provements made in the science are not only thus brought from a variety of sources within the reach of every student, but are, also, by the comprehensive judgment of one mind, and the plastic labor of one hand, reduced to unity of form and method. The various subjects are arranged with particular reference to convenience and clearness, and the analyses are happily applied to practical uses. It is not the work of one mind to build up one vast science by its own original thinking; but it is the work of one mind to digest the mass of details into a convenient compendium for the purposes of education. Men of genius, in different countries and periods, slowly do the first: when that great work is completed, some patient laborer, some man of practical experience and foresight, casts up to do the last. This patient laborer is no less a benefactor than those who have prepared the way for him; and how morally great does he appear, if he voluntarily resign the charm and merit of original investigation for the work of making the labors of others available to the community! He might distinguish himself by inventing a new method, by developing a new and recondite analysis; but he prefers to apply to useful ends what has already been successfully accomplished. There are many cotemporary authors who have successfully and praiseworthily labored in this department. Among these, at least, Prof. Hackley must hold an eminent place. To verify, in some measure, our judgment of the merits of his digest, we shall make a few references.

The whole subject of division is presented with great elegance and clearness. The examples are numerous, and selected with a nice judgment in reference to exercising the skill of the pupil. Here he has introduced many examples with literal exponents and literal coefficients: the law of quotients, when they become infinite series, is given: division by detached co-efficients, and the method of synthetic division by Horner, are presented with great simplicity and beauty. We observe here in the margin, a very neat and concise demonstration of Horner's method. Let any one compare this with the demonstration given in Hutton's Mathematics, and the improvement will be obvious.

The subjects of the greatest common measure, and the least common multiple, are properly placed after division, and treated in a manner to make them easily and perfectly intelligible to the young pupil. The whole subject of radicals, the clear understanding of which is so important to the student in the higher equations, is early introduced, and cleared up most successfully by lucid explanations and appropriate examples. Fractional and negative exponents, which are so apt to embarrass the pupil, are here strip* it as mediu

ped of much of their forbidding aspect, and made an intelligible language to ordinary capacity united with diligence.

On pages 64 and 242 imaginary quantities are resolved with great neatness and clearness.

On pages 100-107 the binomial theorem is determined inductively, according to Newton's method, and at once applied to series and roots. On page 108, after the way has been sufficiently prepared, the demonstration is given. This immediate application of the theorem is a happy conception: it gives interest to the theorem itself, and introduces the pupil naturally to a new and important subject. The demonstration on page 108 is given with a rigor which some mathematicians have regarded as impracticable. We cannot avoid calling attention, in connection with this, to the demonstration of the polynomial theorem on page 109-remarkable alike for its conciseness and elegance. Then follows a neat demonstration of the method of extracting the root of a polynomial.

That most important subject, ratios and proportion, a thorough comprehension of which is essential, indeed, to all mathematical reasoning, is treated of with unusual brevity and transparency, and examples subjoined which serve both to apply the doctrines and to convey useful and interesting information.

We have noticed, as improvements in treating of equations, that a variety of letters are employed to represent unknown quantities, and that elimination by common divisor is introduced in simple equations. The general discussion of equations of the first degree, page 173, is exceedingly satisfactory. In connection with this we have presented new symbols of indeterminate equations, page 177, and extended, page 178, to two or more unknown quantities.

The method of undetermined co-efficients is developed with an important improvement: the exponent is not assumed, as is ordinarily done, but is taken indeterminate, also, and then the relations are afterward proved, and the values afterward deduced.

Next in order, the subject of logarithms is taken up, and very lucidly discussed. Here, at once, a short auxiliary table is given for constructing general logarithmic tables, and its theory and use explained. Then follow a variety of analytic exercises in which logarithms are involved. After this the practical use of the tables is copiously explained, and a specimen page from Callét is given at the end of the volume, as an illustration. Here a variety of exercises are appended. Gauss's system of logarithms, designed exclusively for sums and differences, is then introduced; and finally the calculation of the common and Naperian logarithms by series.

Progressions are next discussed. Under this, what may, perhaps, be fitly called the historic origin of logarithms is explained. The general theory of equations is much improved in the notation and brevity of the demonstration. Sturm's celebrated theorem, which is so blind in late treatises which have attempted to explain it, is restored to its native beauty and transparency by following strictly the author's own method. Prof. Hackley has here evidently consulted the original, instead of relying upon second-hand expositions.

Binomial equations, a subject of great importance, but usually slighted, is introduced and amply treated. It is worthy of notice, also, that cubic and biquadratic equations are treated in a very simple and analytic manner, with trigonometrical solutions of both the reducible and the unreducible case.

