Leo

(fl. Athens, first half of fourth century B.C.)

mathematics.

Leo was a minor mathematician of the Platonic school. All that is known of him comes from the following passage in the summary of the history of geometry reproduced in Proclus’ commentary on the first book of Euclid’s Elements:¹

Younger than Leodamas [of Thasos] were Neoclides and his pupil Leo, who² added many things to those discovered by their predecessors, so that Leo was able to make a collection of the elements more carefully framed both in the number and in the utility of the things proved, and also found diorismoi, that is, tests of when the problem which it is sought to solve is possible and when not.

Proclus’ source immediately adds that Eudoxus of Cnidus was “a little younger than Leo”; and since he has earlier made Leodamas contemporary with Archytas and Theaetetus, this puts the active life of Leo in the first half of the fourth century B.C. It is not stated in so many words that he lived in Athens; but since all the other persons mentioned belonged to the Platonic circle, this is a fair inference.

Leo had been preceded in the writing of Elements by Hippocrates of Chios. His book has not survived and presumably was eclipsed by Euclid’s. He is the first Greek mathematician who is specifically said to have occupied himself with conditions of the possibility of solutions of problems; but the Greek word³ does not necessarily mean that he discovered or invented the subject—and indeed it is clear that there must have been diorismoi before his time. From the earliest days it must have been realized that a triangle could be constructed out of three lines only if the sum of two was greater than the third, as is explicitly stated in Euclid I.22. This was certainly known to the Pythagoreans, and the more sophisticated diorismos in Euclid VI.28 is also probably Pythagorean: a paralelogram equal to a given rectilineal figure can be “applied” to a given straight line so as to be deficient by a given figure only if the given figure is not greater that the parallelogram described on half the straight line and is similar to the defect.⁴ There is also a Diorismos in the second geometrical problem in Plato’s Meno; and if its latest editor, R. S. Bluck, is right in dating that work to about 386-385 B.C., it probably preceded Leo’s studies.⁵ Nevertheless, Leo must have distinguished himself in this field to have been so singled out for mention by Eudemus, if he is Proclus’ ultimate source; it is probable that he was the first to recognize diorismoi as a special subject for research, and he may have invented the name.⁶

NOTES

1. Proclus, in primum Euclidis, Friedlein ed., pp. 66.18—67.1.

2. The Greek word is in the plural.

3. It is ∊υ̒ρ∊ῑν, which in its primary sense means no more than “to find.” T. L. Heath, A History of Greek Mathematics (Oxford, 1921), I, 303, 319, takes it to mean that Leo “invented” diorismoi—which he rightly regards as an error—but this is to read too much into the word in this context.

4. The problem is equivalent to the solution of the quadratic equation

ax - (b/C)x² = A,

which a real root only if

A ≯ (c/b)· (a²/4).

See T.L. Heath, The Thirteen Books of Euclid’s Elements, 2nd ed. (Cambridge, 1925; repr. Cambridge-New York, 1956), II, 257-265.

5. See R. S. Bluck, Plato’s Meno (Cambridge, 1961), pp. 108-120, for the date and app., pp. 441-461, for a discussion of this much-debated problem with full documentation.

6. Primarily signifying “definition,” διoρισμóς came to have two technical meanings in Greek mathematics: (1) the particular enunciation of a Euclidean proposition, that is, a closer efinition of the thing sought in relation to a particular figure; and (2) an examination of the conditions of possibility of a solution. Pappus uses the world in the latter sense only, but Proclus knows both meanings (In primum Euclidis, Friedlein ed., pp. 202.2-5, 203.4, 9-10). It is easy to see how one meaning merges into the other. See T. L. Heath, The Thirteen Books of Euclid’s Elements, I, 130-131; and Charles Mugler, Dictionnaire historique de la terminologie géométrique des grecs, pp. 141-142.

Ivor Bulmer-Thomas

LEO A line of computers, and a company, of historic importance in the British computing industry. J. Lyons & Co (a large firm in the catering industry) initiated in 1947 a project to build a computer to mechanize clerical functions in their own offices. (This decision was almost simultaneous with a similar decision in the US by Eckert and Mauchly, which led to UNIVAC 1.) The project was led by T. R. Thompson, a mathematician, and J. Pinkerton, an electrical engineer. The machine they built, LEO (Lyons Electronic Office), was fully operational at the end of 1953.

In 1954, Leo Computers Limited was founded. The company traded until 1963, when it was merged with the computing division of English Electric. During that time it marketed the LEO III, an extremely advanced commercial machine for its time. See also ICL.

Leo ★★ 2002 (R)

Craving love from his dispirited mom who sees him as all that went wrong in her past, Leo connects with incarcerated Stephen during a class writing project. Through their letters they become each other's lifeline and upon his release Stephen is determined to find him. Wants to give an twisty ending and boasts lots of acting talent but payoff doesn't deliver. 103m/C VHS, DVD . Mary Stuart Masterson, Joseph Fiennes, Sam Shepard, Elisabeth Shue, Davis Sweat, Dennis Hopper, Deborah Kara Unger, Jake Weber, Justin Chambers; D: Mehdi Norowzian; W: Massy Tadjedin, Amir Tadjedin; C: Zubin Mistry; M: Mark Adler. VIDEO

Leo a large constellation (the Lion), said to represent the lion slain by Hercules. Leo is also the fifth sign of the zodiac, which the sun enters about 23 July.

LEO (ˈliːəʊ) Astronautics low earth orbit
• Lyons Electronic Office (early computer built by J. Lyons & Co.)