MATHEMATICS. PLANE GEOMETRY. 



[REMARKS ON BOOK in. 



s in. beo*a*a A B U divided equally in M, and 

 unequally in P, AP-PB + PM 1 



+ H. -MB'.t Add CM 1 to 

 ~ch, /. A P P B + P M> +C JP 

 -MB + CM'; that Is, A T 

 i. PB + PC 1 - BCT* 

 Take P C from each, .'. A P P B 

 B C* P C 1 . But 80*18 

 always the same wherever in tho 

 circumference the point B may 

 be.and PC 1 U alwayi the tame 

 whatever chord be drawn through P, .'. B C 1 P C* 

 U alwayi the tame, .'. A P-P B U always the name, what- 

 ever chord (A B) be drawn through P ; . . if two chords 

 cut one another in any point P. the rectangle con- 

 tained by tho segments of one of them is equal to that 

 contained by the segments of the other. 



Proposition XXXVL Let B be any point without a 

 circle P E F. It U required to 

 prove that whatever line B P F 

 be drawn, cutting the circle, the 

 rectangle B F -B P shall be equal 

 to the square of BE, a lino 

 drawn touching the circle. 



1 III. Find the centre C,* 

 and draw C B, C P, C E. With 

 this same centre C, and radius 

 C B, describe a circle, and pro- 

 duce B F to meet its circum- 

 ference in A. Then (last Proposition) A P-B P = B C s 

 ti. PC 2 = BC EC S = BE'.f Draw the 

 perpendicular C M to A B ; then C M bisects both A B 



s in. and FP,* .-. A F = PB, .-. AP = B F. But 

 it was proved that AP'PB = BE'; .-. BF'BP = 

 B E s , which was to be demonstrated. 



An obvious corollary follows from this proposition, 

 viz. 



COR. 1. Since from any point without a circle two 

 + I7I. lines may be drawn to touch the circle, t these 

 two touching lines must be equal. 



We shall only further observe that, besides the par- 

 ticulars mentioned above, in which slight departures 

 from the authorised text of Euclid have been made, 

 other modifications, chiefly in the diagrams, have been 

 introduced in certain propositions ; but we have nowhere 

 ventured upon any change, where the circumstances of 

 the case did not fully justify and require it. 



EXERCISES ON BOOKS I., It., III. 



1. Prove the converse of Proposition XXII. Book 

 III.; namely, if the opposite angles of a quadrilateral be 

 together equal to two right angles, a circle may be de- 

 scribed about it. 



2. A trapezium may be inscribed in a circle, provided 

 two of the opposite sides are parallel, and that the two 

 non-parallel sides are equal. 



3. If a quadrilateral be described about a circle, that 

 is, if the four sides touch the circle, one pair of opposite 

 sides will always be equal to the other pair. 



4. If from the vertices of the three angles of a tri- 

 angle, perpendiculars be drawn to the opposite sides, 

 they will intersect in the same paint. 



5. If two circles cut one another, the line joining the 

 intersections of tho circumferences, shall be perpendicular 

 to the line joining the centres. 



6. If two circles cut one another, and from one of the 

 points of intersection diameters be drawn, the extremi- 

 ties of these diameters shall be in the same straight line 

 as the other point of intersection. 



7. If any two chords of a circle intersect at right 

 angles, the squares upon their four segments will, to- 

 gether, be equal to the square upon tho diameter. 



8. If two circles touch each other, any straight line 

 through the point of contact will cut off similar segments. 



9. If a quadrilateral be described about a circle, the 

 angles subtended at the centre by one pair of opposite 

 sides, will, together, be equal to those subtended by the 

 other pair ; that is, to two right angles. 



10. Find a point in the prolongation of a diameter of 

 a circle, from which, if a line be drawn to touch the 

 circle, it shall be equal to a given straight line. 



1L Two chords AD, B C, are drawn in a semicircle 

 from the extremities of the diameter A B ; the chords 

 intersect in P. Prove that the rectangles A D 'A P, 

 B C'B P are together equal to the square of the diameter. 



12. If from any point in the diameter of a semicircle 

 two straight lines be drawn to the circumference, one to 

 tho middle of the arc, tho other at right angles to the 

 diameter, the squares upon these two lines always 

 amount to the same . urn, wherever the point be taken. 



13. From a given point without a circle to draw a 

 straight line to cut it, and to terminate in the circum- 

 ference, such that the intercepted chord may have a 

 given length, not greater than the diameter of the circle. 



CHAPTER IV. 



ELEMENTS OF EUCLID. BOOK IV. 



DEFIXITIONS. 



A rectilineal figure Li said to be inscribed 

 in another rectilineal figure, when all the 

 vertices of the former are upon the sides of 

 the latter, each upon each. 



J 



II. 



A rectilineal figure is said to be described about another, 

 when all the sides of the former pais through the vertices 

 of the latter, each through each. 



in. 



A rectilineal figuro is said to be in- 

 cribed in a circle, when all tho vertices 

 of the former are upon the circumference 

 of the circle. 



IV. 



A rectilineal figure is said to bo 

 described about a circle when each side 

 of the former touches the circumference 

 of the circle. 



A circle is said to be inscribed in a 

 rectilineal figure, when the circum- 

 ference of the circle touches each side of 

 the figure. 



vr. 



A circle is said to be described about 

 a rectilineal figure, when its circum- 

 ference passes through all the vertices of 

 that figure. 



VII. 



A straight line is said to be placed in a 

 circle, when the extremities of it are in 

 the circumference of that circle. * 



Sec ante, Book III., Prop. I., III., and IV. 



