own universities they lay dormant, or were taught as a mere matter of course, studied by hardly any, and considered only as matters of idle curiosity by nearly all. We live in better times: yet few of us are sensible of the great obligations which we are under to those modest and unpretending works for the present spirit of inquiry which is now become so general in respect of pure mathematical science and its innumerable applications. As we were, so America is, at this moment. Bowditch was little known to the general scientific world before his present undertaking: yet, so to speak, Bowditch was the child of Diarian nursing. His efforts were first made in those obscure American periodicals which are, except to perhaps half a dozen persons, unknown in this country, and almost as unremembered in their "father-land." Knowing this as we do, we cannot look with indifference upon the attempts which are made in that country to establish a superior class of such publications, dependent for their support on native or adopted talent. Were we to recount the names of those who in this country were made mathematicians by the English Diaries, Repositories, Bees, Correspondents, Companions, Receptacles, &c.,-were we to recount the names of Simpson, Landen, Dalby, Burrow, Lawson, Bonnycastle, Crakelt, Saunderson (George), Robertson (Dr. Abram), Wales, White (Thomas), Wildbore, Vince, Brinkley, Maskelyne, Hutton, Harvey, and a hundred others, already passed away,—were we to recount the names of Gregory, Leybourn, Lowry, Ivory, Wallace, Barlow, Davies, Swale, Mason, Young, Woolhouse, and a hundred others amongst the living, we are sure the importance of these publications would be at once admitted. Can we then but augur well for America from an undertaking like this? Or, still more, from the very superior manner in which this is conducted? Can any one who peruses this single number with attention, fail to be struck with the power of American mind? We think not.

The plan of the work assimilates more with that of Professor Leybourn's Mathematical Repository than with any other type: and we think Professor Gill could scarcely have selected a better model, as to general feature. Should he succeed, even to a partial degree, in effecting by his publication the real good which Mr. Leybourn has done, he will deserve well of his countrymen.

We shall give a brief analysis of this number, and our readers will then see that we may most cordially recommend it to their notice and support, assured that they will find much to compensate them for its cost.

The first article is the investigation of a formula for the longitude when the rhumb is invariable. In this case, the course is the loxodrome, and the expression, for the longitude arrived at is known to be accurately tany, (omitting the spheroidal figure of the earth) x − x = tan. v log. tan y where x and x, are the longitudes of the limits, yy, the polar distances of the same limits, and v the angle of the rhumb; and the formula is well adapted for calculation*. The method has, however, been virtually

* We shall probably have occasion to make some remarks on the various methods which have been proposed for the

1

solution of this problem, as to comparative facility, in a future number of this magazine.

anticipated by several European writers, and we may add that it is not the one most in favour amongst us.

The next is an excellent solution of the problem.—“ In a given ellipse to inscribe the greatest equilateral triangle." It is shown that the triangles vary from the least to the greatest continuously; or, in other words, that there is only one maximum and one minimum value of the side; though obviously there must from the symmetrical form of the curve be four different positions in which they may be drawn.

Thirdly, a new discussion of the problem:-"Upon a horizontal plane a rectilineal path is traced, in which P is constrained to move uniformly. This body is connected by an inflexible and inextensible line with another body, M, which is posited in this plane, and which is supposed to have received some primitive impulse in the direction of this plane. It is required to find the nature of the curve described by the body M, and the other circumstances of the motion, abstracting all consideration of friction." This problem was first considered in connexion with several analogous ones by Clairault in Mém. de l'Acad., 1736, in a discussion with Fontaine and others, who considered the curve to be the Tractory or Equitangential Curve. It has been subsequently discussed by Gergonne, Dubuat, Français, and others, (Ann. des Math. tom. iv.); and by Professor Lowry in Leybourn's Repository, vol. v., as well as in the Ladies' Diary for 1779 by the Rev. Charles Wildbore.

Mr. Gill obtains the equations of motion with great elegance, and differently from anything we have seen. His mode of integrating is similar to that of M. Français. He annexes the discussion of several collateral problems. This paper is on the whole a very instructive one to the young geometer and physicien*.

