S P 1 



fuller growth. The firft fort requires to be cleared of thefe 

 fuckers every two years at farthell. The layers (hould be 

 put down ill the autumn, or in the fpring, and may be taken 

 ©ff and planted as above, in the autumn or fpring following. 

 All the forts may be raifed in this way ; but it is moft pro- 

 per for fuch forts as do not fend off fuckers. The cuttings 

 may be made from the Ihouts of the preceding fummer, and 

 be planted out in a fhady border, in the early autumn. 

 When they have become well rooted, they may be removed, 

 and managed as the others. They fucceed in this way with 

 more difBcuIty than in either of the others. 



All the herbaceous forts may be increafed by feeds, or 

 parting the roots. The feed may be fown in the autumn, 

 or early in the fpring, but the firft; is the better mode, on a 

 bed of fine mould. When the plants appear, they fhould 

 be kept clear from weeds till the autumn, when they may 

 be planted where they are to remain, or in the nurfery for a 

 year or two. The roots Ihould be parted in the autumn or 

 fpring, when the ftems decay, before they fhoot out new 

 ones, being planted immediately where they are to grow. 

 The double-flowered and ftriped varieties can only be pre- 

 ferved in this way. 



They ail afford variety and ornament in the Ihrubbery, 

 and other parts of pleafure-grounds. 



SpiRjEA, African, in Botany. See DtOSMA. 



SPIRAL, in Geometry, a curve line of the circular kind, 

 v/hich, in its progrefs, recedes from its centre ; as in winding 

 from the vertex, down to the bafe of a cone. 



It is called, from its inventor, Archimedes' s fp'tre, or helix, 

 and is thus defcribed : divide the periphery of a circle 

 AP/iA {Plate 'KW. Geometry, Jig. ^.) into any number 

 of equal parts, by a continual bifeftion in the points p. 

 Into the fame number of parts divide the radius C A, and 

 make C M equal to one part, C m to two parts, &c. Then 

 will the points, M, m, m, &c. be points in the fpiral ; 

 which, connected, will give the fpiral itfelf. 



This is more particularly called the frjl fpiral ; and the 

 fpace included between its centre and the point A, the 

 fpiral fpace. 



The firft fpiral may be continued to z fecond fpiral, by 

 defciibing another circle with double the radius of the firft ; 

 and the fecond may be continued to a third, by a third 

 circle, &c. 



Hence, i. Ap is to the periphery, as C m to the radius : 

 wherefore, if the periphery be called p, the radius A C = r, 

 Ap= X, pm —y; then will C m = r — y : confequently, 

 2S p : r :: x : r — y, we ftiall h:xve pr — py = r x. 



2. UCm=y, then will rx=py; which equation the 

 fpiral has m common with the quadratrix of Dinoftrates, 

 and that of Tfchirnhaufen : and, therefore, r" x"" = p" y" 

 will ferve for infinite fpirals and quadratrices. See Qua- 

 dratrix. 



The fpiral line may be conceived to be thus generated • 

 If a right line, as A B (Plate XIV. Geometry, fig. r.j 

 having one end fixed at B, be equally moved round, "^fo as, 

 with the other end A, to defcribe the periphery of a circle ; 

 and, at the fame time, a point be conceived to move for- 

 ward equally, from B towards A, in the right hne A B, 

 fo as that the point defcribes that line, while the line gene- 

 rates the circle ; then will the point, with its two motion--, 

 defcribe the curve B, i, 2, 3, 4, 5, &c. which is called a 

 Jpiral line ; and the plain fpace contained between the fpiral 

 line and the right line B A, is called the fpiral fpace. 



Again, if the point B be conceived to move twice as flow 

 as the line A B, fo that it (hall ^et but half way .ilonf B A, 

 when that line fliall have formed the circle ; and if then you 

 imagine a uew revplutiga to be made of the line carrying the 



S P I 



point, fo that they (hall end their motion at laft together, 

 there will be formed a double fpiral line, as in the figure ; 

 from the manner of which may be eafily drawn thefe corol- 

 laries : 



1. That the lines B 12, B 11, B 10, &c. making equal 

 angles with the firft and fecond fpiral (as alfo B 12, B 10, 

 B 8, &c.) are in arithmetical proportion. 



2. The lines B 7, Bio, &c. drawn any how to the firft 

 fpiral, are to one another as the arcs of the circle inter- 

 cepted betwixt B A and thofe lines ; becaufe, whatever 

 parts of the circumference the point A defcribes, as fup. 

 pofe 7, the point B will alfo have run over 7 parts of the 

 line A B. 



3. Any lines drawn from B to the fecond fpiral, as B 1 8, 

 B 22, &c. are to each other as the aforefaid arcs, together 

 with the whole periphery added on both fides : for at the 

 fame time that the point A runs over 12, or the whole peri- 

 phery, or perhaps 7 parts more, ftiall the point B have run 

 over 12, and 7 parts of the line A B, which is now fup- 

 pofed to be divided into 24 equal parts. 



The area C A B D E of the fpiral of Archimedes 

 (Plate XIV. Geometry, fig. 6. ) is equal to one-third part of 

 the circle, defcribed with the radius C E. 



In like manner, the whole fpiral area, generated by the 

 ray drawn from the point C to the curve, when it makes two 

 revolutions, is the third part of a fpace double of the circle 

 defcribed with the radius 2 C D ; and the whole area, gene- 

 rated by the ray from the beginning of the motion, till 

 after any number of revolutions, is equal to the third part 

 of a fpace that is the fame multiple of the circle defcribed 

 with the greateft ray, as the number of revolutions is of 

 unit. 



Any portion of the area of the fpiral, terminated by the 

 curve C m A, and the right line C A, is equal to one-third 

 of the feftor C A G, terminated by the right lines C A 

 and C G, the fituation of the revolving ray, when the point 

 that defcribes the curve fets out from C. See Maclaurin's 

 Fluxions, Introd. p. 30, 31. See Quadrature of tie 



Spiral of Archimedes. 



Spiral, Logiflic, or Logarithmic. See Logistic and 

 Quadrature. 



Spiral of Pappus, a fpiral formed on the furface of a 

 fphere, by a motion analogous to that by which the fpiral 

 of Archimedes is defcribed in piano. 



This fpiral is fo called from its inventor Pappus. CoUeft. 

 Mathem. lib. ii. prop. 30. 



Thus, if C be the centre of the fphere (fig. 7.), A RB A 

 a great circle, P its pole ; and while the quadrant PM A 

 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 fpherical furface the 

 fpiral PF<7. 



Now if we fuppofe the quadrant P M A to make a com- 

 plete revolution in the fame time that the point, which 

 traces the fpiral on the furface of the fphere, defcribes the 

 quadrant, which is the cafe confidered by Pappus ; then 

 the portion of the fpherical furface terminated by the whole 

 fpiral, the circle A R B A, and the quadrant P M A, will 

 be equal to the fquare of A B. In any other cafe, the area 

 P M A a F Z P is to the fquare of the diameter A B, in the 

 fame proportion as the arc A a is to the wkole circumference 

 A R B A. And this area is always to the fpherical triangle 

 P A a, as the infcribed fquare is to the circle. See Mac- 

 laurin's Fluxions, Introd. p. 31 — 33. 



The portion of the fpherical furface, terminated by the 

 quadrant P M A, the arcs A R, F R, aad the fpiral P Z F, 

 admits of a perfeft quadrature, when the ratio of the arc 



A a to 



