REMAKKS ON BOOK III. ] 



MATHEMATICS. PLANE GEOMETRY. 



571 



4 3 in. the centre F,t and draw 

 F E, F B, F D : then E is a right 

 is in. angle.* And because 

 DE touches, and DC A cuts the 

 + 36HI. circle, A D-D C=DE 2 :f 

 Hyp. but AD-DC= DB 2 ,* /. 

 D E = D B ; also F E = F B, .'. 

 D E, E F are = D B, B F, each to 

 each ; and the base F D is common 

 to the two triangles D E F, D BF, 

 + 81. ' . the angle E = B : t 

 but E was shown to be a right 

 angle, . . B is a right angle : and 

 B P prolonged is a diameter : bu 

 the straight line which is drawn at right angles to a 

 diameter, from an extremity of it, touches the circle, 



16 in. . . D B touches the circle ABC;.', if from a 

 point, <tc. Q. E. D. 



REMARKS ON BOOK HI. 



The third book ia wholly occupied with propositions 

 concerning the circle. In the definitions we have intro- 

 duced two terms, the want of which is very much felt in 

 the Elements the terms arc and chnrd. The absence 

 of this latter term has been the occasion of much loose- 

 ness of expression in certain propositions of Euclid's 

 Third Book : in Proposition I., for instance, as Thomas 

 Simpson has justly noticed, we are directed to "draw 

 within the circle any straight line AB," without the 

 limiting condition that its extremities are to be in the 

 circumference ; and the same want of precision is ob- 

 servable in Propositions III. and IV. By "a straight 

 line in a circle," Euclid certainly means exclusively 

 "a straight lino in a circle, and terminated both ways 

 by the circumference ;" but this should have been ex- 

 pressly stated, and not left to be inferred from obser- 

 vation. Had Definition VII., Book IV. (Simson's 

 Euclid), been placed among the definitions of the third 

 book, the objection here advanced would have been re- 

 moved ; since, according to that definition, "A straight 

 line is said to be placed in a circle, when the extremities 

 of it are in the circumference of the circle ;" but it is 

 better to call such a line a chord, the name given to it 

 in all other geometrical inquiries. 



The demonstration in Proposition I. of Simson's 

 Euclid is imperfect, since all that it proves is, that the 

 centre of the circle must be in the line C E, without de- 

 termining the particular point in that line ; the omission 

 is supplied in the present edition. 



The enunciation of Proposition VI. in Simson is, "If 

 two circles touch one another internally, they shall not 

 have the same centre ;" but as it is plain that each circle 

 cannot be within the other, the word internally should 

 be omitted. If two circles touch externally, then, since 

 eich must be wholly without the other, it is obvious that 

 they cannot have the same centre. In Proposition VII. 

 certain lines are introduced by Euclid namely, the 

 lines E C, E G long before they are wanted ; and the 

 diagram is thus unnecessarily complicated at the outset 

 of the demonstration. We have already stated (page 

 M'.>), with a view to an orderly and systematic arrange- 

 ment of the steps of the argument, that the diagram 

 should not proceed, in advance of the text : the more 

 you depart from this plan, the more you depart from 

 simplicity ; since you thus encumber the diagram with 

 lines that serve no purpose but to distract your atten- 

 tion. In demonstrating this seventh proposition with- 

 out the book, you should commence by exhibiting in 

 your diagram only the lines F B, F C, in addition to the 

 diameter A D : you should then draw E B, for the pur- 

 pose of proving that F A is greater than F B, whatever 

 line, other than F A, F B may be : one part of the pro- 

 position is thus disposed of. The next step is to draw 

 E C, the aid of which is now required but not till now 

 to prove that FB, a lina nearer to that passing 

 through the centre, is greater than any other line F C 

 more remote : another part of the theorem is thus de- 

 monstrated. You may LOW, if you please, introduce 



F G, and then draw E G, for the purpose of proving 

 that F D is shorter than any other line drawn from F. 

