the air; inadequate storage facilities lead to heavy losses through evaporation and seepage; the loss in many refineries still is great; and improper and wasteful uses of petroleum and its products abound. Indeed, an oil expert stated in 1919 that the resource is not made to yield more than 10 per cent of its latent value. And this is the situation in spite of the fact that petroleum is a basic necessity in our modern life, and that the reserve of natural petroleum in known fields in the United States, available by methods of production now in vogue, would be exhausted in some sixteen or eighteen years were the present output to continue.

Space does not permit further illustration of the wasteful use of resources, nor consideration of the remedies which should be applied. Teachers of geography are concerned largely with man's economic adjustment to his physical environment, with his use of earth resources. They should, so far as practicable, point the way to better adjustments and to more effective use of resources. The newer textbooks in geography are giving some attention to these subjects and there is an extended and readily available literature dealing with the problems of conservation. Society is largely responsible for the wasteful use of resources, which it often views with indifference. If teachers of geography throughout the country seize the unique opportunity which confronts them, they will help to make the attitude of the next generation of American men and women a very different one, as a result of which the nation will cease to exploit its resources recklessly and extravagantly, wasting of many things as much as it uses. The future welfare of the nation is the issue at stake.

THE SHENANDOAH CAVERNS.

The exhibition of caverns to the traveling public is noted by the United States Geological Survey as a growing industry in the Shenandoah Valley of Virginia. The famous Valley Pike, now a link in the New York to Atlanta highway, is traversed yearly by thousands of automobile tourists properly intent upon seeing America first, and no one has adequately seen America who has not visited one or more of the caverns in the Shenandoah Valley. Until recently the only caverns that were accessible to the public were the celebrated Luray Caverns, in Page County, and Weyers Caves, in northern Augusta County, near Grottoes. However, within twelve months, the Endless Caverns, near New Market, in Shenandoah County, have been thrown open to the public, and on May 31 another cavern near Mount Jackson, also in Shenandoah County, made its first bid for public favor.

The latest-opened caves have been named Shenandoah Caverns. They are about three miles south of Mount Jackson and two miles west of the Valley Pike, with which they are connected by a macadamized road. They are close to Shenandoah Caverns station, on the Harrisonburg branch of the Southern Railway, and are readily accessible both to the automobilist and to the railway tourist. Commodious rest rooms are provided near the railway. The visitor descends into these caverns by a concrete stairway and soon sees the first stalactites which appear as stout daggers of crystallized lime carbonate, hanging like icicles from points where surface water drips from the limestone roof. At the foot of the stairs is the spacious anteroom to a long chain of high-vaulted chambers connected by narrow passageways, forming in general plan a gigantic letter S, all illuminated by cleverly concealed lights. Attractive natural decorations are found in every room. Here the side walls are covered by fluted veneer done in crystal stucco, there in graceful drapery hang

creamy lambrequins in ruddy-tinted strips. From place to place, singly or in groups, are pendant stalactites and uprising stalagmites-the first inverted narrow cones fed by trickling films of lime-bearing water; the second pillars or columns fed by spattering drops of the water. Giving free rein to fancy the visitor finds resemblance in these cavern deposits to whatever he may choose from the realms of the earth or of the waters under the earth. The beasts and the birds are there, and some of the fishes; silhouette portraits of celebrities, towers and minarets, dungeons and domes, hanging gardens, high cliffs mantled by patterned growths simulating the dainty coralline fungus of moist summer groves. In one room midway down the chain the show piece is a narrow 30-foot cascade of white glittering crystal flanked by twin falls of pale translucent ocher. At the base and to the rear of this diamond cascade, visible by peering between slender columns of oriental alabaster, is the "Fairy's Secret," a tiny pool illuminated in due season by animated torches, presumably carried by a brood of phosphorescent larvae of some insect, perhaps a small fly that is commonly present in such caverns.

As he progresses from room to room the visitor is apt to think each succeeding chamber superior to the last, but whether or not this is true all are likely to agree that the most charming of all is the one that completes the inbound trip. At the end of the developed portion of the cavern a chamber of high vaulted roof suddenly gives place to a low-ceiled room containing a lakelet in which are mirrored a multitude of delicate stalactites a pool of a thousand crystal pendants, the very quintessence of the subterranean charms.

According to A. C. Spencer, of the United States Geological Survey, the caverns of the Shenandoah Valley are far more numerous than the casual visitor would be likely to imagine. The rocks in which this broad trench-like valley has been excavated by water are mainly limestone, and wherever these rocks occur the existence of caverns is indicated by two unfailing signs-the presence of innumerable water sinks and the absence of brooks tributary to the rather regularly spaced creeks. The brookless tracts receive a due share of rainfall and must obviously contribute water to maintain the flow of the creeks and rivers, but their contributions are not delivered by way of the surface drains but through underground channels that supply copious springs in the deep valleys. The sinks are rude funnels, by means of which surface waters are diverted to the subterranean waterways.

