Chapter 11

It would not be difficult to increase greatly the number of the given illustrations of unsolved questions relating directly to the natural numbers. In fact, the well-known greater Fermat theorem is a question of this type, which does not appear more important intrinsically than many others but has received unusual attention in recent years on account of a very large prize offered for its solution. In view of the fact that those who have become interested in this theorem often experience difficulty in finding the desired information in any English publication, we proceed to give some details about this theorem and the offered prize. The following is a free translation of a part of the announcement made in regard to this prize by the Konigliche Gesellschaft der Wissenschaften, Gottingen, Germany:

On the basis of the bequest left to us by the deceased Dr. Paul Wolskehl, of Darmstadt, a prize of 100,000 mk., in words, one hundred thousand marks, is hereby offered to the one who will first succeed to produce a proof of the great Fermat theorem. Dr. Wolfskehl remarks in his will that Fermat had maintained that the equation

x + y = z

could not be satisfied by integers whenever is an odd prime number. This Fermat theorem is to be proved either generally in the sense of Fermat, or, in supplementing the investigations by Kummer, published in Crelle's Journal, volume 40, it is to be proved for all values of for which it is actually true. For further literature consult Hibert's report on the theory of algebraic number realms, published in volume 4 of the Jahreshericht der Deutschen Mathernatiker-Vereinigung, and volume 1 of the Encyklopadie der mathematischen Wissenschaften.

The prize is offered under the following more particular conditions.

The Konigliche Gesellschaft der Wissenschaften in Gottingen decides independently on the question to whom the prize shall be awarded. Manuscripts intended to compete for the prize will not be received, but, in awarding the prize only such mathematical papers will be considered as have appeared either in the regular periodicals or have been published in the form of monographs or books which were for sale in the book-stores. The Gesellschaft leaves it to the option of the author of such a paper to send to it about five printed copies.

Among the additional stipulations it may be of interest to note that the prize will not be awarded before at least two years have elapsed since the first publication of the paper which is adjudged as worthy of the prize. In the meantime the mathematicians of various countries are invited to express their opinion as regards the correctness of this paper. The secretary of the Gesellschaft will write to the person to whom the prize is awarded and will also publish in various places the fact that the award has been made. If the prize has not been awarded before September 13, 2007, no further applications will be considered.

While this prize is open to the people of all countries it has become especially well known in Germany, and hundreds of Germans from a very noted university professor of mathematics to engineers, pastors, teachers, students, bankers, officers, etc., have published supposed proofs. These publications are frequently very brief, covering only a few pages, and usually they disclose the fact that the author had no idea in regard to the real nature of the problem or the meaning of a mathematical proof. In a few cases the authors were fully aware of the requirements but were misled by errors in their work. Although the prize was formally announced more than seven years ago no paper has as yet been adjudged as fulfilling the conditions.

It may be of interest to note in this connection that a mathematical proof implies a marshalling of mathematical results, or accepted assumptions, in such a manner that the thing to be proved is a NECESSARY consequence. The non-mathematician is often inclined to think that if he makes statements which can not be successfully refuted he has carried his point. In mathematics such statements have no real significance in an attempted proof. Unknowns must be labeled as such and must retain these labels until they become knowns in view of the conditions which they can be proved to satisfy. The pure mathematician accepts only necessary conclusions with the exception that basal postulates have to be assumed by common agreement.

The mathematical subject in which the student usually has to contend most frequently with unknowns at the beginning of his studies is the history of mathematics. The ancient Greeks had already attempted to trace the development of every known concept, but the work along this line appears still in its infancy. Even the development of our common numerals is surrounded with many perplexing questions, as may be seen by consulting the little volume entitled "The Hindu-Arabic Numerals," by D. E. Smith and L. C. Karpinski.

The few mathematical unknowns explicitly noted above may suffice to illustrate the fact that the path of the mathematical student often leads around difficulties which are left behind. Sometimes the later developments have enabled the mathematicians to overcome some of these difficulties which had stood in the way for more than a thousand years. This was done, for instance, by Gauss when he found a necessary and sufficient condition that a regular polygon of a prime number of sides can be constructed by elementary methods. It was also done by Hermite, Lindemann and others by proving that epsilon and rho are transcendental numbers. While such obstructions are thus being gradually removed some of the most ancient ones still remain, and new ones are rising rapidly in view of modern developments along the lines of least resistance.

These obstructions have different effects on different people. Some fix their attention almost wholly on them and are thus impressed by the lack of progress in mathematics, while others overlook them almost entirely and fix their attention on the routes into new fields which avoid these difficulties. A correct view of mathematics seems to be the one which looks at both, receiving inspiration from the real advances but not forgetting the desirability of making the developments as continuous as possible. At any rate the average educated man ought to know that there is no mathematician who is able to solve all the mathematical questions which could be proposed even by those having only slight attainments along this line.

