The second part is taken up with a visit to Lemuria, thereby meaning Madagascar and the Mascarene Islands of Bourbon and Mauritius. This visit was made in 1862. The Seychelles were not visited. Some noteworthy details are given of a visit to Madagascar. The home of the Ouvirandra fenestralis is well described, and the account

of a Mauritius hurricane is true to the life. It is mentioned that in the hurricane of March 12, 1868, the iron girders, 200 feet in length, and weighing over 300 tons, were blown from the railway bridge over Grande Rivière, when a force of 100 pounds to the square foot must have been exerted by the wind down the ravine.

In an appendix to the second part, there is a chapter on

the natural history of Madagascar, contributed by Mr. J.

G. Baker, F.R.S., of Kew.

The third part of the volume describes a visit to Columbia along with Capt. Pim and Mr. John Collinson, for the promotion of a transit railway route through Mosquitia and Nicaragua.

The volume will commend itself to the general reader, and the scientific notices mentioned therein will be found very generally interesting and correct.

Modern Metrology. By Lowis D'A. Jackson. (London: Crosby Lockwood, 1882.)

IT is no easy task to give an account of the various systems of weights and measures in use throughout the world, to trace their origin, and to express their equivalents in English and French weight and measure, but this the author has undertaken in one part of his book, and has brought together much valuable and interesting information. This work is so far a cambist or dictionary of weights and measures, both the scientific and commercial equivalents of all foreign units being given. It would have been well, perhaps, if the author had stated for each country the precise authority from which he obtained his equivalent, as works of this kind should as far as possible contain within themselves means for verifying the accuracy of the figures given.

The main object of the work appears, however, to be the discussion of a remedy for the evils of the complex systems of weights and measures which are unfortunately still in use in this country. To provide such a remedy is a serious task, and one well worthy of the attention of a great statesman such as Mr. Gladstone, to whom this work is inscribed.

The author discusses the vexed question of the relative values of standard temperatures at 32° and at 62°, and proposes a new English system based on a cubic foot of 1000 "fluid ounces," at the temperature of the maximum density of distilled water.

The "fluid ounce" is taken as equal to the weight of distilled water contained in a cubical vessel whose dimensions are equal to a "tithe," or tenth part of the linear foot, when weighed and measured also at the temperature of the maximum density of distilled water. The various parts and multiples of the cubic-foot and "foot-weight would be built up by decimal progression, so that a strict correspondence would be always maintained between capacity, linear dimension, and weight. The effect of reducing the temperature of the cubic foot from 62° F. (the present legal temperature) to 39°1 F. (the temperature of the maximum density of water), would bring the weight of the cubic foot more into accordance with modern research, as it would raise its weight from 62 321 lb. to 62.424 lb.

Although we have faint hope of present success in disturbing the deeply-rooted systems of measures now in use by this great commercial nation, or of substituting for the purposes of international science a more acceptable metrology than that based on the metre and gramme, we cannot but recommend this work to the consideration of all interested in the practical reform of our weights and

measures.

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. [The Editor urgently requests correspondents to keep their letters No notice is taken of anonymous communications. as short as possible. The pressure on his space is so great that it is impossible otherwise to ensure the appearance even of communications containing interesting and novel facts.] Sun-Spots and Magnetic Storms

SOME particulars of the remarkable double outburst of sunspots in the latter part of April and of the magnetic disturbances which appear to have been connected with them, as recorded at

Greenwich, may be of interest to your readers. It is in itself a noteworthy fact that there should be on the sun at the same time two spots, one of them double, sufficiently large to be visible to

the naked eye, and this is made still more interesting by the

occurrence of a violent magnetic storm a few days after the appearance of each of these large spots.

The larger of the two spots would seem to have first made its appearance at the east limb on April 11 at about 15h G.M.T., though no photograph of it was obtained till April 14. It must have passed off the disk about April 25 8h., being well shown on a photograph on April 24. It extended over 10° of heliographic longitude, and measured more than 2' of arc in length and breadth.

