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since.142854, and the value of the fraction at the end is five times the, part found, =.714284; whence =.14285714284 =.142857142854. We thus see the necessity of the sequence above stated.

When we multiply (2) by r2, if we represent the tens of r291, "292, &c., by m1, m, &c., we evidenely have

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We here have the key to Mr. Wiley's method; for by the second principle is known at a glance, and by the third principle r, is found by equation (1); whence q, becomes known from (1'), and 93, 94, &c., are successively found from equations (2′), (3′), &c.

(To be continued.)


1. Find the value of x and y in the following equations:

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-Communicated by U. JESSE KNISELY, Pres't and Prof. of Mathematics in Luther College, Newcomerstown, Ohio.

2 Let a regular polygon of 14 sides be described, each of whose equal sides shall be one. Then will the radius of its circumscribing circle, which put=r, be more than two and less than three. Put r = 2 + x; then is a a positive quantity less than one. Let another regular polygon of half the number of sides (7) be inscribed in a circle whose radius is one, and determine one of its equal sides in functions of a expressed in its simplest form. 3. If a line make an angle of 40° with a fixed plane, and a plane embracing this line be perpendecular to the fixed plane, how many degrees from its first position must the plane embracing the line revolve in order that it may make an angle of 45° with the fixed plane?-Communicated by PROF. A. SCHUYLER, Berea, Ohio.

4. A cask contrining a gallons of wine stands on another containing a gallons of water; they are connected by a pipe, through which, when open, the wine can escape in to the lower cask at the rate of c gallons per minute, and through a pipe in the lower cask the mixture can escape at the same rate; also, water can be let in through a pipe on the top of the upper cask

at a like rate. If all the pipes be opened at the same instant, how much wine will be in the lower cask at the end of t minutes, supposing the fluids to mingle perfectly?-Communicated by ARTEMAS MARTIN, Mathematical Editor of Schoolday Magazine, Erie, Pa.

NOTE. To those who use "Nystrom's Mechanics." Nystrom prints "29.869650000+,"

but 29.86960440108+,-U. JESSE KNISELY.

QUERY.-What is the explanation of the phenomena described below? If a ball of cork or other light substance be placed in a vertical jet of water of sufficient force to elevate the ball, it will rise to a point where the force of the ascending jet, or so much of it as is efficient in elevating the ball, is just equal to the weight of the ball, and will there revolve; and its equilibrium will continually be restored, notwithstanding the ball may be disturbed by slight horizontal forces.

Erratum. On page 6, 19th line from bottom, for "12-1 " read 1 +21/1.


Comets and Meteors. By DANIEL KIRKWOOD, L. L. D., Professor of Mathematics in Indiana University, and author of " Meteoric Astronomy." J. B. Lippincott & Co., Philadelphia.

To those who have not yet seen this very interesting book by Prof. Kirkwood, the follow ing quotation from the Preface will serve to indicate its character:

"The origin of meteoric astronomy, as a sciene, dates from the memorable star shower of 1833. Soon after that briliant display it was found that similar phenomena had been witnessed, at nearly equal intervals, in former times. This discovery led at once to another no less importrnt, viz: that the nebulous masses from which such showers are derived revolve around the sun in paths intersecting the earth's orbit. The theory that these meteor-clouds are but the scatered fragments of disintegrated comets was announced by several astronomers in 18677-a theory confirmed in a remarkable manner by the shower of meteors from the debris of Biela's comet on the 27th of November 1872. To gratify the interest awakened in the public mind by the discoveries here named, is the object of this work. Among the subjects considered are, cometary astronomy; aerolites, with the phenomena atteuding their fall; the most briliant star-showers of all ages, and the origin of comets aerolites and falling stars.' Surveying and Navigation, with a Preliminary Treatise on Trigonometry and Mensuration.' By A. SCHUYLER, A. M., Professor of Applied Mathematics and Logic in Baldwin University; author of "Higher Arithmetic," "Principles of Logic," and "Complete Algebra.? Wilson, Hinkle & Co., Cincinnati and New York.

We would be pleased to give an extended notice of this book did our space permit. We must be content to say, however, that, as a text book for the student, and as a manual for the surveyor, we think it admirable, both in plan and execution. The subjects discussed are thoroughly and yet concisely dealt with; and the paper, wood cuts and typography are perfect.

Yates County Chronicle. Persons who are fond of solving mathematical problems and who want something new on that subject every week and a good newspaper besides, will do well to obtain the Yates County Chronicle; published at Penn Yan, New York. DR. S. H. WRIGHT, Mathematical Editor.

Published the First of each Month.

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IOWA SCHOOL JOURNAL.-This periodical starts the new year (1874) with a new volume, and new type and printing material of its own. A neatly. engraved title page and an elegant style of type add very much to its appearance. Its new office is at No. 14, Mills' Block. Address

C. M. GREENE, Publisher,
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In bringing forward the nebular hypothesis to explain the cause of the primitive movements of our solar system Laplace states the following five phenomena which must be considered. 1. The motions of all the planets in the same direction and very nearly in the same plane: 2. The motions of the satellites in the same direction as that of the planets: 3. The motions of rotation of these different bodies and of the sun in the same direction as their projected motions, and in nearly the same planes: 4. The small excentricities of the orbits of the planets and their satellites: 5, and finally the great excentricities of the orbits of comets, while the inclinations of these orbits seem to occur wholly at random.

The only previous hypothesis to which Laplace refers is that of Buffon, the naturalist. Buffon assumed that a comet falling upon the sun had thrown out a torrent of matter, which, uniting at different distances into various globes, had become opaque and solid by cooling, and thus formed the planets and their satellites. This hypothesis would explain the first of the five preceding phenomena stated by Laplace, for all the bodies thus formed would move very nearly in the plane passing through the center of the sun and the direction of the torrent; but the four other phenomena cannot be explained by it. In fact, the smallness of the excentricities of the planetary orbits is directly opposed to this hypothesis; for if a body moving in an ellipse touches the sun it will do so at each of its revolutions, and, although this condition of things might be somewhat modified in Buffon's hypothesis, the chance that the excentricities of the orbits would be small is very slight. Finally, this hypothesis does not account for the comets at all. Among those who preceded Laplace in speculations on the constitution of the universe, after the discovery of the law of gravitation, may be mentioned Kant, the metaphysician, Lambert and Sir William Herschel. In a work published in 1755 under the title

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