tions of Algebra. Every thing, as far as possible, is submitted to the eye, and hence the conclusions have the evident validity of things seen.

Again as all the operations of Algebra have respect, ultimately, to numbers, Prof. Strong keeps this distinctly in view. Each conclusion, however high or abstract, is applied to some operation in numbers, and seen therefore, in its legitimate scope.

Beginners in Algebra are often not aware of the fact, that in this calculus different quantities are compared together merely in view of numerical relations. When, for example, we compare the cube of a with its square, it is not that a cube has any ratio to a square; but merely that the numbers expressing the peculiar units in the one, may be compared with those expressing the units of the other. We repeat-it is with general expressions of numbers that we have to do in Algebra, and with nothing else.

Once more; as we said in our May number, Prof. Strong has added to the before existing stock of knowledge in this department. To his treatise we may justly apply the words of Edward Burke, when speaking of "difficulty," he says of it: "This it has been the great glory of the great masters in all the arts to confront and overcome; and when they had overcome the first difficulty, to turn it to an instrument of new conquests over new difficulties; thus to enable them to extend the empire of science, and even to push forward beyond the reach of their original thoughts the land-marks of the human understanding itself." Prof. Strong has done this, as, in some particulars, we shall farther show; and hence he deserves a niche in the Temple of Fame, among the benefactors of

our race.

We proceed to designate some particulars, wherein we find either improvements upon the usual modes of treating the subjects, or actual additions to the science itself. Of the solution of the irreducible case of Cubic equations we have spoken before. The occurrence of this case under Cardan's formule was inevitable, on account of certain limitations of the auxiliary quantities employed in the general solution. Of this case, Bonnycastle says, (in his Alg., London, 1820,) "The solution of it, except by a table of series, or by infinite series, has hitherto baffled the united efforts of the most eminent mathematicians of Europe." So it has been to this day; and in virtue of this single achievement, Prof. Strong has earned himself the highest credit as a master in his art.

We find another equally ingenious extension of the science, in the method of extracting the roots of numbers of any degree, by a direct process, and without the aid of logarithms.

This depends primarily upon the proposition, that any quantity of the form one, plus b divided by a, can be resolved into any number of factors of the same form, as appears on p. 288, &c.

For the convenient application of this to all numbers, a resolution of the nine digits, as also the figure 10, into factors of the same form, is wanted, as appears in an Article by Prof. Strong in the April No. of the Mathematical Monthly. Altogether, this is one of the most curious feats that we know.

MISCELLANY.

HUMBOLDT'S LETTERS.*-When David Mallet, a native of Scotland, published the posthumous works of Bolingbroke, Johnson, speaking of the latter to Boswell, said, "Sir, he was a scoundrel and a coward; a scoundrel, for charging a blunderbuss against religion and morality; a coward, because he had no resolution to fire it off himself, but left half a crown to a beggarly Scotchman, to draw the trigger after his death!" An equally severe censure some may be disposed to pronounce upon Humboldt for the infidel remarks which for the first time see the light after his decease. But so hard terms are not, in this case, just; for though the remarks are, some of them, flippant, and none of them either new or profound, they disclose nothing concerning the writer, that was not known before. The pantheistic opinions of the author of Cosmos, are sufficiently patent to a discerning reader of that work, and he has been known in Germany to be a free-thinker if not an Atheist. Nor are we inclined to hold the famous naturalist responsible for the publication of these letters which, for many reasons, should never have been printed. They were written, and such of them as emanated from his distinguished correspondents, given, to his life-long friend, Varnhagen von Ense, and placed without reservation at his disposal. But the friend unexpectedly died first, and they fell into the hands of a female relative who gives them to the world, professing that in so doing she is discharging a sacred duty, it being the express wish of Humboldt that they should be published at his death. We do not find sufficient proof of this statement in the evidence which she brings forward to support it. The principal passage cited in proof is from a letter of Humboldt to

*Letters of Alexander Von Humboldt to Varnhagen Von Ense, from 1827 to 1858. With extracts from Varnhagen's Diaries, and Letters of Varnhagen and others to Humboldt. Translated by FREDERICK KAPP. New York: Rudd & Carleton. 12mo. pp. 407.