The group preceding it, consisting of two spots, was first photographed near the east limb on April 11, and was then com

paratively small. Between April 16 and 17 it increased suddenly far exceeding in area any of the spots previously recorded at in size, becoming nearly as large as the other spot group, and Greenwich.

The areas of these groups on the photographs, expressed in millionths of the sun's visible hemisphere, and corrected for foreshortening were as follows:

The total spotted area on 1882, April 17, was 881 for the nuclei, and 4668 for the whole spots, being about double of the greatest spotted area shown on any of the Greenwich photographs previous to this outburst.

On April 16 and 19 violent magnetic disturbances occurred. At Greenwich the declination, horizontal force, and vertical force magnets became violently disturbed on April 16 at 11h. 30m. G.M.T., the first movement for all three being simultaneous and sudden, and the storm movements continued till April 17, 7h. 30m. G.M.T. The magnets remained quiet till April 19, shortly after 15h. 30m. G.M.T., when another sudden and very sharp disturbance commenced, lasting till April 20, 20h. G.M.T.

lations of declination till April 17 oh., the greatest recorded In the magnetic storm of April 16-17, there were large oscilmotion being about 1°, from a diminished declination at 19h. + to an increased declination at 19h. Then until 4h. the register cannot be traced, the motions being either unusually rapid, or the magnet being disturbed by workmen in the upper room. About 7h. there were some sharp motions, after which time the magnet became quiet. The principal fea'ure of the disturbance of horizontal force was at first an increase, followed by a much larger diminution, amounting to about 1-50th of the whole force at 16h., when the trace went off the sheet, and was lost until April 17 oh. There was then a great and rapid increase (in about 40m.) of about 1-40th part. Rapid motions were then shown till 7h. 30m. The whole range of these disturb. ances probably exceeded 1-25th part of the whole force. The vertical force at first decreased somewhat till April 16, 16h., and afterwards very considerably till about 18h., when it was diminished by about 1-100th part, then with the horizontal force it increased till about Ib., when the trace went off the sheet, its value being then

greater than the normal by more than 1-100th part. Finally, the normal value was gradually reached again at 7h. 30m.

In the second magnetic storm, the range of the declination movements was 1° 30', of the horizontal force about 1-50th part of the whole, and of the vertical force about 1-120th part.

Thus the vertical force disturbance on April 19-20 was comparatively small; that of the vertical force on April 16-17 is characteristic of the greatest storms, and since the great disturbances of 1872, February 4, and those of October 4 of the same year, no magnetic storm has been recorded equal to this. Earth-currents were shown throughout both storms.

Royal Observatory, Greenwich, May 8

W. H. M. CHRISTIE

Hypothetical High Tides

I REGRET that I am not able to accept the criticisms of Mr. C. Callaway on my notice of Prof. Ball's lecture. I have studied the effect of tidal and wind waves on many coasts through many years, and my observations do not warrant the statements he makes. Every schoolboy knows the distinction between waves of undulation and translation, and it is in no sense true that I have confused them. With waves of undulation such as occur in mid ocean we have nothing to do in this discussion, but it cannot be unknown to Mr. Callaway that all such waves when reaching a shore, become waves of translation, and more or less powerful denuding agents. If he will have the kindness to refer to my "Report on the Geology of Chio," vol. i. pp. 52, &c., he will find that I have done ample justice to the efficiency of wind waves as agents of geological change. The great tidal current rushing around the earth, with which he credits me, exists only in his own imagination. I have suggested nothing of the kind, but the rapid ebb and flow over the shores of continents of tidal waves several hundred feet in height must necessarily act with great violence upon such shores, and I insist that such tidal waves as pictured by Prof. Ball would have left a very different record from that we find in our Palæozoic rocks. Some of our American Silurian strata were deposited on shores that faced toward the east, where they had an unbroken stretch of several thousands of miles of ocean over which the tidal wave would come to them without obstruction, and there the maximum effect of such tides as Prof. Ball describes would be produced, but no traces of them are found.

I am aware that the reef-building corals of the Devonian were zoologically distinct from any now living, and for that reason conditioned my inference from them; but we have satisfactory evidence that the Devonian coral reefs of Ohio and Kentucky were formed along a shore and in a warm sea, like the reefbuilding corals of the present day, and there are many reasons for believing that their mode of life was the same.

