will have no hesitation whatever in endorsing this

statement!

While a great many notions are introduced into the text in such a way as to make them appear useless, uninteresting, and unintelligible, many of the most important points in a rational system of mathematical education receive little or no attention. Take the broad, general notion of a function, so simple that it can be explained to anyone who is sufficiently unmathematical to understand common sense. It is well illustrated in the case of the senior wrangler who entered the Stock Exchange and began to apply algebraic methods to the money market. He failed because he had

1

AN EGYPTIAN DESERT.

The Geography and Geology of South-eastern Egypt. By Dr. John Ball. (Survey Department of Egypt, Cairo.) Pp. 394. With Maps and many Illustrations.

HE area described in this latest monograph

Here was a case where the result was a function of a number of variables, and he treated some of these variables as constant. The senior wrangler was not sufficient of a mathematician. The mere failure to enumerate all the variables in a function occurring in everyday life represents a national loss of millions per annum. Prof. Perry says:"I must confess, however, that the compilers of modern school algebras must make the gods laugh over the uses to which they put this plotting of functions."

is the southern part of the Eastern Desert-a district little known, and inhabited only by nomad

Arabs of the Ababda and Bisharin tribes.

From north to south, this country is intersected by a mountainous axis which rises to heights of from 3000 to more than 5000 feet, and consists mainly of granite, diorite, gabbro, and other Eastward from this axis a series plutonic rocks. of Wadys run down, somewhat steeply, to the shores of the Red Sea on the east, these shores

We

All this is a very great pity. Most modern mathematical teachers are only too glad to get "formula" questions for their pupils dealing with beams, expansion of steam, flow of water through pipes, electrical resistances, and other practical considerations which familiarise the student in the use of algebraic formulæ and equations. believe they can get the questions they want from this book, and, on the other hand, when it comes to methods of teaching, every teacher naturally prefers his own. But if the substance of the text is a fair indication of what is meant by "practical mathematics," we agree with Prof. Perry's remark (p. xiii.), which, when quoted without its context, reads to the effect that

"The subject of practical mathematics is, I am happy to say, a subject which is not likely to commend itself to such institutions, nor are such text-books likely to be of much use to real students."

Evidently neither "academic" nor "practical" mathematics supplies exactly what is wanted. They both have one fault in common, namely, that they place difficulties before the student without any rhyme or reason. Mathematics is not in itself difficult or uninteresting; a child of three can invent a theory and notation for minus quantities without any assistance, help, or encouragement whatever. What we want is a

which render the coast one of the foulest in the world for shipping. Westward from the mountain axis another series of Wadys lead down more Dr. Ball, who gradually to the basin of the Nile. is an accomplished surveyor as well as a geologist, has been able to add much to our knowledge of the physiography and scenery of this almost unexplored country. While devoting his chief attention to the geological features of the district-his discussion of the petrology being especially full and well illustrated-the author has been able to supply much new and interesting information concerning the antiquities, the plant and animal life, and the inhabitants-their languages, industries, and customs.

From a very early period the district has been credited with the possession of considerable mineral wealth, especially famous having been its gold-mines and emerald workings. Dr. Ball's researches, however, do not give much support to the belief that the district may in the future become a great mining centre.

It is true that very numerous small workings scattered all over the country show how wide and persistent has been the search for gold within the area. There do not appear to have been any alluvial workings, but numerous quartz veins, intersecting all the crystalline rocks of the district, sometimes containing containing calcite with ores of copper and iron, yield minute quantities of gold,

the particles of which are seldom visible. In small handmills of diorite this quartz appears to have been ground up and the gold extracted from it, but this appears to have been only profitable when done by convict labour. Prospectors in modern times, guided by the presence of these old workings, have attempted to carry on the extraction of the gold on a larger scale by modern methods, but only in a few cases has it been found that this can be done with profit, and most of the concessions have been surrendered.

The once-famous emerald-workings of Zabara and Sikait in this district are opened in masses of mica-schist, which alternate with gneiss, and contain crystals of tourmaline and beryl-the clear green varieties of this latter mineral constituting the valuable gem emerald. It would appear,

however, that the ancients were satisfied with specimens which, owing to their clouded or flawed characters, do not appeal to the jewellers of the present day, and, extensive as the old workings undoubtedly were, there appears to be little chance of the industry being revived.

At several points on the Red-Sea coast, deposits of gypsum and anhydrite, with pockets of sulphur, occur, and concessions for the working of the latter mineral have been granted.