The development of extensive underground waterways in limestone formations like those of the Shenandoah Valley hinges upon the two geologic facts that large masses of rock are always cut by joints and that limestone is dissolved by rainwater, which always contains more or less carbon dioxide. Surface water entering fissures, joint cracks, and bedding planes attacks the limestone walls and thus by a process of etching converts close fractures and joints into relatively open crevices. As this process of solution goes on lateral connections will be made from crevice to crevice, and the downward etching of the linked openings will be halted only when the subsurface water channels have become closely adjusted to the water table controlled by surface streams. Thus it is that the caverns of the Shenandoah Valley are formed.

Mr. N. D. Parker, formerly President of the Standard Scientific Company, has recently joined the Cambridge Botanical Supply Company as a general sales manager. Without question the sales of this company will be greatly increased under his splendid management. This firm publishes the Cambasco News, which is sent free to any teacher who asks for it.

MATHEMATICS IN THE UNIVERSITY OF CHICAGO. The following university and collegiate courses in mathematics and mathematical astronomy are announced at the University of Chicago (academic year 1922-1923). All courses meet four times a week for a quarter of twelve weeks. Courses which continue for more than one quarter are indicated with Roman numerals, as I, II, III, orIV. By Professor E. H. Moore: Vectors, Matrices, and Quaternions; Matrices in General Analysis I, II, III, IV; Analytic Geometry. By Professor L. E. Dickson: Theory of Numbers I, II; Solid Analytics; Theory of Equations. By Professor H. E. Slaught: Differential Equations; Elliptic Integrals; Calculus I; Plane Trigonometry. By Professor G. A. Bliss: Definite Integrals; Elliptic Functions; Calculus II, III. By Professor E. J. Wilczynski: Projective Differential Geometry I, II; Functions of a Complex Variable; Calculus I, II; Trigonometry. By Professor F. R. Moulton: Analytic Differential Equations I, II, III; Advanced Ballistics 1, II, III; Descriptive Astronomy, Sidereal Universe. By Professor W. B. MacMillan: Analytic Mechanics I, II, III; Celestial Mechanics; Descriptive Astronomy I, II. By Professor A. C. Lunn: Units and Dimensions; Dynamics of Continuous Media; Canonical Equations and Quantum Theory; Thermodynamics. By Dr. Mayme I. Logsdon: Theory of Algebraic Invariants; Calculus I, II, III; College Algebra; Analytic Geometry. By Professor J. W. A. Young: Limits and Series; College Algebra, Analytic Geometry. By Professor Kurt Laves: Plane Trigonometry; Spherical Trigonometry with Applications to Astronomy and Geodesy; Surveying; Practical Astronomy; Satellites.

THE LINCOLN SCHOOL.

Under the directorship of Dr. Otis W. Coldwell, Lincoln School, Teachers College, Columbia University, this school is making rapid strides toward accomplishing the purpose for which it was established. The appointment of Dr. Caldwell as its head meant success as this gentleman does not know the meaning of the word failure.

During the past summer there was brought together an array of some thirty speakers whose ability and reputation could not be surpassed, to discuss vital subjects of current topics in science. These lectures were given in the auditorium of the Horace Mann School. The results accomplished by this school are being studied by science educators the country over and undoubtedly the work being done here will be a pattern for science and mathematics teaching in the very near future.

FINDS FOSSIL FLOWER EMBEDDED IN ROCK.

Fossil flowers are such rare discoveries in the United States that the finding of a dogwood "flower" in a fragment of rock from the Glenrock coal field, Converse County, Wyo., is of interest. Dr. F. H. Knowlton, a paleobotanist of the United States Geological Survey, identified the fossil as a species of Cornus, a typical genus of the dogwood family.

There are some forty or fifty living species of the genus Cornus, which is widely distributed over three continents of the Northern Hemisphere and has one representative south of the Equator, a species in Peru. The leaves of more than twenty fossil species of Cornus have been found in North America, but the dogwood flower just identified, is the first one yet found in the United States. Species of dogwoods first appeared in the middle of the Cretaceous, the geologic period in which dinosaurs lived; in other words, the genus Cornus seems to have made its first appearance, probably more than four million years ago.

GRINDSTONES AND PULPSTONES PRODUCED IN 1921. The output of grindstones and pulpstones in the United States in 1921 amounted to 26,340 tons, valued at $1,227,322, according to figures reported by the producers to the United States Geological Survey, Department of the Interior. This was a decrease from the output in 1920 of over fifty per cent in quantity and of twenty-eight per cent in value.

