REMARKS ON BOOK I.] 



MATHEMATICS. PLAXE GEOMETRY. 



559 



border than either of the others ; so that if a given sur- 

 face were to be divided into regular compartments, ol 

 equal area, by a network of costly materials, economy, 

 guided by science, would suggest the hexagon as the 

 figure to be chosen. And this is the figure selected by 

 the bee in the construction of the honeycomb. You 

 have seen that of all the figures of geometry, there are 

 but three which can so cover a surface as to leave no 

 waste of room no interstices. The bee chooses one oi 

 //!<!. Of these three, the hexagon is that which most 

 economises material : the bee chooses it. It is necessary, 

 too, as well for compactness and strength, as for the safe 

 lodgment of the grub, that the hexagonal cell should 

 terminate in a solid angle. Her choica of angles, that 

 noulil do, is to be made from an infinite variety ; but 

 among all these, mathematicians have discovered, by a 

 profound analysis, that there is one, and but one, mode 

 of formation by which the object would be attained with 

 the least expenditure of materials ; and this one the beo 

 adopts. She closes her hexagonal tube with an angular 

 termination, formed by three plane faces. Each piano 

 cuts off or excludes a portion of the tubular space ; but 

 the space within the solid angle just makes up for what is 

 thus rejected; and the faces of the solid angle are so 

 shaped, and so inclined, as to fulfil all the mathematical 

 conditions of a >iiiniinu,n of surface ; so that while the 

 angular space just compensates for the tubular space cut 

 off, it effects the compensation not on!y with lean amount 

 of material, but with tha leait amount possible. The 

 compensating space might be secured in an infinite 

 number of ways. Of this endless variety there is om- 

 way more economical than any other, and that one is 

 chosen by the bee. 



You must regard this reference to the architecture of 

 the bee as a digression, into which we have been allured 

 by the preceding examination of Euclid's corollary. But 

 the subject is full of instruction ; and when your mathe- 

 matical knowledge is sufficiently extensive to enable you 

 fully to estimate the science of the honeycomb, you mav, 

 like the ancient geometrician Pappus, feel even additional 

 reverence for the Creator's power, and additional grati- 

 tude that you have been endowed with faculties to com- 

 prehend and admire the exquisite geometry of the bee. 



Proposition XXXVII. maybe omitted. It is include:! 

 in the more general theorem which follows, and which is 

 demonstrated independently of the particular case. What 

 is proved of triangle-; upon cqn-il bases is, of course, 

 proved of triangles on the same base. 



Propositions XXXV. and XXXVI., about parallelo- 

 grams, seem to be related to one another, just as these 

 two propositions abont triangles are related. But there is 

 a difference ; for, although the thirty-fifth is only a par- 

 ticular case of that which follows, yet it is not super- 

 fluous, since the particular case is required in the proof 

 of the general theorem. The thirty-ninth proposition, 

 however, like the thirty-seventh, w superfluous ; as the 

 general theorem which follows does not require its aid. 

 The forty-first, too, might have been replaced by a pro- 

 position of wider generality. You can give it the exten- 

 sion here suggested by putting "equal bases" for "the 

 lame base" in the enunciation, and modifying the con- 

 struction and demonstration accordingly. 



Before passing on to the few remaining propositions, it 

 may be worth while to notice that the term trapezium is 

 used in the thirty-fifth proposition for the first time. 

 This is, in fact, the only place in which it occurs in the 

 Element*, though it is a term frequently employed in the 

 applications of geometry to practical matters, such as 

 mensuration, surveyiiu.', >to. 



Proposition XLII. requires a word or two. The line 

 A E is improperly introduced, in other editions, into the 

 construction ; you will gee, by attending to the details, 

 that the required parallelogram is completed without any 

 aid from this line ; it is in the demonstration, alone that 

 AE becomes necessary, and therefore its introduction 

 should bo deferred till the construction is completed. 



Of the forty-seventh proposition but little need be said 

 here. It is stated to have been discovered by Pythagoras, 

 who is recorded to have sacrificed a hundred head of oxen 

 to the gods on the occasion ; but this is probably a fable. 

 The multiplication table has also been generally attri- 

 buted to the same philosopher ; but the modern French 

 geometrician, M. Chasles, in his historical researches, has 

 shown this to be a mistake. 



The proposition in question is one of the most im- 

 portant, in its practical applications, of all the theorems 

 of geometry. Many forms of demonstration have been 

 given ; but that of Euclid is not to be surpassed in 

 elegance and clearness. 



The last proposition is remarkable only for the pecu- 

 liarity of its demonstration ; it is tha converse of the 

 preceding, and is demonstrated in the direct manner, 

 contrary to Euclid's general practice of demonstrating 

 converse propositions indirectly. As an exercise, you 

 may supply an indirect demonstration yourself : other 

 exercises on the first book are here subjoined. 



EXERCISES OS BOOK I. 



1. Prove that the two diagonals of a parallelogram 

 bisect each other. 



2. If in the sides of a square, at equal distances from 

 the four vertices, four points be taken, one in each side, 

 the figure formed by the four straight lines, joining the 

 four points, will also be a square. 



3. From two given points to draw two straight lines, 

 to meet in a given straight line, and to make equal angles 

 with it. 



4. Prove that the two lines drawn, as in the last exer 

 else, are, together, lest than any other two lines from the 

 [mints meeting in the given line. 



6. To divide a right angle into three equal parts. 



6. The diagonal* of a rhombus intersect each other at 

 right angles. 



7. If the three sides of a triangle be bisected, the per- 

 pendiculars drawn to the sides from the three points of 

 bisection will all meet in the same point. 



8. Tne straight line which bisects two sides of a tri- 

 angle will be parallel to the third side. 



9. If the middle points in the sides of any quadri- 

 lateral figure are joined by four straight lines, the figure 

 so formed will be a parallelogram. 



10. The sum of the angles of a four-sided figure is 

 twice as great as the sum of the angles of a three-sided 

 figure ; the sum of the angles of a five-sided figure, three 

 times as great ; of a six-sided figure, four times as great ; 

 and so on. 



11. Through a given point, between two non-parallel 

 straight lines, to draw a third straight line, terminating 

 in the former, which shall be bisected at the given point. 



12. From whatever point within an equilateral triangle 

 perpendiculars be drawn to the sides, their sum shall 

 always be the same. 



13. If from any point within a parallelogram, lines be 

 drawn to the four vertices, each pair of opposite triangles 

 thus formed will be, together, equal to half the parallelo- 

 gram. 



14. If two triangles have two sides of the one equal to 

 two sides of the other, each to each, and if the angle 

 contained by the two sides of the one, together with that 

 contained by the two sides of the other, make two right 

 angles, the two triangles will be equal in surface, or area. 



1 .1. In the figure to Proposition XL VII. , if F D, G H, 

 E K, be joined, the triangles F B D, G A H, K C E, will 

 all be equal to one another, and to A B C. 



16. Prove the converse of Proposition XXXIV.; 

 namely if the opposite sides of a quadrilateral be 

 equal, or if the opposite angles be equal, the figure will 

 be a parallelogram. 



17. If the points of bisection of the three sides of a 

 triangle be joined, the triangle will be divided into four 

 component triangles, all equal to one another. 



