BOOK i. PROP. XLVII. AND XLVIII.] MATHEMATICS. PL ANE GEOMETRY. 



651 



. . A E is a parallelogram, . . A B 

 + Pr. 34 = D E, and A D = B E ; t 

 but A B = A D, . . the four sides A B, 

 A D, D E, E B are all equal, . . A E is 

 equilateral. Likewise, all its angles are 

 right angles ; for since A D meets the 

 parallels A B, D E, the angles A + A D E 



Pr. 29. = two right angles ; * but A 

 + Const. is a right angle, t .'. A D E is 



a right angle ; but the opposite angles 



Pr. 34 of a parallelogram are equal,* 



.' . B, E are each right angles, . . the figure A E is 

 rectangular, and it was proved to be equilateral : it is 

 .' . a square, nntl it i* described upon the given straight line 

 A B. Which was to be done. 



COR. Hence, every parallelogram that has one right 

 angle, has all its angles right angles. 



NOT* . It mav be easily proved that the figure A E is rectangular, 

 provided it have one right angle A, and that it be equilateral ; for 

 if the diagonal D B be drawn, the figure will be divided into two 

 equal triangles (Pr. VIII.), .-. A = E, .-. the triangles are right- 

 angled Uoaceles triangles, and (Pr. V. and XXXII.) each base angle 

 Is half a right angle, .-. the angles A, B, E, 1) are all right angles. 

 Hence, a square is a fuitr-sided Jiyuti', which has all its sides 

 equal, and one of its angles a right angle ; that the other three are 

 also right angles is demonstrable, at above, and ought not to be 

 assumed in a definition. 



PROPOSITION XLVII. THEOREM. 

 In any right-angled triangle (B A C), the square (B E) de- 

 scribed upon the side (B C) suljtfnding the right angle, is 

 equal to the squares (B G, (' H) described upon the sides 

 containing the right angles. 



Pr. 46. The squares being described,* through A draw 



i. A L parallel to B D or C E ;t draw also A D, 

 Hyp. F C. Then, because B A C is a right angle,* 

 and that B A G is also a rig it angle, C A, A G are in the 

 + Pr. 14. same straight line, t For a like reason, A h, 

 A H ore in the same straight line. Now the angle D B C= 

 F B A, each being a right 

 an"le ; add to each the 

 angle ABC, .-. DBA = 

 F B C ; also the two sides 

 AB, BD = thetwoFB, BC, 

 each to each, .'. the triangle 

 A B D= tlie triangle F B C.* 

 Pr. 4. Now the paral- 

 lelogram B L is double the 

 triangle A B D, because they 

 are on the same base B 1), 

 and between the same paral- 

 + Pr. 41. lels Bl), A L;t 

 and the square B G is double 

 the triangle F B C, because 

 these also are on the same base, and are between the 

 same parallels F B, G C ; but the doubles of equals are 

 themselves equal, . . B L = B G. 



In like manner, by drawing AE, B K, it may be 

 demonstrated that C L = C H, . '. the whole square B K 

 the two squares B G, C H ; that is, the square de- 

 scribed upon B C is equal to the squares described upon 

 A B, A C, . . in any right-angled triangle, &c. Q. E. D. 



PROPOSITION XLVIII. THEOREM. 

 If the square described upon (B C) one of the sides of a 

 triangle (A B C) be equal to the squares described upon 

 the other two sides, the angle (A) contained by these two 

 sides is a right angle. 



From A draw A D at right angles 

 Pr. u. to AC,* and make AD 

 + Pr. 3. = A B,f and draw D C. 

 Then, because A D = A B, the square 

 of A D = the square of A B : to each 

 of these add the square of AC, 

 the squares of A 1), A C= the squares 

 B c of A B, A C. But the square of D C I 



= the squares of A D, A C, because < 



Pr. 47. I) A C is a right angle,* and the square of i 

 B C is by hyp. = the squares of A B, A C, . . the square 

 of 1) C = the square of B C, .-. D C = B C. Hence, in 

 the two triangles ABC, ADC, there are two sides 

 B A, A C in the one, equal to the two DA, A C in the 



other, each to each, and the base B C equal to the base 



pr.s. DC, .-. the angle BAC = DAO.* Bat 



D A C is a right angle, . . B A C is a right angle, . . if 



a square, &c. Q. E. D. 



