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time t, counting from the commencement of motion, may be expressed thus
n- f ()* in which n is the initial rotary velocity of the disk.
If we substitute this expression for vz in the last two equations (3) (p 541, June No.,) and follow a similar process to that by which equations (4) of that paper are deduced, we shall get, for the equations of motion Сп
102 2 MgY sin 20
(cos 0 - cosa) d t2 d 12
A For the sake of simplicity suppose the initial position of the axis be horizontal, or a =90 and the above become dy Сп
AS *r(ed.cos o
If aff'a' represents the cycloidal curve, and aee'e" g' the curve in question, it will be observed that the angular velocity of the axis given by the 2nd equation (6) is the same for both, for equal values of 0, while
dy the value of the horizontal component of that velocity, sino at
с than for the cycloidal curve, by the term
A sin 0 As 0 diminishes, d cos 0 is positive and this term is subtractive and
dy hence for any point e or e' on the descending branch, is less than for
dt the corresponding point f or f' of the cycloid, and the branch aee'e" will be behind the branch aff', and will descend lower.
с At e' the term
A sin 0 curve ascends, 0 increases, and the increments of cos O become negative.
A mi no S.*P(0)d.cos 0, attains its maximum, for as the
* When the retarding force is independent of the velocity, as in the case of friction, the f(t) in the above expression is linear; when this force is dependent upon the velocity, as for the resistance of the air, f(t) will, in general, be an infinite and diverging series in the powers of t; whether the force is due to either, or both combined, of these causes, the above expression for the velocity of rotation may however be used for the present purpose.
But as the values of t on this branch of the curve are nearly double those or equal values of 0 of the descending one, the integral sof(t)d.coso will become zero at some point g', before 8 has regained its initial value, at which point " will be the same as for the corresponding point g of
с the cycloid. Above the point g' the term oS, 1(e)d.cos O bo
A sin comes negative and (with its negative sign) becomes additive and therefore, above g' the values of are always greater than for corresponding
dt points of the cycloid. Hence the angular velocity of the axis can never become zero and consequently the axis cannot rise to its initial elevation and form a cusp, but must make an inflexion and culminate at a, below the initial elevation.
Commencing a second descent from a' with an initial velocity, the succeeding wave will be flattened (as shown in treating the subject of “initial gyratory velocities”), the second culmination a, will not (as a similar train of reasoning to that just gone through for the first undulation proves) be as high as a : and pari ratione, each succeeding wave will be more flattened and extended than the preceding, until they soon virtually disappear, and the curve becomes a descending helix.
After these undulations have disappeared, as the descent is only due to loss of rotary velocity (and consequently loss of deflecting force) measured by f(i), it is evident that the future character of the helix will be determined by this function.
do In fact, as the descending velocity is then very minute compared
dy with the horizontal velocity its
dy equat., (6); and, equating the values of sin 0 deduced from these
dt two equations, we shall have C
2 Mgy A,
cos - sino
A By differentiating both members and making various reductions we get
3 sin2A-2 с
A sino sin 20 А an equation which, after the disappearance of the undulations, gives the value of 0 in terms of t.
As f(t) increases 0 diminishes in the first member, to the limit corresponding to sin2 = which makes the numerator of the fraction in the first member 0, and the denominator a maximum; showing, to that limit, a constant descent of the axis, or a descending helix for the curve.
As the values of f) beyond f(!)n do not belong to the question, there can be no farther descent below that value of 0 which reduces the first member to zero; or beyond sin20=4.
At this elevation, as the deflecting force has vanished entirely with the rotary velocity, it is evident the elevation of the axis must be maintained by the centrifugal force alone, due to the gyratory velocity.
In fact, if we calculate directly the angle to which the axis must fall from a horizontal position, in order that the velocity generated shall be just sufficient, if deflected into horizontal gyration, to exert a centrifugal force adequate to maintain it, we shall find this same value, sin2 0=3*
In reality, the air resists gyration as well as rotation, and hence the descent will continue; but if a gyroscope could be placed in a perfect vacuum, and the slight friction at the point of support be entirely annulled, the axis would descend in a helix until it reached this limit, at which it would forever gyrate, though the rotation of the disk would soon by friction of the axle, entirely cease.
* If the solid of revolution is of dimensions so small that it may be considered concentrated in its centre of gravity, it would require, in the fall of its axis through angle 90°– , the velocity ✓2 gy cos e; and this velocity, deflected into horizontal gy.
cos ration in a circle whose radius is y sin 0, would create a centrifugal force 2 g
cos24 whose component normal to the axis of figure is 29 Equating to this the
sin 0 opposing component of gravity g sin 9, we get sinag=, as in the text.
For finite dimensions of the solid, the direct determination of the limit in question, is more complicated, and it is scarcely necessary to introduce it here.
[The following communication from our correspondent, M. A. Dwight, contains important suggestions in reference to the study of Art.-ED.]
MR. EDITOR.—Professor Dana, in his communication published in your Journal, (No. 12, p. 289,) wishes to correct my “misapprehension" of his opinions advanced in his address, (No. 10, p. 294, of the same work.)