One successful mining industry, however, would appear to be in full operation in the little island. of Tuberged, or St. John's, lying out in the Red Sea, forty or fifty miles from the coast. Here, in the midst of serpentine rocks, numerous beautiful crystals of peridot (oliome) are obtained, and the work of exploitation is being successfully

carried on.

author is well known as the present professor hygiene at Harvard, and former director of the Hygienic Laboratory, Public Health and Marre Service, U.S.A., who has contributed much to the scientific investigation of milk and its bacte logy. From the sanitary point of view the bo is sound, but, in addition, its author shots I knowledge of the subject from the producer's pr of view, a side of the question which is frequent!, overlooked by sanitary reformers in this country Written in simple language, it is a book for the educated public generally, and many striking car toons and diagrams and terse sayings serve to drive home the views enunciated, e.g.

"It (milk) requires scrupulous care from pasture to pail, and from pail to palate."

"The milk problem starts with the cradle, and ends with the grave. Sometimes it leads to an untimely grave."

One or two considerations may be quoted as illustrating the author's appreciation of the producer's point of view. As regards the question ducer's point of view. as to where the blame lies for an unhygienic mi supply, the author says the tendency is for the consumer to blame the producer, for the producer to blame the consumer, for the middleman to blame both of these, and for the health officer to blame all three. Really, society must blame itself, we are suffering the inevitable penalties we must pay for modern conditions of life, and of all those concerned the farmer is least to blame for the situation as it exists, and the consumer in the city should be ever mindful that he has large.. brought the conditions upon himself.

While expressing the opinion that bottled m is the ideal method of distribution, Prof. Rosen... fully recognises its dangers and difficulties—di?“ culties in the cleansing and handling of the botties and in transportation, which, he points out, is expensive, bulky, the breakage is considerable and the return freight adds to the cost. He looks forward to the time when milk will be dispensed in some form of cardboard non-returnable package

On the mixing of milk from several cows, this the author says is desirable, as it furnishes a more uniform product, and tends to dilute infer tion if present; this the framers of Bills in hus country might note.

On the cells present in milk, it is stated that normal milk has relatively few or no leucocys, and, when critically examined, the majority of cells distinctly differs from leucocytes.

The author would ascribe from 5 to per cent. of all human tuberculosis to infection with the bovine bacillus, but he is careful not suggest that this is necessarily derived from milk and quotes Weber's observations (made for the

A long chapter is devoted to pasteurisation of milk, and this process is strongly advocated, under proper safeguards, such as labelling with the time, temperature, and date of pasteurising, as a general method for treating the milk supply if a pure milk cannot otherwise be guaranteed.

Singularly little is said respecting infants' milk depôts. In view of the comparison made by some between the United States and this country in this respect, we should have expected more information had this method of dealing with infant mortality the importance which some would ascribe to it. Altogether the book is a valuable one, and one to read, ponder, and digest.

R. T. HEWLETT.

OUR BOOKSHELF.

Theoretische Astronomie. By Dr. W. Klinkerfues. Neubearbeitung von Dr. H. Buchholz. Dritte verbesserte und vermehrte Ausgabe. Pp. xxxviii+1070. (Braunschweig: F. Vieweg und Sohn, 1912.) Price 50 marks.

THE claims of Klinkerfues and Buchholz are readily admitted by all acquainted with the calcuation of orbits. The classical treatises of

Oppolzer and Watson having run out of print, it was most fortunate that this, the practical, side of gravitational astronomy was taken up by Buchholz, who extended the original work of Klinkerfues to make it fit to take the place of the former works. The plan of dividing the subject-matter into lectures (Vorlesungen) is still continued, although much is unsuitable for verbal exposition. Lecture 14, for example, extends to ninety pages, and gives, besides the general theory of the earth's rotation, all the formulæ necessary for the reduction of observations.

The greater part of this edition agrees exactly with the last, but the additions are well worthy of attention. The new preface runs to twenty pages, and gives a historical treatment of recent advances in the theory of orbits. Although Gylden's work in no way enters into the subject treated in the work, Buchholz has described at some length the claims of the great Swedish astronomer. We believe that posterity will give to Gylden the place which is due to one who did much for the cause of dynamical astronomy as applied to real, in contrast with merely ideal, problems. The criticisms of Bauschinger and others have led to Harzer's method not being developed, hut the vector method of Willard Gibbs is retained. We are glad to see that Leuschner's method has been introduced. The method is

out.

carefully explained, the formulæ are collected, and a considerable number of examples are worked The necessary new table is given, and Oppolzer's M- and N-tables have been reproduced. An appendix gives the known errors in the works of Bauschinger and Oppolzer on the determination of orbits.

