instant of its setting out from the point A, the radius AC turns uniformly about the centre C towards the point E, so that it coincides with CE at the moment in which the right line AG also coincides with it; by the continual intersection of these two lines we shall have a curve AMD, called the quadratrix. From this description it follows that any space AP passed over by the right line AG is to the circular arc AB described in the same u= a time by the extremity of the radius, as any other space AC passed over by this right line is to the corresponding arc ABE, described by the radius. Call AP=I, PM=y, AP=U, AC=a, ABE=90o=c, we shall have Io. mia::«;c:: angle ACB : angle ACE, therefore C... a 2ly. CP : PM::CA : AG, or a-*:y::a: tang u, therefore y tangom; and this will be the equation of the co-ordinates of the quadratrix when the point A is the origin of the abcissæ. 76. But if we place their origin at the centre C, making CP=X, we shall have y= tang (c–= c? x2 c? m? c4 x4 it (a + &c. &c. 8. 4. a? 2. 3. 4. a? CS + -&c. 2. 3. as 2. 3. 4. 5. ao 2.3. as .2.3.4.5.a" therefore when x=0, y which will become the base CD, will have for its expression aa ; consequently if the base of the quadratrix were known, we should immediately have the quadrature of the cir cot a a + 2a3 2a? с a C cle, it is this circumstance which gave the name of quadratrix to this curve. 77. If from the centre C and with the radius CD we describe the quadrant DLK, its length will be equal to the radius CA; for : DLK::a: c, therefore DLK=a. We shall also have PC=the , aa arc LD; for aa : KL:.a : u; therefore KL=I=AP, and PCS LD. с tangan Cut a 78. Let us now take negative abscissæ AP and substitute their value in the first equation. It will become y= _* (a+x) which gives the negative ordinates PM. Therefore the curve has a branch AM', to which we shall find that a right line QN drawn at the distance AQ = a is the asymptote, by supposing y infinite ; for then we shall find that tang and consequently that a. If after having coincided with CE, the right line AG, and the radius CA continue to move, the one descending towards a, the other revolving in the same direction; it is visible that their intersection will describe the part Da of the quadratrix. It is likewise evident that if this curve could be described geome. trically we should immediately be able to assign an angle of any number of degrees, for example of -90°. For this purpose we need only divide AC at the point P, so that AP should be to AC :: 1:m; for then drawing the ordinate PM, and the radius CB, the angle ACB would = 90° since x: a :::0::1:m. m VI. THE SPIRAL OF ARCHIMEDES. C A 79. This name is given to the curve CKMA, described by a point C, moving uniformly along the ra N dins CA, during the M uniform revolution of this radius about В. the centre C; so that when the radius has passed through the entire circumference this point may coincide with the point I A. If after having prolonged the radius CA, we cause it to make a ar second revolution; the point C, still continuing to recede from the origin of its movement, will describe a second spiral, then a third, and so on; or rather all these spirals will form but one and the same curve, whose revolutions may be multiplied indefinitely. 80. This premised, the ordinate CM=y, is to the radius CA (a) as the arc ABN which is the corresponding abscissa and which í call x, is to the entire circumference ABNA which call We have therefore y= for the archimedian spiral. Hence it follows; 1o. That this spiral is a transcendental curve; Illy, That it passes through the centre of the generating circle; III°. That it also passes through the point A ; IV°. That if we make x = " + x', the equation will become y= ax' Q+ ; and therefore that by giving to x, the values between o and , the spiral will make a second revolution, which will be terninated at the extremity of a radius double the first. It will make a third, fourth, &c. if we take x=27 +x", x=3+ +x", &c. 81. To draw a tangent MT to any point M of the curve, conceive the radius Cmn infinitely near to the radius CMN, and after having described a circle with the radius CM, draw CT perpendicular to CM; then by similar triangles Mmr, MTC we shall have CM x Mr. mr : Mr:: CM : CT But CM= ABN, and Cm= a a a ABn; therefore Cm-CM = mr = Nn; and sincea :y:: Nn:Mr ; a a Fy; but a:y::*: the arc OQM=*4; consequently the suba tangent CT must be taken equal to the circular arc OQM. VII. THE PARABOLIC SPIRAL. 112 82. If on any radius CN we take a part NM a mean proportional between the arc AN and a given line p, the curve which passes through N E B VIII. THE HYPERBOLIC OR RECIPROCAL SPIRAL. B 83. Suppose that from the point C, taken as a centre on the indefinite line CP, we describe the arcs AG, QM, PO, &c. equal in IR length, and that through their ex P tremities C, M, O, &c. we cause to passa curve CKGMO. This curve will be the hyperbolic spiral. It is easy to see that if we draw a a right line BR parallel to the axis CP, and distant from it by the quantity CB = AG=QM = PO, A &c. this right line will be the N asymptote of the hyperbolic spiral, E because it can only meet that line when the radius CM is infinite. F 84. Call the radius CA = a, AN=r, CM=y, AG=QM=b; K we shall have x: 6::a: :y hence zy=ab, an equation similar to that of the hyperbola between its asymptotes. If we call + the circumfe. rence whose radius=a, and if for x, we substitute successively the values + x, 2+1,......ma + x, we shall successively have ab ab ab y = y= •y= 25 + x my + x From this we infer that as the abscissa increases, the ordinate diminishes, and that this latter does not become zero till m is infinite. Hence the hyperbolic spiral performs an infinite number of revolutions about its centre before it reaches that centre. 85. Let us now enquire the value of the subtangent CT, and to this end let us suppose the line Crm infinitely near CM, and draw the arc mq; then draw CT perpendicular to CM, and meeting at T the tangent MT. Call Qq=rm=i; we shall have by bi y+i;b::y: Qr= hence rM=b; by y + y +2 bi rm : rM :: Cm : CT, therefore i : ::yti: CT=6. ' ; but yti yti Consequently in the hyperbolic spiral the subtangent is constant, as was also the case in the logarithmic curve. IX. THE LOGARITHMIC SPIRAL. N 86. This spiral is a curve which cuts under the same angle all the radïi CM drawn from its TIL centre C; so that the tangent MT always makes an equal angle with the radius CM, which ever side we suppose it УС to be. This curve bas several D beautiful properties but they K cannot conveniently be explained without the assistance T of the fluxional or differential calculus B The study of curves is perhaps one of the most delightful in the whole range of mathematical science ; besides their use in the research of truths, they may be employed with advantage to works of fancy. The reciprocal and logarithmic spirals have lately been introduced into architecture by Mr. P. Nicholson. The volutes of the Ionic order are easily drawn with compasses by the properties of logarithmic spirals, and when thus formed, possess more elegance and grace than those which have been produced by any other mode that has yet been suggested. The natural form of the reciprocal spiral is well adapted to the configuration of the volutes of the Corinthian capital. Nothing can ever regulate the judgment so as to direct the hand in forming every variety of figure with grace and elegance, so much as a knowledge of mathematical curves. See Nicholson's Architectural Works. |