Allow me to say in reply, that I think no "misapprehension" exists, and that the point of difference between us lies, in the fact that each one entertains opinions on the same subject remotely opposite from the other. Professor Dana says, “the ancients had, it is true, built magnificent temples. But the taste of the architect and that of the statuary or poet, is simply an emanation from the divine breath within man, and is cultivated by contemplation, and only surface contact with nature." I think, on the contrary, that all true art has its foundation in science. In order to correct my “misapprehension" of him, he says, “I was aiming to show, that the ancients had not pursued the study of nature far enough to arrive at any of the profound laws which make the foundation of modern science, and I spoke of their proficiency in architecture and sculpture, as no evidence of such knowledge, as it reached its state of perfection without it."
The educated world acknowledge that the ancient Greek temples are models of architecture, and architects have ever made them a study, endeavoring, if possible, to master the scientific principles on which they were constructed. And that the ancient statues could exhibit such truth to nature, both in form and expression, argues a most profound knowledge of the science of anatomy. If Professor Dana doubts this, let him take the anatomical plate prepared by Fau for the study of artists, exhibiting the muscular development in the statue of the Laocoon, and compare it with the statue itself, and he can not fail to be convinced that the sculptors of that wonderful group have proved their knowledge of the science of anatomy beyond question, and had also obtained“ a deep insight into the profound laws which make the foundation of modern science." It is said of Benjamin West, that when he first saw the Apollo Belvedere, he exclaimed, “How like a Mohawk warrior!” He then described to the bystanders, their education, their dexterity with the bow and arrow, the admirable elasticity of their limbs, how much their life expands the chest, while the quick breathing of their speed in the chase dilates the nostrils with that apparent consciousness of vigor which is so nobly depicted in the Apollo. "I have seen them often," added he, “ standing in that very attitude, and pursuing with an intense eye, the arrow which they had just discharged from the bow.” The Italians present admitted that a better criticism of the merits of the statue had rarely been given." (Gall's Life of West.) Could this "immortal statue,” which is considered the model for students in sculpture have been produced by any artist who had had "only surface contact with nature?" Does not the term art imply the necessity for a combination of knowledge and skill in addition to the “emanation from the divine breath within man and a surface contact with nature ?"
No. 13.-[VOL. V., No. 1.)-20.
Professor Dana also says, " it will show our appreciation of Miss Dwight's views, when we say, that our scheme of a “Scientific School,' printed before that address was delivered, included a Professor of Drawing, (in all its departments,) another of Architecture, another of Æsthetics or the History and Criticism of Art, and this we regarded as merely an initial step toward a wider expansion of the Art department. These topics were associated with the various sciences, so that the art student, according to the contemplated plan, would have an opportunity to acquire that comprehensive acquaintance with modern science which is necessary to equip him for his best and highest efforts.” Allow me here to repeat the question already given, why should art be included as one of the pursuits in a Scientific School, if science is entirely superfluous to the pursuit of it, -as it must certainly be, if Professor Dana is correct in his opinion that "ancient architecture and sculpture reached its state of perfection without it ?" Nothing can be carried further than " a state of perfection;" therefore "the profound laws which make the foundation of modern science," and which the ancients are supposed “not to have reached,” could add nothing to the improvemeut of those arts. Professor Dana in his plan for a scientific school, includes among his corps of professors, one of Æsthetics or the History and Criticism of Art? Modern Art certainly furnishes no subjects for scientific discussion, and if ancient sculpture and architecture “are the result of contemplation and surface contact with nature," what occasion have they for such a professor, as in that case there can be no established principles for him to inculcate, and in criticising works of art, he can have no criterion of merit.*
If the Professor of Drawing at the New Haven Scientific School agrees in opinion with Professor Dana, the scholars there, are, to say the least, in danger of being misled, and we again repeat the opinion already expressed, that if other scientific schools will give prominence to the study of art, “Yale like other shaded plants will begin to dwindle, and her laurels fade.” By the term ART we do not mean surface work, but art having for its basis scientific principles founded in the immutable laws of nature. Pursued on this basis the study is ennobling, and elevating, and expands the whole mental capacity; but as mere surfaco work, it is as unsatisfactory and as unprofitable for all purposes of education, as the most mechanical employment. Those who would take any pleasure in it, or prefer it to the true and better way, would enjoy the music of a hand organ more than the finest concert of instruments and voices. To excel in sculpture and architecture scientific knowledge is indispensable. True, modern artists do not acknowledge this, but do they excel? Most of their works are, at best, but an imitation of the antiques which were produced by the aid of scientific knowledge, and without these models, how would they know where to begin their work? A student who is familiar with general literature will detect the plagiarisms of authors, and those who have studied ancient art, will detect the plagiarisms of artists, as well as the blunders that betray their want of artistic education. If the sculptor of the Beatrice Cenci, had had the training that all artists require, she would not have been so mistaken in the choice of her subject, which is not one for sculpture, because that art being limited to the repre
• My limits do not admit of extracts, and I reser Professor Dana to “Winckleman's History of Ancient Art," also “ Gæthe's Essays on Art.” Fuseli's Lecture on Ancient Art, and those of other artists, Richardson, Barry, Reynolds, Hayden, etc. I think he will find the various authors of these works believed that the ancient artists worked on scientific principles derived from the study of nature.