The Spirit of Mathematical Analysis: And Its Relation to a Logical SystemJ.W. Parker, 1843 - 103 páginas |
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The Spirit of Mathematical Analysis: And Its Relation to a Logical System Martin Ohm Visualização integral - 1843 |
The Spirit of Mathematical Analysis: And Its Relation to a Logical System Martin Ohm Visualização integral - 1843 |
The Spirit of Mathematical Analysis, and Its Relation to a Logical System ... Martin Ohm Pré-visualização indisponível - 2018 |
Palavras e frases frequentes
according to powers according to whole actual logarithm actual numbers actual or imaginary algebraical sum applied binomial series binomial theorem broken number coefficients conceived consequently considered contradiction convergent convergent series correct equation denominate numbers denote determinate number determinate value differences of whole dignand divergent divergent series divisor enunciated equal expressions equal to zero exist follows formula fraction Hence idea indenominate infinite number infinite series proceeding infinitum Instruction-Book laws of operation letters loga m+n-p magnitudes mathematical analysis multiplied natural logarithm negative number negative whole number numerical series obtain p+q.i p+qi particular perfectly positive number positive or negative positive whole number quadratic equation quotient radicand represents result SECTION sense of sect series proceeding according shew signification simplest value single-meaning square root summand symbol transformed unknown expression value in cyphers whole functions whole or broken whole powers
Passagens conhecidas
Página 2 - Calculus, where he has demonstrated that we have p (x + co ) ~p (x) + ccp' (x) + 1 p" (x) -fete., as long as the series is convergent; but it is usually employed without ceremony in all cases. * * * " The theory of infinite series in general rests upon a very bad foundation. All operations are applied to them as if they were finite; but is this permissible ? I think not.* Where is it demonstrated that the differential of an infinite series is found by taking the differential of each term ? Nothing...
Página 2 - J. + cos 3x— &c. an utterly false result, for this series is divergent. " The same remark holds for the multiplication and division of infinite series. I have begun to examine the most important rules which are (at present) esteemed to hold good in this respect, and to shew in what cases they are correct, and in what not so. This work proceeds tolerably well, and interests me infinitely.
Página 4 - Polsson among others has spoken decidedly against their being employed. But that the series which are used, and from which deductions are drawn, ought to be always and necessarily convergent, is a circumstance of which the Author of this Essay has not been at all able to convince himself; on the contrary, it is his opinion that series, as long as they are GENERAL, so that we cannot speak of their convergence or divergence, must always, when properly treated, necessarily and unconditionally produce...
Página 1 - Works (CEuvres completes de NH Abel. Christiania, 1839) and in which he writes thus: " Divergent series are in general very mischievous affairs, and it is shameful that any one should have founded a demonstration upon them. You can demonstrate any thing you please by employing them, and it is they who have caused so much misfortune, and given birth to so many paradoxes. Can any thing be conceived more horrible than to declare that 0= 1 -2"+ 3" -4
Página vii - Ohm means one in which the letters represent neither magnitudes nor numbers, but are considered as perfectly insignificant (inhaltlos).
Página ii - ... little due to the man who sends another forth in search of undiscovered truth; b.ut he has established some claim to the good opinion of his fellow men who, having gone forth in pursuit of the treasure, brings it home and delivers it over to the keeping of all who are willing to receive it at his hands. Of the merits of a work of which truth is the object we cannot have an adequate idea nor a perfect relish without some acquaintance with the errors against which it is levelled and which it is...