Mr. A. J. Ellis on the Systematization of Mathematics. 227 
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 alterimg 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 for 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 
to an independent algebra of areas, in which the generating lines are 
considered immediately. The laws of the relations of lines thus 
discovered, are shown to be identical with the laws of the relations 
of tensors. Consequently, with certain limitations, the whole of the 
algebra of tensors may ‘be interpreted as results in the algebra of 
areas. This leads to a perfect conception of the principle of homo- 
nomy, or dissimilar operations having the same laws, and conse- 
quently the same algebra. 
In dynamical or modern geometry, all lines are considered as in 
construction, having initial and final points. If the initial points of 
any two straight lines are joined to a third, not on either, and the two 
parallelograms be completed, the lines drawn from the point parallel 
to the given lines are dynamically equal to them; if these last 
lie on each other, the first two lines have the same direction; if the 
last have only one point in common and lie in the same straight line, 
the first have opposite directions ; and if the last do not lie in the 
same straight line, the first have different directions, and the angle 
between the last is the angle between the first lines. Similar defini- 
tions can be given of direction in the case of angles and circular 
ares. If from the final point of any line we draw a line equal to 
a second, avd join the initial point of the first with the final point 
of the line thus drawn, we are said to append the second to the first, 
and the joining line is called the appense of the other two. The 