The point made by Mr. Searles Wood in regard to the coal, though objected to by Mr. Callaway, is well taken, for we know that the great coal marshes of America were located just at the sea level, and from time to time were inundated by the sea and covered with sheets of marine limestone. It requires no argument to show that the growth of the Carboniferous forests and the formation of beds of peat-now coal-could not have taken place with tides rising one-half or even one-fourth of the height of those described by Prof. Ball.

On carefully reviewing the facts which suggested any objection to Prof. Ball's theory, I am compelled to reiterate the statement before made, that on the east coast of North America the geological record bears positive and conclusive testimony against the high tide theory, and that at least since the Archæan ages no tides have swept this coast like those required by Prof. Ball's theory, even as modified by Mr. Darwin. New York, April 10

Aurora Borealis

J. S. NEWBERRY

A POLAR aurora of remarkable activity and brilliancy was observable here at 11.15 last night. The centre of energy at the time of origin was under Cassiopiea, a highly luminous streamer shooting upwards from the horizon, and covering, but not hiding the stars in that constellation. Brilliant white flickering streamers and rays, arising from the horizon, quickly spread towards the north-west, and an irregular shaped mass of crimson light, of an altitude of 45°, and many degrees in breadth, appeared, and suffused the northern hemisphere from Cassiopiea to Gemini. Between the horizon and the mass of

crimson-coloured vapour the sky was of a light green hue, and upwards, through this area of greenish light, the rays and streamers shot.

During the period of greatest energy, a beam of vivid light arose towards the north-west, was projected over Gemini, and quite extinguished the light of the two large stars in that constellation. About midnight the meteor faded. Soon afterwards the sky became overcast. Throughout the day the wind had been blowing briskly and coldly from a point in the north-east. Barometer, highest during the day in the shade, 60; aneroid, 302. In the afternoon, curious slender-rayed cirri from the north-west, generally indicative to my mind of auroral disturbanie, crossed the northern sky towards the zenith. To-day, detached clouds and blue sky, and the wind is blowing stiffly and icily from the same point. Barometer and aneroid same as yesterday, the latter inclined to full.

An auroral display of the like splendour and activity is very rare, if not wholly unknown in this locality at this time of the year. X.

Worcester, May 15

AT 10'55 p.m. last night, I observed a very beautiful aurora borealis, consisting mainly of three beams, nearly vertical, and then parallel to the direction of the stars a and of the Great Bear. They moved slowly towards the east, and about 113 faded away. About 11 o'clock they were very bril iant, and the central beam, then quite close to the Polar star presented a slightly purple or violet colour, as represented in the sketch inclosed. Near the earth, and at about 15° to 20° elevation, there was a mass or cloud of suffused light, from which the beams seemed to rise. It seems that an aurora was also seen about a fortnight ago from Dublin. The wind has been easterly, and to-day is somewhat colder than yesterday.

J. P. O'REILLY Royal College of Science, Dublin, May 14

Spectrum of Wells' Comet THE Continuous spectrum of this comet, especially of the nucleus, is remarkably bright. I observed it on April 18 and May 6 and 15. There were at least three bright bands, and perhaps more. I believe the usual three were visible, but they were so indistinct that I did not observe their positions. The middle one was much the brightest, and the only one plainly visible. I never saw a comet in which the bright bands were s faint, relative to the continuous spectrum. May 16

T. W. BACKHOUSE

The Recent Violent Storm

MAY I ask your permission to insert from me a curious circumstance, which came under my notice, soon after the tremendous storm which took place on Saturday, April 29 last. On the glass of some of the windows of the house in which I reside I perce ved a very singular appearance upon them, somewhat resembling a deposit from milk. On looking at it through a microscope I discovered a number of very beautiful crystals, which, without doubt, were caused by the spray from the sea, as on applying it to the tongue, there was a strong taste of salt. The distance the spray must have been carried on this occasion could not have been less than sixty miles, taking into consideration the course of the wind, which was south-west.

