SCIENCEscience.sciencemag.org/content/sci/19/480/401.full.pdf · press on beyond the guide; risk life for the magnificent sensation of a waterspout, a cloudburst, an avalanche, a

SCIENCEA WEEKLY JOURNAL DEVOTED TO THE ADVANCEMENT OF SCIENCE, PUBLISHING THE

OFFICIAL NOTICES AND PROCEEDINGS OF THE AMERICAN ASSOCIATIONFOR THE ADVANCEMENT OF SCIENCE.

FRIDAY, MARCH 11, 1904.

CONTENTS:T'he Amterican Association for the Advance-

ment of Science:-The Message of Non-Euclidean Geometry:PROFESSOR GEORGE BRUCE HALSTED ....... 401

The Society for Plant Morphology and Phys-iology: PROFESSOR WV. F. GANONG ........ 413

Scientific Books:-Still another Memoir on Palcospondylus:PROFESSOR BASHFORD DEAN. Catalogue ofKeyboard Musical Instruments: CHARLESK. WEAD...................................... 425

Scientific Journals and Articles............. 427Societies and Academies:-

The Philosophical Society of Washington:CIIARLES K. WEAD. The Chemical Societyof Washington: A. SEIDELL. The ElishaMitchell Scientific Society: ALVIN S.WHEELER..................................... 428

Discussion and Correspondence:-Convocation Week: PROFESSOR CHARLES E.BESSEY, PROFESSOR GEO. F. ATKINSON, DR.W. J. HOLLAND. The Raphides of CalciumOxalate: DR. H. W. WILEY. The Tlerm'Bradfordian': F. A. R.................. 429

Special Articles:-Notes on Flu,orescence and Phosphores-cence: W. S. ANDREWS .................. 43.5

Paleontological Notes:Pleurocoelus versus Astrondon; The Armorof Zeuglodon: F. A. L................... 436

Fossil Fishes in the American Museum ofNatural History ........................ 437

Scientific Notes and News.................437University and Educational News........... 440

MSS. intended for publication and books, etc.. intendedfor review should be sent to the Editor of SCIENCE, Garri-son-on-Hudson, N. Y.

THE AMERICAN ASSOCIATION FOR THEADI ANCEMENT OF SCIENCE.

THE MESSAGE OF NON-EUCLIDEANGEOMETRY.*

1. MATHEMATICS AND ITS HISTORY.THE great Sylvester. once told me that

he and Kronecker, in attempting a defini-tion of mathematics, got so far as to agreethat it is poetry.But the history of this poesy is itself

poetry, and the creation of non-Euclideangeometry gives new vantage-ground fromwhich to illuminate the whole subject, frombefore the time when Homer describes Pro-teus as finger-fitting-by-fives, or. counting,his seals, past the epoch when Lagrange,confronted with the guillotine and askedhow he can make himself useful in the newworld, answers simply, 'I will teach arith-metic.'Who has not wished to be a magician like

the mighty Merlin, or Dr. Dee, who wrotea preface for the first English translationof Euclid, made by Henricus Billingsley,afterward, Aladdin-like, Sir Henry Bill-ingsley, Lord Mayor of London?Was not Harriot, whose devices in alge-

bra our schoolboys now use, one of thethree paid magi of the Earl of Northum-berland? Do not our every-day numeralsstand for Brahmin and Mohammedan, com-ing first into Europe from the land of thesacred Ganges, around by the way of thepyramids and the Moorish Alhambra?

* Address of the vice-president and chairmanof Section A, American Association for the Ad-vancement of Science, St. Louis meeting, De-cember, 1903.

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The appearance of courses on the historyof mathematics in all our foremost univer-sities is a fortunate and promising sign ofthe times. I had the honor of being thefirst to give such a course in America, atPrinceton, in 1881.

2. GEOMETRY AND ITS FOUNDERS.

But something especially fascinating,pure, divine, seems to pertain to geometry.When asked how God occupies himself,

Plato answered, 'He geometrizes contin-ually.'

It is a difficult, though highly interest-ing, undertaking to investigate the vestigesof primitive geometry. Geometric figuresand designs appear in connection with theprimitive arts; for example, the making ofpottery. Arts long precede anything prop-erly to be called science. The first crea-tions by mankind are instruments for life,though it is surprising how immediatelydecoration appears; witness the sketchesfrom life of mammoth and mastodon andhorses by prehistoric man. But, in a sense,even the practical arts must be preceded bytheoretical creative acts of the human mind.Man is from the first a creative thinker.Perhaps even some of our present theo-retical presentation of the universe is dueto creative mental acts of our pre-humanancestors. For example, that we inevitablyview the world as consisting of distinct in-dividuals, separate, distinct things, is a pre-human contribution to our working theoryand representation of the universe. It isconscious science, as a potential presenta-tion and explanation of everything, whichcomes so late.Rude instruments were made for astron-

omy.The creative imagination which put the

bears and bulls and crabs and lions andscorpions into the random-lying stars madefigures which occur in the Book of Job,more ancient than Genesis itself.

