36 DISSERTATION SECOND. [parti 



able an application of the term negative, as any that can 

 be proposed ; yet, in reality, it means nothing but this in- 

 telligible and simple truth, that x 3 + px 2 + qx -f- r=(x — a) 

 (x -f- b) (x -f- c ;) or that the former of these quantities is 

 produced by the multiplication of the three binomial fac- 

 tors, x — a, x + b, x -{- c. We might say the same nearly 

 as to imaginary roots ; they show that the simple factors 

 cannot be found, but that the quadratic k factors may be 

 found ; and they also point out the means of discovering 

 them. 



The aptitude of these same signs to denote contrariety 

 of position among geometrick magnitudes, makes the fore- 

 going application of them infinitely more extensive and 

 more indispensable. 



From the same source arises the great simplicity intro- 

 duced into many of the theorems and rules of the mathe- 

 matical sciences. Thus, the rule for finding the latitude 

 of a place from the sun's meridian altitude, if we employ 

 the signs plus and minus for indicating the position of the 

 sun and of the place relatively to the equator, is enunciat- 

 ed in one simple proposition, which includes every case, 

 without any thing either complex or ambiguous. But if 

 this is not done, — if the signs phis and minus are not em- 

 ployed, there must be at least two rules, one when the sun 

 and place are on the same side of the equator, and another 

 when they are on different sides. In the more complicated 

 calculations of spherical trigonometry, this holds still more 

 remarkably. When one would accommodate such rules 

 to those who are unacquainted with the use of the algebra- 

 ick signs, they are perhaps not to be expressed in less than 

 four, or even six different propositions ; whereas, if the use 

 of these signs is supposed, the whole is comprehended in 

 a single sentence. In such cases, it is obvious that both 

 the memory and understanding derive great advantage from 



