January 29, 1897.] 



SCIENCE. 



171 



The author in other publications has 

 claimed that this must have been the law, 

 and explained the phenomena as parallel 

 with that which takes place at the begin- 

 ning of every series arising in the Paleozoic 

 and Mesozoic, and also according to Minot's 

 law of growth and other phenomena of the 

 earlier stages in the ontogeny of every 

 animal 



All inferences with reference to the length 

 of time that life has existed upon the earth 

 are consequently defective, since, as far as 

 known to the author, they do not take into 

 consideration the differing rates of evolu- 

 tion at different times in the history of or- 

 ganisms. 



Alpheus Hyatt. 



THE BLACKBOARD TREATBIENT OF PHYS- 



lOAL VECTORS. 

 The tedious part of geometrical reading 

 is the need of searching for the letters which 

 designate the lines. Frequently this is the 

 chief difficulty in the demonstration. In a 

 measure, the same is also true when a ge- 

 ometrical proof is to be written down, par- 

 ticularly where special vector symbols (e. g., 

 the [4-B] of Moebius) are employed. There 

 is, perhaps, no remedy for this in printed 

 work ; but in the classroom, with a black- 

 board available, coplanar vectors may be 

 drawn in great variety at pleasure. I will 

 therefore describe the following method of 

 elementary treatment which, though it con- 

 tains no essential novelty, is new, I think, 

 from a pedagogic point of view, and for 

 this reason not without value. 



Of the four specifications which charac- 

 terize a vector — position, quantity, direc- 

 tion, sign — the first three usually come 

 within the range of indulgence of the aver- 

 age student ; but with the sign he will have 

 nothing to do. Thus it becomes necessary 

 to the author to be simply a mode of expressing a gen- 

 eral fact, or series of facts, that occur everywhere , and 

 in all series more or less through the action of the 

 law of taohygenesis. 



to especially emphasize the latter, and this 

 is done by putting an arrowhead on the 

 proper end of it. A physical vector is thus 

 fully given by an arrow of definite length, 

 originating in a definite point and pointing 

 in a definite direction. "With this laid down 

 insistently, the principle of vector summa- 

 tion is next developed* in the usual way. 

 Here, again, the sign quality needs to be 

 accentuated. The origin of the first arrow 

 is the given point of application. The 

 origin of every other arrow is the point of 

 the preceding, beginning with the first 

 arrow already placed. If two vector sys- 

 tems are equivalent, this implies that if 

 the free tail of each begins at a common 

 point, then the free tip of each system must 

 terminate in the same final point. 



It is simpler to begin with the first kine- 

 matic vector, velocity, rather than with dis- 

 placement. The inherent importance of the 

 space relations is easily pointed out in the 

 course of the development. 



With these customary introductions it is 

 my plan to write down vector equations on 

 the blackboard just like algebraic equations, 

 using for my terms definitely specified ar- 

 rows. Thus I obtain consecutively : 



Sum : The equation reads, for instance, 







!■ 



To change the direction of an arrow is to 

 change the sign of the term. Hence (1) is 

 identical with (2). 



Difference : 



© !-«-=/<' 



or by transposing, 



which may be tested by construction. 

 Again from (3) 



-1= ^ ./ 



* By supposing one of the vectors to be forming on 

 a blackboard moving as specified by the other vector. 



