...the axiomatic results that flow from it ; for instance, we may employ one general and familiar axiom, that " what is true up to the limit is true at the limit ;" an axiom of most extensive and important application in entering upon the higher mathematics. Of...

...construction is indefinitely extended, are properties of the limiting figure. This is expressed by saying, that what is true up to the limit is true at the limit. 21. Similar curvilinear Jigures are those which are the Limits of similar polygons. 22. If several...

...or Hypotheses. The Axiom which \ve employ in introducing this Idea into our reasoning is this : — What is true up to the Limit is true at the Limit. This Idea and Axiom are the bases of all Methods of Limits, Fluxions, Differentials, Variations, and...

...remains. The curve line is the Limit of the polygon ; and in this process we proceed on the Axiom, that "What is true up to the limit is true at the limit." This mode of conceiving mathematical magnitudes is of wide extent and use ; for every curve may be...

...Hypotheses. The Axiom which we employ in introducing this Idea into our reasoning is this: — • What is true up to the Limit is true at the Limit. This Idea and Axiom are the bases of all Methods of Limits, Fluxions, Differentials, Variations, and...

...specific form of co . 23. The next proof which I shall examine depends upon the principle of continuity " that what is true up to the limit is true at the limit." It is as follows : e~a* Since fe~ai Sin xdx = j (Cos x + a Sin x) ; .-. r «-" Sin xdtv = --^— ....

...be got rid of. It bears a resemblance to the postulate assumed sometimes for mathematical purposes, that what is true up to the limit, is true at the limit. To copy a thing, is to pirate the whole ; to abridge or describe it, is piracy of the ideas ; to publish...

...that number be increase'! without limit, and the polygon will "ultimately " become a circle. But since what is true up to the limit is true at the limit, it follows that, in order to keep the same face of the satellite constantly turned towards the centre...

...definitions or Hypotheses. The Axiom which we employ in introducing this Idea into our reasoning is this: — What is true up to the Limit is true at the Limit. This Idea and Axiom are the bases of all Methods of Limits, Fluxions, Differentials, Variations, and...

...by a variable decreasing without limit. But when, page 109, he shows that Whewell's axiom, Whatever is true up to the limit is true at the limit, is unsound, he makes too sweeping a use of the word whatever. The axiom is obviously intended to apply...