THE SCIENCE ABSOLUTE OF SPACE Independent of the Truth or Falsity of Euclid's Axiom XI (which can never be decided a priori). 3Y 3- 1~ BOLYAI 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 NEOMON 2407 Guadalupe Street AUSTIN, TEXAS, U. S. A. 1896 TRANSLATOR'S INTRODUCTION. The immortal Elements of Euclid was already 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 Apollonius, in Pappus; strugglèd in Theon, in Hypatia; 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 centuries, to western Christendom, an Englishman, 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, obtained a Moorish copy of Euclid's Elements, and made a translation from the Arabic into Latin. iv TRANSLATOR'S INTRODUCTION. The first printed edition of Euclid, published in Venice in 1482, was a Latin version from the Arabic. The translation into Latin from the Greek, made by Zamberti 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 threequarters the only printed Greek text of all the books, and from it the first English translation (1570) was made by "Henricus Billingsley," 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 Government, by Oxford, and by Cambridge, no proof of a theorem in geometry will be accepted which infringes Euclid's sequence of propositions. Nor is the work unworthy of this extraordinary immortality. Says Clifford: "This book has been for nearly twenty-two centuries the encouragement and guide of that scientific thought which is one thing with the progress of man from a worse to a better state. TRANSLATrOR 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 eyery 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 with 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 assumptions he said no word. Among Euclid's assumptions is one differing from the others in prolixity, whose place fluctuates 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 (Leipzig, 1883-1888), gives it as the fifth postulate. James Williamson, who published the closest translation of Euclid we have in English, indicating, by the use of italics, the words not in the original, gives this assumption as eleventh among the Common Notions. vi TRANSLATOR' S INTRODUCPTION. 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 fortyeight pages from the other axioms. It is not used in the first twenty-eight propositions 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 twentyseventh. 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 transversal 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 inward 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, alongside 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 understood 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 noticeable blots on his work. viii TRANSLATOR'S INTRODUCUION. Sir Henry Savile, in his Praelectiones on Euclid, Oxford, 1621, p. 140, says: "In puicherrimo Geometriae corpore 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 " To gild refined gold, to paint the lily, To cast a perfume o'er the violet." At the end of that year Gauss from Braunschweig writes to Bolyai Farkas in Klausenburg (Kolozsvar) as follows: [Abhandlungen der Koeniglichen Gesellschaft der Wissenschaften zu Goettingen, Bd. 22, 1877.] "I 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 world thereon I myself have advanced far (though my other wholly heterogeneous 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 core 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 rectilineal 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 let 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 possible 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 instruments 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 indemonstrable. And perhaps Bolyai got nearer than Gauss to the unattainable. In his " 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 translation 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 tranquillity." 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 obscurities 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: "those straights which are produced indefinitely from less than two right angles meet." [This is the form which occurs in the Greek of Eu. I. 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 Stolz 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 triangle'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 recalcitrant 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 astronomical system of Eudoxus, and Hultsch has published the writings of Autolycus, we see that Euclid knew surface-spherics, was familiar 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 meeting; which is real if the triangle's angle-stum is less than a straight angle? Of how naturally the three systems of geometry 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 edition of Clavius, where the fifth postulate is given as Axiom 13. Saccheri says it should not be called an axiom, but ought to be demonstrated. 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 September 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 unexpectedly 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 Febrnary 9th, 1775, at Bolya, in that part of Transylvania (Er TRANSLATOR'S INTRODUCTION. xv dély) called Székelyfold. He studied first at Enyed, afterward at Klausenburg (Kolozsvar), theri went with Baron Simon Kemény 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 workiirg 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 1i 'Q, and in January, 1804, was made professor of mathematics in the Reformed College of Maros-Vasarhely. Here for 47 years of active teaching le had for scholars nearly all the professors and nobility of the next generation in Erdély. 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 College 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-follows: TENTAMEN | JUVENTUTEM STUDIOSAM IN ELEMENTA MATHESEOS PURAE, ELEMENTARIS AC [ SUBLIMIORIS, METHODO INTUITIVA, EVIDENTIA- I QUE HUIC PROPRIA, INTRODUCENDI. | CUM APPENDICE TRIPLICI. | Auctore Professore Matheseos et Physices Chemiaeque! Publ. Ordinario. Tomus Primus. Maros Vasarhelyini. 1832. I Typis Collegii Reformatorum per JOSEPHUM, et I SIMEONEM KALI de felso Vist. | At the back of the title: Imprimatur. M. Vasarhelyini Die I 12 Octobris, 1829. [ Paulus Horvath m. p. j Abbas, Parochus et Censor | Librorum. Tomus Secundus. | Maros Vasarhelyini. 1833. The first volume contains: Preface of two pages: Lectori salutem. A folio table: Explicatio signorumn. Index rerum (I-XXXII). Errata (XXXIII-XXXVII). Pro tyronibus prima vice legentibus notanda sequentia (XXXVIII-L4II). Errores (LIII-LXVI). TRANSLATOR' S INTRODUCTION. xvii Scholion (LXVII-LXXIV). Pluriium errorum haud animadversorum numerous minuitur (LXXV-LXXVI). Recensio per auctorem ipsum facta (LXXVII-LXXVIII). Errores recentius detecti (L X X VXCVIII). Now comes the body of the text (pages 1-502). - Then, with special paging, and a new title page, comes the imnmortal Appendix, here given in English. Professors Staeckel and Engel make a mistake 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 "Curm 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 referred to at all in the words "Cumr appendice triplici." These words, as explained in a prospectus in the Magyar language, issued by Bolyai Farkas, 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 Secundus, 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 German into Latin a condensation of his treatise, of which the principles were discovered and properly appreciated in 1823, and which was given in writing to Johann Walter von Eckwehr 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 twentysix numbered pages, followed by two unnumbered 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 "Tentamen," contributed for the TRANSLATOR'S INTRODUCTION. xix printing of his eternal twenty-six pages, 104 florins 50 kreuzers. That this Appendix was finished considerably before the Vol. I, which it follows, is seen from the references in the text, breathing a just admiration for the Appendix and the genius of its author. Thus the father says, p. 452: Elegans est conceptus simiiumn, quem J. B. Appendicis Auctor dedit. Again, p. 