Capellus. Buxtorf the father announced his system in his Tiberias, c. viii. p. 8—80. Against this Capellus wrote his arcanum punctationis revelatum, to which Buxtorf the son, in his tractatus de punctorum origine antiquitate, &c. rejoined. To this Capellus replied in his vindiciis arcan. punct. revel. which, together with his first work, are reprinted in his Commentaria et notæ criticæ. Amstel fol. 1689. Since this time the learned have made themselves partizans, some of Buxtorf and some of Capellus. Buxtorf's grounds for the antiquity of our punctuation were, 1. The sense of the Old Testament would be uncertain without the vowel points: 2. Vowel points are included under μ xiga, Matt. v. 18. 3. Some passages from the Zohar and the Talmud. Capel. Ius and his followers on the contrary appealed to the ancient Samaritan alphabet on coins, which are without vowel points: 2. To the unpointed manuscripts, and especially to the unpointed Thoras: 3. To the ancient translators of the Bible: 4. To Origen and Jerom: 5. The Talmud and Corpus cabbalisticum. See Tychsen upon the age of the Hebrew points. Repertor. Th. iii. p. 102.

In later times Schultens has, in my opinion, been most successful in the inquiry. He considers, that agreeably to the analogy of the other eastern languages, some vowel points must have been very old, and used perhaps by the writers in the Hebrew tongue. He no where however fully unfolds his system. Schult. instit. Ling. et lib. 3. p. 48. 62. After him Hr. R. Michaelis has treated upon the "Indecisive grounds of the antiquity of our present punctuation." To examine the grounds, which appear to me decisive consult, beside the above named writings, Dupuy's dissertation philologique et critique sur les voyelles de la langue Hebraique, in the histoire de l'academie des inscriptions et belles lettres. T. 36. Paris 1775. Matthew Norburg's dissertat. de Hebræorum vocal. Lond. 1784. 4to. Trendelberg's dissertation in the Repertorium. Th. xviii. n. 2.

ARTICLE 7.

A course of Mathematics in two volumes, for the use of Academies as well as for private tuition. By Charles Hutton, L L. D. F. R. S, late Professor of Mathematics in the Royal Military Academy. From the fifth and sixth London editions. Revised and corrected by Robert Adrain, A. M. fellow of the American Philosophical Society, and Professor of Mathematics in Queen's College, New-Jer sey-New-York: Samuel Cambell, & Co.-8vo. p. 583 and

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598.

ROM Dr. Hutton's preface it appears, that his object was to compose a short and easy course of Mathematics, for the purpose of academical education, to contain the most useful principles, dispos ed in a convenient practical form, with concise demonstrations and examples for illustration. Several editions were published in two volumes a third volume, composed by the joint labours of Dr. Hutton and Dr. Gregory, was added to the edition of 1811, to make it conform to the improved plan of education at the Royal Military Academy at Woolwich, for which the work was more particularly designed. This additional matter has been differently arranged by the American editor, to incorporate it with the similar articles of the other part, and the whole work, with several additions and corrections, is now given to the American public, in two volumes, of a convenient size.

That the reader may form some idea of the work, we shall give a very brief abstract of the several articles, taking them in the same order as in the American edition, and shall add such remarks as may occur in the examination.

1. Arithmetic contains most of the rules usually given in treatises on this subject, with a variety of examples for illustration. Several useful rules are however omitted, such as Practice, Circulating Decimals, &c.

2. Logarithms. This includes a brief explanation of the principles and a method of computing logarithms, with the uses of these numbers in multiplication, division, involution, and evolution.

3. Algebra. This article embraces the usual introductory definitions and rules, the method of solving simple and quadratic equations, and a collection of questions to exercise the learner. Car

dan's rule for cubic equations is given, but without a demonstration and the method of investigating the roots of cubic and higher equations by approximation is explained. Then follow the alge braical formulas for computing simple and compound interest and annuities.

In several of the examples the roots are not all given. Thus in page 253, vol. 1, all the negative roots are neglected. The value x=0, in the 13th example, page 236, is not noted. The 4th and 5th examples, page 243, are indeterminate, instead of having definite values, as we should be led to infer from the answer to the 4th example. These omissions have a tendency to embarrass the student, and they ought to be corrected in an other edition.

The subject of this article is resumed in the second volume (from the third of the English edition) under the title of "nature and solution of equations in general," in which are given several of the well known theorems for computing the powers and products of the roots from the coefficients of the generating equation, the method of transforming equations to others, having roots of difdifferent values, rules for solving quadratic and cubic equations by the table of sines, tangents and secants, and Euler's method of computing the roots of biquadratic equations. These subjects are intimately connected with the article on Algebra in the first volume, and the whole ought to have been arranged together in a methodical manner.

