one, am convinced that no explanation of its principles, however clearly made, and, I may add, however completely understood, will impart a practical knowledge of this most important method. It is as useless to attempt it, as to try to prepare an army for the battle field by a daily lecture, instead of a daily drill; or by explaining tactics instead of practising them. The one thing should be done, but the other must not be left undone. This, I think, touches the grand difficulty in our whole scheme of mathematical instruction. An attempt has been made to reverse the natural order of things in mathematical study in the same manner as has been done in that time-honored custom of trying to learn language by first studying its grammar. There is a natural method in the study of mathematics, just as truly as there is of language, and the employment of it is followed by results equally surprising in both cases. The important processes actually employed in calculus are not so very numerous nor are they especially difficult to acquire. No real use, however, can be made of its methods until these are acquired. It must often happen that the full significance of such processes is not apprehended until long after they are employed with dexterity. Certain it is that such dexterity and familiarity conduce wonderfully to their correct comprehension. Such treatises as those of Todhunter, Williamson and Boole contain invaluable lists of selected exercises from which such dexterity can be obtained. The text of such works affords very little difficulty even to students of moderate capacity provided they have acquired ready skill in the algebraic work. This ready skill is not a rare and difficult accomplishment at all :-no more so than is the ability to perform chemical analysis a rare gift. It simply needs cultivation by some one who has such ready skill himself. But this kind of work is not in accord with the disciplinary spirit and can hardly flourish where that spirit prevails.

A

How much the true spirit of mathematical instruction has been misconceived in classical colleges may be seen from an occurrence which was not, I am led to believe, of an unusual character. gentleman of distinguished classical attainments and linguistic talents, now a valued professor in that department, who had spent several years abroad perfecting himself in his specialty, on returning from Europe accepted the position of tutor at Yale. On assuming the duties of that office what was his surprise and indignation at finding that the department of instruction to which

he was assigned was mathematics, and the subject algebra - a subject wholly foreign to his tastes and special requirements— one indeed which he had not touched since his freshman year. Evidently the only duty expected of him was to hear recitations from a text-book, and give marks for such recitations. The daily marking system is perhaps the most characteristic and most pernicious outward expression of the disciplinary spirit. How have the evils of that system been intensified in our larger and older colleges by the wholesale manner in which the work is done! The work of recitation and instruction can no doubt often be advantageously combined, but what is the probability that valuable instruction will be communicated during the hour to which the exercise is confined when the number of students in the recitation room is thirty, forty or even more? Not more than fifteen or twenty can in one hour's time have separate personal contact with an instructor which is of practical value. What a perversion of the purposes of the noble endowments for higher education, to expend almost the entire energy of the teaching force of the many institutions who adopt this system, in a daily effort to weigh with minutest accuracy the fidelity with which assigned tasks have been committed to memory -I am not willing to say understood! How sadly this fails to accomplish the worthiest ends of education! So entirely has memoriter recitation gained control of the educational field in our colleges and so completely has it stifled free inquiry, that there is no opportunity for a student to ask a single question of his instructor. In my college days, a question from a student during recitation was unknown, and would have been regarded by the student world as overstepping the bounds of propriety. How little opportunity there was out of recitation for questions is evident when the size of the classes is considered.

What other mathematical subjects should be included in the course of the liberal arts it is not my object at this time to attempt to decide. If we but once succeed in getting the instruction upon a healthy basis the question will decide itself in accordance with experience and the needs of science. The most diverse views may be entertained as to whether the college course can embrace analytical mechanics, or the theory of determinants (now so universally used) or whether it can omit vector and quaternion analysis. When I mention subjects so far out of the ordinary range of undergraduate study, I am aware that I shall be met with incre

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SECTION A. ADDRESS BY HENRY T. EDDY.

dulity as to its possibility. When, however, I find that in our small western college, graduating less than a dozen annually, we have now had for years volunteer classes pursuing all these and other subjects annually with success, I do not share in any incredulity as to its practicability or profitableness.