The grindstones produced amounted to 16,310 short tons, valued at $477,259, a decrease of 63 per cent in quantity and 61 per cent in value. The pulpstones produced amounted to 10,030 short tons (2,940 pieces) valued at $750,063, an increase of sixteen per cent in quantity and sixtythree per cent in value. The demand at paper mills, which were very active late in 1920 and early in 1921 and which during and after the war could not renew their supply of English stone, increased the market for domestic pulpstones. If the depression that has followed this activity continues there will probably be a considerable decrease in the output of pulpstones in 1922.

The imports of grindstones and pulpstones were valued at $81,880 as against $77,046 in 1920. The exports of grindstones were valued at $281,976 as against $424,322 in 1920.

PRODUCTION OF PHOSPHATE ROCK IN 1921.

According to conservative estimates made by the United States Geological Survey from the incomplete returns available April 1, the quantity of phosphate rock sold in the United States in 1921 was about 1,968,000 long tons, valued at $10,928,300, as compared with 4,103,982 long tons, valued at $25,079,572 in 1920.

The total production of Florida was approximately 1,675,000 long tons, valued at $9,036,000.

Tennessee followed with an approximate total of 293,000 long tons, valued at $1,892,300, which included a small quantity of brown rock from Kentucky.

The western states were represented by only one producer, and South Carolina dropped out entirely.

The general business depression of 1921 is illustrated in the decline of the production of phosphate rock. The decrease in the selling price of agricultural products, combined with the high freight rates, prevented farmers from purchasing fertilizer, and the low rates of exchange discouraged exporters in the industry.

PRODUCTION OF CALCAREOUS MARL IN 1921.

The output of calcareous marl in the United States in 1921 amounted to 53,730 short tons, valued at $183,743, according to reports made by the producers to the United States Geological Survey, Department of the Interior. The quantity decreased forty-five per cent and the value fortythree per cent as compared with 1920. In 1921 the average value per ton was $3.42; in 1920 it was $3.31. Nearly all the calcareous marl sold in the United States in 1921 was used for liming the soil. Some was used as a filler in patent fertilizers. More than sixty-three per cent of the total output-33,978 short tons-was produced in Virginia and was valued at $105,821. The other producing states were California, Maryland, New York, North Carolina, Ohio, Pennsylvania, South Carolina, and West Virginia. Deposits were developed in Michigan and Wisconsin,

PROBLEM DEPARTMENT.
Conducted by J. A. Nyberg,

Hyde Park High School, Chicago.

This departmeni aims to provide problems of varying degrees of difficulty which will interest anyone engaged in the study of mathematics.

All readers are invited to propose problems and solve problems here proposed. Problems and solutions will be credited to their authors. Each solution, or proposed problem, sent to the Editor should have the author's name introducing the problem or solution as on the following pages.

The Editor of the department desires to serve its readers by making it interesting and helpful to them. If you have any suggestion to make, mail it to him. Address all communications to J. A. Nyberg, 1039 E. Marquette Road, Chicago.

LATE SOLUTIONS.

740. Six original proofs by Edw. A. Ravenscroft, New Trier H. S., Kenilworth, IU.

SOLUTION OF PROBLEMS.

741. Proposed by Harris F. MacNeish, College of the City of New York. Find without using trigonometry the volume of a regular icosahedron in terms of the edge e.

Solution by Irma Luelleman and George Maischaider, Mattoon, Ill. Pass a plane through the icosahedron perpendicular to the axis LM at its midpoint. The section formed will be a regular decagon whose side is e/2. The center, O, of the decagon will also be the center of the polyhedron. DE is a side of the decagon, DO its radius; OF will be perpendicular to the face ABC. After finding OF we obtain the volume of a triangular pyramid which has ABC for a base and OF for altitude. Insert cut figures No. 5416..

BC=e; DE=e/2. OD is the radius of a decagon of which DE is a side; OD=e(1+√/5)/4. DC=e√3/2; DF=e√3/6. OF2=OD2FD2=e2(14+6√5)/48; OF=e(3+√5)/4√3. Area of AABC = e2√3/4. Hence the volume of the icosahedron is 20 × 11⁄2 ×OFe2√/3/4 = 5e3(3+5)/12=2.18c3 approx.

Similarly solved by Michael Goldberg, Philadelphia, Pa.; J. F. Howard, San Antonio, Tex.; H. Lazott, Worcester, Mass.; the class in Solid Geometry, Culver Military Academy, Culver, Ind.; and the following pupils of the Dickinson H. S., Jersey City, N. J.: Ernest Lundt, Erwin Rainer, Henry Siemers, and Richard Slaner. E. Tabor, Upper Lake, Calif., gave two solutions, one based on the theorems: the ratio of the surface of one side of the largest cube inscriptable in a dodecahedron of edge e is to the volume of an icosahedron of the same edge as 6:5c; and the edge of such