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neglected the old; and whether a combination of what is sound and true in both may not rather be needed in order to attain the whole truth. It is doubtful whether supply does not react upon demand as much as demand on supply; whether the consideration of disutility, implied in the conception of cost of production, is not equally important with that of utility, and equally deserving of distinct investigation; whether, in fine, the efforts and exertions of producers to supply wants are not as potent a factor in advancing civilization, and as creative of new wants, as the pressure of wants and desires themselves. The Austrian writers allow so much—though perhaps they here exhibit some lack of distinct statement— to the influence of " cost of production," that they might, it would seem, go a little further, and place it on an equality with the principle of marginal utility. They would then, perhaps, recognize what Prof. Marshall, in his broader,and, as it appears to us, more philosophic, exposition of value, calls the fundamental symmetry of the laws of the forces working on both sides, which is exhibited in the analogy between " marginal utility "and "marginalcost of production," and a law of "diminishing returns" and one of "decreasing utility." They would, in short, without sacrificing altogether the vast amount of trouble bestowed by Ricardo and his followers on one side of the problem,assign a proper, and not an exclusive, emphasis to the side which they had themselves done so much to elucidate. For these reasons we consider Mr. Smart's modest conclusion —that " the last word on value has not been said by the Austrian'school "—to be as sound and as pertinent, as his exposition of their views is clear, pointed, and suggestive.

OUR BOOK SHELF.

Across Thibet. By Gabriel Bonvalot. Translated by C.

B. Pitman. Two Vols. (London: Cassell and Co.,

1891.) After the return of M. Bonvalot and Prince Henry of Orleans from the East, so much was said of their journey that we need not now repeat any of the details of M. Bonvalot's narrative. It may suffice for us to commend the book very cordially to the attention of readers who like to wander in imagination with travellers in remote parts of the world. M. Bonvalot, as his translator says, has those qualities of courage, self-command, tenacity, knowledge of human character, and good humour, which go to make up the successful traveller; and he writes of his achievements so simply and naturally that there is nothing to interfere with the reader's full enjoyment of his story. The travellers, as everyone interested in geographical exploration will remember, started from the frontiers of Siberia, and in the course of the journey which brought them to Tonquin passed right through Tibet. Their route lay to some extent over ground which no European had ever before traversed, and this is, of course, the portion of his subject on which M. Bonvalot writes most carefully and effectively. The work has been translated in a clear and pleasant style, and it is enriched with many interesting illustrations.

Light. By Sir H. Trueman Wood. "Whittaker's Library of Popular Science." (London: Whittaker and Co., 1891.) We have here a popular and interesting account of many of the facts relating to the nature and properties of light. The subject is treated in a way that will induce many readers to glance through its pages, even if they do not

more carefully peruse it ; while many a more advanced student will read the chapters on double refraction and polamation, lenses, and interference and refraction. Of other points touched on, we may mention spectrum analysis, optical instruments, chemical effects of light, fluorescence and phosphorescence—all of which are delightfully treated by the author.

In the appendix will be found a list of the more elementary and popular works on the subject, which should prove useful to those who wish to extend their knowledge.

LETTERS TO THE EDITOR.

[ The Editor does not hold himself responsible for opinions expressed by his correspondents. Neither can he undertake to return, or to correspond with the writers of, rejected manuscripts intended for this or any other part ofNatu RE. No notice is taken of anonymous communications.]

Opportunity for a Naturalist.

Will you allow me to say that the letter which you kindly inserted under this head in your issue of December 24, 1891 (p. 174), has brought me many replies? After considering them, I have made arrangements with Mr. O. V. Aplin (member of the British Ornithologists' Union, and author of "The Birds of Oxfordshire') to proceed to Uruguay in August nexl. Mr. Aplin will reside for six months on an eslaucia in the provinceof Minas, and devote himself primarily to birds, but will also collect insects and plants. P. L. Sclater.

3 Hanover Square, W.

Dwarfs and Dwarf Worship.

