32 CoLOEADO College Studies. 



well approved on at the Universities, in England, Scotland, and 

 Ireland, that it is ordered to be publickly read to their pupils 



Like all English arithmetics of that time, that of Ward con- 

 tains rules and illustrations without reasoning. In the chapter 

 on algebra he first explains the negative sign as meaning al- 

 ways subtraction, just as in arithmetic, but he tacitly intro- 

 duces later on " negative quantities " — thus using the negative 

 sign in a different sense without explaining that fact to the 

 pupil. Until about 'fifteen or twenty years ago all our algebras 

 contained that very defect, of attributing to the minus sign by 

 definition only one meaning, but of actually using it in two 

 senses. 



Ward shows how to expand a binomial with a positive inte- 

 gral exponent " without the trouble of continued involution," 

 and then makes the following interesting remarks: "Now from 

 these considerations it was, that I proposed this method of rais- 

 ing powers in my Compendium of Algebra, p. 57, as wholly new 

 (viz., so much of it as was there useful) having then (I profess) 

 neither seen the way of doing it, nor so much as heard of its 

 being done. But since the writing of that Tract, I find in Dr. 

 Wallis's History of Algebra, p. 319 and 331, that the learned Sir 

 Isaac Newton had discovered it long before." 



The third part, on Geometry; is quite inferior in point of 

 precision and scientific rigor. After the definitions follow 20 

 problems, intended for the excellent purpose of exercising " the 

 young practitioner," and bringing "his hand to the right man- 

 agement of a Kuler and Compass, wherein^I would advice him 

 to be very ready and exact." Then follows a collection of 24 

 " most useful theorems in plane geometry demonstrated." This 

 part is semi-empirical and semi-demonstrative. A few theorems 

 are assumed and the rest proved by means of these. The theo- 

 rem, " If a right line cut two parallel lines, it will make the 

 opposite angles equal to one another " is proved by aid of the 

 theorem that " if two lines intersect each other, the opposite 

 angles will be equal." The proof is based on the idea that 

 " parallel lines are, as it were, but one broad line," and that by 