489: Appendicis Actor, rem acumine singulari aggressus, Geometriam pro omni casu absolute veram posuit; quamvis e magna mole, tantum summe necessaria, in Appendice hujus tomi exhibuerit, multis (ut tetraedri resolutione generali, pluribusque aliis disquisitionibus elegantibus) brevitatis studio omissis. And the volume ends as follows, p. 502: Nec operae pretium est plura referre; quum res tota exaltiori contemplationis puncto, in ima penetranti oculo, tractetur in Appendice sequente, a quovis fideli veritatis purae alumno diagna legi. The father gives a brief resumé of the results of his own determined, life-long, desperate efforts to do that at which Saccheri, J. H. Lambert, Gauss also had failed, to establish Euclid' theory of parallels apriori. XX TRANSLATOR'S INTRODUCTION. He says, p. 490: "Tentamina idcirco quae olim feceram, breviter exponenda veniunt; ne saltem alius quis operam eandem perdat." He anticipates J. Delboeuf's "Prolégoménes philosophiques de la géométrie 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 preceding space only a special case, only a species under a genus, and so requiring a descriptive adjective, Euclidean, this wonderful production 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 University of Giessen. He it was who first did justice publicly to the works of ILobachevski and Bolyai. Incited by Baltzer, in 1866 J. Hoiiel issued a French translation of Lobachevski's Theory of Parallels, and in a note to his Preface says: " M. Richard Baltzer, dans la seconde édition de ses excellents Elenents de Geometrie, a, le premier, introduit ces notions exactes à la place qu'elles doivent occuper," Honor to TRANSLATOR'S INTRODUCTION. xxi Baltzer! But alas! father and son were already in their graves! Fr. Schmidt in the article cited (1868) says: "It was nearly forty years before these profound views were rescued from oblivion, and Dr. R. Baltzer, of Dresden, has acquired imperishable 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 Géométrie élémentaire, has given extracts 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 kezdoknek LElements of the science of space for beginners] (1850-51t, pp. 48. In the College are preserved three sets of figures for this book, two by the author and one by his grandson, 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 unendlich-kleinen Grossen gereinigt, anschaulich und logisch-streng darzustellen. II. In der Geometrie, die Begriffe der geraden 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 geschnittene 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 unabhangige Geometrie abzusondern; 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 gUltig seyen. Nach ein lateinischen Werke von 1829, M. Vasarhely, und eben daselbst gedruckten ungrischen. Maros Vasarhely 1851. 8vo. pp. 88. In this book he says, referring to his son's Appendix: "Some copies of the work published here were sent at that time to Vienna, to Berlin, to Goettingen.... From Goettingen the giant of mathematics, who from TRLANSLATOR'S INTRODUCTION. xxiii his pinnacle embraces in the same view the stars and the abysses, wrote that he was surprised to see accomplished what he had begun, 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 Astronomy at Marburg. He writes: "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 attempts to prove the Euclidean axiom had miscarried. 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 communicated to me thereon I sent to Gauss, who xxiv TRANSLATOR'S INTRODUCTION. then informed me how much farther already had been attained on this way, and later also expressed himself about the great acquisition, which is offered to the few expert judges in the Appendix to your book." The "latest edition" mentioned appeared in 1851, and the passage referred to is: "This proof Lof 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 ail founded on it, remains a hypothesis, whose validity for our life indeed is sufficiently proven by experience, whose general, necessary exactness, however, c o ul d 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 discoverers of the non-Euclidean geometry. But then, since neither of these sufficiently realized the transcendent importance of the matter 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 Lobachévski, who claimed TRANSLATOR'S INTRODUCTION. XXV at once, unflinchingly, that their discovery marked an epoch in human thought so momentous as to be unsurpassed by anything recorded 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 mathematics section of the Magyar Academy. As professor he exercised a powerful influence 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, née, Arkosi Benko 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 demonstrations 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-curriculum, 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 "souslieutenant" and sent to Temesvar, where he was to present himself on the 2nd of September. From Temesvar, on Nov.ember 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 moment 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 would not have written you before receiving it, if 1 had not wished to address to you the letter I ar 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 publish, 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 magnificent 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." Prom 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 Eckwehr (later k. k. General) [in the Austrian Army], a written treatise, which is still in his hands. "On the prompting of my father I translated my treatise into the Latin language, and TRANSLATOR'S INTRODUCTION. xxix it appeared as APfyendix to the Tentamen, 1832." The profound mathematical ability of Bolyai Janos showed itself physically not only in his handling of the violin,.where he was a master, but also of arms, where he was unapproachable. It was this skill, combined with his haughty temper, which caused his being retired as Captain 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 permitted 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 mathematics. He left parts of a book entitled: Principia doctrinae novae quantitatum imaginariarum perfectae uniceque satisfacientis, aliaeque disquisitiones analyticae et analytico- geometricae cardinales gravissimaeque; auctore xxx TRANSLATOR' S INTRODUCTION. Johan. Bolyai de eadem, C. R. austriaco castrensium 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 Society. APPENDIX. SCIENTIAM SPATII absolute veram exhibens: a veritate aut falsitate Axiomatis XI Euclidei (a priori haud unquam decidenda) independentemn. adjecta ad casum falsitatis, quadratura circuli geometrica. Auctore 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 commences 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 wlich 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 Z ABC, and the straights BA, CD not intersecting) means the portion of / ABC comprised between ray BA, sect BC, ray CD; while BACD designates the portion of the plane ABC comprised between the straights AB and CD. I is the sign of perpendicularity. Il is the sign of parallelism. / means angle. rt. / is right angle. st. / is straight angle. is the sign of congruence, indicating ttlat two magnitudes are superposable. AB-CD means / CAB= / ACD. xa means x converges toward the limit a. A is triangle. Or means the [circumference of the] circle of radius r. area (r means the area of the surface of the circle of radius r. THE SCIENCE ABSOLUTE OF SPACE. ~1. If the ray AM is not cut by the ray [3] M p N 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 suc h ray BNV, and only one, passB ing through any point B (taken outside of the straight AM), and that FIG. 1. the sum of the angles BAM, ABN can not exceed a st. Z; for in moving BC around B until BAM+ABC=st. Z, somewhere ray BC first does not cut ray AM, and it is then BC Il AM. It is clear that BNII EM, wherever the point 1 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 constantly CDB=(CBD<NBC); but NBC —0; and so also ADB'-O. 6 SCIENCE ABSOLUTE OF SPACE. ~ 2. If BN II AM, we will have also CN II AM. I N For take D anywhere in MACN. RS If C is on ray