By Erik Gregersen
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Extra resources for The Britannica Guide to the History of Mathematics (Math Explained)
For this we have the mathematicians of ancient Greece to thank. The Pre-Euclidean Period The Greeks divided the field of mathematics into arithmetic (the study of “multitude,” or discrete quantity) and geometry (that of “magnitude,” or continuous quantity) and considered both to have originated in practical activities. Proclus, in his Commentary on Euclid, observes that geometry—literally, “measurement of land”—first arose in surveying practices among the ancient Egyptians, for the flooding of the Nile compelled them each year to redefine the boundaries of properties.
For instance, one group of curves, the conchoids (from the Greek word for “shell”), are formed by marking off a certain length on a ruler and then pivoting it about a fixed point in such a way that one of the marked points stays on a given line. The other marked point traces out a conchoid. These curves can be used wherever a solution involves the positioning of a marked ruler relative to a given line (in Greek such Nicomedes (3rd century BCE) discovered a special curve, known as a conchoid, with which he was able to trisect any acute angle.
3, 4, 5; 5, 12, 13; or 119, 120, 169). From the Greeks came a proof of a general rule for finding all such sets of numbers (now called Pythagorean triples): if one takes any whole numbers p and q, both being even or both odd, then a = (p2 − q2)/2, b = pq, and c = (p2 + q2)/2. As Euclid proves in Book X of the Elements, numbers of this form satisfy the relation for Pythagorean triples. Further, the Mesopotamians appear to have understood that sets of such numbers a, b, and c form the sides of right triangles, but the Greeks proved this result (Euclid, in fact, proves it twice: in Elements, Book I, proposition 47, and in a more general form in Elements, Book VI, proposition 31), and these proofs occur in the context of a systematic presentation of the properties of plane geometric figures.
The Britannica Guide to the History of Mathematics (Math Explained) by Erik Gregersen