Horopter, in Optics, a right line drawn through the point where the two optic axes meet, parrallel to that which joins the centres of the two eyes or pupils. Hyperbola, one of the Conic Sections, being that, made by a plane cutting a cone, through the base, not parallel to the opposite side. Hypothenuse, in a right angled triangle, the side which subtends, or is opposite to the right angle. Imaginary quantities, in Algebra, the even roots of negative quan tities. Impact, the simple or single action of one body upon another to put it in motion. Incidence or Line of Incidence, the direction or inclination in which one body strikes or acts on another. Inclination, the mutual tendency of two lines, planes, or bodies, towards each other. Inclined plane, a plane inclined to the horizon, or making an angle with it. Incommensurable lines, or quantities, such as have no common measure. Increment, the small increase of a variable quantity. Infinitesimals, certain infinitely or indefinitely small parts, also the method of computing by them. Inscribed Hyperbola, one that lies wholly within the angle of its asymptotes. Interscendent, a term applied to quantities, when the exponents of their power are radical quantities; as x√2. Intersection, the cutting off one line or plane by another. Isoperimetrical figures, such as have equal perimeters or circumferences. Lemniscate, the name of a curve in the form of the figure 8. Multiplicand, one of the two factors in multiplication, being the number to be multiplied. Negative, in Algebra, something maked with the sign — Nodes, the two opposite points where the plane of the orbit of a planet intersects the plane of the ecliptic. Nonagon, a figure having nine sides and angles. Oblate, flattened or shortened. Oblique, aslant, indirect, or deviating from the perpendicular. Orbit, the path of a planet or comet, being the curve line described by its centre, in its proper motion in the heavens. Ordinates, right lines drawn parallel to each other, and cutting the curve in a certain number of points. Oscillation, the vibration, or the ascent and descent of a pendulum. Osculatory circle, the same as the circle of curvature. Parabola, a figure arising from the section of a cone, by a plane parallel to one of its sides, Parallax, an arc of the heavens intercepted between the true place of a star and its apparent place. Parameter, a certain constant right line, in each of the three conic sections; called also latus rectum. Pentagon, a figure consisting of five sides and angles. Perimeter, the limit, or outer bounds of a plane rectilineal figure. Periphery, the circumference or bounding line of a curvilineal figure. Polygon, a figure of many sides and angles. Polynomial, a quantity consisting of many terms, called a multinomial. Positive quantities, in Algebra, of a real, or additive nature. Prime numbers, those which may only be measured by unity. Prism, a solid, whose two ends are any plane figures, which are parrallel, equal, and similar, and its sides connecting those ends parallelograms. Pyramid, a solid having any plane figure for its base and its sides triangles, whose vertices all meet in a point at the top, called the vertex. Quadratic Equations, those in which the unknown quantity is of two dimensions. Quadratrix, a mechanical line by means of which right lines are found equal to curves. Quindecagon, a plane figure of 15 sides. Radical sign, the sign or character denoting the root of a quantity. Radix or root, a certain finite expression or function, which being evolved or expanded, according to the rules proper to its form, produces a series. Rational, the quality of numbers, fractions, &c. when they can be expressed by common numbers. Reciprocal, the quotient arising by dividing 1, by any number or quantity. Refrangibility of Light, the disposition of the rays to be turned aside. Root, in Arithmetic and Algebra, denotes a quantity, which being multiplied by itself produces some higher power. Series, a rank or progression of quantities or terms, which usually proceed according to some certain law. Spheroid, a solid body approaching to the figure of a sphere with one of its diameters longer than the other. Spiral, a curve line of the circular kind, which, in its progress, recedes always more and more from a point within called its centre. Terms of a product, of a ratio, &c. the several quantities employed in forming or composing them. Variable, a term applied to such quantities as are considered in a variable or changeable state, either encreasing or decreasing. INTRODUCTION. SOME ACCOUNT OF THE HISTORY OF THE MATHEMATICS, AND OF THE PROGRESSIVE DEVELOPEMENT OF THE PRINCIPLES OF THE MATHEMATICAL SCIENCES. THE student who desires to acquaint himself with the reason, object, and uses of the several departments into which mathematical study is divided, will derive great advantage from becoming acquainted with the origin and progress of the successive inventions by which men have advanced from the power of estimating numbers, to the highest branches of geometrical analysis. Such a history necessarily includes the definitions and reason of every process, and if read together, or when any new branch is entered upon, it will enlarge, correct, and fix the notions of the student better than any abstract definitions. MONTUCLA's great work has served as the basis of all our English histories of the Science; but the present sketch is abstracted chiefly from the able Dissertations by the late Professor PLAYFAIR, prefixed to the Supplements of the Encyclopædia Britannica, and partly from the elegant history of the Abbé BossUT. In nothing was the inventive and elegant genius of the Greeks better exemplified than in their GEOMETRY. The elementary truths of that science were connected by EUCLID into one great chain, beginning from the axioms, and extending to the properties of the five regular solids; the whole digested into such admirable order, and explained with such clearness and precision, that no work of superior excellence has appeared. ARCHIMEDES assailed the more difficult problems of geometry, and by means of the method of Exhaustions, demonstrated many curious and important theorems, with regard to the lengths and areas of curves, and the contents of solids. APOLLONIUS treated of the Conic Sections, the Curves which, after the circle, are the most simple and important in geometry. Another great invention, the Geometrical Analysis, ascribed very generally to the Platonic school, but