BN, ray BD cuts 9Q ray AM, since BN II AM, and so also ray CD cuts ray AM. But c\ if C is on ray B4I take BQ II CD; ^A \^ 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 FIG. 2. Peach 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 M point B of ray AB, a point D p C of ray AM, such that BCM=NAM. c B For (by ~ 1) is granted A Y N BDM>NAM, and so that FIG. 3. MDP=MAN, and B falls in SCIENCE ABSOLUTE OF 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 [4] N AM a point F such that FM BN. For by ~ 1 is granted BCM> CBN; c and if CE=CB, and so EC-OBC; evidently BEM<EBN. The point P is moved on EC, the angle BPM F B always being called u, and the anP gle PBN always v, evidently u is E at first less than the corresponding v, but afterwards greater. Indeed A IG u increases continuously from FIG. 4. BEM to BCM; since (by ~ 4) there exists no angle >BEM and <BCM, to which u does not at some time become equal. Likewise v decreases continuously from EBN to CBN. There is therefore on EC a point F such that BFM=FBN. ~6. If BN II AM and E anywhere in the straight AM, and G in the straight BN; then GN II EM and EM I1 GN. For (by ~ 1) BN II EM, whence (by ~ 2) GN II EM. If moreover FMBN (~5); then MFBN-NBFM, and consequently (since BNII FM) also FMII BN, and (by what precedes) EM II GN. 8 SCIENCE ABSOLUTE OF SPACE. ~ 7. If BN and CP are each II AM, and C N M P not on the straight BN; also BN II CP. For the rays BN and CP do not inD \ tersect (~3); but AM, BN and CP either are or are not in the same plane; and in the first case, AM either / c is or is not within BNCP. B A c Fr,. 5. If AM, BN, CP are complanar, and AM falls within BNCP; then every ray BQ (in NBC) cuts the ray AM in some point D (since BNII AM); moreover, since DM II CP (~ 6), the ray DQ will cut the ray CP, and so BN II CP. But if BN and CP are on the same side of N M AM; then one of them, for example CP, falls between the two other straights BN, AM: but every ray BQ (in NBA) cuts the ray AM, and so also the straight CP. Therefore B C A BNIICP. FIG. 6. If the planes MAB, MAC make an angle; then CBN and ABN have in common nothing but the ray BN, while the ray AM (in ABN) and the ray BN, and so also NBC and the ray AM have nothing in common. But hemi-plane BCD, drawn through any ray BD (in NBA), cuts the ray AM, since ray SCIENCE ABSOLUTE OF SPACE. 9 R N M p BQ cuts ray AM (as BNIIAM). Q Therefore in revolving the hemi-plane i] BCD around BC until it begins to leave the ray AM, the hemi-plane BCD at last will fall upon the hemiA plane BCN. For the same reason this B c same will fall upon hemi-plane BCP. '(.7. 7 Therefore BN falls in BCP. Moreover, if BR II CP; then (because also AM II CP) by like reasoning, BR falls in BAM, and also (since BRIIl CP 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 11 as well to AM as to CP. ~ 8. If BN 11 and - CP (or more briefly BN N M p II -CP, andAM (inNBCP) bisects I Q ' I the sect BC; then BN il AM. // / For if ray BN cut ray AM, also ~/ ray CP would cut ray AM at the same point (because MABNB-A c MACP), and this would be common Fr-. 8. 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 case 2 of ~10. [Author's note.j 10 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 MAP I MAB, and the PM l LZ, which NBD makes with N| ^K NBA (on that side of MABN, S E: where MAP is) is <rt.Z; then 'c MAP and NBD intersect. For let ZBAM=rt.Z, and B -^^: AC BN (whether or not C f falls on B), and CE I BN (in FIG. 9. NBD); by hypothesis LACE <rt.Z, and AF (i CE) will fall in ACE. Let ray AP be the intersection of the hemiplanes ABF, AMP (which have the point A common); since BAM I MAP, ZBAP-ZBAM =rt.Z. If finally the hemi-plane ABF is placed upon the hemi-plane ABM (A and B remaining), ray AP will fall on ray AM; and since AC IBN, and sect AF<sect AC, evidently sect AF will terminate within ray BN, and so BF falls in ABN. But in this position, ray BF cuts ray AP (because BN II AM); and so ray AP and ray BF [6f intersect also in the original position; and the point of section is common to the hemi-planes MAP and NBD. Therefore the hemi-planes MAP and NBD intersect. Hence follows eas SCIENCE ABSOLUTE OF SPACE. 11 ily that the hemi-planes MAP and NBD intersect if the sum of the interior angles which they make with MABN is <st.Z. ~10. If b.oth BN and CPII ^AM; also is ~s ~ BN II -CP. Q RI F o r either MAB i ~ \ \ and MAC make an angle, or they are in a plane. c If the first; let the henmi-plane QDF biD A sect _ sect AB; then FIG. 10. DQ I AB, and so DQ II AM (~ 8); likewise if hemi-plane ERS bisects 1 sect AC, is ER II AM; whence (~ 7) DQ II ER. Hence follows easily (by ~9), the hemiplanes QDF and ERS intersect, and have t~ 7) their intersection FS 11 DQ, and (on account of BN 1 DQ) also FS 11 BN. Moreover (for any point of FS) FB=FA=FC, and the straight FS falls in the plane TGF, bisecting 1 sect BC. But (by ~7) (since FS II BN) also GT IIBN. In the same way is proved GT II CP. Meanwhile GT bisects 1 sect BC; and so TGBNTGCP (~1), and BN II -CP. If BN, AM and CP are in a plane, let (falling without this plane) FS II -AM; then (from 12 SCIENCE ABSOLUTE OF SPACE. what precedes) FS 11 - both to BN and to CP, and so also BN II -CP. ~ 11. Consider the aggregate of the point A, and ail points of which any one B is such, that if BN 11 AM, also BN-AM; call it F; but the intersection of F with any plane containing the sect AM call L. F has a point, and one only, on any straight Il AM; and evidently L is divided by ray AM into two congruent parts. Call the ray AM the axis of L. Evidently also, in any plane containing the sect AM, there is for the axis ray AM a single L. Call any L of this sort the L of this ray AM (in the plane considered, being understood). Evidently by revolving L around AM we describe the F of which ray AM is called the axis, and in turn F may be ascribed to the axis ray AM. 71 ~ 12. If B is anywhere on the L of ray AM, and BN II -AM (~ 11); then the L of ray AM and the L of ray BN coincide. For suppose, in distinction, L' the L of ray BN. Let C be anywhere in L', and CP II -BN (~ 11). Since BN II -AM, so CP Il =AM (~ 10), and so C also will fall on L. And if C is anywhere on L, and CP IlI AM; then CP II =-BN (~ 10); and C also falls on L' (~11). Thus L and L' are the SCIENCE ABSOLUTE OF SPACE. 13 same; and every ray BN is also axis of L, and between all axes of this L, is. The same is evident in the same way of F. ~ 13. If BN II AM, and CP II DQ, andZBAM +ZABN=st.Z; then also ZDCP+LCDQ= st.Z. M SN L SO P 0 For let EA= EB, and EFM= DCP (~ 4). Since ZBAM+ZABN =st. Z= ABN+ F - GB Z ABG, we have Ei i [ _ ZEBG=-ZEAF; nElUf o and so if also BG Fi.11(. -=AF, thenAEBG zEtAF, ZBEG=ZAEF and G will fall on the ray FE. Moreover ZGFM+ZFGN=st. Z (since ZEGB=ZEFA). Also GN 1l FM (~ 6). Therefore if MFRS:PCDQ, then RS II GN (~ 7), and R falls within or without the sect FG (unless sect CD=sect FG, where the thing now is evident). I. In the first case ZFRS is not > (st. /-Z RFIM=zFGN), since RS Il FM. But as RS II GN, also ZFRS is not < ZFGN; and so ZFRS -ZFGN, and ZRFM+ZFRS=ZGFM+Z 14 SCIENCE ABSOLUTE OF SPACE. FGN=st.Z. Therefore also ZDCP+LCDQ =st. -. II. If R falls without the sect FG; then ZNGR-= MFR, 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=/ KFM+ZFGN=st.L; but if R falls within the sect HK, then (by I) LRHL+LZKRS=st.L= LRFM+ ZFRS= LDCP+ ZCDQ. ~ 14. If BN II AM, and CP II DQ, and ZBAM +LABN<st.Z; then also ZDCP+LCDQ< st./. For if ZDCP+ZCDQ were not <st., and so (by ~ 1) were =st.L, then (by ~ 13> also L BAM+ZABN=st. L (contra hyp.). 15. Weighing ~~ 13 and 14, the System of Geometry resting on the hypothesis of the trulh of Euclid's Axiom XI is called,; and the system founded on the contrary hypoth[8] esis is S. Al/ things which are not expressly said to be in î or in S, it is understood are enunciated absolutely, that is are asserted true whether s or S is reality. SCIENCE ABSOLUTE OF SPACE. 15 ~16. If AM is the axis of any L; then L, in z is a straight I AM. N M? For suppose BN an axis from any point B of L; in v, ZBAM+ZABN =st.Z, and so ZBAM=rt.L. And if C is any point of the straight AB, and CPllAM; then B A c(by ~ 13) CP^AM, and so C on L (-. 