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clearly illustrated by the following example.

"Given a circle ABC, fig. 5, a line DE and a point F, to find a point G in DE such that a tangent GC=GF." By supposing it done, then constructing as appears from the fig.; DK=DK'=DL is .. given; hence find the point on the line that will be the centre of the circle through KFK' &c. The problem is impossible, and .. fails, if F be on the line.

If F and K coincide, it becomes true of every point in the line, and.. the question may be proposed in a porismatic form, viz. "Given a circle and a line; a point may be found such that the distance from it to any point in the line may be equal to the tangent to the same point."

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These propositions most frequently may be enunciated either as Theorems or loci, but however they are quite distinct. The ancients sometimes considered them as local Problems, but that they are distinct will appear by the following example. Required to draw a line from a given point G, fig. 6, intersecting a triangle ABC, so that the perpendicular from one angle may be equal to those from the others." Let GS be the drawn line, take H the middle point of AC, then if the perpendicular from B=those from A and C it is equal to twice that from H... BH is bisected at S, .. the point S is given. If G and S coincided, the question would be porismatic, and could not be enunciated as a locus.

From this account of the origin of Porisms they may be defined. 66 Propositions asserting the possibility of finding such conditions as will render a certain Problem indeterminate or capable of several solutions."

We may now proceed to the enunciation of questions. They contain, besides some that perhaps have not heretofore appeared, a collection of the best that could be procured from the most noted geometrical works, and from the English Diaries and Repositories for many years back. They only require on the part of the Student a thorough knowledge of the Elements, and doubtless some portion of that δυναμις αναλυτικη so much prised by the Ancients.

In their arrangement a mutual dependance has been considered as much as the difficulty of the subject would allow. Wherever a figure is alluded to in these questions, the reader will always be able to construct it as directed.

THEOREMS.

I. It may be proved by the first Book that the perpendiculars from the middle points of the sides of a ▲ intersect in one point, hence prove the same of the perpendiculars from the s.

II. In the same ▲, the points of occourse of the perpendiculars from thes, of the bisectors of the sides from thes and of the perpendiculars from the middle points of the sides, lie in directum. (This may be proved by the 1st Book.)

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III. If from any point either inside or outside a rectilineal figure, perpendiculars be let fall on all its sides, + the sum of the 's of the alternate segments made by them will be equal.

IV. If in any trapezium two opposite sides be bisected, the sum of the 's of the other two sides, together with the's of the diagonals, is equal to the sum of the 's of the bisected sides, together with four times the2 of the line joining those points of bisection.

V. Drawing that diameter of the circumscribing that bisects the base, the extremity of it that is next the base, is equidistant from the extremity of the base, the centre of the inscribed and the centre of the O that touches the base and sides produced.

VI. In the isosceles AABC having AB = BC, if the point D be taken in AB, and D C joined. The solid under AC2 and BD is to the difference of the solids under AB, DC2 and AD2.

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VII. The perpendicular from the vertex on the base of an equilateral A, is equal to the side of an equilateral ▲ inscribed in a O, whose diameter is the base.

VIII. If on the sides of a ▲ segments of Os be described similar to a segment on the base, and from the extremities of the base tangents be drawn intersecting their circumferences; the points of intersection and the vertex of the A will be in directum.

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IX. The centre of a O, which touches the semicircles described on the two sides of a right angled A, is in the middle point of the hypothenuse.

X. If on three sides of a right angled ▲ semicircles be described, and with the centres of those described on the sides, Os be described touching that described on the base; they will also touch the other semicircles.

XI. If two Os touch one another internally at a point t, and a line be drawn intersecting the outer in the points a, a' and the inner in the points b, b, the atb b't a' (32. 3. Elr.)

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XII. The sum of the O's of the lines from any point of a to the s of the inscribed 2 is to eight times the of the radius. And if drawn to the s of the 2 inscribed equilateral ▲ they are equal to six times the rad. 2

XIII. Any two radii of a O being drawn, and any point taken in the intercepted arc, and perpendiculars being let fall on the radii from said point; the line joining the extremities of these perpendiculars is to the perpendicular from the extremity of one of the given

radii on the other.

This theorem is of use in trigonometry.

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XIV. Let mon be the perpendicular diameter of the circumscribed O, and bv a perpendicular of the A produced to the circumference, and vs a perpendicular from the point in which this last meets the circumference on the perpendicular diameter, then onXns the 2 of half the difference of the sides.