most successfully cultivated by Apollonius, is one of the most ingenious and beautiful contrivances in the mathematics. It is a method of discovering truth by reasoning concerning things unknown, or propositions merely supposed, as if the one were given, or the others were really true. By this analytical process, therefore, the thing required is discovered, and we are at the same time put in possession of an instrument by which new truths may be found out, and which, when skill in using it has been acquired by practice, may be applied to an unlimited extent. A similar process enables us to discover the demonstrations of propositions, supposed to be true, or, if not true, to discover that they are false. Trigonometry, which had never been known to the Greeks as a separate science, and which took that form in Arabia, advanced, in the hands of REGIOMONTANUs, in the 15th century, to a great degree of perfection, and approached very near to the condition which it has attained at the present day. He also introduced the use of decimal fractions into arithmetic. Cut off in the prime of life; his untimely death, amidst innumerable projects for the advancement of science, is even at this day a matter of regret. He was buried in the Pantheon at Rome; and the honours paid to him at his death prove that science had now become a distinction which the great were disposed to recognise. WERNER, who lived in the end of this century, is the first among the moderns who appears to have been acquainted with the geometrical analysis. He resolved Archimedes's problem of cutting a sphere into two segments, having a given ratio to one another. MAUROLYCUS of Messina flourished in the middle of the sixteenth century, and besides furnishing many valuable translations and commentaries, he wrote a treatise on the conic sections, which is highly esteemed. He endeavoured also to restore the fifth book of the conics of Apollonius, in which that geometer treated of the maxima and minima of the conic sections. In the early part of the seventeenth century, CAVALLERI was particularly distinguished, and made an advance in the higher geome try, which occupies the middle place between the discoveries of Archimedes and those of Newton. For the purpose of determining the lengths and areas of curves, and the contents of solids contained within curve superficies, the ancients had invented a method, to which the name of Exhaustions has been given. Whenever it is required to measure the space bounded by curve lines, the length of a curve itself, or the solid contained within a curve superficies, the investigation does not fall within the range of elementary geometry. Rectilineal figures are compared by help of the notion of equality which is derived from the coincidence of magnitudes both similar and equal. This principle, which is quite general with respect to rectilineal figures, must fail, when we would compare curvilineal and rectilineal spaces with one another, and make the latter serve as measures of the former, because no addition or subtraction of rectilineal figures can ever produce a figure which is curvilineal. It is possible, indeed, to combine curvilineal figures, so as to produce one that is rectilineal; but this principle is of very limited extent In the difficulty to which geometers were thus reduced, it occurred. that, by inscribing a rectilineal figure within a curve, and circum scribing another round it, two limits could be outained, one greater and the other less than the area required. It was also evident, that, by increasing the number, and diminishing the sides of those figures, the two limits might be brought continually nearer to one another, and of course nearer to the curvilinear area, which was always intermediate between them.. In prosecuting this sort of approximation, a result was at length found out, which must have occasioned no less surprise than delight to the methematician who first encountered it; namely, that, when the series of inscribed figures was continually increased, by multiplying the number of the sides, and diminishing their size, there was an assignable rectilineal area, to which they continually approached, so as to come nearer it than any difference that could be supposed. The same limit would also be observed to belong to the circumscribed figures, and therefore it could be no other than the curvilineal area required. A truth of this sort occurred to ARCHIMEDES, when he found that two-thirds of the rectangle, under the ordinate and abscissa of a parabola, was a limit always greater than the inscribed rectilineal figure, and less than the circumscribed. In some other curves, a similar conclusion was found, and Archimedes contrived to show that it was impossible to suppose that the area of the curve could differ from the said limit, without admitting that the circumscribed figure might become less, or the inscribed figure greater than the curve itself. The method of Exhaustions was the name given to the indirect demonstrations thus formed; and though few things more ingenious than this method have been devised, and though nothing could be more conclusive than the demonstrations resulting from it, yet it laboured under two very considerable defects. In the first place, the process by which the demonstration was obtained, was long and difficult; and, in the second place, it was indirect, giving no insight into the principle on which the investigation was founded. A more compendious, and more analytical method, was therefore particularly required. CAVALLERI, born at Milan in the year 1598, is the person by whom this great improvement was made. The principle on which he proceeded was, that areas may be considered as made up of an infinite number of parallel lines; solids of an infinite number of parallel planes; and even lines themselves, whether curve or straight, of an infinite number of points. The cubature of a solid being thus reduced to the summation of a series of planes, and the quadrature of a curve to the summation of a series of ordinates, each of the investigations was reduced to something more simple. It added to this simplicity not a little, that the sums of series are often more easily found, when the number of terms is infinitely great, than when it is finite, and actually assigned. The rule for summing an infinite series of terms in arithmetical progression had been long known, and the application of it to find the area of a triangle, according to the method of indivisibles, was |