The greatest fault of the book is its bulk. For a work intended to assist in the numerical calculation of orbits it is almost essential that the auxiliary tables should be easily manipulated. This work could with advantage be divided into several volumes. J. JACKSON.

New Contour Map of the Near and Middle East (The Land of the Five Seas). (London: G. W. Bacon and Co., Ltd.) Price 7s. 6d. THIS wall-map includes the empires of Babylon, Persia, Parthia, Egypt, and Rome. It shows the routes of Alexander the Great, Pompey, and St. Paul, and illustrates classical history from the earliest times. The scheme of ten colours makes it possible to show with impressive clearness the close relation between land configuration and the spread of civilisation. The scale of the map is ninety-five miles to an inch. In the bottom left corner an inset orographical map of Palestine is provided. The size of the map as a whole-40 by 30 in.-will indicate that it is scarcely large enough for use in big classes, but it should prove of service secondary schools. to individual students in the higher forms of

The Tarn and the Lake. Italian Renaissance. xi + 48.

Thoughts on Life in the By C. J. Holmes. Pp. (London: Philip Lee Warner, 1913.) Price 2s. 6d. net.

reading; originally designed as an introduction to THIS essay of Mr. Holmes forms very pleasant a few studies of Italian painting and sculpture, it overran its intended bounds and became eventually the present little volume. Entertaining analogies are drawn between certain communities of fish and certain societies of men, and though the appeal is more directly to anglers, all readers who appreciate literary expression will enjoy the essay.

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 of NATURE. No notice is taken of anonymous communications.]

Pianoforte Touch.

THE question as to whether what is known as touch is due to any peculiarity, apart from the energy, with which the hammer strikes the strings, and, consequently, with which the fingers strike the keys, would appear to be determinable by direct experiment. Personally I have always held, probably with the majority of people, that touch is dependent on such peculiarities, but the evidence seems to be decidedly against it. A note was struck one hundred times in succession so as to produce sound of as nearly as possible the same intensity; on half these occasions it was struck in a pressing, or caressing, manner; in the other cases it was struck sharply; the different

kinds of blows were intermingled irregularly, and according to a list previously drawn up, and the damper was held up while the note was being struck. As each note was struck a verdict as to its quality was given by a person sitting out of view of the piano, and this verdict was compared with the character of the blow given. The observer, it should be mentioned, was not a trained musician, but was exceptionally appreciative of what is called touch.

Discarding all cases where the note had accidentally been more or less loud than the average, it was found that in 49 per cent. of the cases of uniform loudness the effect on the auditor coincided with the character of the blow given, in 51 per cent. the reverse. In another series of 100 notes the values were 51 and 49 respectively. It is evident, therefore, according to this, that different quality of touch produces no difference in the quality of the sound produced. No doubt further evidence should be obtained by a repetition of such experiments with other performers and other observers, and a further test would be obtained by having two performers alternately striking the same note, and ascertaining whether the observer could detect the difference.

Even if the independence of "touch" on the character of the stroke is fully established, there is no difficulty in explaining the apparent difference in touch of a performance with the fingers and one with a piano-player. At least seven factors may be specified as determining touch:-(1) Differences in loudness of a note (or notes) as compared with that of the preceding notes, (2) deviation from the theoretical value, according to the score, of the duration of the note as compared with that adopted for the preceding and succeeding notes (i.e. different degrees. of staccato or legato); (3) variations in time as compared with that of the preceding notes; (4, 5, and 6) similar differences in loudness, duration, and time of one note in reference to its nominally contemporary notes; (7) the use of the loud pedal. The use of the soft pedal is so specialised that it need not be considered; it is used more for modifying a whole passage than for modifying individual notes, and it practically gives the performer the command of a second instrument; but, perhaps, an eighth factor of a general character should be included, namely the extent to which the above means of emphasising notes is made to harmonise with the rhythm and sense of the music.

