methods of observation designed to avoid confusion in operation on a very active front. The Germans had an extremely efficient flash ranging service, many of the good features of which were copied by the Allies as they became known through captured documents. The flash ranging reported not only the positions and activity of hostile batteries, but also the exact locations of other enemy objectives such as traffic on roads, troop movements, position of observation balloons, etc. Being provided with high-power telescopes, and since observation was obtained from stations on a wide base (from five to eight miles) the flash ranging sections were particularly well suited for observation and ranging in the enemy back areas, and these sections rendered invaluable services both to the artillery and to the army intelligence.

A battalion of five companies (74th Engineers) furnished the ranging troops for an American army. A sound ranging section.

was in the field with the first American division to enter the line (March, 1918) and on the signing of the armistice the entire front of the second American army was covered with both flash and sound ranging sections and a portion of that of the first American army was covered by flash ranging, although the ranging battalion allotted to this army had not yet arrived in France. The ranging service was thus a "going concern " from the very first and was not one of the many which could have delivered results had the war but lasted a little longer.

A flash ranging section consisted of about one hundred men commanded by a lieutenant who was assisted by three other officers and by an exceptionally high grade of non-commissioned officers and men, all of whom had been given a month's intensive training in France. The instruments and methods employed were those suited for accurate survey and present no special features of interest.

A sound ranging section was similar in organization to the flash section except that there were fewer enlisted men (60-70) due to the fact that instruments took the place of living observers to a great extent. The "central" instrument recorded photographically the time of

arrival of the sound of the enemy guns at a series of instruments at surveyed positions near the front line and covering a length of about five miles; this instrument delivered automatically developed and fixed photographic records in less than a minute after the sound of the enemy gun reached the front line and this record could be interpreted by the use of quick graphical methods so that the position of the enemy gun could be telephoned to the friendly artillery in about a minute more. The probable accuracy of the location could be given and also the caliber and target of the piece which had just fired. The service was not interfered with by rain or fog or darkness, though it was rendered less accurate by strong winds. Calculations were rendered difficult by great artillery activity though not impossible except under actual "barrage" conditions.

In ranging the friendly artillery on enemy objectives it was possible to range all the guns of the battery simultaneously, thus effecting considerable time saving over other methods of ranging. If the ranging was being done on an enemy battery which had just fired the accuracy attained was very great (less than twentyfive yards), because of the fact that in this case no wind or temperature corrections need be applied in the calculations.

After the American advances of September and November a careful survey was made of most of the enemy positions which had been located by either the sound or the flash ranging sections on a part of the American front; the result of this survey was that of the locations of the flash ranging about one third were accurate to within fifty yards, another third to within one hundred yards and the other third with errors of more than one hundred yards. In the first third were many extremely accurate locations of guns the positions of which were visible from two or more observation posts; in the last third were mostly locations of concealed heavy caliber distant guns generally more easily located by sound ranging, whose positions could only be inferred from smoke puffs by day or flares in the sky by night.

The survey showed that the estimates of accuracy made by the sound rangers in report

ing a location had been very conservative; a location reported not accurate to within fifty yards was often accurate to within twenty-five yards. In general, the average of a half dozen locations of the same gun taken on different days under differing weather conditions was of a very high order of accuracy; often a matter of but five or ten yards.

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In general a location either by sound or by flash which had been rated fair" when reported to the artillery was found on survey to have been within the unavoidable errors in artillery fire.

An idea may be gained of the amount of artillery information supplied by the ranging sections from the following figures taken from the reports of the artillery information officer of one of the American corps. This officer had at the time the following sources of information: three American sound ranging sections, two American and three French flash ranging sections, aviation and observation balloons. During a period of three weeks of rapid advance when the sound sections were out of operation while moving for a considerable portion of the time 425 separate locations of enemy batteries were made. Of these the two American flash sections reported 64 per cent. the three French flash sections reported 16 per cent. and the three American sound sections reported 21 per cent. In a period of two weeks when the advance had been checked by the Germans the total number of locations were 392, and the percentages were: From the three American flash sections 38 per cent.; from the two French flash sections 8 per cent., and from the three American sound sections 56 per cent.

The following figures taken from another and very active sector are also instructive. For a period of three days preparation for an advance the following locations were made: Sound, 22; flash, 22; balloons, 0; aviation, 0. For a period of sixteen days of rapid advance: Sound, 4; flash, 46; balloons, 30; aviation 77. For a period of four days of stabilization: Sound, 6; flash, 34; balloons, 13; aviation, 15. These figures are characteristic. During preparations for an advance, both the sound and

flash sections are very useful and important sources of information. During rapid advance the sound ranging does not get into action as often or as soon as the flash. In this period the greater part of the information comes from the air observation.

