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o 'pull' air particles apart. On p. 52 we are told hat the motions of particles of a water wave "are always at right angles to the direction of the wave tself." On p. 68 the author corrects this statement, but in doing so, takes occasion to speak of a plane "in," instead of 'passing through' the line of progression. On p. 380 he describes harmonic partial-tones as modifying the quality of their fundamental," though he obviously means the quality of the compound sound due to the fundamental and other partial-tones combined. On p. 387 it is said that the "ratios of frequencies" which characterize particular sounds "are called intervals," and that by dividing one note by another we obtain the intervals between them. Language of this kind is, indeed, hardly misleading, but it is certainly very slipshod.

Before passing from the more generally acoustical, to the more specially musical portion of Prof. Zahm's volume, it is proper to point out one important respect in which it has the advantage over most, or possibly all, the manuals on the same subject which have preceded it. This merit consists in giving a somewhat full account of elaborate experimental researches on beats, combinationtones and quality conducted by Dr. Koenig, the results of which are to a considerable extent at variance with conclusions previously announced by Prof. Helmholtz. In the opinion of our author, Dr. Koenig is "one who, not excepting even the eminent German philosopher just mentioned (Helmholtz), has contributed more than any other person to the advancement of the science of acoustics" (p. 17). A more balanced judgment, while placing great reliance on Dr. Koenig's experimental skill and on the superlative excellence of the apparatus constructed by him, would probably attribute to Helmholtz's opinion a preponderant weight in interpreting and correlating the results of experiment. Be that, however, as it may, Prof. Zahm has done excellent service by popularizing the work so laboriously performed, and so modestly placed on record, by the eminent instrument-maker to whom no one who has put his hand to acoustical work can fail to be under considerable practical obligations.

The specifically musical are decidedly the least The meritorious parts of our author's performance. looseness of phraseology already complained of is here at its worst. On p. 166 we are told that a 'comma,' C) is "the smallest interval used in music." A beginner might easily take this to mean that notes differing by only that interval were actually heard consecutively in a musical phrase-of course an absurd supposition. Very misleading, again, is the statement on p. 388 that tones, ake major and minor tones, that differ from each other only by a comma "are considered in music to have the same value." The only rational meaning to be got out of it seems to be that in the equally tempered scale the distinction between major and minor tones is obliterated

held, and so the reader is left free to suppose e.g. that the tritone, F-B, is a consonance. On p. 390 the 'inversion' of intervals is mentioned without any explanation of its meaning.

Attention may well be called to a process of reasoning which occurs on pp. 388-390. Prof. Zahm abruptly introduces (p. 388) calculation by "frequency-ratios"; assumes, without attempt at proof, that addition of two semitones is performed by squaring the ratio 1, and then remarks (p. 390) "From the foregoing we observe that the sum of two intervals is obtained by multiplying, not by adding their ratios together." An assumption in a particular case is thus made to do duty as a general demonstration.

On p. 396 we read that "so perfectly does the interval of the fifth answer the requirements of the ear that even unpractised singers find it quite natural to take a fifth to a chorus that does not quite suit the pitch of their voice." If, as this passage appears to suggest, practised singers in America find it still more natural to accompany melodies in consecutive fifths, wonderful effects may surely be expected from the choruses to be heard at the Chicago exhibition.

On p. 429 our author describes a diagram by Helmholtz as concerned with the transposition of an interval by an octave, whereas what it really deals with is the enlargement of the interval in question by the addition to it of an octave. On p. 430 he writes down, as constituents of the chromatic scale of C, the notes E, F, B. and Cb.

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On p. 441, he tells us that in listening to such violin players as Joachim, Wilhelmj, and others one can always hear distinctly the Tartini, or beat-tones, that add such richness and volume to violin music."

To gauge the amount of truth contained in this remark it suffices to bear in mind that in the case of most major, and of all minor consonant chords, Tartini's tones cause a decided dissonance. Players who made them 'always distinctly audible' would soon be reduced to permanent inaudibility themselves.

