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sphere, that there is little reason to desire or to expect that it will ever be superseded. Mr. Colburn published also a Sequel to Mental Arithmetic, in which the principles and rules of written arithmetic were deduced from the solution and analysis of ques. tions according to the method adopted in the former treatise. This Sequel was very well executed as far as it went; but it was not full enough for all the wants of the higher classes in our schools. It omitted proportion and progression, the “Rule of Three," and the doctrine of powers and roots. Mr. Adams has undertaken to supply these deficiencies, following mainly in the track of Mr. Colburn, but appearing fully competent also to mark out a path for himself. By this enlargement of plan, he has brought many useful problems in mensuration and mechanics within the scope of his work, and has extended the analysis and induction over much new ground, though many of the new problems are still left to be performed by arbitrary rules.

The first part of Mr. Adams's book consists of exercises in mental arithmetic, arranged under the different arithmetical rules. Where the principles have not been taught in the First Lessons, they are here carefully deduced from an analysis of a number of simple questions, following which are numerous and well selected examples. These examples pass gradually from simple to more complicated questions, so as to give the pupil a thorough training. In the second part, the different processes are arranged in the same order as before ; and when the operations are compli. cated, distinct rules are given, illustrated by examples for practice containing larger numbers than were suitable for the exclusively mental operation. When the operations are simple, and suffi. ciently explained in the analysis and induction contained in the first part, a reference is merely made to that part, and the exam. ples for practice follow, without any enunciation of a rule.

The author's reasoning and explanations are very clear, simple, and concise, his disposition of the different parts judicious, and his selection of examples well suited to exercise the mind of the pupil. As a whole, we prefer this work to any arithmetic we have seen in use. Still, there are a few things in it that we think susceptible of improvement. Among these are the modes of proving the sums in addition and multiplication. In the former case, we prefer the method of Lacroix to that adopted by our author ; and in the latter, Mr. Adams's mode, strictly speaking, is no proof at all. The methods of multiplying and dividing by the factors of a number are nowhere taught in this work, though they are often found extremely convenient. The rule for reducing any number of pence and farthings to the decimal of a pound " by inspection," as it is usually called, is also omitted ;

and the rule for expressing shillings as decimals is not explained so clearly as it might be.

Some other particulars might be pointed out, in which a slight alteration for the better might probably be effected. But, speak. ing generally, the deficiencies are neither numerous nor impor. tant; and we should not allude to them at all, except with a view to stimulate Mr. Adams to make his work still more full, and more deserving of the patronage which it merits, and will un. doubtedly receive.

2. - Sermons. By GEORGE W. BETHUNE, Minister of the

Third Reformed Dutch Church. Philadelphia: Mentz &
Rovoudt. 1846. 8vo. pp. 301.

The author of these sermons deserves well of our literary community. A clergyman of sincere and ardent piety, and zealously devoted to the pastoral charge of a large congregation in Philadelphia, he yet finds time for the assiduous cultivation of good literature, and in that capacity appears frequently, and always honorably, before the public. His occasional verses are very pleasing, from their religious feeling and purity and elevation of sentiment, as well as for their expression of warm domestic af. fection. He has published several anniversary discourses, which are uniformly characterized by a sound literary judgment, a generous enthusiasm in favor of high aims and purposes, a manly independence in the expression of sentiment, and a flowing eloquence of style.

This volume of sermons, whose beautiful exterior does honor to the Philadelphia press, is published, as the author states in a modest and appropriate preface, in compliance with the wishes of some friends, " in the hope that, by the blessing of God, they may do good.” They are fourteen in number, and embrace a variety of subjects both doctrinal and practical. Like all sermons, they are to be regarded in a double aspect; first, as expressions of theological opinion, and secondly, as literary compositions. In their first aspect or capacity, we, of course, do not expect to deal with them, except to say, that, while the writer states his own views with great frankness and earnestness, he does so in no of. fensive or dogmatical spirit.

