is to the greater; as radius to the tangent of an angle greater than 45° and radius is to the tangent of the excess of this angle above 45°; as the tangent of half the sum of the opposite angles, to the tangent of half their difference. In a plane triangle, twice the product of any two sides, is to the difference between the sum of the squares of those sides, and the square of the third side, as radius to the cosine of the angle included between the two sides.

CONIC SECTIONS. 16th Property of the hyperbola. If an ordinate to the axis major be produced to meet the asymptotes, then the rectangle of the segments intercepted between the cones and either asymptote will be equal to the square of the semi-axis minor.

Property 17. If an ordinate to any diameter be produced to meet the asymptotes, the rectangle of the segments intercepted between the curve and one asymptote, will be equal to the rectangle of the segments intercepted between the curve and the other. Property 18. If from any point of the curve a line be drawn to the nearer asymptote, parallel to the other asymptote, the rectangle of this line, and the distance of its intersection with the asymptote from the center, is a constant quantity; and is equal to the square of half the diagonal of the rectangle of the semi-axes. Property 19. If a parallel to either asymptote cut the conjugate hyperbolas, the diameters passing through the points of intersection will be conjugate to each other. Property 20. The square of any diameter is to the square of its conjugate, as the rectangles of its abscissas are to the squares of their ordinates.

Note. Conic Sections form one of the most important parts of mathematics, which is distinguished for elegance, demonstrating with surprising simplicity and beauty, and in the most harmonious connection, the different laws, according to which the Creator has made worlds to revolve, and the light to be received and reflected, as well the ball thrown into the air by the playful boy, to describe its line, until it falls again to the earth.* It has been said, that few branches of mathematics delight a youthful mind so much as conic sections, and that their study begets emotions in the pupil which might be called natural piety. We are disposed to think this remark rather the product of an enthusiastic ardor in their study, than extensive experience in teaching. Still we believe that they may be made highly useful in

* Amer. Ency.

opening the mind to the true grandeur and beauty displayed in the arrangement and motions of the different orders of worlds disposed through infinite space.

MECHANICS. Natural Philosophy is the science which treats of the laws of the material world. The term law, signifies the mode in which the powers of nature act. Natural Philosophy is divided into Mechanics, Optics, Electricity and Magnetism. Body is any collection of matter exist. ing in a separate form. Force is any cause which moves or tends to move a body, or which changes or tends to change its motion. Hydrostatics is that branch of mechanics which treats of the equilibrium and motion of fluids in the form of water: Pneumatics is that which treats of the equilibrium and motion of fluids in the form of air. By particles is meant the smallest parts into which a body may be supposed to be divided by mechanical means. In Geometry, we conceive figures to occupy space without excluding other figures from it, but in Mechanics we take objects such as they occur in nature, viz. not only extended, but impenetrable. All bodies possess gravity and inertia, which properties are intimately connected with the phenomena and laws of motion. The weight of a body is the force it exerts in consequence of its gravity, and is measured by its mechanical effects. Uniform velocity, is when a body describes equal spaces in equal successive parts of time. Accelerated velocity is when the space described by a body in equal successive parts of time continually increases; and retarded velocity is when the space described continually decreases. The fundamental principles of mechanics rest on three kinds of evidence: 1. They are conformable to all experience and observation. 2. They are confirmed by various accurate, experiments. 3. The conclusion deduced from them have always proved true in fact, without an exception. When it is required that the sum of the resolved forces shall be equal to a given quantity, the number of pairs of forces will be limited to the number of triangles which can be described in a semi-ellipse of which the given force is the distance between the two foci. When it is required that the difference of the resolv. ed forces shall be equal to a given quantity, the number of pairs of forces will be limited by the number of triangles which can be described from the foci of an hyperbola, (having their vertices in the curve) of which the given force is the distance

between the foci. A body will be kept at rest if it be acted upon by any number of forces which are represented in quantity and direction by the sides of a polygon taken in order, and conversely. Of three forces which keep a body at rest, the two components and the resultant may severally be represented by the sine of the angle included between the directions of the two others. When a body is supported by a prop placed under its center of gravity, the pressure will be the same whether the whole quantity of matter be uniformly diffused through the space occupied by the body, or whether it be all as it were concentrated in that center of gravity. When two weights are in equilibrio upon a straight lever, placed in a horizontal position, they are to each other inversely as the length of the arms from which they are respectively suspended. In a combination of levers, the opposite forces are in equilibrio, when the power is to the weight as the product of all the arms on the side of the weight is to the product of all the arms on the side of the power. In the wheel and axle, whilst the power descends through a space equal to the circumference of the wheel, the weight ascends through a space equal to the circumference of the axle. In the single fixed pulley, the power and weight move through equal spaces in the same time. The times of descent down similar systems of inclined planes are as the square roots of the lengths of the planes. The point about which the pendulum revolves, is called the center of suspension. The vibration of a pendulum is its motion from a state of rest at the highest point on one side, to the highest point on the other side. The center of oscillation of a pendulum, is such a point that, were all the matter of the pendulum collected in it, the quantity of motion would be equa! to the sum of the momenta of all the parts taken separately. The center of oscillation is below the center of gravity.