Prof. Zahm's volume is creditably free from misprints: the following have, however, been noted: P. 23, 1. 16' period' for 'periods.'

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P. 68, 1. 21 amplitude' for 'amplitudes.'
P. 90, 1. 8 Ajugari' for ‘Agujari.'
P. 142, in diagram, B, for B1.
P. 152, in diagram I, B for B.
P. 388, ll. 11 and 12, G for F.

GERLAND'S ETHNOLOGICAL ATLAS.

Atlas der Völkerkunde. (Berghaus' Physikalischer Atlas, Abth. vii.). Bearbeitet von Dr. Georg Gerland, Professor a.d. Universität in Strassburg. (Gotha : Perthes, 1892.)

Un p. 389 the notes of the diatonic scale, and their ANTHROPOLOGY owes much to Prof. Gerland, whose

relations, in respect to rapidity of vibration "to one another," are set out, and it is added that all but the second and the seventh of the intervals thus indicated are consonant. The essential piece of information, that it is not the intervals formed by these notes with one another," but with the tonic, that are in question, is with

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completion of the two last volumes of the late Prof. Waitz's "Anthropologie der Naturvölker" is a monument of that co-ordinated knowledge of fact which is the source of sound principle. His new "Atlas of Ethnology," while forming part of the great Physical Atlas of Berghaus, may be obtained and used as a separate work by anthro

pologists, to whom it will be of great service in methodizing the vast and growing information with which they have to deal. This application of graphical method, it is true, has difficulties which even the greatest skill cannot altogether overcome, but Prof. Gerland may well be content with his success in making evident at a glance the characteristics of mankind, seen from many points of view. Their distribution over the earth, as thus made evident, inay often lead straight on into theories of origin. The fifteen plates contain nearly fifty maps, each suggesting a principle, or showing where there is room for one.

Plate I. represents on two planiglobes the classifiIcation of human races as to skin and hair. Prof. Gerland does not even combine these two characteristics, and points out in his introductory remarks that any attempt to map out man into defined physical races is impossible, for such division does not exist in nature. Anthropologists of course know this, but care is not always taken to make it clear that racetypes are not so much complete realities as statistical abstractions from partial realities, the various measurable characters of skull, limbs, complexion, hair-form, &c., conbining and blending too intricately for absolute definition. I was struck by meeting lately in a popular book with a confident mention of the four distinct Aryan race-types, and it occurred to me that it would bring the statement down to its bearings to put one of Prof. Gerland's planiglobes before the author, desiring him to define and map out these varieties of mankind. Even in Gerland's broad general distinctions of complexion and hair, an anthropologist not thoroughly special on the anatomical side may find novelty and difficulty. The opinion that all native Americans are similar as to race is here strongly and probably with reason modified by the native Brazilians being separated on the complexion-map from other peoples of North and South America, and placed to match the Tartars and Chinese. What amount of evidence there is for placing the Berbers of North Africa under the same map-colour seems not so clear, but it is to be noticed that the same tint includes several more or less distinct grades in Broca's scale. An attempt is even made to separate the friz-haired negroids into classes according to the arrangement of their corkscrew-tufts of hair on the skin. Plate III., in two maps, classifies man according to his religious beliefs and customs, and here the prevalence of special rites offers instructive generalizations. Thus the American line which limits the smoking of tobacco as a religious ceremony, indicates the spread of this peculiar rite from some religious centre over an enormous area. No doubt it is rooted in nature, from the fact that its narcotic ecstasy brought the priest into direct visionary contact with the spirit-world. But none the less, it proves the religions of savage tribes, separated by great distances on the map, to be bound together by historical connexion. Not less remarkable is the compactness of the districts of Eastern Asia and the opposite Continent of America, where masks are used, apparently originally with religious significance Here again it is evident that we have to do not merely with independent growth from the human mind, but in some way with historical transmission. It must be remembered in using these maps, that they bind their author only to fact, and not to theoretical interpreta