Now, in the piano-player we have the means, either by moving levers or by pressure on the bellows, of altering the loudness (1) or time (3) of a note as compared with that of preceding notes, but such alterations are gross and sluggish in comparison with those possible in finger playing; the inertia of the mechanism has to be overcome, and the result depends directly or indirectly on the pressure of air in the bellows, which cannot be altered instantaneously. The least unsatisfactory operation is a retardation of the time. The duration of the note as compared with that of its neighbours (2), or of its fellow-notes (5), as well as the loudness of it as compared with its fellownotes (4), can only be accomplished by modifications in the slots of the roll, and they are, therefore, stereotyped and always the same; this produces a very unsatisfactory result.

When hearing for the first time a well-played piece with the air strongly brought out in this way, the impression produced is decidedly favourable, but at the second hearing a sense of irritation is felt; we know exactly the degree of emphasis which is going to be placed on each note of the air, and the monotony is even more trying than where no emphasis is attempted. It is a case similar to that of a beautiful landscape which never changes; the charm of music lies largely in its imperfections, or, at least, in the

varying and unexpected degree in which its beautare brought out. An alteration in the time of or temporaneous notes (6) is, I believe, attempted some rolls, but the effect, I should imagine, wo, be even less satisfactory than in the other cases, it use of the loud pedal is a potent defect in the piar player. Needless to say, this pedal is not used in get finger playing only to produce loudness, but m.generally to produce softness, and a smooth fox: sound. In a slow movement a good performer often depress and raise the pedal for nearly every tee and the effect produced depends entirely on the cr rect timing of these movements with the depress of the keys; this is impossible unless the messig. from the brain to the fingers and to the feet are simu taneous. This cannot be so with a piano-plave, where the sound is produced by a separate mechan ism; this sound (either of the particular note in que tion or of its predecessor) must travel to the brain, which then has to interpret it, and to send a messag to the finger which controls the pedal lever; the performer is conscious of an act of thought being nece sary in using the pedal of a player, whereas in finger playing its use appears as if it were instinctive. With the player it is practically only used for producing loudness.

I believe that the thud of the air on the keys another defect in the player. When in the same rote as the instrument, it seems possible to decide after hearing one bar whether the player or fingers are being used, and this is certainly so, as I have ascer tained by trial, and apparently more easily so, when one is in a distant room, the reason of this being that the thud penetrates the walls more easily than the note, and hence attains more relative predominance. I have an instance of this in a striking clo of which the note is inaudible in the next room, while the thud can be distinctly heard.

No doubt many of the defects of the piano-player will be diminished in time, especially by such devices as those of Prof. Bryan. Already one of the best piano-makers is putting on the market an instrument which is a great advance on its predecessors, the chief feature of it being a reduction in the size of the bellows, which admits of much greater control over the sound production. Still, it is a case of play ing with the feet, instead of with ten independent fingers.

When the damper is allowed to act in the ordinary way, it is possible that the effect produced (touch) may be modified by the character of the blow given to the keys, for this blow results in the damper being raised. as well as in the hammer striking the strings, and these two actions may not synchronise to the same extent with blows of different character.

SPENCER PICKERING.

A Danger of so-called "Automatic Stability." FROM time to time devices have been proposed for securing "automatic stability" in aeroplanes by mears of a suspended weight or "pendulum," which operates on rudder-planes governing the motion of the machine A similar device is also in actual use for governing the motion of torpedoes in a vertical plane, in ejunction with a further device for maintaining the torpedoes at a constant depth below the surface, or more strictly at a level where the hydrostatic pressure is constant. "Pendulum" arrangements for automatic stability of aeroplanes have frequently figured in the pages of such journals as The Scientif American, and it must be admitted that such derivs are calculated to appeal strongly to the imagination of readers whose knowledge of dynamical principies is limited in range or nil.

The statement, which stands in my name, to the

effect that such a device increases the number of degrees of freedom of the apparatus with an accompanying increase in the number of possible oscillations and of conditions necessary for stability is, I believe, incontrovertible. One form of dynamical instability that may result in such cases is the setting up of violent oscillations, ever increasing in amplitude, in the pendulum itself, accompanied by flapping of the control planes, in which case this particular method of control becomes worse than useless.

The remedy which naturally suggests itself, in such circumstances, is to damp down the oscillations of the pendulum by means of frictional or other resistances, and it is probable that few university graduates who have taken first-class honours in mathematics would think that such a contrivance could possibly be wrong. The following test case will show how very dangerous it is to attempt to draw conclusions from general considerations.

For the aeroplane or torpedo, we substitute a heavy, rigid body POR, free to rotate without resistance about a horizontal axis through its centre of gravity O, perpendicular to the plane of the paper, and therefore, in the absence of other causes, in neutral equilibrium, and we assume that the moment of inertia of this body is considerable.

