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2310=2 X 3 X 5 X7 X a) or still higher, going up the scale

of multiplied primes as far as you care to go. In each case, you have

to leave out of account all the primes that are factors of the num-

ber base, but will find all other primes crowded into a smaller and

smaller fraction of any finite successive list of numbers.

Here's the way it works as far as I've gone:

Number Base % eliminated % remaining

2 50 50

2X3 = 6 66% 33%

2X3X5 = 30 73% 26%

2X3X5X7 = 210 77% 22%

I refuse to go higher. You can work it out for 2310 or for any

still higher number base yourself.*

Now mind you, the larger you make the number on which you

base your number system, the more inconvenient it is to handle

that system in practice, regardless of how beautiful it may be in

* Since this chapter first appeared, knowledgeable readers have sent me

formulas to use in such calculations. If I had known them I would have had a

lot less trouble.

HO THE PROBLEM OF NUMBERS AND LINES

theory. It is perfectly easy to understand the system for writing

and handling numbers in a 30-based system, but to try to do so

in actual manipulations on paper is a one-way ticket to the booby

hatch at least if your mind is no nimbler than mine.

The gain in prime-concentration in passing to a 30-based sys-

tem (and I won't even talk about a 210-based system or anything

higher) is simply not worth the tremendous loss in manipulability.

Let us therefore stick with the 6-based system, which is not only

more efEcient as a prime-concentrator than our ordinary 10-based

system is, but is actually easier to manipulate once you are used

to it.

Or we can put it another way. It is the 6-based system which is,

in this respect at least, of prime quality.*

* Let there be no groaning in the galleryl

l O EUCLI D S FI FTH

Some of my articles stir up more reader comment than others,

and one of the most effective in this respect was one I once wrote

in which I listed those who, in my opinion, were scientists of the

first magnitude and concluded by working up a personal list of

the ten greatest scientists of all time.

Naturally, I received letters arguing for the omission of one or

more of my ten best in favor of one or more others, and I still

get them, even now, seven and a half years after the article was

written.

Usually, I reply by explaining that estimates as to the ten great-

est scientists (always excepting the case of Isaac Newton, concern-

ing whom there can be no reasonable disagreement) are largely a

subjective matter and cannot really be argued out.

Recently, I received a letter from a reader who argued that

Archimedes, one of my ten, ought to be replaced by Euclid, who

was not one of my ten. I replied in my usual placating manner,

but then went on to say that Euclid was "merely a systematizer"

while Archimedes had made very important advances in physics

and mathematics.

But later my conscience grew active. I still adhered to my own

opinion of Archimedes taking pride of place over Euclid, but the

phrase "merely a systematizer" bothered me. There is nothing nec-

essarily "mere" about being a systematizer.*

For three centuries before Euclid (who flourished about 300

B.C.) Greek geometers had labored at proving one geometric

theorem or another and a great many had been worked out.

What Euclid did was to make a system out of it all. He began

* Sometimes there is. In all my non-fiction writings I am "merely" a

systematizer. Just in case you think I'm never modest.

n8

THE PROBLEM OF NUMBERS AND LINES

with certain definitions and assumptions and then used them to

prove a few theorems. Using those definitions and assumptions

plus the few theorems he had already proved, he proved a few ad-

ditional theorems and so on, and so on.

He was the first, as far as we know, to build an elaborate mathe-

matical system based on the explicit attitude that it was useless to

try to prove everything; that it was essential to make a beginning

with some things that could not be proved but that could be ac-

cepted without proof because they satisfied intuition. Such in-

tuitive assumptions, without proof, are called "axioms."

This was in itself a great intellectual advance, but Euclid did

something more. He picked good axioms.

To see what this means, consider that you would want your list

of axioms to be complete, that is, they should suffice to prove all

the theorems that are useful in the particular field of knowledge

being studied. On the other hand they shouldn't be redundant.

You don't want to be able to prove all those theorems even after

you have omitted one or more of your axioms from the list; or to

be able to prove one or more of your axioms by the use of the re-

maining axioms. Finally, your axioms must be consistent. That

is, you do not want to use some axioms to prove that something is

so and then use other axioms to prove the same thing to be not so.

For two thousand years, Euclid's axiomatic system stood the

test. No one ever found it necessary to add another axiom, and

no one was ever able to eliminate one or to change it substantially

which is a pretty good testimony to Euclid's judgment.

By the end of the nineteenth century, however, when notions

of mathematical rigor had hardened, it was realized that there

were many tacit assumptions in the Euclidean system; that is,

assumptions that Euclid made without specifically saying that he

had made them, and that all his readers also made, apparently

without specifically saying so to themselves.

For instance, among his early theorems are several that demon-

strate two triangles to the congruent (equal in both shape and

size) by a course of proof that asks people to imagine that one tri-
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