As literary compositions, they have some decided merits. Their style is glowing, animated, and stirring. The preacher is at earnest in his work. They flow from an ardent temperament, and have more of fervor and unction than is usual among our

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3. — Tables of Bearings, Distances, Latitudes, Longitudes, foc.,

ascertained by the Astronomical and Trigonometrical Survey of Massachusetts. Published agreeably to a Resolve of the General Court, by John G. PALFREY, Secre. tary of the Commonwealth. Boston: Dutton & Went. worth, State Printers. 1846. 8vo. pp. 73.

In an article upon the trigonometrical survey of this State, * our readers were informed, that “ the legislature had ordered to be printed and distributed to the different towns and clerks of courts the positions and details of the stations throughout the State, as determined by the trigonometrical survey, accompanied by such other matter obtained in executing the work as may be useful in laying out roads, and in the measurement of towns. The preparation of this work, requiring considerable labor and judgment, devolves upon Dr. Palfrey, the Secretary of State." This publication is now before us, and an enumeration of its contents will not be useless, if it helps to call the attention of surveyors, engineers, astronomers, and mariners to what may be in a high degree serviceable to them in their various pursuits.

In constructing the State map, Mr. Borden, “ for greater ac. curacy and convenience," divided the State into five sections.

North American Review, Vol. LXI., pp. 467, 468.

For each of these sections there are four tables, in which are given the relative positions and distances of the primary and secondary points, the latitudes and longitudes of the same stations, and their bearings and distances from an assumed principal point of the triangulation. The explanations necessary to make these tables available to the most unpractised are supplied, together with examples of their use. The “ three-point problem," which may be so advantageously employed in connection with these tables to determine the position of the observer when three sec. tions forming a triangle are in sight, is also explained and applied. The primary and secondary lines are so numerous, and may be multiplied to such an extent by means of the third and fourth tables of each section, that a surveyor may now almost always find a known azimuth for determining the variation of the needle, and the local attraction, if any exists. The method of doing this is fully displayed.

In the introduction to this compilation, we find a formula for computing the bearings and distances where the latitudes and longitudes only are expressed, another for computing the latitudes and longitudes of points not heretofore calculated, and a general formula for computing latitudes and longitudes on the surface. sphere. As a necessary accompaniment to the second formula, tables are added containing the values, in feet and hundredths, for every change of minute in latitude, both on the meridian and on the parallels between 43° 1' and 410 12. The values of the constants in the last formula are also given, by which any person is enabled to use it easily. Mr. Borden's determination of the figure and magnitude of the earth from the data afforded by the Massachusetts survey, and the comparison of these elements with those which M. Bessel deduced from the mean of ten trigonometrical surveys measured in different parts of the world, give ad. ditional scientific interest and value to this part of the work.

We were aware of the labor and judgment necessary to ac. complish this work under the most favorable circumstances; and many unforeseen difficulties have increased both the care and responsibility of the editor. They have been overcome with so much success, that we believe, after careful examination, that no error of any importance is to be found in it, except the omission by the engraver of an important line from the south end of the base to Mount Lincoln.

Dr. Palfrey, however, in his “ Advertisement,” has repeated a statement once made in our own pages, and since found to be en. tirely incorrect. In the paper on the Massachusetts survey already referred to, we said that the base apparatus used by Mr. Borden had been adopted in the United States coast survey. The assertion

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in both instances was made upon the same authority; and it was an authority quite sufficient, if we consider only the position of our informant, and his right or duty to be acquainted with the subject. Therefore, though we were misled, it was not in consequence of any carelessness or indifference. We improve the first occasion to correct the error, and to say, that, when Dr. Bache, the superintendent of the coast survey, publishes a description of the base apparatus designed and used by himself, it will be found to differ in many essential respects from the one invented by Mr. Borden. In saying this, we do not, of course, intend to compare the merits of the two plans.

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