Waddon, Croydon, Surrey, May 15

NATHANIEL WATERALL

The Cuckoo

Is it a normal habit, or only an erratic freak of that quaint bird, the cuckoo, to sing at night. On the night of Monday, the 8th inst., I first heard, at 10 p.m., three or four calls, but took little notice of it, thinking the bird had been startled from his dreams, but in a short time he recommenced, and went on continuously with short intervals of silence, until 12 or 1, precisely as by day.

It was a dark night with only dim starlight. I should like to know whether others have remarked this nocturnal loquacity the cuckoo. This particular bird has usually commenced his song or call at about 4 a.m. J. E. A. BROWN

May 11

The Swedish Fisheries

IN NATURE of April 20 you ask for an explanation of the difference in the figures of Dr. Oscar Dickson and Dr. Lundberg relating to the Swedish fisheries.

1. You mention that Dr. Lundberg, in the "Notizen uber die Schwedischen Fischereien,” 1880, valued the herring fisheries of Sweden at 5,000,000 marks (3,547, 303, p. 27), but you forget that Lundberg's book only refers to the "Ostsee und Sussvasser Fischereien," and Dr. Dickson only speaks of the west coast or North Sea fisheries.

2. About the "millions of barrels representing millions of pounds sterling," the Swedish North Sea Herring Fisheries statistics value the barrel herring now only at 1 to 2 shillings in the first (fisherman's) hands. And you will notice that the millions of barrels mentioned by Dr. Oscar Dickson in the Scotsman, refer to a former period, nearly a century ago, not to the present period, commenced 1877. Pyphis, Sweden, May 3 GERHARD VON YHLEZ

THE OLDEST EGYPTIAN TOMBS AND
TENANTS

ALTHOUGH the existence of mankind in the dawn of civilisation at the Stone age, and using Palæolithic tools is distinctly proved in Northern Africa and Algeria, the specimens discovered on the soil of Egypt are not so unequivocally Palæolithic, although those published by Sir J. Lubbock approach the type. Of the Neolithic stone weapons, numerous examples have been found, some undoubtedly in use at the time of the eleventh and twelfth dynasty, others probably descending to the eighteenth and nineteenth dynasties. The indications, however, of sepulchres of the Paleolithic period are absolutely wanting on the soil of Egypt, and except the stone huts in the Arabian Peninsula, belonging to the more ancient period, there are no remains of contemporary construction.

In the graves around the oldest pyramids Neolithic remains are occasionally found, and there can be no doubt that flint weapons were extensively used at the oldest pyramidal period, which, however, was one of copper and bronze, copper and even iron objects having been found in the air passages of the great Pyramid, and indications of the use of the hollow bronze drill in the stone sarcophagi of the same epoch. The pyramids were arranged chequer-wise in groups, and each separate site belonged to a different dynasty, the kings and other royal personages being inhumed in them, while around the pyramid of each site were arranged the tombs of the courtiers and officers of the court. The arrangement of these tombs differs at the respective sites. At Sakkarah they are arranged in rectangular groups of streets, and the same arrangement prevails at Gizeh. At Abusir the last undulations of the step-shaped hills which crown the pyramids are occupied by some tombs scattered about of the time of the fourth and fifth dynasty. At Dashour there are also some tombs of a very early and unknown period, and at Meidum, tombs of the third dynasty. All these tombs bear a general resemblance to each other, and pass by the general Arab name of mastabas,“ counters,” or "beds." At first sight they look like the pedestals of pyramids, or truncated pyramids, being of rectangular shape, with sloping sides; they are, however, not square, but rectangular, and the angle of the sides is so great that the walls, if produced upwards, would rise to 600 feet, an impossible height for such a construction. Consequently they are not of the class of pyramids, but only show the Egyptian preference for converging lines, instead of purely parallel or rectangular forms; the short sides also in some instances are step-shaped, the layers of stone are squared and laid in horizontal courses, and not polygonal m sses. These rectangular masses of masonry or brick-work, the details of which will be subsequently described, did not contain the sepulchral chamber, for that is always found in the solid rock beneath, the access to which was by a