tion. This same plate maps out the immense distra whose natives have a myth of a deluge, the upheavi the earth, &c., but it cannot, distinguish in North South America, for instance, between regions where de myths are old,and those where they were introduced: Jesuits a few generations ago. Plate IV., mapping regions liable to special diseases, as malarious fes pestilence, cholera, yaws, &c., contains in a conden form a vast collection of knowledge, bearing on an pological arguments as to the relation of race to phy constitution, and thus opening into one of the problems of the history of man. Plate V. classes ot varieties of human food, clothing, dwellings and occ tions. Plate VI. and onward map out the distrib of nations and tribes at different periods as know: history, Plate XIV. being devoted to the distributi languages over the world.

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Anthropologists who keep this atlas at hand as al in their work will by practice find out its merits defects. The representation of the geographical dis tion of arts and customs has long been a feature d Pitt-Rivers Museum, where so far as possible each ser illustrating development and transmission of car is accompanied by a small world-map coloured to the parts of the world it occupies. It is of c impossible to Prof. Gerland to work in such de involving as it would do hundreds of separate He has to indicate his distributions on a moder number of plates and mostly uses planiglobes, a jection which, after being neglected for generati will, in its improved modern arrangement, cera come into more general favour. On these, by ingen devices of tinted patches and streaks, combined ~ lines and dots, he succeeds in giving a more gee. survey of man and civilization than our students have had in their hands before. EDWARD B. Th

OUR BOOK SHELF.

Castorologia; or, The History and Traditions of the adian Beaver. By Horace Martin, F.Z.S. (Lond) Stanford, 1892.) "BEAVER" was once the most important fur in world. In former days the pelt of this Rodent w standard by which all barter in the Dominion of C was regulated, and "beaver" passed as current throughout the whole of North America. Even now quantity of beaver skins brought to England is consc able. Mr. Poland, in his "Fur-bearing Animals us that upwards of 63,000 beaver skins were sold by formerly required a much larger supply than this, Hudson's Bay Company in 1891. But "beavert 1743 it is said that 127,000 beaver-skins were imp into La Rochelle alone. Our "top" hats are now of silk, and beaver has become a fur of seconds importance.

Besides the fur of the beaver many other point interest attached to this animal will be found disc more or less completely in Mr. Martin's volume. before its fur was required for hats castoreum or cas -a substance found in two large glands, situated the base of the beaver's tail-was a much-valued spe in medicine, as spoken of by Hippocrates and f Even at the present time its use is by no means doned, and the "crude article" is "still sold at ou stores at prices varying "from eight to ten do

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ound.” But in past centuries castoreum was considered sovereign remedy for every kind of disease. Many amusing details on this part of the subject are given by Mr. Martin, mostly extracted from the "Castorologia" of Johannes Francus, published in 1685. The wisdom of Solomon himself is attributed by this learned author to the virtues of the beaver. To acquire it, it is only necessary "to wear a hat of beaver's skin, to rub the head and spine with that animal's oil, and to take twice a year the weight of a gold crown piece of castoreum."

At the end of his volume Mr. Martin places a short account by Mr. C. V. Riley, the well-known American entomologist, of Platypsillus castoris, a parasite on the beaver, and one of the most remarkable among the many extraordinary forms of parasitic insects. Mr. Riley correcily refers this creature to the coleoptera, although other naturalists, and, amongst others, its discoverer, Ritsema, have expressed different views on this point He omits, however, to refer to the excellent account of Platypsillus castoris, written by the late John Leconte, and published in the Proceedings of the Zoological Society of London for 1872. Dr. Leconte has here shown that it is necessary to make a special family (Platypsillida) for the reception of this curious parasite, but that it must be unquestionably referred to the coleoptera.