 We say you may introduce F G if you please ; for, in 

 strictness, there is no absolute occasion for it ; F C, or 

 F B, already drawn, would answer the purpose equally 

 well : of the remaining part of the construction nothing 

 need be said here. 



In Proposition VIII. we have actually omitted Euclid's 

 superfluous line the line corresponding to that which, 

 as just noticed, may be dispensed with in the preceding 

 proposition. As the diagram here is somewhat more 

 complicated, it was thought that the suppression of an 

 unnecessary line from D, and consequently of two lines 

 in connection with it from M, would be a desirable relief 

 to it. You will perceive it to be so if you compare the 

 diagram here, with that in other editions of Euclid. 



The demonstration of Proposition IX., given in the 

 present edition, is not that of Euclid, though substi- 

 tuted for it by De Chales, and some other editors. In 

 the earlier copies of Euclid, two different demonstrations 

 of this simple theorem are given. They have both 

 been preserved by Gregory, Stone, Williamson, and 

 Bonnycastle. It is the second of these two only that 

 Simson retains ; but, as remarked by Williamson, there 

 is a defect in the reasoning, which, however, it is scarcely 

 worth while to remove. The demonstration given in the 

 text is so easy and obvious, that, as Austin justly ob- 

 serves, " we cannot suppose Euclid would have over- 

 looked it." It is probable, therefore, that the demon- 

 strations here discarded are not Euclid's own, but the 

 interpolations of some other writer. 



Although it is proved in Proposition XX. that an 

 angle at the centre of a circle is double that at the cir- 

 cumference subtended by the same arc, yet it must not 

 be inferred that to every angle at the circumference there 

 corresponds an angle at the centre, upon the same arc, 

 which is double of the former. If the angle at the 

 circumference stand upon a semi-circumference, there 

 can evidently be no corresponding angle at the centre 

 upon the same arc ; and if the angle at the circum- 

 ference stand upon an arc greater than a semi-circum- 

 ference, the angle at the centre, the sides of which ter- 

 minate in the extremities of the same arc, will stand 

 not upon that arc but upon the opposite part of the 

 circumference. 



The construction of Proposition XXV. is different in 

 the present edition from that given in Euclid, which 

 requires three diagrams : moreover, the "segment of a 

 circle," as Euclid has it, need not bo given, only the arc. 

 Propositions XXVI. and XXVII. differ also in matters 

 of construction from those of Euclid ; for, as they at 

 present stand in Simson, they are insufficiently demon- 

 strated ; as noticed in the text. 



Proposition XXXIIL In all the editions of Euclid 

 that we have seen, there is a superfluous line (the line 

 B E) introduced into the second of the three diagrams ; 

 and in some editions the line B E is also introduced into 

 the third diagram, for what purpose we are unable to 

 conceive ; the tendency, though of course not the inten- 

 tion, is to mislead. 



The demonstration applies equally to both diagrams, 

 tn the first, the line B E, which goes to form the angle 

 E, is quite in keeping with the text, as this is really an 

 angle in the segment spoken of. But in the second of 

 ;hese diagrams, the line is not merely superfluous it is 

 utrony ; since the angle E, of which it is a side, is an 

 angle in a segment different from that to which the text 

 refers. In order that the proper angle should be exhi- 

 >ited in both diagrams, the vertex of it should be at H, 

 as shown by the dotted lines in this edition, but there 

 s no absolute occasion to introduce this angle at all. 



Propositions XXXV. and XXXVI. may be otherwise 

 and more easily established, as follows : 



Proposition XXXV. Let P be any point within a 



:ircle A B D. It is required to prove that whatever 



chord, A B, be drawn through P, the rectangle 



A P'P B will be always the same. 



l ill. Find the contre C ;* and draw C M at right 



f U I. angles to A B ;f draw also C P, C li. Then, 