In conclusion, I wish to call for reform in our mathematical teaching. I call for deliverance from the imprisonment it has suffered, for removal of the bands with which it has been bound. I call for the introduction of a spirit of free inquiry. Let not all mathematical teaching be run in one narrow mould. Let it not be so conducted that he who has neither taste for the study nor special knowledge of it, stands on an equal footing as a teacher with the man of real mathematical insight. Mathematics has been termed the science of necessary conclusions. It furnishes the indispensable groundwork for the study of exact science. It fills a wide field in the intellectual needs of any people of strong intellectual powers. It awakens the interest and stimulates the open ing mind as do few studies. One study of genuine interest, ore worthy and absorbing pursuit is sufficient to drive away lethargy and stamp a manly, vigorous character upon a young man. We cannot afford to let this blade of Damascus rust in its scabbard. Now is a favorable time for revising our estimates of what can and ought to be done in this field. Higher mathematical culture has commenced a new and fruitful growth in this country in various places, but nowhere with more promise than at the Johns Hopkins University, at Baltimore, that wonderful new star in the galaxy of American Universities. To lofty ideals must be joined sound learning, great patience, and an ambition never to rest until we occupy the foremost place in intellectual achievement. Many of our younger mathematicians have won golden opinions for their abilities and zeal in foreign universities. I think we ought to form an association of the mathematicians of this country for the purpose of concerted action in improving the mathematical training in our colleges, in which the fire and enthusiasm of this young blood should be called upon to help on the cause. No learned body among us has a more vital interest in this question than the one I now have the honor of addressing. I therefore need make no apology for attempting, as I have done, to fix your attention upon those phases of it which have been suggested to me by my experience.

PAPERS READ.

PHENOMENA OBSERVED UPON THE SOLAR SURFACE FROM 1881 TO 1884. By Rev. S. J. PERRY, Stonyhurst College Observatory, Whalley, England.

[ABSTRACT.]

THE principal subject contained in this communication was the classification of the faint evanescent markings that are always visible on the sun. The only observer who had previously made any serious study of these objects was E. L. Trouvelot whose researches at the Harvard College Observatory in 1875 were known to the author only through the few lines on the subject in Prof. Young's work on the sun. Trouvelot classed them all as "veiled solar spots," but the author of this paper has thought it more advisable to divide them into three classes.

Class 1. Ill-defined patches of a cold grayish tint, which, individually, never last for more than three minutes. They are scattered over the whole of the surface, and appear suddenly as irregular patches. The sudden appearance of many of these faint objects in the same locality and their rapid development, frequently give rise to a blurred appearance which has been so marked a feature in Dr. Janssen's magnificent solar photographs.

Class 2. Other faint spots appear at first as small, dark and fairly defined dots, and after retaining this appearance for about two minutes, suddenly spread out and cannot, at this stage of their short existence, be distinguished from spots of the first class. These are also found in every heliographic latitude and at all times, and never last more than a few minutes.

Class 3. This last class of ill-defined spots approaches much nearer to the ordinary spots than those already mentioned. They have never been seen outside the spot zones and are generally in the immediate vicinity of spot groups, but they remain indistinct throughout their whole period of visibility. They last sometimes

for a few hours but may be generally seen for two or three days. Their formation is apparently very rapid, but they disappear gradually.

This classification of the faint solar spots was followed by some remarks on the spot maximum of 1882-3; on the position of faculæ with respect to spots, offering an explanation of the apparent lagging of faculæ behind large spots; on abnormal extensions of the penumbra; on dark shadows seen in connection with faculæ ; on the colors seen in spots; and on the motion observed on several occasions when the moving body remained always projected on the solar disk from its first to its last appearance.

DESCRIPTION OF A MODEL OF A RULED CUBIC SURFACE KNOWN AS THE CYLINDROID. By Prof. ROBERT S. BALL, Royal Astronomer of Ireland, Dublin, Ireland.

[ABSTRACT.]

In a work on the "Theory of Screws" published in 18761, I discussed the various kinematical and dynamical relations of the ruled cubic surface of the third degree which is defined by the equation.

z (x2+ y2) — 2mxy = 0

This surface had been discovered by Plücher and he had, the writer is informed, made a model of it to illustrate its application in the theory of the Linear Complex. It does not however appear that Plücher had ever contemplated the significance of this surface except in the region of pure geometry. The dynamical interest of the surface is however very great as will be understood from the following enumeration of a few of its properties.

We first premise the following well-known theorems :

(1) Any movement of a rigid body can be produced by a twist about a screw (meaning by a twist, a rotation around an axis and a translation parallel thereto and equal to the product of the rotation by the pitch of the screw).

1 Dublin, Hodges Foster & Co.