Federal Canals.-The leading United | gravel, and package freight; on the States Government canals in opera-middle Ohio, from Cincinnati to Evanstion are the Hennepin Canal (Illinois), St. Mary's Falls Canal, St. Clair Flats Canal, Port Arthur Canal (Texas), Morgan Ship Canal (Texas), Galveston and Brozos Canal (Texas), and the various river canals around falls or shoals in the Ohio, Columbia, Tennessee, Cumberland, Mississippi, Monongahela, and other rivers.

River Traffic.-The complete traffic statistics of the Allegheny River are not available, but in 1909 the usual quantity of coal, gravel, sand, lumber, timber, and stone were shipped. The traffic of the Ohio River proper consists mainly of coal, but in other respects is various. On the upper Ohio, from Pittsburg to Cincinnati, the traffic consists mainly of coal, logs, sand,

ville, coal, lumber and timber, grain, tobacco, and other farm products, and on the lower Ohio coal, corn, wheat, groceries, live stock, flour, and tobacco. Omitting duplications the total traffic of the Ohio River and its tributaries is about 20,000,000 tons annually; and that of the Ohio River proper about 11,500,000.

The annual tonnage moved on the Columbia River is about 3,500,000 tons, consisting chiefly of grain, flour, lumber, farm produce, logs, machinery, and general merchandise. The Hudson River moves annually about 8,600,000 tons chiefly of building materials, coal, wood, grain, lumber, ice, farm produce, manufactures, and general merchandise.

The total traffic on the Delaware | Ibid., Bulletin No. 21, Commercial ValuRiver in 1906 aggregated 20,577,000 tons. The leading shipments are coal, sand, petroleum, stone, oysters, and fish fertilizers, chemicals, and iron products; the leading receipts are sand, coal, lumber, petroleum, produce and fruit, chemicals, sugar, grain, railroad ties, and fertilizers.

CLEVELAND, F. A., and POWELL, F. W.-
Railroad Promotion and Capitaliza-
tion in the United States. New York,
Longmans, Green & Co., 1909.
COLE, W. M.-Accounting and Auditing.
Minneapolis and Chicago, Cree Pub-
lishing Company, 1910.
HANEY, L. H.-Congressional History of
Railways in the United States. (2
vols.) Madison, Wis., Democrat Print-
ing Company, 1909-10.

HATFIELD, H. R.-Modern Accounting.
New York, 1909.

HOUGH, B. O.-Elementary Lessons in
Exporting. New York, Johnston Ex-
port Publishing Company, 1909.
JOHNSON, E. R.-Elements of Trans-
portation. New York, D. Appleton &
Co., 1909.

RAILWAYS OF THE WORLD

The Archiv für Eisenbahnwesen | cent., since 1904, and of 16,445 miles, publishes the statistics of the railways of the world, bringing them down to the end of 1908, or the nearest date of official reports, which for the United States and Canada is June 30 of that year. The data given show the mileage for each country for each of the five years ending with 1908, the increase for the last four years, and the proportion of railway mileage to area and population. The grand total for the world is 611,478 miles, which is an increase of 61,505 miles, or 11.2 per

or 2.8 per cent., over 1907, which seems moderate progress when we remember that 13,000 miles have been built in the United States in a single year. But it is fully up to the progress of recent years. Of the increase of 61,505 miles since 1904 considerably more than one half was in America, 33,690 miles, and of this total 27,115 miles were in North America. The mileage at the end of 1908 (the United States and Canada to June 30th) was as follows:

XXII. MATHEMATICS AND ASTRONOMY

MATHEMATICS

E. B. WILSON

As one of the oldest sciences, clearly | There will probably result an even somewhat necessary for practical life stronger tendency than at present to and considered valuable for mental introduce the fundamental concep discipline, mathematics, in its canon- tions and methods of differential and ical elementary subdivision of arith- integral calculus earlier than is now metic, algebra, and geometry, has common. The commissioners for the long been taught in essentially un- United States are Profs. D. E. Smith changed form to school children, and (Columbia), W. F. Osgood (Harvard), in the colleges a certain amount of and J. W. A. Young (Chicago). Unanalytical geometry and of calculus der their direction a large number of has been offered in nearly as un- associates are preparing reports on changing form to a few students. In various topics, such as: general elesharp contrast to this fixity of con- mentary schools, normal schools, tent and generality of distribution of mathematical work in American pos elementary mathematics stands the sessions, colleges, technical schools of rapidly increasing domain of higher collegiate grade, universities, etc. mathematics to which special periodicals aggregating thousands of pages per annum are devoted, but which is practically unknown and unknowable except to mathematical specialists; indeed even a professional mathematician finds it difficult to read, to say nothing of appreciating, researches in fields remote from those in which he has concentrated his own studies.

