MATH KM \TICS.-PLANE GEOMETRY. 



[RF.MARKS ow BOOK I. 



that the relative jxwrtioiu of two things can hare no- 

 thing to do with their relative maynitwl-* ; that tho two 

 triangle*, affirmed under certain conditions to be e<fuat, 

 and placed side by aide before yon on the paper, must 

 remain equal, however their relative positions bo altered ; 

 .or one be turned upside down, or be made to over- 

 lap the other, can make no ilitl". mice as to the equality 



-.equality <( the two in nuutnihuk. You will thus 



ee that, for the purpose of instituting an inquiry na to 



tin- triangles, it is quite allowable to 



imagine one to be placed upon, or to be applied to the 



r, with the view of ascertaining whether, by a suit- 

 able adjustment, a complete adaptation may bo nought 

 about. Iti this manner Euclid directs you to apply the 

 triangle A B C to the triangle 1> K F in a certain way 

 namely, so that the point A may be on D, and the strain-lit 

 line A B upon D E. He then affirms that B must coin- 

 cide with E ; and as he never affirms anything without 

 immediately answering the inquiry vhyt he adds, In 

 \ I! is equal to 1) K. The adjustment is thus brought 

 about as far as the sides A K, D E are concerned. I If 

 then asserts that, thin partial adjustment remaining 

 undisturbed, A C must fall upon D F, fcccau.se the angle 

 or opening A is equal to the angle or opening D. If 

 A C fell beyond D F, the angle A would be greater than 

 the angle D ; and if A C fell nhoit of D F, the angle A 

 would be less than the angle D ; Euclid's conclusion, 

 therefore, is irresistible. As A C then necessarily falls 

 upon D F, the point C must as necessarily fall upon or 

 coincide with the point F, because A C is equal to D F. 

 And thus having proved, first, that B coincides with K, 

 and then that C coincides with F, he infers, in virtue of 

 the 10th axiom, that the base B C must coincide with 

 the base E K. Hence the adaptation ia complete ; 

 is perfect coincidence, and therefore perfect equality in 

 every respect. 



Proposition V. is always found to be more or less per- 

 plexing to a learner ; and it is certainly one of the most, 

 if nut thf most knotty of the propositions in the first 

 book. We would recommend a beginner, after completing 

 the construction as directed, to erase the base B C of the 

 ori ^inal triangle, in order that nothing may divert his 

 attention from the two triangles A F C, A G B. The 

 line thus expunged may be restored after these triangles 

 have been proved to be in all respects equal : they are 

 nothing more than the two triangles already considered 

 in Proposition IV. in a different position, one triangle 

 partially overlapping the other. Young students are 

 sometimes deterred from prosecuting the study of Euclid 

 by the length and difficulty of tins proposition. They 

 should be apprised that the propositions are not arranged 

 in the order of their difficulty ; tluit none more trouble- 

 some than this fifth will ever after be met with, and that 

 the last theorem in the book is quite as easy as tho first 

 A great point will be gained, if you master this fifth 

 proposition ; for you may then conclude with confidence 

 that you will find yourself fully adequate to all that 

 follows : but you must not come to this conclusion till, 

 closing the book, you find yourself able to demonstrate 

 the theorem step by step without a reference to it. This 

 mode of testing your progress must be resorted to all 

 along. It is not enough that you read and nnder-tand 

 Euclid's demonstrations you must acquire the ability 

 of furnishing these demonstrations yourself ; you may 

 vary the language, but you must preserve tho rigour of 

 the :> .limning nothing without a reason. 



We do not clearly soe why tho fifth proposition should 

 be called I'nni ainmrnm, or the atfes' bruli/r. They say 

 it is because " tuna*" stick at it ; but we believe it was tho 

 tictntitth proposition that wan so designated by some of 

 the ancients; for Proclus informs us, in his Com 



. that tho Epicureans derided the tweu- 

 pro|.*ition as being manifest "even to as.se.s ;" for 

 if a bundle of hay were placed at one extremity of the 

 bate of a triangle, and an ass at the other, the nniinal 

 would not be such an as* as to take the crooked path to 

 the hay instead of the straight one ; as he would know 

 the direct course to be the shorter : thii was therefore 

 called the attti' brulj,-. 