se

rectangular shaft or well, down which the coffin and mummy were lowered by ropes; the mass of masonry above had only in it the sepulchral chamber and the cell for the sepulchral statue. The general" cemetery," with its street of tombs, was called in Egyptian kher or khel; the individual tombs bore the general name of maha pulchres," which was applicable to any class of tomb, whether those of the kings or used by the inhabitants of the town or city. The term mer was applied to any tombs which had pyramidal construction, as those made of brick with pyramidal tops cut out of the flank of the hill at the time of the eleventh dynasty at the Drah Abu'-1 Neggah, in the western quarter of Thebes; but the term applied to the syringes or hollowed passages and tunnelled tombs at Gournah and elsewhere is asi, a word applied to a plant, perhaps a "reed," but also meaning a chamber, and this word was used to express the so-called mastaba, or quadrilateral sepulchre of the early dynasties. The great necropolis of Sakkarah is supposed to be that of the ancient Ka-Kam, the city of the Black Bull, known to the Greeks under the name of Cochome, and the pyramid there to have been named Ap, the "elevation" or "Mount," and the pyramid was step-shaped, made of unbaked brick, probably when first built in seven stages.

The mastabas were the mausolea of the richer and more important personages of the court hierarchy and Egyptian bureaucracy, for the poor and slaves were not buried with any consideration; they were hustled into superficial graves about three feet deep beneath the soil, and at this distant period of time are found only as skeletons, with no linen wraps remaining or other traces of emblems, and must therefore be regarded as the oldest and most primitive examples of Egyptian interment, and their bodies were unprovided with coffins. Occasionally, perhaps, some of the least poor, or slaves of extraordinary merit, had rectangular vaulted chambers, constructed of brick-work, vaulted and covered with a white coating; in the interior of these rude graves are found small vases or cups of coarse pottery or calcareous stone or alabaster, but uninscribed. These graves recal to mind those of the later Roman period, although the Memphian ones belong to a period long anterior.

The mastabas vary in size and dimension, but their average or normal dimensions are nearly fifty yards long, twelve yards wide, and thirty deep. The chief of these mastabas is the Mastabat-il-Pharaoun, which recent discoveries show to have been the sepulchre of Unas of the fifth dynasty, who was not buried in a pyramid. The mastabas are said to be peculiar to Gizeh, and not to be found elsewhere, and a long interval of civilisation must have preceded the construction of these tombs, as they show a considerable knowledge of architecture by their regular and geometric construction, while the square blocks and regular layers, each vertical joint being overlapped by a square stone, evidence considerable technical experience in the art of masonry. The other mastabas are made of similar masses of brick-work, and the bricks are of two kinds, those of the oldest mastabas, made of rectangular yellow bricks, composed of sand, pebbles, and some Nile mud, their dimensions being 22X11X7 of a metre, and black bricks made of alluvial soil and straw, larger in size, being 38X18X 14 of a metre; these bricks are not older than the middle of the fourth dynasty, while the others are as old as the second line. The black bricks continue till the time of the Romans, and both kinds are sun-dried, no burnt bricks, with the exception of the conical stamped bricks, found at the tombs of the Drah-Ahi' l'Neggah, having been employed in constructions till the age of the Cæsars.

In the masses of brick-work or masonry which stood above the soil over the sepulchral chamber, hewn out of the solid rocks beneath, were constructed the mortuary chapel, for the performance of masses or liturgies to the

