The History of Mathematics: A Brief Course

218 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICS

Consider, in contrast, the case of a physical constant, say the universal gravita-

tional constant, usually denoted Go, which occurs in Newton's law of gravitation:

Here F is the force each of two bodies exerts on the other, Ì and m are the

masses of the two bodies, and r is the distance between their centers of gravity.

The accepted value of Go, given as upper and lower assured limits, is 6.674215 ±

0.000092 Í · m^2 /kg^2 , although some recent measurements have cast doubt on this

value. From a mathematical point of view, Go is determined by the equation

Fr^2

Mm

and its value is found—as Cavendish actually did—by putting two known masses Ì

and m at a known distance r from each other and measuring the force each exerts

on the other. The assertion that the ratio Fr^2 /Mm is the same for all masses

and all distances is precisely the content of Newton's law of gravity, so that two

experimenters using different masses and different distances should get the same

result. But Newton's law of gravity is not deducible from axioms; it is, rather, an

empirical hypothesis, to be judged by its explanatory power and its consistency with

observation. What should we conclude if two experimenters do not get the same

result for the value of Go? Did one of them do something wrong, or is Newton's law

not applicable in all cases? Does it even make sense to ask what the 50th decimal

digit of Go is?

8.2. You can represent y/ab geometrically by putting a line of length b end-to-end

with a line of length a, drawing a circle having this new line as diameter, and then

drawing the perpendicular to the circle from the point where the two lines meet.

To get \fa and \fb, you would have to use Descartes' unit length / as one of the

factors. Is it possible to prove by use of this construction that \fabl = ,/aVb? Was

Dedekind justified in claiming that this identity had never been proved?

8.3. Try to give a definition of real numbers—perhaps using decimal expansions—

that will enable you to say what the numbers y/2, Ë/ÚÚ, and \/6 are, and how

they can be added and multiplied. Does your definition enable you to prove that

y/2y/l = \/6?

8.4. Use the method of infinite descent to prove that \/3 is 'irrational. [Hint:

Assuming that m^2 = 3n^2 , where m and ç are positive integers having no common

factor, that is, they are as small as possible, verify that (m - 3n)^2 = 3(m — n)^2.

Note that m < 2n and hence m-n < n, contradicting the minimality of the original

m and n.]

8.5. Show that s/3 is irrational by assuming that m^3 = 3n^3 with m and ç positive
integers having no common factor. [Hint: Show that (m — n)(m^2 +mn + n^2 ) = 2n^3.
Hence, if ñ is a prime factor of n, then ñ divides either m — ç or m^2 + mn + n^2. In
either case ñ must divide m. Since m and ç have no common factor, it follows that
ç = 1.]

8.6. Suppose that x, y, and æ are positive integers, no two of which have a common
factor, none of which is divisible by 3, and such that x^3 + y^3 = z^3. Show that there
exist integers p, q, and r such that æ — χ = ñ^3 , æ — y = q^3 , and χ + y — r^3. Then,