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From the epoch of Silurian crinoids to the era of the drift and its included traces of hunan arts, is a transition as vast in point of time as the distances in space which the astronomer reduces to definite figures, but which the mind in vain attempts to realize. Compared with such a transition, the lapse of time from the earliest traces of human art to our modern nineteenth century is brief enough; yet the contrast seems scarcely so great between the organic forms of our lower silurian rocks, and the mammals of the drift, as that which separates the first rude evidences of human ingenuity in the latter formation, from such triumphs of mechanical skill as the “Great Eastern ” of the Thames, or the “ Victoria Tubular Bridge” of our own St. Lawrence. The great acbievement of mechanical science and fearless enterprise embodied in the gigantic structure which now spans the wide waters of the St. Lawrence, and has been opened for traffic since last we assembled here, is the crowning feature of that arterial system of railways which well nigh annihilates for us the impediments of time and space and is already revolutionizing our whole relations of commercial and social life.

It is impossible, however, to revert to either of those wonderful triumphs of mechanical science, without also recalling the painful coincidence that, alike in the Great Eastern Steam Ship and the Victoria Bridge, the inventive genius that had planned and directed each, throughout all the stages of its progress towards completion, was snatched away when seemingly on the eve of realizing his most cherished hopes. The death of Robert Stephenson, at the too early age of fifty-one, only a few weeks before the completion of that colossal creation of his genius which constitutes, not for Canada only, but for the world at large, one of the fittest memorials of the great Engineer, has already been referred to in the Annual Report of the Council : for, honored by ranking him among our Honorary Members, the Canadian Institute claims her share in the loss occasioned by the death of him whose remains have been laid amid the royal and noble dead of Westminster Abbey, with marks of distinction and tokens of public sorrow, rarely accorded but to such combinations of genius and great personal worth.

Your attention has been recalled by the interesting communication of Dr. Rae, to the latest results of Arctic discovery, which, while clearing up all mystery as to the fate of the lamented Franklin, ranks him in one sense among those whose loss we have anew mourned during

the past year. Permit me, in thus referring to the honored name of Franklin, to couple with it that of a personal friend, Mr. Henry Goodsir, formerly Curator of the Royal College of Surgeons of Edinburgh, who volunteered his services as Naturalist of the Franklin Expedition, and has doubtless perished, like his chief, though we lack the poor consolation of even learning his fate. I have watched with liveliest interest each detailed account of the relics of that ill-fated expedition, in the hope of recognising traces of one, not the least gifted or worthy among those whom Britain justly mourns. A young, enthusiastic, and highly gifted student of science: Henry Goodsir has fallen on a field more honorable, and striving in a nobler cause than most of those which furnish the laurels of heroes. Yet it is impossible not to revert with mournful regret to the ardent, sanguine votary of science, thus perishing before one desire had been accomplished, or one hope realized; going forth with the accumulated knowledge that constituted his weapons for that dread field, like the young soldier ardent for the strife:

"And lost to life, and use, and name, and fame." It is a duty which generally devolves on the President of a Society like this, to commemorate on such occasions, those whose loss we have to lament during the past year ; for, alas, no year passes over us, in which we have not to mourn some blank which death has made in our own numbers, or in that great Commonwealth of Science and Letters in which we claim to take our humble part. Among the ranks of our own members death has removed some who were wont to take a lively interest in our proceedings; and all of us, I doubt not, have deeply sympathised in the very painful circumstances which attended the loss of one of our number, the only son of His Excellency, Sir Edmund Head : a youth of great promise, and of rare enthusiasm in his early devotion to science. And when we look abroad on that wider circle which our sympathies embrace, we see that the Old World and the New have shared with an impartial equality in death's irrevocable bereavements. Hallam and Prescott, Brunel and Stephenson, De Quincy and Washington Irving, have, during the past year, followed one another to the grave; and it will not, I trust, be deemed an intrusion on the special duties of this occasion, if I turn aside for a moment to refer to another loss which science has recently sustained, but in which I claim a larger personal share. Death has been busy of late among Edinburgh men whom I counted my personal

Vol. V.

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friends. Dr. Samuel Brown, Professor Edward Forbes, and Hugh Miller, have followed one another to the grave within a brief period, and ere the past year drew to a close, Dr. George Wilson was added to the number of those who live only in honored memory. Dying at the early age of forty-one, when a career full of rich promise appeared only opening before him, and his mind seemed to be ripening in many ways for a great life-work : those who knew his capacity and his genius regard all that he had accomplished as insignificant indeed when compared with what he would have done if spared to those years in which men chiefly fulfil the promises of youth. Yet what he did accomplish, amid many and sore impediments to progress, is neither poor nor of small amount. Nor is it a light thing now to remember that one whose years of public life have been so few, and even these encroached on by the ever increasing impediments of failing health, has been laid in his grave amid demonstrations of public sorrow such as have rarely indeed been accorded, in that native city of his, to Edinburgh's greatest men. This was due even more to the genial kindliness and worth of a noble Christian man, than to the unwearied zeal of a popular public teacher, and an enthusiastic student of science. His loss to his university is great, but to his friends it is irreparable. In him the faith of science, and the nobler faith of the Christian, were blended into perfect harmony; for no doubt springing from halfrevealed truths of science ever marred the serene joy of his faith while looking at the things which are not seen. Prejudice and falsehood, ignorance and vice, were felt by him to be the common foes of both ; and pardon me, if I add, that no man I have ever known carried more genially and unobtrusively, yet more thoroughly, his earnest Christian faith into all the daily business and the duties of life.