In the slow course of post in this Protectorate I have just received copies of the Times of September 3 containing Mr. K. G. Halliburton's paper on " Dwarf Races and Dwarf Worship," and of September 14 and 22, containing subsequent correspondence on the same subject. Having crossed the Atlas Mountains at several different points, and approached the district which is indicated by Mr. Halliburton as the original home and hidden sanctuary of his diminutive and venerated people, I have read his paper with much interest and may perhaps be permitted to criticize his conclusions. My chief during my expedition to Morocco, that distinguished traveller Mr. Joseph Thomson, is, I believe, at present in Katanga, anil therefore more inaccessible than I am ; but when he is able to -speak on the subject, his judgment on the case which Mr. Halliburton has very elaborately set up will not, I am confident, be different from mine.

Mr. Halliburton begins with a statement that is at once startling and decisive. The information he has collected puis it, he says, beyond question that there exists in the Atlas Mountains, only a few hundred miles from the Mediterranean, a race of dwarfs only 4 feet high, who are regarded with superstitious reverence or are actually worshipped, and whose existence has been kept a profound secret for 3000 years. Such an emphatic assertion ought to rest on clear and irrefragable evidence ; and I read Mr. Halliburton's paper in constant expectation of the proofs of his remarkable discovery, but reached the end of it without coming on a shred of testimony in support of his contention, of the slightest value to anyone acquainted with Morocco and the Moors. The paper is highly discursive, and abounds in what seem to me far-fetched and irrelevant speculations, on the connection between ancient Moorish poems and Greek mythology, on the derivation of the Phoenician deities, and on the meaning of Moorish habits and customs ; but the only evidence, confirmatory of its thesis, adduced in it and in Mr. Halliburton's subsequent letters, amounts to this: that six Europeans have seen dwarfs in Morocco; that an indefinite number of natives have romanced about dwarfs in their usual way; that there are in Morocco artificial caves—presumably dwellings—of such small size as to suggest that they must have had very short inhabitants ; and that there have come down to us from antiquity traditions as to Troglodytes who dwelt in the Atlas Mountains.

Mr. Halliburton's European witnesses are unimpeachable; and had my friend Mr. Hunot, whose knowledge of the country is extensive and accurate, distinctly said that there is a race of dwarfs in Morocco, I should not have ventured to con

tradict him. But all that Mr. Hunot says, in the long paragraph quoted from his letter, is that he recollects an adult dwarf of about the height ofa boy of ten or eleven years of age who lived and died in Mogador. All that Captain Rolleston says is that he saw in Tangiers a dwarf of about thirty~five or forty years of age 3 to 4 feet in height, and of an unusually light complexion. All that Mr. Carleton says is that he has seen a dwarf at Alcazar. All that Sir John Drummond Hay says is that he hunted up at Tangiers some Sus and Dra people who had seen dwarfs. All that Miss Day says is that she had done the same at Telcmen. All that Mr. Harris says, of his own knowledge, is that he has seen two dwarfs-one at Fez, about 4 feet 2 inches in height, and of a light brown colour ; and the other, about whom no particulars are given, somewhere in the country. All that Miss Herdman, whom I had the pleasure of meeting at Fez, says is that she has never seen a dwarf in Morocco, but that she has heard of one, and has dra\vn out tales about a tribe of dwarfs from her native servants. All that Mr. Halliburton himselfsays to the point is that he has seen and measured a very timid and obliging dwarf of about thirty years of age, 4 feet 6 inches in height, and of a peculiar reddish complexion, in Tangiers. Let me add to Mr. Halliburtorfs list of European witnesses. I have myself seen two dwarfs in Morocco-one in Fez, and the other in some northern town (I cannot for the moment recollect which, and have of course no papers to refer to). The first of these might perhaps have passed as a true dwarf-a man ofsmall size, but well proportioned, like Tom Thumb; but the other was certainly a disease-dwarf, with a large unshapely head and trunk, and little bowed legs, like (fanny Elshie, or the \Vise Wight o' Mucklestane Moor. Rickets are not unknown in Morocco. I have no doubt that that malady is common in certain districts periodically visited by famine or devastated by war, and in which infant feeding is not conducted on scientific principles; and the probability is that men and women of stunted and distorted growth are more numerous in proportion to population in Morocco than they are in England. The wonder is to me that the number of instances of the occurrence of dwarfs in Morocco, which Mr. Halliburton in his longcontinued researches has been able to establish, is so exceedingly small; and that one dwarf, for example he of Fez, has, like a stage army, to do duty several times over. But had he succeeded in identifying ten times the number of dwarfs that he has actually traced out, he would only have proved that dwarfs exist in Morocco as in all other countries, and would not have advanced rt step towards proving his proposition that there is a tribe of dwarfs in the Atlas. Iknuw alittle Scotch town in which there are three dwarfs; but it would be scarcely legitimate to infer from that fact that there is a concealed clan of MaeManikins in the Grnmpians. That the dwarfish condition in the dwarfs described by Mr. Halliburton was an accidental variation, and not a racial characteristic, is rendered more than probable by the fact that two of them-the only two who are reported to have had families-had offspring of normal stature. The native reports about dwarfs and dwarf tribes, which Mr. Halliburton sets forth in much detail, are obvious fictions-of the kind which the professional story-teller pours forth copiously every day in the Soko in scores of Moorish towns and villages, only adapted, of course, to the requirements of an eager English listener. The names of the reporters are not given, nor are the opportunities they possessed of obtaining the information they convey explained ; while some of the practices they attribute to the dwarfs-such as finding of treasure by writing on wood, and the feeding of horses on dates and camels' milk with the view of rendering them swift of pace-I have heard ascribed to tribes in the Atlas that are certainly not composed ofclwarfs. Morocco is the hot-bed of fable, and infested by the cockand-bull, and I can picture to myself the grave delight with which the natives questioned by Mr. Halliburton would stimulate his curiosity and then satisfy it. Mr. Halliburton emphasizes the fact that he is a Q.C., and accustomed to cross~ examination; but British perjury and Moorish mendacity have little in common, and are to be fathomed by entirely different methods. The way in which he measured the Tangiers dwarf, _Iackin (he actually took 2 inches off his height because a native who was present told him that _Iackin had raised his heels to that extent while being measured), casts some doubt on his powers of observation ; while the extracts from his diary show that no process of sifting has been carried out, but that everything favourable to his theory has been thankfully received. I