REMARKS AND COMMENTS ON THE FIRST BOOK OP EUCLID. 



The following observations on the general character of 

 geometrical reasoning, and on the First Book of Euclid 

 in particular, are intended for the guidance and instruc- 

 tion of those whose acquaintance with the subject is 

 limited to what has now been delivered. We think it 

 very probable that, among sujh persons, there may ba 

 some who, however attentively they may have read the 

 portion now completed, have failed to perceive, so clearly 

 as is desirable, the main object and intention of a course 

 of geometrical study. It is true that the demonstrations 

 themselves are so free from obscurity, and so thoroughly 

 convincing, that no doubt can remain on the mind of an 

 attentive reader as to the truth of the several conclusions 

 arrived at ; so that anything added to these demon- 

 strations, by way of elucidation of the steps, or as con- 

 firmatory of the results, would be felt by the merest 

 beginner to be an ineumbrance rather than an aid. 



There is no doubt, however, that Geometry is some- 

 times taken up with erroneous expectations as to what 

 it teaches ; and is read with a pliant docility of mind 

 a passive acquiescence in the dicta of the teacher which 

 Euclid himself would be the first to condemn. It ia 

 chiefly for the purpose of guarding you against such mis- 

 takes that we append the following remarks to the first 

 book of the elements. We should have prefixed them, 

 could we have been quite certain that you would have 

 been familiar with the geometrical terms which would 

 have been employed. We offer them here, in the ex- 

 pectation that you will give the foregoing part a second 

 reading, guided by the additional light here to be given, 

 in reference to the objects and advantages of Geometry, 

 and also as regards the true spirit in which its principles 

 should be studied. 



A youth, destined ultimately for some mechanical or 

 scientific occupation, is told and properly told that, 

 to excel in his calling, he must study Euclid ; this study 

 is usually associated in his mind with a case of mathe- 

 matical instruments, scales, compasses, parallel-rulers, 

 (bo. Now, without expressly forbidding these things, it 

 is of importance that he should be distinctly informed 

 that, for aught that appears to the contrary, Euclid 

 never handled, or even saw, compasses, parallel-rulers, 

 ifcc. , in his life. It U certain that he gives no counte- 

 nance to the use of any such mechanical contrivances in 

 his work. Had Euclid been asked, there is no doubt 

 that he would have declared his inability to describe a 

 circle, and eveu to draw a straight line. What we call 

 practical geometry Euclid was entirely regardless of; in- 

 deed the application of geometry to the practical business 

 of life was viewed by the ancient geometricians rather 

 as a degradation of the purely intellectual science they cul- 

 tivated, than as enhancing ita value ; and we accordingly 

 ind that but few of Euclid's problems, or practical con- 

 structions, are such as a skilful workman would follow. 



Important and extensive as are the practical appli- 

 cations of geometry, it should nevertheless be borne in 

 mind that the Elements would have existed, just as they 

 low do, if these applications had never been thought of. 

 file availability of geometry in practice is merely a con- 

 ingeut and accidental circumstance, uncontemplated by 

 .he geometricians, just as their theory of the conic 

 sections was elaborated without any prospective regard 

 to the future demands of physical astronomy, or to the 

 discoveries of Newton. 



And it is right to contemplate geometry under this 

 purely intellectual aspect, and to study it as a strictly 

 abstract science. Practical operations are all more or 

 less imperfect. There are imperfections of vision, of the 

 hand, of the instruments employed. Pure Euclidean 

 geometry tolerates no such imperfections, however mi- 

 nute or unimportant they may be in a practical point of 

 view. The circle of Euclid is a perfect circle such a 