Both sound and flash ranging have proved their value in the American Expeditionary Forces and are to be retained in the peacetime army; the sound because it is the one source of information when all others fail in foggy weather and because thus far no camouflage has been devised to prevent its working; the flash because of its relatively great mobility and consequent importance in open warfare.

AUGUSTUS TROWBRIDGE

PRINCETON UNIVERSITY

THE AMERICAN MATHEMATICAL

SOCIETY

THE two hundred and third regular meeting of the society was held at Columbia University on Saturday, April 26, extending through the usual morning and afternoon sessions. This being the first eastern meeting since October, the attendance was large, including sixty-seven members, indicating, as it may be hoped, a revival of the conditions preceding the war.

President Morley occupied the chair, being relieved by Professor Kasner. The election of the following persons to membership in the society was announced: Mr. N. W. Akimoff, Philadelphia, Pa.; Dr. Tobias Dantzig, Columbia University; Mr. A. C. Maddox, Guthrie, Okla., High School; Mr. Montford Morrison, Chicago, Ill.; Professor Ganesh Prasad, Central Hindu College, Benares, India; Mr. F. M. Weida, State University of Iowa; Mr. C. L. E. Wolfe, University of California. Two applications for membership were received.

It was decided to hold the coming summer meeting of the society at the University of Michigan in the first week in September. Professors Beman, Bliss, Karpinski, Osgood and the secretary were appointed a committee on arrangements for this meeting. A committee was also provided to prepare nominations for officers to be elected at the annual meeting in December.

Professor E. W. Brown, L. E. Dickson and H. S. White were appointed as representatives of the society in the division of physical sciences of the national research council; and President R. S.

Woodward, and Professors Birkhoff and MacMillan as representatives of the society in the American section of the International Astronomical Union.

The committee on the publication of a mathematical year book presented a preliminary report and was continued and asked to make a further report at a future meeting.

A special feature of the meeting was the reports by Captain Jackson, Dr. Gronwall and Major Veblen on the work in ballistics at Aberdeen and Washington, which occupied the first part of the afternoon session. The titles of these reports are included in the list of papers below.

About fifty members and friends gathered at the midday luncheon; thirty-two attended the dinner at the Faculty Club in the evening. Much satisfaction was expressed at the revival of these pleasant occasions.

The Chicago Section held its regular spring meeting on March 28-29. The San Francisco Section met at the University of California on April 5. The following papers were read at the New York meeting:

C. J. Keyser: "Concerning groups of dyadic relations in an arbitrary field."

J. K. Whittemore: "Certain functional equations connected with minimal surfaces.''

W. B. Fite: "Linear functional differential equations.''

L. B. Robinson: "Note on a theorem due to Wilczynski."'

L. B. Robinson: "A curious system of polynomials, continued.''

O. E. Glenn: "Covariants of binary modular groups."'

O. E. Glenn: "Modular covariant theory of the binary quartic. Tables" (preliminary report). O. E. Glenn: "Invariants of velocity and acceleration.''

F. H. Safford: "Reduction of the elliptic element to the Weierstrass form."

Philip Franklin: "Computation of the complex roots of the function P(z)."

A. R. Schweitzer: "On the history of functional equations" (preliminary report).

E. D. Roe, Jr.: "The irreducible factors of 1+x+ +n-1. Second paper."

...

E. D. Roe, Jr.: "The irreducible factors of a circulant."

Dunham Jackson: "Small are computations and related questions."'

T. H. Gronwall: "Qualitative properties of the ballistic trajectory."

Oswald Veblen: "Progress in design of artillery projectiles.''

G. D. Birkhoff: "Boundary value and expansion problem for differential systems of the first order." G. D. Birkhoff: "Note on the closed curves described by a particle moving on a surface in a gravitational field."

G. D. Birkhoff: "Note on the problem of three bodies."

Edward Kasner: "A characteristic property of central forces."'

J. F. Ritt: "On weighting factor curves for low elevations.''

A. C. Lunn: "Some functional equations in the theory of relativity."

J. R. Kline: "Concerning sense on closed curves in non-metrical plane analysis situs."'

R. L. Moore: "On the most general class L of Fréchet in which the Heine-Borel-Lebesgue theorem holds true."'

H. S. Vandiver: "On the class number of the field (ei) and the second case of Fermat's last theorem.''

F. W. Beal: "On certain points of congruences of circles.''

L. L. Silverman: "Regular transformations of divergent series and integrals."

T. C. Fry: The application of the modern theories of integration to the solution of differential equations."

C. A. Fischer: "Completely continuous transformations and Stieltjes integral equations."'

Arnold Emch: "On closed curves described by a spherical pendulum."