The daring astrologer, whose predictionsforetold eclipses, saw no reason why hisconstructions should not equally fit humanlife. He chose to create a causal relationbetween the geometric configurations of theplanets and the destinies of individuals.This was the way of science, where thoughtprecedes and helps to make fact. No de-scription or observation is possible withouta precedent theory, which stays and sticksuntil some mind creates another to fight it,and perhaps to overshadow it.

That legend of the origin of geometrywhich attributes it to the necessity of re-fixing land boundaries in Egypt, where allwere annually obliterated by the Nile over-flow, is a too-ingenious hypothesis, madetemporarily to serve for history. Somepractical devices for measurement arose inEgypt, where periodic fertility fostered aconsecutive occupancy, whose records, ac-cording to Flinders Petrie, we have formore than nine thousand years.But in the Papyrus of the Rhind, meas-

urements of volume come before those forsurface.Geometry as a self-conscious science

waits for Thales and Pythagoras.We find in Herodotus that Thales pre-

dicted an eclipse memorable as happeningduring a battle between the Lydians andMedes. The date was given by Baily asB. C. 610.

So we know about when geometry, wemay say when science, began; for thoughprimarily geometer, Thales taught thesphericity of the earth, was acquaintedwith the attracting power of magnetism,and noticed the excitation of electricty inamber by friction.A greater than he, Pythagoras, was born

B. C. 590 at Samos, traveled also intoEgypt and the east, penetrating even intoIndia. Returning in the time of the lastTarquin, and finding Samos under the do-minion of the tyrant Polyerates, he went.

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as a voluntary exile to Italy, settled atCroton (as Ovid mentions), and there cre-ated and taught new and sublimer hypoth-eses for our universe. The most diverselydemonstrated and frequently applied theo-rem of geometry bears his name. The firstsolution of a problem in that most subtleand final of ways, by proving it impossible,is due to him; his solution of the problemto find a common submultiple of the hy-pothenuse and side of an isosceles righttriangle, an achievement whereby he cre-ated incommensurability.

It is noteworthy that this making of in-commensurables is confused by even themost respectable of the historians of mathe-matics with the creation of irrational num-bers. But in the antique world there wereno such numbers as the square root of twoor the square root of three. Such num-bers can not be discovered, and it was cen-turies before they were created. TheGreeks had only rational numbers.

3. EUCLID.

Under the Horseshoe Falls at Niagarapress on beyond the guide; risk life forthe magnificent sensation of a waterspout,a cloudburst, an avalanche, a tumblingcathedral of waterblocks! It must end inan instant, this extravagant downpour ofwhole wealths of water. Then out; andlook away down the glorious canyon, andread in that graven history how this mo-mentary riotous chaos has been just so,precisely the same, for centuries, for ages,for thousands of years.

In the history of science a like antithesisof sensations is given by Euclid's geometry.

In the flood of new discovery and richadvance recorded in books whose merenames would fill volumes, we ask ourselveshow any one thing can be permanent?Yet, looking back, we see this Euclid notonly cutting his resistless way through therock of the two thousand years that make

the history of the intellectual world, but,what is still more astounding, we find thatthe profoundest advance of the last twoeenturies has only served to emphasize theconsciousness of Euclid's perfection.

Says Lyman Abbott, if you want an in-fallible book go not to the Bible, but toEuclid.

In 'The Wonderful Century,' AlfredRussel Wallace says, speaking of all timebefore the seventeenth century: "Then go-ing backward, we can find nothing of thefirst rank except Euclid's wonderful sys-tem of geometry, perhaps the most remark-able product of the earliest civilizations."Says Professor Alfred Baker, of the Uni-

versity of Toronto: "Of the perfection ofEuclid (B. C. 290) as a scientific treatise,of the marvel that such a work could havebeen produced two thousand years ago, Ishall not here delay to speak. I contentmyself with making the claim that, as ahistorical study, Euclid is, perhaps, themost valuable of those that are taken upin our educational institutions."At its very birth this typical product of

the Greek genius assumed sway over thepure sciences. In its first efflorescence,through the splendid days of Theon andHypatia, fanatics could not murder it asthey did Hypatia, nor later could that dis-mal flood, the dark ages, drown it. Likethe phcenix of its native Egypt it risesanew with the new birth of culture. AnAnglo-Saxon, Adelhard of Bath, finds itclothed in Arabic vestments in the Moorishland of the Alhambra.

In 1120, Adelhard, disguised as a Mo-hammedan student, went to Cordova, ob-tained a Moorish copy of Euclid's 'Ele-ments,' and made a translation from theArabic into Latin.The first translation into English (1570)

was made by 'Henricus Billingsley,' after-ward Sir Henry Billingsley, Lord Mayorof London, 1591. And up to this very

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year, throughout the vast system of ex-aminations carried on by the British gov-ernment, by Oxford and by Cambridge,to be accepted, no proof of a theorem ingeometry should infringe Euclid's sequenceof propositions. For two millenniums hisaxioms remained undoubted.

4. THE NEW IDEA.

The break from Euclid's charmed circlecame not at any of the traditional centersof the world's thought, but on the eircuin-ference of civilization, at Maros-Vasarhelyand Temesvar, and again at Kazan on theVolga, center of the old Tartar kingdom;and it came as the creation of a willful,wild Magyar boy of twenty-one and aninsubordinate young Russian, who, a poorwidow 's son from Nijni-Novgorod, entersas a charity student the new university ofKazan.The new idea is to deny one of Euclid's

axioms and to replace it by its contradic-tory. There results, instead of chaos, abeautiful, a perfect, a marvelous newgeometry.