Ón account of these and other peculiarities the beaver is unquestionably an animal of great general interest, and Mr. Martin has done well to devote a volume to what is evidently his favourite theme. There is, we must allow, little, if anything, original in it, and the statements on scientific points cannot always be implicitly depended upon. But the author has brought together a large amount of information on the subject, and his book is "popularly written" and "fully illustrated," though we cannot quite agree to his claims to have produced an "exhaustive monograph."

An Allas of Astronomy. By Sir Robert Stawell Ball,
LL.D., F.R. S. (London: George Philip and Son,

1892.)

A NEW book by Sir Robert Ball is always a matter of interest, but the present one naturally lacks the usual characteristics. It is described as "a series of 72 plates with introduction and index." In addition to monthly and general maps of the stars, the atlas reproduces pictures of the sun, moon, planets, and comets, and contains diagrams illustrating their motions and dimensions. As the book is chiefly meant to be a companion to more general works, the introductory matter is purposely brief, but still it has several features of interest. Special attention may be drawn to the excellent description of a simple graphical method of determining the orbit of a binary star.

To the serious student who may possess a small telescope the new atlas will be very useful. Here he may learn how to determine the positions of sun spots, how to find the places occupied by the various planets, and what objects are most likely to be within reach of his instrument. Those interested in selenography will derive much assistance from the twelve plates showing the moon at different phases, which have been specially drawn by Mr. E ger, each being accompanied by an index map. One can only wonder, however, that some of the recent excellent photographs of the moon have not been pressed into the service.

The star maps, on the whole, are excellent, and our only complaint is of the excessive density of the Milky Way, which, in some parts of the maps, is almost sufficient to obliterate the names and numbers of the stars. The monthly maps will be particularly useful to those who are just learning the constellations, a new feature being a belt indicating the track of the planets.

Spectroscopic astronomy is entirely omitted, the author being of opinion that this great branch of work can only

receive justice in a separate atlas. In this we heartily agree, and trust that such an atlas will soon be forthcoming.

The author's large following of readers will no doubt welcome the new commer, but we must express regret that astronomical photographs are not more fully represented. It would be interesting, for example, to reproduce a series of photographs of typical nebulæ, all of which, we believe, are now available. A plate showing the alvantages of photography in the delineation of stars would also add to the interest of the atlas.

LETTERS TO THE EDITOR.

[The Editor does not hold himself responsible for opinions ex· pressed 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 of NATURE. No notice is taken of anonymous communications.]

Vector Analysis.

I FANCIED that, in reply to the voluminous letters of Prof. Willard Gibbs (NATURE, xliii. 511; xliv. 79), I had said in a few words all that is requisite (if, indeed anything be requisite). to show the necessary impotence, as well as the inevitable unwieldiness, of every system of (so-called) Vector Analysis which does not recognize as its most important fea ure the product (or the quotient) of two vectors :-i.e. a Quaternion.

A recent perusal of the first four pages of a memoir by Mr. O. Heaviside (Phil. Trans. 1892):-for so far only could I go:-has dispelled the illusion. For he calls the correspondence just spoken of a "rather one-sided discussion":-a truly particularly desired to read the memoir (which the Author had Delphic delivery :-cleared up, however, by what follows it. I kindly sent me) as I hoped to learn from it something new in Electrodynamics. But, on the fifth page, I met the check-taker as it were-and found that I must pay before I could go. further. I found that I should not only have to unlearn Quaternions (in whose disfavour much is said) but also to learn a new and most uncouth parody of notations long familiar to me; so I had to relinquish the attempt. In the last of the four pages of my progress, I had found that Mr. Heaviside (though, as above stated, he has a system of his own) is an admirer of Prof. Gibbs' system, to such an extent at least that he thinks "his treatment of the linear vector operator specially deserving of notice." There I was content to leave the matter.