Varnhagen, dated the 7th of December, 1841; of which, however, we are furnished only with disjointed clauses. These, in a free but faithful paraphrase, are as follows: "Your last very complimentary letter contains words which I may not misunderstand." "You are hardly content with the exclusive possession of my impieties." "With such property, after my decease, not far distant, you may do what you will. One is bound to open himself during life only to those whom one deeply respects, therefore, to you." Varnhagen, a soldier, diplomatist, and one of the best of the German prose writers of his day, enjoyed Humboldt's unbounded confidence. Rahel, the wife of Varnhagen, was a woman of rare powers and attainments, and with the family, until her decease and after, Humboldt stood in relations of cordial intimacy. It is not unlikely that his intention was to provide materials for a biography of himself to be prepared, at a future day, by his trusted friend, who would know how to use them with discretion. This, at least, is the most charitable hypothesis. But for this solution, we should be obliged to conclude that his vanity, which was not small, degenerated into the vulgar thirst for notoriety, which delights to make a stir at whatever cost, and that instigated by this passion and by petty hatred and resentment towards contemporaries about him, he arranged for the publication of confidential conversations which he was bound in honor to keep private, and of gossiping letters which do no credit either to his head or heart. In every event, one who reads this book is compelled to lower his estimate of the writer's character and intellectual power. He is in correspondence with princes and sets so high a value on their praises, that he preserves and deposits with his friend their flattering epistles, however short and void of interest to any one but himself; at the same time that he ridicules them and bitterly complains, see, for example, what he says of Prince Albert, who had not alluded to Victoria in connection with the Cosmos,-whenever they fail to yield him the desired amount of deference, or to suit him in minor particulars. We are convinced, independently of this volume, that the reputation of Humboldt has exceeded his merits. His extraordinary capacity for acquiring knowledge and his skill in recording his researches in a genial and methodical manner, are undoubted. But he does not, after all, belong in the very first rank of men of science, with Sir Isaac Newton, Kepler, Leibnitz,—with the men who have struck out paths of discovery and whose souls, while engaged in the study of Nature, have been alive to higher truth. It is not to be forgotten that his abilities were aided by remarkable good fortune and uncommon

opportunities for their development and display, and that his fame, which filled the world, was, in a considerable degree, the result of happy circumstances,-of personal qualities which made him attractive to sovereigns, and of extensive explorations on both continents, undertaken at a period when they were comparatively infrequent.

Something may be said in extenuation of Humboldt's unbelief, and of his habit of sneering indiscriminately at the ministers of religion. He was born in the period of the French Revolution; was brought up under the reign of Rationalism, and his situation in the latter part of his life was not propitious for a change of views either upon Christianity or the clergy. In no Protestant state, are the mischiefs of an established religion more palpable than in Prussia. The vast patronage in the hands of the government, which has at its disposal not merely the civil offices but professors' chairs and the schools throughout the kingdom, together with a multitude of parishes, presents a temptation to aspirants for station to conform their thinking to the sentiments prevalent at court. If the court become evangelical, the door to preferment is shut against men of a contrary type of opinion. The consequence is that in the universities and elsewhere, there is much profession of piety which is but half honest, and many in their anxiety to eschew cant and hypocrisy, and to distinguish themselves from the place-hunters, withstand the current views and take the side of unbe

lief. When despotical tendencies rule at court, as of late has been the fact, the spectacle is presented of clergymen zealous for prerogative and vehement in their denunciations of literal notions in politics, or content to enjoy their offices, maintaining silence upon the encroachments of power and the destruction of popular freedom. It is natural that men like Humboldt, a sincere friend of liberty, should look with disgust upon this numerous class of persons, many of whom he was obliged to come in contact with daily. Von Raumer and Hengstenberg, and Bunsen as he was before he lost office, with many other prominent names of men still living, are handled in these letters in a manner which must be anything but pleasing to them. Of the bad taste of what is said of them, if it is supposed to be intended for the public, and the unamiable nature of the observations, even if made to Varnhagan, there ought to be but one opinon. Of the allusions to religion, it is sufficient to say that the author in effect abjures inquiry, as if supernatural things were not an appropriate or possible subject of knowledge, and contents himself with a dogmatic rejection, in a light tone, of the faith of Christendom. Such a course, so at variance with

the spirit of science, not to characterize it more severely, though it cannot be excused, may be regarded as less unworthy when the associations of Humboldt and the peculiarities of the Prussian Court and Church are taken into view.

LETTERS OF JAMES W. ALEXANDER, D. D.*—These volumes were published, it is declared on the title page, to serve as a memoir of the writer, it having been expressly his wish that no formally written life of him should be given to the public. We are not informed how particular or earnest was the injunction that his life should not be written; but we find in this correspondence an expression of his opinions, which is pertinent to the publication of these letters: "When you or I depart this life, the letters of the survivor, (free as they have been about persons who may then be alive,) might prove very mischievous to the surviving party. I think, therefore, we both ought to provide for the return of the letters to the writers or the family of the writers."

But the letters are published, whether with greater or less violence to the supposed or expressed wishes of the author we do not undertake to decide, and they will be variously judged, according to the interest that is felt in the author, and the importance which is attached to his opinions. The reason given for their publication is, that a memoir may be furnished of the growth and characteristics of the inner man, which the editor thinks can be best accomplished by the publication of a familiar and unreserved correspondence from early manhood till death. There can be no doubt that he is correct in this. A still more effectual method would be to give to the public his letters to his wife-all his private and confidential communications concerning family and private interests, and the record of his inmost thoughts and feelings. But a question might arise as to whether the great public has a right to know man's inner life, in such methods, at a cost so great to personal delicacy and private feelings. On this subject we agree with Tennyson, whose scorching words we quote:

"For now the poet cannot die,

Nor leave his music as of old,

But round him, ere he scarce be cold

* Forty Years' Familiar Letters of James W. Alexander, D. D., constituting, with the Notes, a Memoir of his Life. Edited by the surviving correspondent, JOHN HALL, D. D. In two volumes. New York: Charles Scribner. 1860. 8vo. pp. 412, 379.