The fourth and final article, as well as the most elaborate and important one, is a paper on Spherical Geometry. This is a branch of science exclusively English, and of very recent growth in any regular form. It seems to have taken a definite character from the researches of Mr. Davies, concerning the "Nature of the hour-lines on the antique sun-dials," published in the Edinburgh Transactions, vol. xii.; and has been followed up by two other papers on the general principles and equations which are involved in the fundamental idea. An abstract of it has also been published in the appendices to the Ladies' Diary for 1835-6; and several interesting questions concerning spherical loci have been interspersed through the Diaries and Leybourn's Repository, by Messrs. Davies, Woolhouse, and Rutherford. The abstracts above spoken of seem to have turned Mr. Gill's attention to the subject, and his processes are, in some cases, material improvements upon those of his predecessor, though, in others, they fall short of them. This article ought not to be passed over by any one who feels interested in this important branch of geometry.

We feel, however, disposed to take an exception to Mr. Gill's notation. Mr. Davies had employed the Greek* letters to designate spherical co-ordinates, so as to distinguish them completely from the notation of the linear system. Mr. Gill employs, contrary to the analogy which is suggested by the uniform practice of modern writers, the Italic letter. Even in plano, the angle made by the radius vector and the angular origin is written X, 0, or ; but Mr. Gill has employed y to designate the spherical radius vector, and a the polar angle. We are at a loss to account for the motives of this change; and as a uniform system of notation is essential to a successful cultivation of any branch of analytical science, we trust that Mr. Gill will yet accede to that which he found in use, except he can show that it violates some of the essential rules of a good notation, and that his own is free from that charge.

We shall conclude, by referring our readers to the remarks on the present state and probable tendency of spherical geometry to be inserted in our next number; and with cordially recommending Professor Gill's labours to the notice of our American as well as our English readers.

REMARKS ON CRYSTALLIZATION,

BY MR. THOMAS GRIFFITHS.

IN the Journal of the Royal Institution, (vol. xiv.,) I inserted a short notice respecting the method "of colouring alum-crystals," together with a few observations on the best nuclei for them to form on.

I there state, that "coke with a piece of lead attached to it, in order to make it sink in the solution, is the best substance for a nucleus, or if a smooth solid surface be used, it will be necessary to wind it round with cotton or worsted, otherwise no crystals will adhere to it."

This fact was afterwards noticed by Dr. Faraday, in his Chemical Manipulation, as follows:-" Prepare a solution of alum for crystallization by diminution of temperature; hang a thread across it, or leave in it a glass rod with a thread wound round it, and observe the greater tendency to deposition on the one substance than the other."

Now, I believe, that these are the only two published notices respecting the influence which the mechanical texture of a nucleus has on the deposition of crystals from solutions; I have been at work a little on the subject from time to time, and more especially, lately, in preparing illustrations for the lectures in the Chemical school of St. Bartholomew's Hospital, and have obtained several results, which, although

* The employment of the Greek characters has a perfect analogy with the established practice of all good writers on the geometry of co-ordinates, and on physical astronomy. In truth, if the intersection of any locus, linear or superficial, with a sphere concentric with the origin of co-ordinates be sought, we have

| only to take the radius vector constant, and the polar equation of that locus (or those points, when the given locus is a line) becomes identical with the spherical equation of the curve (or points) of intersection. This notation is therefore in perfect keeping with pre-established ones.

probably already known to many experimenters, still have not hitherto been made public, as far as I know.

I put forward no claims to originality; my object in now publishing an account of the experiments, is to furnish the lecturer, the student, and the amateur, with a few striking illustrations of some of the beautiful phenomena of the wonderful subject of crystallization.

It has often been remarked to me by some of the most eminent chemists, that for want of a record of the experiments of the lecturetable, they are either wholly lost or only verbally known, and therefore it is my intention occasionally to describe in this journal, some of the most striking experiments suitable for class-illustrations.

Experiment I. Place a smooth glass rod, and a stick of the same size, in a hot saturated solution of alum; upon examination the next day, the stick will be found covered with crystals, whilst the glass rod is perfectly free from them.

