PROBLEM-IV. To Construct an Equation of any Order by means of a Locus of the same Degree as the Equation proposed, and a Right Line. E F D KAM R Ꮐ As the general method is the same in all equations, let it be one of the 5th degree, as x3-bx1+ acx3—a2dx2+a3ex -a4f0. Let the last term af be transposed; and, taking one of the linear divisors, f, of the last term, make it equal to z, for example, and divide the equation by a*; then x5-bx4+acx3 —a2dx++a3ex will z = at B F On the indefinite line BQ describe the curve of this equation, BMDRLFC, by the method taught in prob. 2, sect. 1, of this chapter, taking the values of x from the fixed point B. The ordinates PM, SR, &c, will be equal to z; and therefore, from the point в draw the right line BA = f, parallel to the ordinates PM, SR, and through the point A draw the indefinite right line кс both ways, and parallel to BQ. From the points in which it cuts the curve, let fall the perpendiculars MP, RS, CQ; they will determine the abscissas BP, BS, BQ, which are the roots of the equation proposed. Those from A towards q are positive, and those lying the contrary way are negative. If the right line ac touch the curve in any point, the corresponding abscissa x will denote two equal roots; and if it do not meet the curve at all, all the roots will be imaginary. If the sign of the last term, af, had been positive, then we must have made z = — f, and therefore must have taken f, that is, below the point P, or on the negative side. BA = EXERCISES. Ex. 1. Let it be proposed to divide a given arc of a circle into three equal parts. Suppose the radius of the circle to be represented by r, the sine of the given arc by a, the unknown sine of its third part by x, and let the known arc be 3u, and of course, the required arc be u. Then, by equa. vIII, IX, chap. iii, we shall have Putting, in the first of these equations, for sin 3u its given value a, and for sin 2u, cos 2u, their values given in the two other equations, there will arise S sin u. cos2 u. sin3 u Then substituting for sin u its value x, and for cos2 u its value r2 x2, and arranging all the terms according to the powers of x, we shall have r2x + 1 ar2 a cubic equation of the form 3 O, with the condition that 2723 > 192; that is to say, it is a cubic equation falling under the irreducible case, and its three roots are represented by the sines of the three arcs u, u + 120°, and น + 240°. Now, this cubic may evidently be constructed by the rule in prob. 3 cor. 3. But the trisection of an arc may also be effected by means of an equilateral hyperbola, in the following manner. Let the arc to be trisected be AB. In the circle ABC draw the semidiameter AD, and to AD as a diame- H ter, and to the vertex A, draw the equilateral hyperbola AE to which the right line AB (the chord of the E G arc to be trisected) shall be a tangent in the point a; then the arc AF, included within this hyperbola, is one third of the arc AB. For, draw the chord of the arc AF, bisect AD at G, so that G will be the centre of the hyperbola, join DF, and draw GH parallel to it, cutting the chords AB, AF, in I and K. Then, the hyperbola being equilateral, or having its transverse and conjugate equal to one another, it follows from Def. 16 Conic Sections, that every diameter is equal to its parameter, and from cor. theor. 2 Hyperbola, that GK KI =AK2, or that GK: AK :: AK: KI; therefore the triangles GKA, AKI are similar, and the angle KAI =AGK, which is manifestly =ADF. Now the angle ADF at the centre of the circle being equal to KAI OF FAB; and the former angle at the centre being measured by the arc AF, while the latter at the circumference is measured by half FB; it follows that AF = FB, or = AB, as it ought to be. Ex. 2. Given the side of a cube, to find the side of another of double capacity. Let the side of the given cube be a, and that of a double one y, then 2a3 =y3, or, by putting 2a=b, it will be a2b=y3; there are therefore to be found two mean proportionals be tween tween the side of the cube and twice that side, and the first of those mean proportionals will be the side of the double cube. Now these may be readily found by means of two parabolas; thus: Let the right lines AR, AS, be joined A at right angles; and a parabola Aмн be described about the axis AR, with the parameter a; and another parabola AMI about the axis As, with the parameter 6: cutting the