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 3 sin . cos2 u. sin 3 u Then substituting for sin u its value x, and for cos2 u its value r2x2 and arranging all the terms according to the powers of x, we shall have x3-r3x+ ar3 =0, a cubic equation of the form x3 px + q = 0, with the condition that p3 > 193; 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 + 120o, and 2 + 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 AE, 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 byperbola 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 Now the angle ADF at the centre of the circle being equal to KAI or 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. =ADF. 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 a2=by3 ; 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: QS M Let the right lines AR, AS, be joined A at right angles; and a parabola ANH be described about the axis AR, with the parameter a; and another parabola AMI about the axis As, with the parameter 6: P cutting the former in M. Then AP = x, R PM = = y, are the two mean proportionals of which y is the side of the double cube required. For, in the parabola ▲мн the equation is y3 = ax, and in the parabola AMI it is x2 by. Consequently a y: yx, and y:x:: x ; b. Whence yx H = ab; or, by substitution, yby ab, or by squaring y3ba2b2; 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, pro. duced, 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: Suppose a point in be conceived to move from the position A, and to generate a line AP, A by a motion any how regulated; and suppose the celerity of the point i, 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 rp, in the given time allowed for the fluxion: then will the said line pp represent the fluxion of the fluent, or flowing line, AP, at that position. 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 B n m P P D point m describes the line AP: also, let the distance Pp be taken, as before, to express the fluxion of the line or base AP; and complete the rectangle reqp. Then, like as Pp is the fluxion of the line AP, so is Pq, the fluxion of the flowing parallelogram AQ: both these fluxions, or increments, being uniformly described in the same time. 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 Pp denote the fluxion of the line AP: Then, like as the rectangle rap, or PQ X rp, de B R notes the fluxion of the flowing rectangle ABQP, So also shall the fluxion of the variable solid, or prism ABERQP, be denoted by the prism PQRrqp, or the plane PR X Pp. And, in both the 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 former 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. 40 scribed scribed by the motion of a Variable Magnitude, whether it be a line or a plane. Thus, let a variable line PQ be carried by a parallel motion along AP; or while a point p is carried along, and describes the line AP, suppose another point, 4 4 4 A PP A PP A e to be carried by a motion perpendicular to the former and to describe the line ra: let pq be another position of PQ, indefinitely near to the former; and draw or parallel to AP. Now in this case there are several fluents, or flowing quantities, with their respective fluxions: namely, the line or fluent AP, the fluxion of which is pp or ar; the line or fluent rq, the fluxion of which is rq; the curve or oblique line Ae, described by the oblique motion of the point, the fluxion of which is eq; and lastly, the surface APQ, described by the variable line ra, the fluxion of which is the rectangle Porp, or rQ X гp. In the same manner may any solid be conceived to be described, by the motion of a variable plane parallel to itself, substituting the variable plane for the variable line; in which case the fluxion of the solid, at any position, is represented by the variable plane, at that position, drawn into the fluxion of the line along which it is carried. 9. Hence then it follows in general, that the fluxion of any figure, whether plane or solid, at any position, is equal to the section of it, at that position, drawn into the fluxion of the axis, or line along which the variable section is supposed to be perpendicularly carried: that is, the fluxion of the figure AQP, is equal to the plane PQ X pp, when that figure is a solid, or to the ordinate rQ X rp, when the figure is a surface. 10. It also follows from the same premises, that in any curve or oblique line aq, whose absciss is AP, and ordinate is ra, the fluxions of these three form a small right-angled plane triangle aqr; for ar=pp is the fluxion of the absciss AP, qr the fluxion of the ordinate ra, and qg the fluxion of the curve or right line AQ. And consequently that, in any curve, square of the fluxion of the carve, is equal to the the |