dead, which, however, must have been of a simpler nature than those in use at a later period. In all these mastabas which, as a rule, face northwards, generally towards the north-east angle, is a kind of stele or sepulchral tablet of limestone, some times like facade, composed of separate pieces, and having two square pillars or columns in front, without capitals or abaci, forming a kind of entrance hall. This part of the mastaba is rarely on the south, never on the west, and the ceiling is always continuous, sometimes slightly vaulted by the stones supporting one another. The tablet is often like a door, with jambs, lintels, and hieroglyphics; sometime the facade or stele has a kind of false door with large figures of the occupant of the tomb and his wife at the sides of the false door, with a semicylindrical tambour above the door and a kind of cornice above that, with a sepulchral dedication to Anubis, never to Osiris, and representations of the person for whom the sepulchre was made, at a repast or some other diversion, as the fowling represented on the mastaba at Meidum. On the portion of the soil covering the serdab or inner niche and the well by which the sarcophagus and its mummy were lowered, are found little vases filled with a coating containing inside the trace of the water with which they were filled. The interior chapel or asi was either single, or had more than one chamber, and the walls of these were covered with pictures and inscriptions engraved in intaglio and brightly coloured, still vivid after 6000 years, but no furniture itself or offerings are discovered in the rooms, which have been long open to the spoiler. The inscriptions refer to the calendar and festivals throughout the year, the titles of the deceased, adoration to Anubis, and tables of food, or menus in use at the period; and the gourmands of the Egyptian aristocracy fared sumptuously every day out of well-filled flesh-pots and jars of wine and beer. The paintings on the walls depict the chase, the farm, the industrial household, the amusement of dances by professional women, games, and other diversions, and were no doubt intended to recal to the spirit of the dead his favourite occupations and his former wealth. Such solaces were reserved for the rich; the poor reposed after death about as indifferently as during life.

When constructed of masonry, the walls of the chapel in the mastaba were often made of rubble revetted, and at the end, at the foot of the false door is often found the stone altar of libations, sometimes with two small obelisks engraved only on one face; at other times, instead of obelisks, two supports in the shape of altars. The stele or sepulchral tablet was at the earliest period made in shape of a facade, but often quite blank, a mere white slab. It is not till after the sixth dynasty that these tombstones were rounded at the top, like those of the present day. When the chapel was ornamented, the tombstones are often blank; when the walls of the chapel-the asi —were unadorned, the tablets were often inscribed. In the most ancient tombs the tombstones are often built up of pieces and are inscribed with hieroglyphics of an early and rude type. The art is bad, and the inscriptions are not in regular lines, but dispersed over the area; the hieroglyphics themselves are often peculiar, executed with more elaborate detail than at the later period of the middle Empire. The object of these early inscriptions is to record the name and titles of the departed, and it is remarkable that at this period persons had the ran ãā, or great name," and the ran-nets, or "little name." A tomb, for example, of the second dynasty, at Sakkarah, was made for a man whose great nime was Sekarkhabau, or "Sochari's rising amongst spirits," whose small name was Hothes--that of a rat or some small animal; and his wife's great name was Atherhotep, and her little name Teps; and this as early as the second dynasty. These chapels now have no doors, if they ever had, and except the vases found strewed here and there on the floor, the

other objects which may have been deposited there have entirely disappeared. Behind the wall, on the south side more often than the north, and on the north more often than the west, was a secret niche, which the Arabs call the serdab, occasionally communicating with the chamber by a square orifice. In this niche was deposited a statue of the deceased. In this statue was supposed to reside his ka, or spirit, a kind of manes, or ghost, which inhabited the tomb, went in and out of the sepulchre and Hades, and to which was attached a priest, who performed the liturgies or litanies, in certain ways, and with peculiar rites. In the earlier inscriptions this ka is not mentioned, but at the close of the twelfth dynasty, all the benefits conferred by deities on the deceased were said to be due to his ka. It was in this chapel and to this image that the ancestral worship was paid, and the ka, which was a kind of idolon of the dead, was supposed to receive the same satisfaction as the dead himself. Most of the statues in the museums of Europe at the time of the fourth and sixth dynasties, came from the serdab of the sepulchre of the period. They were portraits of the dead, and sometimes represented him holding the tools or other marks of his profession. The whole of the mastaba, or chapel, and its mass was superposed on the real sepulchral chamber beneath, which it covered. The descent to this was by a rectangular well or shaft, from six feet six inches, to nearly ten feet square, and this cell passed through the masonry or platform of the mastaba, and then through the living roots of the foundation, and was made of large blocks; it was down this well that the sarcophagus was lowered to the chamber, by a shaft from thirty to seventy-five feet deep. Hence, at the base of the shaft, a short passage led to the rectangular chamber, which was well built, but has only once been found uncemented, and in it was placed the sarcophagus of granite, or calcareous stone, and the mummy, or body. The cell itself was carefully blocked up with rubbish to prevent access to the chamber, and the mouth of it is generally found either in the long axis of the tomb, or else behind the tombstone. The sarcophagus of this period has no resemblance to the later cases in shape of the human form, generally made of wood, which prevailed from the eleventh dynasty, or about 1800 B.C., to the first century, A.D., but are rectangular chests with vaulted cover, with projections at the edges. The bodies found in these chests are distinguished by the absence of linen or wraps in which they may have been embalmed, and bones of the skeleton are only discovered generally, of a brown colour, with a faint odour of bitumen, which is the more remarkable as the mummies found in the pyramid had both linen and indications of bitumen.

