2nd. When two circles, B, C, lie each without the other and do not meet the distance between the centres, DH > (DF+ HK,) the sum of the radii, and conversely, when the distance between the centres, DH > (DF+ HK,) the sum of the radii, the circles lie each without the other and do not meet.

PROP. 9.-THEOR.

If a point be taken within a circle, from which there fall more than two equal straight lines to the circumference, that point is the centre of the circle.

CON. Pst. 1 & 2.

DEM. Def. 17, I. A diam. of a O is any st. line through the cen., and terminated both ways by the Oce.-7, III.

E.1 Hyp.1 In O ABC let D be taken,

2

2 & from D to Oce more than two equal
lines, as D A = DB

=

DC;

3 Conc. then D shall be the cen. of the O.

C. Pst. 1 & 2.

D.1 Def. 17, I.

2 7, III.

3 H. ad imp.

Join DE, & prod. it to the Oce in F & G.

FG is the diam., & in FG is taken . not the cen.

D

.. DG greatest line from D to Oce, DC >DB, and DB > DA;

but they are also equal;-an impossibility;

E not the cen. of O ABC.

So no point but D the cen.

4 Conc.

5 Sim.

6 Conc.

7 Rec.

.. If a point be taken, &c.

Q. E. D.

COR.-1. From any other point than the centre only two equal st. lines can be drawn to the Oce, whether the point (7, III.) be within, or (8, III.) without the circle.

COR.-2. From three points given not in the same st. line the circumference of the circle may be found.

SCH.-This Prop. gives the criterion for determining the centre of a circle it is, that from a point, supposed to be the centre, to the circumference more than two points in the circumference shall be equally distant from the centre.

USE and APP.-By this Proposition the Problems may be demonstrated; 1st, To draw a circle through three given points, A, B, D; 2nd, To find the centre of a given circle ABDE; and 3rd, To determine the centre of ABD, an arc of a circle. The demonstration of the first, is equivalent to the demonstration of the other two.

C. 1 Pst. 1. 10, I. Join ⚫s A,B,D and bis. AB and

211, I.

BD;

at F and G raise perps. FC, GC
meeting in C;

3 Psts. 1. & 3. join CA, CB, CD and with either

as rad. from C desc. a O

Ak

4 Sol.

the passes through A,B and D.

4 Ax.1. &9.III... CA = CB=CD &.. C cen. of through

•s A, BD,

Q. E. F.

PROP. 10.-THEOR.

One circumference of a circle cannot cut another in more than two

points.

two

CON. 1, III. Pst. 1.

DEM. Def. 15, I. 9, III.

5. III. If two circles cut one another they shall not have the same centre.

SUP.-If possible let Oce FAB cut Oce DEF in more than s, as in B, G, F.

SCH.-"Two circles cannot have more than two points in common;" if they coincide in three points they will coincide in every point; or, "only one circle can be drawn through three given points."

PROP. 11.-THEOR.

If one circle touch another internally in any point, the straight line which joins their centres being produced shall pass through the point of contact.

SUP.-If FG produced do not pass through A, let it, if possible, fall as FGDH.

D.1 20,I.Def.15,I.

2

·.· FG + GA > FA, but FA = FH,
.. FG GA > FH,

3 Sub. Ax. 5. take away com. pt. FG.. rem. AG > rem. GH;

D.4 Def. 15, I.

5 Remk.ad imp. 6 Conc.

7 Remk. 8 Rec.

but AG GD.. GD > GH;

i. e. the less

the greater;-an impossibility.

.. FG joining F and G, being produced cannot fall except upon · A;

i.e. FG prod. must pass through A, the of contact. Therefore, If one circle touch, &c.

Q. E. D.

SCH.-When the distance, FK, between F and K, the centres of the two circles ADE and ABC is equal to the difference of the radii, AF and AK, the circles touch internally. For

C. Pst. 1.

Take L, a. in

ADE, and join KL, FL;

D.1 20,I.Def. 15, I. then. in ▲ FKL, FL < FK + KL, but KA = KL;

USE AND APP.-I. By aid of this proposition an oval may be described on any given major axis, as AB.

5 Pst. 3.

6 Pst. 1, 2.

7 Pst. 3. 8 Pst. 3. 911, III. 10 Sol.

three eq. pts. AF

FG= GB;

from F and G with FA and G B desc. Os AEGD and BEFD; from ⚫s of inters. D, E, draw DE, DG, EF, EG, prod. to

H, I, K, L ;

next from D with rad. DH or DI desc. arc MHON,
and from E with rad. E K or EL desc. arc MKPN;

the arcs AH, HI, IB, BL, LK, and K A touch ins H,I,K,L,
and the curve AHCIBLP KA will form an oval.

If the major axis be divided into four or more equal parts, by a similar method ovals more elongated, or with the minor axis in less proportion to the major, may be described.

N.B.-The oval thus described is only an approach to the true ellipse, the method being practically useful, not theoretically correct.

II. It is on the same principle that a Spiral is described, by successive semicircles taken alternately from two common centres A and B; for the line AB which joins them being produced, passes through the points of contact of the successive semicircles, l, m, n, o, p, q, r, s.

From A, the eye of the spiral,s with rad. AB describe the semicircle C; from B, with Bl, the semic. D; from A, with Am, semic. E; from B, with Bn, semic. F; from A, with Ao, G; from B, with Bp, H; from A, with Aq, I; and from B, with Br, K. The spiral may be continued to any extent in the same way.

A Spiral is a curve line making revolutions round the centre or eye of the curve, which do not return into themselves as the revolutions of a circle do. A Plane Spiral is generated by a continually increasing radius, and according to the law or rate of increase spirals differ in their curves. Our older English writers considered the circle and the ellipse to be spirals; but they are excluded by the above definition, inasmuch as their revolutions return into themselves, or to the very point from which they started.

Besides the above, the principal plane Spirals are, the Spiral of Archimedes or Conon, the Hyperbolic or Reciprocal Spiral,-the Lituus, and the Logarithmic Spiral; but they can only be noticed here by way of definition. 1st, When from a given point any number of lines are drawn forming equal angles at that point, and the length of each line increases in succession by an equal quantity, the curve which passes through these points is named the Spiral of Archimedes. 2nd, The Hyperbolic or reciprocal Spiral is a curve passing through the extremities of any number of arcs of circles of equal length measured from a given st. line. 3rd, The Lituus, so named from the crooked staff of the Roman augurs, is a Spiral to be thus described ;-"Let a variable circular sector always have its centre at one fixed point, and one of its terminal radii in a given direction. Let the area of the sector always remain the same, then the extremity of the other terminal radius as it revolves describes the Lituus." 4th, The Logarithmic Spiral, in which the radii make equal angles, and the spiral cuts them all at an equal angle, the length of the successive radii increasing in geometrical progression.

It may be observed that curves are infinite in variety, though only about thirty have received specific names. The Parabola, Hyperbola, Cycloid, Watt's Parallel motion curve, &c., are among them; but it would be out of place to explain them here.