1 2. (~ n ). But in S, no three points A, B, C on L or on F are in a straight. For some one of the axes AM, BN, CP (e. g. AM) falls between the two others; and then (by ~ 14) ZBAM and ZCAM are each <rt.Z. ~ 17. L in S also is a line, and F a surface. For (by ~ 11) any plane _ to the axis ray AM (through any point of F) cuts F in [the circumference of] a circle, of which the plane (by ~ 14) is I to no other axis ray BN. If we revolve F about BN, any point of F (by ~ 12) will remain on F, and the section of F with a plane not I ray BN will describe a surface; and whatever be the points A, B taken on it, F can so be congruent to itse!f that A falls upon B (by ~ 12); therefore F is a uniforin surface. 16 SCIENcE ABSOLUTE OF SPACE. Hence evidently (by ~~ 11 and 12) L is a uniform line.* ~ 18. The intersection with F of any plane, drawn through a point A of F obliquely to the axis AM, is, in S, a circle. For take A, B, C, three points of this section, and BN, CP, axes; AMBN and AMCP make an angle, for otherwise the plane determined by A, B, C (from ~ 16) would contain AM, (contra hyp.). Therefore the planes bisecting I the sects AB, AC intersect (~ 10) in some axis ray FS (of F), and FB=FA=FC. y p Make AH I FS, and reM| s / volve FAH about FS; A will describe a circle of /,' |radius HA, passing,~ |' |through B and C, and situated bot/i in F and in [9] A 3 the plane ABC; nor have FI(T. 13. F and the plane ABC anything in common but O HA (~ 16). It is also evident that in revolving the portion FA of the line L (as radius) in F around F, its extremity will describe O HA. * It is not necessary to restrict the demonstration to the system S; since it may easily be so set forth, that it holds absolutely for S and for 1. SCIENCE ABSOLUTE OF SPACE. 17 ~ 19. The perpendicular BT to the axis BN of L (falling in the plane of L) is, in S, N tangent to L. For L has in ray BT no point except B (~ 14), but if BQ falls in TBN, then the center of the section of the Q plane through BQ perpendicular U -^B to TBN with the F of ray BN Fic. 14. (~ 18) is evidently located on ray BQ; and if sect BQ is a diameter, evidently ray BQ cuts in Q the line L of ray BN. ~ 20. Any two points of F determine a line L (~~ 11 and 18); and since (from ~~ 16 and 19) L is I to all its axes, every Z of lines L in F is equal to the Z of the planes drawn through its sides perpendicular to F. 21. Two L form lines, ray AP and ray P M D BD, in the same F, making with a third L form AB, a sum of interior angles <st.Z, intersect. (By line AP in F, is to be A B-c understood the line L drawn FG. 15. through A and P, but by ray AP that half of this line beginning at A, in which P falls.) For if AM, BN are axes of F, then the hemiplanes AMP, BND intersect (~ 9); and F cuts 18 SCIENCE ABSOLUTE OF SPACE. their intersection (by ~~ 7 and 11); and so also ray AP and ray BD intersect. From this it is evident that Euclid's Axiom XI and all things which are claimed in geometry and plane trigonometry hold good absolutely in F, L lines being substituted in place of straights: therefore the trigonometric functions are taken here in the same sense as in '; and the circle of which the L form radius = r in F, is 2-r,~ and likewise area of Or (in F) = -r2 (by - understanding S0 1 in F, or the known 3.1415926...) ~ 22. If ray AB were the L of ray AM, and C on ray AM; and the /CAB (formed by the M N p straight ray AM and the L form H| L | line ray AB), carried first along [io] Gcl the ray AB, then along the ray FK |BA, always forward to infinity: DF the path CD of C will be the line L of CM. FIG. 16. For let D be any point in line CD (called later L', let DN be II CM, and B the point of L falling on the straight DN. We shall have BN -AM, and sect AC=sect BD, and so DN-C CM, consequently D in L'. But if D in L' and DN 11 CM, and B the point of L on the straight DN; we shall have AM-BN and CM -DN, whence manifestly sect BD=sect AC, SCIENCE ABSOLUTE OF SPACE. 19 and D will fall on the path of the point C, and L' and the line CD are the same. Such an L' is designated by L'IIIL. ~ 23. If the L form line CDF 111 ABE (~ 22), and AB-BE, and the rays AM, BN, EP are axes; manifestly CD=DF; and if any three points A, B, E are of line AB, and AB=n.CD, we shall also have AE=n.CF; and so (manifestly even for AB, AE, DC incommensurable), AB:CD=AE:CF, and AB:CD is independent of AB, and completely determined by A C. This ratio AB:CD is designated by the capital letter (as X) corresponding to the small letter (as x) by which we represent the sect AC. y ~ 24. Whatever be x andy, (~23), Y=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 y=n.x, take x=AC=CG=GH=&c., until we get AH=y. Moreover, take CD III GK 11 HL. We have ((~ 23) X=AB:CD-CD:GK=GK: HL; and so AB (ABI) y HL CDj or Y=Xn=Xx. If x, y are multiples of i, suppose x=nmi, and y=ni; (by the preceding) X=Im, Y=In, consequently n y Y=Xm=-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 ~, for any x, we have X=1, but in S is X>1, and for any AB [ii and ABE there is such a CDF Ili AB, that CDF =AB, whence AMBN-AMEP, though the first be any multiple of the second; which indeed is singular, but evidently does not prove the absurdity of S. ~ 25. In any rectilineal triangle, the circles with radii equal to its sides are as the sines of the opposite angles., \P For take XABC=rt.Z, + '\ \ and AMI BAC, and BN and CP II AM; we shall have CAB lAMBN, and so (since CB "C< /^ 111| BA), CB _ AMBN, consequently CPBN _ 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 reasoning ZCED=-CAB. But (by~21) intheLline A CDE (supposing always here the radius =1), EC:DC=: sin DEC =: sin CAB. SCIENCE ABSOLUTE OF SPACE. 21 Also (by ~ 21) EC:DC=OEC:ODC(in F)=OAC:OBC (~ 18); and so is also OAC: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. ~O For take ZABC=rt.Z, and E,, -` c CED I to the radius OA of the \ sphere. We shall have CED _ AOB, and (since also BOC I FA G 1 BOA), CD 1 OB. But in the FIG. 18. triangles CEO, CDO (by ~ 25) (EC: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, lowingfrom this, is thus established independently of Axiom XI. ~ 27. If AC and BD are 1 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. c _7 D M, For take DE I CA; E - in the triangles ADE, ADB (by ~ 25) OED:O AD:OAB= -- -^A -BN sin: 1: sin v. G F Fr. 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 [12] polygon BFG... inscribed in OAB. Passing through all the sides BF, FG, &c., planes I to OAB we form also a polygonal figure 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+...-(OAB and DH+HK+....-ED. Therefore also (ED: OAB=CD: AB. But we had OED: OAB-sin u: sin v. Consequently CD: AB=sin u: sin v. If AC goes away from BD to infinity, CD: AB, and so also sin: sin v remains constant; but u-'rt. (~1), and if DMII BN, v-z; whence CD: AB1: sin z. SCIENCE ABSOLUTE OF SPACE. 23 The path called CD will be denoted by CD III AB. ~ 28. If BN II -AM, and C in ray AM, and AC=x. we shall have (~ 23) A X=sin: sin v. iB For if CD and AE are I BN, ~F<4 /XE and BF 1 AM; we shall have (as l J;nin ~ 27) C G G ( OBF: ODC=sin u: sin v. D But evidently BF=AE: therefore M O(EA: OCD=sin u: sin v. FIG. 20. 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 X=sin u: sin v. ~ 29. If ZBAM=rt.Z, and sect AB=y, and N p \KvE BNII AM, we shall / K have in S D // / Y=cotan 2 u. F For, if sect AB= /-._. ` sect AC, and CP II C-....yC_-H ~ G Q AM (and so BN II FIG. 21. CP), and ZPCD= ZQCD; there is given (~19) DSray CD, so that DS II CP, and so (~ 1) DT II CQ. Moreover, if BE I ray DS, then (~ 7) DS II BN, and so (~ 6) 24 SCIENCE ABSOLUTE OF SPACE. BN I ES, and (since DT II CG) BQ lI ET; consequently (~1) /EBN=ZEBQ. Let BCF be an L-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-2v. Likewise it is evident BG-2BL-2z. But BC=BG-CG; wherefore y-z-v, and so (~24) Y=Z:V. Finally (~ 28) Z=: sin ~ u, and V=: sin (rt.L- ~ u), consequently Y-cotan nu. ~ 30. However, it is easy to see (by ~ 25) [13 that the solution of the problem of Plane M' TM N Trigonometry, in S, requires the expression of the circle B in terms of the radius; but / this can by obtained by the;/ rectification of L. Let AB, CM, C'M' be I C~ - --- kA ray AC, and B anywhere in - D ray AB; we shall have (~ 25) D s in u:sin v-Op:o y, FIG. 22. and sin ':sin v'-O ':o y'; sin v sin U' and so sinv.