Abstract
After the acceptance of zero as a number, following the adoption of the decimal way of writing numbers (see Sect. 1.2), it became gradually common to think of numbers as points on a straight line: the positive numbers extending, in order of increasing magnitude.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Galileo was the first to point out the one-to-one correspondence between the terms of the two sequences of (5.1). This and a discussion relating to the one-to-one correspondence between the points of line-segments of different length appear in his book Discourses and Mathematical Demonstrations Concerning Two New Sciences.
- 2.
To see this, imagine a net of knots arranged in rows and columns, as in a fishing net spread on a plane. The positive rational numbers are given by a/b, where ‘a’ and ‘b’ are any two positive integers, and can therefore be arranged on the net, ‘a’ deciding the row and ‘b’ the column, if we allow the same number to appear over more than one knots. For example: 1/1 (= 1) will be on (first row-first column) but also on (nth row-nth column) where n is any integer (n/n = 1). Similarly 2/3 will be on (second row-third column) but also on ((2n)th row-(3n)th column) etc. Now if the knots are a countable set, the positive rational numbers (which correspond to a subset of the knots) are also a countable set. We can put the knots of the net into a one-to-one correspondence with the counting numbers by first counting the knots on the perimeter of the smallest square on the net (one of its corners being on (first row-first column)), go on to knots on the next square, then to knots on the square after that, and so on. We are, in this way convinced that the knots are a countable set, and therefore the positive rational numbers are a countable set. And the same, of course applies to the negative rational numbers. Now if two different sets of numbers are countable, their union is countable (to see this assume a correspondence between the members of the first set with the even numbers and a correspondence between the members of the second set with the odd numbers). We therefore conclude that the rational numbers are a countable set.
- 3.
Apparently two other English scientists, Henry Power and Richard Townely had discovered the gas law in 1653. Boyle knew of their experiments and acknowledged their contribution in his monograph of 1662.
- 4.
L. Principe, Robert Boyle: the secret alchemist, (Princeton University Press), 1998.
- 5.
The figure represents a cross section (a thin slice to be identified with the page) of the system under consideration. The system extends above and below the page, but the values of the physical quantities we consider do not change in the direction normal to the page. The same applies to the other figures of this section.
- 6.
Often, and certainly so in the 17th century, air and not vacuum was used as the reference medium. The refractive index of air (relative to vacuum) is not exactly unity and depends to some small degree on the colour (frequency) of light.
- 7.
When a slightly convex lens is placed on a flat glass plate and white light is reflected by the two close surfaces into the observer’s eye at a suitable angle, the point of contact of the lens is seen as a dark spot surrounded by rings of different colours. When monochromatic light is used, the observer sees a dark spot surrounded by bright and dark rings, a phenomenon typical of interference between waves arriving at a point in phase (the maxima of the one wave coinciding with the maxima of the other) producing bright rings, or in opposite phases (the one canceling the other) producing dark rings.
- 8.
G. W. Leibniz, De quadratura, edited by E. Knobloch (Abh. Akad. Wiss., 1993).
The cited extract appears in: Infinity, by B. Clegg, (Constable & Robinson Ltd), 2003.
- 9.
Isaac Newton, The Principia, A new translation by I. Bernard Cohen and Anne Whitman. Preceded by a Guide to Newton’s Principia by I. B. Cohen. (University of California Press), 1999.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Modinos, A. (2014). The Seventeenth Century: The Bloom of Science. In: From Aristotle to Schrödinger. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-00750-2_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-00750-2_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-00749-6
Online ISBN: 978-3-319-00750-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)