But Mr. Heaviside has just published (Electrician, 9/12/92) an elaborate attack on Quaternions, of a kind which is calculated to do real injury to beginners. In answer to his remarks, in which he continues to point to me as the persistent advocate of a system which all right-minded physicists should avoid, I would simply refer him (and his readers, if there be such) to a brief Address which I gave a short time ag to the Physical Society of Edinburgh University (Phil. Mag. Jan. 1890). One or two sentences, alone, I will quote here :

"if we can find a language which secures, to an unparalleled extent, compactness and elegance, and at the same time is transcendently expressive-bearing its full meaning on its face— it is surely foolish, at least, not to make habitual use of it." "For (Hamilton) the most complex trains of formulæ, of the most artificial kind, had no secrets :-he was one of the very few who could afford to dispense with simplifications: yet, when he had tried quaternions, he threw over all other methods in their favour, devoting almost exclusively to their development the last twenty years of an exceedingly active life."

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The main object, however, of my pre-ent letter, is to call attention to a paper by Dr. Knott, recently read before the Royal Society of Edinburgh. Dr. Knott has actually had the courage to read the pamphlets of Gibbs and Heaviside; and, after an arduous journey through these trackless jungles, has emerged a more resolute supporter of Quaternions than when he entered. He has revealed the (from me at least) hitherto hidden mysteries of the Dyadic, and of Prof. Gibbs' strange symbols Pot, Lap, Max, New, &c. The first turns out to be only the linear and vector function; and the others are merely more or less distressing symptoms characteristic of imperfect digestion or assimilation of v. And when, at my request, Dr. Knott

translated into intelligible form the various terms of one of the less formidable formula of Mr. Heaviside's memoir, I was surprized to find two old and very unpretending friends masquerading in one person like a pantomime Blunderbore. In one of his Avatars the monster contains, besides the enclosing brackets, no fewer than 24 letters, 12 suffixes, 3 points, and 5 signs! When he next appears he has still the brackets to hold him together, but although he has now only 18 letters, he makes up his full tale of 44 (or 46) symbols; for he has 9 -suffixes, 3 indices, 3 points, 5 signs, and 3 pairs of parentheses! I used to know him as compounded of 14 separate marks only, viz. :- V2vr+ 25vv1So, :-but, unless I had required to dissect him, I should never have put him in anything resembling his new guise.

Dr. Knott's paper is, throughout, interesting and instructive-it is a complete exposure of the retensions and defects of the (so-called) Vector Systems. "Wer diesen Schleier hebt soll Wahrheit schauen!" I find it difficult to decide whether the impression its revelations have left on me is that of mere amused disappointment, or of mingled astonishment and pity. P. G. TAIT.

Edinburgh, 24/12/92.

Measurement of Distances of Binary Stars. WITH reference to Mr. C. E. Stromeyer's letter on the above subject, which appeared on p. 199, it may be of interest to point out that his plan of determining the distance of a binary star is by no means a new one.

The method was, I think, first suggested by Mr. Fox Talbot at the Edinburgh meeting of the British Association in 1871; but the mere idea was sufficiently obvious as soon as the possibility of determining velocities by the spectroscope had been demonstrated by Dr. Huggins.

The first discussion of the geometrical conditions of the problem was given by Prof. C. Niven in the Monthly Notices, vol. xxxiv. No. 7, where he exhibits the relation connecting the parallax, the relative velocity, and the elements of the orbit of a double star, and computes the value of the product (V) of the parallax and velocity for a small number of binary systems.

In a paper published in the Proceedings of the Royal Irish Academy for May, 1886, I examined the same question from a slightly different point of view, being at the time unaware of Prof. Niven's paper, and was led to similar results. An epitome of this paper was published in your Astronomical Column, vol. xxxiv. p. 206. From the results obtained it appeared that, all things considered, y-Coronæ Australis and a-Centauri were the most likely binaries to yield to this method of eliciting the secret of their parallax, while a-Geminorum, one of the stars selected by Mr. Stromeyer, was shown to be most unfavourable on account of the situation of its orbit.

