1 center of gravity of any number of bodies or particles of matter from a plane given in position, equal to ? When will a body projected describe a parabola? What is the velocity of a projectile at any point of the parabola? Where do the foci of all the parabolas that can be described by a body projected from a given point, lie? How does the random of a projectile vary? How does the greatest altitude vary? How does the time of flight vary? When is the random the greatest? When does a projectile rise to its greatest hight? When is the time of flight greatest? What are the simple mechan. ical powers? How are compound machines formed? What is the lever? What is the fulcrum or prop? What is the power? What is the weight? In treating of the mechanical powers, what is the first inquiry? What is the first axiom of the lever ?--the second ?— the third? How will two forces which are in equilibrio, be to each other, when acting in the same plane, and perpendicular to the extremities of a straight lever? In how many different orders are lev. ers divided? What is a lever of the first order?-of the second ?— of the third? When forces act perpendicularly to the arms of a straight lever, when is an equilibrium produced? How is the weight when compared with the power, in the first and second kinds of levers ?--how in the third ? When will two forces acting at the extrem. ities of the arms of any lever be in equilibrio? How can the weight of a body be found by false balance? What kind of lever is a steelyard? In the wheel and axle, when is an equilibrium produced? What is a pulley? In the inclined plane when is an equilibrium produced? When the power acts parallel to the plane, how are the power, the pressure and the weight to each other ?--how when the power acts parallel to the base of the plane? When will the least power be required to raise or sustain a weight upon a given inclined plane? When is the pressure on a given inclined plane the greatest? What is the screw? In the screw, when is an equilibrium produced? What instruments come under the denomination of the wedge? What is the power of the wedge in overcoming resistance proportioned to? What does an equilibrium imply? What is the measure of a force? What is momentum compounded of? What is the rope ma. chine? When is a given weight in equilibrio, with two given powers which are equal to one another, and which pass over a pulley in the same horizontal line? What is the velocity acquired in falling down an inclined plane the same as? How is the time of descending the whole length of an inclined plane, to the time of falling freely through the hight? How does the time of describing any inclined plane vary? How does the velocity acquired in describing an inclined plane vary? If chords be drawn in a circle from the extremity of that diameter which is perpendicular to the horizon, how are the velocities which bodies acquire in falling through them? and what are the times of describing these chords equal to? When a body moves over a system of planes, what is the ratio of the velocity lost in passing from one plane to the succeeding one proportioned to? What is the ratio of the times of descending through similar curves, similarly situated with respect to the horizon? What is the pendulum? What is a cycloid? How will the times of vibration of pendulums of different lengths acted upon by different accelerative for tes, vary? How will the times of vibration of pendulums of the same length vary? How do the lengths of pendulums vibrating in the same time vary? How are the number of vibrations performed in a given time by pendulums of different lengths, and acted upon by different accelerative forces in respect to the square roots of the forces, and to the square roots of the lengths? SUPPLEMENTARY QUESTIONS.* ALGEBRA. What method of finding the common measure is con tained on 166th page? What are included under infinite series? What are found under the heads infinite series and indeterminate coefficients? What is summation of series? What is recurring series? What is method of differences? What is the remark in ref erence to approximation on 133d page? What is Newton's method for determining the co-efficients? What is the advantage of employ. ing algebraic language in geometrical demonstrations? In mentioning a line or an angle, what concise method may be adopted? If a line in any given directionis to be considered positive, how is it in the opposite direction? When one geometrical quantity is multiplied into another, what is fixed upon as the unit? When is it expedient to consider an entire line an unit? How are the areas of superficial figures found? How is algebra applicable to curves? How are the positions in the several points in a curve determined? What is an ordinate? What is an abscissa? What are the co-ordinates? What is made to compensate for the impossibility of drawing lines to every point in a curve? What is each abscissa equal to ? When will one of the co-ordinates be given? When are ordinates positive? When the abscissas ? At what point does an abscissa vanish? When do the two co-ordinates vanish together? When does an abscissa or an ordinate change from positive to negative, or the contrary? How are lines supposed to be produced? How surfaces? How solids? What is the cube of the abscissa equal to ? How is the curve described? What is the locus of an equation? How are the different orders of lines distinguished? What is an asymptote of a curve? LOGARITHMS. What contrivances were made to shorten mathematical culclations before the invention of logarithms? Who is the inventor of logarithms? When was his first work on the subject published? To whom was it dedicated? For what work has it been recently reprinted? Who translated the work into English? For what other persons has the merit of the invention of logarithms been