IV. In Trigonometry we show 1st, by this Prop., in the last figure but one, that the half chord F B or FA of an arc A HB is perp. to the semidiam. C H, and consequently is the Sine of the half arc HB, or HA; 2nd, that the sides of a triangle (4, VI.) have the same ratio as the sines of their opp. angles. V. In the last figure but one, That part of the perp. to the chord which passes through the centre and is intercepted between the centre and the chord, namely, CF, is called the versed sine (see note to Def. 2, p. 4); and the radius, semichord, and versed sine form respectively the hypotenuse, base and perp. of a rt. angled triangle, and by 47, I., when any two are measured, or given, the third may be found ;-for, rad.semich.2 + vers. sine2; semich. =√rad.2 — vers. sine2; and vers. sine = √rad.2 semich.2 PROP. 4.-THEOR. If in a circle two straight lines cut one another, which do not both pass through the centre, they do not bisect each other. CON.-10, I. 1, III. Pst. 1. DEM.-Def. 15, I. 3, III. Ax. 11. All rt. /s are equal. Ax. 9. The whole is greater than its part. E.1 Hyp.1. Let ABC be a circle; 2 2. & A C, BD two st. lines cutting in E, but not both through cen. F; then AC, BD do not bis. one another; i. e. E not mid.. both of AC & BD. 3 Conc. A SUP. I. Let BD pass through the centre and AC not. C. 10, I. Def. 15, I. | Bis. BD in F, then F cen. of O. of BD; SUP. II. Let neither AC nor BD pass through the centre. E.1 Hyp. 1 If possible let both AE = EC and BE ED. C. 1, III. Pst. 1. Find F the cen. and join FE. D.1 H. 2 3. III. 3 H. 4 3, III. 5 Ax. 11. FE through cen. F, bis. .. FE cuts ACL, and FEA Again, FE through cen. F bis. BD not through F, .. FE cuts BD 1, and FEB is a rt. . 6 Remk. Ax. 9. i. e. a part the whole;-an impossibility; 7 Conc. 8 Rec. Q. E. D. COR.-No parallelogram except a rectangle can be inscribed in a circle. D.1 C. & 34. I... the diags. are diams. .. the diags. bis. each other in their centres. 2 Def. 15, I... the diags. are equal. 3 8, I. 34, I. 4 Conc. And the suppl. s are equal; .. each is a rt. ▲, and all the s are rt. s. i.e. the inscribed fig. must be a rectangle. USE.-The fourth Prop. has been employed to determine the eccentricity of the Sun's apparent path, or of the Earth's orbit described in a year. In an eccentric wheel the distance of the fixed point, or centre of rotation, E, round which the revolution is performed, from F, the centre of the wheel, will be found in the same way. PROP. 5.-THEOR. If two circles cut one another, they shall not have the same centre. CON.-Pst. 1. DEM.—Def. 15, I. Axs. 1 & 9, Join EC, & from E draw a st. line EFGH, E cen. of ABC .. ECEF. = EG. .. E not the com. cen. of Os ABC, CDG. SCH.-"This proposition may be better announced thus: "Concentric circles cannot meet, and that which has the lesser radius will be included within the other.""-LARDNER, p. 94. For. CA < CB .. © A within B. Q. E. D. A B PROP. 6.-THEOR. If one circle touch another internally, they shall not have the same centre. CON.-Pst. 1. DEM.-Def. 15, I. Axs. 1 & 9. E.1 Hyp. Let O CDE touch O ABC inter nally in C; 2 Conc. they have not the same cen. B SUP.-If they have the same centre let it be Join FG, and from F draw a st. line FEB, meeting the Os in F cen. of But FC • s E & B. CDE.. FC FB .. FE FB, = FE; i. e, the less gr.; which is impossible; Q. E. D. SCH.-Props. 5 & 6 may be combined into one; "circles with a common centre do not touch either externally or internally:" for the circle with the less radius will have every point within the circumference of the other, and consequently does not meet the other in any point whatever. PROP. 7.-THEOR. If any point which is not the centre be taken in the diameter of a circle, then, 1st, of all the straight lines which can be drawn from it to the circumference, the greatest is that in which the centre is, and the other part of that diameter is the least; and, 2nd, of any other st. lines, that which is nearer to the line which passes through the centre is always greater than the one more remote; also 3rd, those lines which make equal angles with the diameter are equal; and, 4th, from the same point there can be drawn only two equal st. lines, one upon each side of the diameter. CON.-Pst. 1. 23, I. At a given in a given line, to make a rectl. equal to a given rectl.. 3, I. From the gr. of two given lines to cut off a part equal to the less. DEM.-20. I. Any two sides of a ▲ are together gr. than the third side. 24, I. It two As have two sides of the one equal to the two sides of the C C.1 Pst. 1. 2. FA through cen. E the K H FD the other pt. of diam. AD the least; 3. FB nearer to FA> FC more remote and FC > Third, lines FB, FI making equals with diam. AD. 4. the line FB = the line FI. and Fourth, of lines from the same. F to the Oce, 5. only two eq. lines, FG, FH, one on each side of diam. AD. Join BE, CE, GE. I-A line FA through cen. E > any other line as BF. D.1 20. I. 2 Def.15, I. two sides of a ▲ > third side.. BE + EF > BF, but AE BE.. AE + EF i.e. FA> FB. = II.-The other part of the diam., FD < any other line FG. D.1 Def. 15, I. C.) Again · BE=CE & FE com. to AS BEF, CEF, but BEF> <CEF .. BF > CF; Also GF+ FE > GE, and EG = ED; take away com. pt. FE.. rem. GF > rem. FD. .. Of all st. lines from Fa not the cen. to |