When a man of such genuine kindliness and worth is suddenly called away in his prime, with still so much of his life-work seemingly waiting its accomplishment, it is as when a brave vessel founders in mid-ocean. The wild eddy of the troubled waters gathers around the fatal gulf, and a cry of sympathetic sorrow rises up as the news is borne along to distant shores. But the ocean settles back to its wonted flow where that gallant bark went down, and the busy world soon returns to its old absorbing occupations. But there are those to whom that foundered bark has been the shipwreck of a life's hopes'; and to me the loss of my life-long friend and brother will make life's future years wear a shadow they could never wear before.

But, Gentlemen, I trespass on the privileges of this chair. Let it be my apology to you that the event I mourn is—from accidental cir. cumstances,--peculiarly associated with this meeting and your choice of me as your President. Permit me, in closing an address already too protracted, in which I have aimed at indicating some of those lines of abstract thought whereby science is enlarging our views and widening our sphere of knowledge, to invite you, as in a sense the selfconstituted acolytes in this temple of Canadian science, to enter with renewed energy and devotion on the work of another year : remembering, each one of us, that we know not how few our years of work may be. We may indeed—in a far more absolute and literal sense than Newton could,--say, after all our work is accomplished, that we "seem to have been only like a boy playing on the sea-shore, and diverting himself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before us.” But yet let us remember this at least, that that great ocean of truth does lie before us, and even those pebbles which our puerile labours gather on its shore, may include here and there a gem of purest ray; and meanwhile the search for truth, and even the play along the pleasant shores of its great unexplored ocean, will bring to each one of us his own exceeding great reward.

RESOLUTION OF ALGEBRAICAL EQUATIONS,

(Continued from the last Number of the Journal.)

BY THE REV. GEORGE PAXTON YOUNG, M.A.,
PROFESSOR OF LOGIC AND METAPIYSICS IN KNOX'S COLLEGE, TORONTO.

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PROPOSITION VI. If all the cognate functions (not necessarily unequal) off (), an integral function of a variable p, be, 01, 02, 03,

(1) and if

X = (30-01) (x-02) ...... (x-om)
= m + A, 3m-1 + A, xm-2 +

+ A
;

(2) then the coefficients A1, A2, &c., may be exhibited as rational ex. pressions, that is, (see Def. 1), rational functions of p.

m

For take $, the general symbol under which are included all the particular terms in the series (1); and let the nth power of 0, (n being a whole number), arranged so as to satisfy the conditions of Def. 8, be, on = a + aj ti + a2 t2 + &c. ;

(3) where the coefficients, a, ai, &c., are rational ; and each of the terms, ti, t2, &c., is either some power of an integral surd, or the continued product of several such powers. Suppose yi to be one of the factors of tı; the index of the surd yi being - ; and let the

1-1 several " roots of unity be, 1, 2, ?, ...... ,

Then, from (3), 01 = a + 21 01 + a2 V2 + &c.,

r

th

n

[blocks in formation]

n

n

n

n

n

om = a + aj W1 + a2 w2 + &c.; where vi, UI, &c., are what tı becomes in passing from $ to 01, 02, &c.; and so of the other terms. Therefore, (6") = $1 + 02 + + om +ai (21 tu1 +...+1) + &c, + ai 3 (ti) + &c.;

(4) where, just as 3 (on) represents the sum of the terms, 01, 02,

Am, so 3 (ti) represents the sum of the terms, vi , 1, ... , Wi. Now, in the series, vi, U, &c., if any term vi be fixed upon,

there are a terms, including vi , of the forms,

V1, 201, z2 01 The sum of these is zero. Strike these terms out of 3(ti); and then, in the same manner, whatever term among those remaining in 3 (ty) be considered, it may be demonstrated to be one of a group whose sum is zero. And so on.

Therefore 3 (ti) is zero In like manner all the terms on the right hand side of equation (4), except the first, or ma, must vanish. Consequently, søn) is rational. If now we put

Si = 01 + 02 + + om,
S2 = 01 + 02 + + om,

21-1 vi:

2

2

[blocks in formation]

3

S3 = $1 + $2+

+ om,

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