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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 

 continue to coincide (by sliding), or to have one point only in 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 

 rotation not exceeding a semi-revolution. Proceeding to examine 

 the relations of triangles and parallelograms, we discover the opera- 

 tion of taking a fraction of a straight line, and therefore of a triangle 

 and of any rectilineal figure. We see that this operation is, in fact, 

 the same as that of altering a third line into a fourth, so that the 

 multiples of the third and fourth, when arranged in order of magni- 

 tude, should lie in the same order as those of the first and second 

 when similarly arranged. The relation of two magnitudes, with 

 respect to this order, we term their ratio, and the equality of ratios 

 proportion. The inversion and alternation of the four terms of a 

 proportion are now investigated. The operation of changing any 

 magnitude into one which bears a given ratio to it, is called a tensor. 

 The laws of tensors, being investigated, are shown to be the same as 

 those of fractions. They, however, furnish the complete conception 

 of infinite and infinitesimal tensors, by letting one or other of the mag- 

 nitudes by which the ratio is given become infinite or infinitesimal. 

 Thence is developed the law, that tensors differing infinitesimally are 

 equal fop all assignables. Consequently tensors may be represented 

 by convergent series of fractions. The algebra of tensors allows of 

 the inversion of a sum with the same limitation as in the case of 

 fractions, the complete inversion of a product of tensors, and the 

 practical inversion of a power with a constant integral index. This 

 algebra applied to geometry allows of the investigation of all statical 

 relations, that is, of all the geometry of the ancients, in which 

 magnitudes alone were considered, without direction. In respect to 

 areas, the consideration of the parallelogram swept out by one straight 

 line translated so as to keep one point on another straight line, leads 