former in м. Then AP = PM=y, are the two mean proportionals, of which y is the side of the double cube required. P R M H For, in the parabola AMH the equation is y2= ax, and in the parabola AMI it is rby. Consequently ay yx, :: and y: x :: x: b. Whence yx = ab; or, by substitution, y by ab, or, by squaring y3ba262; or lastly, y3 = a2b =2a3, as it ought to be. = THE THE DOCTRINE OF FLUXIONS. DEFINITIONS AND PRINCIPLES. Art. 1. In the Doctrine of Fluxions, magnitudes or quantities of all kinds are considered, not as made up of a number of small parts, but as generated by continued motion, by means of which they increase or decrease. As, a line by the motion of a point; a surface by the motion of a line; and a solid by the motion of a surface. So likewise, time may be considered as represented by a line, increasing uniformly by the motion of a point. And quantities of all kinds whatever, which are capable of increase and decrease, may in like manner be represented by geometrical magnitudes, conceived to be generated by motion. 2. Any quantity thus generated, and variable, is called a Fluent, or a Flowing Quantity. And the rate or proportion according to which any flowing quantity increases, at any position or instant, is the Fluxion of the said quantity, at that position or instant and it is proportional to the magnitude by which the flowing quantity would be uniformly increased in a given time, with the generating celerity uniformly continued during that time. 3. The small quantities that are actually generated, produced, or described, in any small given time, and by any continued motion, either uniform or variable, are called Increments. 4. Hence, if the motion of increase be uniform, by which increments are generated, the increments will in that case be proportional, or equal, to the measures of the fluxions: but if the motion of increase be accelerated, the increment so generated, in a given finite time, will exceed the fluxion: and if it be a decreasing motion, the increment, so generated, will be less than the fluxion. But if the time be indefinitely small, so that the motion be considered as uniform for that instant; then these nascent increments will always be proportional, or equal, to the fluxions, and may be substituted instead of them, in any calculation. 5. To illustrate these definitions : Sup pose a point m be conceived to move from m P the position A, and to generate a line AP, A P P by a motion any how regulated; and suppose the celerity of the point m, at any position P, to be such, as would, if from thence it should become or continue uniform, be sufficient to cause the point to describe, or pass uniformly over, the distance PA, in the given time allowed for the fluxion: then will the said line ph represent the fluxion of the fluent, or flowing line, AP, at that position. B C n m D 6. Again, suppose the right line mn to move, from the position AB, continually parallel to itself, with any continued motion, so as to generate the fluent or flowing rectangle ABQP, while the point m describes the line AP: also, let the distance P be taken, as before, to express the fluxion of the line or base AP; and complete the rectangle PQgp. Then, like as p is the fluxion of the line AP, so is Pg the fluxion of the flowing parallelogram AQ; both these fluxions, or increments, being uniformly described in the same time. E Ꭱ B P 2 C 7. In like manner, if the solid AERP be conceived to be generated by the plane PQR, moving from the position ABE, always parallel to itself, along the line AD; and if P denote the fluxion of the line AP: Then, like as the rectangle Pagh, or pq × pɲ, denotes the fluxion of the flowing rectangle ABQP, so also shall the fluxion of the variable solid, or prism ABERQP, be denoted by the prism pqRrqh, or the plane PRX PR. And, in both these last two cases, it appears that the fluxion of the generated rectangle, or prism, is equal to the product of the generating line, or plane, drawn into the fluxion of the line along which it moves. 8. Hitherto the generating line, or plane, has been considered as of a constant and invariable magnitude; in which case the fluent, or quantity generated, is a rectangle, or a prism, the fomer being described by the motion of a line, and the latter by the motion of a plane. So, in like manner are other figures, whether plane or solid, conceived to be deVOL. II. Rr scribed |
