PROPOSITIONS. PROP. 1.-PROB. To find the centre of a given circle. Sol.-Pst. 1. Let it be granted that a st. line may be drawn from any one point to any other point. 10, I. To bisect a given finite st. line. 11, I. To draw st. line at rt. Zs to a given st. line from a given point in the same. Pst. 2. A terminated line may be produced to any length in a st. line. DEM.-Def. 15, I. A O is a plane figure contained by one line, which is called the circumference, and is such that all st. lines drawn from a certain point within the figure to the Oce are equal to one another. 8, I. If two As have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal, the angle which is contained by the two sides of the one, shall be equal to the angle con tained by the two sides equal to them of the other. Def. 10, I. When a st. line standing on another st. line makes the adj. Zs equal to each other, each of these Zs is called a rt. L ; and the st. line which stands on the other is called a perpendicular to it. Ax. 1. Things which are equal to the same thing are equal to one another. E. 1 Dat. 2 Quæs. Let ABC be the given O; to find its centre. C.1 Pst. 1.10,1. Draw any chord AB, and bis. it in D; 2 11. I. at D draw DOI AB; 3 Pst. 2. 10,1. prod. CD to E, and bis. CE in F; 4 Sol. then F is the cen. of O ABC. 5 Assum. If not, take G as the cen; 6 Pst. 1. and join GA, GD, GB. D. 1, C.1,Def.15,1 In As ADG, BDG : DA=DB, GA=GB & G D com. 2 8,1. Def.10,1 . L ADG=L BDG, :. BDG is a rt. L. 3 C 2. Ax. 1. But :: FDB is a rt. L.._FDB=L BDG, i.e. the less = the gr.; 4 ad imposs. an impossibility .. G is not the cen. 5 Sim. So, no point out of CE is the cen. 6 And :: CE is bis. in F, 7 Def. 15, I. .. any other point in CE is not the cen., 8 Conc. .. No point but F is cen. of O ABC. Q. E. F. C. 3, CoR.--If in a 0 a st. line, CE, bisects another, AB, at rt. 48, the cen. of the O is in the line, C E, which bisects the other. Sch.-1. When thc · Gʻ is taken in the diam. CE, the demonstration holds good only if Gʻ coincides with F; should this not be the case, it is evident that GʻCEĞ'E.. Gʻ is not the cen. 2. The rigour of the reasoning would have been greatly promoted, if Euclid, previously to the above Problem, had established the following proposition ; Any point, D, fig. 1, F, fig. 2. being assumed within a O, a rt. line, HD or HF, drawn through it and produced indefinitely in both directions, will meet the in two points, and not in more ; and every point of the line between these two points of intersection will be within the O, and every point beyond them without it. LARDNER's Euclid. p. 91 I. Let H D through D, also pass through the cen. C. Fig. I. D Also from C make CD, CH each H A II. Let H F through F not pass through the cen. C. FIG. 2. rad. CA..CD<CA. i. e. CA? – CD?, H rad. :. CA=CB = rad. H 4 5 ... the ·S A & B each on the Oce. the O .8 G & H are without the O. points. Q. E. D. .. the USE.-1. Practically the centre of a O is found, by bisecting any chord, A B with a perp. CE, terminated in the Oce; and C E being bisd. in F, F is the centre. 2. The First Prop. bk. III. is applicable to all cases in which the centre of any circular object, as of the horizontal section of a tree, may be required. A circular disk of metal, a wheel, a flower-bed, any object possessing the circular form will have its centre found in the same way. PROP. 2.—THEOR. If any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle. Con. 1, III. Pst. 1 & 2. DEM. Def. 15, I. 5, 1. The angles at the base of an isosc. A are equal to each other ; and if the equal sides be produced, the Zs on the other side of the base shall be equal. 16, I. If one side of a A be produced the ext. _ is greater than either of the int. opp. angles. 19, 1. The gr. L of every A is subtended by the gr. side, or has the gr. side opposite to it. E1 Hyp. | Let A B C be O, and A B any two s in the Oce; 2 Conc. the st. line from A to B within the O. Sup.-If not within, and it is possible, let it be without, as A E B. F; . 3 Pst. 1 & 2. join D F, and prod. it to meet A B in E. A E B D. 1 Def. 15, I. 5, I. Then :: D A = DB, .LDAB = _ DBA; 2 C. & 16. I. &:: AE in A DAE is prod. to B; .. ext. L DEB > int. & opp. _ DAE. 3 D. 1. But _ DAE= _ DBE; ii. DEB > LDBE; 4 19. I. and .:. DB > D E. 5 Def. 15, I. ad imp. Now D B=DF .. DF> DE; an impossibility; 6 Conc. -. the line from A to B not without the O. 7 Sim. So, AB does not fall upon 8 Conc. .. A B is within the O. 9 Recap. . If any two points be taken 8:c. Q. E. D. the Oce; Cor. 1.-A st. line, A B, cannot cut the Oce of a 0 in more points than two; for, every st. line joining any two points in the O ce falls within the o, neither co-inciding with any other points in the Oce, nor meeting it except in the two given points. Cor. 2.-A st. line which touches a circle meets it only in one point. Cor. 3.-A circle is concave towards its centre. SCH.-Instead of the ez absurdo demonstration this Proposition, a direct method of proof, founded on Axiom A, bk. III. was given by COMMANDINE, who lived between A.D. 1509 and 1575 ; he applied himself to mathematics at Verona, and in 1572, at Pesaro, published Euclid's Elements in fifteen books, in Latin. to B, 3 10, I. C. 1 Assum. In the given line A B, take any · E; D E, D B. .: LDAB= _DBA; 2 C. and in A AED, A E is prod. LDAE; 4 D. 1. but / DAE=LDBE; ... Z DEB > LDBE. 5 D. 4. 19, 1. But _ DEB > LDBE..DB > DE; 6 Remk. i. e. D E the dist. of E from D< DB the rad. 7 Ax. A. III. ... the . E is within the circle. 8 Sim. between A and B; 9 Conc. .. the line AB, joining A and B falls within the O. Q. E. D. So is every USE.—On this proposition are grounded those which show, that a circle touches a st. line in only one point; for if the st. line touched two points of the Oce the st. line would be drawn from one point of the Oce to the other, and consequently would fall within the circle, contrary to the very definition of such a line, that it does not cut the circumference. THEODOSIUS of Tripolis, a mathematician who lived some time after the reign of Trajan, compiled a work on the Properties of the Sphere and of the circles described on its surface, an edition of which was published at Oxford in 1675: he used Prop. 2. bk. III. to demonstrate that a Globe resting on a plane surface cannot touch the plane in any but a single point; otherwise the plane would enter the globe. PROP. 3.-THEOR. If a straight line drawn through the centre of a circle bisect a straight line in it which does not pass through the centre, it shall cut it at right angles; and conversely, if it cuts it at right angles, it shall bisect it. Con. 1, III. Pst. 1. DEM. Def. 15, I. 8, I. Def. 10, I. 5. I. 26, I. If two As have two Zs of the one equal to two Zs of the other, each to each, and one side equal to one side, viz., either the sides adj. to the equal Zs in each, or the sides opp. to them, then shall the other sides be equal, each to each, and also the third of the one to the third L of the other. |