It is a matter for considerable regret that Fermat, who cultivated the theory of numbers with so much success, did not leave us with the proofs of the theorems he discovered. In truth, Messrs Euler and Lagrange, who have not disdained this kind of research, have proved most of these theorems, and have even substituted extensive theories for the isolated propositions of Fermat. But there are several proofs which have resisted their efforts.
Once you've learned to think you can't stop. And an enormous number of people devote their lives to keeping their minds busy and feel extremely uncomfortable with silence.
So long as all is ordered for attack, and that alone, leaders will instinctively increase the number of enemies that they may give their followers something to do.
Whether if soul did not exist time would exist or not, is a question that may fairly be asked; for if there cannot be someone to count there cannot be anything that can be counted, so that evidently there cannot be number; for number is either what has been, or what can be, counted.
In some corner of the world they are probably still holding regular meetings of the Flat Earth Society. We derive no comfort because important people, vocal people, or great numbers of people agree with us. Nor do we derive comfort if they don't.
The more elevated a culture, the richer its language. The number of words and their combinations depends directly on a sum of conceptions and ideas; without the latter there can be no understandings, no definitions, and, as a result, no reason to enrich a language.
Among civilized and thriving nations, on the contrary, though a great number of people do no labor at all, many of whom consume the produce of ten times, frequently of a hundred times more labour than the greater part of those who work; yet the produce of the whole labour of the society is so great, that all are often abundantly supplied, and a workman, even of the lowest and poorest order, if he is frugal and industrious, may enjoy a greater share of the necessaries and conveniencies of life than it is possible for any savage to acquire.
This great increase of the quantity of work which, in consequence of the division of labour, the same number of people are capable of performing, is owing to three different circumstances; first, to the increase of dexterity in every particular workman; secondly, to the saving of the time which is commonly lost in passing from one species of work to another; and lastly, to the invention of a great number of machines which facilitate and abridge labour, and enable one man to do the work of many.
Far be it from us to doubt that all number is known to Him 'Whose understanding is infinite' (Ps. 147:5). The infinity of number, though there be no numbering of infinite numbers, is yet not incomprehensible by Him Whose understanding is infinite. And thus, if everything which is comprehended is defined or made finite by the comprehension of him who knows it, then all infinity is in some ineffable way made finite to God, for it is comprehensible by His knowledge.
By rejecting the authority of the individual and replacing it by the numbers of some momentary mob, the parliamentary principle of majority rule sins against the basic aristocratic principle of Nature.
Germany suffered most as a consequence of this Peace Treaty and the general insecurity which was bound to arise from it. The unemployment figure rose to a third of the number usually employed in the nation, which means, however, that by counting the families of the unemployed as well there were 26 million people in Germany out of a population of 65 millions faced by an absolutely hopeless future.
Look out into the July night, and see the broad belt of silver flame which flashes up the half of heaven, fresh and delicate as the bonfires of the meadow-flies. Yet the powers of numbers cannot compute its enormous age,—lasting as space and time,—embosomed in time and space.