falls on RS (1); and so ZRFM+ZFRS=ZKFM+ZFKO=Z KFM+ZFGN=st.Z; but if R falls within the sect HK, then (by I) ZRHL+^KRS=st.Z = ZRFM+ZFRS=ZDCP+ZCDQ. 14. If BN II AM, and CP II DQ, and ZBAM +ZABNv (23), Y=X X . For, one of the quantities x, y is a multiple of the the other (e. g. y of x), or it is not. If^=n.^, take*=AC = CG=GH=&c., until we get AH=jF. Moreover, take CD ||| GK ||| HL. We have ((23) X=AB:CD = CD:GK=GK: HL; and so AB = f AB y HL~ [ CD or Y=X n =X x . If x } y are multiples of /, suppose and y^ni; (by the preceding) X=I m , Y=P, consequently n _y Y-X m =X x 20 SCIENCE ABSOLUTE OF SPACE. The same is easily extended to the case of the incommensurability of x and y. But if q=y-x, manifestly Q=Y'X. It is also manifest that in J, for any x, we have X=l, but in S is X>1, and for any AB [ and ABE there is such a CDF ||| AB, that CDF -AB, whence AMBN^AMEP, though the first be any multiple of the second; which in- deed is singular, but evidently does not prove the absurdity of S. 25. In any rectilineal triangle, the cir- cles with radii equal to its sides are as the sines of the opposite angles. ^ For take ZABC=rt.Z, and AMlBAC, and BN and CP II AM; we shall have CAB 1 AMBN, and so (since CB_|_ BA), CBlAMBN, conse- quently CPBNl AMBN. Suppose the F of ray CP FIG. 17. cuts the straights BN, AM respectively in D and E, and the bands CPBN, CPAM, BNAM along the L form lines CD, CE, DE. Then (20) ZCDE^the angle of NDC, NDE, and so = rt.Z; and by like reason- ing Z CED = Z CAB. But (by 21) in the L line A CDE (supposing always here the radius =1), EC:D=l:sin DEC =1: sin CAB. SCIENCE ABSOLUTE OF SPACE. 21 Also (by 21) EC:DC=OEC:ODC(inF)=OAC:OBC (18); and so is also 0AC:OBC = l:sin CAB; whence the theorem is evident for any triangle. 26. In any spherical triangle, the sines of the sides are as the sines of the angles opposite. For take ZABC=rt.Z, and CED 1 to the radius OA of the sphere. We shall have CED \_ AOB, and (since also BOC \_ BOA), CDlOB. But in the no. is. triangles CEO, CDO (by 25) OEC:OOC:ODC=sin COE : 1 sin COD=sin AC : 1 : sin BC; meanwhile also ( 25) OEC : ODC=sin CDE : sin CED. Therefore, sin AC : sin BC=sin CDE : sin CED; but CDE= rt.Z = CBA, and CED = CAB. Consequently sin AC : sin BC=1 : sin A. Spherical trigonometry , flowing from this, is thus established independently of Axiom AY. 27. If AC and BD are J_ AB, and CAB is carried along the straight AB; we shall have, designating by CD the path of the point C, CD : AB = sin u : sin v. 22 SCIENCE ABSOLUTE OF SPACE. For take DElCA; in the triangles ADE, ADB (by 25) OED : OAD : OAB= sin u : 1 : sin v. G F FIG. 19. In revolving BACD about AC, B describes OAB, and D describes OED; and designate here by sOCD the path of the said CD. Moreover, let there be any [123 polygon BFG. . . inscribed in OAB. Passing through all the sides BF, FG, &c., planes J_ to OAB we form also a polygonal fig- ure of the same number of sides in sOCD, and we may demonstrate, as in 23, that CD : AB =DH : BF=HK : FG, &c., and so DH+HK &c. : BF+FG &c. : =CD : AB. If each of the sides BF, FG . . . approaches the limit zero, manifestly BF+FG+ . . . =0 AB and DH+HK+...=OED. Therefore also OED : OAB=CD : AB. But we had OED : OAB=sin u : sin v. Conse- quently CD : AB=sin u : sin v. If AC goes away from BD to infinity, CD : AB, and so also sin u : sin v remains constant; but ^=rt. Z (1), and if DM II BN, v=z; whence CD : AB=1 : sin z. SCIENCE ABSOLUTE OF SPACE. 23 The path called CD will be denoted by CD illAB. 28. If BN II ^ AM, and C in ray AM, and AC=x: we shall have ( 23) X=sin u : sin v. For if CD and AE are _L BN, *<\F / f^ and BF_L AM; we shall have (as in 27) OBF : ODC^sin u : sin v. ButevidentlyBF=AE: therefore OEA : OCD=sin u : sin v. But in the F form surfaces of AM and CM (cutting AMBN in AB and CG) (by 21) OEA : ODC=AB : CG=X. Therefore also M FIG. 20. 29. X=sin u If ZBAM=rt. sin v. and sect ABjy, and BNIIAM, we shall have in S Ycotan ^ u. For, if sect AB- sect AC, and CPU AM (and so BNll ^ CP), and ZPCD = ZQCD; there is given (19) DS_j_ray CD, so that DS II CP, and so ( 1) DT II CQ. Moreover, if BE 1 ray DS, then ( 7) DS II BN, and so ( 6) FiG. 21. 24 SCIENCE ABSOLUTE OF SPACE. BNnES, and (since DT II CG) BQllET; con- sequently (1) ZEBN=ZEBQ. Let BCF be an 1,-line of BN, and FG, DH, CK, EL, L form lines of FT, DT, CQ and ET; evidently (22) HG=DF=DK-HC; therefore, CG=2CH=2z>. Likewise it is evident BG=2BL=2^. ButBC=BG-CG; wherefore y=z~v, and so (24) Y=Z:V. Finally ( 28) Z = l : sin y 2 u, and V=l : sin (rt.^~X ^)> consequently Y=cotan ^ u. 30. However, it is easy to see (by 25) that the solution of the problem of Plane Trigonometry, in S, requires the expression of the circle in terms of the radius; but this can by obtained by the rectification of L. Let AB, CM, C'M' be _L ray AC, and B anywhere in ray AB; we shall have ( 25) sin u : sin v=Qp : Qy, and sin u' : sin v 1 ' =Qp' : FIG. 22. and so sm v sin v SCIENCE ABSOLUTE OP SPACE. 25 But (by 27) sin v : sin v' =cos u : cos u ; ,, sin u _ sin ' consequently- .Qy= COS U COS U or OjF : Oy' =tan u' : tan u=tan w : tan w' . Moreover, take CN and C'N' II AB, and CD, C'D' L-form lines j_ straight AB; we shall have also (21) Qy : Qy' =r : r' , and so r : r =tan w : tan w' . Now let p beginning from A increase to in- finity; then w=z, and w '=z ' , whence also r : r tan z : tan z ' . Designate by i the constant r : tan z (independent of r) ; whilst y=0, r i tan z , ^ -. -=- =1, and so y y -2^i. From 29, tan z=y 2 (Y-Y- 1 ); tan z therefore -^_^=i^ or (24) ^_L.i. ~\ i~ But we know the limit of this expression (where y=0) is /I -. Therefore nat. log I 26 SCIENCE ABSOLUTE OF SPACE. - T =*, and nat. log I 1=0=2.7182818..., which noted quantity shines forth here also. If obviously henceforth i denote that sect of which the 1=0, we shall have ri tan z. But ( 21) Gy=2-r/ therefore =2* tan z=i T-Y-' = * (by 24). nat. log Y 31. For the trigonometric solution of all right-angled rectilineal triangles (whence the resolution of all triangles is easy , in S, three LI equations suffice : indeed (a, b denoting the sides, c the hypothenuse, and , ,3 the angles opposite the sides) an equation expressing the relation 1st, between a, c, a; 2d, between a, , ; 3d, between a, b, c; of course from these equations emerge three others by elim- ination. FIG. 23. From 25 and 30 1 : sin a=(C-C~ l ) : (A-A- ] ) = f_^r j (equation for c, a and ). SCIENCE ABSOLUTE OF SPACE. 