5. HOW THE NEW DIFFERS FROM THE OLD.

Euclid had based his geometry on cer-tain axioms or postulates which had in alllands and languages been systematicallyused in treatises on geometry, so that therewas in all the world but one geometry.The most celebrated of these axioms wasthe so-called parallel-postulate, which, in aform due to Ludlam, is simply this: 'Twostraight lines which cut one another can notboth be parallel to the same straight line.'Now this sane Magyar, John Bolyai, and

this Russian, Lobachevski, made a geom-etry based not on this axiom or postulate,but on its direct contradiction. Wonder-ful to say, this new geometry, founded onthe flat contradiction of what had beenforever accepted as axiomatic, turned outto be perfectly logical, true, self-consistent

and of marvelous beauty. In it many ofthe good old theorems of Euclid and ourown college days are superseded in a sur-prising way. Through any point outsideany given straight line can be drawn aninfinity of straight lines in the same planewith the given line, but which nowherewould meet it, however far both were pro-duced.

6. A CLUSTER OF PARADOXES.

In Euclid, Book I., Proposition 32 isthat the sum of the angles in every recti-lineal triangle is just exactly two rightangles. In this new or non-Euclideangeometry, on the contrary, the sum of theangles in every rectilineal triangle is lessthan two right angles.

In the Euclidean geometry parallelsnever approach. In this non-Euclideangeometry parallels continually approach.

In the Euclidean geometry all pointsequidistant from a straight line are on astraight line. In this non-Euclidean geom-etry all points equidistant from a straightline are on a curve called the equidis-tantial.In the Euclidean geometry the limit ap-

proached by a circumference as the radiusincreases is a straight line. In the non-Euclidean geometry this is a curve calledthe oricycle. Thus the method of Kempe'sbook 'How to draw a straight line,' wouldhere draw not a straight line, but a curve.

In the Euclidean geometry, if threeangles of a quadrilateral are right, thenthe fourth is right, and we have a rect-angle. In this non-Euclidean geometry, ifthree angles of a quadrilateral are right,then the fourth is acute, and we never canhave any rectangle.

In the Euclidean geometry two perpen-diculars to a line remain equidistant. Inthis non-Euclidean geometry two perpen-diculars to a line spread away from eachother as they go out; their points two

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inches from the line are farther apart thantheir points one inch from the line.

In the Euclidean geometry every threepoints are either on a straight line or acircle. In this non-Euclidean geometrythere are triplets of points which are nei-ther costraight nor concycltic. Thus threepoi-nts each one inch above a straight line areneither on a straight line nor on a circle.

7. MISTAKE OF THE INEXPERT.

These seeming paradoxes could be multi-plied indefinitely, and they form strikingexamples of this new geometry. Theyseem so bizarre, that the first impressionproduced on the inexpert is that the tradi-tional geometry could easily be proved, asagainst this rival, by careful experiments.Into this error have fallen Professors An-drew W. Phillips and Irving Fisher, ofYale University. In their 'Elements ofGeometry,' 1898, page 23, they say: "Lo-bachevski proved that we can never get ridof the parallel axiom without assuming thespace in which we live to be very differentfrom what we know it to be through ex-perience. Lobachevski tried to imagine adifferent sort of universe in which theparallel axiom would not be true. Thisimaginary kind of space is called non-Euclidean space, whereas the space inwhich we really live is called Euclidean,because Euclid (about 300 B. C.) firstwrote a systematic geometry of our space. "Now, strangely enough, no one, not even

the Yale professors, can ever prove thisnaive assertion. If any one of the possiblegeometries of u'oniform space could ever beproved to be the system actual in our ex-ternal physical world, it certainly couldnot be Euclid's.

Experience can never give, for instance,such absolutely exact metric results as pre-cisely, perfectly two right angles for theangle sum of a triangle. As Dr. E. W.Hobson says: "It is a very significant fact

that the operation of counting, in connec-tion with which numbers, integral andfractional, have their origin, is the one andonly absolutely exact operation of a mathe-matical character which we are able toundertake upon the objects which we per-ceive. On the other hand, all operationsof the nature of measurement which we canperform in connection with the objects ofperception contain an essential element ofinexactness. The theory of exact measure-ment in the domain of the ideal objects ofabstract geometry is not immediately de-rivable from intuition."

8. THE ARTIFICIALLY CREATED COMPONENTIN SCIENCE.

In connecting a geometry with experi-ence there is involved a process which wefind in the theoretical handling of any em-pirical data, and which, therefore, shouldbe familiarly intelligible to any scientist.The results of any observations are al-

ways valid only within definite limits ofexactitude and under particular conditions.When we set up the axioms, we put in placeof these results statements of absolute pre-cision and generality. In this idealizationof the empirical data our addition is at firstonly restricted in its arbitrariness in somuch as it must seem to approximate, mustapparently fit, the supposed facts of ex-perience, and, on the other hand, must in-troduce no logical contradiction. Thus ouractual space to-day may very well be thespace of Lobachevski or Bolyai.