THE ABORIGINAL ROCK-STENCILLINGS OF NEW SOUTH WALES

BY DR. CHAS. B. DAVENPORT

COLD SPRING HARBOR, N. Y.

IN a number of places in eastern Australia curious aboriginal markings are found on the faces of the sandstone cliffs. A good idea of them is given by the photographs. These came from Wolgan Gap near Wallerang in the Blue Mountain region of New South Wales. They are found on overhanging rocks that have served as shelters or camping places for the aborigines and which doubtless have protected their works of art.

These stencillings are made by a sort of spatter work, something like that in vogue a generation ago in this country, using leaves, etc., as forms. The rocks at Wolgan Gap are a coarse sandstone stained almost black by an iron oxide derived from included bands of ironstone. These black surfaces were selected by the artists. Nearby in the rock is a band of shale which had disintegrated at its exposed edge to a white powder. The native artist put some of this white powder in his mouth, placed his hand or foot upon the rock, and blew the moistened powder upon and around his outstretched fingers or toes. When he removed them they were outlined on the rock. Since the sandstone is coarse and deeply pitted, the moist powder was blown into minute cavities where it has remained despite the erosive activities of some generations. The presence of the powder is shown on the photographs as a sort of halo around the object. The hands are either right or left, and, in some cases, both hands seem to have been stencilled at once. Sometimes the whole arm and hand are stencilled together, and in one of the photographs a boomerang is shown. The age of these stencils is not known. They were first discovered at Wolgan Gap about sixty years ago, but others have been known for a longer time, for instance, those at Greenwich, Parametta River, near Sydney.

The significance of these stencillings has been the subject of some controversy. The natives may have been induced to make them as boys carve their names on benches or even rocks. The materials for making the stencillings were present and, the example once having been set, others would emulate it. It is interesting that similar stencillings of the hands were made by cave men on the walls of some of the European caves, as, for instance, those of Aurignac in southern France. Evidently spatter work is no modern pastime.

THE PROGRESS OF SCIENCE

SUBSTITUTES FOR WAR

THIS war, beyond measure disastrous to civilization, is a trial also of our democracy. We may hope that it is an old-world war and an old-men's war, repugnant to the genius of our newer life. The statements of some of our public men and the contents of some of our newspapers can not be read without discouragement. But it is also true that there has perhaps not appeared a cartoon in any American newspaper tending to glorify war, and no legislation has so far been enacted in preparation for war. There is good reason to believe that the people have not been infected by the contagion of blood.

As Professor Patrick argued in a recent issue of the Monthly, man is by genetic inheritance a fighting and a playing animal, not an animal delighting in steady work. The ape and the tiger will be exterminated elsewhere in nature before they will be suppressed in man. It is a slow process, but surely proceeding.

The writer of this note has determined the proportion of each century in which the leading nations have been engaged in war. The curve thus found has no great reliability; for it does not take into account the percentages of the peoples concerned, but its course clearly indicates that even under circumstances as they have been, wars will come to an end. And there is good reason to believe that the newer condition--universal education and universal suffrage, democratic control, improved economic conditions of living for the people, the scientific attitude--will tend to bend the curve more rapidly toward the base line of permanent "peace on earth and good will to men."

While man has inherited instincts which exhibit themselves in playing and fighting, the same instincts may by social control be diverted to playing the games of art or science, to fighting disease and vice. It is rarely wise or feasible to attempt to suppress instincts; they should be directed so as to provide desirable conduct. Loyalty to family, to group, to neighborhood and to nation can not be lightly cast away for an abstract cosmopolitanism. But it can be expressed otherwise than by seizing everything in sight by cunning or by violence.

William James, the great psychologist, in one of his brilliant essays published in The Popular Science Monthly for October, 1910, tells us that history is a bath of blood; we inherit the war-like type; our ancestors have bred pugnacity into our bone and marrow; showing the irrationality and horror of war does not prevent it; but a moral equivalent can be found by enlisting an army to toil and suffer pain in doing the hard and routine work of the world. It is doubtful, however, if the "gilded youths" to whom James refers would accept "dish-washing, clothes-washing and window-washing, road-building and tunnel-making, foundries and stoke-holes," as a substitute for war, and for the great mass of the people there is more than enough of these things. It is to escape from them that we seek excitement and adventure, intoxication by drugs and war.

