KEMARKS OX BOOK II. 1 



MATHEMATICS. PLANE GEOMETRY. 



563 



that the rectangle contained by the two halves of a 

 straight line, is greater than the rectangle contained by 

 any two unequal parts of it, since it shows that to this 

 latter rectangle something must be added, to make up the 

 square of half the line. 



In attempting the propositions in this book, without 

 reference to the text, you will find it useful to keep in 

 remembrance, that in all the constructions, up to Propo- 

 sition VIII. inclusive, the greatest square mentioned in 

 the enunciation is always to be described first ; and that 

 in each proposition after the fourth, the lines employed 

 in that fourth are always to be introduced. After this 

 partial construction, the completion of the diagram, in 

 each case, will readily present itself. 



It may, too, be deserving of notice, that the demon- 

 strations in Propositions V. and VI. may, with advan- 

 tage, be conducted rather differently from the method of 

 Euclid. Thus, in Proposition V., having proved, as in 

 the text, that the projecting rectangle A L is equal to the 

 maryinal rectangle D F, we may proceed as follows : 

 Take each of these rectangles from the entire figure ; 

 then the remainder C F will be equal to the remainder 

 AH + LG; that is, to the rectangle AD-DB (for DB 

 = D H), and the square of C D, since L H = C D : and 

 this completes the demonstration. 



In like manner, in Proposition VI., having proved, 

 as in the text, that the projecting rectangle A L is equal 

 to the marginal rectangle H F, take each rectangle from 

 the entire figure ; then the remainder C F is equal to 

 the remainder A M + L G ; that is, C D 2 = A D D B + 

 C B*, which completes the proof. 



Proposition VIII. is usually found, by a beginner, to 

 be a little perplexing; and Propositions IX. and X., 

 though very elegantly demonstrated by Euclid, are felt 

 to be somewhat lengthy. We shall, therefore, here show 

 how these three propositions may be otherwise demon- 

 strated. 



Prop. VIII. Raring made 



B D = C B, the reasoning may be as fallows : 



By Prop. IV., 



AD' = AB s -|-BD'-f2AB-BD; 

 that U, A D J =. A B 1 -f C B< + 2 A B C B. 



Alio, Prop. VII., 



AB J + CB 2 = 2AB-CB-4-AC; 



. . A D' = 2 A B C B 4- A C' + 2 A B C B ; 

 thatis, AD= = 4 A B C 8 -f- A C J . 

 Which was to be demonstrated. 



Prop. IX. By Prop. IV., 



A D< = A C 2 + C D 2 + 2 A C C D j 

 .-. A D + D B 2 = A C 2 + C D 2 + 2 A C-C D + D B ; 

 that i, AD 1 -4-DB J = BC'- r -CD 2 + 2BC-CD4-DB 2 . 



But, Prop. VII., 



B C 2 + C D 1 = 2 B C-C D + D B 2 ; 



that i, AD-- r -DB 2 



Which was to be demonstrated. 



Proposition X. The demonstration of this may be 

 made to depend on the preceding proposition, as follows: 



Prolong C A to H, 



making A H = B D ; - I - I - I - 



then H C = C D, and HA C B D 



HB = DA. 



And since H D is divided equally in C, and unequally 

 in B, .'. by Proposition IX., 



that ia, A D 3 + B D 2 = 2 C D 2 + 2 A C J . 

 Which was to be demonstrated. 



Although the foregoing demonstrations occupy less 

 (pace than those of Euclid, we would by no means re- 

 commend them to your preference. Those of Euclid are 

 among the finest of his specimens of clear and consecu- 

 tive reasoning, and you will observe that they are alto- 

 gether independent of second-book propositions. They 

 are thus simpler, though longer than those given above. 

 There is, too, a beauty in the reiterated appeals to the 

 forty-seventh of the first book, that more than compen- 

 sates for the length of the argument. The demonstra- 

 tions above, however, may be useful as exercises on the 

 application of second-book propositions. 



There are but two problems in this second book ; and 



each of these, as given in the editions of Simson and 

 others, exemplifies the remark made in the commentary on 

 the preceding book, in reference to Euclid's apparent indif- 

 ference as to the neatness and finish of his constructions. 



In the editions alluded to, the whole of the diagram in 

 Proposition XI. , with the exception of the prolongation 

 of G H to K, is made to appear as essential to the con- 

 struction of the problem ; that is, to the determination 

 of the point H. You have seen, however, that to the 

 discovery of this point, the actual construction of the 

 two squares AD, A G is not necessary; they are wanted 

 only as a part of the machinery in the demonstration that 

 the point H, previously found, divides the proposed line 

 as required. 



In Proposition XIV., too, the drawing of G H is made 

 to enter into the construction of the problem, instead of 

 being postponed for use in the demonstration, as is done 

 in this edition. 



You will not fail to notice the connection between 

 Propositions XII. and XIII., and Proposition XLVII. 

 of the first book. The three together furnish certain 

 corresponding relations between one side of a triangle 

 and the other two, whether the angle opposite that one 

 be right, obtuse, or acute. 



It may not, perhaps, be out of place here to notice, 

 that the word equal, employed so frequently in this and 

 in the preceding book, in reference to rectangles, tri- 

 angles, <t-c. , is used by Euclid in two somewhat different 

 senses. In the earlier propositions of the first book the 

 fourth, eighth, and twenty-sixth, for instance the test 

 of equality is perfect coincidence, as the result of super- 

 position ; but in Propositions XXXV., XXXVI., <fcc., 

 as also in most of the propositions of this second book, 

 the condition of coincidence is excluded, and the figures 

 are declared to be equal if they are proved to inclose the 

 same extent of surface. Legendre, a distinguished French 

 geometer, has proposed to discriminate between these 

 two kinds of equality : figures which, though equal in 

 surface, do not admit of coincidence, he prefers to call 

 not equal figures, but eq\nvalent figures. We think the 

 distinction an appropriate one ; and while upon these 

 minor matters of mere phraseology, we would venture 

 further to suggest, that instead of speaking of lines as 

 greater and leu, the more explicit and restrictive terms 

 longer and shorter would be preferable ; and that part of 

 a line, or portion of a line, are, either of them, designa- 

 tions that might appropriately supply the place of segment 

 of a line. 



EXERCISES ON BOOKS I. AND II. 



1. In any triangle, the squares of the two sides are 

 together equal to the square of half the base, and of the 

 straight line from the vertex to the middle of the base. 



2. The squares of the four sides of a parallelogram, 

 are together equal to the squares of the two diagonals. 



3. If from any point lines be drawn to the four 

 vertices of a rectangle, the squares of those drawn to 

 opposite vertices will, together, be equal to the squares 

 of the other two. 



4. In every parallelogram, the squares of the four 

 sides are, together, equal to the squares of the two 

 diagonals. 



6. If from any point whatever, lines be drawn to the 

 four vertices of a parallelogram, twice the sum of their 

 squares will be equal to the squares of the diagonals, 

 together with eight times the square of the line drawn 

 from the given point to the intersection of the diagonals. 



6. From the last exercise prove that the following 

 curious property exists ; namely : If from, the point of 

 intersection of the diagonals of a parallelogram, a circle 

 t>e described with any radius, the squares of the lines 

 drawn from any point in the circumference to the four 

 corners of the parallelogram, will always amount to the 

 same sum. 



7. In any quadrilateral, the sum of the squares of the 

 'our sides is equal to the sum of the squares of the 

 diagonals, together with four times the square of the 

 ine joining the middle points of the diagonals. 