If anything could be proved or disprovedabout the nature of space or geometry byexperiments, by laboratory methods, thenour space could be proved to be the spaceof Bolyai by inexact measurements, theonly kind which will ever be at our dis-posal. In this way it might be known tobe non-Euclidean. It never can be knownto be Euclidean.

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9. DARWINISM AND GEOMETRY.

The doctrine of evolution as commonlyexpounded postulates a world independentof man, and teaches the production of manfrom lower forms of life by wholly naturaland unconscious causes. In this statementof the world of evolution there is need ofsome rudimentary approximative practicalgeometry.The mighty examiner is death. The

puppy, though born blind, must still beable to superimpose his mouth upon thesource of his nourishment. The little chickmust be able, responding to the stimulusof a small bright object, to bring his beakinto contact with the object so as to graspand then swallow it. The springing goat,that too greatly misjudges an abyss, doesnot survive and thus is not the fittest.

So, too, with man. We are taught thathis ideas must in some way and to somedegree of approximation correspond to thisindependent world, or death passes uponhim an adverse judgment.But it is of the very essence of the doc-

trine of evolution that man's knowledge ofthis independent world, having come bygradual betterment, trial, experiment, ad-aptation, and through imperfect instru-ments, for example the eye, can not bemetrically exact.

If two natural hard objects, susceptibleof high polish, be ground together, theirsurfaces in contact may be so smoothed asto fit closely together and slide one on theother without separating. If now a thirdsurface be ground alternately against eachof these two smooth surfaces until it ac-curately fits both, then we say that eachof the three surfaces is approximatelyplane. If one such plane surface cutthrough another, we say the commonboundary or line where they cross is ap-proximately a straight line. If threeapproximately plane surfaces on objects

cut through a fourth, in general they makea figure we may call an approximate tri-angle. Such triangles vary greatly inshape. But no matter what the shape, ifwe cut off the six ends of any two suchand place them side by side on a planewith their vertices at the same point, thesix are found, with a high degree of ap-proximation, just to fill up the plane aboutthe point. Thus the six angles of any twoapproximate triangles are found to be to-gether. approximately four right angles.Now, does the exactness of this approxi-

mation depend only on the straightness ofthe sides of the original two triangles, oralso upon the size of these triangles?

If we know with absolute certitude, asthe Yale professors imagine, that the sizeof the triangles has nothing to do with it,then we know something that we have noright to know, according to the doctrineof evolution; something impossible for usever to have learned evolutionally.

10. THE NEW EPOCH.

Yet before the epoch-making ideas ofLobachevski and John Bolyai every onemade this mistake, every one supposed wewere perfectly sure that the angle-sum ofan actual approximate triangle approachedtwo right angles with an exactness depend-ent only on the straightness of the sides,and not at all on the size of the triangle.

11. THE SLIPS OF PHILOSOPHY.

The Scotch philosophy accounted for this-absolute metrically exact knowledge byteaching that there are in the human mindcertain synthetic theorems, called intui-tions, supernaturally inserted there. Dr.McCosh elaborated this doctrine in a bigbook entitled 'The Intuitions of the MindInductively Investigated.' One of thesesupernatural intuitions was Euclid's par-allel-postulate! Voila!

'Yet,' to quote a sentence from Wenley 's

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criticism in SCIENCE, of McCosh's discipleOrmond, 'we may well doubt whether athinker, standing with one foot firmlyplanted on the Rock of Ages and the otherpointing heavenward, has struck the atti-tude most conducive to progress.'

K:ant, supposing that we knew Euclid'sgeometry and Aristotle's logic to be trueabsolutely and necessarily, accounted forthe paradox by teaching that this seem-ingly universal synthetic knowledge was inreality particular, being part of the appa-ratus of the human mind itself.But now the very foundations are cut

away from under the Kantian system ofphilosophy by this new geometry which isin simple and perfect harmony with ex-perience, with experiment, with the prop-erties of the solid bodies and the motionsabout us. Thus this new geometry hasgiven explanation of what in the old geom-etry was accepted without explanation.

12. WHAT GEOMETRY IS.

At last we really know what geometry is.Geometry is the science created to giveunderstanding and mastery of the externalrelations of things; to make easy the ex-planation and description of such relations,and the transmission of this mastery.Geometry is the most perfect of the sci-ences. It precedes experiment and is safeabove all experimentation.The pure idea of a plane is something we

have made, and by aid of which we seesurfaces as perfectly plane, over-ridingimperfections and variations, which them-selves we can see only by help of our self-created precedent idea. Just so thestraight line is wholly a creation of ourown.

13. ARE THERE ANY LINES?I was once consulted by an eminent

theologian and a powerful chemist as towhether there are really any such thingsas lines. I drew a chalk-mark on the

blackboard, and used the boundary idea.Along the sides of the chalk-mark is therea common boundary where the white endsand the black begins, neither white norblack, but common to both?

Said the theologian, yes. Said the chem-ist, no.Though lines are my trade, I sympa-

thized with the chemist.There is nothing there until I create a

line and then see it there, if I may say Isee what is an invisible creation of mymind.