27 II. From 27 follows (if ,?M II rN) cos : sin ,3=1 : sin 2/y but from 29 1 :sin ^ therefore cos a sin ^ (equation for , /? and an d A 9 ' an d rr'll aa/ ( 27), and ,3'a>' J_aa'; manifestly (as in 27) sin , or * ~-5 if 5. ~^ (equation for #, <5 and c). If rfl=rt.^, and /35_L^; O<^ ' O<^=1 * sin , and Qc : Q(d=pd\= 1 : cos a, and so (denoting by O^ 2 ? for any x, the product manifestly. But (by 27 and II) Qd=Q6.^(A+A~ 1 ), consequently fL -_?} ~_ I / f 5L -^1 2 f b -b^ ~ f a_ -a^ 2 another equation for #, <5 and c (the second 28 SCIENCE ABSOLUTE OF SPACE. member of which may be easily reduced to a form symmetric or invariable). [15 Finally, from -^(A+A" 1 ), and ~ -^(B+B" 1 ), we get sin p sin a (by III) COt a COt fi=% (equation for , /5, and c. 32. It still remains to show briefly the mode of resolving problems in S, which being accomplished (through the more obvious exam- ples), finally will be candidly said what this theory shows. I. Take AB a line in a plane, and y=f(x] its equation in rectangular co- ordinates, call dz any increment of z, and respectively dx, dy, du the increments of x, of y, and of )A the area u, corresponding to FIG. 24. BH this dz; take BH III CF, and ex- press (from 31) -^ by means of y , and seek ax* the limit of - when dx tends towards the dx limit zero (which is understood where a limit of this sort is sought) : then will become known also the limit of ^., and so tan HBG; and SCIENCE ABSOLUTE OF SPACE. 29 (since HBC manifestly is neither > nor <, and so =rt. Z), the tangent at B of BG will be de- termined by y. II. It can be demonstrated ~ Hence is found the limit of -, and thence, ax by integration, z (expressed in terms of x. And of any line given in the ^concrete, the equation in S can be found; e. g., of L. For if ray AM be the axis of L; then any ray CB from ray AM cuts L [since (by 19) any straight from A except the straight AM will cut L] ; but (if BN is axis) X=l:sin CBN (28), and Y-cotan y> CBN (29), whence or y the equation sought. Hence we get ax and =1 : sin CBN-X; and so ax dy (X 3 -!)^- BH~ l 30 SCIENCE ABSOLUTE OP SPACE. ^=X(X 2 -l), and =X 2 (X 2 -1), whence, by inte- ctx gration, we get (as in 30) ^=^(X 8 -l)*=*cot CBN. III. Manfestly dx dx which (unless given in y) now first is to be ex- pressed in terms of y; whence we get u by integrating. D If AB=/, AC=?, CD=r, and CABDC=s/ we might show (as in II) that -= =r, which = FIG. 25. aq and, integrating, s=%pi ^f_ ~f This can also be deduced apart from inte- gration. For example, the equation of the circle (from 31, III), of the straight (from 31, II), of a conic (by what precedes), being expressed, the SCIENCE ABSOLUTE OF SPACE. 31 areas bounded by these lines could also be ex- pressed. We know, that a surface t, \\\ to a plane fig- ure/* (at the distance q], is to/> in the ratio of the second powers of homologous lines, or as f 4 -\ 2 I^ |^i_?i J : 1. It is easy to see, moreover, that the calcula- tion of volume, treated in the same manner, requires two integrations (since the differen- tial itself here is determined only by integra- tion) ; and before all must be investigated the [in volume contained between p and /, and the ag- gregate of all the straights A-p and joining the boundaries of p and t. We find for the volume of this solid (whether by integration or without it) f 2q ^ The surfaces of bodies may also be deter- mined in S, as well as the curvatures, the involutes, and evolutes of any lines, etc. As to curvature; this in S either is the curv- ature of L, or is determined either by the radius of a circle, or by the distance to a straight from the curve ||| to this straight; since from what precedes, it may easily be shown, that in a plane there are no uniform lines other than L-lines, circles and curves ||| to a straight. 32 SCIENCE ABSOLUTE OF SPACE. IV. For the circle (as in III) - area G , whence (by 29), integrating, dx V. For the area CABDC=^ (inclosed by an M N L, form line AB=r, the III to this, CV=j>, and the sects AC=BD=^) =y; and ( 24) y re^, and so D * (integrating) ^=r/ M_^T If x increases to infinity, then, in FIG. 26. -2* S, ^i=^0, and so u=ri. By the size of MABN, in future this limit is understood. In like manner is found, if p is a figure on F, the space included by^> and the aggregate of axes drawn from the boundaries of p is equal to Y*pi. VI. If the angle at the cen- ter of a segment z of a sphere is 2u, and a great circle is^>, and x the arc FC (of the angle ); (25) l:sin ^=/:QBC, and hence OBC=/> sin u. Meanwhile *=^, and dx^^- 2^ 2* FIG. 27. SCIENCE ABSOLUTE OF SPACE. 33 Moreover, -=OBC, and hence ax . j /2 -j-==- sin u, whence (integrating) au 2- _ver sin u , 2 ~2* The F may be conceived on which P falls (passing through the middle F of the seg- ment) ; through AF and AC the planes FEM, CEM are placed, perpendicular to F and cut- ting F along FEG and CE; and consider the L form CD (f rom C 1 to FEG), and the L form CF; (20) CEF=^, and (21) gg= ver sin * and so ^=FD./. P 47T But (21)/=^.FGD; therefore ^=r:.FD.FDG. But (21) FD.FDG=FC.FC; consequently ^=7r.FC.FC=areaoFC, in F. Now let BJ=CJ=r/ (30) 2r=*(Y Y- 1 ), and so (21) area O2r (in F) =^ 2 (Y- Y- 1 ; 2 . Also (IV) area therefore, area O2^ (in F) =area Q2y 1 and so the surface z of a segment of a sphere is equal to the surface of the circle described with the chord FC as a radius. 