Tho sixth proposition you will find very easy after the 

 fifth it is wnat is called tho eonrtrit of tho first part of 

 tho former proposition. A theorem is said to be the 

 converse of another, when tho hypothesis and the 

 sequence, in that other, change places. The I 

 in Proposition V. is, that two ndtt of t are 



i-.ju-il ; the inference or consequence is, that the two 

 an>jle* opposite to thorn are equal. Tho hypothesis in 

 i VI. is, that two until?* of a triangle arc 

 equal ; and the consequence is, that the two t'nl .1 oppo- 

 site to them are equal : the one proposition is th. > 

 the converse of the other. In general, Kurliil demon- 

 strates the convene of a previous theorem imtirrrr' 

 by what is called the reJurtin <ul iifimiriium method ; that 

 is, he commences with a denial of tho truth stated ; 

 and, reasoning from the contradictory statement, as if 

 it were tru 'hat an absurdity, or impossibility, 



is tho unavoidable consequence ; thus proving that the 

 thing contradicted cannot be otherwise than true. 

 Throughout the whole of this book, the last proposition 

 is the only converse theorem that is not demonst 

 in this indirect manner. It is not every theorem that 

 is true both directly and conversely. You should take 

 note of those that Euclid proves to be ctmrertiblr, and 

 endeavour to discover for yourself which of his propo- 

 sitions hold conversely, though only proved directly. 

 For example, Proposition X X \ I V. pnnes that if tho 

 opposite sides of a quadrilateral are parallel, they are 

 likewise equal. It is also true conversely, that if the 

 opposite sides are equal, they are likewise parallel, as 

 you may prove for yourself, after the direct proposition 

 has been established. 



Proposition VII. is merely subsidiary to the proposition 

 next following ; it is what in some geometrical writings 

 would bo called a Lemma,. You see that the demon- 

 stration of it rests almost entirely on Proposition V. 

 In some modern books on geometry this proposition is 

 with, and the eighth established independently 

 of it ; but, as an intellectual exercise, Proposition VII. 

 is as useful as any in the book. Besides, a proposition, 

 though manifestly introduced as merely subsidiary to 

 something else, may yet possess intrinsic excellence of 

 its own sufficient to justify its retention in tho system. 

 For instance, the proposition before us teaches us this 

 fact, which is certainly not without interest ; namely, 

 that a physical triangle, supposed to have its sides freely 

 movable about joints at its vertices, cannot possibly be 

 thrust out of shape by any force whatever. You may 

 break the bars forming tho framework, but you cannot 

 make the frame itself assume another shape. 



Proposition VIII. is the teeond proposition, in the 

 geometry of triangles, which proves that two triangles 

 are equal in every respect that is, that eaeli is but an 

 exact copy of the other provided three things in one are 

 respectively equal to throe corresponding things in tho 

 other. Tho three things may bo two sides ami the in- 

 cluded angle, as we learn by Proposition IV., or the 

 three sides, as the present proposition teaches. And wo 

 may as well observe here, that there is only one other 

 proposition in the Elements where the like equality of 

 two triangles is inferred from au equality of three t!. 

 in one to three corresponding things in the other : it is 

 Proposition XXVI. On these tin tions tho 



praetical part of plane tr!i/<ni<: imded A tri- 



angle, in tho language, of trigonometry, is said to have 

 six fxirt* the three sides and il. and when 



f these an- nT of the sets, 



namely, mentioned in Propositions IV., V 1 1 1 , and 

 XXVI. the remaining thnv, whieh we see by tl 

 propositions must bo fixed and invariable, liec.uno de- 

 tiTininable, and are matters of computation. 



And hero it may not be an \ a word or two 



: the form of expression eontinualh i by 



Kurlid, when roiupui . i..r the purpose 



their equality. He always speaks of two 

 ides or angles of tha one being equal to two si<! 

 angles of the other. It. Learners aro a; 



this qualifying condition, "each to, t the 



frequent repetition of thuso words were only so much 