In the Monthly Notices for March, 1890, I again drew attention to the subject in view of the accuracy of the results ob tained by the photographic method in the hands of Prof. Pickering and Prof. Vogel. In this paper I gave an extended list of binaries with the usual geometrical and dynamical elements, and in addition the two elements A and B on which the relative velocity depends. I also gave the greatest value which TV can attain in each case and the velocity to be expected in the case of those stars whose parallaxes had been determined.

Again in Mr. J. E. Gore's valuable catalogue of Binary Star Orbits, published in the Proceedings of the Royal Irish Academy for June, 1890, columns 18 and 19 are devoted to the constants A and B computed from my formulæ (which I may say ought more properly to be called Prof. Niven's formulæ on account of the priority of his paper) for eighty-one different orbits.

The subject has also been discussed by Miss Clerke in "The System of the Stars," pp. 199-201, where references to most of the original publications will be found.

I may perhaps add that the inverse problem of determining the elements of the orbit from spectroscopic observations alone has also been investigated by me in the Monthly Notices, vol. li. No. 5, where I have deduced the principal elements of the orbit of B Auriga, a spectroscopic double which no telescope can divide.

I have been disappointed that astronomers engaged on spectroscopic determinations of stellar velocities have not devoted more attention to observations of already known binaries, which

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appear to me to offer a promising field of work, and have d regretted that at this observatory we have not the mea undertaking the investigation, and if Mr. Stromeyer's leter no other effect than to bring the subject once more forw will have done good service, but I should like to point ou the second of the stars selected by him ought on no accou be taken as a test of the feasibility of the method, sing a accurate discussion of the conditions shows that unless this s exceptionally remote system the velocity must be very indeed. For instance, assuming Johnson's parallax, viz the relative velocity of the components amounted last only o'6 miles per second.

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In the northern hemisphere the most favourably st binaries are 70 Ophiuchi, -Ursa Majoris, and, if Peters represents the real motion of the pair, 61 Cygni; while for: southern hemisphere special attention ought to be dire a Centauri and y-Coronæ Australis.

In Mr. Gore's Catalogue, referred to above, will be fort. the materials for determining when to observe any known b most favourably in this respect, and for deducing its par from the measures obtained, and it ought to be borne in t before letting the subject sink back once more into obi that, other things being equal, this method is most liker succeed in the case of the most distant systems, whee parallax is so small that the ordinary trigonometrical necessarily fails us, and that when the micrometer, the he meter, and the stellar photograph break down, the spectrosc. will sound the further depths with ever-increasing facility. Dunsink Observatory, co. Dublin. ARTHUR A. RAMI December 30.

December Meteors (Geminids).

THESE meteors were moderately abundant on the nigh December 12, which appears to have been a very favourable in regard to weather. The chief radiant point was observe. the normal position very close to a Geminorum, and there strong contemporary shower from a centre ea-t of 8 Gemin

At Ioh. Iom. December 12, a fireball estimated to be as brilliant as Venus was observed by Mr. Booth at Leeds moved rather slowly from 150°+43° to 188°+41°, and divi” into two pieces at the finish.

Mr. Wm. Burrows, of Small Lane, Ormskirk, writes to with reference to a meteorite which he observed to fall ata hour on the same night. He says the time was 6.52 10 (December 13), and refers to the phenomenon as folla "Seeing the meteor was coming to the earth I crossed the e to where it appeared to be falling, and it fell about two ye from me. When it struck the earth it made a noise like report of a gun; it also went black instantly. While descent, it had a tail of fire about a foot long. It is 1 inch in diame one way, and 1 inch another, and one inch thick."

Mr. Burrows sends drawings of the object, and it being = in his possession it is hoped the matter may be suitably ins gated. Should it prove a veritable meteorite one interesting cumstance in connection with it will be that its descent t place concurrently with the shower of Geminids.