(Dy= sinv.(Y' C!1~,. ~y'.1" J SCIENCE ABSOLUTE OF SPACE. 25 But (by ~ 27) sin v: sin v'=cos u: cos u'; sin u sin u' consequently. y=,.y; cos u cos u or 0y: Oy=:tan u':tan u=tan w:tan w'. Moreover, take CN and C'N' II AB, and CD, C'D' L-form lines i straight AB; we shall have also (~21) Qy: ~oy'r: r', and so r: r'=tan w: tan w'. Now let p beginning from A increase to infinity; 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-O, r i tan z r=- itan 1, and so Y Y Y - i. From ~29, tan z= (Y-Y-1); tan z therefore 2y - i or (~ 24). 2y.I 2y --- But we know the limit of this expression (where y 0) is. Therefore nat. log I 26 SCIENCE ABSOLUTE OF SPACE. =i, and nat. log I I=e=2.7182818..., which noted quantity shines forth here also. If obviously henceforth i denote that sect of which the I=e, we shall have r=i tan z. But (~ 21) oy=2.r,' therefore OQy=2ri tan z= i (Y-Y-l)= [eT — y tY g(Y-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 [14] equations suffice: indeed (a, b denoting the sides, c the hypothenuse, and a, a the angles opposite the sides) an equation expressing the relation 1st, between a, c, a; 2d, between a, ~, /; ra 3d, between a, b, c; of course from these equations / ' M emerge three others by elim-, b N ination. FIG. 23. From ~~ 25 and 30 1: sin a=(C-C'): (A-A-l)= = e - f i -i1 (equation for c, a and a). e —e j [ eei-eT SCIENCE ABSOLUTE OF SPACE. 27 II. From ~ 27 follows (if pM II rN) cos a:sin j3=1: sin u, but from ~ 29:sin u=- (A+A-'); therefore cos a:sin = -(A+A-1) -I (e+ei (equation for a, p and a). III. If aa' IJ_~a, and pp' and rY'i aa' (~ 27), and i 1'ar' I aa'; manifestly (as in ~ 27);=S -_- = 2 (A+A-1); rr sin u rr = S(B+B-1; and/i and ' (C+C-1); consequently 2 (C+C-l) = (A+A-1). 2 (B+B-1), or r c -cl| _ ( f a -a( b -b I ei+ei - 2 Leï+ei) Le+eiJ (equation for a, b and c). If r"a=rt., and paL s; OC: 'a=1 'sin a, and Oc: (d=: s-=l: cos a, and so (denoting by Ox2, for any x, the product Ox. Ox) manifestly Oa2+Od20-c2. But (by ~ 27 and II) O d=. 2 (A+A-1), consequently c -c 2 a -a1 2 b -b b 2 a -a 2 ei-eiJ ) =4 Lei+e IJ L e-e' J ei-ei J another equation for a, b 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 COS a cos = 1(A+A-1), and C~S-(B+B-), we get sin f3 sin a(by III) cot cot/[- (e+e (equation for a, î, and c. ~ 32. It still remains to show briefly the mode of resolving problems in S, which being accomplished (through the more obvious examples), finally will be candidly said what this theory shows. I. Take AB a line in a plane, and y=f(x) H B its equation in rectangular coN - \ ordinates, call dz any increment of.z, and respectively dx, dy, du y \ the increments of x, of y, and of _A the area u, corresponding to FIG 24 this dz; take BH l CF, and express (from ~ 31) BH by means of y, and seek dx the limit of dy 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 dy, and so tan HBG; and BH' SCIENCE ABSOLUTE OF SPACE. 29 (since HBC manifestly is neither > nor <, and so =rt. ), the tangent at B of BG will be determined by y. II. It can be demonstrated dy2+BH 1 Hence is found the limit of dz, and thence, dx 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 L1; but (if BN is axis) X=1:sin CBN (~28), and Y=cotan 2 CBN (~ 29), whence Y=X+<X2-1. or T -[16] O'r ^e^T=ei +Vle'1, the equation sought. Hence we get dy X(X2-1) 2; dx and BH1: sin CBN=X; and so dx dy 1 B,(X2-1); l~l 30 SCIENCE ABSOLUTE OF SPACE. 1+ dy2 l+l.-X2(X2-l)-1, BH2 dz (X -1, and -X(X2-1)2, and BH dz. X2(X2 1), whence, by intedx gration, we get (as in ~ 30) z=i(X2-1) i cot CBN. III. Manfestly du, HFCBH dx dx which (unless given in y) now first is to be expressed in terms of y; whence we get u by integrating. c _ D If AB=p, AC=q, CD=r, and E - CABDC=s; we might show (as in II) that __ ds_'r, which = e i i i Fio,25. Bdq Lel-e and, integrating, s-S ri [e-ei This can also be deduced apart from integration. 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 expressed. We know, that a surface t, 111 to a plane figure p (at the distance q), is to _ in the ratio of the second powers of homologous lines, or as q -q) 2 4e_-eiJ:1. It is easy to see, moreover, that the calculation of volume, treated in the same manner, requires two integrations (since the differential itself here is determined only by integration); and before all must be investigated the 117] volume contained between p and t, and the aggregate of all the straights I and joining the boundaries of p and t. We find for the volume of this solid (whether by integration or without it) r 2q -2q1 8fpi ei -e i + pq. The surfaces of bodies may also be determined in S, as well as the curvatures, the involutes, and evolutes of any lines, etc. As to curvature; this in S either is the curvature of L, or is determined either by the radius of a circle, or by the distance to a straight from the curve 111 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 111 to a straight. 32 SCIENCE ABSOLUTE OF SPACE. IV. For the circle (as in III) dareaox dx 0x, whence (by ~ 29), integrating, area =x-I2 [e -2+e. V. For the area CABDC=u (inclosed by an M N L form line AB=r, the I1 to this, CD=y, and the sects AC=BD=x) ~du,~~ ~-x d-d-y,; and (~ 24) y-rei, and so. cJ____iD dx (integrating) u=ri [ 1- ) AL Bg If x increases to infinity, then, in FIG. 26. -x S, e -O, 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 byp and the aggregate of axes drawn from the boundaries of p is equal to 'pi. ~-N\ c VI. If the angle at the cen// ~, 1% ter of a segment z of a sphere G- __- -A / is 2u, and a great circle isp, \' E D and x the arc FC (of the angle \> uf); (~25) FIG. 27.:sin u=p: OBC, and hence OBC=P sin u. [18] Meanwhile x-P-, and dx-=d'.l 2^ 2= SCIENCE ABSOLUrE OF SPACE. 33 Moreover, dz, BC, and hence dx dze' sin u, whence (integrating) du 27r ver sin m,2 27r The F may be conceived on which P falls (passing through the middle F of the segment); through AF and AC the planes FEM, CEM are placed, perpendicular to F and cutting F along FEG and CE; and consider the L form CD (from C I to FEG), and the L form CF; (~ 20) CEF=u, and (~ 21) PD ver sin mu FD ver sin U and so z=FD.P. p 2f But (~ 21) p-=.FGD; therefore z=Z.FD.FDG. But (~ 21) M FD.FDG=FC.FC; consequently z=7r.FC.FC=area OFC, in F. Now let BJ=CJ=r; (~ 30) A c --- C 2r=i(Y-Y-1), and so (~ 21) area 02r (in F) =-i2(-Y-)2. FIG. 28. Also (IV) area ( 2y =i(Y2-2+Y-); therefore, area 02r (in F) =area 02y, 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 =area FG=-FDG.p=-, 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 c s_ - _) i 023(X2X-2)-2 i 2X; E the surface generated by the revq olution of the line CD about AB 7- Q-2) A P B and the body described by CABDC FIG. 29. =- fP -Q-1)2. But in what manner al! things treated from (IV) even to here, also may be reached apartfrom integration, for the sake of brevity 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 simplyfor ' (and so for the hypothesis of no i), if indeed the equations do not become identical. But beware lest you understand to be supposed, 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 conducted to an absurdity. Supposing therefore that, in such an expression, the letter i, in case S is reality, designates that unique quantity whose I=ze; but if z is actual, the said limit is supposed to be taken in place of the expression: manifestly all the expressions originating from the hypothesis of the reality of S (in this sense) will be true absolutely, although it be completely unknown whether or not z 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 z, Ox=27.X; from I (~ 31) suitably treated, follows 1: sin a=c: a, but from II COS a =s-1, and so sin,? a+~=rt.Z; the first equation in III becomes identical, and so is true in z, although it there determines nothing; but from the second follows c2 2. c 2=a2+b. These are the known fundamental equations of plane trigonometry in z. 36 SCIENCE ABSOLUTE OF SPACE. Moreover, we find (from ~ 32) in 2, the area and the volume in III each =q;, from IV area Ox=x2; (from VII) the globe of radius x =37X3, etc. The theorems enunciated at the end of VI are manifestly true unconditionally. ~ 33. It still remains to set forth (as promised in ~ 32) what this theory means. I. Whether z or some one S is reality, remains undecided. II. All things deduced from the hypothesis of the falsity of Axiom XI (always to be understood 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 atone really re- [20] mains unknown; and so where remain unknown solely the absolute magnitudes in the expressions, but where a single known case would manifestly fix the whole system. But spherical trigonometry is established absolutely in ~ 26. (And we have, on F, a geometry wholly analogous to the plane geometry of 2.) III. If it were agreed that z exists, nothing more would be unknown in this respect; but SCIENCE ABSOLUTE OF SPACE. 