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THE last English edition of Hutton's Course of Mathematics, in three polumes octavo, may be considered as one of the best systems of Mathematics in the English language. Its great excellence consists in the judicious selection made by the authors of the work, who have constantly aimed at such things as are most necessary in the useful arts of life. To this may be added the easy and perspicuous manner in which the subject is treated—a quality of primary importance in a treatise intended for beginners, and containing the elements of science.

The third volume of the English edition having been but lately published, is scarcely known at present in this country, it is but justice to its excellent authors to state, that they have collected in it a great number of the most interesting subjects in Analytical and Mechanical Science. Analytical Trigonometry, Plane and Spherical, Trigonometrical Surveying, Maxima and Minima of Geometrical Quantities, Motion of Machines and their Maximum Effects, Practical Gunnery, &c. are among the most important subjects in Mathematics, and are discussed in the volume just mentioned in such a manner as not only to prove highly useful to pupils, but also to such as are engaged in various departments of Practical Science.

As the work, after the publication of the third volume, embraced most subjects of curiosity or utility in Mathematics, it it has been thought unnecessary to enlarge its size by much additional matter. The present edition however, differs in

several VOL. I.


several respects from the last English one ; and it is presumed, that this difference will be found to consist of improvements. These are principally as follows:

In the first place, it was thought adviseable to publish the work in two volumes instead of three; the two volumes being still of a convenient size for the use of students.

Secondly, a new arrangement of various parts of the work has been adopted. Several parts of the third volume of the English edition treated of subjects already discussed in the preceding volumes ; in such cases, when it was practicable, the additions in the third volume have been properly incorporated with the corresponding subjects that preceded them; and, in general, such a disposition of the various departments of the work has been made as seemed best calculated to promote the improvement of the pupil, and exhibit the respective places of the various branches in the scale of science.

In the third place, several notes have been added; and numerous corrections have been made in various places of the work : it were tedious and unnecessary to enumerate all these at present; it may suffice to remark the few following :

In pages 58, 59, vol. 1, a note is added on the reduction of fractions to the least common denominator ; and for common cases an easier rule is given, than has been before presented to the public.

In page 169. vol. 1, a useful note is added respecting the degree of accuracy resulting from the application of logarithms. This note will appear the more necessary, when we observe such oversights committed by authors of experience.

In several places of the last or seventh London edition, the corrections made in the first American edition have been adopted. The definition of Surds which had been improperly given in the fifth and sixth London editions, is now correctly in the seventh ; agreeably to the mode prescribed in the first American edition. This erroneous definition of Surds is still retained in the large Algebra of Bonnycastle, pub

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lished in London, in 1820. The true definition is given in the small work of the same author, by the editor, Mr. Ryan, in the New York edition of 1822.

The erroneous computation of the value x in the equation **= 100, which was pointed out and corrected in the first American edition, is expunged from the seventh London edition. The solution of this problem was subject to the same error in both the treatises of Algebra by Bontycastle. The American edition of 1822 is correct, but the larger Algebra, published in London, in 1820, still retains the error.

In the second volume, page 24, American edition, a very simple solution was given to the problem in the sixth example. This problem was solved by a cubic in the sixth London edition; and in the seventh, the solution is reduced from a cubic

quadratic ; but notwithstanding this improvement of the solution, it is still inferior to that given in the first American edition.

There are in the last London edition several errors continued from the sixth edition, which had been corrected in the first American edition. Among these we may notice the demonstration given to the third theorem in Spherics. The demonstration is founded on the assumption, that an angle of a spherical triangle is greater than the angle contained by the chords of the sides containing the spherical angle.

See on this subject the note page 555, vol. II.

In the mensuration, page 411, vol. 1, a remark is added respecting the magnitude of the earth. Dr. Hutton has commonly used a diameter of 7957| English miles, merely because it gives the round number 25,000 for the circumference : in a few places he has used a diameter of 7930. Having some years ago discovered the proper method of ascertaining the most probable magnitude and figure of the earth, from the admeasurement of several degrees of the meridian, I found the ratio of the axis to the equatorial diameter, to be as 320 to to 321, and the diameter, when the earth is considered as a globe, to be 7918.7 English miles.

In the additions immediately preceding the Table of Logarithms in the second volume, a new method is given for as

certaining certaining the vibrations of a variable pendulum. This problem was solved by Dr. Hutton, in his Select Exercises, 1787, and he has given the same solution in the present work, see page 537, vol. 2. The method used by the Doctor appears to me to be erroneous ; but in order that such as would judge for themselves on this abstruse question, may have a fair opportunity of deciding between us, the Doctor's solution is given as well as my own.

It may be proper to observe, with respect to the new solution, as well as Dr. Hutton's that the resulting formula does not show the relation between the time and any number of vibrations actually performed ; but merely gives the limit to which this relation approaches, when the horizontal velocity is indefinitely diminished. If therefore we would use the new formula as an approximation in very small finite vibrations, the times must not be extended without limitation.

Besides the numerous corrections in this third American edition, there is added to the second volume an elementary treatise on Descriptive Geometry, in which the principles and fundamental problems are given in a simple and easy manner, with a select number of useful applications, in Spherics, Conics, Sections, &c.


Columbia College, New York.

May 1, 1822.

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