34 SCIENCE ABSOLUTE OF SPACE. Hence the whole surface of the sphere and the surfaces of spheres are to each other as the second powers of their great circles. VII. In like manner, in S, the volume of the sphere of radius x is found the surface generated by the rev- olution of the line CD about AB A T~ B and the body described by CABDC FIG. 29. = i/_v2^/r) Q~M 2 But in what manner all things treated from (IV] even to here, also may be reached apart from integration, for the sake of brev- ity is suppressed. It can be demonstrated that the limit of every expression containing the letter i (and so resting upon the hypothesis that i is given), [19] when i increases to infinity, expresses the quantity simply for I (and so for the hypoth- esis of no i), if indeed the equations do not be- come identical. But beware lest you understand to be sup- posed, that the system itself may be varied (for it is entirely determined in itself and by itself) ; but only the hypothesis, which may be SCIENCE ABSOLUTE OF SPACE. 35 done successively, as long as we are not con- ducted to an absurdity. Supposing therefore that, in such an expression, the letter i, in case S is reality, designates that unique quan- tity whose \~e; but if r is actual, the said limit is supposed to be taken in place of the expression : manifestly all the expressions or- iginating from the hypothesis of the reality ofS (in this sense] will be true absolutely, although it be completely unknown whether or not I is reality So e. g. from the expression obtained in 30 easily (and as well by aid of differentiation as apart from it) emerges the known value in J, from I ( 31) suitably treated, follows 1 : sin aC : a; but from II COS sin - ^ = 1, and so the first equation in III becomes identical, and so is true in I, although it there determines nothing; but from the second follows These are the known fundamental equa- tions of plane trigonometry in I. 36 SCIENCE ABSOLUTE OF SPACE. Moreover, we find (from 32) in r, the area and the volume in III each =pq; from IV area O#=-^; (from VII) the globe of radius x =%xx s , etc. The theorems enunciated at the end of VI are manifestly true unconditionally. 33. It still remains to set forth (as prom- ised in 32) what this theory means. I. Whether I or some one S is reality, re- mains undecided. II. All things deduced from the hypothesis of the falsity of Axiom XI (always to be un- derstood in the sense of 32) are absolutely true, and so in this sense, depend upon no hypothesis. There is therefore a plane trigonometry a priori, in which the system alone really re- mains unknown; and so where remain un- known solely the absolute magnitudes in the expressions, but where a single known case would manifestly fix the whole system. But spherical trigonometry is established abso- lutely in 26. (And we have, on F, a geometry^wholly an-" alogous to the plane geometry of J.) III. If it were agreed that - exists, nctthing more would be unknown in this respect; but SCIENCE ABSOLUTE OF SPACE. 37 if it were established that I does not exist, then ( 31), (e. g.) from the sides x, y> and the rectilineal angle they include being given in a special case, manifestly it would be impossible in itself and by itself to solve absolutely the triangle, that is, to determine a priori the other angles and the ratio of the third side to the two given; unless X, Y were determined, for which it would be necessary to have in concrete form a certain sect a whose A was known; and then i would be the natural unit for length (just as e is the base of natural logarithms). If the existence of this i is determined, it will be evident how it could be constructed, at least very exactly, for practical use. IV. In the sense explained (I and II), it is evident that all things in space can be solved by the modern analytic method (within just limits strongly to be prdised). V. Finally, to friendly readers will not be unacceptable; that for that case wherein not I but S is reality, a rectilineal figure is con- structed equivalent to a circle. 34. Through D we may draw DM II AN in the following manner. From D drop DB J_ AN ; from any point A of the straight AB erect AC IAN (in DBA), and let fall DC1AC. We 38 SCIENCE ABSOLUTE OF SPACE. will have ( 27) OCD : OAB = 1 : sin z, pro- M vided that DM II BN. But sin z is not >1; and so AB is not >DC. Therefore a quad- - rant described from the cen- A B S FIG. so. ter A in BAG, with a radius DC, will have a point B or O in common with ray BD. In the first case, manifestly ^=rt.^; but in the second case ( 25) (OAO-OCD) : OAB=1 : sin AOB, and so ^=AOB. If therefore we take ^=AOB, then DM will be II BN. 35. If S were reality; we may, as follows, draw a straight _L to one arm of an acute angle, [211 which is II to the other. Take AMI BC, and suppose AB=BC so small (by 19), that if W e draw BN II AM ( 34), ABN > the FIG. 31. . . given angle. Moreover draw CP II AM (34); and take NBG and PCD each equal to the given angle; rays BG and CD will cut; for if ray BG (fall- ing by construction within NBC) cuts ray CP in E; we shall have (since BN^CP), ZEBC< ZECB, and so ECAABD (con- tra hyp.). 