It is significant that December 9-13 constitutes a well-def ærolitic epoch, rendered memorable by the fail at Wold tage, Thwing, Yorkshire, on December 13, 1795, and by others, such as that at Mässing, Bavaria, December 13, 180) Weston, Connecticut, U. S. A., December 14, 1807; at Wibers Finland, December 13, 1813; at Ausson, France, Dece 9, 1858; at Baudong, Java, December 10, 1871, &c. Bristol, January 1. W. F. DENNIN

The Earth's Age.

As Dr. Wallace (NATURE, p. 175) trusts "that on consideration" I shall "admit that" my "objection is in it is evident that I have failed to make clear to him my argu showing that his data do not warrant his conclusion. He overlooks the fact that a thickness of 177.200 fe sedimentary rocks is, standing alone, a perfectly ind quantity; to make it definite it must have a definite ares As he mentions no area for it we are justified in assum he means the land area of the globe, whereas his calc is made as though area were not of the essence of the pr in short, as if the formation of a pile of sediment 177,200 thick, of no matter what area, were the problem.

In Sir A. Geikie's calculation and all other similar ones with hich I am acquainted, the thickness of the sedimentary rock tacitly assumed to be their thickness all over the land area of e globe.

Dr. Wallace's calculation leads to the absurd result that coninents are growing nineteen times as fast as materials are produced to supply their growth.

Leaving the question of the conclusions to which Dr. Walace's data logically lead, I may say that I am not responsible, and do not hold him to be responsible, for the absurd theory as to the thickness of sedimentary rocks on which they are based.

In order to arrive at a scientifically accurate result, what we require to know is the present actual thickness in every part of the world, plus all the thickness which has previously existed in, hat since been denuded away from, every area. The existing thickness in geologically explored areas can perhaps be ascertained within certain limits of error from geological maps and memoirs. For instance where the surface consists of Torridon Sandstone overlying Archæan gneiss of igneous origin, the thickness of sedimentary rock is that of the Torridon Sandstone only, if we assume that the gneiss there is part of the metamorphosed original crust of the earth, for the existence of which Rosenbusch has recently argued.

It is easily demonstrable, first, that in many places the existing thickness of each formation, where undenuded, is far from being the maximum thickness, and, secondly, from the thinning out in some directions, or merging, near the old shoreline, into conglomerates, that some formations were never deposited over certain areas; indeed, the very existence of a sedimentary deposit necessarily implies that of land undergoing denudation and not receiving deposit, although it may well be doubted whether the land area was always nineteen times the area receiving deposit.

Reasoning from the deposits preserved as to those removed by denudation, it is highly improbable that any considerable area ever received either the complete series of deposits, or on the average anything like the maximum thickness of the deposits it actually received. In addition to this, some formations usually considered to be successive may be really contemporaneous, so that the figures representing maximum thicknesses usually taken in calculating the earth's age are probably far above the truth for the purpose in question.

The immense labour involved in calculating the existing thickness of sedimentary rocks in each area, and the thickness which there is any reasonable ground for supposing to have been at any time denuded from that area, as well as the uncertainty of the results, has probably deterred geologists from attempting the task, especially as large areas are very imperfectly known. BERNARD HOBSON.

Tapton Elms, Sheffield, December 24.

THE first part of Mr. Hobson's letter alone requires notice from me, as the latter part characterizes as absurd the views of those eminent geologists who have estimated the total thickness of the sedimentary rocks, and seems to assume that such writers as the late Dr. Croll and Sir Andrew Ramsay overlooked the very obvious considerations he sets forth.

As regards myself, he reiterates the statement that when geologists have estimated the total thickness of the sedimentary tocks at 177,200 feet, they mean that this amount of sediment has covered the whole land surface of the globe; that, for example, the coal measures, the lias, the chalk, the greensand, the London clay, &c., &c., were each deposited over the whole of the continents, since it is by adding together the thicknesses of these and all other strata that the figure 177,200 feet (equal to 33 miles) has been obtained.

