this latter intercepted by the lines will be of a given magnitude.

III. Let A and B be two given points without a circle, given in magnitude and position; a point C may be found within the circle, such, that if any right line be drawn through it, cutting the circle in F and G, and AF, BG be joined, the square of AF will have to the square of BG, a ratio which is compounded of the ratio of FC to CG, and a certain given ratio; which ratio is also to be found.

IV. Let a circle be given by position, and let A and B be two given points; a point C may be found, such, that if any right line be drawn through it, meeting the circle in Ď and E, and AD, BD, also AE, BE be joined; there is a certain given space, which mean proportional between AD2+BD2 and

is a

AE2+BE2.

V. Let a circle be given by position, and let A and B be two given points, a point C may be found, such, that if any right line be drawn through C to meet the circle in Ď and E, and AD, BD, AE, BE be joined ; AD+BD: AE+EB2:: DC : CE.

VI. Having given any number of points A, B, C, D, &c. in the periphery of a given circle, it is required to find another point P (in the periphery,) such, that joining PA, PB, PC, &c. the sum of the lines PA, PB, PC, &c, may be equal to a given line.

VII. A circle and a point being given, another point may be found, such that the straight lines drawn from them to any point in the circumference shall have a ratio which will be given.

VIII. If from two points in the diameter of a circle equally distant from its centre, straight lines be inflected to the extremity of any arc, another point may be found in the diameter, from which a straight line drawn to the termination of the double arc shall contain with the radius, a rectangle equivalent to that under the inflected lines.

IX. A circle and a straight line being given, is position, a point may be found, such that any straight line drawn from it to the given line, shall be a mean proportional between the segments intercepted by the given circumference.

X. A point being given in the diameter of a given circle, another point in the same extension may be found, such that the angle contained by two straight lines drawn from it to the extremities of a chord passing through the given point, shall be bisected by the dia

meter.

XI. A point being given in the circumference of a circle, another point may be found, so that two straight lines inflected from them to the opposite circumference, shall cut off on a given chord, extreme segments, whose alternate rectangles shall have a given ratio.

XII. Two points and two diverging lines being given in position, straight lines, inflected from those points to one of the diverging lines, intercept segments on the other, from points that may be found, and containing a rectangle, which will be also assignable.

XIII. Three diverging lines being given in position, a fourth may be found, such that straight lines can be drawn intersecting all these, and divided by them into proportional segments.

XIV. Let A and B be two given points, and CDE a circle given by position, there is a point P in the right line joining A, B, such that if from P any right line whatsoever be drawn, meeting the circle in C, D, the points A, B, C, D shall be in the circumference of the circle.

XV. Let there be a circle given in position, and let A, B be two given points: a point C may be found within the circle such that if through it there be drawn any line meeting the circle in D, E and AD, BD, AE, BE, be joined; the rectangle ADB will be to the rectangle AEB:: CD: DE.

XVI. If two Os be given in magnitude and position, and tangents DI, DH, 'be drawn from any point to the Os so as to have a given ratio; two points P and R may be found, such, that the right lines PD, RD being drawn from those points to the intersection of the tangents, shall always have the same ratio as the tangents DI, DH.

XVII. Two right lines being given in position, a circle may be found, such, that if another be described upon any radius thereof as a diameter, the chord of the arc

intercepted by this latter line given in position shall be of a given length.

XVIII. Let there be two given points in the circumference of a circle given in magnitude and position: a circle may be found, given also in magnitude and position, such, that if from any point in the circumference straight lines be drawn to the given points, and a tangent to the given circle; the 2 of the tangent shall have to the rectangle of the lines drawn to the given points a given ratio.

XIX. Let a circle be given in position, and let A and B be two given points; a point Ċ may be found, such, that if any right line be drawn through it, meeting the in D and E; and AD, BD, also AE, BE be joined; there is a certain given space which is a mean proportional between AD2+BD2 and AE2+BE2.

PROBLEMS.