41. Equivalent triangles ABC, DEF have the sums of their triangles equal. ,L H F -^ or ^ et MN bisect M ^rV- s ^TTT" AC and BC > and P Q 4-/1Q bisect DF and FE; \l \ and take RS III MN, ~s- E and TO III PQ ; the per- pendicular AG to RS will equal the perpendicular DH to TO, or one for example DH will be the greater. In each case, the ODF, from center A, has with line-ray GS some point K in common, and (39) AABK=AABC=ADEF. But the A AKB (by 40) has the same angle-sum as ADFE, and (by 39) as AABC. Therefore also the triangles ABC, DEF have each the same angle-sum. In S the inverse of this theorem is true. For take ABC, DEF two triangles having equal angle-sums, and ABAL=ADEF; these will have (by what precedes) equal angle-sums, 44 SCIENCE ABSOLUTE OF SPACE. and so also will AABC and AABL, and hence manifestly BCL+BLC+CBL=st. Z. However (by 31), the angle-sum of any tri- \M\ angle, in S, is r in F ( 21) = area s (by 32, VI) if the chord CD is called s. If now, bisecting at right angles the given radius 5 of the circle in a plane (or the Iy form radius of the circle in F), we construct (by 34) DB|^CN; by dropping CA 1 DB, and erecting CM 1 CA, we shall 1 get z; whence (by 37), assum- ing at pleasure an L form radius for unity, tan 2 ^ can be determined geometrically by means of two uniform lines A_ of the same curvature (which, 3 their extremities alone being given and their axes con- FIG. 41. SCIENCE ABSOLUTE OF SPACE. 47 structed, manifestly may be compared like straights, and in this respect considered equiv- alent to straights) . Moreover, a quadrilateral, ex. gr. regular = n is constructed as follows: Take ABC=rt.Z, BAC=i rt. Z, ACB= rt. Z, and BC=*. By mere square roots, X (from 31, II) can be expressed and (by 37) constructed; and having X (by 38 or also 29 and 35), x itself can be determined. And octuple A ABC is manifestly = n , and by this a plane circle of radius s is geometrically squared by means of a recti- linear figure and uniform lines of the same species (equivalent to straights as to compari- son inter se) ; but an F form circle is plani- fied in the same manner: and we have either the Axiom XI of Euclid true or the geomet- ric quadrature of the circle, although thus far it has remained undecided, which of these ' two has place in reality. Whenever tan 2 ^ is either a whole number, or a rational fraction, whose denominator (re- duced to the simplest form) is either a prime number of the form 2 m +l (of which is also 2=2+l), or a product of however many prime numbers of this form, of which each (with the 48 SCIENCE ABSOLUTE OP SPACE. exception of 2, which alone may occur any number of times) occurs only once as factor, we can, by the theory of polygons of the illus- trious Gauss (remarkable invention of our, nay of every age) (and only for such values of z) , construct a rectilineal figure =tan 2 ^n = area Q5. For the division of n (the theorem of 42 extending easily to any polygons) mani- festly requires the partition of a st. Z, which (as can be shown) can be achieved geomet- rically only under the said condition. But in all such cases, what precedes con- ducts easily to the desired end. And any rec- tilineal figure can be converted geometrically into a regular polygon of n sides, if n falls under the Gaussian form. It remains, finally (that the thing may be completed in every respect), to demonstrate the impossibility (apart from any supposition), of deciding a priori, whether , or some S (and which one) exists. This, however, is re- served for a more suitable occasion. APPENDIX I. REMARKS ON THE PRECEDING TREATISE, BY BOLYAI PARKAS. [From Vol. II of Tentamen, pp. 380-383.] Finally it may be permitted to add something appertaining to the author of the Appendix in the first volume, who, however, may pardon me if something I have not touched with his acute- ness. The thing consists briefly in this: the form- ulas of spherical trigonometry (demonstrated in the said Appendix independently of Euclid's Axiom XI) coincide with the formulas of plane trigonometry, if (in a way provisionally speak- ing) the sides of a spherical triangle are ac- cepted as reals, but of a rectilineal triangle as imaginaries; so that, as to trigonometric formulas, the plane may be considered as an imaginary sphere, if for real, that is accepted in which sin rt. Z. \. Doubtless, of the Euclidean axiom has been said in volume first enough and to spare: for 50 SCIENCE ABSOLUTE OP SPACE. the case if it were not true, is demonstrated (Tom. I. App., p. 13), that there is given a cer- tain /, for which the I there mentioned is =0 (the base of natural logarithms), and for this case are established also (ibidem, p. 14) the formulas of plane trigonometry, and indeed so, that (by the side of p. 19, ibidem) the formulas are still valid for the case of the verity of the said axiom; indeed if the limits of the values are taken, supposing that 2=^=00; truly the Euclidean system is as if the limit of the anti- Euclidean (for /=oo). Assume for the case of i existing, the unit = ij and extend the concepts sine and cosine also to imaginary arcs, so that, p designating an arc whether real or imaginary, !_ -^ _ is called the 2 cosine of p, and _ ~ e _ is called the sine of p (as Tom. L, p. 177). Hence for q real Q -q e e e 1). SCIENCE ABSOLUTE OP SPACE. 51 q _q q s Zl. N IIi Q N Z4.\^i e -f e e +e ~~ ~~ =cos( ?v if of course also in the imaginary circle, the sine of a negative arc is the same as the sine of a positive arc otherwise equal to the first, except that it is negative, and the cosine of a positive arc and of a negative (if otherwise they be equal) the same. In the said Appendix, 25, is demonstrated absolutely, that is, independently of the said axiom; that, in any rectilineal triangle the sines of the circles are as the circles of radii equal to the sides opposite. Moreover is demonstrated for the case of i existing, that the circle of radius y is = ~i i^__^ j ' which, for /=!, becomes *(*_*-*). Therefore (31 ibidem], for a right-angled rectilineal triangle of which the sides are a and b, the hypothenuse c, and the angles oppo- site to the sides a, b, c are , ,9, rt. Z,(for /=!), in I, lisin a=*(^-^):*(^-O; and so c __0- c e*e~* TT ri 1 l:sin = - -- : -- Whence 1 : sin 2v_i ' 2v_i ' 52 SCIENCE ABSOLUTE OF SPACE. = sin cV3): sin '~) have ?/=0, while /^=oc. For all the terms which follow are divided by i*; the first term will be 2 ; and any ratio <-^; and though the ratio everywhere should remain this, the sum would be ^Tom. I., p. 131), /v I -i A/ C which manifestly =?