Mr. Hobson concludes with what he seems to think is a reductio ad absurdum-" Dr. Wallace's calculation leads to the absurd result that continents are growing nineteen times as fast as materials are produced to supply their growth."

But the apparent absurdity arises from the absence of any definition of the "growth of continents," and also from suppoing that the growth of continents is the problem under disCusion. The question is, as to the growth in thickness, of sedimentary deposits such as those which form the geological series. These deposits are each laid down on an area very much smaller than the whole surface of the continent from the denudation of which they are formed. They are therefore necessarily very

much thicker than the average thickness of the denuded layer, and the ratio of the area of denudation to the area of deposition, which I have estimated at 19 to 1, gives their proportionate thickness. If Mr. Hobson still maintains that he is right, he can only prove it by adducing evidence that every component of the series of sedimentary rocks has once covered the whole landsurface of the globe; not by assuming that it has done so, and characterizing the teaching of all geologists to the contrary as absurd. ALFRED R. WALLACE.

Ancient Ice Ages.

MR. READE in his letter (NATURE, p. 174) refers to the striations on the pebbles forming the conglomerates at Abberley and the Clent Hills.

Following the late Sir Andrew Ramsay, he considers the deposits to be of glacial origin, but goes further than that distinguished geologist in citing them as proof of a former ice

age.

It is but reasonable to suppose that glaciers existed in past ages in places where the conditions-such as high altitude and abundant precipitation-were favourable.

Before, however, the existence of a former glacial period can be established, we must have evidence of contemporaneous deposits of undoubtedly glacial origin, and extending over widespread areas-say half a hemisphere. J. LOMAS. University College, Liverpool, December 31.

Printing Mathematics.

THE use of the solidus in printing fractions has been advocated by authorities of such weight that it seems almost a heresy to call it into question. Yet I venture to think that there is a good deal to be said against it. In such matters the course preferred by mathematical writers and their printers is apt to take precedence over that which is most convenient for the great body of those who will read their work. It is tacitly assumed by those who prefer this notation that the getting of mathematical formulæ into line with ordinary printing is an unmixed advantage. No doubt it is easier to set up the work in type thus, but with the consequent rapidity and cheapness of printing the advantage ends. Most people will agree that it is much pleasanter to read a mathematical book in which the letterpress is well spaced, so that the formulæ stand out clearly from the explanatory language, than one in which the two run together in an unbroken stream: just as a book divided into paragraphs is more readable than one which is not. The old style is more restful to the mind and eye, and one can more readily pick out the salient features of the demonstration.

Another aspect of the question seems to me more important. In making any calculation mentally it is much easier to visualize fractions, more especially if complicated, as written in the ordinary way than as written with the new-fashioned notation. The component parts of the mental picture are imagined as spread over a plane instead of being arranged along a line, and can be thought of separately with less confusion. From a similar point of view it will be admitted that it is inconvenient to write mathematical expressions in one form and to print them in another.

Then, again, I doubt whether the assumption that the solidus notation conduces to accuracy is justified. No doubt the printer makes fewer original errors; but whereas with the old notation his frequent glaring errors are more readily detected by the proofreader (or, if missed by him, by the ordinary reader), with the new notation the misplacement or omission of a solidus is, from the simplicity of the error, likely to be overlooked. In general, no one will be the poorer if a little more trouble is taken with the printing, and a little more paper is used for the book.

The symbol has advantages over its equivalent, and to its restricted use, such as is made by Sir G. Stokes, one can hardly object; it matters little how such expressions as a as a/b or dy/dx are printed. But it is the thin end of the wedge; and one is under a debt of gratitude to Mr. Cassie for showing, in your issue of November 3, to what it may lead. May it be a long time before we have to learn to substitute for the harmless expression, bb its newest equivalent, | ¿ \ 1 /2 | | c | d + e \ 31 !

c(d + e)

I trust that no one will interpret the final note of exclamation as
a factorial symbol.
M. J. JACKSON.
D. I. Sind College, Karachi, November 23.

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