To avoid misconception, it should be borne in mind thatinfinitesimals are not regarded as being actual quantities in the ordinary acceptation of the words, or as capable of exact representation.
I have at last become fully satisfied that the language and idea of infinitesimals should be used in the most elementary instruction--under all safeguards of course.
As a mathematician, I prefer the method of infinitesimalsto that of limits, as far easier and less infested with snares.
It appears that Fermat, the true inventor of the differential calculus, considered that calculus as derived from the calculus of finite differences by neglecting infinitesimals of higher orders as compared with those of a lower order.
There is also reason to think that Newton had become dissatisfied with the arguments about infinitesimals on which his calculus was based.
The tract occasioned a controversy which had the important consequence of making it plain that all arguments about infinitesimals must be given up, and the calculus must be founded on the method of limits.
In Scotland the rules of descent differ from the above in several particulars.
Descent is traced, as in England before the Inheritance Act, to the person last seized.
Failing collaterals, the inheritance ascends to the father and his relations, to the entire exclusion of the mother and her relations.
But when the deceased leaves an elder and a younger brother (or their issues), the elder brother takes the conquest, the younger takes the heritage.
The self-sacrifice of the men thus engaged enabled their commander to "see," and to mass his reserves opposite a selected point, while little by little the enemy was hypnotized by the fighting.
When heritage of the deceased goes to an elder brother (as might happen in certain eventualities), the younger of the elder brothers is preferred.
Next to the lineal descendants, and failing them, come the brothers and sisters, and their issue as collaterals.
These are merely the infinitesimals of unity and they grow less in size and consequence as the divisions increase in number.
The so-called infinitesimalsare Nature’s real, natural finites.
We know that mathematicians distinguish between infinitesimals of different orders and that those of the second order are infinitesimal, not only in an absolute way, but also in relation to those of the first order.
It is not difficult to imagine infinitesimals of fractional or even of irrational order, and thus we find again that scale of the mathematical continuum which has been dealt with in the preceding pages.
The continuum of the third order, to which the consideration of the different orders of infinitesimals leads, is itself not useful enough to have won citizenship, and geometers regard it only as a mere curiosity.
They are, in mathematical phrase, "infinitesimals of the second order," discontinuous with the interests of magnitude great enough to attract the attention of the law.
Application of the method of infinitesimals to the determination of the radius of curvature of certain curves geometrically defined.
He gave good measure, not prodigal, for he was loyal to his employer, but putting a very moderate strain on the ribbon, and letting the thumb-nail slide with a contempt of infinitesimals which betokened a large soul in its genial mood.
This is a desperate affair," said the Count, "and I fear that yourinfinitesimals will do her very little good.
These, I say, assert there are infinitesimals of infinitesimals of infinitesimals, &c.
The above list will hopefully give you a few useful examples demonstrating the appropriate usage of "infinitesimals" in a variety of sentences. We hope that you will now be able to make sentences using this word.
