# 公设 - 译自《希腊化时代的科学与文化》（3）

《几何原本》中最令人惊讶的是欧几里德对公设的选择。当然，在这方面亚里士多德是他的老师，亚里士多德对数学原理给予了很大的关注，指出公设的不可避免性和将公设降到最小程度的必要性，然而公设的选择却是欧几里德所做的。

- 如果一条线段与两条直线相交，在某一侧的内角和小于两直角和，那么这两条直线在不断延伸后，会在内角和小于两直角和的一侧相交。

Postulates

The most amazing part of that is Euclid’s choice of postulates. Aristote was, of course, Euclid’s teacher in such matters ; he has devoted much attention to mathematical principles, has shown the unavoidability of postulates and the need of reading them to a minimum ; yet the choice of postulates was Euclid’s.

In particular, the choice of postulate 5 is, perhaps, his greatest achievement, the one that has done more than any other to immortalize the word « Euclidean ». Let us quote it verbatim :

- If a straight line falling on to straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

A person of average intelligence would say that the proposition is evident and needs no proof ; a better mathematician would realize the need of a proof and attempt to give it; it required extraordinary genius to realize that a proof was needed yet impossible. There was no way out, then, from Euclid’s point of view, but to accept it as a postulate and go ahead.

The best way to measure Euclid’s genius as evidenced by this momentous decision is to examine the consequences go it. The first consequence, as far as Euclid was immediately concerned, was the admirable concatenaion of his Elements. The second was the endless attempts that mathematicians made to correct him ; the first to make them were Greeks, like Ptolemy (II-1) and Proclos (V-2), the Jew, Levi ben Gerson (XIV-1), and, finally, Girolamo Saccheri (1667-1733) of San Remo in his Euclides ab omni naecovindicatus (1733), the Swiss, Johann Heinrich Lambert (1728-77), and the Frenchmann, AdrienMarie Legendre (1752-1833). The list could be lengthened considerably, but these names suffice, because they are the names  of illustrious mathematicians, representing many countries and many ages, down to the middle of the last century. The third consequence is illustrated by the list of alternatives to the postulates and succeed in doing so, but at the cost of introducing another one (explicit or implicit) equivalent to it. For example,

If a straight line interests one of two parallels, it will intersect the other also. (Proclos)

Given any figure, there exists a figure similar to it of any size (John Wallis)

Through a given point only one parallel can be drawn to a given straight line (John Playfair)

There exists a triangle in which the sum of the three angles is equal to two right angles. (Legendre)

Given any three points not straight line, there exists a circle passing through them. (Legendre)

If I could prove that a rectilinear triangle is possible the content of which is greater than any given area, I would be in a pisition to prove perfectly rigorously the whole of geometry. (Gauss, 1799)

All these men proved that the fifth postulate is not necessary if one accepts another postulate rendering the same service. The acceptance of any of those alternatives (those quoted above and many others) would increase the difficulty of geometry teaching, however, the use of some of them would seem very artificiel and would discourage young students. It is clear that a simple exposition is preferable to one which is ore difficult; the setting up of avoidable hurdles would prove the teacher’s cleverness but also his lack of common sense. Thanks to his genius, Euclid saw the necessarily of this postulate and selected intuitively the simplest form of it.

There were also many mathematicians who were so blind that they rejected the fifth postulate without realising that another was taking its place. They kicked one postulate out of the door and another came in through the window without their being aware of it !

1】乔治·萨顿（George Sarton）与《希腊化时代的科学与文化》

2】张卜天译本，兰纪正、朱恩宽译本。

http://blog.sciencenet.cn/blog-2322490-1309778.html

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