Geometry is in structure a system oftheorems deduced in pure logical wayfrom certain unprovable assumptions pre-created by auto-active animal and humanminds.

14. THE REQUIREMENT OF RIGOR INREASONING.

Some unscientific minds have a personalantipathy to 'a perfect logical system,''deduced logically' from simple funda-mental truths.' But as Hilbert says:"The requirement of rigor, which has be-come proverbial in mathematics, corre-sponds to a universal philosophic necessityof our understanding; and, on the otherhand, only by satisfying this requirementdo the thought content and the suggestive-ness of the problem attain their full effect.Besides, it is an error to believe that rigorin the proof is the enemy of simplicity.On the contrary, we find it confirmed bynumerous examples that the rigorousmethod is at the same time the simplerand the more easily comprehended. Thevery effort for rigor, forces us to find outsimpler methods of proof."Let us look at the principles of an-

alysis and geometry. The most suggestiveand notable achievements of the last cen-tury in this field are, as it seems to me,the arithmetical formulation of the concept

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uf the continuum, and the discovery ofnon-Euclidean geometry."

The importance of the advance they hadmade was fully realized by John Bolyaiand Lobachevski, who claimed at once, un-flinchingly, that their discovery or creationmarked an epoch in human thought so mo-mentous as to be unsurpassed by anythingrecorded in the history of philosophy orscience, demonstrating, as had never beenproved before, the supremacy of purereason, at the very moment of overthrow-ing what had forever seemed its surestpossession, the axioms of geometry.

15. THE YOUTH LOBACHEVSKI.

Young Lobachevski at the University ofKazan, though a charity student, and, asseeking a learned career, utterly dependenton the authorities, yet plunged into allsorts of insubordination and wildness.Among other outbursts, one night at elevenO clock he scandalized the despotic Russianauthorities of the Tartar town by shootingoff a great skyrocket, which prank put himpromptly in prison. However, he con-tinued to take part in all practical jokesand horse-play of the more daring stu-dents, and the reports of the commandantand inspector are never free from bittercomplaints against the outrageous Loba-chevski. His place as 'Kammerstudent 'he lost for too great indulgence toward themisbehavior of the younger students at aChristmas festivity. In spite of all, he,ventured to attend a strictly forbiddenmasked ball, and what was worse, in dis-cussing the supp.osed interference of Godto make rain, etc., he used expressionswhich subjected him to the suspicion ofatheism. From the continual accusing re-ports of the commandant to the Rektor,the latter, took a grudge against thetroublesQme Lobachevski, and reported hisbadness to the curator, who, in turn, withexpressions of intense regret that Loba-

chevski should so tarnish his brillianitqualities, said he would be forced to in-form the minister of education. Loba-chevski seemed about to pay dear for hisyouthful wantonness. He was to come upas a candidate for the master's degree, butwas refused by the senate, explicitly be-cause of his bad behavior. But his friend,the foreign professor of mathematics, nowrallied the three other foreign professorsto save him, if he would appear before thesenate, declare that he rued his evil be-havior, and solemnly promise completebetterment.

This was the mettle of the youth, thedare-devil, the irrepressible, who startledthe scientific sleep of two thousand years,who contemptuously overthrew the greatLegendre, and stood up beside Euclid, thegod of geometers; this the Lobachevski whoknew he was right against a scornful world,who has given us a new freedom to explainand understand our universe and ourselves.

16. THE BOY BOLYAI.

Of the boy Bolyai, joint claimant of thenew world, we have a brief picture by hisfather. "My (13 + 1) year old son, whenhe reached his ninth year, could do nothingmore than speak and write German andMagyar, and tolerably play the violin bynote. He knew not even to add. I beganat first with Euclid; then he became fa-miliar with Euler; now he not only knowsof Vega (which is my manual in the col-lege) the first two volumes completely, buthas also become conversant with the thirdand fourth volumes. He loves differentialand integral calculus, and works in themwith extraordinary readiness and ease.Just so he lightly carries the bow throughthe hardest runs in violin concerts. Nowhe will so,on finish my lectures on physicsand chemistry. On these once he alsopassed with my grown pupils a public ex-amination given in the Latin language, an

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examination worthy of all praise, where inpart others questioned him ad aperturam,and in part, as opportunity served, I lethim carry out some proofs in mechanicsby the integral calculus, such as variablemotion, the tautochronism of the cycloid,and the like. Nothing more could bewished. The simplicity, clearness, quick-ness and ease were enrapturing even forstrangers. He has a quick and compre-hensive head, and often flashes of genius,which many paths at once with a glancefind and penetrate. He loves pure deeptheories and astronomy. He is handsomeand rather strongly built, and appearsrestful, except that he plays with otherchildren very willingly and with fire. Hischaracter is, as far as one can judge, firmand noble. I have destined him as a sacri-fice to mathematics. He also has conse-crated himself thereto."His mother, nee Zsuzsanna Benk6 Arkosi,

wonderfully beautiful, fascinating, of ex-traordinary mental capacity, but alwaysnervous, so idolized this only child thatwhen in his fifteenth year he was to besent to Vienna to the K. K. Ingenieur-Akademie, she said it seemed he should go,but his going would drive her distracted.And so it did.Appointed 'sous-lieutenant,' and sent to

Temesvfar, he wrote thence to his father aletter in Magyar, which I had the goodfortune to see at Maros-Vatsarhely:MIy Dear and Good Father:

I have so much to write about my new inven-tions that it is impossible for the moment to enterinto great details, so I write you only on onefourth of a sheet. I await your answer to myletter of two sheets; and perhaps I would nothave written you before receiving it if I had notwished to address to you the letter I am writingto the Baroness, which letter I pray you to sendher.