=0, while /= And from 54 SCIENCE ABSOLUTE OP SPACE. f ^ (a+b) -(a+b) a-b -(a-b) ") \ + e i + e i + e i J follows (for w, z;, A taken like ^) And hence C 2 =- which =a 2 +l> 2 . APPENDIX II. SOME POINTS IN JOHN BOLYAl's APPENDIX COMPARED WITH LOBACHEVSKI, BY WOLFGANG BOLYAI. [From Kurzer Grundriss, p. 82.] Lobachevski and the author of the Appendix each consider two points A, B, of the sphere- limit, and the corresponding axes ray AM, ray BN ( 23). They demonstrate that, if , ,?, r designate the arcs of the circle limit AB, CD, HL, separated by r segments of the axis AC=1, AH x, we have Mi).' Lobachevski represents the value of - by 0~ x , e having some value >1, dependent on the unit for length that we have chosen, and able to be supposed equal to the Naperian base. The author of the Appendix is led directly to introduce the base of natural logarithms. 56 SCIENCE ABSOLUTE OP SPACE. If we put ^=d, and r , r r are arcs situated at the distances y, i from , we shall have -=a y =Y, -,=^=1, whence Y=IT He demonstrates afterward ( 29) that, if u is the angle which a straight makes with the perpendicular y to its parallel, we have Y=cot \u. Therefore, if we put z=-~u, we have Y = tan 1 tan z tan j whence we get, having regard to the value of tan J=Y~ 1 , tan z^ (Y-Y^NiJl'-J '] (30). If now jv is the semi-chord of the arc of /^ circle-limit 2^, we prove ( 30) that - tan z constant. Representing this constant by i, and making y tend toward zero, we have =1, whence 2r 2y T ' _1 2=2 * tan z=i - - i SCIENCE ABSOLUTE OP SPACE. 57 or putting -^-=k, \el, / being- infinitesimal at the same time as k. Therefore, for the limit, 1 = 1 and consequently I=& The circle traced on the sphere-limit with the arc r of the curve-limit for radius, has for length 2-r. Therefore, OjK=2rr=2-* tan z=*i (Y Y" 1 ). In the rectilineal A where a, p designate the angles opposite the sides a, b, we have ( 25) sin a:sin ,i=Qa:Qt>=-i(AA- 1 ): -i(B B" 1 ) =sin (^^l) :sin (fc I). Thus in plane trigonometry as in spherical trigonometry, the sines of the angles are to each other as the sines of the opposite sides, only that on the sphere the sides are reals, and in the plane we must consider them as imaginaries, just as if the plane were an imaginary sphere. We may arrive at this proposition without a preceding determination of the value of I. 0* If we designate the constant by q, we tan z shall have, as before =*q (Y-Y- 1 ), 58 SCIENCE ABSOLUTE OF SPACE. whence we deduce the same proportion as above, taking for i the distance for which the ratio I is equal to e. If axiom XI is not true, there exists a de- terminate i, which must be substituted in the formulas. If, on the contrary, this axiom is true, we must make in the formulas i= oo. Because, in this case, the quantity -=Y is always =1, the sphere-limit being a plane, and the axes being parallel in Euclid's sense. The exponent \ must therefore be zero, and consequently i= GO. It is easy to see that Bolyai's formulas of plane trigonometry are in accord with those of Lobachevski. Take for example the formula of 37, tan // (#)=sin B tan //(/), a being the hypothenuse of a right-angled tri- angle, p one side of the right angle, and B the angle opposite to this side. Bolyai's formula of 31, I, gives 1 : sin B=(A-A- 1 ):(P-P- ] ). Now, putting for brevity, i/7 (Jc)=k' , we have tan 2p ' : tan 2a ' (cot a ' tan a ' } : (cot p ' / = A-A- 1 P-P- 1 ^! : sin B. APPENDIX III. LIGHT FROM NON-EUCLIDEAN SPACES ON THE TEACHING OF ELEMENTARY GEOMETRY. BY G. B. HALSTED. As foreshadowed by Bolyai and Riemann, founded by Cayley, extended and interpreted for hyperbolic, parabolic, elliptic spaces by Klein, recast and applied to mechanics by Sir Robert Ball, projective metrics may be looked upon as characteristic of what is highest and most peculiarly modern in all the bewildering range of mathematical achievement. Mathematicians hold that number is wholly a creation of the human intellect, while on the contrary our space has an empirical element. Of possible geometries we can not say a priori which shall be that of our actual space, the space in which we move. Of course an ad- vance so important, not only for mathemat- ics but for philosophy, has had some m^taphy- sical opponents, and as long ago as 1878 I mentioned in my Bibliography of Hyper- 60 SCIENCE ABSOLUTE OP SPACE. Space and Non-Euclidean Geometry (American Journal of Mathematics, Vol. I, 1878, Vol. II, 1879) one of these, Schmitz-Dumont, as a sad paradoxer, and another, J. C. Becker, both of whom would ere this have shared the oblivion of still more antiquated fighters against the light, but that Dr. Schotten, praiseworthy for the very attempt at a comparative planimetry, happens to be himself a believer in the a priori founding of geometry, while his American re- viewer, Mr. Ziwet, was then also an anti-non- Euclidean, though since converted. He says, ' ' we find that some of the best Ger- man text books do not try at all to define what is space, or what is a point, or even what is a straight line." Do any German geometries de- fine space? I never remember to have met one that does. In experience, what comes first is a bounded surface, with its boundaries, lines, and their boundaries, points. Are the points whose definitions are omitted anything different or better? Dr. Schotten regards the two ideas "direc- tion" and "distance" as intuitively given in the mind and as so simple as to not require definition. When we read of two jockeys speeding SCIENCE ABSOLUTE OF SPACE. 