GNOMON, a term originally used to mean an instrumen for allowing one to know the time fror mon, fro yeyvoaKFLy, gignoskein, to know). In its simple and primitiv form it seems to have been a stick placed vertically on a plan surface, and later upon the concave surface of a hemisphere This second form is seen in pocket sundials still used in certaii parts of the world. That the term was at one time substan tially synonymous with "vertical line" is seen in an expres sion of Oenopides of Chios (c. 465 B.e.), for Proclus (c. 460 says that he called "the perpendicular in the archaic manne `gnomon-wise' (Kara yyo,uova), because the gnomon is also a right angles to the horizon." From this early use it came to repre sent a figure like a carpenter's square, but usually with equa arms. Seeking, as the Pythagoreans especially did, to relate num ber to geometric forms, the early Greek mathematicians imagine( squares as built up of gnomons added to unity. For example, the: saw that 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, and so on, are squares, an that the odd numbers in a figure like this were related to thy geometric gnomon. Such numbers were, therefore, themselves called gnomons. The early idea of a geometric gnomon was ex tended by Euclid (q.v.; c. 30o B.e.) to include a figure consisting of two parallelograms forming an L. Four or five centuries late] Heron extended the term still farther, using it to mean that which added to any number or figure, makes the whole similar to that t( which it is added. This usage is also found in the writings of Theor of Smyrna (c. 125) in connection with figurate numbers (q.v.) For example, the pentagonal numbers are 1+4, 1 +4+ 7 + 7 +10, . . ., and the gnomons in this case are 4, 7, 10 . . . ; i.e., they constitute an arithmetical series with a common difference of 3. In the same way gnomons are developed with respect to hexagonal and higher polygons. The gnomon with respect to the square was used by early writers in the finding of square roots, and may still be seen in various elementary arith metics and algebras.
As to the sundial, with a gnomon as a vertical needle, this ih said to have been introduced into Greece by Anaximander (q.v.; c. 575 B.c.), and Herodotus states that it came from the Babylon ians.
BIBLIOGRAPHY.-A brief historical treatment of the subject may b€ Bibliography.-A brief historical treatment of the subject may b€ found in Sir Thomas Heath, A History of Greek Mathematics, vol. i. pp. 77-83 (Oxford, 1921) ; Pauly-Wissowa, Realencyclopadie del Classischen Altertumswissenschaf t (Stuttgart, 1894 seq.) ; D. E. Smith History of Mathematics, vol. ii., pp. 6oi, 603, 669, 671 (Boston, 1925), and various other histories. (D. E. S.)