Of course, the ethnological question here arises, to what race of men did these old Egyptians of the period of the second and subsequent dynasties belong; they have been referred to the Caucasian races, and some of the skulls show a high intellectual development, even frontal sutures occurring. Their colour is painted in the sculptures, and on their statues, either red or copper, the female yellow, but their profile is not Semitic, and shows, as at the period of their history, traces of African blood. Some of the servants are dolicocephalic, and are supposed to be the indigenous race, similar to the Libyans of Northern Africa, who, however, at a later period, are classed amongst the white races.

It is, however, in vain to look for the origin of Egyptian civilisation, either in Aethiopia or Nubia, or south of Egypt, or on the northern coast of Africa, which lies to the west, for there is no evidence of races in these parts having ever attained a nascent civilisation, such as the Egyptian might have started from. Recent discoveries in Southern Mesopotamia, however, show a similar civilisation, almost, if not as old as the Egyptian, with a form of written language developing from the ideographic to that of the conventional type, into which the original

picture invariably declines. The physical type, too, of the Babylonian statues from Tel-lo, approaches the Caucasian rather than the Semitic type.

ON SOME RECENT AMERICAN MATHEMA

TICAL TEXT-BOOKS

66

N NATURE (vol. xvi. p. 21) we drew attention to a bones" of the mathematical text-books in common use in American colleges and schools, and upon the analysis we then furnished of a few works before us we ventured to predict a speedy awakening of mathematical life. Our prognostications have been quickly fulfilled, and we now propose to submit an account of five recent books, some of which are quite fitted to hold their own, in our opinion, with English text-books on the same subjects.

"The Elements of the Integral Calculus, with a Key to the Solution of Differential Equations," by Dr. W. E. Byerly (Boston, 1881), is a sequel to the volume on the "Differential Calculus," previously noticed by us. This work is founded upon Bertrand's classical treatise, supplemented by free use of the allied treatises by Todhunter, Boole, and Benjamin Peirce. The opening chapters give a clear exposition of the use of symbols of operation and of imaginaries. So early an introduction to these subjects is novel to us in this connection, but it shows how the subject of quaternions is coming to the front, and the passage from the subjects of these chapters to quaternions is but a short one. The main portion of the book calls for no special comment. In Chapter XIV. we have a treatment of mean value and probability, founded upon the able contributions of Prof. M. W. Crofton, F.R.S., to Mr. Williamson's treatise.

The novelty of the book is Chapter XV., entitled "Key to the Solution of Differential Equations." This key is based upon Boole's work, and is a collection of concise, practical rules for the solution of these equations. An idea of its form will be best conveyed to some persons by saying that it resembles the analytical key so frequently prefixed now-a-days to handbooks of the British (and other) flora. By a series of references we run the particular equation to ground. Thus, taking the example, (1 + x) y d x + (1 − y) xdy = 0, it is a single equation, this sends us to a number; it involves ordinary derivatives, this advances us a stage; it contains two variables, is of the first order, and finally of the first degree. The upshot is we arrive at the form Xdx+ Ydy = 0, under which head we learn how to solve the equation. Under this last head, as throughout the book, are given numerous illustrative exercises for practice.