37 if it were established that ' 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 praised). V. Finally, to friendly readers will not be unacceptable; that for that case wherein not z but S is reality, a rectilineal figure is constructed equivalent to a circle. ~ 34. Through D we may draw DM II AN in the following manner. From D drop DB I AN; from any point A of the straight AB erect AC IAN (in DBA), and let fall DC AC. We 38 SCIENCE ABSOLUTE OF SPACE. will have (~ 27) OCD: OAB: sin z, proc D M vided that DM Il BN. But sin z z is not >1; and so AB is o not >DC. Therefore a quadA E \-SBS ---- rant described from the cenA BO N FIG. 30. ter A in BAC, with a radius =DC, will have a point B or O in common with ray BD. In the first case, manifestly z=rt.L; but in the second case (~ 25) (OAO-OCD): OAB=1 sin AOB, and so z=AOB. If therefore we take z=AOB, then DM will be II BN. ~ 35. If S were reality; we may, as follows, draw a straight i to one arm of an acute angle, [211 which is II to the other. N MP LT Take AM BC, and s \\ suppose AB=BC so / /. \\ small (by ~ 19), that B- F H__ clif we draw BN II AM FA I (~ 34), ABN > the r.3. 31. given angle. Moreover draw CP Il AM (~ 34); and take NBG and PCD each equal to the given angle; rays BG and CD will cut; for if ray BG (falling by construction within NBC) cuts ray CP in E; we shall have (since BN-CP), LEBC< ZECB, and so EC<EB. Take EF=EC, EFR SCIENCE ABSOLUTE OF SPACE. 39 =ECD, and FS II EP; then FS will fall within BFR. For since BN II CP, and so BN II EP, and BN II FS; we shall have (~ 14) ZFBN+ ZBFS < (st. Z =FBN+BFR); therefore, BFS <BFR. Consequently, ray FR cuts ray EP, and so ray CD also cuts ray EG in some point D. Take now DG=DC and DGT=DCP=GBN; we shall have (since CDGD) BN-GT -CP. Let K (~ 19) be the point of the L-form line of BN falling in the ray BG, and KL the axis; we shall have BN-KL, and so BKL=BGT=DCP; but also KL-CP: therefore manifestly K fall on G, and GT 1I BN. But if HO bisects I BG, we shall have constructed HO II BN. ~36. Having given the ray CP and the s R plane MAB, take CB I the M N /P plane MAB, BN (in plane '\ \ BCP) -BC, and CQ II BN (~ 34); the intersection of ray CP (if this ray falls within \ -- B - BCQ) with ray BN (in the FIG. 32. plane CBN), and so with the plane MAB is found. And if we are given the two planes PCQ, MAB, and we have CB to plane MAB, CR I plane PCQ; and (in plane BCR) BN IBC, CS' CR, BN will fall in plane MAB, and CS in plane PCQ; and the 40 SCIENCE ABSOLUTE OF SPACE. intersection of the straight BN with the straight CS (if there is one) having been found, the perpendicular drawn through this intersection, in PCQ, to the straight CS will manifestly be the intersection of plane MAB and plane PCQ. ~ 37. On the straight AM II BN, is found such N Q T p an A, that AM-BN. If (by[22] /I \ ~ 34) we construct outside of the plane NBM, GT II Gc BN, and make BG1GT, B << 1/^ GC-GB, and CPIIGT; A and so place the hemiFIG. 33. FG.33 plane TGD that it makes with hemi-plane TGB an angle equal to that which hemi-plane PCA makes with hemi-plane PCB; and is sought (by ~ 36) the intersection straight DQ of hemi-plane TGD with hemiplane NBD; and BA is made I DQ. We shall have indeed, on account of the similitude of the triangles of L lines produced on the F of BN (~ 21), manifestly DB=DA, and AM-BN. Hence easily appears (L-lines being given by their extremities alone) we may also find a fourth proportional, or a mean proportional, and execute in this way in F, apart from Axiom XI, all the geometric constructions made SCIENCE ABSOLUTE OF SPACE. 41 on the plane in i. Thus e. g. a perigon can be geometrically divided into any special number of equal parts, if it is permitted to make this special partition in 2. ~ 38. If we construct (by ~ 37) for example, // M NBQ-= rt.Z, and make (by./. 1 ~ 35), in S, AMI ray BQ and Il Bt --- —[Q BN, and determine (by ~37) IM=BN; we shall have, if IA FIG. 34. =X, (~ 28), X=1: sin ~ rt. Z =2, and x will be constructed geometrically. And NBQ may be so computed, that IA differs from i less than by anything given, which happens for sin NBQ=-/e. ~ 39. If (in a plane) PQ and ST are 11 to the straight MN (~27), and AB, CD are equal perpendiculars to MN; manifestly ADECK F CI- _ ABEA; and so the angles PAS // Q (perhaps mixtilinear) ECP, M -\ / I N EAT will fit, and EC=EA. \If, moreover, CF=AG, then A~- --- T AACF-ACAG, and each Fro. 3.. FG. 35 is half of the quadrilateral FAGC. If FAGC, HAGK are two quadrilaterals of this sort on AG, between PQ and ST; their equivalence (as in Euclid) is evident, as also 42 SCIENCE ABSOLUTE OP SPACE. the equivalence of the triangles AGC, AGH, standing on the same AG, and having their vertices on the line PQ. Moreover, ACF= CAG, GCQ-CGA, and ACF+ACG+GCQ= st.Z (~ 32); and so also CAG+ACG+CGA= [23] st.; therefore, in any triangle ACG of this sort, the sum of the three angles =st.. But whether the straight AG may have fallen upon AG (which II MN), or not; the equivalence of the rectilineal triangles AGC, AGH, as well of themselves, as of the sums of their angles, is evident. ~ 40. Equivalent triangles ABC, ABD, C c- Q- (henceforth rectilineal), having one side equal, have the TM ~ -^f sums of their angles equal. 5-D For let MN bisect AC and z / BC, and take (through C) FIG. 36. PQIII MN; the point D will fall on line PQ. For, if ray BD cuts the straight MN in the point E, and so (~ 39) the line PQ at the distance EF=EB; we shall have AABC-=ABF, and so also AABD —AABF, whence D falls at F. But if ray BD has not cut the straight MN, let C be the point, where the perpendicular bisecting the straight AB cuts the line PQ, and SCIENCE ABSOLUTE OF SPACE. 43 let GS=HT, so, that the line ST meets the ray BD prolonged in a certain K (which it is evident can be made in a way like as in ~ 4); moreover take SR=SA, RO IIST, and O the intersection of ray BK with RO; then ZABR -AABO (~39), and so zABC>AABD (contra hyp.). ~ 41. Equivalent triangles ABC, DEF have the sums of their triangles equal. L F Po For let MN bisect w c-S0 s AC and BC, and PQ ~G~ ~ S -bisect DF and FE; and take RS Il MN, A B D -EandTO IlPQ; theperFIG. 37. 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-, \ABC-= DEF. But the AAKB (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=-DEF; 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- [24] angle, in S, is <st.Z. Therefore L falls on C. ~ 42. Let u be the supplement of the anglesum of the AABC, but v of ADEF; then is AABC: DEF=u: v. F For if p be the area of each of the triangles ACG, GCH, HCB, DFK, KFE; and \ / ABC=m.-.p, and ADEF= DK E A G H B n.p; and s the angle-sum of FIG. 38. any triangle equivalent top, manifestly st. Z- -un.s-(m-l1)st. Z =st. Z-m(st. Z-s); and u=-m(st.Z-s); and in like manner v= n(st.Z-s). Therefore AABC: ADEF=P-n n-=:v. It is evidently also easily extended to the case of the incommensurability of the triangles ABC, DEF. In the same way is demonstrated that triangles on a sphere are as the excesses of the sums of their angles above a st.<. If two angles of the spherical A are right, the third z will be the said excess. But SCIENCE ABSOLUTE OF SPACE. 45 (a great circle being called p) this A is manifestly Z sp2 27 2L (~ 32, VI); consequently, any triangle of whose angles the excess is z, is z2 4:. ~ 43. Now, in S, the area of a rectilineal A is expressed by means of the sum of its angles. I' M N' If AB increases to infinity; (~ 42) AABC:(rt._-u-v) /will be constant. But A ABC -/i ' BACN (~ 32, V), and rt.Z / -/u-v —z (~ 1); and so,...., BACN: z = ABC: (rt. ZA D ^u-v)=BAC'N':z'. ~~ —D Moreover, manifestly (~ 30) FIG.39. BDCN: BD'C'N'=r: r' tan z: tan z'. But for y''o, we have BD'C'N' tan z' BAC'N' 1' and also ' — 1; consequently, BDCN: BACN=tan z z. But (~ 32) BDCN=r.i=i2 tan z; therefore, BACN=z.i2. 