Professor Cannon, of Harvard University, proposes international football and other athletic contests as substitutes for war. The adrenal glands, whose secretions excite the combative and martial emotions, must function, and their activity, he argues, can be directed in this way. Mr. Bryan has just now made the proposal that we build six great national roads by which armies might be collected for defence; the secretary of the navy has founded a Naval Inventions Board; the postmaster general has suggested that aeroplanes be used to deliver mail in order that we may have an aerial corps ready for service. There may be an element of the absurd in some of these proposals, as there would be in using submarines to catch cod fish, so that there might be practise in building and managing such crafts for peaceful pursuits. There is, however, psychological justification for aiming to direct the emotions so that their discharge is not destructive, but of benefit to the nation and to the world. Such would be the development of our national resources, the construction of railways, roads, waterworks and the like; social and political reforms; progress in the care of public health, in education and in scientific research. It is proposed that the next congress should spend half a billion dollars on the army and navy. It is possible that on a plebiscite vote, exactly under existing conditions, a majority would vote to make the department of war a department of public works, military defence being only one of its functions, and to spend the sum proposed on public works useful in case of war, but not an incitement to war.

NATIONAL WEALTH AND PUBLIC INDEBTEDNESS

WHILE the lives and the wealth of the European nations are being sacrificed on a scale hitherto unparalleled, it is well in the interests of those nations, as well as of our own, that we conserve the lives and wealth of our own people. The greatest wealth of a nation is its children, its productive workers, its scientific men and other leaders, its accumulated knowledge and social traditions. These are immeasurable, but the Bureau of the Census has recently prepared a report on the material wealth and indebtedness, according to which it is estimated that the total value of all classes of property in the United States, exclusive of Alaska and the insular possessions, in 1912, was $187,739,000,000, or $1,965 per capita. This estimate is presented merely as the best approximation which can be made from the data available and as being fairly comparable with that published eight years ago. The increase between 1904 and 1912 was 75 per cent., for the total amount and 49 per cent. for the per capita. Real estate and improvements, including public property, alone constituted $110,677,000,000, or 59 per cent. of the total, in 1912. The next greatest item, $16,149,000,000, was contributed by the railroads; and the third, $14,694,000,000, represented the value of manufactured products, other than clothing and personal adornments, furniture, vehicles and kindred property.

The net public-indebtedness in 1913 amounted to $4,850,461,000. This amount was made up as follows: National debt, $1,028,564,000, or $10.59 per capita; state debt, $345,942,000, or $3.57 per capita; county debt, $371,528,000, or $4.33 per capita; and municipal debt, $2,884,883,000, or $54.27 per capita. Thus the average urban citizen's share of the net federal, state, county and municipal debt combined was $72.76; and the average rural citizen's share of the net federal, state and county debt combined was $18.49.

The total federal debt in 1910 was $2,916,205,000, of which amount $967,366,000 was represented by bonds, $375,682,000 by non-interest-bearing debt (principally United States notes or "greenbacks"), and $1,573,157,000 by certificates and notes issued on deposits of coin and bullion. Against this indebtedness there was in the treasury $1,887,641,000 in cash available for payment of debt, leaving the net national indebtedness at $1,028,564,000, or $10.59 per capita. The increase in the net indebtedness between 1902 and 1913 amounted to 6 per cent., but for the per capita figure there was a decrease of 13 per cent. The burden due to the national debt is thus very light in comparison with that imposed by the indebtedness of other great nations.

The state debt, however, rests still more easily on the shoulders of the average citizen, being only one third as great as that of the nation. The total state indebtedness in 1913 was $422,797,000, and the net debt--that is, the total debt less sinking-fund assets--was $345,942,000, or $3.57 per capita. The net debt increased by 44.5 per cent. between 1902 and 1913, and the per capita net debt by 18 per cent.

The total county debt in 1913 amounted to $393,207,000, of which amount $371,528,000, or $4.33 per capita, was net debt. The net indebtedness increased by 89 per cent. between 1902 and 1913, and the per capita net indebtedness by 55 per cent. By far the greatest item of indebtedness in this country is that of municipalities. This amounted in 1913 to an aggregate of $3,460,000,000, of which $2,884,883,000, or $54.27 per capita, represented net indebtedness. The rate of increase in net indebtedness between 1902 and 1913 was 114 per cent.

While the nations of Europe are involving themselves in the toils of debts, we should use our vast surplus wealth to pay the national, state and municipal debts, even those contracted for public improvements. We save every year about $100 for each adult and child of the country and waste about an equal sum. It would be well if this wealth could be invested for the benefit of each, and education and scientific research are the most productive of all investments.

SCIENTIFIC ITEMS

WE record with regret the death of Karl Eugen Guthe, professor of physics in the University of Michigan and dean of the Graduate School, in Hanover, Germany; of John Howard Van Amringe, long dean of Columbia College and professor of mathematics; of Carlos J. Finlay, known for his advocacy of the theory that yellow fever is transmitted by mosquitoes; of A. J. Herbertson, of Wadham College, Oxford, professor of geography in the university; of Julius von Payer, the distinguished polar explorer and artist, of Vienna, and of Guido Goldsehmiedt, professor of chemistry in the University of Vienna.