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would undertake to collect in Morocco in a month's time native testimony in support of the existence of a tribe of giants in the Atlas, or of a tribe of men with six digits on each hand, quite as specious and convincing as that which Mr. Halliburton has accumulated in favour ofthe existence of a tribe of dwarfs. Even if the natives interrogated hy Mr. Halliburton had no wish to deceive or to please him, much would depend on the intelligence and honesty of his interpreter, and on the exact terms employed, Only those who have tried can realize how ditticult it is to get precise information on any subject out of natives of Morocco. If the caves in Morocco are to be regarded as at one time the dwellings of dwarfs, then it is clear that dwarfs must at one time have been in complete possession of the country, for such caves are to be found all over it. The most remarkable of them which I have visited at Tassimet, about two days’ journey from Demrtat-caves whichliuropeans had never before explored, and which were excavated in a rock by the side of a waterfallwere in many instances too small even for the accommodation of dwarfs ; and as they yielded to our digging fragments of bone and of pottery, it seemed probable that they had been places of sepulture and not of habitation. Such caves have also undouhtedly been used sometimes for the storage of grain, like the underground metamors; and the invariable answer retumed to our inquiries about their origin was that they had been made by the ]\’omz', or Christians. Never on any occasion did I hear them ascribed to dwarfs. The classical tradition that there were dwarfs in the Atlas is unworthy of serious consideration in the absence of any observation suggesting that it had other than an imaginative foundation. “ Nearly all the myths of Greece," says Mr. Halliburton, “ are laid in Mount Atlas," and monsters more extraordinary than dwarfs must have dwelt there if these myths are to be received as of historical authority. I have tried to prove that the evidence given in favour ot' the existence of a tribe of dwarfs in the Atlas is utterly trivial and untrustworthy; and I shall now endeavour to show that the evidence that can he called to discredit that hypothesis is cogent and convincing. The dwarfs are described by Mr. Halliburton as brave, active, agile, swift~footed, as possessing a vigorous breed of ponies, as experts in the pursuit of the ostrich, and as trading in the Sahara and al"I`assamalt. Is it to be believed that being all this, and being very numerous-there are, Mr. Halliburton says, about |500 of them in Ait Messad, about 1500 at Akdeed, about tooo at Ait Messal, about 500 at Ait Bensid, and about 4o0in three Akka villages-is it to be believed, I ask, that these swarming and enterprising dwarfs would have allowed themselves to be bottled up in a cleft in the Atlas Mountains, so that only half-a-dozen specimens of them have found their way to tbe great towns to the north of the Atlas, where are to be found numerous representatives of all the other Atlas tribes? Is it to be believed again, that the existence of such a peculiar and notorious tribe, known, Mr. Halliburton tells us, to all Moors, should have been concealed from all theinquisitive travellers who have penetrated into the interior of Morocco, to be revealed to Mr. Halliburton standing at its outer gateway? Leo Africanus, whose account of Morocco is marvellously minute and accurate, and who enumerates its tribes, has not a word to say about dwarfs. De Foucauld, who visited Akka, is equally silent about them; and so is Rohlfs, who explored the valley of the Dra. Not one traveller in Morocco has ever heard even a rumour or dark hint relating to them. Thomson and I spent some mouths in the Atlas in constant communication with natives of every class, and in all the strange legends, histories, and adventures narrated to us by the camp brazier, in the jmdak or the knsbtz, there was never a distant reference to a Moorish Liliput; and be it remembered our servants knew that we had a keen eye and ear for curios, human and inhuman. ln all our wanderings in the Atlas we never met a d\varf, and indeed, at a great gathering of people at which we were present, at the feast of Aid el Assir at Glawa we were much struck by the height of the men. Mr. Aissa, who is quoted by Mr. Halliburton as having seen one of the tribe of dwarfs east of Demnat, was our interpreter for three months, and conversetl with us with the tttmost freedom on all conceivable subjects, and he never adverted to this dwarf story. Ihave had several long talks with Mr. Hunot, whotn Mr. Halliburton also quotes-conversttions covering a wide range of topics, amongst them the origin of the caves already alluded to~ and he certainly at that time had no belief that they had ever

