From Algebra and Geometry combined sprang Trigonometry, or the arithmetic of sines. In this beautiful branch of Mathesis the chief machine for producing results consists of the three words sine, tangent, and secant. These words stand for numerical quotients obtained from dividing one side of a right triangle by another side. Their value evidently depends on the shape of the triangle, and not upon its size. By means of these quotients arranged in a table for every variety of right triangle, the calculation of all figures which can be subdivided into triangles is rendered exceedingly simple. The essence of Analytical Geometry consists in conceiving lines and surfaces as the dwelling-place of points, and then expressing by algebraic language the whereabouts of a point in a manner which shall be at once loose enough to allow the point to be in any part of the line or surface, and yet strict enough to forbid its being anywhere else. The Differential and Integral Calculus imagines the point as wandering about in its prescribed limits, and from the law of its motion at each instant of its path would detect the peculiar character of the line in which it moves, or from the character of the line deduce the law of the motion of the point. No person would infer from these definitions of wellknown branches of mathematics the true extent of their field, or the energy and yet versatility of their powers. Let none therefore think slightingly of Hamilton's Quaternions from the description which we will now attempt to give. A line in its full signification does not signify merely length, but direction also. The distance from Boston to Albany is not only 200 miles, but 200 miles west. In like manner, from Albany to New York is 160 miles south. The sum of the two journeys is, therefore, 240 miles southwest. In like manner, a Bostonian, wishing to go to New York, and going to Albany, would have subtracted 200 miles of westing from his 240 miles southwest journey, and would have a remainder of 160 miles southern journey to perform. These are the new significations which Quaternions impose upon the words Addition and Subtraction. Two sides of a triangle are in this calculus equal to the third, and the difference of two is likewise equal to the third. But it is the new interpretation of Division that gives a name to Hamilton's new mathematical engine. A quotient is the representation of the ratio or relation of two quantities to each other. But the relation of two sides of a triangle to each other requires for its complete expression at least four (quater) numbers. One number is required to express the relative length of the two sides; another to express the angle which they make with each other; and two more are required to express the position of the plane of the triangle, what angle, for instance, it makes with the meridian and what with the horizon. It is simply the patient working out of this fundamental idea, that a line has direction as well as length, that has built this eternal monument to the glory of Hamilton of Dublin, which must command the admiration even of those who appreciate more easily the labors of Hamilton of Edinburgh. ART. IX.-Life Pictures: from a Pastor's Note-Book. By WE have not taken this book as our text, because it is in every, or even in any, respect its author's most noteworthy book. But it is his latest, and he has been so long and so usefully before the literary world, that we should do ourselves injustice in not availing ourselves of the opportunity he now of fers us of rendering him justice, though tardy. His "Christ in History," which we characterized in a "critical notice" of half a dozen lines, might have afforded an easier theme for a journal precluded, equally by our own sense of fitness and the just expectations of our public, from discussions which trench upon the grounds of controversial theology. In that work there was not a leading view or pervading sentiment, and hardly a subsidiary thought, which could fail of the entire assent of all Christian readers, while in the work now in hand the author's own creed and ritual, though not obtrusively urged, are necessarily made somewhat prominent. The first thought that suggests itself concerning Dr. Turnbull's style is, that it eludes criticism. Apart from the ideas and sentiments which it expresses, it has no striking traits that demand either praise or censure; unless this negative statement be in itself the highest praise. His diction is never inflated, or ambitious, or redundant; for he evidently never writes for the sake of airing his vocabulary, or with the purpose of attracting notice or acquiring reputation merely as On the other hand, there are no marks of negligence and no sins against good taste; for these cannot be even in the rapid composition of a man whose native powers and liberal culture conspire to give him free command of language. In fine, Dr. Turnbull's sole aim evidently is the direct and intelligible expression of his thought, the transfer by the simplest means possible of what lies in his own mind to the mind of his reader. In the singleness of this aim, he perhaps sometimes omits legitimate and graceful modes of persuasion and impression, glances from a topic before he has exhausted its wealth of argument or motive, and makes less than he easily might of openings for appeal to the fancy and the emotional nature. We are occasionally annoyed by the brevity of a strain which we would gladly have prolonged, and especially by the very cursory treatment of scenes or events in themselves deeply pathetic; and yet we are inclined to think