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to the form of a numerical integer ; secondly, dynamically, con- 

 sidering the effect of a variation in the integer employed. This 

 leads to the conception of a formation (Lagrange's "analytical 

 function "), as a combination of a fixed and independently variable 

 integer. Such a combination is, therefore, also itself dependently 

 variable. The inversion of formations, whereby the independent 

 variable is expressed as a formation of the dependent variable, imme- 

 diately engages our attention. The inversion of a sum leads to a 

 difference, with the limitation that the minuend should be greater 

 than the subtrahend. The inversion of a product leads to a quo- 

 tient, with the limitation that the dividend should be a multiple of 

 the divisor. The inversions of a power lead to the root and loga- 

 rithm, with increasing limitations. The study of discontinuous ob- 

 jects then allows the application of these inversions to the solution 

 of problems in common life. 



The operation by which any group in the arithmetical scale already 

 described is formable from any other group in the same scale, leads 

 to the conception of & fraction, necessarily expressible, according to 

 the general laws of operation, as the quotient of two integers. The 

 operands of such operations must admit of being separated into 

 certain numbers of equal parts, or rather, in order that they may 

 admit of any fractional operation, into any number of equal parts. 

 Thus discontinuous approaches continuous succession. The laws of 

 fractions are the same as the laws of integers, provided the indices 

 used are all integers. The object of the statical algebra of fractions 

 is to reduce all combinations of numerical fractions to numerical 

 fractions. The inversion of formations is less limited than before. 

 There is the same limitation respecting differences, but none respect- 

 ing quotients. The attempt to convert all fractions into radical 

 fractions (whose denominators are some powers of the radix of the 

 system of numeration), leads to the conception of convergent infinite 

 series, and hence allows an approximation to the inversion of a power 

 with a constant index. 



In Geometry, the notion of continuous succession or extension is 

 derived from the motion of the hand, which recognizes separable but 

 not separated parts. This motion gives the conception of surfaces, 

 which by their intersections two and two, or three and three, give 

 lines and points. Recognizing a line as the simplest form of exten- 



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