If I may be allowed the expression, the crystals appear to have a preference for the fibrous surface of the wood, it affords them good hold; they cling to it in quantity; they have none for the smooth surface of the glass rod.

When solutions are suffered to crystallize in a tall glass vessel, it very rarely happens that any crystals adhere to its sides, but as fast as they form at the surface of the solution, they fall to the bottom of the vessel; in a tall wooden vessel the case is different, for the whole of its sides and bottom become studded with crystals.

Experiment II. With a file roughen the surface of a glass rod at certain intervals, and then place it as a nucleus in a hot saturated solution of alum; all the crystals will adhere to the rough surfaces, leaving the smooth surfaces perfectly bright and clear.

Experiment III. Tie a few threads of lamp-cotton at certain intervals around a clean and polished glass rod, and employ it as a nucleus in a similar solution of alum; the threads will be covered with crystals, whilst the polished parts of the glass rod are perfectly free from them; and thus it is not difficult to obtain six or eight distinct bunches of crystals.

Experiment IV. Tie some threads of lamp-cotton, here and there, around a copper wire, (or a glass rod,) and then place it in a hot saturated solution of sulphate of copper (blue vitriol), the threads will be covered with crystals.

Coke is an excellent nucleus for alum-crystals on account of its very porous nature, affording them plenty of secure hold; but gas-coke very often has a smooth, shining, and almost metallic surface, and if a piece of it be placed in a solution of alum, it will be found that the crystals avoid the smooth surface, and form only on the most irregular and porous parts. Is it not possible that some action of this sort may be the reason why, in crystalline minerals, we so often find single, welldetermined crystals, adhering to certain spots of foreign matters?

In making crystals of alum on coke nuclei, I have found it best to employ a boiling saturated solution of alum, and to bore a hole through the coke, so as to pass a string through, by means of which it may be suspended in the solution; it will of course float, and therefore the

string should be left so slack, that when the coke becomes saturated with the solution, and loaded with crystals, it may sink to about the middle of the solution; this is better than loading it with lead, and the finest crystals will always be found on the lower surface, because they have formed quietly and undisturbed by the fall of smaller crystals from the upper part of the solution.

If powdered turmeric be added to the hot saturated solution, the resulting crystals will be of a bright yellow; if litmus be used instead, they will be of a bright red; logwood will yield them of a purple, and common writing-ink of-a black tint; and the more muddy the solution, the finer will be the crystals, hence no filtration is necessary.

But in all cases of coloured alum-crystals, they are much more brittle than pure alum, and the colours are to some extent fugitive; the best way of preserving them is under a glass shade containing a saucer of water; this keeps the atmosphere constantly saturated with moisture, the crystals never get too dry, and their texture and colour undergo but little change. The same plan may be adopted with many other crystals, especially those of sulphate of copper. Those beautiful blue "crystal baskets," now so common in the bazaars and toy-shops, are made with sulphate of copper, and if one of them be kept for a day or two on the mantel-shelf, it loses its beauty, becoming pale, dry, and brittle; but keep it under a glass shade with water, as just directed, and it retains its original beauty unimpaired.

Wire is a bad substance for a nucleus, for two reasons; first, if it is very smooth, crystals will adhere to it with great difficulty, or not at all; and secondly, supposing it has attracted crystals, they are apt to split off, on account of the expansions and contractions of the wire by alternations of temperature.

A very striking experiment showing how the colour of a crystal very often depends upon water of crystallization, consists in carefully drying a crystal of sulphate of copper in a crucible, until it becomes perfectly white; then drop it into water, and it instantly becomes of its original blue colour by absorbing water.

If a crystal of the ferrocyanuret of potassium (prussiate of potassa,) be similarly dried, its yellow colour vanishes, but reappears directly upon being dropped into water.

I have commenced a set of experiments on the instantaneous crystallization of concentrated solutions of Glauber's salts, in vessels secured from the access of air; and if they tend in any way to elucidate this mysterious subject, I shall make them known through the medium of this Journal.

Chemical School, St. Bartholomew's Hospital,

October 13, 1836.