been tenanted by dwarfs, or that there was any dwarf tribe in the country. It is especially noteworthy that Du Bekr, the confidential agent of the British Government at the Court of Morocco, replied to Sir William Kirby Green that no Moor had ever heard of a race of dwarfs in the country. Sir William knew rnw to interrogate a Moor, and as he accepted Du Bekr's statement, I have no doubt that Du Bekr was speaking the truth.

Until the existence of a race of dwarfs in the Atlas Mountains is proved, it is idle to indulge in guesses at the reasons which have led to the fact of its existence being jealously kept secret; so I shall not follow Mr. Halliburton in the argument by which he seeks to show that the race has been regarded with superstitious reverence, and so kept apart. In all countries, at all limes, I believe dwarfs and deformed persons have been looked at askance by the ignorant and superstitious. In Scotland they were regarded as fairies of a brutal and malignant type; and in Morocco I have no doubt they have been credited with the possession of the evil eye and of other pernicious powers. But to maintain that a tribe of them has ever been held sacred and worshipped in the heart of a Mahometan country that is fiercely fanatical is to do violence to our fundamental conceptions of Islam.

Mr. Halliburton's statements about the origin and habits of his supposed tribe of dwarfs are not more worthy of discussion than his theory of the causes which have led to their concealment. They are derived from native sources of the most tainted description, and are either pure inventions, or concoctions of truth and falsehood. We are told that a tribe of acrobats—the Ait Sidi Hamed O Moussa (the tribe of the son of Moses)—is an offshoot of the Aglimien dwarfs, living between the Dra and Akka; that they are a rather small race with a light red complexion; and that dwarfs perform with them in Southern Morocco, but avoid the coast towns where Europeans are; and lhat they are smiths and tinkers. Now, the paragraph setting forth these statemenls contains just as much error and confusion as it is possible to cram into so many words. The Sidi Hamed O Moussa are not a tribe at all, but the followers of a saint whose Kuba is nit far from Taradant. Their troupes are made up of men rir.iwn from various parts of the country; and it would be as correct to regard the Jesuits as a tribe, and describe their ethnic characteristics, as it is to assign distinctive features to the Sidi Hamed O Moussa. Then, as a matter of fact, they are not unusually small men, they are not smiths and tinkers, and they never have <l« nrfs performing wilh them either in town or country. I saw several troupes of them in Southern Morocco, andean testify that they are of average size and of the usual Moorish tint; that ihey follow a more profitable trade than that of tinkering ; aril that they have no dwarfs among them.

