Art. XIII.—The Mathematical Diary, containing New Research es and Improvements in Mathematics ; with Collections of Questions, &c. &c. In Quarterly Numbers. Conducted by ROBERT ADRAIN, LL. D. Professor of Mathematics and Natural Philosophy in Columbia College, New-York. Nos. I, II. New-York. James Ryan.* 1825. . We have attentively examined the first and second numbers of the above little work, and feel convinced that it will abundantly deserve all the success which, under the auspices of the accomplished mathematician who conducts it, it has a very fair prospect of attaining. Its object is to furnish to all those who have a fondness for mathematical pursuits, a convenient vehicle for their inquiries, speculations, and, at the same time, to excite the curiosity, and stimulate the industry of the contributors by proposing, in each number, a variety of interesting questions adapted to all capacities and tastes. These appeals to the ingenuity of students are known to have a powerful influence in promoting application and inquiry; and in this respect, mathematics has an obvious advantage over every other branch of human science. . In order that the method of mutual interrogation may be useful, it is absolutely necessary that the questions should be rendered interesting ; and this is never so well effected, as when they are such as lead to definite and demonstrative solutions. Questions of this sort have all the characteristic interest of enigmas and games of calculation, with the great additional attraction derived from a consideration * We take this opportunity to recommend to mathematical instructors, Mr. Ryan's Elementary Treatise on Algebra. It appears to be prepared with great care and discrimination, from the very best sources, preserving such of the materials of his authorities, as are best calculated to render the book at once practically useful, and theoretically accurate; and keeping out of view such .speculations, as would only lead the student from the line of progress, rendered necessary by our modes of education. Besides a great deal of valuable matter, not to be found in the treatises generally in use, Mr. Ryan's book contains an appendix, by Dr. Adrain, exhibiting an algebraic method of demonstrating the propositions in the fifth book of Euclid's Elements, according to the text and arrangement in Simson's edition. Notwithstanding all that has been said of the force and perspicuity of geometrical demonstration, we cannot help believing that the experience of teachers will convince them, that the fundamental principles of ratio and proportion, may be far more effectually taught by the simple system here laid down, than by the tedious and verbose methods commonly pursued. of the value of their results. Accordingly we find, that in proportion to the number who cultivate the science, more mathematical journals have been well supported than any other kind of periodical work whatever. The English Ladies Diary was commenced in 1704, and has continued without interruption to the present time. The Gentlemen's Diary, the Mathematical Repository of Leybourn, the Miscellanea Curiosa of Dr. Hutton, the Mathematical Companion, and the Belfast Almanac, are also instances of successful journals, supported by the peculiar interest which mathematical speculations possess, for those whose tastes and habits incline them to such studies. Even in this country, where abstract science has never been much cultivated, several publications have from time to time appeared, which have fallen through, rather from mismanagement, or want of perseverance, than from want of adequate support. We are, therefore, glad to see another effort made to facilitate the prosecution and general dissemination of mathematical science, because we are convinced that there exists a natural demand for this kind of knowledge, quite sufficient to meet the proposed supply. The articles in No. I. of the Mathematical Diary, are, first, an Essay on the Rectification and Quadrature of the Circle, a curious and by no means unimportant subject, which the editor of the journal (for to him we venture to ascribe the paper) has discussed with great clearness and ability. The historical account of the successive efforts made by geometricians to square the circle, as it is called, will prove interesting even to the uninitiated reader. We look with much impatience for the promised continuation of this essay, and particularly for the new method of approximation, to which the present paper seems intended as an introduction. The second article is a review of Venturoli's Theory of Mechanics, giving an outline of its plan, and a concise enumeration of its merits. The writer points out, in the fifth chapter of the book, a curious oversight, which seems to have escaped the notice of the translator. These articles are followed by a list of twenty well selected questions of various degrees of difficulty, and in different departments of the science. Number II. contains the solutions to the questions of No. I. with eighteen new questions to be resolved in No. III. The solutions to the thirteenth of No. I. requiring the greatest rectangle inscribable in one of the nodi of a given lemniscate, (a very pretty problem, by the way, and which we are very glad to see extended to the higher orders of lemniscates,) are by VOL. 1. 19 Professor Strong, of Hamilton College, and Dr. Bowditch, of Boston. The answer, by the editor, to the sixteenth question, is recommended to the attention of teachers of navigation. It shows conclusively that the fundamental stating of plane sailing is erroneous, when the actual figure of the earth is taken into consideration, or else, that the term, difference of latitude, must be wrested from its only true and appropriate meaning. Professor Strong and Dr. Bowditch sent the only solutions to the nineteenth question, (by Professor Adrain,) requiring the nature of the curve described by a body projected obliquely along a given inclined plane, when the resistance is uniform.