WILSON AND LEWIS. — UKLATIVITY. 405 



Cor. The area of any irctanf-ie is the product of the intervals of 

 two adjoining sides. 



We ina\- therefore obtain from XIX the theorem 



XXI. The square of the interval of the hypotenuse of a right 

 triangle is equal to the difference in tlie squares of the intervals of the 

 other two sides. 



Cor. The perpendicular from a point to a line has a greater interval 

 than any other line of the same class drawn from the given point to 

 the given line. 



Ha\ing now gi\en a final definition of the measure of area, we may 

 define the unit of angle. The radius of the pseudo-circle, in advancing 

 by rotation over equal angles, necessarily sweeps out equal areas 

 (by 16°). Hence l)y the familiar argument sectorial areas in any 

 pseudo-circle are proportional to the angles at the center. The unit 

 angle will be taken as that angle which, in a pseudo-circle of unit 

 radius, encloses a sectorial area of one-half the unit area. 



Vectors and Vector Algebra. 



13. Translation or the parallel-transformation leads at once to 

 the consideration of vectors. \Ye have shown that when a translation 

 carries A into B and A' into B' the directed segments AB and A'B' 

 are parallel and congruent (Cor. to II). Hence a translation may be 

 represented by a vector, that is, by any directed segment laid of from 

 any origin and having the same interval and direction as AB. The 

 succession of two translations is represented by the sum of their 

 corresponding vectors. The addition and subtraction of vectors and 

 their multiplication by scalars follows the usual laws (by §§ 5-7). 



If two vectors a and b are laid off from a common origin, the paral- 

 lelogram constructed on the vectors is called their outer product a>;b, 

 and the magnitude of this product will be taken numerically equal to 

 the area of the parallelogram.^^ We must bear in mind that not this 

 magnitude (nor yet a vector perpendicular to the plane), but the 

 parallelogram itself is the outer product. We may, however, repre- 

 sent the outer product by any other closed figure of equal area, pro- 

 vided that it is taken with the same sign. The sign attributed to an 



17 Our vector notation will be based upon that of Gibbs, and is identical witli 

 that omploycd bj^ Lewis (Four dimensional Vector Analysis, These Proceodiiig-!. 

 46, 16.3-181) except in the designation of the inner product which we shall 

 define as in that paper, but represent by a*b insteail of ab; tlie latter form will 

 be reserved to denote the dyad. The scalar magnitude of a vector will be 

 represented by the same letter in italic type. 



