An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author. - Softcover

Michigan Historical Reprint Series

Softcover

ISBN 10: 1418183806 ISBN 13: 9781418183806

Publisher: Scholarly Publishing Office, University of Michigan Library, 2005

Synopsis

Excerpt from An Introduction to the Study of the Elements of the Differential and Integral Calculus
Real numbers and functions of real numbers.
The conceptions of space and of number are the subject matter of mathematical investigations. These investigations accordingly diverge into two main branches: Geometry and Analysis. It thus appears that Mathematics are of fundamental importance to all our knowledge of Nature: for our representations of space contain the simplest properties which are common to all things in the surrounding world; and accurate comparison or measurement of quantities leads always to concrete numbers of the units employed: in order to understand the result, we require a knowledge of numbers and of their combinations.
Nature in its phenomena is perpetually exhibiting change; the simplest changes we perceive externally are changes of place, motions. The representation of motion is necessarily combined with that of continuity, i. e. of an uninterrupted connexion in space and of an uninterrupted sequence in time. To describe thoroughly the phenomena of motion is to assign every circumstance in numbers of concrete units: so that if the series of numbers is also to enable us to describe motion, it must contain a continuous series of quantities. Thus the first problem of Analysis is: to develope the conception and the properties of the continuous series of numbers.
First Chapter.
Rational numbers.
1. The natural series of numbers, which arises by adding on a thing to others in counting, advances always by unity; each number is defined by the preceding number and by unity. This series of integers starting from unity can be continued on indefinitely. Now as each several number is a sum of repeatedly added units, such a sum of units can be composed of different given numbers. This arithmetical operation, merely a continued reckoning up of groups of units, is called Addition; it embraces all other operations, from it all others arise.

Edition details

• Publisher: Scholarly Publishing Office, University of Michigan Library
• Publication date: 2005
• Language: English
• ISBN 10: 1418183806
• ISBN 13: 9781418183806
• Binding: Paperback
• Number of pages: 415