* Dr. Bowditch's solution is a master-piece of science, taking up the question on its fundamental principles, without the aid of formulæ previously demonstrated, and pursuing it with admirable ingenuity through all its modifications, varieties, and consequences. The prize solution to a very elegant question in plane geometry, proposed by Mr. Fleming, is also by Dr. Bowditch, and is a fine specimen of skilful geometrical analysis. The questions in No. II. are marked by the same variety of difficulty, as those of No. I. Some of them are so easy as to be within the reach of the least experienced analyst, while others rise to the more elevated regions of the science. Among the latter, we may remark the extension of the thirteenth question of No. I. to a higher order of lemniscates, and a beautiful prize question by Professor Adrain. A mathematician's genius, it is said, is sometimes as well shown by his questions as by his solutions. Certainly no one, who has any pretensions to science, will read the question we have last mentioned, without being forcibly struck with the speculative ingenuity of the proposer. We need scarcely repeat, that the Mathematical Diary has our best wishes for its success. Art. XIV.-The Duties of an American Citizen. Two dis courses delivered in the first Baptist Meeting House in Boston, on Thursday, April 7, 1825, the day of public fast. By Francis WAYLAND, jun. Pastor of first Baptist Church in Boston. Boston. James Loring, 1825. These discourses are an example of those speculations, which are daily growing more and more common, both in this country * It is a little singular that this case of uniform resistance, (of friction for example,) although it is the only one of any practical use, has been altogether overlooked by the English mathematicians, and in Europe, respecting the intellectual, political, and moral progress of the age. The great impulse communicated to society at the revival of letters, and which has never ceased to act, but has continually increased in activity and strength, is now grown so rapid and powerful, as to make itself felt by the most inattentive observer. Formerly we were in the habit of comparing large portions of time with each other—the ancients with the moderns, the days of Grecian and Roman civilization with the middle ages, and these again with the age which has succeeded them. Now we do not content ourselves, with even comparing century with century,—we cannot be satisfied without noting the changes of society from year to year. The denomination of Baptists, to which Mr. Wayland belongs, although numbering among its divines some very learned and able men, has been accused, and with some appearance of reason, of neglecting, and even of discouraging, the acquisition of general literature among its clergy. To this charge, if it be well founded, Mr. Wayland is an honorable exception. It is delightful to see such a man laying aside, for a time, the peculiar dialect of his sect, and entering into the discussion of subjects, connected with the general welfare and happiness of the community. This he has done in the present instance with great ability. From considering, in the first place, the present intellectual and political condition of the nations of Europe, he passes to the examination of our own relations with the old world, and thence deduces the duties, which, as citizens of the United States, we owe to our country, and to the present age. Mr. Wayland rightly divides the inhabitants of Europe into two great classes—those who support a government of will, and those who desire a government of law—the enemies and the friends of political and religious toleration. It seems to us, however, that he has fallen into some inaccuracy, in denominating the former the Catholic, and the latter the Protestant party. It is not always among the Protestants that the conscience is left free, nor is the enlightened Catholic the necessary enemy of toleration. No particular set of doctrines is certain to protect a people against civil or religious despotism. The tyranny which oppresses the Catholics of Ireland is as wicked and inexcusable, as that which crushes the liberals of Spain. Art. XV.--Scena Quarta dell' Atto Quinto di Adad, Poema Drammatico, del Signor Giacomo A. Hillhouse. Tradotta in Verso Italiano da L. Da Ponte. New-York. Stampatori Gray e Bunce. 1825. This is an Italian translation of the fourth scene of the fifth act of Mr. Hillhouse's last drama. Mr. Da Ponte has shown great taste and judgment in the selection of this scene, because, besides its intrinsic merit and independent interest, it exhibits in one view most of the characteristic features of the author's style and manner. Our readers will recollect that this is the part where Hadad, with all the ardor and eloquence of love, urges Tamar to desert her father's court, and fly with him to some “ far peaceful shore.” The whole of this scene is rendered with remarkable fidelity, fluency, and force. The design of the original is fully seized by the translator, and its spirit extremely well sustained. What is remarkable, and very unusual in versions from the English, there is something very Italian in the whole effect of this translation. This is partly to be ascribed, to what was probably an unintentional approach on the part of Mr. Hillhouse, to the Italian dramatic style, and partly to the address of the translator, who has so well availed himself of this resemblance, that scarcely a trace of foreign manner is left in what he has done. The lines enumerating the unearthly splendors with which Hadad seeks to tempt the wondering and agitated girl, are beautifully rendered ; and, indeed, throughout, Mr. Da Ponte shows himself completely master of that portion of his art, which consists in rising and subsiding with the inequalities of his author. Hadad's description of the progress of his passion, has all the fire and fervid vehemence of the original. . “Adad. Alle porte '. |