Remarks on Arithmetic.

ARITHMETIC is studied by two classes of persons, and for objects which differ very essentially from each other. The one seek from the contents of this small volume all the mathematical knowledge they expect ever to acquire; and who, of course, wish to attend to those branches that are to be of frequent use in the common transactions of business. The

other study it only as the first of a long series that are to follow-not so much for any practical purpose, as to prepare the way for the successful pursuit of the several branches which are to succeed. Objects so different cannot be attained to advantage without a corresponding difference in the pursuits. It is necessary, therefore, that arithmetics should be varied, in order to meet the several wants, and we find them to be such, for there is probably no branch of education that has had so many treatises written upon it, as Arithmetic. The little which this volume contains on the subject, is designed for those who are pursuing a liberal education, consequently, but few of the immense multitude of problems which may be found in the various books are solved. There is scarcely any question of difficulty in common arithmetic which cannot be solved with much greater facility by some rule in Algebra. We should therefore perform no important office for the Algebraist by answering for him difficult arithmetical questions, when he can readily answer them for himself by an obvious application of some familiar rule. As for those persons whose entire mathematical knowledge lies in Arithmetic, it would be an almost hopeless task to attempt solutions for all, owing to the multitude of Arithmetics. The proposition, that difficult arithmetical questions can be solved by applying the rules in the higher branches, will appear evident by an example. The 214th page of Daboll's Arithmetic contains the most difficult question in the book, whose statement occupies near half a page, but which might be expressed by asking the question, Into what two numbers can 2000 be divided so that their product will be 960,000? Now to give a solution according to any rule furnished by Daboll, would be a tedious.labor; but by Af fected Quadratic Equations, we have only to let x represent one of the parts and we have the following equation which is easily solved. xx (2000-x)=960000. In the small collection given in the fore part of this volume, there is contained merely a specimen of such as commonly occur. my opinion, Arithmetics for those who intend to study Algebra might very advantageously be expurgated of many of their more difficult parts, which make unreasonable demands upon the time and labor of the student. It is true that they possess some advantages in furnishing mental dis. cipline, but even here they ought to yield to subsequent branches. 31*

In

Algebra, problem 47, Art. 197. Let the four places be represented by A, B, C, D; and let x=the distance from A to B, therefore, by the conditions of the problem the distance from C to D=3x÷2. Moreover, by the problem the distance from A to B+ the distance from C to D= three times the distance from B to C; that is x÷4+3x÷4 three times the distance from B to C=x. But x=the distance from A to B; therefore the distance from A to B= three times the distance from B to C. Hence x+x÷3+3x ÷2=34, and x=12. The distances, therefore, are severally 12, 4, 18.

Mensuration, example 2, Art. 27. By problem II, 249128 X7=1743896, which divided by 22=79268, and this multiplied by .0087266-69.17401288=the length of a degree on the earth at the equator.

MENSURATION page 27. Example 16. By Art. 30, 19× 19.7854 283.5294 the area of the smaller circle, and this added to 1202.64 1486.1694 the area of the larger circle, which divided by .7854=1892.2452, whose square root 43.5 very nearly the diameter of the larger circle.

Example 17. The inveniendum, is the area of the segment of the circle. This is to be obtained (Art. 35.) by finding the area of a sector whose arc has the same chord as that of the segment; and the area of a sector is found (Art. 34.) by multiplying the raidus into half the length of the arc. Now the radius is (Art. 29.) 25, but the length of the arc is not given, and this is found by Art. 26 when the number of degrees in the arc is given: But as the number of degrees is not given, the length of the arc is to be determined by Art. 28.

COMPUTATION OF THE CANON.

Canon, says Hutton, in reference to mathematics is a general rule for resolving all cases of a like nature with the present inquiry. Thus the last step of every equation is such a canon, and if translated into words, becomes a rule to resolve all cases or questions of the same kind with that proposed. Hence, tables of sines tangents, &c. are canons. To compute the natural sine and cosine of a given arc, is a problem on which it was intended to bestow considerable attention, and when this is clearly solved, the tangents, &c.