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Exercises from chapter 1 — Primitive notions and propositions

Exercises and resolutions Exercise 1 — Classify as true (V) or false (F) An infinite number of straight lines pass through a point. VF Verdadeiro. Take the point $P$ and infinite points outside it: each one determines with $P$ a different straight line. A straight line passes through two distinct points. VF Verdadeiro. It is the postulate of determination. A straight line contains two distinct points. VF Verdadeiro. By the postulate of existence, it contains infinities. Two distinct points determine one and only one straight line. VF Verdadeiro. Existence and uniqueness: postulate of determination. A single straight line passes through three given points. VF Falso. Only if they are collinear. Three non-collinear points are not on the same straight line. ↻ Recomeçar

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Competing straights

Two different intersecting lines have exactly one point in common. It seems obvious — and that’s precisely why it took so long to be written. Who discovered Greek geometers used the fact all the time, but Euclid did not include it among his postulates. Some older editions of The Elements added the phrase “two straight lines do not enclose a space” to cover the gap; the philologist J. L. Heiberg, who critically edited the work at the end of the 19th century, considered this excerpt to be a later addition. The modern version comes from David Hilbert’s incidence axioms (1899): if two distinct straight lines had two common points, they would contradict the postulate that two points determine a single straight line.

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Primitive propositions

If point, line and plane are not defined, someone needs to tell the rules of the game. These rules are the postulates. Who discovered The idea of ​​starting a theory with statements accepted without proof is Greek. Aristotle (4th century BC) already discussed why every demonstrative science needs initial principles, and Euclid put this into practice in The Elements, opening the work with five postulates — the first says that it is possible to draw a straight line from one point to another. In 1899, David Hilbert completed the list with about twenty axioms, including rules that Euclid used without stating.

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Primitive notions

Point, line and plane are the pieces with which all geometry is assembled — and, interestingly, they have no definition. Who discovered Nobody “discovered” the point or the straight line: they are ideas as old as the habit of measuring land and drawing plans. The first person who attempted to define them in writing was Euclid of Alexandria, around 300 BC, in the The Elements: “point is that which has no parts” and “line is length without width”. More than two thousand years later, in 1899, the German David Hilbert proposed the solution adopted today: leaving the point, line and plane undefined and describing only how they relate to each other.

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A little history

Further reading The word geometry comes from the Greek: geo (earth) and metry (measure) — measuring the earth. Egyptians, Babylonians, Sumerians, Hindus and Persians already used geometry to distribute fertile land after floods, build houses and temples and follow the sky. But it was the Greeks who were the first to treat it rigorously, as a deductive science, with definitions, postulates and theorems. The Greek tradition: knowledge organized in books and schools. Euclid’s The Elements, the most famous work in geometry. Frontispiece of the first English edition of The Elements (1570). According to tradition, Tales of Miletus took geometry from Egypt to Greece. After him came two schools that studied figures and numbers through analysis and deduction: the Pythagorean, by Pythagoras of Samos, and the Platonic, by Plato. The culmination of this period is the collection The Elements, written by Euclid in Alexandria around 300 BC. It deals with plane figures, circles, triangles and quadrilaterals — and it is from this that the method we follow in this book comes from.

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