August 11, 1911] 



SCIENCE 



181 



DISCUSSION AND COBSESPONDENCE 



THE PYTHAGOREAN THEOREM 



To THE Editor of Science: In your iournal 

 for December 16, 1910, Dr. Northrup asks 

 whether a dynamical investigation which he 

 there gives is a proof of the Pythagorean 

 theorem; and in the number for March 24, 

 the question is discussed by Professor Deimel 

 and Mr. Hersey. Looking at the question 

 from the point of view of vector-analysis, 

 or rather of the algebra of space, I would 

 answer. Tea. Dr. Northrup starting from 

 the principle of kinetic energy and certain 

 other principles of dynamics deduces two ex- 

 pressions for the kinetic energy of the system 

 shown in his diagram; and from the equiva- 

 lence of these expressions he deduces the 

 forty-seventh proposition of the first book of 

 Euclid, commonly called the Pythagorean the- 

 orem ; but he could with ease deduce the more 

 general proposition (Euclid II., 12 and 13) 

 which expresses the side of any plane triangle 

 in terms of the other two sides and their in- 

 eluded angle. His proof is merely the re- 

 verse of the following reasoning. I look upon 

 the X, y, B, r and — r of his diagram as 

 vectors. The kinetic energy of the first mass 

 is im{xWy = imW'x^ ; and similarly that of 

 the second mass is imW^jf. But 



and 



a;= = jR= + r= 4- 2 cos Mr 



i/= = JJ=-f ( — ry — 2 cos Br 



where cos Br denotes the rectangle formed by 

 B and the projection of r along B. Hence 

 I mW'{x' + 2/^) = * 2m(i?= + r=) W% 



= i 2mB-W- + i 2mr-W' 



Here we pass from the one to the other ex- 

 pression for the kinetic energy of the system 

 by means of the extended Pythagorean the- 

 orem; on the other hand, Dr. Northrup can 

 deduce from the two expressions for the ki- 

 netic energy of the system the truth of this 

 geometrical theorem. 



This same principle that E ==■ Jmu" has an 

 important bearing on the fundamental prin- 

 ciples of vector-analysis: it places the ortho- 

 dox quaternionist in a corner from which 



there is no escape. Because E is assumed in 

 mathematical analysis to be positive and im 

 is positive, it follows from the established 

 principles of analysis that v' must be positive; 

 consequently, to hold that the square of a 

 simple vector is negative is to contradict the 

 established conventions of mathematical analy- 

 sis. The quaternionist tries to get out by say- 

 ing that after all v is not a velocity having di- 

 rection, but merely a speed. To this I reply 

 that 



E = cos f mvdv = i mv', 



and that in these expressions v and dv are 

 both vectors having directions which in gen- 

 eral are different. 



-',e- 



Eecently (in the Bulletin of the Quaternion 

 Association) I have been considering what 

 may be called the generalization of the Pytha- 

 gorean theorem. Let A, B, G, D, etc. (Fig. 1), 

 denote successive vectors having any direc- 

 tions in space, and let B denote the vector 

 from the origin of A to the terminal of the 

 last vector; then the generalization of the 

 Pythagorean theorem is 



E' = A^ + B- + C^ + D' + 



-f 2{cos^-B -f- aoa AC + coaAV -f } 

 + 2(cos BC + cos BD +-^ 

 + 2(cos CI» -I-} 

 + etc. 



where cos AB denotes the rectangle formed by 

 A and the projection of B parallel to A. The 

 theorem of Pythagoras is limited to two vec- 

 tors A and B which are at right angles to one 

 another, giving B'^A'-\-B-. The exten- 

 sion given in Euclid removes the condition 