Mr. Halliburton s'rangly advises European travellers and touriss to abstain from any attempt to enter the districts of Morocco inhabited by the dwarfish race, as they would inevitably, while doing so, be murdered or robbed, whether Moslems, Jew?, or Christians. The advice is judicious, for open-mouthed travellers of any persuasion, in quest of dwarfs, are not unlikely to be murdered or robbed in any part of Morocco except in those coast towns to which Mr. Halliburton has apparently confined his own wanderings in the country. European travellers of another sort, however—resolute, incredulous men, explorer.-', and pioneers of trade and commerce—will certainly before long penetrate all those regions where the dwarfish race has been located by Mr. Halliburton. Remembering what I have heard on good authority of the resources of some of those regions, and the indications I have- seen of the mineral wealth of that region to the south of the Atlas where Mr. Halliburton has placed the original home of his dwarfs, I feel disposed to exclaim, like the old sailor in Millais's famous picture "The Norlh-West Passage": "It can be done, and England ought to do it I" When, however, these regions are opened up, I feel >ure that, amongst much that is wonderful in them, there will be found no tribe of dwarfs hemmed in by religious sentiment.

To those interested in the generation and growth of myths in modern times, and under Congress culture. Mr. Halliburton's dwarf-story cannot but afford an instructive study.

Harold Crichton-browne.

Mncloustie Camp, Bechuanaland. November 15. 1891.

Sun-spots and Air-temperature.

It is now widely believed by meteorologists that a certain relation exists between the solar sun-spot cycle and the air-tem

perature of the earth, such lhat to a minimum of sun-spots corresponds, approximately, a maximum of air-temperature, and vice versA. From the comprehensive researches of Dr. Koppen on the subject some time ago, it appeared that this relation is most clearly proved in the case of the tropics, the evidence becoming less as we go north and south. Mr. Blanford showed recently in Nature (vol. xliii. p. 583) that the evidence in the case of India has of late years greatly increased in force.

In a climate so variable as ours it is not, perhaps, to be expected that the existence of such a relation should be very patent and obvious. And there may be some legitimate doubt whether its existence has yet been demonstrated. It is in the hope of possibly advancing the matter somewhat that the following facts are presented.

If we decide to take for our consideration a part of the year instead of the whole, we shall naturally select the hotter part; the part in which the solar action is greatest (just as we might expect to find, and dn find, better proof of the relation in tropical than in cold countries). I select the four months June to September. The data used are, Mr. Belleville's observations of Greenwich mean temperature from 1812-1855, which are, it should be noted,.reduced to sea-level (see Quart, journ. of the R. Met. Soc, January 1888, p. 27), and thereafter the ordinary Greenwich figures. The average difference (about half a degree) does not materially affect the purpose here set.

Taking the mean temperature of those four months, and smoothing the values by means of five-year averages, we gel the second, thick line curve in the upper diagram herewith. The dotted line curve is that of sun-spots, inverted (i.e. minima above and maxima below). The vertical scales for these are both 10 the left.

There is evidently a correspondence between these curves as far as about 1870; maxima of temperature lagging a little, as a rule, behind minima of sun-spots, and minima of temperature behind maxima of sunspots. Since about 1870, the correspondence appears to fail. We look for a temperature-maximum about 1879, and we do not find it.

A consideration of the rainfall here seems instructive. The smoothed curve of rainfall in those four months (third in the diagram; Chiswick to 1869, thereafter Greenwich) is, in the main, roughly inverse to the temperature-curve, as we might expect. Yet it is difficult to trace a very definite relation between it and the sun-spot curve. Thus, consider the three most salient "crests" in it. The first (in height as well as time), in 1829, is close before a sunspot maximum, 1830. The second (least salient of the three), in 1861, is close after a sunspot maximum, i860. The third, in 1879 and 1880, is close after a sunspot minimum, 1878. These rainfall variations, indeed, seem to be under some different law, and it will be observed that the last crest comes (the first example in the whole period) just about wheie we should expect, from previous experience, to find a temperature-maximum. The regular variation of this curve in one direction for several years is a noteworthy feature recently (in 1880 to 1885, and again in 1885 to 1S89). Is the curve now near a maximum which will I e found to coincide with a further obliteration of the normal correspondence between sun-spots and temperature?

We have thus far considered the group of four months, and ihey seem to me to support the view underconsideraiion. May we further look for the relation in individual months?

Suppose we see reason in doing so, and make a selection. The most likely month would perhaps seem to be July, as having the maximum temperature ; or June, as that month in which the sun is highest.