61 around a track in opposite directions, and also on page 87 of Richardson's Euclid, 1891, read, ' ' The sides of the figure must be pro- duced in the same direction of rotation ; . . . going round the figure always in the same direction," we do not wonder that when Mr. Ziwet had written: "he therefore bases the definition of the straight line on these two ideas," he stops, modifies, and rubs that out as follows, "or rather recommends to eluci- date the intuitive idea of the straight line possessed by any well-balanced mind by means of the still simpler ideas of direction" [in a circle] "and distance" [on a curve]. But when we come to geometry as a science, as foundation for work like that of Cayley and Ball, I think with Professor Chrystal: " It is essential to be careful with our definition of a straight line, for it will be found that vir- tually the properties of the straight line de- termine the nature of space. * ' Our definition shall be that two points in general determine a straight line." We presume that Mr. Ziwet glories in that unfortunate expression "a straight line is the shortest distance between two points," still occurring in Wentworth (New Plane Geom- etry, page 33), even after he has said, page 5, 62 SCIENCE ABSOLUTE OF SPACE. "the length of the straight line is called the distance between two points." If the length of the one straight line between two points is the distance between those points, how can the straight line itself be the shortest distance? If there is only one distance, it is the longest as much as the shortest distance, and if it is the length of this shorto-longest distance which is the distance then it is not the straight line itself which is the longo-shortest distance. But Wentworth also says: "Of all lines joining two points the shortest is the straight line." This general comparison involves the meas- urement of curves, which involves the theory of limits, to say nothing of ratio. The very ascription of length to a curve involves the idea of a limit. And then to introduce this general axiom, as does Wentworth, only to prove a very special case of itself, that two sides of a triangle are together greater than the third, is surely bad logic, bad pedagogy, bad mathematics. This latter theorem, according to the first of Pascal's rules for demonstrations, should not be proved at all, since every dog knows it. But to this objection, as old as the sophists, Simson long ago answered for the science of SCIENCE ABSOLUTE OF SPACE. 63 geometry, that the number of assumptions ought not to be increased without necessity ; or as Dedekind has it: " Was beweisbar ist, soil in der Wissenschaft nicht ohne Beuueis geglaubt werden" Professor W. B. Smith (Ph. D., Goettingen), has written: " Nothing could be more unfor- tunate than the attempt to lay the notion of Direction at the bottom of Geometry. ' ' Was it not this notion which led so good a mathematician as John Casey to give as a demonstration of a triangle's angle-sum the procedure called " a practical demonstration " on page 87 of Richardson's Euclid, and there described as * ' laying a * straight edge ' along one of the sides of the figure, and then turn- ing it round so as to coincide with each side in turn." This assumes that a segment of a straight line, a sect, may be translated without rota- tion, which assumption readily comes to view when you try the procedure in two-dimensional spherics. Though this fallacy was exposed by so eminent a geometer as Olaus Henrici in so public a place as the pages of 'Nature,' yet it has just been solemnly reproduced by Pro- fessor G. C. Edwards, of the University of California, in his Elements of Geometry: Mac- 64 SCIENCE ABSOLUTE OF SPACE. Millan, 1895. It is of the greatest importance for every teacher to know and connect the commonest forms of assumption equivalent to Euclid's Axiom XL If in a plane two straight lines perpendicular to a third nowhere meet, are there others, not both perpendicular to any third, which nowhere meet? Euclid's Axiom XI is the assumption No. Playf air's answers no more simply. But the very same answer is given by the common assumption of our geometries, usually unnoticed, that a circle may be passed through any three points not costraight. This equivalence was pointed out by Bolyai Farkas, who looks upon this as the simplest form of the assumption. Other equivalents are, the existence of any finite triangle whose angle-sum is a straight angle; or the existence of a plane rectangle; or that, in triangles, the angle-sum is constant. One of Legendre's forms was that through every point within an angle a straight line may be drawn which cuts both arms. But Legendre never saw through this mat- ter because he had not, as we have, the eyes of Bolyai and Lobachevski to see with. The same lack of their eyes has caused the author of the charming book " Euclid and His Modern SCIENCE ABSOLUTE OF SPACE. 65 Rivals," to give us one more equivalent form: "In any circle, the inscribed equilateral tetra- gon is greater than any one of the segments which lie outside it." (A New Theory of Parallels by C. L. Dodgson, 3d. Ed., 1890.) Any attempt to define a straight line by means of "direction" is simply a case of "ar- gumentum in circulo." In all such attempts the loose word "direction" is used in a sense which presupposes the straight line. The directions from a point in Euclidean space are only the oc 2 rays from that point. Rays not costraight can be said to have the same direction only after a theory of parallels is presupposed, assumed. Three of the exposures of Professor G. C. Edwards' fallacy are here reproduced. The first, already referred to, is from Nature, Vol. XXIX, p. 453, March 13, 1884. "I select for discussion the 'quaternion proof " given by Sir William Hamilton. . . . Hamilton's proof consists in the following: "One side AB of the triangle ABC is turned about the point B till it lies in the continuation of BC; next, the line BC is made to slide along BC till B comes to C, and is then turned about C till it comes to lie in the