(50 



iuaries may be made visible realities rather than symbolic results based 

 upon certain assumptions. 



When dealing with steps not limited to the plane of the paper, then 

 ( O A, n°) may be taken as the symbol of a number that turns any step that 

 is perpendicular to O A, n° round'O A as axis, counter clockwise to an ob- 

 server at A, and lengthens in the ratio of the length of O A to the unit 

 length. This is a quaternion. Quaternions whose angles are o° or 1S0° 

 are ordinary positive and negative numbers, and are called scalars. Qua- 

 ternions whose angles are 90° are called vectors. The square of a vector is 

 a negative scalar. The ordinary rules of algebra hold except that factors 

 are not interchangeable without altering the product. A quaternion, also, 

 cannot multiply a step that is not perpendicular to its axis. It can be geo- 

 metrically represented only by two steps. A vector (O A, 90°) or briefly 

 (O A) may be represented by the step ( ) A. The value of this representa- 

 tion is expressed by the equations : 



(OB)4-(OA)=-(OB-rOA) 



(OB) : ( A ) - OB : O A. 

 The calculus of quaternions is superior for all purposes of investigation to 

 analytical geometry, and as its results can be immediately turned into ana- 

 lytical formulas, it is likely to be very much used and developed in the 

 future. It is especially valuable in mathematical physics. An account of 

 the -system by Sir Wm. Rowan Hamilton, the inventor, was first presented 

 to the Royal Irish Academy in 1843. The first book upon the subject, 

 " Hamilton's Lectures," appeared in 185o. 



II. 



Let a x^ r b x^ + c x + d = o be an equation with general imaginary 

 co-efficients. Divide this by x — r: the quotient is a x- + (a r -f b) x + 

 (a r- -r b r + c) and the remainder is a r ' -j- b r- -[- c r + d. The co-effi- 

 cients of the quotient, and final remainder are best found by synthetic di- 

 vision, which shows the general method of forming each co-efficient by 

 multiplying the last by r and adding the next coefficient of the original 

 equation. The process is identical with the reduction of the compound 

 number (a, b, c, d) whose radix is r. The test of a root is that the remain- 

 der should be zero. 

 The steps that represent these numbers may be constructed as follows : 

 Take in the plane of the paper steps O A, A B, B C, C D, representing 

 the numbers a, b, c, d. Take any point A', and let A' k: A be the r we 