First of all I reply to you in regard to thebinomial.

,o * * * * * * *

Now to something else, so far as space permits.

I intend to write, as soon as I have put it intoorder, and when possible to publish, a work onparallels.At this moment it is not yet finished, but the

way which I have followed promises me with cer-tainty the attainment of the goal, if it in generalis attainable.

It is not yet attained, but I have discovered suchmagnificent things that I am myself astonished atthem. It would be damage eternal if they werelost. When you see them, my father, you yourselfwill acknowledge it.Now I can not say more, only so much: that

from nothing I have created another wholly newworld.

All that I have hitherto sent you compares tothis only as a house of cards to a castle.

P. S.-I dare to judge absolutely and with con-viction of these works of my spirit before you, myfather; I do not fear from you any false interpre-tation (that certainly I would not merit), whichsignifies that, in certain regards, I consider youas a second self.

Nor was the young Magyar deceived.The early flashings of his genius culmin-ated here in a piercing search-light pene-trating and dissolving the enchanted wallsin which Euclid had for two thousandyears held captive the human mind.The potential new universe, whose crea-

tion this letter announces, afterward setforth with master strokes in his 'ScienceAbsolute of Space,' contains the old asnothing more than a special case of thenew.Already all the experts of the mathe-

matical world are his disciples.

17. SOLVING THE UNIVERSE.

Henceforth the non-Euclidean geometrymust be reckoned with in all culture, in allscientific thinking. It shows that the rid-dle of the universe is an indetermninateequation capable of entirely different setsof solutions. It shows that our universeis largely man-made, and must be oftenremade to be solved.

In SCIENCE for November 20, 1903, page643, W. S. Franklin, under a heading for

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410 SCIENCE.

which he shows scant warrant, expresseshimself after the following naive fashion:A clear understanding of the essential limita-

tions of systematic physics is important to theengineer; it is I think,equally important to thebiologist, and it is of vital importance to thephysicist, for, in the case of the physicist, toraise the question as to limitations is to raise thequestion as to whether his science does after alldeal with realities, and the conclusion which mustforce itself on his mind is, I think, that his sci-ence, the systematic part of it, comes very nearindeed to being a science of unrealities.

Of course, we deeply sympathize withthis seemingly sad perception, with itsaccompanying 'simple weeps, ' 'trailingweeps' and 'steady weeps,' but are temptedto prescribe a tonic or bracer in the formof a correspondence course in non-Euclid-ean geometry.At least in part, space is a creation of

the human mind entering as a subjectivecontribution into every physical experi-ment. Experience is, at least in part,created by the subject said to receive it,but really in part making it.

In rigorously founding a science, theideal is to create a system of assumptionscontaining an exact and complete descrip-tion of the relations between the element-ary concepts of this science, its statementsfollowing from these assumptions by puredeductive logic.

18. GEOMETRY NOT EXPERIMENTAL.

Now, geometry, though a natural sci-ence, is not an experimental science. Ifit ever had an inductive stage, the experi-ments and inductions must have been madeby our pre-human ancestors.

Says one of the two greatest livingmathematicians, Poincare, reviewing thework of the other, Hilbert's transcendentlybeautiful 'Grundlagen der Geometrie':What are the fundamental principles of geom-

etry? What is its origin; its nature; its scope?These are questions which have at all times eni-

[N. S. VOL. XIX. No. 480.

gaged the attention of mathematicians and think-ers, but which took on an entirely new aspect,thanks to the ideas of Lobachevski and of Bolyai.For a long time we attempted to demonstrate

the proposition known as the postulate of Euclid;we constantly failed; we know now the reason forthese failures.

Lobachevski succeeded in building a logical edi-fice as coherent as the geometry of Euclid, but inwhich the famous postulate is assumed false, andin which the sum of the angles of a triangle isalways less than two right angles. Riemann de-vised another logical system, equally free fromcontradiction, in which this sum is on the otherhand always greater than two right angles. Thesetwo geometries, that of Lobachevski and that ofRiemann, are what are called the non-Euclideangeometries. The postulate of Euclid then can notbe denionstrated; and this impossibility is as abso-lutely certain as any mathematical truth whatso-ever. * * *The first thing to do was to enumerate all the

axioms of geometry. This was not so easy as onemight suppose; there are the axioms which onesees and those which one does not see, which areintroduced unconsciously and without being no-ticed.

Euclid himself, whom we suppose an impeccablelogician, frequently applies axioms which he doesnot expressly state.

Is the list of Professor Hilbert final? We maytake it to be so, for it seems to have been drawnup with care.