DR. JACQUES LORE, of the Rockefeller Institute for Medical Research, has been elected a foreign fellow of the Linnean Society, London.--Dr. David Bancroft Johnson, president of Winthrop Normal and Industrial College, of Rockhill, S. C., has been elected president of the National Education Association, in succession to Dr. David Starr Jordan, chancellor of Stanford University.

A MEMORIAL to Johann C. Reil, the anatomist, has been erected in Halle. It stands in front of the university clinic, the seat of his labors until called to Berlin in 1810. He died in 1813, aged fifty-five years.--A bronze bas-relief--the work of Mr. S. N. Babb--is about to be erected in St. Paul's Cathedral in memory of Captain Scott and his companions who perished in the Antarctic. At the request of the committee responsible for the memorial an inscription has been written by Lord Curzon, which reads as follows: "In memory of Captain Robert Falcon Scott, C.V.O., R.N., Dr. Edward Adrian Wilson, Captain Lawrence E. G. Oates, Lieut. Henry R. Bowers and Petty Officer Edgar Evans, who died on their return journey from the South Pole in February and March, 1912. Inflexible of purpose, steadfast in courage, resolute in endurance in the face of unparalleled misfortune. Their bodies are lost in the Antarctic ice. But the memory of their deeds is an everlasting monument."

THE SCIENTIFIC MONTHLY

NOVEMBER, 1915

PAPUA, WHERE THE STONE-AGE LINGERS

BY DR. ALFRED GOLDSBOROUGH MAYER

WITH their undaunted spirit for braving the wilds, the English entered New Guinea in 1885. For centuries the great island had remained a mere outline upon the map the fever-haunted glades of its vast swamps and the broken precipices of its mountain ranges having defied exploration, more than the morose and savage character of its inhabitants. Even in the summer of 1913, Massy Baker the explorer, discovered a lake probably 100 miles or more in shore-line, which had remained hidden in the midst of the dark forests of the Fly and Strickland River regions, and here savages still in the stone age, who had never seen a white man, measured the potency of their weapons against the modern rifle.

To-day there are vast areas upon which the foot of the white man has not yet trodden, and of all the regions in the tropical world New Guinea beckons with most alluring fascination to him to whom adventure is dearer than life.

Far back in the dawn of European exploration, the Portuguese voyager Antonio de Abreu, may have seen the low shores of western New Guinea, but it is quite certain that sixteen years later, in 1527, Don Jorge de Meneses cruised along the coast and observed the wooly-headed natives whom he called "Papuas." The name "New Guinea" was bestowed upon the island by the Spanish captain, Ynigo Ortz de Retes, in 1515, when he saw the negroid natives of its northern shores.

Then there came and passed some of the world's greatest navigators. Torres wandering from far Peru, to unknowingly discover the strait which bears his name; Dampier, the buccancer-adventurer, and, in 1768, the cultured, esthetic Bougainville, who was enraptured by the beauty of the deep forest-fringed fjords of the northeastern coast. Cook, greatest of all geographers, mapped the principal islands and shoals of the intricate Torres Strait in 1770; and a few years later came Captain Bligh, the resourceful leader of his faithful few, crouching in their frail sail boat that had survived many a tempest; since the mutineers of the Bounty had cast them adrift in the mid-Pacific. In the early years of the nineteenth century the scientifically directed Astrolabe arrived, under the command of Dumont D'Urville, and, later, Captain Owen Stanley in the Rattlesnake, with Huxley as his zoologist, Then, in 1858, came Alfred Russel Wallace, the codiscoverer of Darwinism, who, by the way, is said to have been the first Englishman who ever actually resided in New Guinea.

The daring explorers and painstaking surveyors came and went, but the great island remained a land of dread and mystery, guarded by the jagged reefs of its eastern shores, and the shallow mud flats, stretching far to sea-ward beyond the mouths of the great rivers of its southern coast. So inaccessible was Papua that even the excellent harbor of Port Moresby, the site: of the present capital, was not discovered until 1873. One has but to stifle for a while in the heavy air that flows lifeless and fetid over the lowlands as if from a steaming furnace, or to scent the rank odors of the dark swamps, where for centuries malaria must linger, to appreciate the reason for the long-delayed European settlement of the country. But those who blaze the path of colonial progress are not to be deterred by temperatures or smells; let us remember that Batavia, "the white man's graveyard," is now one of the world's great commercial centers; and Jamaica, the old fever camp of the British army, is now a health resort for tourists.