The International Congresses. It may have been this growing isola tion of mathematics as a whole and of the individual mathematician in his science which led to the organization of quadrennial international congresses of mathematicians, meeting for the purpose of interchang ing ideas with others in the same or in related fields. The last congress, which met at Rome in 1908, through its designation of Profs. F. Klein (Göttingen), H. Fehr (Geneva), and Sir G. Greenhill (London) to organize and supervise an international commission on the teaching of mathematies, took a step which ultimately may aid materially in modernizing and improving instruction in elementary and collegiate mathematics.

Striking New Work. Some few years ago the late H. Minkowski (Göttingen), building a mathematical theory upon the researches of H. Lorentz and A. Einstein on electromagnetic theory, set forth and de veloped the idea that the material universe should be regarded as four dimensional by adding time as a fourth dimension to the usual three dimensions of space. This theory was immediately taken up by many stu dents of mathematics and physics, and the past year has seen the publication of a large number of contributions to the theory. At the present rate of progress it will not be long before the mathematical and physical significance of Minkowski's work. which was unfortunately interrupted by his early death, will be pretty thoroughly understood.

A very noteworthy contribution to American mathematics is the monograph Introduction to a Form of General Analysis, by Prof. E. H. Moore (Chicago), in the volume called The New Haven Mathematical Colloquium, published by the Yale University Press (1910). The great breadth of view of the author in this

and Methods of Geometrical Optics
(Macmillan Co., 1910), by Prof. J. P.
C. Southall (Alabama Polytechnic),
and a German translation of the In-
troduction to Higher Algebra (Mac-
millan Co., 1907), by M. Bôcher (Har-
vard). In France two noteworthy
treatises have appeared: Les Systèmes
d'Equations aux Dérivées partielles
(Gauthier-Villars, 1910), by C. Ri-
quier, and Leçons sur le Calcul des
Variations, vol. i (A. Hermann et Fils,
1910), by J. Hadamard.
- The re-

work is well shown by his own statement: the existence of analogies between central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features. The great generality and abstractness of the problem and the use of the symbolism of mathematical logic to give brevity to the formulas and statements will probably make the investigation hard reading, and may therefore hinder its University Instructors. passage into current mathematical tirement of Prof. O. Bolza from the doctrine, but the monograph bears University of Chicago and his return the earmarks of a great work, and to Germany, his native land, have deone is tempted to look back forty prived the university and this country years to Benjamin Peirce's mono- of an eminent mathematician, espegraph on Linear Associative Algebra cially known for his exhaustive treato find a sufficiently important sin- tise on the Calculus of Variations, gle American contribution to pure printed in German by B. G. Teubner. mathematics with which to rank it. The university has reinforced its deBibliography. The Carnegie Insti- partment by calling Prof. E. J. Wiltution has published (1909) the elab- czynski from the University of Illinois, orate and valuable work of Prof. D. and thereby maintains, under the H. Lehmer (California) on a Factor leadership of Prof. E. H. Moore, its Table for the First Ten Millions. exceptionally able and active faculty The work is an important event in of mathematics. Another loss is in the science of higher arithmetic. the return of Prof. J. H. Jeans Among important advanced texts (Princeton) to England. Profs. J. may be mentioned: A Treatise on the H. Van Amringe, head of the departDifferential Geometry of Curves and ment at Columbia University, and Surfaces (Ginn & Co., 1909), by Prof. L. A. Wait, head of the department L. P. Eisenhart (Princeton); Theory at Cornell, have retired; Profs. J. of Elliptic Functions, vol. i (Wiley Stringham, head of the department & Sons, 1910), by Prof. H. Hancock at the University of California, E. A. (Cincinnati), Non-Euclidean Geome- Bowser (Rutgers), H. B. Newson try (Oxford Press, 1909), by Prof. (Kansas), and J. E. Wright (Bryn J. L. Coolidge (Harvard); Principles | Mawr) have died.

ASTRONOMY

MABEL LOOMIS TODD

General Survey.-Especially note- cal have been issued; many wellworthy was the year 1910 in astro- known astronomers have died. Three nomical annals; true to prediction attained world-wide eminence, Sir Halley's famous comet returned, and William Huggins, of London; SchiaMars came to opposition in Sept. parelli, of Milan; and Newcomb, of of the year before, discussion of Washington. In France, Fraissinet, this puzzling planet's appearance and Bouquet de la Grye, and Charlois, possibilities extending through many discoverer of a hundred planets; in months. But besides that, work of Denmark, Thiele; in Austria, von exceptional interest and importance Gothard; in Australia, Sir Charles has been prosecuted at the Harvard, Todd; in Ireland, Gore; in Mexico, Yerkes, Lick, Mount Wilson, and Valle; in India, Molesworth, keenother observatories, and at similar eyed observer of planetary surfaces; institutions throughout the world; in America, Hough and Pritchett, publications both popular and techni- | both specialists on Jupiter.