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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 7. 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' a + ; and therefore that by giving to x, the values between o
the spiral will make a second revolution, which will be ter. ninated 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
Mr mr : Mr:: CM : CT
But CM= ABN, and Cm=4
ABn; therefore Cm-CM = mr = Nn; and sincea:y:: Nr:Mr
Mr we shall have
, and the subtangent CT=
For ; but a : 9::1; the arc OQM= Y; consequently the subtangent CT must be taken equal to the circular arc OQM.
VII. THE PARABOLIC SPIRAL. .
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 all the points M thus determined will be the parabolic spiral.
M Let then AN=x, CM=y, AB=a,
B В and we shall have y=a-vpx; an
a equation which shews that substitu
F ting atit, 27+*, &c. in place of x, the curve will make an infinitely number of revolutions about the centre, C, and consequently that it belongs to the class of spirals.
VIII. THE HYPERBOLIC OR
83. Suppose that from the point C, taken as a centre on the in. definite line CP, we describe the arcs AG, QM, PO, &c. equal in
IR length, and that through their ex
P tremities C, M, 0, &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 = RM = PO,
A &c. this right line will be the
N asymptote of the hyperbolic spiral, because it can only meet that line
E when the radius CM is infinite.
F 84. Call the radius CA = a, AN=r, CM=y, AG=QM=b;
к we shall have x :
: a : y hence wy=ab, an equation similar to that
T of the hyperbola between its asymptotes. If we call – the circumfe. rence whose radius=a, and if for x, we substitute successively the values r + x, 27 +1,......ma + x, we shall successively have ab ab
mu + 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 mg; then draw CT perpendicular to CM, and meeting at T the tangent Mt. Call Qq=rm=i; we shall have y+i; b::y:Qr= by
bi rm : rM :: Cm : CT, therefore i :
::yti: CT=. Consequently in the hyperbolic spiral the sublangent is constant, as was also the case in the logarithmic curve.
IX. THE LOGARITHMIC SPIRAL.
86. This spiral is a curve which cuts under the same angle all the radii CM drawn from its centre C; so that the tangent MT always makes an equal angle with the radius CM, which ever side we suppose it 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 calculua
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.
1. In this branch of analysis, quantities are usually distinguished into constant and variable.
A constant quantity, is that which retains always the same value. A variable quantity is that which may increase or decrease ; and which, while it changes from one magnitude to another, passes through all the intermediate states.
Thus, in trigonometry, the radius of a circle is a constant quantity ; while an arc of the circle, as also its sine, tangent, and secant are variable quantities.
It is usual to denote constant quantities by the first letters of the alphabet, a, b, c, &c., and variable quanties by the last letters, I, y, 2, &c. So that in the equation y'=ax + bxx, x and y are to be regarded as variable, and a and b as constant.
2. Let us now suppose that any variable quantity z receives a finile augmentation or increment e, so that after having received this increase, its new state may be expressed by x+e; then our next step must be to ascertain what will be the corresponding increment of any other function of x.
First, it is evident that if x becomes x+e, its square x? will become t+2exte?; and therefore the ratio of the two increments will be
But as e diminishes, this ratio will augment, and 2er te 2x te
1 approach nearer and nearer to that of However it will only
2x equal this latter ratio when e vanishes, or at least becomes indefinitely
1 small. Consequently the ratio is the limit of all the ratios which
2.1 any simullaneous finite increments of x and xx can bear to one another. Similarly when x becomes x +e, x will become x3 + 3e x: + 3e* x +e",
1 and the ratio of the increments will be
3er? + 3er +63 3.x2 +213x+e) now here also, as e diminishes, this ratio approaches towards, and
at last coincides with
which in this case is the limiting ralio.
And by a similar process we shall find the limiting ratio of the
1 increments of x" and x to be
Now the indefinitely small increment of any variable quantity is called its fluxion ; the variable quantity itself is termed a fluent, and it is the first business of the doctrine of Auxions to find in all cases such limiting ratios between the indefinitely small increments of any function, and of the variable quantity upon which it depends.
If the quantity be represented by a single letter as x, the fluxion is denoted by *; and if it be a compound quantity as x", the fluxion will be denoted by flux. (x*).
From what we have just shewn, it appears that flux. (x) or :: flux (x"):: 1; nxt; and therefore that the fluxion of m*=nx. The quantity nal=
Aux (x"), is called the fluxional coefficient. As an example of the mode of determining these limits, let it be required to draw a tangent to the point M of the curve AMm; or,
S which amounts to the same, let it be required to determine the subtangent PT.
Suppose that the absciss AP=r, is increased A by a finite quantity Pp=e; draw the ordinate PM=y, and determine the ordinate mp, by P substituting x+e, in place of x, in the equation of the curve. Whatever may be the value of this ordinate, we may always represent it by y+Pe+Qe* + Re+&c. (P, Q, R, &c. being functions of x); therefore we shall have for the expression of rm, the corresponding increment of PM, the quantity Pe +Qe? +Red+ &c. This premised, let there be drawn the secant SMm, and the line
y Mr parallel and equal to Pp; we shall have PS=;
P+Qe + Re + &c. PS Mr (because
P+Qe + Re’ +&c. Now supposing the point p to approach nearer and nearer to the point P, the point m will come nearer to M, and the point S nearer to the point T; but we shall always have PS=>
y P+Qe + Re® + &c