400 



A General Theory of Mathematical Form. [June 18, 



§§ 280—312. Algebras. 



The genesis of algebras is here treated of. We have three system?, 

 replicas of each other, of which the units are respectively multipliers, 

 multiplicands, and products. We have also a system the units of 

 which are primitive equations, such as 



ab — c, 



arrived at by regarding as units the triads of which one unit is a 

 multiplier, one a multiplicand, and one a product. We have another 

 system, of which the units are primitive algebras, arrived at by regard- 

 ing as units certain collections of primitive equations. The deriva- 

 tion of compound algebras is considered, and the way in which 

 algebras arise from special systems. 



§§ 313— 32r. Quadrates. 



Quadrates are units of a special form of system which may be 

 represented by n asterisks and r?—n dots arranged in n rows and n 

 columns, there being n— 1 dots and one asterisk in each row and each 

 column, and the order of the rows and columns being immaterial. 

 Systems of this form give rise to every species of linear associative 

 algebra. Quadrates are considered by Professor Pierce in his annota- 

 tions of his father's memoir on " Linear Associative Algebras." 



§§ 328 — 331. Isolated Collections — Residuals — Satisfied Collections. 



§§ 332 — 343. Some Isolated Triad Systems — Family No. 1. 



§§ 344 — 349. Some Isolated Triad Systems — Family No. 2. 



We have bere a discussion of some interesting systems composed 

 of isolated collections, i.e., collections which are such that each com- 

 ponent unit is unique with respect to the rest of the collection. 



Systems of Family No. 2 give rise to self-distributive algebras, i.e., 

 algebras such that 



a . bc—ab . ac. 



§§ 350 — 351. Geometry in General. 



In most geometrical investigations the units compose a system of a 

 high order of multiplicity ; we have points, straight lines, conies, &c, 

 unified collections of these, operators such as quaternions, &c, &o. It 

 is, however, thought to be sufficient, for the purpose of illustration, to 

 refer briefly to some comparatively simple systems. 



§§ 352 — 361. System of Points and Straight Lines. 

 We have here a double system. The pairs connecting the two 