it the more manly part for the author to leave his readers to do their own weeping, instead of playing Melpomene at every hand's turn. Dr. Turnbull is a strong and earnest worker in the highest department of thought and effort. He has faith in Christianity, not only as the Gospel of redemption to the isolated soul, but as a regenerating force, whose development has given and will give the key-note to all history, and whose workings, whether traced in the great heart of humanity along the ages, or analyzed in the life-experience of the individual, are the surest evidence of its Divine source and mission. He is earnestly solicitous to exalt this force to its due place in the esteem of influential minds, and to imbue with its spirit those whose action is to mould the nearer, and thus the more distant, future of our country and our common Christendom. This aim, so broad and high, is of course incompatible with sectarianism, or with other than the most comprehensive views and the most catholic sympathies. His books are thus adapted to win the interest, sustain the faith, and stimulate the courage, of Christians of every denomination, and can in no sense be deemed the exclusive concern or property of those appertaining to his own section of the Church. And here we ought to explain what we said with reference to the incidentally denominational character of the book now under review. It is a record of facts and conversations that have fallen within his own knowledge and experience; and, as he is a Calvinistic Baptist, of course his walks of social intercourse and professional duty have for the most part either lain within or led to the fold to which his own pastorate belongs. His specific purpose in this work we can best define by quoting a portion of his Preface. "The object of the following 'Life Pictures,' including narratives, conversations, letters, and so forth, is to bring out, in concrete form, the true idea of the inner or divine life. "It is designed especially for inquiring minds, haunted, perhaps, by the prevalent scepticism, or by other forms of doubt. "The spiritual life is ever a struggle with opposing elements; and in almost every mind there comes a period of doubt and conflict, always painful, sometimes appalling. This is the case especially with strong, contemplative natures, who revolve the problem of life in its deeper significance. "Such are to be found in all spheres, whether of professional or of business life; and often, when we least suspect it, in the bosom of our families. "Doubt, indeed, is the disease of this inquisitive, restless age. It is the price we pay for our advanced intelligence and civilization. It is the dim night of our resplendent day. But as the most beautiful light is born of darkness, so the faith which springs from conflict is often the strongest and best. "From an intimate acquaintance, during a pastorate of a quarter of a century, with a great variety of minds, many of them reclaimed from scepticism, the author supposed he might prepare a work which would be useful in this direction. The experience of some of these reclaimed sceptics is given in this volume. "Other sketches and illustrations are added, to bring out, as vividly as possible, the various phases of Christian experience, from its beginning in regeneration to its consummation in glory. The author believes that it is our privilege and duty, as Christians, to rise into a higher plane of life than is generally hoped for in the present day. He is persuaded, also, that the true idea of Christianity, as a vital power, is the best antidote to infidelity. To aid in the attainment of that 'perfect love,' which casteth out' both doubt and fear, is one aim of this work; for in this consists the essence of the true life, which is ever a vital divine force or germ, the invariable tendency of which must be to beauty, fruitfulness, and joy. As it comes from God, it ascends to God. Hence it is fitly described as a life in God,' ever blessed and immortal." - pp. iii. - V. In pursuance of this design, the author has evidently kept himself within the limits of fact, not only as regards the main outlines of each sketch, but as to the details in which fancy might often have striven to usurp the place of memory. The characters are not idealized, but are such as we have all met with, some of them commonplace characters, except as brought out in strong individuality by peculiar religious experiences, others of them persons of rare gifts and merits, yet like the eminently good whom we have known, and not like the saints of fiction whom we never wish to know. The conversations in which the writer professes to have borne part are such as we can have no doubt took place; and so far are they from being compressed into the prim, artificial moulds in which book-dialogues are often cast, that we can hardly conceive of such talks being written down except from memory. Objections and difficulties are met in these colloquies with great skill indeed, but in the off-hand way in which a man of the author's large resources might have met them without specific preparation, not with the elaborate and exhausting treatment which they would have received in his study. The book would have been more exciting, had Dr. Turnbull drawn upon his imagination for characters and incidents; but to us it is immeasurably more interesting and valuable, as a transcript of fact and experience, than had it been merely his own opinions and speculations dramatized. We are thankful to Dr. Turnbull for not having given us another volume of religious fiction. We believe works of |