226 ) Royal Society pak! 
the algebra of integers, first, statically, in order to reduce all results 
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 a 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 limi‘ed 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- 
sion, we distinguish the straight lines, which coincide when rotated 
about two common points, from the curves, which do not. These 
straight lines are shown to be fit operands for the integer and fraction 
operations. By moving one coinciding line over another so as to 
coutinue to coincide (by sliding), or to have one point only im com- 
mon (by rotating), or no points in common (by translation), we 
obtain the conceptions of angles and parallels, which suffice to show 
that the exterior angle of a triangle is equal to the two interior and 
opposite, and that two straight lines meet or not according as the 
exterior angle they make with a third is not or is equal to, the 
interior angle. Angles are then considered statically as amounts of 
