Geometry

Let no one ignorant of geometry enter here.
— Alleged to have appeared over the door of Plato’s Academy.

For God’s sake, please give it up. Fear it no less than the sensual passions because it, too, may take up all your time and deprive you of your health, peace of mind, and happiness in life.
— Letter from Wolfgang, the father of Janos Bolyai, to his son, advising him to avoid the pursuit of non-Euclidean geometry.

And since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art …

But when great and ingenious artists behold their so inept performances, not undeservedly do they ridicule the blindness of such men; since sane judgment abhors nothing so much as a picture perpetrated with no technical knowledge, although with plenty of care and diligence. Now the sole reason why painters of this sort are not aware of their own error is that they have not learnt Geometry, without which no one can either be or become an absolute artist; but the blame for this should be laid upon their masters, who are themselves ignorant of this art.
— Albrecht Dürer (1471 – 1528), Course in the Art of Measurement, 1525.

 

Euclid’s axioms

- Things equal to the same thing are equal to each other.
- If equals are added to equals, then the sums are equal.
- If equals are subtracted from equals, then the remainders are equal.
- Things which coincide with each other are equal to each other.
- The whole is greater than the part.
— Euclid (fl. 300 BCE)

 

Euclid’s postulates

- A straight line can be drawn joining any two points.
- Any straight line segment can be extended indefinitely in a straight line.
- Given any straight line segment, a circle can be drawn having the segment as a radius, and one end of the segment as a centre.
- All right angles are equal to each other.
- Given a straight line and any point not on this line, there is through that point, one and only one line that is parallel to the given line. (Or you can put it like this: If two lines are drawn which intersect a third line in such a way that the sum of the inner angles on one side is less than two right angles, then sooner or later, if you extend the two lines, they will meet each other, somewhere.)

Proposition 1: If a surface is cut by a plane always passing through a certain point, and if the section is always a circumference of a circle whose centre is that point, the surface is that of a sphere. [This was needed for the next proposition.]

Proposition 2: The surface of any fluid at rest is the surface of a sphere whose centre is the same as that of the Earth.

Proposition 3: Solids of equal density with a fluid will, if let down into the fluid, be immersed so that they do not project above the surface but do not sink lower …

Proposition 4: A solid lighter than a fluid will, if placed in it, not be completely submerged, but part of it will project above the surface …

Proposition 5: Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced …

Proposition 6: If a solid lighter than a fluid is forcibly immersed in it, the solid will be driven upwards by a force equal to the difference between its weight and the weight of the fluid displaced …

Proposition 7: A solid heavier than a fluid will, if placed in it, descend to the bottom of the fluid, and the solid will, when weighed in the fluid, be lighter than its true weight by the weight of the fluid displaced. [Archimedes’ Principle]
 

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