A History of Greek Mathematics and Astronomy

Page 4

To take an example outside the Elements, how can a mathematician properly understand the term latus rectum used in conic sections unless he has seen it in Apollonius as the erect side (ορθια πλευρα {orthia pleura}) of a certain rectangle in the case of each of the three conics?[3] The word ordinate can hardly convey anything to one who does not know that it is what Apollonius describes as 'the straight line drawn down (from a point on the curve) in the prescribed or ordained manner (τεταγμενως κατηγμενη {tetagmenôs katêgmenê})'. Asymptote again comes from ασυμπτωτος {asymptôtos}, non-meeting, non-secant, and had with the Greeks a more general signification as well as the narrower one which it has for us: it was sometimes used of parallel lines, which also 'do not meet'.

[3] In the case of the parabola, the base (as distinct from the 'erect side') of the rectangle is what is called the abscissa (Gk. αποτεμνομενη {apotemnomenê}, 'cut off') of the ordinate, and the rectangle itself is equal to the square on the ordinate. In the case of the central conics, the base of the rectangle is 'the transverse side of the figure' or the transverse diameter (the diameter of reference), and the rectangle is equal to the square on the diameter conjugate to the diameter of reference.

Again, if we take up a textbook of geometry written in accordance with the most modern Education Board circular or University syllabus, we shall find that the phraseology used (except where made more colloquial and less scientific) is almost all pure Greek. The Greek tongue was extraordinarily well adapted as a vehicle of scientific thought. One of the characteristics of Euclid's language which his commentator Proclus is most fond of emphasizing is its marvellous exactness (ακριβεια {akribeia}). The language of the Greek geometers is also wonderfully concise, notwithstanding all appearances to the contrary. One of the complaints often made against Euclid is that he is 'diffuse'. Yet (apart from abbreviations in writing) it will be found that the exposition of corresponding matters in modern elementary textbooks generally takes up, not less, but more space. And, to say nothing of the perfect finish of Archimedes's treatises, we shall find in Heron, Ptolemy and Pappus veritable models of concise statement. The purely geometrical proof by Heron of the formula for the area of a triangle, Δ{D}=√{s(s-a)(s-b)(s-c)}, and the geometrical propositions in Book I of Ptolemy's Syntaxis (including 'Ptolemy's Theorem') are cases in point.

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T. L. Heath
A History of Greek Mathematics and Astronomy

Learned Freeware

T. L. Heath A History of Greek Mathematics and Astronomy

Learned Freeware

T. L. Heath
A History of Greek Mathematics and Astronomy