On examining the smoothed curves of mean temperature for each of tho-e four months, we find that June and .September show a large amount of the correspondence with the sun-spot curve, while the two others do not show much correspondence. These two curves (June and September) are given in the lower diagram, superposed ; the two vertical scales being at the left. June, it will be noticed, presents a wave crest fairly corresponding with each of the six, or seven, sun-spot minima. In the case of September there is a pronounced failure at the sun-spot minimum in 1878.

As a possibly good reason why September might show the relation, while July and August do not (or not so well), I would suggest the fact that September is the month with least cloud. Between May and September, cloud increases to a small secondary maximum in July.

The absence of a miximum of temper,-'!—e in September corresponding to the sun-spot minimum in 1878 may, perhaps, be connected, as in the case of the four months' curve, with the rainfall. A smoothed curve of humidity for September rises, I find, to a high maximum in 1880. The June humidity curvedoes the same, and if it be therefore asked, why we should not have a similar failure in that month's curve, I would invite attention to the fact that the rise to the maximum in the humidity curve for June is a rapid one from the absolute minimum (reckoning from 1858) in 1876; while the rise in September is economize your space, I only gave the skeleton of the argument, but I hoped I had said enough to indicate at least the general outline of my logical views. But as this seems not to have been quite the case, may I now explain myself a little more fully?

J predicted time, given in Marth's ephemeris {Monthly Notices, I March 1891), is 5(1. 32'6m., so that the spot was 104m. late, and this means a decided slackening in its motion of rotation during the present apparition. On August 7, 1891, I saw the , spot pass the central meridian at nh. 32m., or only 2-3m. af:er the time indicated in Marth's ephemeris. In the interval of 5 months, during which 362 rotations were performed, the mean period has been 9h. 55m. 42s., which is nearly 1 second greater than the rotatio 1 period of this marking as observed here daring

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I may remove a slight misunderstanding at once. I said our knowledge of our own continuous existence in the present is to each of us a necessary truth. Dr. Mivart reads this as if I had written "our continued existence in the future "! That we cannot be annihilated while we know that we are existing is, as I shall presently show, not a mere consequence of the law of contradiction. If this law is of any use at all in proving the conclusion, it would certainly be useless without a second premiss, viz. that we are existing; and this latter is the premiss which is a necessary truth.

I suppose everyone will acknowledge that a definition is essentially an arbitrary assertion, and that therefore a definition can by itself give no real information. But a well-understood term does not consist of a definition alone. Its definition may be laid down, as a list of items of connotation (or denotation), and the other part of its meaning, which may be called its import, that is its denotation (or connotation) must be discovered by experience; and the knowledge so acquired is real, not only verbal, knowledge. Now it is possible from a number of definitions alone to deduce a series of propositions. These, like the definitions from which they were deduced, give by themselves only verbal information—they are all truisms—and before they can be made of any practical use, certain real assertions, assigning real import to the terms, and so expressing real knowledge, mast be added to the premisses. Thus, if we wish to determine whether any given proposition is a truism, or conveys real information, we have only to examine the definitions of its terms. If these are found to be inconsistent with each other, the proposition is a contradiction in terms, and must be rejected. If the definitions are not inconsistent, but are independent of each other, the proposition can only be intended to assert the identity of the import of its terms—it therefore conveys real information, which may either be true or false. Lastly, if the definitions can be shown to be dependent on each other, the proposition is equally true whatever import its terms may have, or even if they have no conceivable import at all. It is a truism. If, however, by the aid of other real propositions any real import can be given to its terms, it may have objective, or subjective, applications; but the objectivity or subjectivity is introduced by those other propositions, and is not a property of the original truism.

Take, for example, the proposition, "Everything must either 1 be' or 'not be ' "; or the proposition, "Twice two is four." The truth of either of these propositions depends solely on the definitions of its terms, as I pointed out in my last letter, and this is why I cannot regard them as objective truths. Of course 1 do not doubt that if I had lost an eye I should not remain in the same condition as I was before. But, although "no man out of Bedlam would suppose a statement of a general law would inform us about a concrete thing," this is precisely what Dr. Mivart does if he regards the above proposition as dependent solely on the law of contradiction. Does he not see that he added the objective element to that law in the phrase, "if he had lost an eye"? "Much virtue in If." The status of the proposition, "Two straight lines cannot inclose a space," similarly depends on the definitions of its terms; but, as I pointed out in my last letter, these terms may be denned in two different ways—either by dependent definitions, so making the proposition a truism, or independently so as to make it a real assertion, in which case it might conceivably be false. Dr. Mivart apparently takes the former set of definitions, and then implies that I deduced the latter result from them, which, if he reads my letter again, he will find not to have been the case.