46 SCIENCE ABSOLUTE OF SPACE. Designating henceforth, for brevity, any triangle the supplement of whose angle-sum is z by A, we will therefore have A =z.2. M S. Hence it readily flows // %\ that, if OR11AM and o T ROII AB, the area comprehended between the B A straights OR, ST, BC[25] (which is manifestly the absolute limit of the area of rectilineal triangles increasing without bound, or of A for z-Lst. ), is =i2= area Oi, in F. This limit balng denoted by o, moreover (by ~ 30) 7r2=tan2z. o area or in F (~ 21)= area os (by ~32, VI) if the chord CD is called s. If now, bisecting at right angles the given radius s of the circle in 4 plane (or the L form radius of the circle in F), we construct (by ~ 34) DB iCN; by dropping CA i DB, and M N erecting CM - CA, we shall / get z; whence (by ~ 37), assuming at pleasure an L form /Z radius for unity, tan2z can be determined.geometrically by means of two uniform lines A of the same curvature (which, c -~~-~i their extremities alone being FIG. 41. given and their axes con SCIENCE ABSOLUTE OP SPACE. 47 structed, manifestly may be compared like straights, and in this respect considered equivalent to straights). Moreover, a quadrilateral, ex. gr. regular = is constructed as follows: - c Take ABC=rt.Z, BAC=1 rt. Z, ACB=- rt. Z, and BC=x. A By mere square roots, X (from _i42 ~ 31, II) can be expressed and (by FIG.42. 37) constructed; and having X (by ~ 38 or also ~~ 29 and 35), x itself can be determined. And octuple A ABC is manifestly = o, and by this a plane circo of radius s is geometrically squared by means of a rectilinear figure and uniform unes of the same species (equivalent to straights as to comparison inter se); but an F form circle is planified in the same manner. and we have either the Axiom XI of Euclid true or the geometric quadrature of the circle, although thus far it has remained undecided, which of these two.has place in reality. Whenever tan2z is either a whole number, or a rational fraction, whose denominator (reduced to the simplest form) is either a prime number of the form 2m+1 (of which is also 2=2~+1), or a product of however many prime numbers of this form, of which each (with the 48 SCIENCE ABSOLUTE OF 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 illustrious Gauss (remarkable invention of our, nay of every age) (and only for such values[26] of z), construct a rectilineal figure -tan2zo= area os. For the division of o (the theorem of ~ 42 extending easily to any polygons) manifestly requires the partition of a st. Z, which (as can be shown) can be achieved geometrically only under the said condition. But in all such cases, what precedes conducts easily to the desired end. And any rectilineal 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 ï, or some S (and which one) exists. This, however, is reserved for a more suitable occasion. APPENDIX I. REMARKS ON THE PRECEDING TREATISE, BY BOLYAI FARKAS. [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 acuteness. 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 speaking) the sides of a spherical triangle are accepted 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=1. Doubtless, of the Euclidean axiom has been said in volume first enough and to spare: for 50 SCIENCE ABSOLUTE OF SPACE. the case if it were not true, is demonstrated (Tom. I. App., p. 13), that there is given a certain i, for which the I there mentioned is =-e (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 i-'; truly the Euclidean system is as if the limit of the antiEuclidean (for i' o). Assume for the case of i existing, the unit =i, and extend the concepts sine and cosine also to inmainary arcs, so that, p designating an arc whether real or imaginary, p~ —I -P,</ e_ +e; is called the 2 cosine of, and Pes-i — P i e -e is called the sine of p (as Tom. I., p. 177). Hence for q real q -q -q ---.Z — q'r-1.~-_ e-e e -e = sin(-û\-1) 2_- - 2'-n-1 =-sin(yv-l). SCIENCE ABSOLUTE OF SPACE. 51 q - - qvr.l. qy-z.4hz So e +e e +e 2 -_ --- _ ---- =cos(-q-) -cos(qV_1); 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 radish equai to the sides opposite. Moreover is demonstrated for the case of i existing, that the circle of radius y is - LeY_-e y which, for i=1, becomes r(eY-e-Y). Therefore (~ 31 ibidem), for a right-angled rectilineal triangle of which the sides are a and b, the hypothenuse c, and the angles opposite to the sides a, b, c are a, p, rt., (for i 1), in I, 1:Sn1 (=i (e- e-e).(ea-e-a); and so e e — ea —a 1: sina = 2- Whence 1: sin a 2<-i - 52 SCIENCE ABSOLUTE OF SPACE. =-sin (c^_ -):-sin (ai). And hence 1: sin a=sin (ci(_): sin (al_ 1). In II becomes cos a:sin p=cos (a<i-_):1; in III becomes cos (cV-1)=cos (a-i-).cos (b4-l). These, as all the formulas of plane trigonometry deducible from them, coincide completely with the formulas of spherical trigonometry; except that if, ex. gr., also the sides and the angles opposite them of a right-angled spherical triangle and the hypothenuse bear the same names, the sides of the rectilineal triangle are to be divided by <-1 to obtain the formulas for the spherical triangle. Obviously we get (clearly as Tom.,II., p. 252), from I, 1: sin a=sin c sin a, from II, 1: cos a=sins: cos a; from III, cos c=cos a cos b. Though it be allowable to pass over other things; yet I have learned that the reader may be offended and impeded by the deduction omitted, (Tom. I., App., p. 19) [in ~ 32 at end]: it will not be irrelevant to show how, ex. gr., from c -c_-0 a -a z b -b z e+e'- e+e j e'+e i follows SCIENCE ABSOLUTE OF SPACE. 53 C2=a2+b2. (the theorem of Pythagoras for the Euclidean system); probably thus also the author deduced it, and the others also follow in the same manner. Obviously we have, the powers of e being expressed by series (like Tom. I., p. 168), k k. k k k4 i=3+ 1 4 + 4 i 12 2.3. i 2.3.4. * *' * k k k2 k3 k4 1-1 +,2- l3 34... and so 2i2 2.3.i3 2.3.4.i4 k -k2 k4 k6 e +e 1-2+k+ + ee 3.4. 4 3.4.5.6.i6 =2+ - 2, (designating by Zu k"2 - the sum of all the terms after j; and we have — 0, while i' oC. For all the terms which follow -, are divided by i2; the first kq4 k2 term will be 4 and any ratio <-2; and 3.4ij2 though the ratio everywhere should remain this, the sum would be kTom. I., p. 131), k 4. k2 _ k4 34^2 I __ _ _ 3.4.i2 t j J 3.4. (2 2)' which manifestly o-0, while iA o, And from 54 SCIENcE ABSOLUTE OF SPACE. -c -_ ((a+b) -(a+b) a-b -(a-b) ) ee i+ e - +e e +e i +e i J follows (for w, v, À taken like u) c2+w_, (a+b)+v+2+ (a+b)2 +_)7 2+ 2+ +2+ z 2 Ii2 i2 And hence 2 a2+2ab+b2+a 2-2ab+b2+v+~-w which -a2+b2. which a2'a.+b APPENDIX II. SOMFE POINTS IN JOHN BOLYAI 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 sphereM ' p limit, and the corresponding axes H- L ray AM, ray BN (~ 23). Gc K They demonstrate that, if., p, cl D k r designate the arcs of the circle limit AB, CD, HL, separated by _A B segments of the axis AC-1, AH Fis. 43. =X, we have r ) r~ l[J. Lobachevski represents the value of r by e-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 OF SPACE. If we put =a, and r, r' are arcs situated at the distances y, i from a, we shall have ^-=,Y-Y, -, =,s-=I, whence Y=l i. 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 lu. Therefore, if we put z2-u, we have Y=tan (z+u)- tan z+tan -u' 1-tan z tan u whence we get, having regard to the value of tan -u- Y-', tan z-1 (Y-Y-') I- Ij (~30). If now y is the semi-chord of the arc of circle-limit 2r, we prove (~30) that - tan z constant. Representing this constant by i, and making y tend toward zero, we have 2r _1, whence 2y 2y I i Y SCIENCE ABSOLUTE OF SPACE. 57 or putting -=k, I=el, y kI iekl-1=kt (1+A), À being infinitesimal at the same time as k. Therefore, for the limit, 1=/ and consequently ïI —e. The circle traced on the sphere-limit with the arc r of the curve-limit for radius, has for length 2zr. Therefore, (Dy=27r=2.i tan z-=i (Y-Y-1). In the rectilineal A where (, j designate the angles opposite the sides a, b, we have (~ 25) sin: sin,=~a:Ob-=ri(A-A1): 7i(B-B-1) =sin (av-1):sin (b —1). 