H. S. White: "An explicit formula for two old problems."

L. P. Eisenhart: "Triply conjugate systems with equal point invariants."'

SIR JOSEPH HOOKER1

With the passage of time the importance attached to persons and events becomes strangely altered. History, to be of value to posterity, must be both more and less than a faithful chronicle of the past. Less, if only to bring it within intelligible limits; more, because it must see causes in relation to effects, emphasizing the inconspicuous beginnings of new developments. For such reasons, the judgment of posterity will nearly always differ from that of contemporaries; not necessarily because posterity is endowed with superior wisdom, but rather because the basis of judgment is different. Sir Joseph Hooker and his father, Sir William Hooker (1785-1865), were both botanists of the highest eminence, their combined activities covering more than a century. As we review their careers, we do not know which to admire most. The son, without the slightest false modesty, always insisted on his father's preeminence, giving good reasons for his judgment. It was William Hooker who, with extraordinary energy and enthusiasm, had created great botanical centers, first at Glasgow, and then for the whole British Empire at Kew. When the work was most difficult and recognition hardest to obtain, he had won support and respect; and had laid the foundations on which his son was to build. It is difficult for us, to-day, to realize the labor and vision required to build up the establishment at Kew, in the face of ignorance and opposition. It is difficult for posterity to do full justice to the elder Hooker, just because we can no longer clearly visualize the environment in which he lived. His work, everywhere woven into the fabric of modern botany, has few outstanding or picturesque features. In the case of Sir Joseph Hooker, the imagina1 Life and Letters of Sir Joseph Dalton Hooker. By LEONARD HUXLEY. 2 vols. New York, D. Appleton & Co. 1918.

tion is more easily quickened. Aside from his great merits as a master of technical botany, he will always live in the pages of history as one of the group of men immediately associated with Darwin. The personalities of Darwin, Hooker, Huxley and Wallace stand out in the history of biological science in such a manner that they are never likely to be forgotton. On the contrary, because they will be taken as typical of a movement and a period, they will increase rather than diminish in the estimation of mankind. They will have the value of a moral force; veritable saints of science, patterns for all later generations. In strict equity, it may be that Hooker should not stand on so high a pedestal as we shall place him, but we are concerned rather with our needs than his deserts.

Under these circumstances, an authoritative and full account of the life of Sir Joseph Hooker becomes a necessity. This work, written by Leonard Huxley, and based on materials collected and arranged by Lady Hooker, has just been published in two volumes, and is the subject of this notice. It is the story of a long life of incessant activity; devoted to the classification and description of plants, the administration of a great botanical establishment, and to explorations in distant regions. In 1837 Hooker published the descriptions of three new mosses; in 1911 he published a number of new species of Impatiens. Such a record is surely unique. It seems strange to think that this man, whose living presence is still vividly in our mind, knew four of the founders of the Linnean Society, and talked with Humboldt. There is an amusing account of his first meeting with Humboldt in Paris:

On putting up here I sent in my card with Mr. Brown's books to Baron Humboldt; he was not at home, but sent his flunkey (Scotice Footman) to my bedroom at eight o'clock yesterday morning to say his master wished to see me at nine o'clock. Ten minutes after his Lord had grown impatient and sent to say he was all ready, so I went in and saw to my horror a punchy little German, instead of a Humboldt. There was no mistaking his head, however, which is exceedingly like all the por

traits, though now powdered with white. I expected to see a fine fellow six feet without his boots, who would make as few steps to get up Chimborazo as thoughts to solve a problem. I can not now at all fancy his trotting along the Cordillera as I once supposed he would have stalked. However, he received me most kindly and made a great many enquiries about all at Kew and in England.

Later ou, Hooker was able to emulate Humboldt in the exploration of mountains, but on the other side of the world-in the Himalayas. The story of his Indian work is well known, but is of perennial interest. He did much more than explore new regions and find new plants. India is indebted to him for much of her scientific development and material prosperity. When he went there, he found the government singularly apathetic as regards science. He went out with Lord Dalhousie, the governor-general, who took a fancy to him and treated him very kindly, but had no interest in botany. In a letter home he relates:

I find Lord Dalhousie an extremely agreeable and intelligent man in everything but natural history and science of which he has a lamentably low opinion, I fear. He is a perfect specimen of the miserable system of education pursued at Oxford, and as ignorant of the origin and working of our most common manufacturing products and arts as he is well informed on all matters of finance, policy, etc. I very carefully drop a little knowledge into him now and then; but I can not awaken an interest or any sympathy in my pursuits: he is much pleased at my being busy, and especially with my carrying on my meteorological register three times a day. Lady Dalhousie shares her husband's apathy, but is otherwise a kind hearted creature. In the desert I brought them the gum arabic Acacia, which I thought must interest the late president of the board of trade; but he chucked it out of the carriage window: and the rose of Jericho, with an interest about it of a totally dif ferent character, met no better fate.

On his return from India and indeed while he was still there, he contemplated a "Flora Indica," to contain descriptions of all known Indian plants. In a letter to his father he wrote:

It is easy to talk of a "Flora Indica," and Thomson and I do talk of it, to imbecility! But