The Elements of a New Arithmetical Notation: And of a New Arithmetic of Infinites:in Two Books ... with an Appendix Concerning Some Properties of Perfect, Amicable, and Other Numbers...Hurst, Robinson, and Company,Cheapside; and A. Constable and Company, Edinburgh, 1823 - Arithmetic - 151 pages |
Common terms and phrases
ad infinitum added aggre arithmetical progression bers commensurable cube decagons demonstrated denominator distributed form distributed value divided by 1+1 dividend division divisor duced enneagons equivalent evident exceeds the double expansion expres expression figurate numbers finite number following instances former series fourth term fractional reciprocals fractional series geometrical series gives the series half Hence incommensurable infi infinite series infinitesimal excepted infinitum last term latter series less likewise Lord Brouncker mensurable method of notation metic modern mathematicians multi multiplied number of terms parag partially perfect pentagonal perfect numbers perfectly amicable numbers perimeter posi position with reference powers prime numbers produce the series quadruple quotient ratio reference to unity remainder rest root second term series is equal series is produced series of whole series++++++ series+++++++ similar manner sion sixth square 12+1 subtracted sum arising take place taken separately terms are taken third term tion triple whole numbers ㅎㅎ
Popular passages
Page i - THE ELEMENTS OF A NEW ARITHMETICAL NOTATION, and of a New Arithmetic of Infinites : in Two Books : in which the Series discovered by modern Mathematicians, for the Quadrature of the Circle and Hyperbola, are demonstrated to be aggregately Incommensurable Quantities : and a Criterion is given, by which the Commensurability or Incommensurability of Infinite Series may be accurately ascertained.
Page 69 - In every series of terms in arithmetical or geometrical progression, or in any progression in which the terms mutually exceed each other, the last term is equal to the first term added to the second term, diminished by the first; added to the third term, diminished by the second ; added to the fourth term, diminished by the third; and so on. And if the number of terms be infinite, the last term is equal to the . series multiplied by 1 — 1.
Page 112 - Now this is demonstrated to be the case in these numbers, the parts of each are generative of each other according to the nature of friendship. Ozanam, a French mathematician, AD 1710, gives examples in his " Mathematical Recreations " of such Amicable Numbers. He remarks that 220 is equal to the sum of the aliquot parts of 284, thus...
Page 74 - ... square, then does the pyramid consist of an infinite number of such squares, whose sides, or roots, are continually increasing in arithmetical progression, beginning at the vertex or point, its base being the greatest term, and its perpendicular height the number of all the terms : but the last term multiplied into the number of terms will be triple the sum of all the series, equal the solid content of the pyramid.
Page i - The Elements of a new Arithmetical Notation, and of a new Arithmetic of Infinites...
Page v - Commensurabilityorlncommensurability of Infinite Series may be accurately ascertained. With an Appendix, concerning some Properties of Perfect, Amicable, and other Numbers, no less remarkable than novel.
Page 114 - An inspection of the above shows that the sum of the first and last terms of an arithmetical series, multiplied by the number of terms, is equal to twice the sum of all the terms.
Page 98 - Hence it follows that infinite series of polygonous numbers will be to each other in the ratio of the natural series of numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, &c. 20. The fractional reciprocals also of these polygonous numbers will be to each other as 1 , |-, $, T» y
Page 49 - For they say that the latter of these series is •$• of the square of the periphery of a circle whose diameter is 1.