Draw (Case 7) a O thro. B to touch the constructed line at N, and constructed at M.

Let O be centre of this ; let ON cut ED in R; let OA cut A in P; and let OB cut ✪ B in Q.

centre O and rad. OP goes thro. Q and R, and touches given Os and line at P, Q, R.

The four solutions of Case 7 will give four solutions here: four more will come from drawing || to ED on same side as B.

Thro. A draw a to touch these last two constructed Os (Case 6) in M, N respectively.

Let BM, CN meet in O; and join OA.

Let OA cut OA in P; OB cut O B in Q; and OC cut O C in R.

and touches given Os in

O, centre O radius OP, goes thro. Q and R;

P, Q, R,... OA, OB, OC are lines of centres.

It will easily appear that there are eight solutions.

EXERCISES ON THE TANGENCIES.

1. For certain relations among the data, some of the preceding constructions fail: investigate the necessary modifications to be made, when in—

Case (3), AB is parallel to XY; or A is in XY:

Case (4), given lines are parallel; or A is in one of them:

Case (5), A and B are equidistant from centre of given circle; or A is on given circle:

Case (6), given circles are equal.

2. In the figure of Euc. i. 1, describe a circle to touch the given line, and the two circles of construction.

3. In Case (7) there are generally two circles which can be drawn to have external contact with the given circle: if A is supposed variable, find its Locus under the condition that these two circles touch each other.

4. Given three circles, describe another to touch two of them, and—

1o, bisect the circumference of the third;

2o, cut the third orthogonally.

5. If we consider a point as an infinitely small circle, and a line as an infinitely large circle; show that Cases (3) to (10) may all be solved by the following construction (Gergonne's)—Let A, B, C be three given circles; O their orthogonal circle: let the chords of intersection of O with A, B, C meet an axis of similitude of A, B, C in P, Q, R; and from P, Q, R draw pairs of tangents to A, B, C respectively: then the two circles through the six points of contact of these tangents will touch A, B, C: also, since there are four axes of similitude, there will be eight circles of contact.

SECTION vi-INVERSION.

Def. Every two points P and Q, on a diameter of a circle (centre C) such that the rectangle under CP, CQ is equal to the square on the radius, are called inverse points with respect to that circle. Also the circle is called the circle of inversion; and its centre is called the centre of inversion.

Note-Any fixed circle may be taken as the circle of inversion.

Def. The inverse of a Locus is the Locus of the inverses of all points on it.

Thus if to every position of a point P on a Locus we take the corresponding inverse p, then the Locus of p is the inverse of the Locus of P.

Note-In what follows the radius of the circle of inversion is denoted by R.

THEOREM (1)—The inverse of a line is a circle through the centre of inversion.

Cor. A line is the radical axis of its inverse and the circle of inversion.

THEOREM (2)-The inverse of a circle through the centre of inversion is a line.

Note-We can now give the theory of the Peaucellier movement: for, referring to the figure on p. 5, by the symmetry of the instrument, P, O, Q will always be in one line; and if ANB cuts this line in N, the Ʌs at N are right, and N is the mid point of the diags. of the rhombus.

.. P, Q are inverse points; and, since Q moves on a thro. the centre of inversion O, P will move on a st. line.

THEOREM (3)—The inverse of a circle not through the centre of inversion is a circle.

a

Let A be centre of given O; P any point on it.

From C, the centre of inversion, draw CP, and let it cut the circle again in Q. Let p, q be respective inverses of P, Q.

Cor. (1). C is a centre of similitude of the OS.

Cor. (2). If CT is tang. to A, rad. of ✪ a: rad. of A

THEOREM (4)-Each point of intersection of two Loci is the inverse of a

point of intersection of their inverse Loci.

For the inverse of a pt. of section of the Loci must be a pt. on each of the inverse Loci :

i. e. must be a pt. where they intersect.

Def. If one of the points of section of a secant of a circle is made to move up to the other, then the limiting position of the secant (to which it constantly approaches, and which it ultimately assumes, when the points are brought indefinitely near together) is called a tangent to the circle.

Note (1)—This definition of a tangent will be seen to amount to the same as Euclid's, if we consider that it may be put thus-a tangent is a secant through two coincident points, that is through one point.

Note (2)—The angle between a line and a circle, is the angle made by the line with the tangent at the point where the line cuts the circle; and the angle between two circles, is the angle between their tangents at a point of section.

Def. A variable line from a fixed point to a fixed circle is called a radius vector of the circle, with respect to that point as origin.