But just here this gives us a startlingincident: the two greatest living mathe-maticians both in error. In my own classa young man under twenty, R. L. Moore,proved that of Hilbert 's 'betweenness' as-sumptions, axioms of order, one of the fiveis redundant, and by a proof so simple andelegant as to be astonishing. Hilbert hassince acknowledged this redundancy.The same review touches another funda-

mental point as follows:Hilbert's Fourth Book treats of the measure-

ment of plane surfaces. If this measurement canbe easily established without the aid of the prin-ciple of Archimedes, it is because two equivalentpolygons can either be decomposed into trianglesin such a way that the component triangles of theone and those of the other are equal each to each(so that, in other words, one polygon can be con-

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verted into the other after the manner of theChinese puzzle [by cutting it up and rearrangingthe pieces]), or else can be regarded as the dif-ference of polygons capable of this mode of de-composition (this is really the same process, ad-mitting not only positive triangles but alsonegative triangles).But we must observe that an analogous state of

affairs does not seem to exist in the case of twoequivalent polyhedra, so that it becomes a ques-tion whether we can determine the volume of thepyramid, for example, without an appeal more orless disguised to the infinitesimal calculus. It is,then, not certain whether we could dispense withthe axiom of Archimedes as easily in the measure-ment of volumes as in that of plane areas. More-over, Professor Hilbert has not attempted it.

Max Dehin, a young man of twenty-one, in Mathematische Annalen, Band 55,proved that the treatment of equivalenceby cutting into a finite number of partscongruent in pairs, can never be extendedfrom two to three dimensions.

Poincare's review first appeared in Sep-tember, 1902. But on July 1, 1902, I hadalready presented, before this very section,a complete solution of the question orproblem he proposes, the determination ofvolume without any appeal to the infini-tesimal calculus, without any use of theaxiom of Archimedes.

19. THE TEACHING OF GEOMETRY.

As Study has said: "Among conditionsto a more profound understanding of evenvery elementary parts of the Euclideangeometry, the knowledge of the non-Euclid-ean geometry can not be dispensed with."How shall we make this new creation,

so fruitful already for the theory ofknowledge, for kenlore, bear fruit for theteaching of geometry ? What new waysare opened by this masterful explosion ofpure genius, shattering the mirrors whichhad so dazzlingly protected from percep-tion both the flaws and triumphs of theold Greek 's marvelous, if artificial, con-struction ?

One advance has been safely won andmay be rested on. There should be a pre-liminary course of intuitive geometrywhich does not strive to be rigorouslydemonstrative, which emphasizes the sen-suous rather than the rational, giving fullscope for those new fads, the using of padsof squared paper, and the so-called labo-ratory methods so well adapted for thefeeble-minded. Hailmann, in his preface,sums up 'the purpose throughout' in thesesignificant words: 'And thus, incidentally,to stimulate genuine vital interest in thestudy of geometry.'

I remember Sylvester 's smile when hetold me he had never owned a mathematicalor drawing instrument in his life.His great twin brother, Cayley, speaks

of space as 'the representation [creation]lying at the foundation of all external ex-perience.' 'And these objects, points, lines,circles, etc., in the mathematical sense ofthe terms, have a likeness to, and are rep-resented more or less imperfectly, andfrom a geometer's point of view, no matterhow imperfectly, by corresponding phys-ical points, lines, circles, etc.'But geometry, always relied upon for

training in the logic of science, for teach-ing what demonstration really is, must bemade more worthy the world 's faith.There is need of a text-book of rationalgeometry really rigorous, a book to giveevery clear-headed youth the benefit of hisliving after Bolyai and Hilbert.

20. THE NEW RATIONAL GEOMETRY.

The new system will begin with stillsimpler ideas than did the great Alex-andrian, for example, the 'betweenness'assumptions; but can confound objectorsby avoiding the old matters and methodswhich have been the chief points of objec-tion and contest. For example, says Mr.Perry, 'I wasted much precious time ofmy life on the fifth book of Euclid.' Says

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[N. S. VOL. XIX. No. 480.

the great Cayley: 'There is hardly any-thing in mathematics more beautiful thanhis wondrous fifth book.'For my own part, nothing ever better

repaid study. But the contest is over, fornow, at last, without sacrificing a whit ofrigor, we are able to give the whole matterby an algebra as simple as if only approxi-mate, like Euclid, including incommen-surables without even mentioning them.

Again, we shall regain the pristinepurity of Euclid in the matter of whatJules Andrade calls 'cette malheureuse etillogique definition' (Phillips and Fisher,§7): 'A straight line is a line which is theshortest path between any two of itspoints.'As to this hopeless muddle, which has

been condemned ad nauseam, notice that itis senseless without a definition for thelength of a curve. Yet, Professor A.Lodge, in a discussion on reform, says:" I believe we could not do better thanadopt some French text-book as our model.Also I., 24, 25, being obviously related toI., 4, are made to immediately follow itin such of the French books as define astraight line to be the shortest distancebetween two points." Professor Lodge,then, does not know that the Frenchthemselves have repudiated this nauseouspseudo-definition. Of it Laisant says (p.223):This definition, almost unanimously abandoned,

represents one of the most remarkable examplesof the persistence with which an absurdity canpropagate itself throughout the centuries.

