Now that practical skills have developed enough to provide adequately for material needs, one of these sciences which are not devoted to utilitarian ends [mathematics] has been able to arise in Egypt, the priestly caste there having the leisure necessary for disinterested research.

The whole is more than the sum of its parts.

The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.

The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.

If this is a straight line [showing his audience a straight line drawn by a ruler], then it necessarily ensues that the sum of the angles of the triangle is equal to two right angles, and conversely, if the sum is not equal to two right angles, then neither is the triangle rectilinear.

It is not once nor twice but times without number that the same ideas make their appearance in the world.

The whole is more than the sum of its parts.

The body is most fully developed [at] from thirty to thirty-five years of age, the mind at about forty-nine.

Hippocrates is an excellent geometer but a complete fool in everyday affairs.

But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.

Our account does not rob mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraceable. In point of fact they do not need the infinite and do not use it. They postulate any that the finite straight line may be produced as far as they wish.

We cannot ... prove geometrical truths by arithmetic.

The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.

There are things which seem incredible to most men who have not studied mathematics.

... while those whom devotion to abstract discussions has rendered unobservant of the facts are too ready to dogmatize on the basis of a few observations.

... the so-called Pythagoreans, who were the first to take up mathematics, not only advanced the subject, but having been brought up in it, they thought its principles of mathematics were the principles of all things.

A nose which varies from the ideal of straightness to a hook or snub may still be of good shape and agreeable to the eye.

Such an event is probable in Agathon's sense of the word: 'it is probable,' he says, 'that many things should happen contrary to probability.'

Accordingly, the poet should prefer probable impossibilities to improbable possibilities.

The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.

That which we must learn to do, we learn by doing.

. . . if the consequences are the same it is always better to assume the more limited antecedent, since in things of nature the limited, as being better, is sure to be found, wherever possible, rather than the unlimited.

The continuum is that which is divisible into indivisibles that are infinitely divisible.

Physics

Education is the best provision for old age.

He who has never learned to obey cannot be a good commander.

The roots of education are bitter, but the fruit is sweet.

What is a friend? A single soul dwelling in two bodies.

To Thales the primary question was not what do we know, but how do we know it.

It is the mark of an educated mind to be able to entertain a thought without accepting it.

Those who educate children well are more to be honored than parents, for these only gave life, those the art of living well.

It is the mark of an instructed mind to rest assured with that degree of precision that the nature of the subject admits, and not to seek exactness when only an approximation of the truth is possible.

The gods too are fond of a joke.