Dr. A. S. Hardy's "Elements of Quaternions" (Boston, 1881) is intended to meet the wants of beginners. In addition to the works of Sir William R. Hamilton and Prof. Tait, the author has consulted the memoirs or works of Bellavitis ("Calcolo dei Quaternione" and the "Exposition de la Méthode des Equipollences" in Laisant's translation); Hoüel's “Quantités Complexes; " Argand's "Essai" (1806); Laisant's 66 Applications mécaniques du Calcul des Quaternions," and one or two other books and papers in the American Journal of Mathematics, vol. i. p. 379. It is a good introduction to such a work as Prof. Tait's, the originality and conciseness of which, however, Dr. Hardy thinks to be "beyond the time and need of the beginner."

Cur next book is "An Elementary Treatise on Mensuration," by G. B. Halsted (Boston, 1881). Dr. Halsted is already known to mathematicians here as the author of a very full "Bibliography of Hyper-space and nonEuclidean Geometry," in the American Journal of Mathematics, vol. i., Nos. 3, 4. This treatise on Metrical Geometry is "the outcome of work on the subject while teaching it to large classes," so that it is no hastily prepared book, but has been founded on actual teaching

experience. The methods have a German “smell,” and this is justified by the author's residence, we presume as a student, at Berlin. There are eight chapters: (1) on the measurement of lines (triangles, method of limits, rectification of the circle; (2) on the measurement of angles; (3) of plane areas; (4) of surfaces (he uses Mantel for lateral surfaces, also Steregon and Steradian in connection with a solid angle); (5) of volumes (Quader two chapters the solids discussed are the prism, cylinder, pyramid, cone, and sphere; an article is also devoted to Pappus's theorem. (6) The applicability of the prismoidal formula; (7) approximative methods, as Weddle's method; (8) on the mass-centre, with a paragraph on the mass-centre of an octahedron, which gives Clifford's construction (see Proc. Lond. Math. Soc., vol. ix. p. 28). There are numerous exercises, these we have not tested. The book is most effectively got up," the printing, figures, and paper being, to our mind, excellent.

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Our last two books are by Prof. Simon Newcomb, so well known as the author of "Popular Astronomy." The first, "Algebra for Schools and Colleges" (New York, 1881), has already reached its second edition. It is a capital book; indeed we are disposed to rank it as the best manual on the subject that we have seen for school purposes. It is divided into two portions, "the first adapted to well-prepared beginners, and comprising about what is cominonly required for admission to colleges, and the second designed for the more advanced general student." We shall perhaps best serve the end we have in view in noticing this work by giving an analysis of the author's preface. The principles of construction are (1) that an idea cannot be fully grasped by the youthful mind unless it is presented in a concrete form. Hence numerical examples of nearly all algebraic operations and theorems are given-so numbers are preferred to literal symbols in many cases. The relations of positive and negative algebraic quantities are represented by lines and directions at the very earliest stage. "Should it appear to any one that we thus detract from the generality of algebraic quantities, it is sufficient to reply that the system is the same which mathematicians use to assist their conceptions of advanced algebra, and without which they would never have been able to grasp the complicated relations of imaginary quantities." Principle (2) is that all mathematical conceptions require time to become engrafted upon the mind, and the longer, the abstruser they "It is from a failure to take account of this fact, rather than from any inherent defect in the minds of our youth, that we are to attribute the backward state of mathematical instruction in this country, as compared with the continent of Europe." Prof. Newcomb considers the true method of meeting this difficulty is to adopt the French and German plan of teaching algebra in a broader way, and of introducing the more advanced conceptions at the earliest practicable period in the course. A third feature is the minute subdivision of each subject, and the exercising the pupil on the details before combining them inta a whole. This remark especially applies to the solution of the exercises. Some subjects have been omitted (as G. C.D. of polynomials, square roots of binomial surds, and Sturm's theorem), as they have no application "in the usual course of mathematical study, nor advance the student's conception of algebra," and in studying them there is a waste of power. "Thoroughness" has been our author's aim rather than "multiplicity of subjects." There are other points of interest in this preface which show that the writer is a very experienced teacher, and which we commend to the consideration of teachers here, but we must pass on to indicate the contents of the two parts.

are.

Part I. embraces algebraic language and operations, equations, ratios and proportion, powers and roots, equations (quadratic), progressions, seven books in all.