4. Geometry. In this part, in the compass of less than a hundred pages are inserted the most useful propositions of Euclid's Elements of Geometry, interspersed with other useful matter.

We are strongly induced to believe, that an abridgment of this kind would be found more useful in our schools and universities, than the larger work of Euclid. For the scholar could find in it sufficient to qualify himself to enter upon the other branches of that science, without the trouble of learning several propositions of not much practical utility.

5. Application of Algebra to Geometry. This part contains a few examples, producing simple and quadratic equations. The subject is again taken up in the second volume under the title of "nature and properties of curves and the construction of equations," in which are given an abridged account of the properties of curves of the third and fourth orders, methods of finding the equation of a

curve or a curve surface from its genesis, and the construction and equations of a few of the most remarkable curves.

6. Plane Trigonometry, &c. contains the three fundamental canons, by which the different cases of plane trigonometry are solved in elementary treatises. The articles that follow, shew the application of those rules to the computation of heights and distances; mensuration of planes, solids, Artificer's works, and common sur. veying. Ninety-two pages of the first volume are appropriated to these subjects, which are explained in an ample manner.

7. Conic Sections. This is rather an abridged account, though the most useful properties of these curves are given-such as the general relations of the absciss and ordinate about any diameter, properties of the foci, drawing tangents, areas of the circumscribed parallelograms, asymtotes &c. The demonstrations of the ellipsis and hyperbola are arranged in exactly the same manner, which tends considerably to simplify the subject.

8. Elements of Isoperimetry. In this article the geometrical method of demonstration is used. The theorems are merely elementary, such as finding the greatest polygon, that can be made with sides given in number and magnitude, &c. None of those difficult problems, which have so much exercised the genius of mathematicians since the days of Newton, and which have tended so much to improve modern analysis, could be expected in a work of this nature.

The first volume closes with a collection of problems relative to the division of fields, the construction of geometrical problems, and questions in mensuration, which would have been more naturally connected with the similar articles, in the preceding part of the work.

9. Plane Trigonometry considered Analytically. This article is from the third volume of the English edition, and is one of the most important additions to the work. The whole theory of Plane Trigonometry is deduced from one fundamental equation, shewing the relation between the sides and the cosines of the angles at the base of the triangle. The algebraical symbols, by which this equation is expressed, tend very much to facilitate the computations made from it, on account of the great symmetry between the expressions of the sides and their opposite angles, which, though at first it might seem an object of very small importance, is practi

cally very far from being so. The equation thus found, combined with that well known property of a plane triangle, that the sum of the three angles is equal to two right angles, constitutes the basis of all the computations in this article, in which, besides the demonstrations of the three fundamental canons of trigonometry, are given various theorems for computing the sines, cosines and tangents, of the sums and differences of two given arcs, the resolution of products of sines and cosines of arcs to terms containing the sums and differences, the sines and cosines of multiple arcs, the sum of any series of sines or cosines of arcs in arithmetical progression, and the uses of these series in forming a table of natural sines, &c.

Among the problems given to illustrate the uses of these formulas, the sixth in vol. 2, page 23, deserves notice on account of the elegant solution by the American Editor. That which is given in the English edition is made to depend on a cubic equation, whereas by rejecting the common factors, which strike the eye at the first glance, it might be reduced to a quadratic. Notwithstanding this defect, the author speaks with great confidence of the neatness and brevity of his method, observing that any direct solution to this curious problem, except by the means he has used, "would be exceedingly operose and tedious," when in fact the direct mode, by other formulas, is by far less laborious, as the American editor has shown in his neat and simple solution, in vol. 2, page 25; where the problem is reduced to an equation of the first degree, instead of a cubic. This solution is partly geometrical, and perhaps the author of the work might object to it, on the ground that the whole article was designed to exhibit the analytical method of computation, and that it was not proper to introduce any facilities, arising wholly from the geometrical manner of treating the subject: this objection would be of some weight if the same simple result could

• The present method of notation (which Dr. Hutton has judiciously adop ted) is to name the sides of the triangle by the small letters a, b, c, and the opposite angles respectively by the capital letters A, B, C; so that the angle A is opposite to the side a, B to b, C to c. Supposing therefore, c to be the base of the triangle, the two segments formed by a perpendicular let falb from the vertex on the base will be represented by b.cosA, and a.cosB, their sum gives the base c=a.cosB+b.cos A. By a simple change of the letters composing this equation into the terms corresponding to the base a or b, we may obtain similar expressions for a and b.