In reply to Miss Jones, I may point out—

(1) It most certainly is merely a verbal convention when Miss Jones says, "A and a 'are not applicable to the same thing.'" She had herself just before laid down the convention in question, in the phrase, "If A signifies the negation of a (whatever A may stand for)."

I do not know why Miss Jones should imagine that I think that "assertions (or denials) of the 'existence' of particular objects are the only real propositions," but perhaps she will understand my view better when she has read this letter.

(2) I certainly hold that "inductions have no logical justification whatever," if by "logic" is to he understood formal, or,

as I prefer to call it, symbolic, reasoning. The essence of induction, in my opinion, is the assumption (at first arbitrarily) of an hypothesis to account for observed facts—that is, ultimately, of directly apprehended sensations. The full significance of the hypothesis is elucidated by symbolic reasoning, and the enumeralio simplex is applied to the results of this reasoning, and does not, therefore, appear quite in the simple form exhibited by Miss Jones. But it remains equally true that no induction can ever lead to a necessary truth.

(3) Miss Jones's view of mathematical reasoning is exactly that which I wish to combat. We do not, in mathematics, conclude a universal proposition from a single concrete instance. A mathematical formula does not imply the existence of any instance whatever of its application, any more than a definition implies the reality of the thing defined. The formula is deduced from what may logically be regarded as definitions, and one or any number of applications may indeed be found afterwards, but only by the aid of additional real premisses. It is difficult to exemplify this in the case of geometry, because the accepted geometrical methods are so very imperfect, and geometrical conclusions are not always deduced from definitions alone. As I implied in my former letter, some of them are founded on induction. But it must be evident that the truth of, say, De Moivre's Theorem, does not depend on our having seen that it was true in any one instance.

(4) If Miss Jones reads her own paragraph (4) again carefully, I think she will see that it is not I who have contradicted myself. I showed that if the definitions of the terms of a certain proposition were altered, the proposition might no longer be true, and that if they were not altered it would always be true. Argal, the truth depends on the definitions, and on nothing else.

1 did not maintain that it could ever be to anyone a necessary truth that he was writing with a lead-pencil. That would be an objective proposition, such as I was careful to insist could only be proved by induction. It might, however, be a necessary truth to anyone that he thought he was writing with a leadpencil. As to mathematical truths, so far from believing that "in as far as 'real' they are obtained by induction," I expressed my opinion that they are not "real" at all, but all truisms. Any reality in their applications must be added from outside, by real assertions which are not " mathematical." I object to calling truisms "necessary," not because they are possibly false, but because their truth is only arbitrary. On the other hand, when I call "the apprehension of a present fact" a necessary truth, I mean something more than that it is certain—namely, that its contradictory is unthinkable.

Edward T. Dixon.

Trinity College, Cambridge, January 8.

FRESH EVIDENCE CONCERNING THE DISTRIBUTION OF ARCTIC PLANTS DURING THE GLACIAL EPOCH.

LAST summer (1891) I spent some weeks in Western Russia and Northern Germany, in order to ascertain whether the glacial fresh-water deposits of those countries contained any remains of the vegetation which lived there immediately after the inland ice had melted away. The results of my journey being favourable, I have thought it desirable to communicate them to the readers of Nature ; but before doing so it might be convenient to give a brief summary of previous investigations on the same subject.

The first discovery of fossil Arctic plants was made in England by Mr. W. Pengelly, who found in i860, at Bovey Tracey, in Devonshire, leaves of the dwarf birch {Betula nana), together with leaves of some willows, as Salix myrtilloides, S. cinerea, S. sp. indet. The leaves were identified and described by the late Prof. Heer,1 who pronounced the opinion that the presence ot Betula nana was conclusive evidence of "a colder climate than Devonshire has at the present day." The significance of this discovery was, however, but little appreciated until the researches mentioned below again

1 Philosophical Transactions. 1862, p. 1039 In this paper Heer mentions Salix repens (?>, but this determination was subsequently altered to S. myrtilloides.

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