SPIRAL

, in Geometry, a curve line of the circular kind, which, in its progress, recedes always more and more from a point within, called its centre; as in winding from the vertex of a cone down to its base.

The first treatise on a Spiral is by Archimedes, who thus describes it: Divide the circumference of a circle App &c into any number of equal parts, by a continual bisection at the points pp &c. Divide also the radius AC into the same number of equal parts, and make Cm, Cm, Cm, &c, equal to 1, 2, 3, &c of these equal parts; then a line drawn, with a steady hand, drawn through all the points m, m, m, &c, will trace out the Spiral.

This is more particularly called the first Spiral, when it has made one complete revolution to the point A; and the space included between the Spiral and the radius CA, is the Spiral space.

The first Spiral may be continued to a second, by describing another circle with double the radius of the first; and the second may be continued to a third, by a third circle; and so on.

Hence it follows, that the parts of the circumference Ap are as the parts of the radii Cm; or Ap is to the whole circumference, as Cm is to the whole radius. Consequently, if c denote the circumference, r the radius, then there arises this proportion r : c :: x : y, which gives for the equation of this Spiral; and which therefore it has in common with the quadratrix of Dinostrates, and that of Tschirnhausen: so that will serve for infinite Spirals and quadratrices. See QUADRATRIX.

The Spiral may also be conceived to be thus generated, by a continued uniform motion. If a right line, as AB (last fig. above) having one end moveable about a fixed point at B, be uniformly turned round, so as the other end A may describe the circumference of a circle; and at the same time a point be conceived to move uniformly forward from B towards A, in the right line or radius AB, so that the point may describe that line, while the line generates the circle; then will the point, with its two motions, describe the curve B, 1, 2, 3, 4, 5, &c, of the same Spiral as before.

Again, if the point B be conceived to move twice as slow as the line AB, so that it shall get but half way along BA, when that line shall have formed the circle; and if then you imagine a new revolution to be made of the line carrying the point, so that they shall end their motion at last together, there will be formed a double Spiral line, as in the last figure. From the manner of this description may easily be drawn these corollaries:

1. That the lines B12, B11, B10, &c, making equal angles with the first and second Spiral (as also B12, B10, B8), &c, are in arithmetical progression.

2. The lines B7, B10, &c, drawn any how to the first Spiral, are to one another as the arcs of the circle intercepted between BA and those lines; because whatever parts of the circumference the point A describes, as suppose 7, the point B will also have run over 7 parts of the line AB.

3. Any lines drawn from B to the second Spiral, as B18, B22, &c, are to each other as the aforesaid arcs, together with the whole circumference added on both sides: for at the same time that the point A runs over 12, or the whole circumference, or perhaps 7 parts more, shall the point B have run over 12, and 7 parts of the line AB, which is now supposed to the divided into 24 equal parts. |

4. The first Spiral line is equal to half the circumference of the first circle; for the radii of the sectors, and consequently of the arcs, are in a simple arithmetic progression, while the circumference of the circle contains as many arcs equal to the greatest; therefore the circumference is in proportion to all those Spiral arcs, as 2 to 1.

5. The first Spiral space is equal to 1/3 of the first or circumscribing circle. That is, the area CABDE of the Spiral, is equal to 1/3 part of the circle described with the radius CE. In like manner, the whole Spiral area, generated by the ray drawn from the point C to the curve, when it makes two revolutions, is 2/3 of the circle described with the radius 2CE.

And, generally, the whole area generated by the ray from the beginning of the motion, till after any number n of revolutions, is equal to n/3 of the circle whose radius is <*> X CE, that is equal to the 3d part of the space which is the same multiple of the circle described with the greatest ray, as the number of revolutions is of unity.

In like manner also, any sector or portion of the area of the Spiral, terminated by the curve CmA and the right line CA, is equal to 1/3 of the circular sector CAG terminated by the right lines CA and CG, this latter being the situation of the revolving ray when the point that describes the curve sets out from C. See Maclaurin's Flux. Introd. pa. 30, 31. Se also QUADRATURE of the Spiral of Archimedes.

Spiral

, Logistic, or Logarithmic. See Logistic and Quadrature.

Spiral of Pappus, a Spiral formed on the surface of a sphere, by a motion similar to that by which the Spiral of Archimedes is described on a plane. This Spiral is so called from its inventor Pappus. Collect. Mathem. lib. 4 prop. 30. Thus, if C be the centre of the sphere, ARBA a great circle, P its pole; and while the quadrant PMA revolves about the pole P with an uniform motion, if a point proceeding from P move with a given velocity along the quadrant, it will trace upon the spherical surface the Spiral PZFa.

Now if we suppose the quadrant PMA to make a complete revolution in the same time that the point, which traces the Spiral on the surface of the sphere, describes the quadrant, which is the case considered by Pappus; then the portion of the spherical surface terminated by the whole Spiral, and the circle ARBA, and the quadrant PMA, will be equal to the square of the diameter AB. In any other case, the area PMAa FZP is to the square of that diameter AB, as the arc Aa is to the whole circumference ARBA. And this area is always to the spherical triangle PAa, as a square is to its circumscribing circle, or as the diameter of a circle is to half its circumference, or as 2 is to 3.14159 &c. See Maclaurin's Fluxions, Introd. pa. 31—33.

The portion of the spherical surface, terminated by the quadrant PMA, with the arches AR, FR, and the spiral PZF, admits of a perfect quadrature, when the ratio of the arch Aa to the whole circumference can be assigned. See Maclaurin, ibid. pa. 33.

Parabolic Spiral. See Helicoid.

Proportional Spiral, is generated by supposing the radius to revolve uniformly, and a point from the circumference to move towards the centre with a motion decreasing in geometrical progression. See LOGISTIC.

From the nature of a decreasing geometrical progression, it is easy to conceive that the radius CA may be continually divided; and although each successive division becomes shorter than the next preceding one, yet there must be an infinite number of divisions or terms before the last of them become of no finite magnitude. Whence it follows, that this Spiral winds continually round the centre, without ever falling into it in any finite number of revolutions.

It is also evident that any Proportional Spiral cuts the intercepted radii at equal angles: for if the divisions Ad, de, ef, fg, &c, of the circumference be very small, the several radii will be so close to one another, that the intercepted parts AD, DE, EF, FG, &c, of the Spiral may be taken as right lines; and the triangles CAD, CDE, CEF, &c, will be similar, having equal angles at the point C, and the sides about those angles proportional; therefore the angles at A, D, E, F, &c, are equal, that is, the spiral cuts the radii at equal angles. Robertson's Elem. of Navig. book 2, pa. 87.

Proportional Spirals are such Spiral lines as the rhumb lines on the terraqueous globe; which, because they make equal angles with every meridian, must also make equal angles with the meridians in the stereographic projection on the plane of the equator, and therefore will be, as Dr. Halley observes, Proportional Spirals about the polar point. From whence he demonstrates, that the meridian line is a scale of log. tangents | of the half complements of the latitudes. See Rhumb, Loxodromy, and Meridional Parts.

Spiral Pump. See Archimedes's Screw.

Spiral

, in Architecture and Sculpture, denotes a curve that ascends, winding about a cone, or spire, so that all the points of it continually approach the axis.

By this it is distinguished from the Helix, which winds in the same manner about a cylinder.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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SPHERICAL
SPHERICITY
SPHERICS
SPHEROID
SPINDLE
* SPIRAL
SPORADES
SPOTS
SPOUT
SPRING
SQUARE