continuation of AC. 66 SCIENCE ABSOLUTE OF SPACE. * * It is now again made to slide along CA till the point B comes to A, and is turned about A till it lies in the line AB. Hence it follows, since rotation is independent of translation, that the line has performed a whole revolution, that is, it has been turned through four right angles. But it has also described in succession the three exterior angles of the triangle, hence these are together equal to four right angles, and from this follows at once that the interior angles are equal to two right angles. ' ' To show how erroneous this reasoning is in spite of Sir William Hamilton and in spite of quaternions I need only point out that it holds exactly in the same manner for a triangle on the surface of the sphere, from which it would follow that the sum of the angles in a spherical triangle equals two right angles, whilst this sum is known to be always greater than two right angles. The proof depends only on the fact, that any line can be made to coincide with any other line, that two lines do so coincide when they have two points in com- mon, and further, that a line may be turned about any point in it without leaving the sur- face. But if instead of the plane we take a spherical surface, and instead of a line a great SCIENCE ABSOLUTE OP SPACE. 67 circle on the sphere, all these conditions are again satisfied. 4 * The reasoning employed must therefore be fallacious, and the error lies in the words printed in italics; for these words contain an assumption which has not been proved. "O. HENRICI." Perronet Thompson, of Queen's College, Cambridge, in a book of which the third edi- tion is dated 1830, says: '''Professor Playfair, in the Notes to his 'Elements of Geometry' [1813], has proposed another demonstration, founded on a remark- able non causa pro causa. "It purports to collect the fact [Eu. I., 32, Cor., 2] that (on the sides being successively prolonged to the same hand) the exterior angles of a rectilinear triangle are together equal to four right angles, from the circum- stance that a straight line carried round the perimeter of a triangle by being applied to all the sides in succession, is brought into its old situation again; the argument being, that be- cause this line has made the sort of somerset it would do by being turned through four right angles about a fixed point, the exterior 68 SCIENCE ABSOLUTE OF SPACE. angles of the triangle have necessarily been equal to four right angles. "The answer to which is, that there is no connexion between the things at all, and that the result will just as much take place where the exterior angles are avowedly not equal to four right angles. * 'Take, for example, the plane triangle formed by three small arcs of the same or equal circles, as in the margin; and it is manifest that an arc of this circle may be car- ried round pre- cisely in the way described and re- turn to its old sit- uation, and yet there be no pre- tense for infer- ring that the exterior angles were equal to four right angles. "And if it is urged that these are curved lines and the statement made was of straight; then the answer is by demanding to know, what property of straight lines has been laid down or established, which determines that what is not true in the case of other lines is SCIENCE ABSOLUTE OP SPACE. 69 true in theirs. It has been shown that, as a general proposition, the connexion between a line returning to its place and the exterior angles having been equal to four right angles, is a non sequitur ; that it is a thing that may be or may not be; that the notion that it re- turns to its place because the exterior angles have been equal to four right angles, is a mis- take. From which it is a legitimate conclu- sion, that if it had pleased nature to make the exterior angles of a triangle greater or less than four right angles, this would not have created the smallest impediment to the line's returning to its old situation after being car- ried round the sides; and consequently the line's returning is no evidence of the angles not being greater or less than four right angles." Charles L. Dodgson, of Christ Church, Ox- ford, in his " Curiosa Mathematica," Part I, pp. 70-71, 3d Ed., 1890, says: "Yet another process has been invented quite fascinating in its brevity and its ele- gance which, though involving the same fal- lacy as the Direction-Theory, proves Euc. I, 32, without even mentioning the dangerous word 'Direction.' 70 SCIENCE ABSOLUTE OP SPACE. 1 ' We are told to take any triangle ABC; to produce CA to D; to make part of CD, viz., AD, revolve, about A, into the position ABE; then to make part of this line, viz., BE, revolve, about B, into the position BCF; and lastly to make part of this line, viz., CF, revolve, about C, till it lies along CD, of which it originally formed a part. We are then assured that it must have revolved through four right angles: from which it easily follows that the interior angles of the triangle are together equal to two right angles. ' ' The disproof of this fallacy is almost as brief and elegant as the fallacy itself. We first quote the general principle that we can not reasonably be told to make a line fulfill two conditions, either of which is enough by itself to fix its position: e. g., given three points X, Y, Z, we can not reasonably be told to draw a line from X which shall pass through Y and Z: we can make it pass through Y, but it must then take its chance of passing through Z; and vice versa. "Now let us suppose that, while one part of SCIENCE ABSOLUTE OP SPACE. 