Thus far we have considered Prof. Hackley's work simply as a digest. In this point of view, indeed, we wish mainly to consider it; for it is in this that its great value consists, and in which the author intended it to consist. But so astute a mathematician must, in spite of his own modest intention, cause his work to be pervaded by lines of original thought, as well as give original modifications to the thoughts of others. We should do injustice to the author, therefore, did we not touch upon his performance in this point of view likewise. We would call attention, therefore, briefly to several particulars. The introduction of Horner's method of synthetic division is an important improvement in itself, but the explication of it is the author's own; we have already alluded to this, and we mention it here again for the sake of remarking its originality.

Indeterminate analysis of the first degree, page 186, and indeterminate analysis of the second degree, page 240, are very creditable examples of the author's original analytical power. The examples subjoined to the first are happily selected-they are not merely curious, but embrace solutions of practical utility. We refer also to maximum and minimum values, page 242. On pages 244-6, the method of Mourey for avoiding imaginary quantities is most ingeniously and clearly explained. In permutations and combinations new forms of notation, exceedingly convenient, are introduced, together with various modifications of the ordinary problems and formulas. The application of this subject to a variety of others, and especially to the calculation of probabilities, is also worthy of notice. The explanation of Gauss's formulas, involving sums and differences, is, we believe, new in an English dress, having been hitherto confined to German works.

After logarithms and progression there is given a very full set of formulas and rules for interest and annuities in which logarithms are applied. These tables are of great practical utility..

The important subject of interpolation is treated of in the best manner we have ever met with: also, that every equation has a root, page 203; and the subject of conjugate equations.

Another improvement is the application of Horner's method of synthetic division to the depression of roots of equations, page 316; and, in connection with Sturm's theorem, to approximations to the roots of higher equations, very rapidly, to any required number of decimal places, pages 334-338. The determination of the imaginary roots of the higher equations, page 384, and the theory of vanishing fractions, are new in an elementary treatise, and ingeniously expounded.

Our author has made improvements on the theory of elimination in higher equations, by Labatie, worthy of remark.

We call attention, also, to difference series, and a most ingenious method of applying them to determining the places of roots in the higher equations, pages 416-18; to the subject of variation, pages 425-7; to the elimination of symmetrical functions, page 436; to a new method of solving the cubic equation, by a young American; and to a simple, but very complete, exposition of the diophantine analysis, page 457.

The work concludes with an article on the theory of numbers, in which an attempt is made to give a brief explication of this extensive subject, both as treated by Legendre and others, and according to the peculiar method of presenting it adopted by Gauss. The nature of primitive roots is explained, and the Gauss method, depending on them, of solving binomial equations of all degrees. Full references are here given to larger works.

The subjects here referred to will be found generally to contain much that is new to ordinary students, extracted from eminent mathematicians of different countries, and pervaded by the author's original conceptions and modifications.

The critique we have ventured to make on Prof. Hackley's Algebra, we confess, aims rather to point out its excllences than to seek for its defects. This last and less gracious work we will leave to other hands. We believe that where a work has commanding merits, a greater favor is done to the public on its first introduction by leading them properly to appreciate it, than by engaging their attention to curious criticisms upon doubtful points, or by making a parade of the reviewer's skill in noticing narrowly those defects which are incidental to the best attempts. We are

decidedly of opinion, also, that a candid and thorough examination of the work will bring before the mind so much to admire and commend, that, as in our own case, there will be little disposition to mark faults which the author's own judgment and skill are adequate to correct in subsequent editions.

ART. VI.-Twenty-seventh Annual Report of the Missionary Society of the Methodist Episcopal Church.

WERE We called upon to designate that event upon which future ages are likely to look back as vastly the most important in the history of the last hundred years, we should refer to the revival and new development of missionary enterprise. This opinion we should announce without hesitation; having at the same time a lively recollection of other stupendous facts, which have made the period referred to one of the most memorable of the great historical epochs embraced in the annals of our race-of the struggle of our forefathers with the power of Britain, which gave birth to a great nation, and ushered in a new political, social, and religious economy-of the French Revolution, which swept with volcanic fury over half the civilized world, overturning and rearing thrones, subverting and re-constructing human society throughout enlightened empires-of the Reform Bill and Catholic emancipation, which have resulted in making essentially popular the essentially aristocratic government of the most wealthy and powerful nation on earth-of the abolition of slavery in the West Indies, by which eight hundred thousand bondmen were made free-of the extension of the British East Indian empire over a population of one hundred and twenty millions-of the introduction of China into the family of nations, and the free advent into the bosom of her incredible population, which may thus be given to social, moral, and economic meliorations, hitherto unknown and impossible to her narrow, bigoted civilization. We by no means affect to undervalue the importance of these great events, which have produced radical and durable, and, we verily believe, beneficial changes in the lot of the largest portion of the human family. We are quite satisfied that ours is, all things considered, the happiest of countries. We think it demonstrable that Frenchmen of the present generation, as well as the people of several other European nations, are enjoying the good fruits of their bloody revolution. We are believers in human progress, and wait hopefully to see the emancipated slaves of Ja