The theorem is here referred to partly on account of its bearing on the theory of imaginariesin geometry.
Poncelet throughout his work makes continual use of the foregoing theories of imaginaries and infinity, and also of the before-mentioned theory of reciprocal polars.
Bearing in a somewhat similar manner also on the theory of imaginaries in geometry (but the notion presents itself in a more explicit form), there is the memoir by L.
But to discuss adequately their philosophical import is difficult to me, since I am unacquainted with any satisfactory philosophy of imaginaries in pure Algebra.
For all the fruitful uses of imaginaries, in Geometry, are those which begin and end with real quantities, and use imaginaries only for the intermediate steps.
This is the case with imaginaries in Cayley's sense, and their use in Geometry, great as are its technical advantages, and rigid as is its technical validity, is wholly destitute of philosophical importance.
That no contradictions arise from the geometrical interpretation of imaginaries is not wonderful; for they are interpreted solely by the rules of Algebra, which we may admit as valid in their interpretation to imaginaries.
All the fruitful uses of imaginaries, in Geometry, are those which begin and end with real quantities, and use imaginaries only for the intermediate steps.
And those who invented imaginaries hardly suspected the advantage which would be obtained from them for the study of the real world, of this the name given them is proof sufficient.
It was because Maxwell was accustomed to 'think in vectors,' and yet it was through the theory of imaginaries (neomonics) that vectors were introduced into analysis.
The above list will hopefully give you a few useful examples demonstrating the appropriate usage of "imaginaries" in a variety of sentences. We hope that you will now be able to make sentences using this word.
