they depend upon the distances, they must be inversely proportional to the squares of these dimensions, in order that on the whole, from the masses and distances combined, they may be simply dependent on the first power of the distances, dimensions, and velocities of the bodies. Hence the law of gravity, and conversely corresponding motions of revolution or oscillation in two similar systems must, by the law of gravity, have the same periods, since the dimensions of the paths or orbits of these motions, their velocities, and their changes of velocity, are all proportional to the dimensions of the systems. Hence all measures of time, whether by periods of orbital, of rotary, or of oscillatory motions, are by the law of gravity independent of the dimensions of the material universe; and if the solar sytem had been constructed on the scale of a common planetarium, it would still have moved, by virtue of the forces inherent in matter, pari passu, through the same phases of motion and configuration, with the same periods as now. - W. 3. Develop the Naperian logarithm of x into a series.— dx х dx dy = dy But by division y+1=dy-ydy + y2dy — y3 dy + y^dy &c. By integration log x = y-y2 + } y3 — { y2 + fy0 &c. Restoring the value of y=x-1, we get log х = (x — 1) — 1 (x − 1 )2 + } ( x − 1 )3 — † (x — 1)1 &c. - ARTEMAS MARTIN, Franklin, Pa. 4. Note on Right-angled Triangles. I have for many years kept on hand for my own convenience a list of the fifty two right-angled triangles described by Professor HOYT in the May number of the Monthly. I prepared the series by using the formula (a2 + b2)2 = (a2 — b2)2 + (2 a b)2, which is easily seen to be true, and in which any numbers whatever may take the places of a and b. If for these letters we substitute the natural numbers in succession we shall obtain thirty sets of numbers no one of which will exceed a hundred; fourteen of these however are equimultiples of some of the others, and twenty-two other multiple sets may be found within the same limit. We thus find, as Professor Horr has done, fifty-two right-angled triangles whose sides are expressed by integral numbers not exceeding one hundred, and sixteen of which are dissimilar in form. I cannot now call to mind where I found the formula given above. Prof. E. S. SNELL, Amherst College, Mass. - 5. Note on Equal Temperaments. It is assumed that the number of vibrations in a given time, producing a musical tone, is to the number producing its octave as 1 is to 2; that the numbers in like manner corresponding to a note and its "fifth" are to each other as 1 to 1.5; and that in a scale of equal temperament the numbers corresponding to the successive tones are in geometrical progression. Required, the number of equal intervals into which an octave must be divided, so as to have one of the tones approximate nearly to a "fifth." Let the ratio in the geometrical progression, y = the number of intervals approximating the fifth. x = the number of intervals in the octave. Then which, by the method of continued fractions, gives the approximate values z = 1.14871; and by (1) the fifth, 2 = 1.557. 5; whence by (2) This error of .0157 and 2 1.49830. is too great to be tolerated by the musical ear. would give y=7, x = 12, z = 1.05946, This is the usual chromatic scale, and as the error in the "fifth" is but 0017 it satisfies the ordinary ear. The approximation 1 would give as the ratio of the fifth 2" = 1.50042, an error of only .00042. The approximation would, like, give a flatted fifth; its ratio would be " 1,49994, and this error of .00006 would probably be quite imperceptible even to the nicest ear. An instrument to play 53 notes in an octave would, however, probably be difficult of construction; nor can we expect voices to move with certainty through such small intervals.-M. H. DOOLITTLE, Sophomore Class, Antioch College, Ohio. Additional Note by Rev. THOMAS HILL, President of Antioch College. — Mr. DOOLITTLE has calculated the values of the approximation as follows: 1.0000, 1.1487, 1.3195, 1.5157, 1.7411, 2.0000. The nearest notes to these, in the twelve-semitone scale, are C, D, F, G, B-flat, C, which will instantly be recognized as the scale of B-flat, with the seventh and fourth omitted, that is, the scale of the old Scotch melodies. Thus it appears that this old-fashioned scale approximates rudely towards a division of the octave into five equal divisions. The simple fractions which approximate most nearly to these values are,,,, and perhaps the Scotch singer may follow these simpler divisions when singing without a keyed or fretted instrument. If, with Mr. POOLE, we admit the prime seventh into our musical theories, this is a point perhaps worthy of investi gation. Mr. DOOLITTLE'S 5-note scale differs from the Scotch in taking C instead of B-flat as the tonic. If we add to it the note A, retaining C as the tonic, it becomes identical with the Irish scale, in which, for example, “Huggamur pene on Sambhruliun" is written. THE MOTIONS OF FLUIDS AND SOLIDS RELATIVE TO THE EARTH'S SURFACE. [Continued from Page 97.] SECTION Vİ. ON THE MOTIONS OF THE OCEAN. 72. BESIDES the actions of the sun and moon which give rise to the tides, there are only two causes which can produce any sensible motions on the waters of the ocean. One of these is the action of the atmosphere upon the surface of the ocean, and the other, the difference of density between the water near the equator and that towards the poles, arising from a difference of temperature. The general motions of the atmosphere at the surface of the ocean have a tendency to cause a westward motion of the water in the torrid zone, and an eastward motion in the middle and higher latitudes; and from what we know of the effects of strong winds upon the ocean, we have reason to think that these general motions of the atmosphere are adequate to produce sensible motions, since, after the inertia of the water is once overcome, which, however small the force, is only a question of time, the only force necessary is that which is adequate to overcome the resistance of friction, which is very small where the velocity is small. The difference of density between the equator and the poles causes a slight interchanging motion of the water between them, and consequently, where not interrupted by continents, it produces a system of motions in the ocean similar to those of the atmosphere. Hence these two causes of oceanic disturbance, whatever their relative weight, both act in the same directions, and conjointly cause the observed westward motion of the ocean near the equator, and eastward motion towards the poles. 73. The westward motion of the water of the ocean in the torrid zone was first observed by Columbus, and is now well established; and observations also show that there is a motion towards the east in the higher latitudes. A bottle thrown into the ocean near Cape Horn was picked up three and a half years afterward at port Philip, Australia, a distance of 9000 miles, which makes the eastward velocity in that latitude more than 7 miles per day. And Sir James Ross, when sailing eastward near Prince Edward's Island, found himself every day from 12 to 16 miles by observation in advance of his reckoning. (Voyage to the Antarctic Seas, Vol. II. p. 96.) But a westward motion being established in the torrid zone, an eastward motion in the higher latitudes must be admitted; for, as was shown in the case of the atmosphere (§ 35), the one cannot exist without the other. 74. It has generally been supposed that the equatorial westward current of the ocean is caused principally by the action of the westward winds there; but Professor GUYOT thinks that "it is too deep and rapid to admit of being explained by their action alone," and that "the difference of temperature between the regions near the equator and those near the poles controls all other causes by its power and the constancy of its action." (Earth and Man, pp. 189, 190.) The torsive or deflecting force which causes the westward motion of the atmosphere and the ocean in the equatorial regions, and the eastward motion in the higher latitudes, has been shown to be as the velocity of the interchanging motion between the equatorial and the polar regions; and hence if this motion in both were similar, the relative amount of this force in each must be as the |