9 
MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
SECTIONS. PAGE. 
222-239.—Groups—Circuits. 35—37 
240-252.—Groups containing from one to twelve units .. 37M?3 
253-255.—Some general forms of groups. 43 
256-269.—A family of groups. 43—47 
270-274.—R-adic groups . 47 
275-279.—Substitutions . 48 
280-312.—Algebras . 48—54 
313—327.—Quadrates. 54-57 
328-331.—Isolated collections—Residuals—Satisfied collections. 57, 58 
332—343.—Some isolated-triad systems.—Family No. 1 .... 58-61 
344—349.—Some isolated-triad systems.—Family No. 2 .. 61 
350-351.—Geometry in general . 62 
352-357.—System of coplanar points and straight lines ... 62, 63 
358-359.—Coplanar points, lines, and conics. 63 
360-391.—Logic. 63-70 
Scope of the Memoir. 
1. My object in this memoir is to separate the necessary matter of exact or 
mathematical thought from the accidental clothing—geometrical, algebraical, logical, 
&c.—in which it is usually presented for consideration ; and to indicate wherein 
consists the infinite variety which that necessary matter exhibits. 
2. The memoir is confined to the exposition of fundamental principles, to their 
elementary developments, to their application to such a variety of cases as will 
vindicate their value, and to a description of some simple and uniform modes of 
putting the necessary matter in evidence. I have been unable to ascertain that the 
principles here set forth have been previously formulated. 
Fundamental Principles. 
3. Whatever may be the true nature of things and of the conceptions which we 
have of them (into which points we are not here concerned to inquire), in the 
operations of reasoning they are dealt with as a number of separate entities or units. 
4. These units come under consideration in a variety of garbs—as material objects, 
intervals or periods of time, processes of thought, points, lines, statements, relation¬ 
ships, arrangements, algebraical expressions, operators, operations, &c., &c., occupy 
various positions, and are otherwise variously circumstanced. Thus, while some units 
are incapable of being distinguished from each other, others are by these peculiarities 
rendered distinguishable. For example, the angular points of a square are distin¬ 
guishable from the sides, but are not distinguishable from each other. In some 
instances where distinctions exist they are ignored as not material. Both cases are 
included in the general statement that some units are distinguished from each other 
and some are not. 
5. In like manner some pairs of units are distinguished from each other, while 
others are not. Pairs may in some cases be distinguished even though the units 