XV. If two Os touch one another, and also touch a right line; the part of the line between the points of contact is a mean proportional between the diameters of the Os.

XVI. If the opposite sides of a quadrilateral figure inscribed in a be produced, the of the line joining the occurses the sum of the 'rs the tangents from the same points.

XVII. When any number of Os pass through the same two points, tangents drawn from any point in either of the two external circumferences to the remaining Os shall be to one another in a given ratio.

XVIII. If from any point in the periphery of a per

pendiculars be let fall on the three sides of the inscribed A; the feet of those perpendiculars lie in directum.

XIX. P is a point in the diameter of a O, and PB a perpendicular to the diameter; draw any chord PA and a tangent AB meeting PB in B, and draw BD and CE (C the centre) perpendicular to PA; then PE=DÀ.

XX. Within a given ▲ suppose another A to be inscribed, by joining the middle point of its sides; and again within this A suppose another A to be inscribed by joining the middle point of its sides, and so on ad infinitum, what will be the limit of the aggregate, of the sum of the 's of the sides of all the As so formed?

XXI. If the circumference of a O be divided into any number of parts, and perpendiculars from the points be let fall on any diameter; the sum of the perpendiculars on one side of the diameter is = to the sum of those on the other.

XXII. Prove that if from the extremities of the side of a pentagon inscribed in a straight lines be drawn to the middle of the arc subtended by the adjacent side, their difference is to the radius; the sum of their 's is three times the □ of the radius; and the rectangle contained by them is to the of the radius. XXIII. If four Os touch, either internally or externally three sides of any quadrilateral, their centres are always on the same circumference.

XXIV. If the sides of a circumscribed quadrilateral to a touch at the s of an inscribed one, their diagonals cut in the same point.

XXV. The diagonals of a quadrilateral in a O, are as the sums of the rectangles of the sides that terminate in them.

XXVI. The O's of the side of an equilateral A in a O, is = the area of a regular dodecagon in the same.

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XXVI. The 2 of the side of an equilateral ▲ = 3 0 of radius of circumscribing O (12. 2. El.)

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XXVIII. The of the side of a regular pentagon in a the sum of the 's of the sides of a regular hexagon and decagon, in the same O.

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XXIX. Taking a point in the periphery of a tween any two s of an inscribed equilateral A; the sum of the lines to it from the adjacents is to the line from the opposite. (Cor. 16. 6. Elr.)

XXX. The solid under three sides of a A divided by

four times the area, is to the radius of the circumscribing O. (Cor. 16. 6. Elr.)

XXXI. If equilateral As be described on the three sides of a A, and a, a', a" be the centres of their inscribed Os, the Aa a' a" formed by joining those points is an equilateral A.

XXXII. The distance between the centres of the inscribed and circumscribing Os to a A, is a mean proportional between the radius of the circumscribing and the difference between that radius and the diameter of the inscribed. (Cor. 6. 2, and 4 and 16 B. 6).

XXXIII. Draw the tangents a m, bn at the ends of the diameter of a semicircle, and any tangent mpn crossing them, join an, bm intersecting in o, join po and produce it to the diameter at v; then pov is at rights to the diameter, and bisected at o.

XXXIV. If through the middle point of any chord of a two chords be drawn; the lines joining their extremities will intersect the first chord at = distances from the middle point.

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XXXV. Let F E, the diameter of a O, described about any plane ▲ ACB, bisect the base AB perpendicularly in H and let EP, FS be drawn perpendicular to AC; with the radii CH, CS and centre C describe Os CQG and CSD meeting AC in G and S; and let S be the nearest point to C, then from P and G draw the tangents PQ, GD and the rectangle PQXGD will be = to the area of the ▲ ACB.

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XXXVI. In any plane A, the difference between the of half the sum of the sides and the of the line bisecting the base, multiplied into the difference between the 2 of the line bisecting the base and of half the difference of the sides, is to the 2 of the area of the A.

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XXXVII. The sum of the three radii of the Os that touch one side, and the other two produced of a ▲ is = to four times the radius of the circumscribing the radius of the inscribed O.

XXXVIII. If equilateral As be described on the three sides of a ▲, the lines joining their vertices with the opposites of the A, all pass through one point.

XXXIX. If a O touch two given Os, the line joining

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