0/ 



Laboratory and field work on the phosphate of alumina. By H. A. 

 HrsTON. 



Recent methods for the determination of phosphoric aciu. By H. A. 

 HrsTON. 



The digestibility of the pentose carp.ohydrates. By W. E. Stone. 



The action of phenyl-hydrazin on rrRFUROi.. By W. E, Stone. 



A graphical solution for equations of hiCtHer de(.ree. for both real 



AND IMAGINAItY ROOTS. By A. S. HaTHAWAY. 



•1. Preliminary on imaginary numbers. 



The usual idea of imaginary numbers, as presented in our text books of 

 algebra, is that they are symbols introduced for the sake of making the 

 laws of algebra formally complete. It is implied in the name given to 

 these numbers that they have no actual meaning. This is a mistake. The 

 failure to mean anything in ordinary cases is not the fault of the numbers, 

 but results from the nature of the concrete quantities with which they are 

 generally used. Like difficulties are experienced with real numbers under 

 similar circumstances. Let us go briefly over the list of numbers and em- 

 phasize this point. 



First, the numbers 1, 2, 3, 4, that denote repetitions of a concrete quan- 

 tity. If the quantity be incapable of the indicated repetition the result is 

 imaginary. Thus: Three spaces of four dimensions. This may be com- 

 prehensible to a different order of beings, but not to us. 



Second, the numbers I, };, \, that denote partitions of a concrete quan- 

 tity. Nevertheless, a space of I a dimension, a school of 1 a student, are 

 impossibilities. 



Third, the number — 1, This number must be used with quantities of 

 two kinds such that two of equal magnitude and different kinds give, when 



*NOTE.— This preliminary ou the graphic representation of imaginary numbers was 

 not presented to the Academy. It is a simple and direct presentation of the subject 

 without the use of analytical geometry, and on that account may be interesting to 

 mathematicians; at the -^ame time, it places the whole article upon an elementary basis, 

 and makes it available to a larger circle of readers. 