In the first place, the idea expressed is incom-prehensible to beginners, since it presupposes thenotion of the length of a curve; and further, it isa vicious circle, since the length of a curve canonly be understood as the limit of a sum of recti-linear lengths; moreover, it is not a definition atall, since, on the contrary, it is a demonstrableproposition.

As to what a tremendous affair thisproposition really is, consult Georg Hamel

in Mathematische Annalen for this veryyear (p. 242), who employs to adequatelyexpress its content the refinements of theintegral calculus and the modern theory offunctions.

Moreover, underneath all this even isthe ass-umption of the theorem, Euclid, I.,20: 'Any two sides of a triangle are to-gether greater than the third side'; uponwhich proposition, which the Sophists saideven donkeys knew, Hilbert has thrownbrilliant new light in the Proceedings ofthe Lontdon Mathematical Society, 1902,pp. 50-68, where he creates a geometry inwhich the donkeys are mistaken, a geom-etry in which two sides of a triangle maybe together less than the third side, exhib-iting as a specific and definite example aright triangle in which the sum of the twosides is less than the hypothenuse.Any respectably educated person knows

that in general the length of a curve isdefined by the aggregate formed by thelengths of a proper sequence of inscribedpolygons.The curve of itself has no length. This

definition in ordinary cases creates for thecurve a length; but in case the aggregateis not convergent, the curve is regarded asnot rectifiable. It had no length, and evenour creative definition has failed to endowit with length; so it has no length, andlengthless it must remain.

If, however, it can be shown that thelengths of these inscribed polygons form aconvergent aggregate which is independentof the particular choice of the polygons ofthe sequence, the curve is rectifiable, itslength being defined by the number givenby the aggregate.

21. GEOMETRY WITHOUT ANY CONTINUITY

ASSUMPTION.

Euclid in his very first proposition andagain in I., 22, 'to make a triangle fromgiven sides,' uses unannounced a contin-

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uity assumption. But nearly the whole ofEuclid can be obtained without any con-tinuity assumption whatever, and this greatpart it is which forms the real domain ofelementary geometry.

Continuity belongs, with limits and in-finitesimals, in the C.alculus.

Professor W. G. Alexejeff, of Dorpat, in'Die Mathematik als Grundlage der Kritikwissenschaftlich- philosophischer Weltan-schauung' (1903), shows how men of sci-ence have stultified themselves by igno-rantly presupposing continuity. He callsthat a higher, standpoint which takes ac-count of the individuality of the elements,and gives as examples of this discrete ordiscontinuous mathematics the beautifulenumerative geometry, the invariants ofSylvester and Cayley, and in chemistry theatomic-structure theory of KekuLe and theperiodic system of the chemical elementsby Mendelejev, to which two theories, bothexclusively discrete in character, we maysafely attribute almost entirely the presentstandpoint of the science.

Still more must discontinuity play thechief role in biology and sociology, dealingas they do with differing individuals, cellsand persons. How desirable, then, thatthe new freedom should appear even asearly as in elementary geometry.

After mathematicians all knew thatnumber is in origin and basis entirely in-dependent of measurement or measurablemagnitude; after in fact the dominanttrend of all pure mathematics was itsarithmetization, weeding out as irrelevantany fundamental use of measurement ormeasurable quantity, there originated inChicago from the urbane Professor Dewey(whom, in parenthesis, I must thank for hisamiable courtesy throughout the article inthe Educational Review which he devotedto my paper on the 'Teaching of Geom-etry'), the shocking tumble or reversal that

the origin, basis and essence of number ismeasurement.Many unfortunate teachers and pro-

fessors of pedagogy ran after the newdarkness, and even books were issued try-ing to teach how to use these dark lines inthe spectrum for illuminating purposes.

There is a ludicrous element in theparody of all this just now in the domainof geometry.After mathematicians all know of the

wondrous fruit and outcome of the non-Euclidean geometry in removing all thedifficulties of pure elementary geometry,there comes another philosopher, a Mr.Perry, who never having by any chanceheard of all this, advises the cure of thesetroubles by the abolition of rational geom-etry.

Just as there was a Dewey movement sois there a Perry movement, and books ongeometry written by persons who neverread 'Alice in Wonderland' or its com-panion volume, 'Euclid and his ModernRivals.'But the spirits of Bolyai and Loba-

chevski smile on this well-meaning strenu-osity, and whisper, 'It is something toknow what proof is and what it is not; andwhere can this be better learned than in ascience which has never had to take onefootstep backward?'

GEORGE BRUCE HALSTED.KENYON COLLEGE.

THE SOCIETIY FOR PLANT MORPHOLOGYAND PHYSIOLOGY.

THE seventh regular annual meeting ofthis society was held, in conjunction withthe meetings of several other scientific so-cieties, at the University of Pennsylvania,Philadelphia, Pa., December 28-30, 1903.In the absence of the president and vice-president, the most recent past president,Dr. Erwin F. Smith, presided. Thoughnot large in point of numbers the meeting

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THE MESSAGE OF NON-EUCLIDEAN GEOMETRYGEORGE BRUCE HALSTED

DOI: 10.1126/science.19.480.401 (480), 401-413.19Science

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