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. If we designate the constant r by q, we tan z shall have, as before rynq (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 determinate, 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 -Y must therefore be zero, and consequently i= oo. 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 In (a)=sin B tan In (p), a being the hypothenuse of a right-angled triangle, f 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-):(P-P-'). Now, putting for brevity, 2n (k)=k', we have tan 2f': tan 2a'= (cot a'-tan a'): (cot f' -tan ')=(A-A-'):(P-P-'):sin B. APPENDIX III. LIGHT FROM NON-EUCLIDEAN SPACES ON THE TEACHING OF ELEMENTARY GEOMETRY. BY G. B. HALS'rED. 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 advance so important, not only for mathematics but for philosophy, has had some metaphysical opponents, and as long ago as 1878 I mentioned in my Bibliography of Hyper 60 SCIENCE ABSOLUTE OF 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 reviewer, Mr. Ziwet, was then also an anti-nonEuclidean, though since converted. He says, " we find that some of the best German 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 define 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 " direction" 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 produced 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 elucidate 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 virtually the properties of the straight line determine 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 Geometry, 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 measurement 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, solZ in der Wissenschaft nicht ohne Beweis geglaubt werden." Professor W. B. Smith (Ph. D., Goettingen), has written: " Nothing could be more unfortunate 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 turning 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 rotation, 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 Professor 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 XI. 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. Playfair'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 Parkas, 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 matter because he had hot, 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 tetragon 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 "argumentum 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 2oo 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 isin 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 common, and further, that a line may be turned about any point in it without leaving the surface. But if instead of the plane we take a spherical surface, and instead of a line a great ScIENCE ABSOLUTE OF SPACE. 67 circle on the sphere, all these conditions are again satisfied. "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 edition is dated 1830, says: "Professor Playfair, in the Notes to his 'Elements of Geometry' [1813], has proposed another demonstration, founded on a remarkable 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 circumstance 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 because 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 cti ~ that an arc of this A!'\ ~ circle may be car'/ \ ried round precisely in the way;~/ ~\ described and re~/ ~\.~ turn to its old situation, and yet r \-. there be no prey^~/ t ~ tense for inferring 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 OF 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 returns to its place because the exterior angles have been equal to four right angles, is a mistake. From which it is a legitimate conclusion, 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 carried 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, Oxford, in his "Curiosa Mathematica," Part I, pp. 70-71, 3d Ed., 1890, says: "Yet another process has been inventedquite fascinating in its brevity and its elegance-which, though involving the same fallacy as the Direction-Theory, proves Euc. I, 32, without even mentioning the dangerous word 'Direction.' 70 SCIENCE ABSOLUTE OF SPACE. "We are told to take any triangle ABC; to ~C — o G produce CA to D; to make part of CD, viz., X \0 AD, revolve, about A, into the position A BE; 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 OF 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 inclined to AE. " That is, you are asserting, 'A pair of lines which are equally inclined to a certain transversal, are so to any transversal.' LDeducible from Euc. I, 27, 28, 29.]" MATHEMATICAL WORKS BY GEORGE BRUCE HALSTED, A. M. (PRINCETON); PU. D. (JOHNS HOPKINS); EX-FELLOW 0F PRINCETON COLLEGE; TWICE FELLOW OF JOHNS HOPKINS UNIVERSITY; INTERCOL LEGIATE PRIZEMAN: SOMETIME INSTRUCTOR IN POST GRADUATE MATHEMATICS, PRINCETON COLLEGE; PROFESSOR OF MATHEMATICS, UNIVERSITY OF TEXAS, AUSTIN, TEXAS; MEMBER OF THE AMERICAN MATHEMATICAL SOCIETY; MEMBER OF THE LONDON MATHEMATICAL SOCIETY; MEMBER OF THE ASSOCIATION FOR THE IMPROVEMENT OF GEOMETRICAL TEACHING; EHRENMITGLIED DES COMITES DES LOBACHEVSKY-CAPITALS; MIEMBRO DE LA S0 -CIEDAD CIENTIFICA "ALZATE " DE MEXICO; SOCIO CORRESPONSAL DE LA SOCIEDAD DE GEOGRAFIA Y ESTADISTICO DE MEXICO; SOCIETAIRE PERPETIJAL DE LA SOCIETE MATI:EMATIQUE DE FRANCE; SOCIO PERPETUO DELLA CIRCOLO MATEMATICO DI PALERMO; PRESIDENT 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. IOth St., New York. Chapman & Hall. London. Synthetic Geometry. 2nd Ed. 1893. $1.50. John Wiley & Sons. 53 E. 10th St., New York. Lobachévski'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 Lobachévski. 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 LOBACHIVSKI. BY A. VASILIEV. 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 revolutionized geometrical science, but has attracted the attention of physicists, psychologists and philosophers." " Without question the best and most authentic source of information on this original thinker." From a two-column Review in the Nation, April 4, 1895, by C. S. Pierce: "Kazàn was not the milieu for a man of genius, especially not for so profound a genius as that of Lobachévski." "All of Lobachévski's writings are marked by the same highstrung logic." I have read it with intense interest. By issuing this translation 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 Lobachévski. It is a most fortunate thing for us in the rank and file that you have maintained such an interest 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 instrumentality in spreading the news in America. Very sincerely, A. L. DANIELS, Professor of Mathematics, UJniversity of Vermont. STAUNTON. VA., October 13th, 1894. The history of the life and work of such a man as Lobachévski 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, BETILEIIHEM, PA., October 22nd, 1894. 1 have read the Lobachévski with much pleasure and-what is better-profit. Yours very truly. C. L. DOOLITTLE, Professor of Mathematics in the University of Pennsylvania. HALLE a. S., LAFONTAINESTR., 2; 23,10, '94. Hochgeehrter Herr: Auf der Naturforscherversammlung in Wien lernte ich Prof. Wasilief aus Kasan kennen, der mir erzaehlte, das Sie seine Rede bei der Lobatschefsky-Feier uebersetzen wollten. Diese Nachricht war mich sehr willkommen, da die russische mir unverstandlich ist. Nun erhalte ich heute von Ihnen diese Uebersetzung zugesandt und sage Ihnen dafuer meinen verbindlichsten Dank. Sie haben mit der Uebersetzung dieser interessanten Rede sich den Anspruch auf den Dank der mathematischen Welt erworben! Hochachtungsvoll Ihr ergebener, STAECKEL. STANFORD UNIVERSITY, PALO ALTO, CAL., October 19th, 1894. 1 have read the Lobachévski with the greatest interest, and rejoice 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 Geornetry we have. Yours truly, A. P. CARMAN, Professor of Physics 'L1eland 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 ELEMIENTS OF GEOMETRY WITH A COMPLETE THEORY OF PARALLELS. BY N. I. LOBACHEVSKI. Translated from the Russian by DR. GEORGE BRUCE HALSTED. [Thougli this is Lobachévski's greatest work, it has never before been translated out of the Russian into any other language whatever.]