71 AE, viz., BE, revolves into the position BF, another little bit of it, viz., AG, revolves, through an equal angle, into the position AH; and that, while CF revolves into the position of lying along CD, AH revolves and here comes the fallacy. "You must not say * revolves, through an equal angle, into the position of lying along AD, ' for this would be to make AH fulfill two conditions at once. ' ' If you say that the one condition involves the other, you are virtually asserting that the lines CF, AH are equally inclined to CD and this in consequence of AH having been so drawn that these same lines are equally in- clined to AE. "That is, you are asserting, 'A pair of lines which are equally inclined to a certain trans- versal, are so to any transversal. ' [Deducible fromEuc. I, 27, 28, 29.]" MATHEMATICAL WORKS BY GEORGE BRUCE HALSTED, A.M. (PRINCETON); PH. D. (JOHNS HOPKINS); EX-FELLOW OF PRINCETON COLLEGE; TWICE FELLOW OF JOHNS HOPKINS UNIVERSITY; INTERCOL LEGIATK PRIZEMAN; SOMETIME INSTRUCTOR IN POST GRADUATE MATHEMATICS, PRINCETON COLLEGE; PROFESSOR OF MATHEMAT- ICS, UNIVERSITY OF TEXAS, AUSTIN, TEXAS ; MEMBER OF THE AMERICAN MATHEMATICAL SOCIETY; MEMBER OF THK LON- DON MATHEMATICAL SOCIETY; MEMBER OF THE ASSO- CIATION FOR THE IMPROVEMENT OF GEOMETRICAL TEACHING; EHRENMITGLIED DES COMITES DES LOBACHEVSKY-CAPITALS ; MIEMBRO DE LA So- CIEDAD ClENTIFICA "ALZATK" DE MEXICO', SOCIO CORRESPONSAL DE LA SOCIEDAD DE GEOGRAFIA Y ESTADISTICO DE MEXICO; SOCIETAIRE PERPETUAL DE LA SOCIETE MATHE- MATIQUE DE FRANCE; SOCIO PERPETUO DELLA ClBCOLO MATEMATICO DI PALERMO; PBESIDENT OF THE TEXAS ACADEMY OF SCIENCE. Mensuration. 4th Ed. 1892. $1.10. Ginn & Co. Boston. U. S. A., and London. Elements of Geometry. 6th Ed. 1893. $1.75. John Wiley & Sons. 53 E. 10th St., New York. Chapman & Hall. London. Synthetic Geometry. 2nd Ed. 1893. $1.50. John Wiley & Sons. 53 E. 10th St., New York. Lobachevski's Non-Euclidean Geometry. 4th Ed. 1891. $1. G. B. Halsted. 2407 Guadalupe St., Austin, Texas, U. S. A. Bolyai's Science Absolute of Space. 4th Ed. 1896. $1.00. G. B. Halsted, 2407 Guadalupe St., Austin, Texas, U. S. A. Vasiliev on Lobachevski. 1894. 50c. G. B. Halsted, 2407 Guadalupe St., Austin, Texas, U. S. A. Sent postpaid on receipt of the price. VOLUME ONE OF THE NEOMONIC SERIES. NICOLAI IVANOVICH LOBACHEVSKI. BY A. VASIL.IEV. Translated from the Russian by GEORGE BRUCE HALSTED. From a six-column Review of this Translation in Science, March 29, 1895: "Non-Euclidian Geometry, a subject which has not only revo- lutionized geometrical science, but has attracted the attention of physicists, psychologists and philosophers." "Without question the best and most authentic source of infor- mation on this original thinker." From a two-column Review in the Nation, April 4, 1895, by C. 8. Pierce: "Kazfin was not the milieu for a man of genius, especially not for so profound a genius as that of Lobachevski." "All of Lobachevski's writings are marked by the same high- strung logic." I have read it with intense interest. By issuing this transla- tion you have put American readers under renewed obligation to you. FLORIAN CAJORI. I have read with great interest your translation of the address in commemoration of Lobachevski. It is a most fortunate thing for us in the rank and file that you have maintained such an in- terest in the history of this non-Euclidian work; for while you have conquered for Saccheri, Bolyai and the rest the share of fame that is their due, you have made it impossible for American teachers of any spirit to shut their eyes to the " hypothesis anguli acuti." Very truly yours, G. H. LOUD, Professor of Mathematics in Colorado College. BURLINGTON, VT., October 19th, 1894. I am astonished to find these researches of such deep,, philo- sophical import. You many congratulate yourself on your in- strumentality in spreading the news in America. Very sincerely, A. L. DANIELS, Professor of Mathematics, University of Vermont. STAUNTON. VA., October 13th, 1894. The history of the life and work of such a man as Lobachevski will be a grand inspiration to mathematicians, especially with such a leader as yourself, in the important field of non-Euclidean Geometry. Very truly yours. G. B. M. ZERR. BETHLEHEM. PA.. October 22nd, 1894. I have read the Lobachevski with much pleasure and what is better profit. Yours very truly. G. L. DOOLITTLE, Professor of Mathematics in the University of Pennsj'lvania. HALLE a. S., LAFONTAINESTR., 2; 23,10, '94. Hochgeehrter Herr: Auf der Naturforscherversamrnlung in Wien lernte ich Prof. Wasilief ans Kas;ln kennen, der mir erzaehlte. das Sie seine Kede bei der Lobatschefsky-Feier uebersetzen wollten. Diese Xach- richt war mich sehr willkommen. da die russische mir unver- standlich ist. Nun erhalte ich heute von Ihnen diese Uebersetzungzugesandt und sage Ihnen dafuer meinen verbindlichsten Dank. Sie haben mit der Uebersetzung dieser interessauten Rede sich den Ans- pruch auf den Dank der mathematischen Welt erworben ! Hochachtungsvoll Ihr ergebener, STAECKEL. STANFORD UNIVERSITY, PALO ALTO, CAL., October 19th, 1894. 1 have read the Lobachevski with the greatest interest, and re- joice that you. "in the midst of the virgin forests of Texas." are able to do this work. And. by the way, I have heard at different times a number of professors speak of your Geometry (Elements). All who have examined it. and whom I have heard speak of it, seem to think it the best Geometry we have. Yours truly. A. P. CARMAN, Professor of Physics, Leland Stanford. Jr.. University. READY FOR THE PRESS. VOLUME FOUR OF THE NEOMONIC SERIES. THE LIFE OF BOLYAI. From Hungarian (Magyar) sources, by Dr. George Bruce Halsted. [Containing the Autobiography of Bolyai Farkas, now first translated from the Magyar.] VOLUME FIVE OF THE NEOMONIC SERIES. NEW ELEMENTS OF GEOMETRY WITH A COMPLETE THEORY OF PARALLELS. BY N. I. LOBACHEVSKI. Translated from the Russian by DR. GEORGE BRUCE HALSTED. [Though this is Lobachevski's greatest work, it has never be- fore been translated out of the Russian into any other language whatever.] U. C. BERKELEY LIBRARIES