claimed? and with what reason? Who wrote the preface to Wright's translation? What are some of the peculiarities of Napi. er's tables? What intercourse was had between Napier and Briggs? What was Brigg's third improvement? By whom were they suc ceeded in the cultivation of logarithms? What are some of the best early additions of logarithmic tables? How is the common logarithm of any number obtained when we have the logarithms of the num bers in any other system given? The answers to the remaining questions are interspersed among the solutions and in the Appendix. MENSURATION. In what does the problem of the quadrature of the circle consist? What approximations have been made in this? Of what does the Gothic arch consist? APPENDIX. ARITHMETIC. What is position? Of how many kinds? What is single position? What is double position? ALGEBRA. When is an expression called an infinite series? How may a fraction be expanded into an infinite series? How may equa. tions of any degree be produced from simple equations? TRIGONOMETRY. What is the product of radius and the sine of the sum of two arcs equal to ? What is the product of radius and the sine of the difference of two arcs equal to ? What is the product of radius and the cosine of the sum of two arcs equal to ? What is the product of radius and the cosine of the difference of two arcs equal to? What ratio has the sum of the sines of two arcs or angles to the difference of those sines? In any plane triangle, of the two sides that include a given angle, what ratio has the less to the great. er? and what is the ratio of radius to the tangent of the angle above forty five degrees? In a plane triangle, what is the ratio of the product of twice any two sides to the difference between the sum of the squares of these two sides, equal to ? CONIC SECTIONS. What is the 17th ? is the 20th? What is the 16th property of the hyperbola? MECHANICS, What is natural philosophy? What does the term law signify. Into what is natural philosophy divided? What is a body? What is force? What is hydrostatics? What is pneumat. ics? What is meant by particles? In geometry, how do we conceive figures ?-how in mechanics? What do all bodies possess? What is the weight of a body? What is uniform velocity? What is accelerated velocity? What is retarded velocity? What are the three kinds of evidence on which the fundamental principles of mechanics rest? When it is required that the sum of the resolved forces shall be equal to a given quantity, to what will the number of pairs of forces be limited? When is it required that the differences of the resolved forces shall be equal to a given quantity, to what will the number of pairs of forces be limited? When a body is acted upon by any number of forces which are represented in quantity and direction by the sides of a polygon, what will be the state of the body? Of the three forces which keep a body at rest, how may the two components and the resultant be severally represented? When a body is supported by a prop placed under its center of gravity, how will be the pressure? When two weights are in equilibrio upon a straight lever, placed in a horizontal position, how are they in respect to each other? In a combination of levers, when are the opposite forces in equilibrio? In the wheel and axle, whilst the power descends through a space equal to the circumference of the wheel, what space does the weight descend through? In what kind of pulley does the power and weight move through equal spaces in the same time? How do the times of descent down similar systems of inclined planes vary? What is the point called, about which the pendulum revolves? What is the vibration of a pendulum? What is the center of oscillation? Where is the center of oscillation in respect to the center of gravity? GENERAL PRINCIPLES OF MATHEMATICS. MATHEMATICS is the science of quantity, which is any thing that can be multiplied, divided or measured, as a line, weight, time, &c. Mathematics are based on arithmatic, algebra and geometry, and are either pure or mixed. In pure mathe. matics, quantities are considered independently of any substances actually existing; but in mixed mathematics, the relations of quantities are investigated in connection with some of the properties of matter, or with reference to the common transaction of business. Definitions are explanations of terms in which are laid the foundation of all mathematical knowledge. A complete definition distinguishes the thing defined from every thing else. Propositions which are self evident, are called axioms; those which require proof are called theorems. A problem is something to be done. Demonstration is either direct or indirect. A lemma is a proposition which is demonstrated, for the purpose of using it in the demonstration of some other proposition. A corollary is an inference from a preceding proposition. A postulate is something to be done, the process of which is so easy, as to require no explanation. A quantity is given when known directly or conditionally. In geometry, a quantity may be. given in position, or magnitude, or both. Two propositions are contrary to each other, when one affirms what the other denies, and both can never be true. One proposition is the converse of another when the order is inverted; so that, what is given or supposed in the first, becomes the conclusion in the last and what is given in the last, is the conclusion in the first. The practical application of the mathematics are numerous. Mathematical principles are necessary in mercantile transactions, in navigation, in surveying, in civil engineering, in mechanics, in architecture, in fortification, in gunnery, in optics, in astronomy, in geography, in history, in the con. cerns of government, &c. Mathematics are peculiarly fit. ted for the enlargment and improvement of the reasoning powers. |