A superscript is denoted by ^{x}. For example, y^{4} indicates y to the power 4. A subscript is denoted by {x}. For example, a{ik} indicates the ik-th element of array a. Algebraic expressions in the original text usually had the variables in italic. The italic underscore markup has been removed for readability, so for example: _dx_{i}_ = λ′_d_′_x_{i}_ + λ″_d″x_{i}_ has been changed to: dx{i} = λ′d′x{i} + λ″d″x{i} Obvious typographical errors and punctuation errors have been corrected after careful comparison with other occurrences within the text and consultation of external sources. Except for those changes noted below, all misspellings in the text, and inconsistent or archaic usage, have been retained. For example, co-exist, coexist; every-day, everyday; connexion; assertorial; apodeictic; premisses. § 82, 'so Erdmannn confidently' replaced by 'so Erdmann confidently'. § 150 Footnote [157], 'Delboeuf' replaced by 'Delbœuf' for consistency. § 152, 'one subdivison must' replaced by 'one sub-division must'. § 159 Footnote [167], 'Delboeuf' replaced by 'Delbœuf' for consistency. § 204, 'and homogenous;' replaced by 'and homogeneous;'. End of the Project Gutenberg EBook of An essay on the foundations of geometry, by Bertrand Russell