A circle touching a line or lines, a circle or circles, or passing through a point or points, affords in all ten problems.

I. To describe a circle through three points.

a circle to touch three given lines. to pass through a point and touch two lines. to pass through two points and touch a line. to pass through two points and touch a circle. to pass through one point and touch a line and a circle.

to touch two lines and a circle.

to touch two circles and a line.

to touch two circles and pass through a point.

to touch three circles.

XI. Given two unequal Os and a line in position; to find a point such, that if tangents be drawn from it, m times the of one tangent + or 2 n times the D2 of the other may have a given magnitude; or that the tangents may have a given ratio.

XII. Given a tangent to a O in position, to inflect a line between it and the O, parallel to a line given in position, so that the tangent cut off may be to it in a given ratio.

XIII. Through a given point within the greater of two given Os to draw a line, so that the part thereof intercepted by the less may have to the part intercepted by

the greater a maximum ratio.

XIV. The area of a rectangle being given to describe it, so that its sides shall pass through four given points.

XV. A ▲ being given in position, and two points being also given; it is required to draw lines through those points to intersect one another in one of the sides of the A, and meet the other sides, so as to cut off segments adjacent to the given points in them, having to each other a given ratio.

XVI. A polygon being given in magnitude and position, it is required to describe a polygon of an equal number of sides, having its several s on the several sides of the given polygon, and such that the perimeter shall be the least possible?

XVII. Three Os being given by position, it is required to draw a tangent to each of them, so as to intersect in the same point, and have the segments between that point and the points of contact in a given ratio to one

another.

XVIII. Four right lines being given in a position, it is required to describe a A whose s may be upon three of the lines, and whose sides may make given s with the remaining line?

XIX. Three right lines being given in position, and three points being also given, it is required to describe a rectilineal figure, whose sides may pass through the given points, and whose s may be upon the lines given in position.

XX. A ▲ being given by position, and two points being also given, it is required to draw lines through these points to intersect one another in one side of the A, and meet the other sides, so as to cut off segments adjacent to the given points in them, such, that the rectangle con⚫ tained by those segments may be to a given space.

XXI. Three points being given in position, it is required to construct a A, the sides of which may be given in magnitude, and pass through the given points.

XXII. From a given point P in the periphery of a given semicircle APR, it is required to draw the right

line PB meeting the diameter AR in B, so that BC being drawn perpendicular to AR meeting the periphery in C; BC2 BP shall be of given magnitude.

XXIII. Let two Os have the same centre, and let A, B, be two given points in the circumference of one of them. It is required to inflect AC, BC, to the circumference of the other O, so that ACX CB may be of a given magnitude.

XXIV. Any number of points A, B, C, D, &c. being given in the periphery of a given O, it is required to find another point P (in the periphery), such, that joining PA, PB, PC, &c. the sum of those lines may be to a given

line.

XXV. From two given points in the diameter of a given semicircle, it is required to draw two right lines to meet each other at some point in the periphery, so that the sum or difference of the respective rectangles under those two lines, and the two given ones (M and N) may be to a given magnitude.

XXVI. To construct a trapezium, there is given one of the sides, both the diagonals and the formed by their intersection, also the made by either diagonal, and the side opposite the given one.

XXVII. Given the positions of the centres of gravity of three As, formed by lines drawn from the angular points of a A to the centre of the inscribed circle; to determine the position of the A.

XXVIII. In a given trapezium to inscribe another, such that three of its sides may pass through a given point, and the remaining side be of a given length.

XXIX. Let there be any number of right lines AP, BP, CP, DP, &c. given in position, and let Z be a given point without them; it is required to draw a through the points P and Z to intersect the given lines in A, B, C, D, &c. so that the sum of all the intercepted parts AP, BP, CP, DP, &c. may be to the given line.

XXX. Let two Os, one of which is included within another, be given in a plane, and let there be given a point situated within both the Os, in the line joining their centres; it is required to describe a O, from the given point as a centre, to meet the peripheries of the given Os,