The nature of capital and income

Bibliographic data

Monograph

Identifikator:: 102659555X

URN:: urn:nbn:de:zbw-retromon-82920

Document type:: Monograph

Author:: Fisher, Irving http://d-nb.info/gnd/118533541

Title:: The nature of capital and income

Place of publication:: New York

Publisher:: The Macmillan Company

Year of publication:: 1923

Scope:: XXI, 427 Seiten

Digitisation:: 2019

Collection:: Economics Books

Usage license:: Get license information via the feedback formular.

APPENDIX TO CHAPTER XIII 395 year will bring the radius vector to the position OD, at which time the same interest, DFE, may be detached, and so on in- definitely, the result being a toothed wheel. Each tooth being +5 of the radius 04, the sum of the twenty-five teeth will be exactly equal to the radius. In case the interest is reckoned semi-annually, the teeth will be fifty in number instead of twenty-five, but each will be half as large; and so on until, for “continuous” reckoning, we have an infinite number of teeth each of infinitesimal size; but the sum of the whole number in the complete revolution will still be exactly equal to 0A. In case no income is received, the accumulation of capital is represented by increasing the length of the radius vector, the end of which thus traces a spiral. The radius vector re- volves around the spiral once every twenty-five years, and the ratio between the “amount,” 04', after a com- plete revolution, and the original “principal” 04, will be e, that is, 2.718, provided the rate of interest is reckoned continuously. The same spiral represents the accumulation of capital, whatever may be the rate of interest. For a complete revolution does not represent a definite length of time, but the purchase period ; and it is clear that a more rapid rate of interest is repre- sented by a more rapid turning of the radius vector. Thus, if the rate is not 49, but 89, the radius vector swings around through the same spiral once in twelve and a half years instead of once in twenty-five years. Again, if the rate is 29, the revolution is fifty years. The spiral is what is known as the “equiangular spiral,” and has the property that the tangent at any point is inclined at a constant angle to the radius vector. The angle is in this case such that its tangent is Zr. This : SP angle is 80° 57". The equation of this spiral is =e 0 which p represents the radius vector, 8 the angle of revolution, and p, the initial radius vector; e and = are, of course, the magnitudes ordinarily represented by these letters, namely, the base of the Napierian system of logarithms and the ratio of the circumference of a circle to its diameter.

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