We next imagine a light, small pendulum OQ to be fixed in bearings in the body POR, so that it can turn about the same axis, but we suppose that a

frictional couple is called into play between the large body and the pendulum at these bearings. The pendulum being light, this frictional couple exerts no appreciable effect on the large body POR, but the friction is sufficient rapidly to damp out the oscillations of the pendulum itself. The effect of a rudder plane controlled by the pendulum we represent by the assumption that the pendulum operates some mechanism which impresses on the large body a couple proportional to the angle QOP, tending to make it revolve towards OQ, the object of this couple being to bring that body into a position of rest in which OP is pointing vertically downwards.

When the large body is rotating in the counterclockwise direction (as in the figure) the small pendulum assumes a position of equilibrium OQ on the right-hand side of the vertical, and inclined to the vertical at a certain angle a, the moment of its weight then just balancing the frictional couple. When the body begins to swing backwards the pendulum swings with it until both have described an angle 24, so that the pendulum occupies the position OQ', now making an angle a on the opposite side of the vertical. During this portion of the motion the controlling mechanism impresses on the body a constant angular acceleration, because the angle QOP remains constant. Consequently in the new position the body is rotating with a certain angular velocity set up by this acceleration. In the subsequent motion the pendulum remains at rest in the position OQ', and the body performs a simple harmonic rotation about OQ', but owing to its initial angular velocity it does not come to rest until its angular distance from 00' is greater than the angle QOP. It follows by this reasoning that the oscillations increase in amplitude, and this effect owes its existence to the frictional couple.

G. H. BRYAN.

The Structure of the Diamond.

We have applied the new methods of investigation involving the use of X-rays to the case of the diamond, and have arrived at a result which seems of considerable interest. The structure is extremely simple. Every carbon atom has four neighbours at equal distances from it, and in directions symmetrically related to each other. The directions are perpendicular to the four cleavage or (111) planes of the diamond; parallel, therefore, to the four lines which join the centre of a given regular tetrahedron to the four corners. The elements of the whole structure are four

directions and one length, the latter being, in fact, 1.52 x 10-8 cm. There is no acute angle in the figure. These facts supply enough information for the construction of a model which is easier to understand than a written description.

If we proceed from any atom, using only standard directions, to the next but one, the straight line joining the first to the last is a diagonal of a face of the cubical element of structure; if we move in the same way through four stages, using all four standard directions in turn, the straight line joining the first and the last is a cube edge. Starting from any atom we can return to it after six stages, using three standard directions twice each. In this way we always link together rings of six carbon atoms.

If the structure is looked at along a cleavage plane it is seen that the atoms are arranged in parallel planes containing equal numbers of atoms, but separated by distances which alternate and are in the ratio 3:1 (actually 1.52 x 10-8 cm. and 0.51 x 10-8 cm.). It is a consequence of this arrangement that no second order spectrum is reflected by the (111) planes, although spectra of the first, third, fourth, and fifth orders are found. It was this fact that suggested the structure described above. Several other tests, however, may be applied, and all are satisfied.

Zincblende appears to have the same structure, but the (111) planes contain alternately only zinc and only sulphur atoms. In this way the crystal acquires polarity and becomes hemihedral.

Leeds, July 28.

Artificial Hiss.

W. H. BRAGG. W. L. BRAGG.

REPLYING to the inquiry of Lord Rayleigh (in NATURE of May 29, vol. xci., p. 319) as to the way in which an artificial hiss may be produced with a moderate pressure of air, I suggest that a current of air directed against a sharp edge of a knife held somewhat obliquely may answer his purpose.

In this connection it is interesting to note that for the formation of the hissing sound in our mouth the presence of saliva seems necessary. If I dry the tongue and the other parts which are needed for the pronunciation of the hissing "s," it is almost impossible to produce an audible "s," and the tongueinstinctively, as it were-makes an effort to gather some saliva and to wet itself.

I would therefore suggest that Lord Rayleigh wet the end of the rubber tube with which he experimented. FRED J. HILLIG. Kioicho 7, Kojimachi, Tokyo, July 1.

IT had occurred to me also that the moisture of the mouth might play a part in the production of a hiss, but I do not find that such drying as I can give makes an important difference.

I have to thank several correspondents for suggestions. In particular, Mr. G. Beilby sent me two pipes suitable for a 4 in. water pressure, which gave a better effect than anything I had then tried, but still, in my estimation, much short of a well-developed